ABSTRACT
Title of dissertation: ISOTOPE SHIFT SPECTROSCOPY OF
ULTRACOLD STRONTIUM
Neal Carden Pisenti
Doctor of Philosophy, 2019
Dissertation directed by: Dr. G.K. Campbell and Professor S.L. Rolston
Joint Quantum Institute,
National Institute of Standards and Technology,
and
Department of Physics, University of Maryland
We describe the design, construction, and performance of a laser system to
probe the ultra-narrow (Γ/2π ≈ mHz) clock transition 1S0 → 3P0 in strontium.
We present the first reported spectroscopy of this transition in two of the bosonic
isotopes, 84Sr and 86Sr. Furthermore, we measure the complete set of isotope shifts
between all four stable isotopes on the clock line and the narrow intercombination
line 1S 30 → P1, permitting a King plot analysis of the isotope shifts. Complications
arising from the unambiguous determination of a line center in 87Sr 3P1 prevent
us from making claims about the King linearity, but we provide a statistical boot-
strap analysis of the isotope shifts 88−84Sr and 88−86Sr to compute a field shift ratio
F698/F689 = 0.9979, with a 95% confidence interval [0.9952, 1.0008]. The intercept
term K698 − (F698/F689) K689 is similarly determined to be -2.0 GHz-amu, with a
95% confidence interval [−3.9,−0.3] GHz-amu. Finally, we describe the design of a
next-generation apparatus that will enable improvements on the results described
here, as well as other studies that involve coherent manipulation of strontium atoms
on the clock line.
ISOTOPE SHIFT SPECTROSCOPY OF
ULTRACOLD STRONTIUM
by
Neal Carden Pisenti
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2019
Advisory Committee:
Dr. Gretchen K. Campbell, Advisor
Professor Stephen L. Rolston, Chair
Professor Christopher Jarzynski, Dean’s Representative
Dr. James (Trey) Porto
Professor James Williams
©c Copyright by
Neal Carden Pisenti
2019

Dedication
For my parents, who have always pushed me to do more and pursue my interest
in physics, and for Pascal, who will pull me into the future.
ii
Acknowledgments
It is hard to look back over six years to the person I was entering graduate
school; so much has changed since then, and I have grown much as a physicist. In
hindsight, I didn’t realize quite how lucky I was stumbling into the JQI, with its
incredibly talented and diverse set of researchers. The opportunities it provided to
work and interact with some of the world’s foremost physicists is hard to match
elsewhere, and for this I am tremendously grateful.
The people who have directly or indirectly helped me along the way is perhaps
innumerable, and I am bound to forget some of them by mistake. A number of
undergraduates made valuable contributions to the lab — Eliot Bohr and Peter Zhou
in particular helped with everything from vacuum assembly to electronics design.
In the early days, Joe Pechkis helped jumpstart the lab for six months as a post-doc
while we were still in the building phase. Other graduate students passed through
our lab as well, either as visiting students or as a waypoint to other experiments
at the JQI and elsewhere. Avinash Kumar, Erin Marshall, Madison Anderson, and
Swarnav Banik all helped for some time on the strontium apparatus, either by
building electronics or designing optics, but remained close for moral support and
the occasional tip when we were facing a challenging technical problem. Alexander
Hesse was with us for an entire year from Heidelberg, designing and building a Bitter
coil electromagnet with attendant high-current drive electronics. He also provided
much-needed manpower in the construction phase of the new strontium lab in the
PSC. Junyi Jhang spent a summer with us, and built a very robust 679nm repump
iii
laser to replace the finicky one we had been using, and explored a novel coupled-
resonance locking scheme for the two repumps. Alessandro Restelli has been a
mainstay of the strontium group, and taught me everything I know about electronics
design. He always has a clever idea to work around a difficult technical issue, and
brought some fantastic outside perspective to the physics monoculture. The new
graduate students, Peter Elgee and Ananya Sitaram, and now too Nick Mennona,
are bringing new life to the strontium experiment. They’ve worked tirelessly to
bring up the new apparatus, and particularly in the case of Peter, extend the life
of the old one. Much of the isotope shift data in this paper wouldn’t have been
possible without his help. Hiro Miyake joined the lab as a post-doc a little over a
year ago, and in that time I’ve been really grateful for all of his help and expertise,
and moral support when everything was breaking and it seemed there was no light
at the end of the tunnel. This experiment is as much his now as it is mine. Finally,
I don’t quite have the words to describe how much I appreciate Dan Barker and
Ben Reschovsky, the two previous graduate students in the strontium lab. They
raised me, and taught me most of what I know about working in an AMO lab. Dan
seemed to have a truly encyclopedic knowledge of the field, and was always happy to
answer my dumb questions. Ben had a supernatural talent getting lasers to behave,
and often helped me to refocus and just get to work when I had a tendency to
overthink before doing. They both have some excellent experimental experimental
chops, which I hope to someday be able to match.
Beyond the immediate strontium group, I’m grateful for the camaraderie and
collaboration of other members of the JQI, and the “CPR” research group — Matt
iv
Reed, Zach Smith, Aftaab Dewan, Dalia Ornelas, Sandy Craddock, Mary Lyon,
Varun Vaidya, Creston Herold, Joe Tiamsuphat, Tsz-Chun Tsui, Sarthak Sub-
hankar, Sean Li, Ana Valdes Curiel, Ryan Price, Dan Campbell, Dimitrios Try-
pogeorgos, Andika Putra, Prasoon Gupta, Elizabeth Goldschmidt — I could say
something about each of them, but Zach Smith in particular was always there for
moral and technical support. He has a knack for bridging the gap between physics
and the real world. The other members of the Campbell research group made many
things possible, both for their friendship and help when the experiment stopped
working — Monica Gutierrez, Avinash Kumar, Swarnav Banik, Madison Anderson,
and Hector Sosa. Also thanks to the two former sodium post-docs, Fred Jendrzejew-
ski and Steve Eckel, who was always very helpful. Ken Wright in particular helped
me through the ups and downs of graduate school, both at Town Hall and sharing
the brutal commute to College Park.
I’m very grateful to Gretchen Campbell, who did much more than just pay
our salaries. She was always there with advice and help working through a tough
problem, but was also happy to let us all experiment with new ideas in the lab
without breathing down our necks. I think we are all lucky to have her as an
advisor. Steve Rolston and Trey Porto also provided some useful guidance at times
as part of the extended research group, as did Luis Orozco who was selflessly there
with support both scientific and beyond.
Finally, none of this would’ve been possible without Sarah’s loving and patient
support. I am grateful to her in ways that extend beyond graduate school, and am
lucky to have her as my partner.
v
This work was supported by the NSF through the PFC at the JQI.
vi
Table of Contents
Dedication ii
Acknowledgements iii
List of Tables ix
List of Figures x
List of Abbreviations xii
1 Introduction 1
1.1 Strontium laser cooling primer . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1 Repumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.2 Narrow line magneto-optical traps . . . . . . . . . . . . . . . . 8
1.2 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Experimental apparatus 13
2.1 Vacuum systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Laser Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.1 Blue 461nm lasers . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.2 Red 689nm lasers . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.3 Repumping lasers . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.4 Dipole trap laser . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Magnetic field coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4 Computer control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5 Custom electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5.1 Laser diode current controller . . . . . . . . . . . . . . . . . . 36
2.5.2 Optical beatnote servo . . . . . . . . . . . . . . . . . . . . . . 40
3 Design and construction of a narrow-linewidth clock laser 46
3.1 Motivation and target specifications . . . . . . . . . . . . . . . . . . . 47
3.2 Pound-Drever-Hall: noise analysis and limitations . . . . . . . . . . . 49
3.2.1 Index of refraction . . . . . . . . . . . . . . . . . . . . . . . . 50
vii
3.2.2 Mechanical length stability . . . . . . . . . . . . . . . . . . . . 51
3.2.3 Thermal length stability . . . . . . . . . . . . . . . . . . . . . 63
3.2.4 Servo noise & electronics . . . . . . . . . . . . . . . . . . . . . 68
3.3 Final design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.4 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4 Publication: An ultra-low noise, high-voltage piezo driver 85
4.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.3 Circuit Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.3.1 Flyback regulator . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.3.2 Low-noise stabilization and fast modulation . . . . . . . . . . 91
4.3.3 Digital control and slow modulation . . . . . . . . . . . . . . . 93
4.4 Noise model & analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.7 Future developments . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5 Isotope shift spectroscopy 107
5.1 Isotope shifts: theory and practice . . . . . . . . . . . . . . . . . . . . 109
5.1.1 Mass shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.1.2 Field shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.1.3 King plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.1.4 Hyperfine effects . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2 Precision measurement of strontium isotope shifts . . . . . . . . . . . 120
5.2.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2.1.1 Spectroscopy of 689nm intercombination lines . . . . 122
5.2.1.2 Spectroscopy of 698nm clock line . . . . . . . . . . . 125
5.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.2.2.1 Systematics . . . . . . . . . . . . . . . . . . . . . . . 136
5.2.2.2 Detailed analysis of the clock transition lineshape . . 141
5.3 King plot analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.4 Discussion and future directions . . . . . . . . . . . . . . . . . . . . . 148
6 SrII and future directions 150
6.1 Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.1.1 AOSense atomic beam source . . . . . . . . . . . . . . . . . . 153
6.1.2 Science chamber . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.2 Laser systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.3 Magnetic field coils and high-NA imaging . . . . . . . . . . . . . . . . 160
6.4 Computer Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.5 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Bibliography 171
viii
List of Tables
1.1 Summary of strontium lines . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Dipole trap waists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Frequency offsets for 461nm beam paths . . . . . . . . . . . . . . . . 21
3.1 Summary of ULE mechanical properties . . . . . . . . . . . . . . . . 54
5.1 Summary of isotope shift measurement . . . . . . . . . . . . . . . . . 134
5.2 Absolute frequency measurements for 698nm transition . . . . . . . . 134
5.3 Absolute frequency measurements for 689nm transition . . . . . . . . 135
5.4 Hyperfine A and B coefficients . . . . . . . . . . . . . . . . . . . . . . 135
5.5 Summary of 689nm systematics . . . . . . . . . . . . . . . . . . . . . 140
5.6 Systematic frequency shifts on the clock and intercombination line
transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.7 Atomic and nuclear structure coefficients . . . . . . . . . . . . . . . . 147
6.1 Zeeman slower simulation data from AOSense. . . . . . . . . . . . . . 154
6.2 Bakeout time constants for various flange sizes . . . . . . . . . . . . . 155
6.3 Summary of optical access on SrII science chamber . . . . . . . . . . 158
ix
List of Figures
1.1 Strontium Level Diagram . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Narrow-line cooling force . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Operation of the fermionic red MOT . . . . . . . . . . . . . . . . . . 12
2.1 Schematic of the SrI vacuum system . . . . . . . . . . . . . . . . . . 14
2.2 SrI chamber-side optics (horizontal) . . . . . . . . . . . . . . . . . . . 16
2.3 SrI chamber-side optics (vertical) . . . . . . . . . . . . . . . . . . . . 18
2.4 Spectroscopy for 461nm . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 461nm slave lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6 689nm laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.7 Repump laser systems . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.8 Dipole laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.9 Current controller analog subcircuits . . . . . . . . . . . . . . . . . . 37
2.10 Current controller noise power-spectral-density . . . . . . . . . . . . . 41
2.11 Beatnote lock sample trace . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1 Clock Zeeman level diagram . . . . . . . . . . . . . . . . . . . . . . . 48
3.2 ULE cavity and FEA geometry . . . . . . . . . . . . . . . . . . . . . 53
3.3 Lab vibrational power spectral density . . . . . . . . . . . . . . . . . 53
3.4 Cavity length fluctuation ∆L due to mirror tilt . . . . . . . . . . . . 55
3.5 Viton rod support cavity FEA . . . . . . . . . . . . . . . . . . . . . . 57
3.6 Final design for Fabry-Pérot cavity mount . . . . . . . . . . . . . . . 58
3.7 FEA mesh and boundary model comparison . . . . . . . . . . . . . . 59
3.8 Summary of clock cavity FEA . . . . . . . . . . . . . . . . . . . . . . 62
3.9 Clock cavity zero-crossing in αCTE . . . . . . . . . . . . . . . . . . . . 65
3.10 PDH closed-loop noise analysis . . . . . . . . . . . . . . . . . . . . . 69
3.11 Schematic of PDH lock . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.12 PDH error signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.13 Clock laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.14 Clock laser beatnote . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.15 Atomic spectroscopy of clock transition in Sr87 and Sr84 BEC . . . . 84
4.1 Schematic of the high-voltage piezo driver . . . . . . . . . . . . . . . 88
x
4.2 Quench circuit for piezo driver . . . . . . . . . . . . . . . . . . . . . . 91
4.3 Noise model for piezo circuit . . . . . . . . . . . . . . . . . . . . . . . 96
4.4 Piezo circuit noise model and measured noise power spectral density . 99
4.5 Time-domain traces of high-voltage output from piezo driver . . . . . 101
4.6 Piezo driver temperature coefficient . . . . . . . . . . . . . . . . . . . 102
4.7 Piezo driver modulation input transfer function . . . . . . . . . . . . 103
5.1 Finite-size nucleus potential energy . . . . . . . . . . . . . . . . . . . 113
5.2 Magnetic field calibration . . . . . . . . . . . . . . . . . . . . . . . . 127
5.3 Detailed lineshape curves for 698nm clock transition . . . . . . . . . 129
5.4 Bootstrap analysis of 689nm transition . . . . . . . . . . . . . . . . . 131
5.5 Bootstrap analysis of 689nm isotope shifts . . . . . . . . . . . . . . . 132
5.6 Bootstrap analysis of 698nm clock lines . . . . . . . . . . . . . . . . . 133
5.7 Thermal line pulling for clock transition . . . . . . . . . . . . . . . . 146
5.8 King Plot (all isotopes) . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.9 Bootstrap distribution of King plot parameters . . . . . . . . . . . . . 148
6.1 Zeeman slower simulation data from AOSense. . . . . . . . . . . . . . 152
6.2 CAD rendering of new SrII apparatus . . . . . . . . . . . . . . . . . . 157
6.3 SrII 461nm laser system . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.4 SrII 689nm laser system . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.5 Bitter coil electromagnet . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.6 SrII high-NA objective . . . . . . . . . . . . . . . . . . . . . . . . . . 166
xi
List of Abbreviations
AMO Atomic, Molecular, and Optical physics
AR Anti-reflection
BEC Bose-Einstein condensate
bMOT Blue MOT
CF ConFlat (vacuum flange)
CPR “Campbell-Porto-Rolston”
CSS Computer & Space Sciences Building
ECDL External cavity diode laser
EMI Electromagnetic interference
EOM Electro-optic modulator
FDG Fermi-degenerate gas
FEA Finite-element analysis
FMC FPGA Mezzanine Card
FORT Far-off-resonance (dipole) trap
FP Fabry-Perót cavity
FPGA Field-programmable gate array
GUI Graphical User Interface
IDE Integrated developemt environment
JILA JILA
JQI Joint Quantum Institute
LAN Local area network
LDO Low-dropout voltage regulator
MOT Magneto-Optical Trap
OSI Open-systems interconnection model
PDH Pound-Drever-Hall lock
PHY Physical layer of the OSI model
PI Proportional-integral feedback control
PID Proportional-integral-derivative feedback control
PLL Phase-locked loop
PM Polarization-maintaining fiber
PSC Physical Sciences Complex
RBW Resolution Bandwidth
RGA Residual Gas Analyzer
rMOT Red MOT
SI International System of Units (or, Système International d’unités)
SoC System on Chip
TCP/IP Transmission Control Protocol/Internet Protocol
UI User interface
UV Ultra-violet
xii
VCO Voltage-controlled oscillator
xiii
Chapter 1: Introduction
The modern history of atomic physics, in many ways, is intimately intertwined
with the history of the laser. Since their conception and first demonstration in the
mid-20th century, lasers have enabled us to coherently probe and manipulate the
quantum-mechanical nature of atomic structure. With the development of laser
cooling techniques, scientists routinely achieve samples of dilute atomic gasses at
temperatures many orders of magnitude lower than would otherwise be possible.
This ultra-cold regime has opened a window onto new physical phenomena that
emerge at such energy scales — Bose-Einstein condensation (BEC) and its fermionic
counterpart, synthetic crystal lattice structures, new phase transitions, topological
matter — all become tools at the disposal of science.
Recent efforts in the cold-atom community seem to fall into into one of two
categories: using atomic systems for precision metrology, or using atomic systems
for quantum simulation or information processing. In a sense, the latter category
is only possible because of the former — precision measurement enables precision
control and the ability to engineer near-defect-free quantum systems for further
study. Strontium, which is the element in focus for the entirety of this dissertation,
is a particularly interesting candidate for both kinds of scientific investigation. It has
a range of optical transitions with natural linewidths spanning over nine orders of
magnitude; three of its isotopes are bosons with a wide range of scattering lengths,
and one isotope is a fermion with an abnormally high nuclear spin I = 9/2. The
narrowest line, connecting the ground 1S0 state to the excited 3P0 “clock” state
1
(see Fig. 1.1), is a leading candidate for the anticipated redefinition of the second
based on optical atomic clocks. Fermionic atoms in either the ground or clock state
exhibit, to a very high degree, SU(N) symmetry in the nuclear spin subspace [1].
This makes strontium a good system to simulate Hamiltonians which carry this
symmetry, where it is now possible to achieve any N ≤ 2I + 1 = 10. Combining
this SU(N) symmetry with the orbital degree of freedom provided by the 1S0 – 3P0
clock transition allows one to simulate other Hamiltonians of interest, such as the
Kondo lattice or Kugel-Khomskii models [2].
Beyond simulating these models directly, which often involves atoms pinned in
a crystal lattice structure created by a deep three-dimensional (3D) optical lattice,
there has been recent interest in constructing lattice models which utilize synthetic
dimensions [3–5]. Again, strontium and other alkaline-earth(-like) atoms such as
ytterbium feature prominently in many of these proposals because of the coherence
offered by their narrow clock transitions, the presence of isotopes with high nuclear
spin, and two easily accessible J = 0 orbital states with no magnetic moment.
Another set of proposals use the external momentum states of atoms trapped in
a uniform potential as a synthetic lattice dimension, realizing an exactly-solvable
model with SU(N) symmetry using thermal atoms [6], or a method for estimating
the eigenspectrum of the nuclear spin density matrix via Ramsey spectroscopy [7].
In this thesis, we will focus on experiments made possible by the narrow tran-
sitions in strontium. In particular, I will discuss the construction of a clock laser to
address the ultra-narrow 1S 30 – P0 transition, and the first reported spectroscopy
of this transition in 84Sr and 86Sr. We further measure all isotope shifts on both
the clock and the narrow intercombination-line transition 1S0 – 3P1; this permits
a King plot analysis which is the subject of Chapter 5. However, before we get to
that point, we will set the stage by discussing some important experimental details
for the laser cooling and trapping of strontium gasses.
2
(5s6s) 3S
1
(5s5p) 1P γ
1  ˜  610 s -1
γ ̃  10
1st stage cooling 5 s -11 3
“bMOT” (5s4d) D2 γ (5s5p) P˜  2
461 nm 1
2
0 s -1 (5s5p) 3γ ~   32 MHz P1
(5s5p) 3P
0
2nd stage cooling
“rMOT”
689 nm
γ ~  7.4 kHz
698 nm
γ ~  mHz
“clock” transition
(5s2) 1S
0
Figure 1.1: Low-lying electronic structure of strontium. Linewidths and decay rates are taken from
Refs. [8–12]. Hot atoms are first cooled and captured into a magneto-optical trap (MOT) operating
on the broad 461 nm transition 1S0 – 1P1 (bMOT). Typical temperatures achieved after this first
cooling stage are ∼ mK. Atoms are then transferred to a frequency-modulated “broadband” MOT
operating on the much narrower intercombination transition 1S – 30 P1 at 689 nm (rMOT). The
intercombination MOT is recoil rather than Doppler limited, and we can achieve temperatures as
low as . 1 µK. This is cold enough to load directly into a far-off-resonance dipole trap (FORT)
and proceed with evaporative cooling to quantum degeneracy. A slow leak from the broad 461 nm
transition necessitates repumping during bMOT operation; several repumping schemes exist [13],
however we follow the practice at JILA and repump through 3S1 with 707 nm and 679 nm light.
Atoms decaying to 3P1 return to the ground state in ∼ 21.5 µs, while atoms in 3P0,2 are meta-stable
with lifetimes & 100 s. Repumping in 87Sr is complicated by the hyperfine splitting of 3P2 and
3S1; full repumping on the 707 nm transition requires at least five tones spread over several GHz.
We achieve this by frequency-modulating a fiberized electro-optic modulator (EOM), as described
in the text. The “clock” transition at 698 nm is weakly dipole-allowed in 87Sr via hyperfine state
mixing, while in the bosonic isotopes it must be mixed in, e.g., with a static magnetic field. The
broad array of transitions with varying linewidths makes strontium appealing for applications in
metrology and quantum simulation.
3
707 nm
γ ˜ 7    10
6 s -1
679 nm
γ ˜ 1.4 ×   106 s -1
688 nm (Depump)
γ ˜ 4.3    10
6 s -1
× ×
1.1 Strontium laser cooling primer
There are many good pedagogical introductions to laser cooling and trapping,
see for example Ref. [14].1 Here, I will review the important aspects as they pertain
to creating degenerate gasses of strontium. The basic idea behind all laser cooling is
to engineer a situation whereby an atomic system preferentially absorbs photons that
slow its motion. This is readily achieved via the Doppler effect — by detuning your
laser red of a resonant transition, atoms moving towards the light see it blue shifted
into resonance, and atoms moving away see it red shifted further from resonance.
This produces a velocity dependence to the scattering rate (and thus a velocity-
dependent force, since each scattering event on average produces a momentum kick
dpscat = ~k), and the atomic motion is damped.
To understand this quantitatively, we start with the scattering rate Γscat for a
two-level system. This can be written
( )
Γ s0
Γscat = , (1.1)
2 1 + s0 + (2∆/Γ)
2
where Γ is the transition linewidth (related to the excited state lifetime Γ ≡ 1/τ if we
can neglect branching to other electronic states), ∆ is the detuning from resonance,
and s0 is the on-resonance saturation parameter. For the remainder of this thesis,
we adopt a convention (unless otherwise clear from context) that capitalized Greek
letters for frequency units are implicitly angular frequency, and carry a factor of 2π.
We relate s0 to the Rabi rate Ω, which characterizes the strength of the optical
drive, via
2|Ω|2 I
s0 = ≡ . (1.2)
Γ Isat
1Our discussion here follows the discussion in this reference.
4
Transition Γ/2π (MHz) Isat (mWcm−2) λ (nm)
1S 10 – P1 (blue MOT) 30.2 40.4 460.9
1S 30 – P1 (“intercombination”) 7.48× 10−3 3.0 × 10−3 689.4
1S – 30 P0 (“clock”) 10−9 10−10 698.4
Table 1.1: Summary of strontium lines. Linewidths and saturation intensity calculations use
recommended values from Ref. [15]. Because the details are isotope-dependent (see discussion in
future chapters), we do not list exact values for the clock transition 1S0 – 3P0; however, the order
of magnitude is Γ/2π ≈ mHz, I ≈ 10−10sat mWcm−2.
In Eq. 1.2, we introduce the saturation intensity Isat, which sets the relevant intensity
scale for driving a particular transition. It can be conveniently written
πhcΓ
Isat = , (1.3)
3λ3
with h representing Planck’s constant, c the speed of light, and λ the wavelength of
the transition. Table 1.1 lists these numbers in strontium for easy reference.
Given the scattering rate in Eq. 1.1 and the assumption that spontaneous
emission is isotropic, one can write the time-averaged radiation force as dp/dt =
~kΓscatt, where each photon imparts a momentum dp = ~k, with ~ = h/2π, and
k = 2π/λ. To achieve confinement in addition to cooling, one must engineer a
spatial dependence to the scattering rate, such that atoms experience a restoring
force towards the center of the trap. This is done with a magnetic field gradient,
whereby the detuning ∆ in Eq. 1.1 picks up a spatial dependence from the Zeeman
effect. A magneto-optical trap (MOT) is generated with six beams (to provide
confinement in all three dimensions) in a quadrupole magnetic field, where the
beam polarizations are set to address Zeeman lines that are shifted onto resonance
as an atom leaves the trapping region, thus providing a restorative force to keep the
atom in the MOT. In the appropriate limit, this can be modeled semi-classically
as a damped harmonic oscillator, where the equilibrium temperature is set by the
Doppler limit TD = ~Γ/2. Most MOTs operate in this regime.
5
The exact scheme for laser cooling strontium is informed by its electronic
structure, which is shown in Fig. 1.1 [16]. Initial laser cooling is done on the broad
(Γ/2π = 30.2MHz) 1S – 10 P1 transition at 461nm. A Zeeman slower [14] decel-
erates atoms in a thermal beam, and they are captured into a 3D MOT dubbed
the “blue MOT” (or “bMOT”). For the 461nm transition, the Doppler temperature
TD = 1.1mK, which is still too hot to be loaded into a far off-resonant dipole trap
(FORT) [17], which for our experiment has a typical trap depth ≈ 15 µK. To fur-
ther cool beyond the 461nm Doppler limit, we operate a second stage MOT on the
much narrower (Γ/2π = 7.5 kHz) intercombination line 1S0 – 3P1 at 689nm (the
“red MOT”, or “rMOT”). The MOT dynamics in this regime are very different than
the traditional broad-line MOT dynamics, since the recoil shift ω 2R = ~k /2m is
roughly equal to the transition linewidth. In this regime, the relevant energy scale
and thus the dynamics are set by the relative size of the detuning ∆, the power-
√
broadened linewidth Γ′ = Γ 1 + s0, and the recoil shift ωR [18]. Furthermore, the
gravitational force Fgrav = mg can provide a non-negligible perturbation to tradi-
tional MOT dynamics, which must be accounted for. We discuss these details in
more depth in Sec. 1.1.2.
Because the ground state of strontium is non-magnetic, the final stages of cool-
ing are carried out in an optical dipole trap. To achieve good spatial mode matching,
we load into a flattened, “pancake” trap formed by a gaussian elliptical beam with
a roughly 1:10 aspect ratio, with the tight trap direction oriented along gravity.
A second “crossing” dipole beam provides additional confinement along the weaker
axes, enhancing the density and thus the elastic scattering rate during evaporative
cooling. Our lab has brought 84Sr, 86Sr, and 87Sr to quantum degeneracy [19, 20];
achieving Bose-Einstein Condensation (BEC) in 88Sr requires sympathetically cool-
ing with 87Sr, because the s-wave scattering length in 88Sr is near zero (as = −2a0)
and it does not re-thermalize during evaporative cooling. Additionally, only small
6
BECs of 88Sr are possible before the negative interaction energy causes it to collapse.
It is possible to load isotopic mixtures by shelving the first species in the meta-
stable 3P2 state, which is magnetically trapped by the quadrupole field of the MOT,
while the second species is loaded. One can then repump to the ground state and
proceed with laser cooling on the narrow intercombination-line transition. For a
comprehensive review of laser cooling in strontium, see Ref. [13].
1.1.1 Repumping
As indicated in Fig. 1.1, there is a small branching ratio out of the otherwise
cycling transition at 461nm into the 3P manifold. Atoms entering 3P1 return quickly
to the ground state (τ = 21.3 µs), but the 3P2 state is metastable and must be
depumped. In our lab, we do this by applying a laser at 707nm, which optically
pumps population via 3S 3 31 into both P1 and P0. Atoms that end up in 3P0 must
also be depumped, which is achieved by a second laser at 679nm. For 87Sr, this
scheme is complicated by the hyperfine structure of both 3P 32 and S1, which are
split into five and three distinct F manifolds, respectively. These hyperfine manifolds
are separated by several GHz, and full repumping requires addressing all five 3P2,
F = {13/2, 11/2, 9/2, 7/2, 5/2} levels with 707nm light.
The original recipe from JILA [21] involved driving a modulation onto the
current and piezo actuator of the 707nm repump laser to span the whole hyper-
fine spectrum. This technique has obvious drawbacks – one cannot simultaneously
lock the laser and frequency broaden in this manner, and in practice, the mode-hop
free tuning range is smaller than the desired modulation bandwidth. The result is
a somewhat uncontrolled repumping scheme which requires continuous monitoring
and careful tuneup by hand to achieve good results. More recently, groups using
the 707nm/679nm repump scheme have migrated to using a fiber EOM to gen-
erate sideband tones to address each line separately [22, 23]. More details on our
7
implementation can be found in Sec. 2.2.3.
1.1.2 Narrow line magneto-optical traps
The operation of a MOT on a narrow-line transition deviates from the damped
harmonic motion discussed above for a broad-line MOT [14]. In the case of stron-
tium, the 689nm linewidth is Γ/2π = 7.5 kHz. This has a few important conse-
quences. First, the narrow linewidth implies a very low saturation intensity, in
our case 3 µWcm−2. For typical MOT volumes, the beam waist is several mm, so
even µW of optical power will saturate the transition. We thus typically consider
√
the power-broadened linewidth Γ′ = Γ 1 + s0 as the relevant scale parameter. We
capture atoms from the bMOT into the rMOT with about 700 µW of power, with
rMOT beam waists ≈ 2.5mm. Thus, s ′0 ≈ 2400 and Γ ≈ 370 kHz.
Given this saturation-broadened linewidth, the velocity class ∆v that is res-
onant with our MOT beams is only ≈ 40mms−1, while√the Doppler-limited tem-
perature in the bMOT implies a most-likely velocity v = 2kBT/m ≈ 370mms−1,
with kB Boltzmann’s constant. Furthermore, the spatial extent of the MOT cooling
force is set by the magnetic field gradient, such that at some distance wz the atom is
Zeeman-shifted by one linewidth. For 3P1, the Zeeman shift is 2.1MHzG−1; typical
rMOTs operate at a field gradient of ≈ 1Gcm−1 to 5Gcm−1, and we find the best
performance at 1.5Gcm−1. This implies a resonance “shell” that is at most 1.2mm
wide, which is also smaller than the typical size of the bMOT.
To achieve efficient transfer from the bMOT to the rMOT under these con-
ditions, we modulate the frequency of the rMOT beams. This has the effect of
generating a comb structure, where regardless of the velocity or position of an atom
in the bMOT, there is some cooling light that is on-resonance with that atom. Once
transfer to the rMOT is complete, we increase the magnetic field gradient slightly
to compress the rMOT volume, ramp the detuning of the rMOT light closer to res-
8
0.2 (a) 0.6 (b)
0.4
0.1 0.2
0.0 0.0
δ/2π = -100 kHz -0.2
-0.1 δ/2π = -500 kHz
-0.4
δ/2π =-1000 kHz
-0.2 δ/2π =-1500 kHz -0.6
-6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6
Horizontal Position (mm) Vertical Position (mm)
Figure 3.5: (a) Force as a function of horizontal position for various detunings. For large detunings,
 ⌧ sat, the cooling force has a very localized eect. For smaller detunings,  ⇡ sat, the force has
a dispersive shape. (b) The rMOT potential energy curves in the vertical direction. The eect of gravity,
which causes a linear tilt to the curves, causes the position of the atoms to sag for large detunings. These
plots were generated using Eq. (3.5) with s = s0 = 36, v = 0, and dBz = 1.5 G/cm.
Conversely, the small linewidth means that we can achieve much lower temperatures
in the rMOT. The Doppler temperature TD = ~/2kB is only ⇡ 180 nK. In fact, this
Fteimgupreer1a.t2u:reIlilsusstoraltoiwontohfat(aw)ethmeuMstOcTonfosridceeranadno(tbh)ecrofrurnesdpaomndeinntgalpcooteonltiinagl elinmerigtykn(ionwclundainsg
tthhee erffand aec
ecotilotfegmrapveitrya)tufroer.sTevheerarledcioffielrteenmt dpeetruantuinrgesisinrethlaetendartroowth-leinteemrMpeOrTat,uwreithcosr0re=sp3o6,nvdi=ng0,magnetic quadrupole field gradient of 1.5Gcm−1. Unlike a broad-line MOT, which has a
dtoispaenrsaivveerfaogrceevaecltoincgityovtehratthae esnintigreletrsappopnintagnveooluusmee,mtihsesinoanrreovwe-nlitnewMouOldTgoinvley aaffneacttosma ,foorrce
over narrow shells where the atom is Zeeman shifted into resonance. Additionally, gravity adds a
nTon-neg=lig(~ibkle)2p/emrtkurb.aFtioorntthoetrhMe dOyTn,amT ics, ca⇡us4i6ng0 antKomasntdotbhuenfcuhnndeaamr tehnetablocttooomlinogf tlhime MitOisT
vorelucomile (c). FiguresBreproduced from Refr.ec[o2i0l ].
given by Trecoil/2 ⇡ 230 nK [33,34]. Since the saturation intensity of the 689-nm transition
is so low (Isat = 3.0 µW/cm2) it is very common to operate in a highly saturated regime.
p
In this case, an important parameter is the saturation broadened linewidth sat =  1 + s0.
Instead of exerting a damping force on the atoms within the volume of an ellipsoid,
as for the bMOT, the rMOT force only acts on the atoms over a thin region of position
or velocity space. This localized force is illustrated in Fig. 3.5(a). The width of this
region has a characteristic width of wz = sa9t/(µ0dBz) in position space and wvz = sat/k
in velocity space. For s0 = 1, these widths are only wz ' 33 µm and wvz ' 7 mm/s.
Even for our maximum available laser intensity s0 = 2400 the widths are wz ' 1.2 mm
and wvz ' 250 mm/s which are smaller than the spread of positions and velocities in the
33
Force (ℏkΓ)
Potential Energy (mK)
onance, and then switch from the “broadband” modulated rMOT stage to a single-
frequency MOT. Finally, we decrease the rMOT beam powers to achieve a lower
temperature. Empirically, these parameters are set by optimizing the peak density
of the cloud (which is typically equivalent to maximizing the total atom number,
although we get a better signal-to-noise looking at the fitted peak density).2
Because the gravitational force Fgrav begins to compete with the cooling force
Fcool = ~kΓscatt, the location of the rMOT is set by the force-balance between these
two effects. This is illustrated in Fig. 1.2.3 Final loading into the dipole trap is
achieved by adjusting the magnetic field shim coils to move the rMOT equilibrium
position on top of the dipole beam.
The 87Sr fermionic rMOT operates on the 3P1 (F = 9/2→ F ′ = 11/2) transi-
tion, and is complicated by the hyperfine structure in both the ground and excited
states. The ground state magnetic field sensitivity is set only by the nuclear magnetic
moment, while the excited state field sensitivity is set primarily by the electronic
spin magnetic moment — this leads to a large differential Zeeman shift between the
ground and excited states, shown schematically in Fig. 1.3. As a result, it is possible
to induce a σ+ (or conversely σ−) transition anywhere in the MOT volume. This
is a terrible situation to find oneself, because we are no longer guaranteed to only
absorb photons that give a restorative kick back towards the center of the MOT.
To make matters worse, because the linewidth is so small it is possible to exist in
a region where the scattering rate is much larger for the “wrong” polarization than
for the “right” polarization, depending on the particular mF level the atom might
exist in.
The way around this problem is to add a second laser operating on the
2Ref. [20] gives more details of our exact experimental parameters.
3As an aside, we would have benefitted greatly from launching our vertical rMOT beams from
beneath the chamber, because the end of the rMOT cooling cycle is primarily determined by this
beam. Losses going through the optics before retro-reflection decreases the available power
10
F = 9/2 → F ′ = 9/2 transition, which quickly randomizes the ground state mF
populations. This way, for a particular location in the rMOT, you are guaranteed
to end up in a state which scatters favorably towards the MOT center. This process
is aided by the Clebsch-Gordon weights for particular σ+/σ− transitions. With the
addition of this second “stirring” laser, the fermionic rMOT dynamics are stabilized
and we get a time-averaged restoring force as is required for the trap to operate.
But the performance is degraded somewhat from the performance of the bosonic
rMOT, and optimization is more complicated because of the second laser.
1.2 Outline of thesis
The remainder of this thesis is as follows. In the next chapter, I describe the
experimental apparatus used to carry out the experiments described here. Chapter 3
describes the construction of a narrow linewidth laser to address the 698nm clock
transition in strontium, and initial tests of its performance. Chapter 4 contains an
edited reprint of a paper published in Review of Scientific Instruments, detailing an
ultra-low noise, high bandwidth piezoelectric driver. Chapter 5 contains the first
reported measurements of the clock transition in 86Sr and 84Sr, and the results of a
detailed isotope shift measurement between all isotopes on the 689nm intercombina-
tion line and 698nm clock line; these results seed a King plot analysis of strontium
to extract details of the nuclear structure. We also discuss improvements to our
experiment which might enable enough precision to constrain theories of physics be-
yond the standard model, which predict deviations from King linearity. Finally, in
Chapter 6, we describe the design and construction of a next-generation strontium
apparatus, and future directions for the physics we intend to study.
11
m (a) mF F (b)
-7/2 7/2
-5/2 5/2
-3/2 3/2
-1/2 1/2
1/2 -1/2 55 45 35 28 20 15 10 6 3 1
3/2 δ -3/2
5/2 -5/2 1 3 6 10 15 20 28 35 45 55
7/2 Γ -7/2
9/2 + - -9/2σ σ -11/2 -11/2  -9/2  -7/2  -5/2  -3/2  -1/2   1/2   3/2   5/2   7/2   9/2   11/211/2 mF
Position
m (c) (d)F mF
-7/2 7/2
-5/2 5/2
-3/2 3/2
-1/2 1/2
1/2 -1/2
3/2 δ δ -3/2
5/2 -5/2
7/2 Γ Γ -7/2
9/2 + - -9/2
11/2 σ σ -11/2
Position Position
Figure 3.8: Fermionic rMOT Zeeman shifts. (a) Position dependent Zeeman shifts for + (purple) and 
F(gigreuerne) 1tr.a3n: sitSiocnhse.mDautiec toilltuhsetrlatrgioenmoisfmthatechbebhetawveioern oµfe tahned fµergm, siomniec orfMthOeTs.loIpnes( a)r,e wnegasthivoewaanldl
psomsseibalereσp+os(itpivuerpfoler)eanchd pσo−lar(igzraetieonn). tTrahnesreitfioorne,s,thwehreMreOtThedomesFnolatbperlovciodrereaspconfidns intog tfohreceexfocritaeldl
sZteaetme.anSusubbplleovtesls(.cA) asmndall(dve)rtsihcoalwshtihftehsaasmbeedniiangtroadmu,cebdutotthe linessloaprees wfoerigclhatreitdy.b(yb)tThheerseqleuvaarendt
Cllebsch-Gordaonncfoaectocrisenftrsomcor(rbe)s.poTnhdeinrge itso athcelea+r panredferentcreanfsriotimontshoefCthlebFsch=-g9o/r2do!n wFeig=ht1s1/t2o
smcatntieforldfr.omThethsetroconrgreescttM+O(T)bteramns,itbiountsraergeafrodrlethsse ohfigwhhesatt(lgorwouesnt)d-mstFatvealmueFs. le(vc)elanthde(da)tosmhowis tihne,
tshaemre Ziseeamraengioshnifwtshaesreinit(aw) iflolrstchaetter+ anpdhotontrafnrosimtiotnhserewsrpoenctgivMelyO.TFobre(acm) aanndd(db)ethpeusthiecdknoeusst of
tthe tlirnaeps. aTreopgreotpoartoiounadl tohitsheaprpelaarteivnet liisnseues,traensgetchosn. dDlauseetrootpheravtairnioguosnlintheestFren=gt9h/s,2 t→he F+=(9/2)
tbreaanmsitwioilnl prarenddoomminizaenstlythsecamtteFr mpopreulpahtoiotonnssinfrotmhethgerolaurngde (sstmatael.l) TmhFust,ataens taotothme lwefiltl(rbieghpt)umofptehde
tMoOaTn cmenFterl.eTvehlertehfoarte,sicfawtteercsanprraepfiedrleynrtainadlloymfirzoemthethaetomcoircrpeoctpuMlatOioTnsbaemaomn,gssttalbl itlhieziZnegemthaen sMtaOteTs,
dthyenaCmleibcssc.hF-Gigourdreanrecporeodcuiecnedts fwroilml prRoevfi.de[2f0o]r. stable operation of the rMOT. This state randomization is
provided by a second laser operating on the F = 9/2 ! F = 9/2 transition.
1329
Energy Energy
Energy
Chapter 2: Experimental apparatus
Most neutral atom experiments require a few key experimental systems to
function. First, because the cold atomic gasses we create are fragile, they need to
be well isolated from the chaotic laboratory environment. This is achieved by per-
forming experiments in an ultra-high vacuum environment — the exact pressures
required depend on the lifetime required of your cold atomic sample, but systems
regularly achieve pressures in the low 10−11 torr. Second, our only mode for inter-
acting with the atoms comes from electromagnetic fields. Again, the details vary
by species and specific experimental requirements, but laser systems (for resonant
and off-resonant interactions), a set of magnetic field coils, and possibly electric field
control (which can be important for the highly susceptible Rydberg states) are uni-
versal. And finally, because the natural timescale for atomic processes is typically
in the µs regime, real-time computer control systems are necessary to facilitate ex-
perimentation, and other classical control systems to stabilize laser frequencies and
powers need to be designed and implemented. In the next few sections, I address our
solution to this generally posed set of experimental requirements, with references to
additional details in previous students’ theses.
2.1 Vacuum systems
A detailed description of the SrI apparatus can be found in previous the-
ses [19,20]. Here, I briefly summarize the key design features and reproduce some
figures to highlight changes that have been made since Refs. [19] and [20].
13
Dierential Pumping Tubes 5 in
Atomic Beam
Shutter Gate Valves
Ion Pumps
TSP
Oven
Zeeman Slower
Tube Zeeman
Vertical Slower
Horizontal TC Beams Main Chamber Beam
TC Beams
Ion Gauges Pump Body
Figure 2.1: Schematic of the SrI vacuum system, reproduced from [20]. An effusive oven produces a
hot, collimated atomic beam of strontium, which passes through a split stage of transverse cooling
before entering a variable-pitch Zeeman slower. The oven, transverse cooling, and main chamber
are separated by differential pumping to reduce the impact of the hot oven on vacuum pressures
in the main science chamber.
The vacuum system consists of an oven, a transverse cooling stage, a variable-
pitch “spin-flip” Zeeman slower,1 and a science chamber (Kimball Physics 8 in “spher-
ical square”, MCF800-SphSq-G2E4C4A16). The main science chamber is separated
from the oven by two stages of differential pumping, each with a 75L/s ion pump
(Gamma Vacuum 75S), and a pump body adjoins the main chamber opposite the
Zeeman slower. This pump body contains a third 75L/s ion pump and a titanium
sublimation pump, along with an ion gauge and sensor package for a residual gas
analyzer (RGA). In routine operation, we find the titanium sublimation pump does
not help our pressure,2 which holds a steady-state value of ≈ 5× 10−11 torr, near
the sensitivity limit of the ion gauge (Agilent UHV-24P).
The oven contains two sections: a capped half-nipple “reservoir”, which holds
1This is a misnomer, since strontium has a J = 0 ground state and thus no spin to flip. In this
context, we mean the magnitude of the magnetic field goes through zero partway along the axis of
the slower.
2However, during the initial pumpdown, the titanium sublimation pump decreased the vacuum
pressure by roughly a factor of two.
14
≈ 10 g strontium, and a nozzle containing 496 hypodermic needles clamped together
in a V-shaped groove. Our design is based on the one presented in Ref. [24], and
further details can be found in Ref. [20]. The reservoir is heated to 600 ◦C by an
out-of-vacuum clamshell heater (Mellen Company Inc. 11C-1302-TC), and the noz-
zle is kept 75 ◦C hotter than the reservoir to prevent clogging. The entire assembly
is wrapped in several layers of hardened ceramic insulation (Cotronics #372-2), and
the temperature is monitored at several points by high-temperature thermocouples.
The temperatures at the reservoir and nozzle are each controlled by a proportional-
integral-derivative (PID) temperature controller from Omega (CNi16D44-EIT), which
is programmed via ethernet and actuates a solid state relay to apply or disconnect
power from the clamshell heaters.3 The advantage of this design is its mechan-
ical simplicity, however several oven designs exist in the literature that heat in-
vacuum [25], or utilize a 2D MOT and are much more compact [26]. In Chapter 6,
we discuss the next-generation strontium apparatus, which uses a compact commer-
cial beam source from AOSense.
The hot atomic beam is blocked by a pneumatic linear actuator (MDC 662006)
attached to a sheet-metal flag.4 We open the beam shutter only during the bMOT
load to reduce strontium buildup on the Zeeman slower viewport. Atoms are decel-
erated by the Zeeman slower to . 60m/s and captured into the bMOT, from which
point additional stages of laser cooling bring the dilute gas to ∼ nK temperatures
or to quantum degeneracy, if desired. A schematic of the vacuum assembly can be
3These temperature controllers seem ubiquitous at the JQI, however we have encountered a
rather frequent but intermittent bug in one which causes a programmed temperature ramp to
jump immediately to the end value. Power cycling the control box fixes the issue, but it remains
disconcerting.
4After years of operation, the rubber o-ring which seals the upper and lower pneumatic chambers
disintegrated. However, after a few minutes on the phone with MDC, we discovered this is trivial
to replace — the top cap unscrews from the pneumatic actuator, and the piston can be unscrewed
to access the o-ring. A few months after this failure, the pneumatic solenoid valve also failed, and
MDC was only able to replace it with a DC-operated equivalent. This solenoid is driven at 24V
by a monolithic NPN darlington transistor array (Texas Instruments ULN2803A) with an internal
free-wheeling diode.
15
Figure 2.2: Horizontal schematic of the SrI chamber-side optics system, reproduced from [20] with
modifications reflecting changes since its publication. The clock/689 nm spectroscopy laser is either
collimated (waist w ≈ {1.25mm × 1.71mm}), or focused (w ≈ {332 µm × 461 µm}) through the
chamber, by removing lens L1 (f = 250mm) which forms a relay telescope with the final 1" lens
(f = 250mm) before the chamber.
seen in Fig. 2.1.
The chamber-side optics can be seen in Figures 2.2 and 2.3. The rMOT and
bMOT beams are combined on dichroic mirrors and enter through the 2.75" CF
viewports. Their polarization is set with a two-color λ/4 waveplate (689nm and
461nm), and are retro-reflected on the opposite side of the chamber. Four 1064nm
dipole trapping beams, the “main” pancake, “cross” pancake, vertical beam, and
horizontal lattice, each enter along various axes and are combined as indicated with
dichroic mirrors or polarizers. The “loose” vertical dipole trap in Fig. 2.3 is not used
for the results discussed here, however previous results on the production of 86Sr
16
Beam Waist (µm)
wx wy wz
Main pancake 310 — 31
Cross pancake — 270 16
Vertical cross/Lattice 105 105 —
Horizontal lattice — 100 100
Table 2.1: Summary of dipole trapping beams and their waists.
BEC required this beam to avoid large three-body collisional losses during evapo-
ration [19, 20]. Each dipole beam passes through a Glan polarizer, and is intensity
locked through a near-normal-incidence pickoff or backside polished mirror.5 The
vertical lattice beam (see Fig. 2.3) passes through a liquid-crystal variable waveplate
(Meadowlark) on the underside of the chamber. By rotating the polarization, we
can dynamically ramp from dipole to lattice configuration by changing the amount
of power that is retro-reflected after passing through a polarizing beamsplitter. The
waists for each dipole beam can be found in Table 2.1.
2.2 Laser Systems
After the vacuum chamber, the next key ingredient to a cold-atom experiment
is the laser system used to trap and manipulate the atoms. Strontium, given its
rich electronic structure, is no exception and at times we have needed as many as
twelve separate laser systems to run an experimental cycle. In the next sections, I
describe some of these laser systems and provide some forward-looking advice for
improving the lock stability as the experiment transitions to the next generation
(see Chapter 6).
5We find the Glan polarizer crucial for achieving good intensity lock performance. Due to the
small difference in reflectivity of s– versus p–polarized light from the wedged pickoff, polarization
rotation can appear to the lock as intensity modulation and thus be written back onto the intensity
of the light.
17
Figure 2.3: Vertical schematic of the SrI chamber-side optics system, reproduced from [20] with
modifications reflecting changes since its publication. The “loose” vertical dipole crossing beam
is not used for the results discussed here, but is necessary for evaporation of 86Sr to avoid large
three-body loss.
18
2.2.1 Blue 461nm lasers
As we describe in Sec. 1.1, the broad 461nm transition is used for the first
stages of laser cooling, and for imaging. While the exact setup has changed period-
ically over the history of this apparatus, we currently employ three separate lasers
for generating light at 461nm — a frequency-doubled Toptica system (the TA-SHG
Pro), a Toptica ECDL diode laser (the DL Pro), and an injection locked diode from
Nichia (NDB4261).
The 461nm DL Pro is locked to a hollow-cathode lamp (Hamamatsu L2783-
38NE-SR6) via Doppler-free polarization rotation spectroscopy [28], see Fig. 2.4.
Because 88Sr is vastly more abundant than the other isotopes, and the 461nm isotope
shifts are small compared to the Doppler-broadened lineshape, there is only one
easily-resolved absorption feature for all four isotopes corresponding approximately
to the 88Sr resonance position. To provide additional flexibility, we reference the
Toptica SHG-Pro to this spectroscopy laser via beatnote. This is done with a
modified version of the circuit described below in Sec. 2.5.2. External frequency
control of the SHG-Pro is done via a slow piezo actuator, making it impossible to
phase lock these two lasers. However, we low-pass filter the output of the beatnote
circuit at ≈ 1Hz, such that we slave the Toptica SHG-Pro frequency to the reference
laser at long timescales and rely at shorter times on the good frequency stability of
the SHG-Pro. Because the linewidth of this transition is 32MHz, the lock is rather
forgiving. We notice performance in the MOT to depend marginally at the level of
5MHz offsets from the nominal operating frequency.
6When making this purchase, we were informed that Hamamatsu will no longer be manufactur-
ing these hollow-cathode lamps. We are hedging against this future on a few fronts: Peter Zhou,
an undergrad who has worked with us for the past few years, is building a spectroscopy setup
inspired by the dispenser design in [27]. We have also explored transfer cavity locks, whereby we
would reference both the 461 nm lasers and 679 nm/707 nm repump lasers to the stable 689 nm
system. Finally, the AOSense source described in Chapter 6 has optical access to do spectroscopy
transverse to the atomic beam.
19
λ/2
To AOM
Beatnote 110 MHz
461-nm To
Master Wavemeter
Toptica
DL pro HP
Wedged PDint. lock
Window λ/4
PDbalanced
λ/4 λ/2 λ/2
Hollow Cathode Lamp
λ/2
Figure 2.4: Schematic of the 461 nm DL Pro spectroscopy laser, reproduced from [20]. Light for
the polarization rotation spectroscopy is intensity locked after passing a cleanup cube to convert
polarization drift into intensity drift, which can be servoed. We find it crucial to have very clean
polarizations for our spectroscopy lock, and place a Glan polarizer after the polarizing beam cube
that splits light for the pump and probe arms. Additionally, a µ-metal shield is placed around the
hollow cathode lamp to reduce the coupling of magnetic field noise onto the lockpoint.
20
Beampath Frequency (MHz) Notes
bMOT 160 Double-passed from 80MHz
Transverse Cooling 182
Zeeman Slower 411
Imaging 211
Table 2.2: Summary of the frequency offsets used for each of the main 461 nm beam paths.
This laser locking setup suffers from a few drawbacks. First, while the beat-
note provides a great deal of flexibility, bootstrapping a low-bandwidth frequency
lock from a circuit nominally designed for high-bandwidth phase locking has two
performance drawbacks. First, the 1Hz low-pass filter between the lock circuit and
the laser limits the frequency slew rate. This filter is necessary because of the very
high voltage-to-frequency gain at the laser input, but sweeping the laser far enough
to lock to a different isotope requires a few seconds. Second, also due to the high
voltage-to-frequency gain, we must further attenuate the low-passed feedback sig-
nal. This limits the lock capture range, such that we are unable to fully sweep from
88Sr to 84Sr without manually recentering the lock. Both of these issues could be
resolved by implementing a feed-forward, or by moving to a fully digital solution.7
Most of the blue light for the experiment is generated by the Toptica SHG-
Pro laser system, shown in Fig. 2.5. Acousto-optic modulators (AOMs) provide
the necessary frequency offsets from the main laser; see Table 2.2 for the detunings
we use. We typically operate with ≈ 20mW to 22mW in the transverse cooling
and ≈ 8mW to 10mW per beam in the MOT (derived from a three-way PM fiber
splitter from Evanescent Optics), however the power out of this laser has been
slowly degrading since its installation. There are three imaging systems, two in
7For this kind of lock, a digital solution might provide a lot of benefits. We have discussed
building a frequency counter with an FPGA, perhaps using ideas from http://hamsterworks.co.
nz/mediawiki/index.php/High_Speed_Frequency_Counter, combined with the PYNQ platform
(http://www.pynq.io/) for easy integration into computer control. Briefly, a gray code counter
is synthesized in gateware, which is driven by the (digitized) beatnote. The output of this counter
sampled at the main FPGA clock frequency, and some arithmetic performed to extract the counted
frequency.
21
the horizontal direction (the low-magnification “Flea”, and the high-magnification
“Grasshopper”) and one in the vertical direction (the high-resolution “PIXIS”). Light
for these imaging paths is derived from two AOMs; the Grasshopper exists stand-
alone, but the Flea and PIXIS cameras require a flipper mirror to switch between
two fiber launches coupled from the same AOM. All beam paths also have shutters
(Stanford Research Systems SR475) for full optical extinction.
Light for the Zeeman slower is generated by injection locking a medium-
power laser diode (Nichia NDB4261). Approximately 10mW of 461nm light is
fiber-coupled from the main Toptica SHG-Pro system (see Fig. 2.5) and passed
through the output rejection port of an optical isolator (Thorlabs IO-5-461-HP).8
This injection-locked diode delivers ≈ 50mW to 60mW of light to the experiment,9
and is located on the vacuum system table to avoid losses due to fiber coupling.
Because the injection current window is small, we employ a crude injection
current locking scheme inspired by the system presented in Ref. [29]. Briefly, some
of the light is picked off on a polarizing beam cube and sent to a scanning Fabry-
Pérot cavity. When the laser is stably injection locked and single mode, the cavity
peaks have a characteristic height. As the laser drifts towards unlocking, the peak
height decreases and one begins to see secondary frequency modes creep in from
below. In Ref. [29], they use a microcontroller to monitor the peak height, and
stabilize the current to keep the laser locked.
Our system is essentially the same, however our laser does not exhibit the same
gradual decrease in peak height near the high-current edge of the injection locking
window. Rather, we find this lasers response to be more symmetric in current tuning,
8We had previously used ConOptics isolators for this purpose (711C-3), and still do for our red
injection-locked lasers. However, in two separate instances at 461 nm, we had an issue with the
Faraday crystal spontaneously cracking. This is quickly discovered, as the output mode turns to
speckle, and you can visually see discoloration and hairline fractures extending into the crystal
bulk. ConOptics was nice enough to fix our isolators in both instances, but we suspect some issue
with this model at shorter wavelengths.
9When this laser was first set up, it delivered closer to 70mW of power.
22
Toptica TA-SHG pro
PD Tolock Wavemeter
PDmon To
Transverse
λ/4 Cooling
λ/2
AOM
80 MHz
λ/4 Power Meter
λ/2 λ/4
λ/2 λ/2 AOM λ/2
AOM 183 MHz
211 MHz
λ/4 AOM AOM
λ/2 411 MHz 211 MHz
λ/2
To
To λ/2 bMOT
Zeeman
To Slave To
Beatnote Grasshopper 
To Imaging
PIXIS λ/2
Imaging To
Flea Imaging
Nichia 461 nm To Scanning FPI
Zeeman ToHigh Power Diode 
Slower Current
Feedback
λ/2 λ/2 From
Toptica To
TA-SHG Wavemeterλ/2
Figure 2.5: Schematic for the core 461 nm laser systems, reproduced from [20] with modifications.
23
and quicker to unlock. An Arduino Due, mounted to a Eurocard carrier card and
housed in a custom Eurocard rack,10 handles all aspects of the injection lock. The
Arduino Due is based on the SAM3X chipset from Atmel, and has two integrated
12-bit digital-to-analog converters (DACs), as well as several 12-bit analog-to-digital
converters (ADCs). Thus, with this single board, we can generate a ramp voltage
for the Fabry-Pérot piezo (which passes through an external op-amp to extend the
voltage range to 0V to 15V) and a feedback voltage for the diode current controller
(Thorlabs LD202C), and also read the photodiode voltage to calculate the cavity
peak height.11 The user can set the target peak height, the “unlocked” peak height,
the number of scans to average before implementing the feedback algorithm, and the
feedback stepsize. When the lock is engaged, the Arduino either does nothing if the
peak height is above the target threshold, or if this is not the case, it will step the
current by some amount. On the next scan, if the peak height decreased, the Arduino
will reverse the sign of its current step. This continues until the peak height is above
the target threshold. While crude, we find this locking algorithm to be sufficient for
most purposes. There is an auto-relock implemented as well (whereby a secondary
“unlocked” threshold targets larger current excursions), however we find this to not
work particularly well and have not invested the time to improving the performance.
Code and Eagle CAD files can be found on GitHub.12
Future improvements might include improving the locking algorithm, and in-
tegrating the circuit into the planned Eurocard “smart backplane”, such that the
Arduino talks directly to our custom current controller (see Sec. 2.5.1) rather than
via analog voltage to a Thorlabs current controller. Because the injected current
10This rack currently just supplies ±15V to the circuit, but eventually it will enable integration
with custom current controller electronics residing in separate modules, see Sec. 2.5
11The photodiode voltage is sampled quickly via the SAM3X Direct Memory Access (DMA)
engine, activated on a threshold condition. Thus, during a given scan of the cavity, the photodiode
voltages are written directly to a buffer in memory without directly involving cycles from the CPU.
12https://github.com/JQIamo/injection-ffwd
24
window for this diode is so small, the hysteresis in the Thorlabs current controller
frontpanel dial can be very frustrating when adjusting the diode to lock. It would
also be beneficial to have a mechanism for reporting the “unlocked” or “nearing
unlocked” state to the computer control system, such that a cycle could be auto-
matically suspended while corrective action is taken. We currently work around this
laser unlocking by closely monitoring the fluorescence of the bMOT, which drops
considerably when the Zeeman slower laser is not operating properly.13
2.2.2 Red 689nm lasers
The red 689nm laser system is centered around a DL-Pro which is locked
via the Pound-Drever-Hall technique [30] to a passive Fabry-Pérot cavity made of
ULE glass. This laser is close to resonance (up to an AOM shift) with the 87Sr
F = 9/2 – 11/2 transition, with the offset between the ULE cavity resonance and
the atomic frequency provided by double-passed 1GHz, high-frequency AOM (Brim-
rose).14 Light is divided between an injection lock (a few mW) and a network of fiber
beamsplitters which provide reference light to several beatnote-locked slave lasers.
The primary set of electronics for the PDH lock are provided by Toptica: a FALC
analog PID, and a PDD-110 module for demodulating the PDH error signal. We
also use a slow integrator from SRS to keep the fast current feedback signal from the
FALC centered, by feeding back to the laser piezo. We find this to work better than
the built-in “infinite integrator” in the FALC, which seemed to invariably cause a big
mode hop whenever the laser happened to come out of lock. The linewidth of this
13Other groups have had similar issues keeping this non-AR coated 461 nm diode happily injec-
tion locked, and have switched to using the more expensive AR coated version instead [22]. In this
configuration, the effective power gain of the diode is much lower, but because the lasing threshold
current of the AR-coated diode is so high, the mode is set exclusively by the frequency of the
injected light.
14This AOM is very temperamental and polarization sensitive, which makes double-passing
difficult. As discussed in Chapter 6.2, a much more versatile option would be to lock one laser
directly to the cavity (or through a phase-modulated sideband generated by a wide-bandwidth
EOM, as is done in Chapter 3.3) and beatnote-reference slave lasers as needed.
25
laser was measured via a heterodyne beatnote with a temporary 689nm diode locked
to the clock laser cavity described in Chapter 3, and found to be . 200Hz which is
more than sufficient for addressing the 7.5 kHz intercombination line transition.
In addition to the injection-locked diode mentioned above, which generates the
rMOT light for 87Sr, we primarily use light from the “long Steck” laser for rMOT
operation (see Fig. 2.6). Specifically, this laser provides the trapping light for all
bosonic isotopes, and the stirring light for the fermionic rMOT (see discussion in
Sec. 1.1.2). It is a home-built ECDL based on the design from [31]. The diode is
a cheap, low-power, single mode diode (HL6738MG) that we have anti-reflection
(AR) coated in the University of Maryland FabLab. The coating recipe is the one
suggested in Ref. [32]. We remove the hermetic cannister and deposit a layer of
Al2O3 to bring the facet thickness to ≈ λ/2, followed by a final layer of HfO2 to
bring the thickness to λ/4 (or an odd multiple thereof). The deposition is monitored
in situ by actively scanning the current of the laser diode around threshold, and
monitoring the response on the integrated photodiode. As the front facet becomes
a better reflector, the threshold current decreases. When we observe the threshold
current start to increase again, we halt the deposition of Al2O3 and begin depositing
HfO2 until the threshold current is pushed as high as possible. Deposition is done
in an electron “E-beam” evaporator, and the diode is held (along with up to four
other diodes) in an aluminum mounting clamp affixed to the top of the deposition
chamber. Electrical signals for monitoring the photodiode and scanning the laser
current are provided via vacuum feedthrough. We find this technique to be rather
robust, and has generated several successful AR coatings for diodes at 689nm and
679nm.
26
EOM
λ/4 λ/2 20 MHz
50 mm 100 mm λ/2
ULE Optical Cavity PDPDH
PD in Vacuummon
λ/2 λ/2
Toptica DL pro
S λ/2
P RI DS To Slave Faraday
689 Injection Rotator
P
1 D10H FALC To Beatnote AOM
Network
Toptica 1 GHz
Electronics
Long Steck Short Steck 689 Injection
To Beatnote To Beatnote To
Network Network Wavemeter
λ/4
λ/2 λ/2 λ/2
To From
rMOT To Master To
AOM Exp. AOM AOM rMOT
80 MHz 100 MHz 80 MHz
λ/2 λ/2 λ/2
Figure 2.6: Intercombination line laser systems, reproduced from [20] with modifications. A Top-
tica DL Pro (the “red master”) is locked to a ULE cavity, and is used to reference several slave
lasers highlighted in green. The beatnote network (not shown) is a network of butt-coupled fiber
splitters which takes reference light from the red master and distributes to the several beatnote
locks. It also splits light to the fiber switch (shown in Fig. 2.7) for monitoring on the wavemeter.
27
2.2.3 Repumping lasers
As discussed in Section 1.1.1, efficient laser cooling of strontium requires re-
pumping out of the metastable 3P2 level populated during operation of the bMOT.
Several repump schemes exist for strontium [13,33–35], which use some combination
of 3S, 3P, or 3D states. We follow the scheme used at JILA and elsewhere, pumping
out of 3P2 via the 3S1 level at 707nm. Unfortunately, 3S1 can decay to any of the
5s5p 3P states, and so a second repump at 679nm is required to repump out of the
metastable 3P0 level as well. Other repump schemes address states which can only
decay to 3P2 or 3P1 via selection rules, and so only require one laser. However, the
additional complexity of a second repump is somewhat mitigated by the availability
of cheap, single-mode diodes at the 679nm and 707nm wavelengths.
Both repump lasers are homebuilt Littman-Metcalf ECDL systems. The
707nm diode was wavelength selected and AR-coated from Toptica, while the 679nm
diode was AR-coated by our lab as described above. Light from the 707nm ECDL is
coupled to a secondary injection-locked system to increase the available power, and
passed through a fiber EOM for reasons discussed below. Light from the 679nm
laser passes through a 200MHz AOM and is brought to the experiment via SM
fiber. The reason for the AOM on this repump arm is to eventually gate the re-
pumping of atoms from 3P0 for imaging both ground– and excited-state atoms in
experiments with the 698nm clock transition. Both lasers are locked to a wavemeter
(High Finesse, WS7) via custom LabView codes.
Repumping of the fermionic isotope is complicated by the hyperfine interac-
tion, which splits 3P2 into five manifolds and 3S1 into three manifolds, as discussed
in Sec. 1.1.1. We initially used a technique inspired by an old JILA method, whereby
the 707nm laser was made multimode by driving a voltage ramp on both the piezo
and current modulation ports. By monitoring the bMOT fluorescence while adjust-
28
From Wavemeter
Lasers High Finesse
Labview
DAC
WS7 VI
Locking Electronics
16x1 Fiber Switch
679 nm Repump
AOM
λ/2
To
To Exp.
Wavemeter
707 nm Repump Master 707 nm
Injection Slave
λ/2
To λ/4
Wavemeter
λ/2 λ/2
To To
Exp.
λ/2 Wavemeter
Fiber EOM
Figure 2.7: Repump laser systems, reproduced from [20] with modifications. The 707 nm repump
master seeds a secondary injection locked diode, which produces rempump light to the apparatus.
For experiments using 87Sr, a wide-bandwidth fiber phase modulator adds sidebands to bridge the
few-GHz hyperfine splitting in 3P2. Both the 707 nm and 679 nm lasers are frequency stabilized to
a wavemeter.
29
...
ing the center of both the laser current and piezo, decent repumping efficiency could
be achieved. However, this technique was not very robust or repeatable, and as the
laser drifted, so did the repumping efficiency. To circumvent this issue, we now use
a wideband fiber EOM (Jenoptik, PM705) to address the repumping transitions in
a controlled manner.
Several groups use some variation of the EOM repump scheme for the fermion [22,
23], but we follow the method outlined in Ref. [22] after discussions with the author
at DAMOP 2015. A high-frequency DDS evaluation board from Analog Devices
(using the AD9914 chip, which is a 12-bit, 3.5 GSPS DDS) is controlled by an
Arduino-based microcontroller via SPI. Using the built-in frequency profiles in the
AD9914, the microcontroller pre-programs the DDS with a series of frequencies and
amplitudes (which can optionally be changed before each experimental cycle). It
then toggles the profile-select pins to change the output frequency. We find decent
repumping efficiency locking the laser frequency at 423.913THz, and cycling the
EOM frequencies between 655MHz, 1240MHz, 1510MHz, and 1720MHz with a
dwell time of 20ms per frequency. This is slightly different than the values reported
in Ref. [22], which were similarly determined via empirical methods. The efficacy of
this technique is measured by steady-state bMOT fluorescence, and we find we are
limited primarily by the amount of optical power that can safely be passed through
the fiber EOM.15
15We initially damaged one of these EOMs, which then had to be sent back for repair. It turns
out the damage threshold for CW power is wavelength-dependent, and the value in the datasheet
was optimistic. We currently operate with between 4mW to 5mW through the fiber. It is also
worth noting that the mechanical mate between the PM fiber pigtails attached by Jenoptik and
the standard FC connector is not good, and the butt-coupling efficiency is maximized when the
connector is slightly unscrewed. This would benefit greatly from a dedicated free-space launch
between fibers, or launching directly into the fiber EOM, but we currently have enough CW power
to make this less important.
30
2.2.4 Dipole trap laser
The optical dipole trap, used in the final stages of the experimental cycle,
is formed with light from a 30W fiber laser at 1064nm (IPG YLR-30-1064-LP-
SF). A series of polarizing beam cubes splits light from the laser head into four
arms, each of which pass through an AOM (either IntraAction AOM-402AF4 or
AOM-302AF4) and are coupled into a large-mode-area (LMA) single mode fiber
(Coastal Connections S- FAmkFAmk-10nx/130/3-7.6).16 The AOM provides power
stabilization and fast turn-on/turn-off, while a slower mechanical shutter built off
the design in Ref. [36] provides full optical extinction.17 A schematic of this laser
system is shown in Fig. 2.8.18
The power stabilization servo is the same FPGA-based design in Ref. [37].
Light from a backside-polished mirror or wedged pickoff is incident on a battery-
powered photodiode. The transimpedance conversion is achieved with a fixed BNC
stub terminator (either 5 kΩ or 10 kΩ), and the resulting voltage signal is passed
to a custom-built op-amp subtractor circuit. This generates an error signal by sub-
tracting the desired setpoint voltage from the computer-controlled DACs. Finally,
16Many groups use photonic crystal fibers in this situation, but we’ve had mostly good luck with
these LMA fibers. The advantage is they are very cheap by comparison, however we have burned
the fiber tip on two separate occasions. One of these was a very perverse situation — an AOM
was turned off, which diverted more power to a beam dump near an upstream mirror of a separate
arm. Heat from the beam dump was enough to steer the beam off the fiber core and burn the tip.
Care must be taken to align these at low power, and prevent high-power back reflections on lattice
arms from burning the output fiber face as well.
17These shutters worked nicely for about one to two years, before they started sticking intermit-
tently and generally falling apart. We are considering building a replacement based on the very
tiny bi-stable solenoid shutter from Takano (BOS10/15). A polished sheet metal flag will replace
the anodized flag that comes with this device to avoid excessive heat dissipation in the shutter
itself.
18One piece of lab lore which is worth passing along: typically, the crystal in an AOM is angle-
cleaved to prevent a reflected acoustic wave. However, this can never be perfect — in our devices,
it is ≈ −50 dBc. The result is light which has been shifted to lower frequency appearing in the
+1 order of the AOM. In most cases this will have no effect, but if two arms of your dipole trap
are driven on opposite AOM orders at close to (but not the same!) frequency, you end up with a
running wave which heats atoms quickly out of the trap.
31
λ/2 AOM λ/2 λ/2 AOMAOM
45 MHz 35 MHz30 MHz
λ/2 AOM
40 MHz
To V.
Lattice/Cross
From To Main To H.
Fiber Pancake Lattice
Laser To Cross
Pancake
Figure 2.8: 1064nm dipole laser system, reproduced from [20].
the error signal is passed to one of the digital servo inputs. Using a separate sub-
traction circuit allows us to run four dynamically-controlled power servos on two of
the FPGA-based boxes.
In Chapter 5, we discuss a dithering procedure used to spectroscopically probe
the atomic sample in the absence of AC stark shifts from the dipole trap. This is done
with an Teensy Arduino microcontroller mounted onto a custom breakout circuit,
which provides override logic for the TTL state of the AOM driver on each dipole
arm. When triggered by the computer control system, the microcontroller toggles
the sample-and-hold input of the FPGA servo in conjunction with the output TTL
control of the AOM driver with a programmable duty cycle.
2.3 Magnetic field coils
Beyond the appropriate laser systems, which generate optical fields to probe
and manipulate our atomic samples, magnetic fields are another handle we have
to interact with the atoms via the Zeeman effect. Indeed, well-controlled magnetic
32
fields are essential to the proper function of a magneto-optical trap. Our apparatus
is no exception, and includes wire-wound coils for two field regimes: low-field shim
coils, and high field coils that can be switched from (nearly) Helmholtz to (nearly)
anti-Helmholtz configuration.
Details on the original coil setup can be found in Ref. [19]. Low-field shim coils
are wound in pairs directly onto the vacuum flanges and are driven by an Agilent
E3614A current supply, which can be dynamically controlled via analog voltage from
the computer control system. These coils can handle a few amperes of current, and
generate fields up to a few gauss. Their primary purpose is to cancel stray fields in
the lab, and during the final stages of laser cooling, shim the equilibrium position
of the red MOT to spatially overlap the atoms with the dipole trap. The vertically-
oriented z-shim coil is also used to apply small quantization fields (up to ≈ 6G)
during spectroscopy.
The high-field coil pair is constructed from hollow-core copper tubing wound
onto a form and inserted into the recessed vacuum viewports to get closer to the
atoms, and produces a field gradient of 0.96Gcm−1 A−1 in the anti-Helmholtz config-
uration. They are driven by a TDK-Lambda GEN40-125 power supply (in constant
voltage mode), and a bank of high-power MOSFETs controls the current flowing
to the coils. The gate volate of these MOSFETs is controlled by a simple PI servo
developed by other groups at the JQI, and includes a fast turnoff feature which
shunts the gate to ground. A snubber circuit shunts the inductive load into 5 Ω,
which gives an ≈ 75 µs time constant for turnoff. This fast switching is important,
because the rMOT and bMOT operate at very different field gradients (≈ 60Gcm−1
versus ≈ 1.5Gcm−1). A closed-loop chiller forces cooled water through the coils to
keep them from overheating.
We have recently added the ability to switch the relative polarity of our two
MOT coils, to go from an anti-Helmholtz configuration to a Helmholtz configura-
33
tion. This allows us to use these coils to apply very large fields, up to ≈ 200G,
when working with the clock line in the bosonic isotopes of strontium (see Chap-
ter 5). This is achieved with an H-bridge built out of high-current Contactor relays
(LEV200A4NAF). The relays have a switching time of ≈ 50ms to 75ms, which must
be interlocked with the coil drive to avoid switching at high current. The interlock
and control of these relays is provided by a chip-programmable logic device (CPLD),
which serves two functions: first, it will not change the state of the H-bridge unless
the high-current servo output is disabled, and second, it holds the output of that
servo to off until the H-bridge has had sufficient time (100ms, for safety) to fully
switch. The CPLD implements this logic in a programmable fabric similar to an
FPGA, but which doesn’t require re-initializing after a power cycle.
2.4 Computer control
Computer control of the apparatus is achieved by a heterogeneous collection of
equipment coordinated by a centralized pulse generator to synchronize the timings.
This structure is common amongst neutral atom experiments, since control is done
on a per-shot basis and is almost always able to be pre-compiled into a sequence of
steps that must happen at a specific time. In our lab, the experimental sequence
is programmed in a custom LabView environment that has been built up over the
years by various groups at the JQI and NIST, and is collectively known as SetList.
Documentation and source code can be found on Github [38].
Each shot of the experiment involves a number of steps, defined per line of the
SetList, which specifies the output state of every device connected to the control
computer. Examples of devices include National Instruments DAQ cards, which
provide analog and digital I/O, Arduino-controlled DDS synthesizers, CONEX linear
actuators, scientific CMOS cameras, and SRS function generators, although the
advantage of this architecture is any device can (in theory) be added, as long as
34
it can receive a timing trigger and be programmed to execute something specific
on that trigger pulse. The “pseudo-clock” timing device is a PulseBlaster, which
provides 24 digital outputs that can be used either as standalone TTL outputs or
as triggers for other devices. The rising edge of a trigger pulse can be specified with
10 ns resolution, and the device outputs a 50 ns pulse per trigger event. We optically
isolate and buffer all digital outputs, and buffer all analog outputs with BUF634
amplifiers.
An experimental cycle involves programming all devices, starting the pseudo-
clock triggers, and acquiring an image at the end. Processing of these images is
done in IGOR to extract relevant properties of the atomic gas at the end of the
experimental cycle.19 The setlist is parametrized by any number of variables defining
the state of a device on a particular line. The top-most VI of SetList wraps the
individual shot engine with a sequence generator, allowing randomized or linear
scans of any number of variables. It also has the ability to accept a set of variables
from some outside program (such as the analysis program) to achieve closed-loop
operation, however we have never actually used this feature.
2.5 Custom electronics
Over the course of my degree, I have had occasion to design several custom
circuits for laboratory instrumentation. Here, I describe a subset of them, especially
as they pertain to laser control. One such design for an ultra-low noise, high-voltage
piezoelectric driver, was published in Rev. Sci. Instr. and appears as Chapter 4 of
this thesis. Other circuits which I have had considerable involvement with the design
and construction, but which are not covered here, can be found on Github.20 We
19Our analysis scripts are available at https://github.com/JQIamo/sr-scripts, however we
have started to migrate some of these to Python for future work.
20These include a full-featured AOM driver (https://github.com/JQIamo/aom-driver) for
which a technical note is also available (https://github.com/JQIamo/aom-driver/blob/master/
35
have also made heavy use of Arduino-based microcontrollers, particularly the Teensy
fork of the Arduino project,21 for easy integration of new hardware peripherals with
our computer control system. Firmware, hardware, and associated design files can
be found on the JQI organization GitHub account [39].
2.5.1 Laser diode current controller
External-cavity diode lasers have three frequency control points: diode tem-
perature, diode current, and the length or grating angle of the external cavity [32].
Thus, under typical operating conditions, one needs three active control drivers, one
for each frequency knob. Here, I discuss our solution to the second of these con-
trols, namely, the laser diode current. The atomic physics community has gotten
substantial mileage out of a design originally due to Libbrecht and Hall [40], core
pieces of which have even found their way into commercial drivers (e.g., the one from
Vescent Photonics). Recently, several groups have made improvements and updates
to the original analog design to incorporate digital control [41] and improve noise
performance near the current limits [42]. Inspired by these efforts and the desire to
port the design to a Eurocard form-factor, we re-implement the key aspects of the
design with some modifications discussed here. The ultimate noise performance is
similar to that reported in the literature [41], and outperforms many commercial
drivers.
The basic analog design is shown in Fig. 2.9(a). A low-noise op-amp senses
the voltage drop across a precision resistor, Rsense, and adjusts the gate voltage of
a MOSFET such that the voltage drop matches the voltage Vset. Thus, the output
aom_driver.pdf), and a linear temperature controller (https://github.com/JQIamo/Linear-
Temperature-Controller based on the Wavelength Electronics WTC3243 chip.
21See the project website, https://www.pjrc.com/teensy/, for more information.
36
Vsupply To current
Gain adjust mod. inp.
U2
Vset POT
U1
FFWD
-FFWD
Iout = (Vsupply - Vset)/Rsense
To laser diode
(a) (b)
Figure 2.9: Key analog subcircuits for the low-noise current controller. In (a), we show the
basic current controller originating from the Libbrecht-Hall design. The voltage across a precision
resistor, Rsense, is sensed by an op-amp and controlled by feeding back to the gate of a low-noise
MOSFET. The current setpoint is controlled by the voltage Vset, which is referenced to the supply
voltage Vsupply. In this way, noise on the supply becomes common-mode, greatly reducing the noise
that appears on the output current. In (b), we show our current feed-forward circuit. A voltage
from the piezo driver (see Chapter 4) is mirrored across a potentiometer. Tuning the wiper of the
potentiometer allows continuous tuning of the gain from G = −1 to 1. This feed-forward current
is summed with an external modulation input and the DC current provided by the schematic in
(a) to generate the final current output.
current is given by
Vsupply − Vset
Iout = . (2.1)
Rsense
The Libbrecht-Hall design also includes a very clever current modulation circuit,
which we will not discuss here, but instead refer the reader to the original paper [40]
or to the excellent discussion in [41].
The setpoint voltage, Vset, is controlled by a 16-bit DAC (AD5541) with excep-
tional noise performance. To reduce the differential-mode noise between the DAC
and the laser current supply regulator, the DAC is referenced to the supply voltage
via an ovenized Zener diode reference (LM399H).22 An op-amp buffers the nega-
tive terminal of the Zener reference, and provides a floating virtual ground for the
22This particular voltage reference appears ubiquitously in similar electronics designs. However,
we have noticed it to still be rather sensitive to mechanical and temperature perturbations. In
future iterations, I suggest looking into the LTC6655 or LTZ1000 devices from Linear Technology
as possible alternatives.
37
Rsense
DAC. To handle the reference level translation between this IC and the onboard
microcontroller, which digitally controls the voltage setpoint, a pseudo-differential,
quad-channel op-amp shifts the microcontroller SPI lines to the floating ground
reference. The microcontroller also handles user interaction from the frontpanel, in-
terfacing with a digital rotary encoder, switch, and a low-profile LCD display. The
rotary encoder includes a pushbutton, which enables simple menu navigation, e.g.,
to set the maximum current range.
In contrast to previous designs, we have elected to use a newer, programmable
output low-noise voltage regulator from Texas Instruments (TPS7A4700) rather
than the old, albeit low-noise, LM317. This new LDO voltage regulator can supply
up to 1A, has a digital enable/disable, and the operating voltage can be adjusted by
the jumper array J3 (see the schematic on GitHub [43]). Thus, one can reconfigure
the operating voltage with a convenient overhead relative to the laser diode voltage
drop to avoid dissipating too much heat through the MOSFET. It has a very low
output noise (4 µV RMS, 10Hz-100 kHz), which we reduce further by filtering across
50 µF.
The primary upgrade we have made over previous designs is to implement an
adjustable-gain feed forward from the voltage supplied by a piezo driver in the same
rack. The lasing frequency of an ECDL is determined by mode competition between
the external cavity, the “internal” cavity formed by the (imperfectly anti-reflection
coated) diode front facet, and the intrinsic gain profile of the diode itself. Physi-
cally, this manifests itself in a mode-hop free tuning range of the laser, that is, the
frequency over which the laser can be continuously scanned. Because the “inter-
nal” and “external” cavities are controlled by two separate knobs (the former by the
diode temperature/current, and the latter by the voltage applied to a piezoelectric
actuator controlling the external cavity length), one must scan both concurrently to
remain on the same lasing mode. Typically, this is done by adjusting the laser cur-
38
rent proportional to the external cavity piezo voltage such that the internal cavity
resonance tracks the external cavity resonance. For a good pedagogical discussion
of this effect, I refer the reader to the recent discussion in [44].
Implementation of this feed forward is complicated in our lab by two effects:
first, the feed-forward gain is not necessarily constant across diodes, even of the
same type. Second, we have both Littrow and Littman-Metcalf style ECDL lasers,
which have an opposite sign in the action of their external cavity.23 To address both
design challenges simultaneously, we implement the schematic shown in Fig. 2.9(b).
An op-amp (U1 in Fig. 2.9(b)) inverts the externally-supplied feed-forward voltage
and drives the opposite end of a trimpot. By adjusting the trimpot wiper, one can
continuously adjust the feed forward gain between ±1. The signal from the trimpot
is buffered, and sent to the analog modulation input summation node of the current
controller, and is added in with an optional external current modulation signal. If
one does not wish to use the feed-forward feature, the node can be shunted to ground
with a low-profile jumper to avoid introducing additional noise from EMI pickup in
the lab. This trimpot is exposed on the frontpanel of our circuit, making it easy to
adjust after installation with a laser.
The whole circuit is packaged into a standard 12HP-wide Eurocard module,
and is designed to reside in a rack with the piezo driver described in Chapter 4.
A divided down monitor of the piezo voltage can be routed via the backplane to
the current controller to drive the FFWD node in Fig. 2.9(b). One can either tune
the feed-forward gain with it engaged, by slowly scanning the grating piezo of the
ECDL and adjusting the gain trimpot to maximize the mode-hop free tuning range,
or offline by plotting the necessary current corrections to remain on the proper laser
23Specifically, we use an old JILA design for the Littman-Metcalf lasers, which at this point shold
be considered a “legacy” model and only houses our less-sensitive repump diodes. Newer lasers,
and all of our narrow-linewidth systems, implement the design from Daniel Steck’s group [31].
Between these two models, the frequency tuning is inverted for a positive voltage change ∆V on
the piezo.
39
mode as the grating piezo voltage is increased. The slope of the resulting line gives
the appropriate feed-forward gain; see, for example, Ref. [44].
The final performance of our driver is shown in Fig. 2.10. The spike near 50 kHz
is likely spurious, due to pickup from the CRT display of the SR780, however we
have not made a thorough investigation of this yet.24 The total RMS-integrated
noise from 1Hz to 10 kHz is 67.2nA. Below ∼ 100Hz, we see the performance
√
dominated by 1/f flicker noise, with a power spectral density of ≈ 20 nA/ Hz at
√
1Hz. Above ∼ 100Hz, the noise has a white spectrum with roughly 1 nA/ Hz noise
spectral density.
2.5.2 Optical beatnote servo
The design for this board originated with the circuit presented in Ref. [45].
Electronics design files, CAD files for rackmount enclosures, and firmware for our
implementation can be found on Github [46]. The basic idea is to use a ubiquitous
phase-locked loop (PLL) chip from Analog Devices as a phase/frequency discrimi-
nator, which generates an error signal for a tunable proportional-integral (PI) loop
filter that then feeds back to a laser. In this configuration, the RF signal fed to the
PLL is generated by an optical beatnote between a reference laser and the laser to
be locked; in effect, we are treating the laser as a VCO which has been demodulated
from optical to RF frequencies with a reference laser.
Our design makes several improvements upon the design in Ref. [45]. First,
we utilize a limiting amplifier (Texas Instruments ONET1191P) to perform signal
conditioning and reduce the sensitivity of the lock to power in the beatnote.25 The
24In chatting with students from Daniel Steck’s group at DAMOP 2016, one of them mentioned
they saw a very similar feature when using that spectrum analyzer. We have not had a chance to
verify this hypothesis, however.
25This chip provides up to ≈ 40 dB gain, while clamping the output at a customizable maximum
power, roughly −5 dBm in our case, to protect RF components further down the signal chain. It is
reasonably quick, up to ≈ 11GHz, and has the advantage of driving a differential output which is
40
G = 981
R
meas
Spectrum Analyzer
Figure 2.10: Measured noise power spectral density for our current controller. The output is passed
through a 50 Ω resistor, Rmeas, and the voltage drop across Rmeas is AC-coupled and monitored
by a low-noise instrumentation op-amp (AD8421) with a gain of G = 981, see inset. The signal is
then sent to a spectrum analyzer (Stanford Research Systems SR780), which records the voltage
noise power spectral density. This is converted to a current noise power spectral density by Ohm’s
law. The large peak around 50 kHz was mysterious, since we had not seen it previously. However,
there is some indication it may be due to spurious pickup from the CRT display of the SR780 (see
text).
41
board layout is such that reconfigurable components to change loop gain parameters
are soldered on the bottom side, making them easy to swap. To account for long-
timescale drifts, a slow integrator is implemented which feeds back to the piezo of
an ECDL (as in [45]), however we add the option to reverse the sign of the slow
integrator response with respect to the fast current modulation output. This is
useful in our lab, as we have a combination of Littrow and Litman-Metcalf ECDLs
which have oppositely signed piezo response. A small IDC header adapts to a
microcontroller, which handles user input from the frontpanel to reprogram the
PLL. Our board is pin-compatible with a line of Analog Devices PLL chips covering
50MHz to 7GHz.26 The reference frequency is supplied by a custom-built DDS
based off the AD9910 chip.27 In combination with a fixed multiplier from the PLL,
the DDS can be used to sweep or hop the frequency of the beatnote between or
during experimental cycles.
To generate the optical beatnote, we have traditionally used the fiber-connected
biased photodetectors from thorlabs (e.g., DET025AFC), into a 50 Ω termination and
then to a Minicircuits low-noise amplifier (e.g., ZX60-P103LN+) to pre-amplify the
signal before sending it to the beatnote circuit. A directional coupler located just
after the amplifier gives a monitor signal, which is displayed on a spectrum ana-
lyzer (either individually or multiplexed with other beatnote signals, see Fig. 2.11).
However, we have recently discovered a much superior solution. Hamamatsu sells
the preferred signal input mode for the phase-locked loop chips we use. There is also a loss-of-signal
detection feature, which is not currently used, but might be useful in future iterations to detect
low beatnote power.
26These chips are the ADF4110-4113 series, plus the ADF4107. Analog Devices sells PLLs that
operate as high as 18GHz, and one might be able to adapt this design to work at those frequencies
with some more careful consideration of RF PCB layout. The distinct advantage of this approach
is the feedback loop filter does not need to work at high frequencies; rather, the PLL demodulates
the beatnote signal to generate an error signal which can be fed into a standard PID controller.
27For files, see https://github.com/JQIamo/ad9910-dds. This board was designed by Carlos
Bracamontes and Alessandro Restelli, and provides digitally-synthesized RF waveforms up to ≈
400MHz.
42
60 〈Me〉an-square phase error:
φ 2 = 0. 100
70
80
90
100
110
120
1460 1462 1464 1466 1468
Frequency (MHz)
(a) (b)
Figure 2.11: Sample trace for optical beatnote lock. On the left, we see the typical performance
(power averaged over five traces, 1 kHz RBW, 10MHz span). The mean-square phase error, an
indication of the lock performance, is computed to be 〈φ2〉 = 0.100 (see text). The lock bandwidth,
as indicated by the position of the servo bumps, is roughly 1MHz in this trace. It is possible to push
the peak of the servo bump out to a few MHz, however the laser performance degrades as measured
by atom number in the rMOT and similarly the mean-square phase error increases. We find the
optimal performance corresponds to adjusting the gain such that the servo peaks are flattened but
close in to the carrier. On the right is a screenshot from our web-based monitoring program. A
small webserver written in Python directs a multiplexed RF switch to alternate between monitor
channels, which are then sent to a Rigol spectrum analyzer. The program synchronously pulls
traces from the spectrum analyzer over TCP/IP to display in the web browser. This allows us to
simultaneously monitor several beatnotes from anywhere in the lab.
a GaAs metal-semiconductor-metal (MSM) photodetector with a 30 ps risetime, in
a convenient SMA-connected package (part number G4176-03).28 The bias voltage,
≈ 7V, is supplied through a bias-tee mounted inside the box with the remaining
beatnote circuitry. The only disadvantage is the photodiode is not fiber-coupled;
however, one can easily rig a fiber launch assembly and clamp the photodiode us-
ing standard optical mounting components. Alternatively, and I suggest this for
others wishing to implement this setup, a custom machined mounting plate could
be designed and ordered from any of the fast-turn CNC prototype companies for a
nominal price.29 Using this photodiode and a beatnote board sporting the speedy
28See their website for details, https://www.hamamatsu.com/us/en/product/category/3100/
1025/G4176-03/index.html. At the time of our purchase, they were selling for $ 174, which is
cheaper than the much slower version from Thorlabs.
29We have had good success, on both price and quality/leadtime, with Protolabs. However, their
sales representatives are perhaps over-eager and will call you more than necessary about “future
projects”!
43
Beatnote power (dBm)
ADF4113 PLL, we have easily locked a beatnote around 2.6GHz.30
We show the typical performance of our circuit in Fig. 2.11. The performance
figure of merit, in this case, is the the mean-square phase error 〈φ2〉, defined via the
equation [45,47]
( )
exp −〈φ2〉 ∫Pcarrier= . (2.2)
P (ν)dν
Physically, it is a measure of how much power is in the delta-function peak, Pcarrier,
relative to the amount of power in the rest of the beatnote spectrum integrated over
frequency ν. Ultimately, the lock quality is decided by the atomic response. Here,
we primarily use this circuit to offset lock lasers addressing the narrow (≈ 7.5 kHz)
intercombination transition in strontium. We do notice some degradation of the
MOT performance if the beatnote servo bumps are too pronounced; presumably, this
corresponds to the onset of an oscillation. This behavior is strongly determined by
where your laser is in {piezo, current} space. After locking both the high-frequency
current feedback and the low-frequency piezo “infinite integrator”, increasing the
laser current tends to reduce the effective volt-to-frequency gain (and vice versa) as
the piezo is driven to a new operating point to compensate. For the “Long Steck”
laser (see Fig. 2.6), we tend to operate with the beatnote gain turned all the way up
but then push the servo bumps low and closer in to the carrier by increasing the laser
diode current after the system is locked. For the “Short Steck” lasers, depending
on the frequency, the beatnote gain does not need to be turned very high, but can
also benefit from adjusting the current after the laser is locked.31 In an attempt to
quantify this heuristic, we have recently added a measure of 〈φ2〉 to our beatnote
30This is as high as we’ve had any reason to go, however, the Rubidium Rydberg lab uses this
board with the ADF4107 PLL and a fast commercial photodiode to lock at 6.8GHz.
31While not very scientific or repeatable in an absolute sense, this is just one of many personality
quirks one becomes accustomed to in dealing with particular lasers! Enough time spent staring at
the beatnote monitor spectrum gives you a sense of what a good lock looks like in terms of MOT
atom number and temperature.
44
spectrum multiplexing software, see Fig. 2.11b. A useful feature to implement in
the future is a long-term logger of 〈φ2〉, which could be correlated back to atom
number to say something more quantitative about the lock performance.
While these beatnote circuits are generally pretty robust, there are a few
improvements which should be made in any future version. First, integrating the
microcontroller on-board (such as in the high-voltage piezo driver, described in [48]
and Chapter 4) will greatly simplify the box assembly, and enable a more tightly
integrated electronics package. Second, I suggest moving to a tunable PI controller,
like the one implemented in [49].32 We have found the “default” PLL loop filter
components suggested in Ref. [45] to work well, but depending on the laser, the
overall loop gain is easily too low or too high. And the current method, to adjust
the final gain by attenuation through a potentiometer, is problematic from a signal-
to-noise perspective but also because, in the case of the unlimited piezo integrator,
one often wants a very small gain but then is hampered by a very small dynamic
range.
There are a few layout improvements which will be useful for diagnosing prob-
lems. First, placing probe/testpoints after the limiting amplifier could rule out
many signal integrity issues, which can arise especially if the PCB is populated by
hand. Second, better attention to how this board will plug together with other RF
components in the final box will reduce the headache in assembly. However, both
of these are cosmetic modifications.
32This, in turn, is based off the implementation of Joe Tiamsuphat in the famous Joe-PI, see
https://github.com/JQIamo/analog-pi.
45
Chapter 3: Design and construction of a narrow-linewidth clock laser
As discussed in the introduction, strontium is an appealing system for atomic
physics because of its somewhat unique combination of atomic properties. While
it doesn’t have nearly as many stable isotopes as ytterbium, it does have three
bosons with scattering lengths that span almost the entire spectrum from near
zero (a88 ' −2a0) to near unitarity (a86 ' 823), and a fermion with an unusually
high nuclear spin I = 9/21. The optical electronic transitions similarly span a
wide range, from the broad-line cooling transition at 461nm (Γ/2π ≈ 32MHz)
to the narrow-line intercombination transition at 689nm (Γ/2π ≈ 7.5 kHz) to the
mHz-wide clock transition at 698nm. These disparate transition linewidths provide
handles for manipulating strontium with varying levels of coherence and control, and
arise from dipole selection rules that are broken by terms in the Hamiltonian coupling
the “forbidden” level to levels with fully dipole-allowed transitions. For example,
spin-orbit induced state mixing between 1P1 and 3P1 creates a small admixture
of 1P1 character in the true 3P1 eigenstate, which we’ve labeled with the Russel-
Saunders (LS) coupling term symbol for convenience. Similarly, hyperfine state
mixing between states with the same F quantum number creates a small admixture
of 3P1 character in the 3P0 eigenstate for 87Sr, making the strictly dipole-forbidden
J = 0→ J ′ = 0 transition very weakly allowed. [50] No such hyperfine state mixing
exists in the bosonic isotopes, and so we must artificially mix these levels with an
1This is not so unusual from a nuclear physics perspective, but certainly unusual for species
used in laser-cooled atomic physics experiments.
46
external magnetic field.
In this chapter, I discuss the design and construction of a “clock laser” to
address the mHz clock transition in strontium. Target specifications, general con-
siderations for achieving narrow linewidths, and the ultimate performance will be
presented in the sections that follow.
3.1 Motivation and target specifications
As its name suggests, the “clock” transition in strontium was initially explored
for the purpose of making next-generation atomic clocks, which are now reaching
resolutions . 1× 10−18 [51–55].2 To get an idea of the precision indicated by this
order of magnitude, the age of the universe is roughly 4.3× 1017 s, meaning a clock
with state-of-the-art precision would have lost . 100ms since the big bang. This
level of precision is enabled by recent advances in stable laser systems, which transfer
the stability of a Fabry-Pérot cavity in a very carefully engineered environment to
the optical domain [57, 58]. The applications for stable laser systems extend far
beyond pedestrian atomic clocks, to include spectroscopic observation of quantum
many-body phenomena, gravitational wave detection, searches for dark matter, and
even deep-space navigation (see [58] and references therein).
For our purposes, we will not need to push the frequency stability of our narrow
laser to the cutting edge, although it is worth noting that several companies now
offer sub-Hz laser systems for sale, optionally in conjunction with a super-continuum
frequency comb to transfer that stability elsewhere in the electromagnetic spectrum.
Instead, we wish to ultimately use this laser as a spectroscopic probe of nuclear
spin dynamics in 87Sr. The differential Zeeman splitting between 1S0 and 3P0 in
2The program to redefine the SI second, of course, relies on both the precise measurement of
frequency but also on the careful consideration of systematic frequency shifts. A number of candi-
date systems are under consideration, from trapped ions to neutral atom lattice clocks, including
neutral strontium. For a recent review, which is now already out of date with state-of-the-art
stable laser systems, see the 2015 Rev. Mod. Phys. publication from Ludlow et. al. [56].
47
Differential zeeman splitting
(109 Hz/Gauss)
(5s5p) 3P
0
(5s2) 1S
0
m -9/2 -7/2 -5/2 -3/2 -1/2 1/2 3/2 5/2 7/2 9/2
F
Figure 3.1: Level diagram for the clock transition in 87Sr. Zeeman sublevels are split by 109Hz/G,
shifting adjacent levels off-resonance from the spectroscopy laser. In this manner, individual
ground-state sublevels can be addressed and manipulated for state preparation and readout.
87Sr is mF × 109Hz/G [59], meaning a laser with ≈ 100Hz stability and a modest
magnetic field will energetically resolve individual Zeeman transitions between the
ground and clock-state manifolds as schematically illustrated in Fig. 3.1. If we can
reliably prepare and read out the nuclear spin state of our atomic gas, we can begin
to explore some interesting proposals in quantum simulation using alkaline-earth
atoms [2, 6, 7]. With upgrades made on the new strontium experiment described in
Chapter 6, we hope to pursue some of these theoretical proposals. In the meantime,
we use this laser to perform isotope shift spectroscopy of the clock transition, as
described in Chapter 5.
For the remainder of this chapter, we will assume a target linewidth of 100Hz,
which at 698nm represents a fractional stability
δν ∼ 100Hz (3.1)
ν 429.228THz
≈ 1× 10−13 .
48
With this target specification in mind, we now discuss some of the engineering
challenges required to achieve one part in 1013 fractional stability.
3.2 Pound-Drever-Hall: noise analysis and limitations
As briefly mentioned in the previous section, narrow-linewidth laser systems
achieve their stability by transferring the excellent mechanical stability of a reference
cavity to the light via a Pound-Drever-Hall (PDH) lock [30,60]. While care must be
taken not to introduce noise via the lock electronics, in principle the PDH technique
can achieve shot-noise-limited laser linewidths. Indeed, state-of-the-art laser systems
are limited instead by Brownian noise in the cavity spacer, mirror substrate, and
mirror coating [58, 61]. In Section 3.2.4 we discuss considerations surrounding the
PDH lock itself, but first we turn our attention to mechanical considerations in
designing an optical resonator capable of delivering a 100Hz laser.
Inherent in the PDH laser stabilization scheme is the assumption that an
optical Fabry-Pérot cavity is capable of supporting a resonant mode whose center
frequency has the same fractional stability that we seek to achieve for our laser.
Quantitatively, we express the cavity resonance condition ν = mc/(2nL), where ν is
the resonance frequency, c is the speed of light, L is the cavity length, n is the index
of refraction, and m is the longitudinal mode number. This comes from requiring
that the optical phase be unchanged modulo 2π after one round-trip in the cavity.
Considering the response of the cavity resonance, δν, to uncorrelated perturbations
δL and δn in the length and index of refraction (respectively), we find
( )2 ( )2
∂ν ∂ν
(δν)2 = ( δL + δn∂L ) ∂n (3.2)mc 2 ( mc )2
= δL2 + δn2 ,
2nL2 2n2L
Taking the linear response to perturbations in each quadrature separately, and ex-
49
pressing it as a fraction, we find that the fractional stability scales like
δν ∼ δL δn, . (3.3)
ν L n
To achieve δν/ν ∼ 10−13, we must stabilize both δL/L and δn/n at the
same order. For the 100mm cavity we will use, this means δL . 10−14 m.3 This is
a rather stringent requirement, and requires careful consideration of perturbations
that couple into length changes δL or changes in the index of refraction δn. We will
treat each in turn.
3.2.1 Index of refraction
From the Lorenz-Lorentz equation, we know that the index of refraction n is
related to the mean polarizability α and density ρ of a gas by
n2 − 1 4π
= αρ . (3.4)
n2 + 2 3
Using the standard equation of state for an ideal gas and solving for n, we find
√√√√ ( )√1 + 2( β ) PkBTn = (3.5)
1−( βkBT)P
≈ 3 β1 + P , (3.6)
2 kBT
where P is the pressure, T is the temperature, kB is the Boltzmann constant, and
β = 4πα/3. The second equality in Eq. 3.6 is valid in the limit (β/kBT )P  1.
3As a point of comparison — this is equivalent to stabilizing the length of our cavity to ten
times the width of a proton. Clearly, the length isn’t well-defined at that level; the light samples
a “large” region of the mirror (given by the mode area), and thus the length δL is in some sense
averaged over the transverse extent of the optical field.
50
For air, one can calculate
( )
3 β ≡ β ∼ 10−90 . (3.7)
2 kBT
In this limit, we relate n = 1 + β0P and take β0 as our small parameter to obtain
δn β0δP
= ≈ β0δP . (3.8)
n 1 + β0P
Equation 3.3 requires δn/n . 10−13. Taking β ∼ 10−90 , this means we must suppress
δP . 10−4 Pa. The easiest way to reduce pressure fluctuations is to place the
cavity under vacuum, which has the added benefit of partially decoupling other
environmental perturbations from the cavity (such as temperature and acoustic
fluctuations, discussed later). In convenient units, 10−4 Pa ∼ 10−7 torr, which is
easily achieved using standard vacuum techniques. The pressure in our vacuum
canister is in the mid-10−8 torr. This is higher than expected, given the somewhat
lower pressure ≈ 1× 10−8 torr in the vacuum canister housing the 689nm cavity
(see Sec. 2.2.2), but is workable for our purposes. We suspect that either a slightly
longer bakeout needed to occur, or that the KF “Kwik” flange attaching the ion
pump to the canister has introduced a slow leak.4
3.2.2 Mechanical length stability
Equation 3.3 also requires the fractional length stability δL/L . 10−13. Our
lab had previously purchased a Fabry-Perot cavity from Advanced Thin Films (now
part of IDEX) formed by two high-reflectivity mirrors optically contacted to a pas-
sive 100mm spacer (∆νFSR = 1.5GHz) made of ULE glass; see Fig. 3.2(d) for
4The KF flange is only rated to ≈ 10−8 torr.
51
more details.5 With this cavity, length fluctuations of the spacer must be sup-
pressed to . 10−14 m. Damping of high-frequency vibrations can be achieved with
mounts made of elasto-polymers such as viton, which absorb mechanical energy in
this regime, and low frequency vibrations can be suppressed with a vibration iso-
lation platform. We purchased one such isolation stage, the model 100BM-8 from
Minus-K, which provides up to 60 dB of attenuation at frequencies ∼ 100Hz and
has a natural frequency ∼ 0.5Hz. Figure 3.3 shows the measured vibrational power
spectral density in the lab (before isolation with the Minus-K stage), along with the
manufacturer-specified performance of the vibration isolation stage.
In addition to these vibration-damping techniques, a clever mounting config-
uration can make the cavity length first-order insensitive to elastic deformation due
to vibrational acceleration [63]. One can study this problem via numerical simula-
tion, using a strategy known as finite-element analysis (FEA). FEA is a numerical
modeling technique used to solve complex boundary-value problems, where ana-
lytical solutions are unknown due to the complexity of the shape being modeled.
It involves dividing the larger system into smaller “finite elements,” approximating
the differential equation on that discretized region such that it can be solved, and
then stitching together the results to find a global solution. There are many FEA
solvers available, both commercial and open-source, with varying degrees of pre–
and post-processing capabilities. We chose to use the MultiPhysics FEA engine by
COMSOL to model the Fabry-Pérot cavity under various mounting geometries to
aid in the design of a mount which minimizes the sensitivity of the cavity to “bad”
5We in fact purchased two such identical cavities, with high-reflectivity mirror line coatings
at 689 nm and 698 nm. The plan was to gain experience working with high-finesse cavities for
the intercombination-line rMOT, which could be used in the design of the clock laser. This was
certainly true, however in hindsight this cavity is overkill for stabilizing the rMOT and perhaps
“underkill” for the clock application, as will be discussed below. That said, having the enhanced
stability for our 689 nm lasers has worked out well, since we do not need independent spectroscopy
on that line and it has facilitated the study of intercombination-line photoassociation resonances,
described in [62].
52
(a) (b) (c)
100 mm
2X Mirrors 1mm X 45 6 mm
25.4mm CHMFR (vent)
50 mm
Planar 50 cm
ROC
12.7 mm
6 mm
TYP
(d)
Figure 3.2: (a) Reduced CAD model of the ULE cavity. Appropriate boundary conditions are
defined on the planes of symmetry to reduce the computational complexity of the FEA simula-
tion. (b) Meshed geometry. (c) Visualization of how the cavity deforms under a uniform vertical
acceleration. Color scale shows displacement along the cavity axis, which is the figure of merit for
∆L/L. (d) Blueprints for the ULE cavity, provided by Advanced Thin Films.
BM-8 VERTICAL
10 20
Elevated 1/2" Breadboard
1 0
Optics Table
.1 -20 10 4
.01 -40
.001 -60
5
BM-8 HORIZONTAL 10
10 20
1 0
10 6
.1 -20
.01 -40
.001 -60 10 1 100 101 102
.3 .4 .5 .6 .8 1 2 3 4 5 6 7 8 10 20 30 40 60 80 Frequency (Hz)
FREQUENCY (Hz)
Figure 3.3: At left, the performance curves for 100BM-8 vibration isolation platform from Minus-
K. At right, we show the mechanical noise spectral density in our lab, measured with a Wilcox
accelerometer. Total integrated RMS noise (1Hz – 200Hz) is < 10−3 m/s2; with the additional
damping provided by the Minus-K stage, we expect to remain below the 10−3 m/s2 RMS noise
threshold required to achieve our 100Hz linewidth (see text). It is worth noting the higher mechan-
ical noise floor of the elevated 1/2" breadboard; the assumption that mounting these on rubber
feet would further damp vibrations is not borne out in the data.
53
TRANSMISSIBILITY
Acceleration ASD (g / Hz1/2)
DECIBELS
Mechanical Property Symbol Value
Density ρ 2.21× 103 kg/m3
Young’s Modulus E 67.6 GPa
Poisson’s Ratio ν 0.17
Table 3.1: Summary of mechanical properties for ultra-low expansion glass (ULE). These are used
in performing a finite-element analysis simulation of the Fabry-Pérot cavity, see text.
elastic deformation. The simulations themselves were conducted according to the
methods outlined in [63]. A model of the Fabry-Pérot cavity was constructed using
the COMSOL model builder, and the material properties of ULE were used (see
Table 3.1). We additionally leveraged the cylindrical symmetry of the cavity to
reduce the computational complexity of the model. The CAD model for the final
design can be seen in Fig. 3.2, along with a visualization of the elastic deformation
under uniform vertical acceleration.
As discussed in Ref. [63], we are primarily concerned with analyzing the me-
chanical response at low frequencies (. 100Hz). Higher Fourier components can be
easily attenuated using traditional vibration isolation techniques, such as mounting
the cavity vacuum chamber on rubber feet, placing the entire setup on a floated
optical table, and enclosing the setup in a box lined with acoustic-damping foam.
In general, perturbations & 50Hz are caused by acoustic noise sources, while those
∼ 1Hz to 50Hz are usually seismic [64]. With this in mind, we consider designs that
minimize the coupling between external mechanical noise and the cavity length. At
vibrational frequencies f much smaller than the lowest normal mode of the cavity,
we are able to greatly simplify the problem by modeling the mechanical response as
a quasi-static deformation; that is, vibrations which are close to DC compared to the
cavity’s first normal mode can be treated as uniform accelerations. Reference [63]
does a back-of-the-envelope calculation to show that we expect the response at fre-
quencies . 35 kHz to be well approximated by a quasi-static treatment, and our
FEA model confirms this.
54
(a)
L
∆L/2
θ (b)
∆L/2
∆L/2 ∆L/2
D (c)
Figure 3.4: Length fluctuation ∆L due to mirror tilt. (a) shows the basic cavity geometry. In (b)
and (c), the mechanical axis is denoted with the dashed black line, while the optical cavity axis is
shown in red. Tilt modes with even symmetry only affect the optical cavity length at first order if
there is a rotation between the mechanical and optical axes, shown in (b). Odd-mode mirror tilt
affects the cavity length at first order for an optical axis which is displaced from the mechanical
axis, shown in (c). Symmetry considerations predict odd tilting modes for accelerations transverse
to the cylindrical axis, while accelerations along the axis of the cylinder induce even tilting modes.
55
We simulate the cavity response to a uniform vertical acceleration under a
variety of mounting configurations and boundary conditions. The figure of merit
is the fractional path length change of the optical mode in question (here, the
Gaussian TEM00 mode). At lowest order, this is the fractional axial displacement
of the mirrors, ∆L/L. Mirror tilt also couples to the optical length of the cavity
at first order if the optical and mechanical axes are not coincident, see Fig. 3.4.
Since the degree of misalignment between the mechanical and optical axes is hard
to estimate a priori, we will simply put bounds on the displacement and rotation
of the cavity axis relative to the mechanical axis such that the tilt-induced response
to a uniform acceleration has the same order of magnitude as a longitudinal shift
under the same acceleration.
We first consider the so-called “rod” mounting geometry, used by our group
and others to hold a stabilization cavity for the 689nm intercombination line laser,
as well as by some groups for pre-stabilization of a 698nm clock laser [65]. In
this configuration, an aluminum block supports two viton rods at a particular angle
from the vertical symmetry plane, which in turn support the ULE cavity. Figure 3.5
shows the resulting (longitudinal) linear response coefficient Ka as a function of θ,
the central half-angle. The linear response coefficient is defined via
∆L/L = Ka · a , (3.9)
where a is a uniform acceleration. As expected, the closer you are able to support
the cavity to a plane of symmetry (θ → 90◦), the less susceptible it becomes to
deformation. However, central half-angles greater than ∼ 50◦ to 60◦ become precar-
ious, as the vertical support is reduced and the cavity is at risk of falling. There is
also no true zero-crossing in Ka using this configuration.
Our ultimate choice in mounting geometry, which we discuss next, is inspired
56
Rod mounting geometry
θ
Viton supports
Figure 3.5: FEA analysis for “rod” support method. Fractional length stability under a 1m/s2
vertical acceleration is shown at left, for various central half-angles. The geometry is illustrated at
right.
by other groups’ efforts to find a zero-crossing in Ka [63, 66,67]. Heuristically, a
transverse acceleration compresses the cavity, thereby inducing an elongation along
the cavity axis where the magnitude of this effect is governed by Poisson’s ratio.
However, a competing effect can act to compress along the cavity axis — by support-
ing it at two points displaced from the center of the cavity, a transverse acceleration
causes the middle part to “droop,” inducing stresses that point inward. A proper
balance of these two effects can in principle cancel first-order longitudinal shifts in
∆L/L. The “cutout” cavity offered by Stable Laser Systems and deployed in several
labs takes advantage of this [66,67]. In our case, we are constrained by the existing
cavity geometry shown in Figure 3.2(d), which uses an ordinary cylindrical spacer.
Our simulation and ultimate design assumes a configuration in which the cavity is
supported by four viton pads at a particular angle from the central symmetry plane.
The exact angle, width, and position of these pads is varied such that we minimize
Ka. The final geometry is shown in Fig. 3.6.
Unfortunately, we found that the exact assumptions made in modelling the
boundary conditions at the support points can produce large and qualitative dif-
ferences in the final value of Ka, as seen in Fig. 3.7. The mounts themselves are
specified in COMSOL as small boundary regions, which are constrained to zero dis-
57
4 de g
60°
z
y
x
(cavity axes) 10.0
37.0
6.1
1.0
0.606
120°
19.6 20.0
9.0
5.7
28.1
52.366
Figure 3.6: Final design for the cavity holder. Viton pads rest in the four pocket cutouts, and
provide some additional damping. A base plate, also suspended by viton rods, fixes the cavity axis
at a 4◦ tilt to avoid introducing parasitic etalons between the cavity and the vacuum viewports.
All dimensions are in mm.
58
Normal Mesh 1.0
0.4 Fine Mesh
Finer Mesh
0.2 Extra Fine Mesh 0.5
Extremely Fine Mesh
0.0
0.2 0.0 Constrain  
Constrain  r ,
0.4
Constrain  z ,
0.6 0.5 Fixed constraint
0.8
1.0
1.0
26 28 30 32 34 36 38 40 26 28 30 32 34 36 38
Mount position (mm) Mount position (mm)
Figure 3.7: Meshing and boundary model comparison. At left, we show the spread in fractional
length stability for different mesh granularities (specified in COMSOL) for a zero-size, fixed-
boundary constraint. There is no systematic trend, but the spread gives a sense of the simulation
uncertainty at the ∼ 10−11 level. At right, we show the effect of varying boundary constraints.
The color shading, from light to dark, indicates different widths along φ̂ of 0.5mm, 1mm, and
2mm. The specific boundary model used makes a large and qualitative difference in the simu-
lation results, see discussion in text. Plots assume a mounting angle θ = 60◦, and a pad width
w = 10mm.
placement in different dimensions. Unlike in [67], we did not find much dependence
on the exact size of the constrained region, but we do reproduce their observation
that Ka depends strongly on the constraints placed on displacement degrees of free-
dom at the mountpoint. The design in [67] uses four rigid support rods placed under
the shoulder of a cutout in the cavity spacer. In that configuration, it is obvious
what the constrained displacement is, and indeed they find the best match between
simulation and experiment when the model only constrains the vertical displacement
to zero.
In our case, the mount is a viton pad, which provides a compliant mechanism to
transfer force between the aluminum mounting block and ULE cavity. Rather than
deal with the complex physics involved in simulating this boundary constraint, we
model the quasi-static force exerted by the viton mounting pad as a body load along
the x̂, ŷ, or ẑ direction while the contact region between viton and ULE remains
fixed at zero displacement. We are assuming that the friction between the viton
and ULE supplies the necessary force in all dimensions for a no-slip, steady state
solution. Under these assumptions, we fix the displacement of the boundary region
59
L/L  (×10 10)
L/L  (×10 10)
between the viton pad and ULE spacer. A visualization of these different boundary
models and their effect on Ka is summarized in Fig. 3.7. We also investigate the
effects of finite mesh size on our simulation. Figure 3.7 showsKa for various meshing
parameters ranging from “Normal” to “Extremely Fine”. As can be seen from the
plot, meshing errors in the simulation arise at the ∼ 10−11 level, and can be ignored
relative to other systematic errors in the simulation, namely, those arising from
ambiguities in the proper boundary constraints.
In terms of mirror tilt, we found that there is a zero crossing in tilt sensitivity
at a particular pad position along the cylindrical axis, which was very insensitive to
any other varied geometrical parameter. This is unsurprising, as there should exist
an Airy point for our geometry. For a straight beam of length L under uniform load,
√
the Airy points exist a distance L/ 3 apart. Supporting the beam at these points
guarantees the two ends to be vertical (i.e., zero tilt of the end faces). For a 100mm
cavity, the Airy points should in theory lie 28.9mm from the cavity midpoint; we
find the optimal central mount position to be between 28mm and 29mm, in good
agreement with this prediction.
Following Ref. [68], we fit the results of the FEA simulation to interpolate
between modeled conditions. The fit is done up to quadratic order, and can be
represented as a matrix equation
  1   
θ
 Θ 
   =  a11 a12 a13 a14 a15 a16  d ·  , (3.10)Ka a21 a22 a23 a24 a25 a26 θ · d θ2
d2
where Θ is the fractional tilt sensitivity and Ka is the cavity length susceptibility
60
discussed previously. The variables θ and d parameterize the mounting pad half-
angle and position, respectively, as defined in Fig. 3.8. This can be written more
compactly as
S = M ·X , (3.11)
where now X and S are both matrices whose rows represent the simulated mounting
conditions and associated deformations. While the matrices involved are not strictly
invertible, one can solve this equation numerically by least-squares minimization.
We do this, given simulation data for θ ∈ [45◦, 90◦] and d ∈ [25mm, 40mm]. The
results can be seen in Fig. 3.8.
Given all the results from the numerical FEA model, we choose the mounting
configuration shown above in Figure 3.6. The pads are 10mm wide, positioned
37mm from the center axis and at a 60◦ azimuthal angle. A central half-angle of
60◦ gives a large enough safety margin for supporting the cavity against gravity while
still allowing a zero-crossing in Ka. Furthermore, this zero crossing was preserved
for all pad widths investigated, from w = 10mm to w = 18mm. Since the exact
width does not seem to have a great impact on the final result, we use a pad width
of 10mm for convenience. We expect the longitudinal linear response coefficient
Ka ≈ 10−11/(m s−2), and the tilt sensitivity to be ≈ 2 nrad/(m s−2). As discussed
above and illustrated in Fig. 3.4, for a tilt sensitivity at this level, we can tolerate
a displacement of ∼ 0.25mm between the cavity and optical axis before mirror tilt
begins to dominate over longitudinal displacement.
Given the expected linear response coefficient Ka ∼ 10−11/(m s−2), we must
ensure vibrations are damped below 10−2 ms−2. In Fig. 3.3, we showed the ampli-
tude spectral density of vibrations in our lab, measured on a Wilcox accelerometer
61
Tilt sensitivity Length sensitivity
1.0 3
0.5
0.0 2
0.5 1
1.0 0
1.5
2.0 1
384036 m)
40
34 m 3436
38 m)
40 32 on 
( 40 32 on 
(m
50
Mou 60
50
n 28
30
siti M 60 28
30
t s
iti
 ha 70 o o 70 olf 26 unt 26-ang 80  
p  h
l t ae l f( -D un ang
80 t p
eg l. o e)  (M Deg.) Mo
un
2.5
38 0.6
2.0 a uniform acceleration), for varioCusenmtreaslh siynmg mpaertarymeters ranging from “Normal” to
plane
36 φ̂
.05.3 “Extrem
r̂ ely Fine”. As can be seen from the plot, meshing errors in the simulation
1
34 arise at the ∼ 10−11.0 level, and can be ignored relative to other systematic errors inTh1e spectrum was recorded on a PicoScope 4262 digital oscilloscope, using 4096 frequency
0.0 Hz bandwidth. The Wilcoxthheassiamfurelqautieonncy. corner at ∼ 1Hz; additionally, low-32 frequenc0y.5spectral components in Fiẑgure ?? likely su er from window leakage at DC (spectrum
reported with Blackman windowing) and over-estimate the vibrational noise power.
30 - 0.0
w (10mm)
0.3
θ φ̂
28 r̂ d
26 5 -1.0-0.
The spectrum was reco-1rd.5ed on a PicoScope 4262 digital oscilloscope, using 4096 frequency
40 50 60 70 Hz bandw8id0th. The WilcFoxEAha sGaeformeqeuternycy 3co1rner at ∼ 1Hz; additionally, low- ẑMount half-anfgrelqeu (eDnceygsp.)ectral components in Figure ?? likely su er from window leakage at DC (spectrum
Coef. / Tilt (nrad) ∆L/L (×10−10)
const. −8.96× 10−1 −1.54
θ −1.04× 10−1 1.10× 10−1
d 7.63× 10−2 −2.92× 10−1
θ · d 2.33× 10−3 1.48× 10−3
θ2 1.21× 10−4 −9.36× 10−4
d2 5.32× 10−4 4.18× 10−3
Final sensitivity −1
(sim.) 2.03 −1.03× 10
Figure 3.8: Summary of results from the clock cavity FEA. At top, we show the points and
quadratic fit to both tilt and length sensitivity as a function of angle θ and position d. The contour
plot shows the same fitted surfaces, with the zero-crossing contours highlighted in red. Dashed
lines correspond to the length sensitivity, while solid lines correspond to the tilt sensitivity. The
blue dot is the experimentally realized mounting configuration, and the green shaded region is what
we consider “safe” mounting angles. The table shows the fit coefficients, and the final sensitivity at
the experimentally realized configuration is reported in the last line; see text for further details.
62
Mount position (mm)
Tilt  (×10 9 rad.)
L/L  (×10 10)
-0.6
-0.9
-1.2
-1.5
-1.8
-2.1
(model 728A, connected through a P703B transducer).6 The total RMS noise (1Hz
to 200Hz) is 1× 10−4 m/s2 (3× 10−4 m/s2) measured directly on the optics table
(elevated 1/2-inch breadboard), respectively. This is tolerable by itself, however
the Minus-K stage will provide an additional 60 dB of attenuation, and floating our
optical table will further reduce seismic noise. Finally, a well-designed box around
the cavity and laser will provide shielding from acoustic noise sources.
3.2.3 Thermal length stability
In addition to mechanical deformation, temperature fluctuations can couple
into the length of the cavity through the coefficient of thermal expansion, αCTE,
according to
δL
= αCTE δT (3.12)
L
The cavity is made of Corning ultra-low expansion (ULE) glass, which has αCTE .
2×10−8/K [70]. In addition, ULE has a zero-crossing in αCTE somewhere near room
temperature, at which point the length becomes insensitive to first order changes
in temperature. State of the art stable laser systems take great care to tune the
temperature to this zero crossing [71], and depending on the complexity, employ
multiple temperature-stabilized heat shrouds in vacuum to reduce thermal coupling
via blackbody radiation [57]. For our purposes, we actively stabilize the temperature
of the vacuum housing with a resistive heater.
To find the zero crossing of αCTE, we locked a 689nm laser to the “clock”
cavity and monitored a beatnote between this laser and our frequency-stabilized
6The spectrum was recorded on a PicoScope 4262 digital oscilloscope, using 4096 frequency bins
over a 200Hz bandwidth. The Wilcox has a frequency corner at ∼ 1Hz; additionally, low-frequency
spectral components in Figure 3.3 likely suffer from window leakage at DC (spectrum reported with
Blackman windowing) and over-estimate the vibrational noise power. We have divided by 1.73 to
normalize the power spectral density with respect to the Blackman windowing function [69].
63
rMOT laser as the temperature setpoint was slowly increased. After each change
in setpoint, we either wait sufficiently long for the whole system to thermalize (as
measured by the beatnote reaching a steady-state value), or fit the beatnote to an
exponential and extracted the steady-state value and time constant. Figure 3.9
shows the results of this analysis. Near the zero-crossing temperature T0 of αCTE,
we can express it to first order as
αCTE = β(T − T0) . (3.13)
Plugging this expression into Eq. 3.12, and recalling that δν/ν = −δL/L, we get
[ ]
δν = β (δT )2 ν (3.14)
where δT = T − T0, and β is the linear susceptibility of αCTE to changes in temper-
ature about T0. From the parabolic fit in Figure 3.9, we extract T0 = 30.34(6) ◦C
and β = −2.48(7)× 10−10 /K2.
Temperature controllers, such as the WTC3243 from Wavelength Electronics,
can stabilize temperatures at ∼ 10−3 K.7 Taking a conservative estimate of the
temperature stability, δT ≈ 10mK, we can easily achieve a fractional stability
δν/ν . 10−13. The short-term stability is further improved by the effective low-
pass filter created by the small thermal conductance between the environment and
the cavity itself. Thus, with our setup, we are able to achieve sufficient temperature
stability for the target linewidth.
Finally, it is worth mentioning that even if one operates the cavity at the zero-
7We have developed a custom temperature controller to use in our lab based on the evaluation
board for this chip. It is capable of controlling up to four devices, and can be configured to drive
thermo-electric coolers (TEC) or resistive heaters. The feedback gain and integrator time constant,
as well as setpoint, temperature, and output current monitor, are exposed via microcontroller
interface to a frontpanel display. For Eagle design files, see the GitHub repository https://
github.com/JQIamo/Linear-Temperature-Controller.
64
0.1
0.0
0.1
533
532
531
530
529
528
527 Fit (red. χ2 = 0. 72)
526
T0 = 30. 34(6) C
525
22 24 26 28 30 32 34
Temperature (C)
Figure 3.9: Temperature dependence of the clock laser cavity resonance, used to find the temper-
ature T0 where αCTE = 0. We lock a 689 nm laser to the clock cavity, and track the beatnote
between that laser and the narrowed rMOT “master” laser as we adjust the temperature of the
clock cavity. Thermalized, steady-state beatnote values are extracted by fitting the beatnote to an
exponential and finding the infinite-time asymptote. Here, we neglect the point at 23 ◦C in the fit
because it increases the reduced χ2 per degree of freedom to χ2/6 ≈ 3, which we can reject with
α ≈ 0.006 in a χ2 test. We suspect that, far from the zero-crossing temperature, the functional
form of αCTE deviates from the linear approximation used here. Errors on each point are taken
as the 1σ confidence interval in the steady-state value extracted as described above, and used to
weight the parabolic fit to find the zero-crossing temperature, T0. Performing a χ2 test on the
remaining datapoints gives a reduced χ2/5 = 0.72, which is slightly smaller than the ideal value
χ2/5 = 0.87, but can be accepted with an α = 0.6. The curvature β (see text) is found to be
β = −2.48(7)× 10−10 /K2, and T0 = 30.34(6) ◦C. From these parameters, a conservative temper-
ature stability δT ∼ 10mK around T0 is enough to stabilize the fractional length to . 10−13. The
very low thermal conductance between the vacuum housing and the cavity itself further serves to
low-pass filter any temperature fluctuations, so the short-term thermal stability will be improved
even beyond the estimates here.
65
Beatnote freq. (MHz) Residuals
crossing of αCTE, Brownian motion induced by thermal fluctuations in the cavity
spacer, mirror substrates, and mirror coatings lead to perturbations in the cavity
length [61]. This thermal noise can be limiting in some systems [72], and has driven
efforts to build next-generation cavities operated at cryogenic temperatures [57,58].
Here, we verify that thermal fluctuations are below other noise sources in our system.
Initial treatments of thermal noise in optical cavities were motivated by the
long-standing effort to build gravitational wave interferometers [61, 73]. The key
physical insight comes from the Fluctuation-Dissipation theorem: thermal fluctu-
ations of a system in equilibrium are related to the dissipative linear response of
that system to an external drive [74].8 Quantitatively, we can write the fluctuation-
dissipation theorem as [61,75]
4kBT
GX(ω) = − Im [H(ω)] , (3.15)
ω
where GX(ω) is the single-sided power spectrum of the fluctuation X at angular
frequency ω = 2πf , kB is the Boltzmann constant, T is the temperature, and
HX(ω) is the (complex) linear-response transfer function of the system, given by
≡ X(ω)H(ω) (3.16)
F (ω)
for some generalized force F . The imaginary part of H gives the dissipative portion
of the response. What Eq. 3.15 implies is that, by measuring the mechanical linear-
response of a system, we can make a statement about the power spectrum of thermal
fluctuations in equilibrium.
Equation 3.15 is a normal-mode representation of the fluctuation-dissipation
theorem, however computationally it can be difficult to achieve high precision when
8Brownian motion, first theoretically described by Albert Einstein in 1905, and Johnson-Nyquist
noise in electrical circuits, are both applications of the more general fluctuation-dissipation theo-
rem.
66
the elastic material is inhomogeneous [75]. This is true in our case, where the
high-reflectivity coating at the surface of the mirror locally damps the mechanical
response of the whole cavity assembly. Additionally, we care about the response of
X in a spatially localized region, coinciding with the extent of the cavity optical
mode (which is ≈ 200 µm). Thus, a normal mode decomposition requires summing
many modes for adequate convergence.
Given these computational difficulties, is is standard practice to use finite-
element analysis (FEA) to measure the dissipation, Wdiss, under a periodic drive
delivered to the localized region sampled by our optical field. Specifically, one applies
an oscillatory pressure
P (~x, t) = F0 cos(ωt)u(~x) (3.17)
to the mirror surface, where F0 is the amplitude and u(~x) is a gaussian “form factor”
which weights the spatial response by the gaussian profile of the laser. The dissipated
power Wdiss, measured via FEA, can be related back to the transfer function H(ω)
by
2W
Im diss[H(ω)] = . (3.18)
F 20ω
Thus, one obtains the power spectrum GX(ω) by combining Eqs. 3.15 and 3.18 to
get
8kBT Wdiss
GX(ω) = . (3.19)
ω2 F 20
In Ref. [61], Numata et. al. measure the mechanical quality factor of several
common cavity materials by monitoring their mechanical ringdown after delivering
an impulse to the material. They then use these results in a finite-element analysis
67
model to obtain the power spectrum of displacements of the cavity length via the
fluctuation-dissipation theorem. Case No. 1 (Table 1 from Ref. [61]) corresponds
closely to our our cavity, with ULE spacer √and mirror substrate. They find the
displacement noise spectral density, Sx(f) ≡ GX(2πf), to be
√
≈ 6× 10
−17 m/ Hz
SX(f) . (3.20)
f
Thus, for our 100mm cavity, we can expect thermal fluctuations to contribute to
√
the frequency noise spectral density of our clock laser at the order of 260mHz/ Hz
at 1Hz, falling off with 1/f at higher frequencies. Given the other limitations of our
system, we do not expect to hit this thermal limitation to the laser linewidth.9
3.2.4 Servo noise & electronics
Having addressed perturbations to the cavity length, we turn to the remaining
noise consideration: performance of the PDH lock itself. To understand the closed-
loop locking behavior of the laser, it is helpful to work in the Laplace domain, where
iω → s. Figure 3.10 shows a schematic of the PDH locking setup, where each block
(the laser, cavity/PDH discriminator, and servo) has its own transfer function and
we also allow each block to introduce additional noise.10
9Converting a frequency noise power spectral density to the traditional laser “linewidth” mea-
sured in an experiment is not straightforward. For a white noise power spectrum |Sν(f)|2 = h0,
which is independent of f , one gets an analytic relationship for the Lorentzian linewidth ∆νFWHM =
πh0 via the Wiener-Khintchine theorem. In the case of random-walk, or “red” noise, which is the
case for the√thermal fluctuations discussed here, |Sν(f)|
2 = h −20f and the linewidth technically
diverges. One then introduces a low-frequency cutoff, fc, and obtains a Gaussian lineshape with
FWHM = (8 ln 2)h0fc. For a more thorough accounting, with approximate formulas for more
complicated noise spectra, see the excellent discussion in Ref. [76] (and the references therein).
10This noise analysis follows the treatment of [70,77,78].
68
Figu
quadrature, since each noise sourceTihseassesucomneddtteormbe, huonwcoervreerl,atiseda. nTohisue
rtee 6: Closed-loop servos, wremh.avTehis is more properly
scthreeamteadtiicn.
Analyzing around the loopLa,sewe have Y (s) = D(s) [Y0 − Y (s)] +quadrature, sincre each noise source is assumed to bePDuHncorrelated. Thus, we have
K [Hz/V] D [V/Hz]
an expression for tAhneanloyizsieng arounUd(st)h=e lGoo(ps,)Ew(es)ha+veSG
Y |(Ks) = 2K(s)U(s) + S2 L. For large closed-+loop gaiSnG, |Y ≈+Y0 as dSeLs|ired.2 | |2 2 | |(216)| | 21 +KGD 2 |1K+GKSGDD| KSG SD|
The second term, however, is a noise terwmhU.er(Tesh)iYs=|,SisEm|(,os=ar)en[Ydp0rU− aYrn | operl
(yestt)rh]ee+atoeSduDtipnYu(ts, )e=rroK+r,(as)nUd(cso)n+trol signals, respectively. The
1 +KGD|2 |1 +KGD|2 |1 +
quadrature,WsiencseeeatchhatnobiysemsoaukricnegisthaessluomopterdagUtanoi(snsfbe)elra=urfgnuGecno(ecrsntr)ieooEluangt(shesd,)a.+wtTeeShacucashn, wbseulophcpakrvesasrenogisiveen by D−(PDH/cavity discriminator),G E(s) = D(s) [Y
Servo 0
We(→sseer0eviont)h,tahtaenbldyarKgmea-g(klaianinsgGel rt[iV)mh,/eVita]l.noodSpimSgDiala,inrlSyl,Gasr,geSrevLoenaorueghth, eweascsaonciasutepdprensosisneoisspeectral density
fluctua U(s) = G(s)E(s) +arec0tot.ihonns|2etHroib
SwuuLettvpoieuofrtn,t,shn.eeor(ilrsSaoesereei,rnFatbrnyiogdu1u/rccGeoend6→t)rb.oy0l tishnigetnhPaeDlslH,arrgees-pgaeicntilvimeliyt.. TSihmeilarly, servo
KSG S |2G
transfer functions at eaSchG bislosc+ukpparreessegdivbeyn 1/G → 0. HKGDS|o2 lving for Y , w2beyfinDd (PD
(H16/o)cwaevvietry, ndoisiscerimintirnoadtuocre)d, by the PDH
wh|1er+e Y , E|, and U are the output, error, and control signals, respectively. The
Figure 3.10: Block schematic representing the closed-loop behavior of a PDH lock. The laser (aka,
the “plant” in control theory literature) outputs some signal Y (s) = K · U(s). Similarly, the PDH
cavity acts as a frequency discriminator, converting the laser frequency Y (s) to an electrical error
signal E(s) via E(s) = D · Y (s). Finally, the servo closes the loop by computing a feedback signal
U(s) = G · E(s). Each stage in the feedback loop has a particular frequency-dependent gain (K,
D, G), and might introduce some additional noise with spectral density SK , SD, SG. See text for
more details.
69
Analyzing around the loop, we find
Y (s) = K(s)U(s) + SL (3.21)
E(s) = D(s) [Y0(s)− Y (s)] + SD (3.22)
U(s) = G(s)E(s) + SG , (3.23)
where Y , E, and U are (respectively) the output, error, and control signals, and Y0
is the “command” frequency to which we want to lock (given by the cavity resonance
frequency νc). The transfer functions for each block are given by D (PDH/cavity
discriminator), G (servo), and K (laser), and SD, SG, SL are the associated noise
spectral density contributions.
Solving for Y , we find
( )
KGD KGSD +KSG + SL
Y (s) = Y0 + . (3.24)
1 +KGD 1 +KGD
The first term is the standard closed-loop transfer function — ie, Y tracks Y0 as
Y GT
H(s) ≡ = , (3.25)
Y0 1 +GT
whereGT (s) ≡ KGD is the (total) closed-loop gain. We see that forGT  1, H → 1
and Y ≈ Y0 as desired. The second term in Eq. 3.24, however, is a noise term. This
is more properly treated in quadrature, since each noise source is assumed to be
uncorrelated. Thus, we have an expression for the noise given by
| |2 |KGS |
2
D |KS 2G| |SL|2
Sn = 2 + 2 + 2 , (3.26)|1 +KGD| |1 +KGD| |1 +KGD|
where Sn is the total noise spectral density, related to the final laser linewidth by
∆νL = π |S |2n (assuming a white-noise spectrum) [76,77].
70
We see that by making the servo gain G large enough, we can suppress noise
fluctuations SL of the laser by 1/G→ 0 in the large-gain limit. Similarly, servo noise
SG is suppressed by 1/G → 0. However, noise introduced by the PDH frequency
discriminant, SD, goes as
|KGS |2D |S 2D|
lim 2 = 2 . (3.27)G→∞ |1 +KGD| |D|
Thus, as might be expected from the previous discussion, our final laser linewidth
is dominated by the noise of our frequency discriminator — essentially, noise at the
output of the discriminator is indistinguishable from noise in the laser itself [77].
Subsequent servo electronics try to compensate for this noise, but ultimately end
up writing SD back onto the laser output. The noise SD itself has several sources —
“fundamental” sources such as shot noise in the photodetector, technical noise in the
photocurrent amplification and demodulation to generate the error signal, as well as
noise associated with the “true” cavity resonance frequency (induced by vibrations,
etc., addressed previously).
We now return to Equation 3.27 for the noise associated with the PDH fre-
quency discriminant. We see there are two “knobs” for reducing the size of this term:
we can increase the gain D, or decrease the noise spectral density SD. Ultimately,
the gain D will be limited by stability of the lock, since making it too large will
drive the system into oscillation. Similarly, SD cannot be made arbitrarily small,
as we push against fundamental noise sources (like shot noise) which are physically
beyond our control. To achieve optimal performance, then, we must ensure that SD
is limited by these fundamental noise sources and not technical noise. To state this
another way, we want SD to be shot-noise limited. For the remainder of this section,
we discuss the details required to operate at the shot-noise limit.
The Pound-Drever-Hall locking scheme, described briefly above and shown
71
Phase-modulation
Mixer
sidebands λ/4 (demodulate error sig.)
EOM Low-pass
filter
Φ
Phase delay to
Ω select proper demodulation 
quadrature
Modulation Error signal
frequency
Loop Filter
Laser (PID)
Lock point
Figure 3.11: Schematic illustration of a PDH lock. Optical (phase-modulated) sidebands are
imprinted on the laser light with an electro-optic modulator, and the promptly reflected light
captured on a fast photodiode. By demodulating this photodiode signal at the phase modulation
frequency, one recovers an error signal with a sharp slope around the cavity resonance. This can
be used to feed back to the laser to stabilize its frequency to the cavity resonance.
72
schematically in Figure 3.11, involves the phase-modulation of a coherent laser field
coupled into a Fabry-Perot cavity. We can write down the electric field, Einc, incident
on the cavity as [30]
E = E e[i(ωt+β sin Ωt)inc 0 ( )] (3.28)
≈ E J (β)eiωt + J (β) ei(ω+Ω)t − ei(ω−Ω)t0 0 1 , (3.29)
where we have kept the first two terms of the Jacobi-Anger expansion. Here, E0
is the amplitude of the electric field, β is a parameter describing the modulation
depth, Ω/2π is the modulation frequency, and J0, J1 are Bessel functions. To obtain
the electric field reflected from the cavity, Eref, we use the transfer function [79]
( )
Eref r 1− eiω/∆νFSR
F (ω) = = , (3.30)
E 2 iω/∆νinc 1− r e FSR
where r is field amplitude reflection coefficient for the mirror and ∆νFSR is the free
spectral range, which for our cavity is 1.5 GHz.11 Using Eq. 3.30, we can rewrite
the reflected field amplitude as
[ ]
Eref = E
iωt
0e J0(β)F (ω) + J1(β)F (ω + Ω)e
iΩt − J1(β)F (ω − Ω)e−iΩt . (3.31)
11The cavity, shown in the reduced CAD model of Fig. 3.2, is formed by a flat mirror and a
mirror with a 500mm radius of curvature, optically contacted onto a 100mm spacer.
73
The reflected power, proportional to |E |2r , thus becomes
{
Pref = P0 |J0F (ω)|2 + |J 2 21F (ω + Ω)| + |J1F (ω − Ω)|
[ ]
+ J0J1 [F (ω)F ∗(ω + Ω)e−iΩt + F ∗(ω)F (ω + Ω)eiΩt ]
− J0J1 F (Ω)}F ∗(ω − Ω)eiΩt + F ∗(ω)F (ω − Ω)e−iΩt (3.32)
+ [2Ω terms] ,
where we use the shorthand Jn(β) → Jn, and P 20 ∝ |E0| . Rewriting complex
exponentials as sines and cosines and collecting like terms, expression (3.32) reduces
to
{
P 2ref = P0 |J0F (ω)| + |J F (ω + Ω)|21 + |J1F (ω − Ω)|2{ }
+ 2J J ∗0 1 Re{F (ω)F (ω + Ω)− F ∗(ω)F (ω − Ω)}cos Ωt
− 2J0J1 Im F}∗(ω)F (ω + Ω) + F ∗(ω)F (ω − Ω) sin Ωt (3.33)
+ [2Ω terms] .
What is left is a DC term, two terms oscillating with Ω (formed by the beat between
the carrier and each sideband), and terms oscillating at 2Ω formed by the sidebands
beating with each other. You can see the two Ω terms plotted in Fig. 3.12.
At this point, we make the assumption that ∆νFSR  Ω/2π  δνc, which
is valid for our cavity (which has a linewidth δνc ∼ 6 kHz, and Ω/2π ∼ 20MHz,
∆νFSR ∼ 1.5GHz). This lets us write F (ω ± Ω) ≈ 1 when the laser frequency
ν = ω/2π is near a cavity resonance νc (that is, when ∆ν = ν − νc → 0). In
that limit, the cosine term goes to zero while the sine term is maximized and anti-
symmetric around the cavity resonance — see Figure 3.12. The “lockable” error
signal is the quadrature of Pref in phase with sin Ωt, which can be extracted by
74
1.00 Sin Quadrature
Cos Quadrature
0.75
0.50
0.25
0.00
0.25
0.50
0.75
1.00
0.4 0.2 0.0 0.2 0.4
/ FSR
Figure 3.12: Both quadratures of the PDH error signal. We demodulate to obtain the signal in
phase with sin Ωt, finding a steep discriminant at ∆ν = 0. This plot assumes Ω/2π = 0.2×∆νFSR,
and r = 0.99.
75
Signal (arb.)
demodulating at frequency Ω and taking the in-phase component.
To better understand the near-resonance behavior of this error signal, we ex-
pand Eq. 3.30 about the resonance condition ωc = 2πνc. To first order in ∆ν, we
find
∣∣
F (ω)∣ ≈ − ∆ν r2πi , (3.34)
ω→ωc ∆ν 2FSR 1− r
with r the field amplitude reflection coefficient. The cavity finesse, given by
F ∆νFSR ≈ πr= , (3.35)
δν 2c 1− r
can be used to reduce Eq. 3.34 to
∣∣
F (ω)∣ ≈ − ∆ν2i . (3.36)
ω→ωc δνc
Plugging this into the sine term of Eq. 3.33 and again taking F (ω ± Ω) ≈ 1, we
arrive at the linearized error signal , related to ∆ν by a cavity transfer function
DPDH according to
 = DPDH∆ν (3.37)
8J0(β)J1(β)
DPDH = P0 (3.38)
δνc
where again δνc is the cavity linewidth and P0 is the power incident on the cavity.
Returning to our earlier discussion, the dominant technical noise contribution
to the final laser linewidth is given by the noise introduced at the PDH discriminant,
which goes as |SD|2/|D|2. To minimize this term, we want high PDH gain D and
low spectral noise density SD. From Eq. 3.38, we can maximize D by maximizing
the coefficient J0(β)J1(β) (which occurs for β = 1.08) and by increasing the power
76
P0 incident on the cavity. Clearly narrower cavity linewidths also correspond to
higher gain in the PDH transfer function, but this parameter is already fixed by the
ULE cavity. Ideally, we want to keep P0 low because the power buildup in a high-
finesse cavity can start to approach the damage threshold for the mirror coatings,
and additionally cause heating-induced length fluctuations which negatively impact
the stability of the resonator.
Practically speaking, the error signal given in Eq. 3.37 has units of optical
power and must undergo further processing in order to obtain a useful signal to pass
into the servo block G. Specifically, we must convert  into a voltage error signal E,
using a high-speed photodiode and transimpedance amplifier. For simplicity, we use
a fixed-gain amplified photodiode (Thorlabs PDA10A).12 In a locked configuration,
F (ωc) = 0 and from Eq. 3.32 the average power Pref reflected from the cavity is
P 2ref ≈ 2P0J1(β) (3.39)
arising from the two promptly reflected sidebands at ωc±Ω. This DC optical power
gives rise to a fundamental shot noise spectral density in the photocurrent given
by [77]
√ √
Ssn = 2 2eI , (3.40)
√
where e is the electron charge and I is the photocurrent. The extra factor of 2
arises from the demodulation required to generate an error signal, which selects shot
noise contributions at both +Ω and −Ω fourier frequencies.
The optical power can be converted to a photocurrent via the diode responsiv-
12Another possibility, e.g. as done in Ref. [80], would be to build a custom transimpedance
amplifier with a nice photodiode such as the S5971 or G4176-03 from Hamamatsu. One could
then optimize the signal-to-noise ratio in the desired modulation band. Analog devices has a very
nice tool to aid in such designs, and can be found at http://www.analog.com/designtools/en/
photodiode/.
77
ity — for Si-based photodiodes near 698nm, the responsivity is RPD ≈ 0.45AW−1.
Combining Equations 3.39 and 3.40 yields
√
Ssn = 8eRpdJ1(β)2P0 . (3.41)
This photocurrent noise spectral density represents the shot-noise limit for a par-
√
ticular input power; in convenient units, it is roughly Ssn ≈ 35 pA/ Hz for 1 µW
input power, and scales as the square root of optical power. The datasheet for the
√
PDA10A photodiode gives a noise-equivalent power (NEP) of 3.5× 10−11 W/ Hz.
This is specified at the peak of the diode’s optical response, which happens to be
near our wavelength of interest.
It’s worth noting that, since setting up the clock laser locking electronics, Thor-
labs has discontinued the PDA10A in favor of an upgraded version, the PDA10A2,
√
which features a NEP of 5× 10−14 W/ Hz, nearly three orders of magnitude smaller
than the PDA10A. One could also use the DET36A2, which is slightly slower
√
(25MHz bandwidth versus 350MHz) but has a NEP of only 1.6× 10−14 W/ Hz.
The above derivations can be extended to allow for a very versatile, tunable-
offset PDH lock [81]. The idea, which we will employ in our system (see the final
setup in Fig. 3.13), involves adding a second modulation frequency Ωoffset that can be
tuned to provide an arbitrary frequency difference between the cavity resonance νc
and the laser frequency. This is critical, since we have no way to tune the resonance
frequency of the ULE cavity and need to bring the laser into resonance with the
atomic transition. The incident electric field, then, can be expressed as
E = E ei(ωt+β sin Ωt+βoffset sin Ωoffsett)inc 0 (3.42)
which contains sidebands at Ω, Ωoffset, and Ωoffset ± Ω. Requiring Ωoffset  Ω,
this expression can be reduced via the same optical transfer function (Eq. 3.30)
78
and simplifications above to give the dual-sideband frequency discriminant, DDSB,
as [81]
2 8J0(β)J1(β)DDSB = J1(βoffset) P0 (3.43)
δνc
= J 21(βoffset) DPDH . (3.44)
The intuition here is we are using the modulation frequency Ωoffset to generate a
sideband which can be tuned onto resonance with the cavity; this sideband in turn
has the same PDH sidebands at Ωoffset ± Ω as in the standard PDH setup, and can
be used to generate an error signal as before. However, the amplitude of that error
signal is decreased by the amount of power we can push into the first-order sideband
at Ωoffset. It is maximized for a modulation depth βoffset = 1.84, but is roughly one-
third the size of the standard PDH error signal for equivalent laser power. However,
the degraded signal-to-noise of the dual-sideband PDH can be compensated for by
increasing the optical power P0. One must also be careful using this technique,
since the additional sidebands can generate spurious features at frequencies far from
the desired lockpoint, but with a wavemeter and a little care any confusion can be
avoided.
3.3 Final design
The initial layout of this laser included a pre-stabilization cavity, which nar-
rowed the laser linewidth to . 10 kHz before being sent to the high-finesse ULE
cavity. The rationale for this was twofold: first, a spectrally narrowed light source
would reduce the demands for the second-stage servo responsible for the PDH lock
to the ULE cavity, and second, during initial alignment, it would be helpful to see
a well-resolved PDH error signal on the ULE cavity. In addition, the final lock
to the high-finesse cavity would be done by feeding back to a voltage-controlled-
79
18 MHz
To PDH (PDH)
PD (transmission mon.) lock
Intensity mon.
Power splitter/
combiner
λ/4
∼ 800 MHz
(From PLL)
ULE Mode matching
Fiber EOM
FP cavity
Minus-K vibration stage
Camera (mode mon.)
To wavemeter
Isolator
80 MHz
AOM
Clock laser λ/2 λ/2
To experiment
To frequency comb
Figure 3.13: Final layout of the clock laser system. An earlier iteration of the layout included a two
stage locking scheme, whereby the laser was locked to a home-built pre-stabilization cavity (finesse
F ≈ 1200). The final lock to the ULE cavity was achieved via feedback to a voltage-controlled
oscillator (VCO), which controlled an AOM to remove residual phase noise from the pre-stabilized
light. At longer timescales, the length of the pre-stabilization cavity was servoed with a piezo such
that the VCO control voltage stayed close to its central value. This locking scheme was abandoned
in favor of the one presented above for reasons described in the text. Currently, a single Pound-
Drever-Hall lock stabilizes the laser to a passive Fabry-Perót cavity made from ULE. The cavity
is mounted in such a way to minimize sensitivity to mechanical vibrations, held in a temperature-
stabilized vacuum canister, and rests on a vibration isolation stage (BM-8 from Minus-K). The
entire ULE breadboard is enclosed in an acoustically damped box to further minimize mechanical
noise. Because the ULE cavity resonances do not match up with the atomic transition frequencies,
a broadband fiber EOM (Jenoptik, PM705) is used to span the difference; we do not temperature
stabilize this device or perform active fiber phase-noise cancellation because of the shortness of the
fiber pigtails, but this might be necessary in the future. A variable RF frequency is summed on an
RF power splitter/combiner with the 18MHz dither used to generate the PDH error signal; both
tones are then amplified and sent to the EOM. The optical sideband generated by the variable
RF drive is made to be resonant with the cavity, and the resulting PDH error signal is passed
through the Toptica FALC analog PID and fed back to the current of the laser diode to stabilize
its frequency. A final AOM is used for fine frequency scans of the clock light, and for precise
pulse shaping. The 1st (+1) order of this AOM is fiber-coupled and sent to the experiment; in the
future, it may be necessary to include fiber phase noise cancellation via this AOM. The 0th order
is fiber-coupled and sent to the wavemeter for monitoring, and a polarizing beam splitter picks off
a few mW before the AOM to be counted against an optical frequency comb.
80
oscillator (VCO), which has the added advantage of linearity and a lack of mode
hops. This was the original design of the clock laser system at JILA [82], which was
used successfully for a number of years.
The pre-stabilization cavity was used for the initial alignment and evaluation
of the laser system at 689nm. It consisted of a homebuilt invar spacer (length
L = 100mm) with backside polished mirrors and a custom high-reflectivity line
coating from CVI at 698nm.13 The mirrors form a modified hemispherical cavity
(the curvatures are r1 = 500mm, r2 = ∞, while the length is L = 100mm). We
measured the finesse to be F ≈ 1225 at 689nm, which yields a cavity linewidth
γc ≈ 1.2MHz. With a good servo, one can reasonably expect to split the cavity line
by a factor of ∼ 100 or more, which gets us to something on the order of the ULE
cavity linewidth. The pre-stabilization cavity is hermetically sealed in a custom box
machined from aluminum, and three Peltier coolers stabilize the temperature of the
enclosure.
We ultimately chose to abandon the pre-stabilization cavity, again for several
reasons. First, the additional complexity of the servo means more things have the
possibility to come unlocked.14 The linearity and lack of hysteresis in the VCO/AOM
frequency control was nice, but even in double-passed configuration it is difficult to
completely remove amplitude modulation coupled to changes in drive frequency. We
attempted to compensate for this with an intensity lock, but it was unclear if the
bandwidth was high enough.
Second, because the laser cooling group had purchased a Toptica DigiLock
servo a few years back that was sitting on the shelf, we opted to only buy a sin-
gle Toptica FALC servo and use the DigiLock for the (nominally less demanding)
13The mirrors were quoted with a reflectance R & 0.995 at 698 nm, but in private communication
with CVI, they expect the coating to perform ∼ 0.999 even though it could not be guaranteed. The
finesse we measured at 689 nm corresponds to a reflectance of 0.997, which seems to be consistent
with this statement.
14And in an ultracold strontium lab, the number of servos is already too damn high!
81
second-stage lock to the ULE cavity. This digital servo, in retrospect, was a poor
choice. The bandwidth performance was never fantastic, the control software was
buggy, and it required a computer to be plugged in to the module via USB to relock
when needed. We were able to achieve a passable lock with the 689nm diode used
to characterize the system, but when we switched to the 698nm diode, we could not
push the bandwidth larger than ∼ 100 kHz without oscillations.
Our experience with the DigiLock, compared to the FALC, made us rethink
the wisdom of a pre-stabilization cavity entirely. We had successfully operated a
direct PDH lock to an identical cavity for the 689nm laser systems without trouble,
and in conversations with the Menlo field technicians, they hinted that very few
labs use pre-stabilization cavities anymore. The reduced complexity meant more
laser power for the experiment, which to drive the clock transition in the bosonic
isotopes was initially a real concern (see discussion in 5.2.1.2), and after switching
to a direct lock with the FALC, the laser would stay locked for several hours at a
time compared to the tens of minutes to an hour in the previous scheme.
3.4 Performance
We analyze the performance of the clock laser in two separate ways. First,
during construction and initial coupling into the ULE cavity, we used a laser at
689nm locked to a pre-stabilization cavity15 and then to the ULE cavity via feedback
to a double-passed AOM. This 689nm laser could then be measured against our
master 689nm laser used for the narrow-line rMOT. We generate a heterodyne
beatnote between the two systems, and measure a joint FWHM Lorentzian linewidth
of 289(14)Hz, as shown in Fig. 3.14. This puts an upper bound on the linewidth of
15This cavity, made from an Invar spacer with a piezo-actuator, was originally planned for the
final 698 nm system; however, better lock stability was achieved by locking directly to the ULE
cavity via PDH. In aligning to a very high finesse cavity, however, it was useful to have the
pre-narrowed laser as this allowed us to see an actual error signal as we swept across the cavity
resonance.
82
0.25
0.00
0.25
1.0
0.8
0.6
0.4
0.2
0.0
1000 500 0 500 1000
Frequency (Hz)
Figure 3.14: Heterodyne beatnote between two independent lasers at 689 nm. One laser is locked
to the “clock” cavity, while the other laser is the master oscillator for the narrow-line rMOT. The
joint FWHM Lorentzian linewidth is 289(14)Hz (solid orange line). This is likely dominated by
the linewidth of the less well-isolated Fabry Pérot cavity used to lock the rMOT master laser,
however from this beatnote measurement one cannot disentangle the relative contributions from
each laser.
our 698nm system, and is likely dominated by the linewidth of the less well-isolated
689nm Fabry-Pérot cavity.
In addition to the heterodyne beatnote at 689nm, we can also use the atoms
themselves to measure the performance of the laser. Unfortunately, we do not have a
magic wavelength dipole trap on this apparatus (see Chapter 6 for discussion of the
next-generation system, which will have magic wavelength lattices designed in from
the beginning). The resulting lineshape becomes temperature-dependent, and we
can use the framework from Ref. [83] to estimate the width of a carrier transition in
a deep, non-magic wavelength lattice. Specifically, the full-width-at-half-maximum
(FWHM) Γc is given by
( ) ∣∣∣√ ∣′ − ∣k α /α 1∣B
Γc ≈ T , (3.45)
h 0.295
83
residuals
Beatnote amplitude (arb.)
0.2
0.1
0.0 0.0
0.1
0.2
Gaussian FWHM=60.7(4) kHz Gaussian FWHM=7.4(9) kHz
1.1
1.0
1.0
0.9
0.8
0.8
0.6 0.7
0.6
0.4
0.5
300 200 100 0 100 200 20 10 0 10 20
Detuning (kHz) Detuning (kHz)
Figure 3.15: Atomic spectroscopy of the clock transition in 87Sr (left) and a 84Sr BEC (right).
For the 87Sr spectroscopy, each point is the mean of five measurements and the error bar taken
as the standard deviation; the fit gives a reduced χ2/20 = 0.99. The FWHM are displayed, and
correspond to a 1/e width of 36(2) kHz and 4.4(5) kHz respectively.
where h, kb are respectively Planck’s and Boltzmann’s constant, T is the atom
temperature, and α′/α is the ratio of polarizabilities in the excited and ground
states. At 1064nm, this polarizability ratio is ≈ 0.7. This translates to a thermally
broadened carrier linewidth of roughly 11.5 kHzµK−1. This is much larger than the
. 1Hz linewidths one expects from an optical clock transition.
Despite this difficulty achieving a truly narrow clock lineshape, we show some
spectroscopic loss features out of a dipole trap in Fig. 3.15. Clearly, the spectra
are inhomogeneously broadened beyond the . 200Hz linewidth we might expect
from the laser; in the case of 87Sr, we see a thermal lineshape with a width ≈
40 kHz. In the case of the 84Sr BEC, the lineshape is much more complicated but
its characteristic width is set by the chemical potential. The narrowest features we
observed were in a BEC, on the order of 4 kHz to 5 kHz.
84
Atom number (norm.) Resid.
Atom number (norm.) Resid.
Chapter 4: Publication: An ultra-low noise, high-voltage piezo driver
Over the course of my PhD, I had the opportunity to work on several electron-
ics design projects to enhance the instrumentation and control of our experiment.
Here, I include the pre-print arXiv version of a paper that came out of one such
project, to create a low-noise, high-bandwidth piezoelectric driver well-suited for
controlling ECDL lasers. In conjunction with the several improvements and modifi-
cations made to the Durfee/Hall/Librecht current controller and the optical phase-
locked-loop beatnote servo (both described previously in Sec. 2.5), this piezo driver
completes a set of high-performance laser control electronics that have afforded a
great deal of versatility when working with narrow-line transitions in strontium.
The published version of this chapter appears in Review of Scientific Instru-
ments 87, 124702 (2016) [48].1 What appears below has been edited and reformat-
ted for increased clarity, and includes an additional section at the end discussing
aspects which did not make it into the published version, and future improvements
that could be made to the design and its firmware.
4.1 Abstract
We present an ultra-low noise, high-voltage driver suited for use with piezoelec-
tric actuators and other low-current applications. The architecture uses a flyback
1The design was created in discussion and collaboration with Alessandro Restelli, the JQI
electronics guru, to whom I am indebted for much of my electronics expertise. I performed the
PCB layout and collected the reference data (with occasional assistance from Alessandro), and
wrote the manuscript. All authors contributed critical edits to the manuscript, and provided
insights and support throughout the process.
85
switching regulator to generate up to 250V in our current design, with an output of
1 kV or more possible with small modifications. A high slew-rate op-amp suppresses
the residual switching noise, yielding a total RMS noise of ≈ 100 µV (1Hz–100 kHz).
A low-voltage (±10V), high bandwidth signal can be summed with unity gain di-
rectly onto the output, making the driver well-suited for closed-loop feedback appli-
cations. Digital control enables both repeatable setpoints and sophisticated control
logic, and the circuit consumes less than 150mA at ±15V.
4.2 Introduction
Many instrumentation applications in the modern laboratory require agile,
low-noise voltage sources capable of supplying hundreds of volts or more. For exam-
ple, piezo-actuated mirrors and diffraction gratings play an important role in atomic
physics experiments (used, e.g., in Fabry-Pérot cavities [84, 85] and external-cavity
diode lasers [32]), while avalanche photodiodes and photomultiplier tubes require
large bias voltages for proper operation. In the realm of biophysics, electrokinetic
separation methods such as free-flow or capillary electrophoresis [86] require large
electric field gradients, and the recent push to develop lab-on-a-chip devices could
benefit from miniaturized high-voltage sources. [87]
Laboratory devices are often operated in a closed feedback loop, where small
voltage changes on top of a large DC voltage are necessary to stabilize the output of
a particular system. For example, the frequency of an extended-cavity diode laser
can be locked by feeding back to a piezo-actuated diffraction grating or mirror, which
in turn supplies optical feedback to the diode. Commercially available piezoelectric
drivers typically provide a modulation input for such closed-loop applications, but
the input voltage is often gained such that it spans the entire output range of
the device. Other designs separate high- and low-voltage control pathways, which
can extend the bandwidth to ≈ MHz, but the low-voltage control is AC-coupled
86
to the output. [88] While these designs have advantages, many applications would
benefit from an architecture that provides a unity-gain, DC-coupled feedback path
to the high-voltage output. This low-gain modulation input could make closed-loop
systems less susceptible to noise contributions from the servo controller, which we
often find in our laboratory to be a limiting factor in laser lock stability.
Instrumentation electronics capable of supplying high voltages traditionally
fall under one of two architectural umbrellas: DC-DC switching converters, and
linear-type amplifiers. While DC-DC converters are efficient and can work at very
high voltages, they suffer from switching noise and limited control bandwidths.
Linear-type devices are typically constructed from a high-voltage operational am-
plifier (op-amp), powered either from a high-voltage linear regulator or more typi-
cally from a secondary switching converter. While the op-amp can provide 100dB
or more of power-supply noise rejection [89], linear regulators must handle any ex-
cess voltage by dissipating heat and so may be more cumbersome to deploy in the
laboratory.
We present a circuit with a hybrid architecture. The high voltage is gen-
erated by a galvanically isolated DC-DC converter, while a low-noise, high-slew-
rate op-amp simultaneously removes noise at the output and provides a low-gain,
high-bandwidth (& 100 kHz) modulation input for closed-loop feedback applica-
tions. This architecture is able to achieve extremely low noise (≈ 100 µVRMS) over
the entire output range, draws very little current, and fits comfortably onto a small-
footprint printed circuit board (PCB). Additionally, the high-voltage output remains
single-ended and referenced to ground, allowing it to drive piezo actuators with a
grounded terminal. The schematic is presented in Sec. 4.3, with a noise analy-
sis in Sec. 4.4 and characteristic performance data in Sec. 4.5. Complete design
files, including the schematic, bill of materials, and board layout, can be found on
87
Passive Filtering
DRV2700 Flyback Regulator f 30 Hz Input nodes5V c ~ Key circuit nodes
VHV CHV (1nF) Low-noise stabilization Vout
D1
HV output
Vdd SW 5V Cout (1uF)
R3 C3 (22pF) V R
DRV2700 MG 1 C1R9
Cmod
R11
gnd FB V C V U2
Vmod
FB 4 FG
R R4 (HV floating gnd)10 U1 R2
R R7 modVctl
R DAC5
R6 R8
Switchable lowpass filter
Figure 4.1: Schematic of the high voltage supply and stabilization. The voltage VHV is generated
using a Texas Instruments DRV2700 high voltage driver in flyback configuration. A high slew-rate
op-amp (U2) senses the output voltage across R1 and R2, and controls it by modulating the mid-
ground node VMG. The DC control signal for this op-amp, Vctl, is supplied by a digital-to-analog
(DAC) converter, which is passed through a switchable low-pass filter. This design allows for very
heavy filtering of the DAC 1/f noise during steady-state operation, but the corner frequency can
be increased if the output needs to be scanned more quickly. The Vctl gain is set by (1 +R1/R2),
while the modulation gain is set by −R1/Rmod. The op-amp U2 removes residual switching noise
and stabilizes the DC output according to the transfer function given in Eq. (4.3). A MOSFET
quench circuit, shown in Fig. 4.2, connects at nodes Vout and VMG.
GitHub.2 The board manufacture and component cost is less than $200, making it
a cost-effective alternative to commercial options.
4.3 Circuit Design
The design principles discussed below show how we leverage the characteris-
tics of a galvanically-isolated switching regulator without sacrificing the low-noise
requirements of many laboratory applications. Our design targets a 250V output,
but we discuss straightforward modifications to the schematic that make it possible
to tailor the gain and output range to a specific application. The entire electronics
package fits into a compact Eurocard rack module (with the high-voltage section
taking only a fraction of the PCB), and draws less than 150mA at 15V. The high-
voltage output current will be limited by the switching regulator and by the LM7171
2https://github.com/JQIamo/hv-piezo-driver
88
op-amp used for low-noise stabilization (U2 in Fig. 4.1, which can supply at most
≈ 100mA [90]), but is sufficient for nearly all piezoelectric applications.
Fig. 4.1 shows an overview of the circuit schematic. A flyback regulator
(Sec. 4.3.1) controls the potential between the high-voltage (VHV) and floating-
ground (VFG) circuit nodes, while the low-noise stabilization circuitry (Sec. 4.3.2)
controls the output node Vout relative to the true circuit ground. A digital-to-analog
converter (DAC) generates a voltage setpoint, Vctl, which is sent to the high-voltage
flyback regulator and to the low-noise stabilization circuit. The DAC is controlled
by an integrated microcontroller, and can be programmed to output slow (≈ 10Hz)
rail-to-rail voltage ramps in addition to setting the DC operating point (Sec. 4.3.3).
To improve the large-amplitude slew rate, a MOSFET “quench” circuit (Fig. 4.2)
is included to reduce the RC time constant of the high-voltage node Vout when
needed. A low-pass filter with a switchable corner frequency can be engaged dur-
ing DC operation to reduce 1/f noise from the DAC (discussed in Sec. 4.4). Fast
output modulation between ±10V can be achieved through the input node Vmod.
This node is DC-coupled to the high-voltage output, and is useful for closed-loop
feedback control.
4.3.1 Flyback regulator
The high-voltage DC-DC converter used here is based on the Texas Instru-
ments DRV2700 piezo driver.3 This integrated circuit can be operated as a boost
converter to drive an on-chip differential amplifier up to 100V, or as a flyback con-
verter up to 1 kV or more. In flyback configuration, the internal-boost switch of the
DRV2700 (pin SW in Fig. 4.1) drives a step-up transformer. When the switch closes,
current begins to flow through the primary coil of the transformer and induces a
3The identification of commercial products in this paper is for information only and does not
imply recommendation or endorsement by the National Institute of Standards and Technology.
89
corresponding voltage across the secondary coil. In this state, the output diode D1
is reverse-biased, and the capacitor (CHV in Fig. 4.1) holds its charge. When the
switch opens, the voltage across the secondary coil is inverted, putting the diode
into conduction and charging the capacitor. By changing the switching duty cycle,
the DRV2700 is able to regulate the voltage across the galvanically isolated output
(nodes VFG and VHV in Fig. 4.1).
The DRV2700 implements output voltage control by comparing the feedback
input pin at node VFB with an internal (1.3V) reference. The resistors R9 and
R10 are chosen such that pin FB is at 1.3V when the output of U1 is at ground:
R10/(R9 + R10) = 1.3V/5V ≈ 0.26. The op-amp U1 subtracts VFG and G · Vctl
from the voltage at node VHV, ensuring the DRV2700 regulates the output voltage
such that
VHV − VFG = G · Vctl , (4.1)
where the gain G is set by the resistor ratio R3/R4 ≡ R5/R6, and Vctl is the control
voltage set by the DAC. The capacitors C3 and C4 are chosen such that C3 = 22 pF
and
C4 R3
= || , (4.2)C3 R4 R7
as suggested by the DRV2700 datasheet [91], where R6 = R7 = R8, and Ri || Rj ≡
RiRj/(Ri + Rj). In our implementation, we choose a gain G ≈ 50 (R3 = R5 =
499 kΩ; R4 = R6,7,8 = 10 kΩ), allowing a 5V control signal Vctl to span 250V at the
output. A different DAC and/or a different gain factor could be chosen to adjust
the maximum output voltage.4 The transformer (ATB3225, 1:10 step-up winding),
diode, and RC feedback network are all based on values suggested in the DRV2700
4Note that the passive components must be rated for the chosen output voltage.
90
Vout
+15V +15V
RMOS
VTH
Q1
100k
VMG 8.2k 499R
100pF
-15V
Figure 4.2: MOSFET “quench” circuit. When the mid-ground node VMG (also shown in Fig. 4.1)
goes below the threshold value set at VTH, the op-amp puts the HV MOSFET Q1 into conduction.
When engaged, the quench time constant is given by τ ≈ RMOSCout, which for our circuit is set
to 1ms. The capacitor Cout = 1 µF is shown in Fig. 4.1, and details of this circuit are discussed in
Sec. 4.3.3.
datasheet. [91, 92]
The output of the flyback regulator is passed through a four-pole, low-pass
RC filter. The corner frequency fc ≈ 30Hz is chosen to be high enough that a slow
(≈ 10Hz) rail-to-rail triangle ramp can be applied by the DAC at Vctl (for, e.g.,
sweeping over a resonance in spectroscopy), but low enough that the ≈ 100 kHz
switching noise is substantially attenuated. Additional capacitors on both the VHV
and VFG resistor networks shunt high frequency noise to ground. This filter topology,
modeled on a lossy transmission line, is sufficient for our application, but other
corner frequencies or topologies could also be used.
4.3.2 Low-noise stabilization and fast modulation
The low-noise stabilization circuit is crucial to the performance of the design,
as it is responsible for removing noise at the output of the flyback converter. To
91
2.7k 15k
8.2k
1k
accomplish this, a high slew-rate op-amp (Texas Instruments LM7171, 4100V/µs;
see U2 in Fig. 4.1) drives the galvanically isolated floating ground of the flyback
converter. This op-amp senses the voltage Vout referenced to true circuit ground,
and adjusts its output such that
( ) ( )
R R
V 1 1out = 1 + || Vctl − Vmod . (4.3)R2 Rmod Rmod
Here, Vmod is the applied modulation, which can vary between ±10V and is in-
verted before being summed onto the output. We choose R1 = Rmod = 1MΩ and
R2 = 20.5 kΩ such that the DC gain ∆Vout/∆Vctl is ≈ 50 and the modulation gain
∆Vout/∆Vmod is unity. Depending on the application, other gain configurations
could work equally well provided the non-inverting gain of U2 closely matches the
gain of the flyback regulator (since they both derive from Vctl).
The op-amp U2 controls Vout via two different feedback pathways. At low
frequencies, it modifies the floating ground reference VFG of the flyback converter,
and the DRV2700 in turn modifies VHV to maintain a constant voltage between VFG
and VHV. At higher frequencies, U2 is decoupled from VFG by the passive filtering
network. In this regime, Cout provides a low-impedance path between U2 and the
output, such that high-frequency switching noise can be directly compensated for
by the op-amp. We chose a value Cout = 1 µF, which is a compromise between
component size and the desire for a large capacitance. In addition, a small resistance
R11 = 50 Ω is inserted between U2 and Cout to ensure stable operation. Smaller R11
and/or larger Cout might provide better performance, but this has not been tested.
The choice of components for resistors R1 and R2 is crucial for the low-noise
performance of the system. Because this resistive divider is responsible for accurately
sensing the voltage Vout, noise introduced by these resistors cannot be corrected by
the op-amp. In general, resistors are fundamentally limited by Johnson noise, in
92
which thermal fluctuations contribute to a white noise power spectrum given by
4kBTR, where T is the temperature and kB is Boltzmann’s constant. [93] However,
resistors also exhibit 1/f current noise caused by equilibrium fluctuations of the
resistance. [94, 95] This excess noise depends on the applied voltage, and therefore
is an important consideration in a high-voltage circuit. It is also highly dependent
on the resistor composition and varies from manufacturer to manufacturer. Seifert,
et. al. [96] characterized 1/f noise in a variety of resistors, and found that the
Vishay TNPW 0.1%-series resistors showed a noise spectrum almost consistent with
Johnson noise down to 1Hz. Our current design uses this series in a 1206 package,
but we noticed low-frequency noise correlated with varying strain on the PCB,
potentially due to the relatively large footprint of this package. Future boards
might instead use three TNPW 0.1% 0603 resistors in series for both R1 and R2 to
minimize strain-induced output noise.
The value of capacitor C1 is a tradeoff between two competing design consid-
erations. On the one hand, a larger C1 extends the frequency range where switching
noise from the DRV2700 is suppressed. However, large values of C1 limit the band-
with of Vctl. We empirically settled on C1 = 1 nF, which is large enough to saturate
the feedback gain in the 40 kHz–100 kHz range where switching noise dominates,
but not so large that it limits the bandwidth of Vctl below the corner set by the
switchable low-pass filter described in Sec. 4.3.3. Once C1 was chosen, capacitor
Cmod was calculated to match the impedances R1 || C1 = Rmod || Cmod. For our
circuit, this means Cmod = C1.
4.3.3 Digital control and slow modulation
The high-voltage setpoint (absent voltages summed in at Vmod) is controlled
by a low-noise DAC. This has two advantages: digital control enhances setpoint
repeatability, and simplifies the integration with computerized control electronics or
93
sophisticated servo loops. While the modulation input Vmod has a limited range of
±10V, larger voltage swings can be achieved by reprogramming the DAC.
As discussed below in Sec. 4.4, without modification the DAC would domi-
nate the noise performance of Vout. Therefore we add a single-pole, low-pass filter
between Vctl and the non-inverting node of U2 to bring the DAC noise contribution
below other noise sources in the circuit. This filter has a switchable corner frequency
(between 165Hz and 0.8Hz) to optimize noise performance at DC while still per-
mitting AC modulation when needed. It consists of a 20.5 kΩ resistor and 47 nF
capacitor, with a secondary 10 µF capacitor that can be switched in to operate with
the lower corner frequency.
One downside of the flyback regulator presented above is that while the switched
transformer can quickly charge the output capacitors, the discharge time τ is limited
to ≈ 1 s by the RC time constant of the circuit. To get around this limitation, we
have added an auxiliary MOSFET “quench” circuit5 to quickly shunt Vout to ground
(see Fig. 4.2). This circuit works by monitoring the voltage at VMG, the mid-ground
node controlled by op-amp U2. If VMG drops below a threshold set by VTH, the
comparator op-amp in Fig. 4.2 changes the gate voltage of the MOSFET to put it
into conduction. The time constant for this configuration is given by τ ≈ RMOSCout.
For our circuit, this changes τ to ≈ 1ms, allowing Cout to be quickly discharged.
The threshold is VTH = −10.4V, but other values could be chosen depending on
the design requirements.
The high-voltage design presented here has the flexibility to exist as a stan-
dalone circuit or be integrated with other electronics, and we have included several
auxiliary features to make this more convenient. For example, the analog modula-
tion input is differentially buffered to break ground loops (not shown in Fig. 4.1),
5The quench circuit presented here is based on the pulldown FET discussed in the DRV2700
datasheet, with some additional modifications to suit our purposes.
94
and the digital portion can be interfaced with other devices in the lab to expand
conceivable control scenarios. Secondary features include a divided-down output
that can be used as a monitor or fed forward to a low-noise laser diode current
controller like the one in Erickson, et. al. [41] Of course many variations are pos-
sible, and we refer the reader to our GitHub page for more details on our specific
implementation.6
4.4 Noise model & analysis
To better understand the circuit performance and the measured output noise
reported in Sec. 4.5, we introduce the noise model shown in Fig. 4.3. A summary of
each noise contribution (op-amp, DAC, Johnson-Nyquist, and residual ripple from
the DRV2700, all calculated at the node Vout) is shown in Fig. 4.4, along with the
cumulative root-mean-square (RMS) noise estimates in different frequency bands.
We will neglect noise appearing at node Vmod due to the external modulation because
its exact character depends on the external drive.
To facilitate the noise analysis, we calculate the voltage and current (tran-
simpedance) gains from the input nodes of U2 to the output, Vout. Starting with
the non-inverting node, we find the voltage gain to be
( )
(v) 1 1
G+ = 1 + Z1 + , (4.4)R2 Zmod
where Z1 and Zmod are the equivalent impedances of R1 || C1 and Rmod || Cmod,
respectively. The transimpedance gain, (i)G+ , follows by multiplying
(v)
G+ by the
impedance, Z+, seen from that node. Thus,
[ ]
(i) (v) RLP
G+ = G+ , (4.5)1 + 2πifRLPCLP
6https://github.com/JQIamo/hv-piezo-driver
95
Vout
Noise from DRV2700
(v
v n,DRV
 )
n
R1 C1
Op-amp Cmod
U2 in Vmod
V en vnU2 R2 Rmod
in
Modulation noise
Intrinsic op-amp noise Vctl
R DAC voltage noiseLP vn (v
C n,DAC
 )
Switchable low-pass filter LP
Figure 4.3: Noise model, including contributions from the op-amp, DAC, DRV2700, and modula-
tion input. The element labeled CLP consists of a 47 nF capacitor in parallel with a 10 µF capacitor
connected to ground through a switch, such that the corner frequency of the filter can be changed
depending on the mode of operation (see text). Though not drawn in the figure, we also consider
the Johnson noise contributions from all resistors.
where the bracketed term is Z+, f is the Fourier frequency, and i is the imaginary
unit.
We now calculate gains from the inverting node of U2. Because any currents
appearing at this node will be cancelled by the feedback of U2, the transimpedance
gain (i)G− is simply the impedance Z1, given by
(i) R1
G− = Z1 ≡ . (4.6)1 + 2πifR1C1
96
With these expressions in hand, we can calculate the output noise contribution from
each source in our model.
As shown in Fig. 4.3, the op-amp noise is parametrized by two noise contri-
butions: en, the input voltage noise spectral density, and in, the input current noise
√ √
spectral density. For the LM7171 at 10 kHz, en = 14 nV/ Hz and in = 1.5 pA/ Hz
with a 1/f noise character below this frequency. [90] The voltage noise appears at
the non-inverting input, while the current noise is present at both inputs. By mul-
tiplying each source by the appropriate gain, we obtain the equivalent output-noise
power spectral densities (PSD),
2 ∣∣ ∣(v)∣2e 2v,U2 = ∣G+ ∣ en ∣ ∣ (4.7)
e2 = ∣ (i)G− ∣2i2 + ∣ (i) 2i,U2 n G ∣ 2+ in ,
where e2 2v,U2, ei,U2 are the output-referred voltage and current noise PSD, respectively
(plotted in Fig. 4.4).
Next, we calculate the DAC noise contribution. The voltage gain from the
node Vctl is given by
[ ]
(v) 1 (v)
Gctl = G , (4.8)1 + 2πifRLPC +LP
where the bracketed term represents the contribution to the transfer function from
the switchable low-pass filter. Without the addition of this low-pass filter, the
voltage noise of the DAC would dominate both the low- and high-frequency noise
performance of the circuit. A simple solution would be to place a fixed-corner filter
at this node, but this would severely restrict the AC performance of Vctl. Thus,
we use a switchable low-pass filter (as described in Sec. 4.3.3) to achieve low-noise
performance during DC operation, while still permitting the DAC to modulate Vctl
more quickly when needed. The non-zero resistance of the switch contributes a zero
97
to the transfer function at ≈ 23 kHz, but has negligible effect on the performance.
The DAC voltage noise contribution can now be calculated as
∣∣ ∣e2 (v)∣2= G v2n,DAC ctl n,DAC , (4.9)
where vn,DAC is the frequency-dependent output voltage noise of the DAC as reported
in the datasheet. [97] In subsequent calculations, we take the DC-mode operation
(fc = 0.8Hz) for the switchable low-pass filter.
The Johnson noise contribution can be calculated by modeling each resistor
with a parallel current noise given by i2R = 4kBT/R. Resistors R1, R2, and Rmod all
contribute current noise at the inverting node of U2, which as discussed previously
has a transimpedance gain to the output given by (i)G− . The resistor RLP contributes
current noise at the non-inverting node, which sees a transimpedance gain (i)G+ .
Thus, the total Johnson noise is
e2n,JN [=∣ 4kBT×∣ ∣∣ ( ) ∣∣ ∣ ](i)1 1 1 G ∣2 (4.10)(i) 2 +G− + + + .R1 R2 Rmod RLP
The low-noise stabilization circuit is limited in its ability to reject residual
switching noise from the DRV2700 regulator (after the passive filtering network) by
the total loop gain analyzed from node Vout. The LM7171 has a reported open-loop
gain of 85 dB (≈ 1.8× 104), with a dominant pole at ≈ 10 kHz. [90] We can model
the open loop gain, GOL, as
1.8× 104
GOL = . (4.11)
1 + i(f/10 kHz)
98
10 4
50V
100V
10 5 200V
SR780 Noise Floor
Noise Model
10 6
10 7
10 8
10 9100 101 102 103 104 105
Frequency (Hz)
Noise source RMS Voltage RMS Voltage(1Hz – 10Hz) (10Hz – 100 kHz)
op-amp voltage (ev,U2) 26 µV 31 µV
op-amp voltage (ei,U2) 51 µV 73 µV
DAC (en,DAC) 28 µV 8 µV
Johnson-Nyquist (en,JN) 3 µV 14 µV
DRV2700 (en,DRV) 15 µV 43 µV
total (calculated) 66 µV 92 µV
Figure 4.4: Output voltage noise. The graph on the left shows the calculated noise contributions
from each source in our model, along with the total calculated noise. The graph on the right shows
the measured output noise at several voltages, with the total noise model from the lefthand graph
superimposed for reference. Power spectral density (PSD) is referred to the high-voltage output,
and the table shows the integrated RMS noise due to each noise source in different frequency bands.
The total RMS noise (summed in quadrature) over the entire 1Hz – 100 kHz range is calculated to
be 113 µV, with the residual DRV2700 switching noise measured at 100V as described in the text.
Frequency-dependent noise spectra for the DAC and op-amp were extracted from the datasheets.
The measured noise in the right-hand plot was taken on an SR780 spectrum analyzer, AC-coupled
with a corner frequency . 1Hz. The integrated RMS noise (1Hz − 100 kHz) is {138, 80, 101} µV
measured at {50, 100, 200} V, with no output load.
The feedback network contributes a gain
R2 || Zmod
GFB =
Z1 +R2 ||
, (4.12)
Zmod
arising from the voltage partition between Vout and the inverting node of U2. From
these, we write down the closed-loop gain seen from Vout,
G(DRV)
1
= . (4.13)
1 +GOLGFB
99
PSD (VRMS/ Hz)
The contribution to the output from residual switching noise, vn,DRV, is then
∣ ∣
e2 = v2 ∣G(DRV)∣2n,DRV n,DRV . (4.14)
For R1 = 1MΩ and C1 = 1 nF, vn,DRV is attenuated by as much as 76 dB at 6.3 kHz.
We estimate the noise spectral density vn,DRV by monitoring the node VU2 in
Fig. 4.3, since the output of this op-amp represents the control signal required to
cancel voltage fluctuations at Vout. The trace for en,DRV plotted in Fig. 4.4 is derived
from the results of this measurement. Because the measured vn,DRV drops below the
noise floor of our spectrum analyzer at ≈ 10 kHz, we only plot the trace out to this
frequency.
Given the noise model discussed above, our circuit is dominated by the op-
amp’s intrinsic current noise at lower frequencies, and voltage noise at higher fre-
quencies. The op-amp current noise contribution could be suppressed by using lower
resistances R1 and Rmod, however one must be careful about power and current lim-
itations when dealing with such high voltages. Each noise source is tabulated and
plotted in Fig. 4.4, and the total voltage noise (1Hz − 100 kHz, Vout = 100V) is
calculated to be 113 µVRMS .
4.5 Results
The measured performance of the high-voltage piezo driver is shown in Fig. 4.4,
where we plot the noise power spectral density measured at several different output
voltages. These traces were taken on a Stanford Research Systems SR780 spectrum
analyzer, with the high-voltage output coupled through a 0.5Hz high-pass filter
and without any capacitive load. This represents a worst-case scenario, as larger
capacitances at the output will reduce the noise. The integrated noise (1Hz −
100 kHz) was measured to be {138 µVRMS, 80 µVRMS, 101 µVRMS} for {50V, 100V,
100
(a)
(b)
Figure 4.5: Time-domain traces of the high-voltage output (100V, color online). (a) Short-time
scatterplot, measured on a PicoScope 5442B (AC coupled). The points have been down-sampled
for clarity, and colored based on a normally-distributed probability of occurence in each 100-µs
timeslice (scaled to the most probable voltage). The color thus provides a visual estimation of the
RMS width. (b) Long-term trace, measured on a Keithly 2010 digital multimeter. A 100V DC
offset is subtracted from the plotted values. The top panel shows the lab temperature during the
same time period.
200V} outputs. This matches roughly with the expected total RMS noise calculated
in Sec. 4.4. The difference in noise performance at different output voltages can be
traced back to the residual ripple of the DRV2700, which is larger for lower output
voltages. Indeed, one can see the characteristic shape change in Fig. 4.4 between
100V and 50V as the residual ripple en,DRV begins to dominate at low frequencies.
Fig. 4.5 shows the performance at both short- and long-time scales. At long
times, voltage fluctuations on the order of a few mV can be observed. This is due
generically to 1/f noise, but also correlates with the external temperature. A cross-
correlation between the measured temperature and output voltage yields an effective
temperature coefficient of −24 ppm/◦C at 100V, see Fig. 4.6. The short-term trace
was taken on a PicoScope 5442B (AC-coupled, 100V output), and downsampled for
101
Temperature 22.0
4
21.5
2 21.0
20.5
0
20.0
2
19.5
4 19.0
0 5 10 15 20
Time (h)
Covariance / Temp Voltage
Temp 1.15 −2.4× 10−3
Voltage −2.4× 10−3 6.5× 10−6
Temp. Coeff: −21.1 ppm/◦C
Figure 4.6: Output voltage fluctuation, measured during a period of laboratory temperature in-
stability. The graph at top shows deviations in the output voltage (centered at ≈ 100V, plotted in
blue), with the temperature plotted in yellow-orange over the same time period. From this data,
we compute the covariance matrix, shown at bottom. The diagonal elements represent the vari-
ance σ2, while the off-diagonal elements represent the covariance COV(T, V ) = 〈(T − T )(V − V )〉
between the temperature T and voltage V . At ≈ 100V output, this represents a temperature
coefficient of −21.1ppm/◦C, given by COV(T, V)/(σ2T〈V 〉).
clarity. Points are binned into 100-µs slices, and colored based on their normally-
distributed probability of occurrence. The color scale is normalized to the most
probable voltage in each bin.
Fig. 4.7 shows the measured frequency response under different load conditions.
The unloaded bandwidth is as high as a few megahertz, while a 1 µF capacitive load
can still be driven at ≈ 100 kHz. Several mechanical resonances can be seen with a
700nF piezoelectric load, as expected. In a laboratory setting, these resonances can
be mitigated by using a digital feedback controller with notch filters tuned to match
the exact resonance frequencies observed in the system [98], thereby extending the
usable bandwidth out to ≈ 100 kHz.
102
Voltage (mV)
Lab Temperature (C)
0
10
Unloaded
700nF PZT
1uF capacitor
20
180
90
0
90
180
101 102 103 104 105 106 107
Frequency (Hz)
Figure 4.7: Modulation input transfer function (color online). The solid blue line indicates the
unloaded frequency response, with the phase plotted in the lower panel. The dash-dotted orange
trace shows the response with a 700 nF piezoelectric actuator (Thorlabs PN AE0505D08F). Several
mechanical resonances above ≈ 50 kHz are clearly visible. The dashed green trace shows the
response under a 1 µF capacitive load. Loaded response bandwidth is ≈ 100 kHz, and the unloaded
gain is flat within 0.1 dB out to 1MHz where the phase is −20◦.
4.6 Conclusion
We have designed, built, and characterized a high-voltage piezoelectric driver
optimized for use in a modern atomic physics laboratory. It is based on a flyback
configuration switching regulator, but is able to achieve very low noise performance
by active stabilization from a high slew-rate op-amp. This hybrid architecture makes
it small and easy to deploy in a variety of situations, without requiring an external
high-voltage power supply. The design principles discussed here can be adapted to
fit the exact application, and all design files are freely available on GitHub for others
to use and modify.
103
Phase Magnitude (dB)
4.7 Future developments
This circuit has functioned nicely in our lab for over a year now, however
there are a few design modifications that should be made before fabricating a new
version. First, the onboard microcontroller7 has an ADC which is connected to a
divided-down pickoff of the high-voltage output monitor. The idea here was to allow
the microcontroller to read the actual output voltage, and in conjunction with its
own knowledge of what the programmed setpoint is, infer the voltage supplied by
the analog modulation input. From this reading, the microcontroller can adjust the
internal setpoint to keep the external modulation input centered around a particular
value.
Unfortunately, when we went to implement this feature, it was discovered that
the voltage read by the microcontroller ADC was very non-linear with the output
voltage, particularly at the low end, which came as a bit of a surprise! However,
on further inspection of the datasheet, we discovered an important footnote: the
analog source resistance should be kept as low as possible (less than a few ohms),
and the input impedance RC time constant should be . 1 ns. Neither of these is
presently true for our design; the equivalent source resistance is ≈ 2.3 kΩ, and given
parasitic capacitances can be ∼ pF, the source RC time constant will be greater
than 1 ns. Luckily, this is quickly resolved by buffering the voltage monitor before
sending it to the ADC.
Currently, we work around this nonlinearity by implementing a “calibration”
procedure. The microcontroller creates a lookup table at several voltages over the
full output range of the device, and thus can map the ADC measured values to the
7We are using the MK20DX256VLH7, an ARM Cortex M-4 from Freescale, along with the
Teensy bootloader for easy programming and integration with the plethora of Arduino-based li-
brary firmware. Not every board has this bootloader populated, however; instead, we have a
“programming board” with the bootloader chip mounted, which plugs into the header J5 and
performs the same functionality.
104
actual output voltage, linearly interpolating between calibration points. Unfortu-
nately, this procedure seems to only be good to ≈ 2−3V accuracy. Thus, the digital
feed-forward functionality is a bit limited. We also have not yet implemented the
slow-scan feature, but its addition will be straight forward using the API laid out
in src/piezo_api.h [99].
The firmware, which can be found under the code/ directory of the main
repository [99], has a good deal of inline documentation. However, I will mention a
few high-level design choices here, along with some frustrating workarounds in the
Arduino build system. First, piezo.ino is the top-level Arduino sketch. Individual
modules, which implement different functionality, are broken into their components
in src/*.h. In the main loop() function, calls are made to process input in the
individual modules (e.g., to process input from the frontpanel rotary encoder). There
is a stub module for implementing the slow scan mentioned above.
Unfortunately, the Arduino build system comes close to being an excellent
tool, but misses the mark on several fronts. The IDE, apart from having a difficult
UI for any project of reasonable size, does not provide a good way to package code
on a per-project basis. We instead use a custom make script, in conjunction with
a build options file (build.options.json), to facilitate compilation. This specifies
the local lib/ folder as the libraries path. After compilation, one can grab the HEX
file generated in bin/, and use the Teensy loader 8 to flash the microcontroller. In
other projects (e.g., the AOM driver found in [100]), we also use this same code
structure, and pull the current git commit hash just before compilation to allow
firmware traceability (something which should be ported to the piezo codebase as
well). More details about the Arduino builder tool can be found on Github9, and a
quick google search reveals many helpful StackExchange answers.
8https://www.pjrc.com/teensy/loader.html
9https://github.com/arduino/arduino-builder
105
I will conclude by mentioning a long-standing plan to integrate this circuit,
along with other Eurocard form-factor circuits, into a local-area network (LAN)
ecosystem of controlled devices. One might imagine controlling the current/piezo of
a diode laser, either programmatically or by hand from the main computer, without
needing to walk to the back of the lab. This could be useful, for example, in our
wavemeter-based repump laser locking scheme. Software running on the computer
controlling the wavemeter could remotely send packets to the laser control electron-
ics, instructing them to update their output based on some computed frequency
error signal.
To facilitate this future integration, all recent Eurocard-based circuits im-
plement an I2C bus connection to the backplane.10 A rear-facing module in the
Eurocard rack, which could be as simple as an Arduino-based WIZNET module or
as fancy as an FPGA or SoC such as the Zynq, exposes rack modules to the lab
network. Commands sent to the rack are parsed and passed to individual modules
to query or change their behavior. I would’ve liked to see this planned integration
through to completion, but it continually fell down the list of priorities.11
10For the backplane standard we have settled on, see [101], particularly the in-progress branch
at the time of this writing.
11It is worth noting that Alessandro Restelli has been experimenting with some good success
with the PYNQ ecosystem, which integrates a python front-end with an FPGA/Linux backend
running on a Zynq SoC. Using this as a model, one could imagine an easy way to expose low-level
hardware behavior in a convenient pythonic frontend. Xilinx sells a rather cheap but performant
evaluation board for the Zynq, the MicroZed (http://zedboard.org/product/microzed), which
has built-in ethernet PHY and a high-pincount FMC connector on the bottom. In the future,
it could be used to quickly add realtime capability to custom-built electronics, while providing a
python interface via the PYNQ project.
106
Chapter 5: Isotope shift spectroscopy
Recently, there has been a push to use the unprecedented precision offered by
cold atomic systems to investigate new physics beyond the standard model. For a
good summary of those efforts, including precision measurements to detect variations
in fundamental constants, see the recent review by Safronova et. al. [102]. Here,
however, we focus on measuring isotopic shifts in atomic energy levels as a probe of
many-body electron correlations and nuclear effects.
There is a long history of these kinds of measurements, dating back to the
mid-1900s [103, 104]. In particular, W.H. King recognized that one could plot the
isotope shifts, scaled by a factor related to the difference in mass, of two separate
atomic transitions and the points would fall along a line. This relationship is the
eponymous “King Linearity,” and the plots are known as “King Plots.” The slope
of the line gives the ratio of the field shifts of the two transitions, and the intercept
is related to the mass shift. The theoretical underpinnings of these shifts will be
discussed in Sec. 5.1.
Deviations from King Linearity are known in some systems, and hint at the
complexity of atomic and nuclear structure calculations [105]. For example, relativis-
tic corrections to electron wavefunctions, which give rise to fine structure splitting,
also contribute to a J-dependence of the field shift. Any attempts to attribute
deviations from King Linearity to new physics beyond the standard model will
require careful theoretical study to rule out any complicated standard model con-
tributions [106]. Furthermore, any attempt to constrain new physics models using
107
isotope shift spectroscopy requires high-precision measurements establishing King
Linearity, and the assumption that no fortuitous cancellations between standard
model effects and new physics occur.
In addition to placing constraints on beyond-standard-model theories that pre-
dict deviations from King linearity, isotope shift spectroscopy is a very sensitive tool
to benchmark state-of-the-art atomic structure calculations. In particular, recent
data from photon recoil spectroscopy of the D2 line in Ca+ suggest a several-sigma
disagreement between experiment and theory [107]. Specifically, the authors of
Ref. [107] perform several different ab initio calculations of the field shift ratio on
the D1 and D2 lines, the closest of which still differs by 3σ from the experimental
uncertainty.1 By contrast, a recent experiment in francium found excellent agree-
ment with the ab initio theory [108]. Thus, there is significant utility to performing
similar analyses in another independent system to benchmark these ab initio meth-
ods. Strontium is particularly appealing because it presents several very narrow
optical transitions which allow a precise determination of the isotope shifts.
One distinct drawback, however, is the lack of a fourth stable bosonic isotope.
As discussed below, complications arising from hyperfine interactions make a clean
theoretical analysis of the fermionic isotope, 87Sr, challenging. The remaining two
isotope shifts of 84Sr and 86Sr, relative to the most abundant isotope, 88Sr, make the
King plot overconstrained — we are reduced to fitting a line between two points.
However, measurement of all isotope shifts including those involving 87Sr will provide
high-precision experimental data for theorists to compare in a proper accounting of
hyperfine effects. Despite this drawback, one can still extract values for the field
shift ratio and other interesting quantities, with uncertainties propagated from the
underlying experimental uncertainties in each datapoint.
1In private communication with M. Safronova, there is reason to think this discrepancy is even
larger (closer to 6σ), given known issues with some of the ab initio methods.
108
The rest of this chapter is laid out as follows: in Sec. 5.1, I will describe some
theoretical machinery necessary for analyzing and understanding the isotope shift
data. In Sec. 5.2, I describe details of the isotope shift measurements we undertook
on the 689nm intercombination transition and the 698nm clock transition. Results
are presented in Sec. 5.2.2, and Sec. 5.3 presents a King plot analysis. We end with
a discussion of future extensions to this work.
5.1 Isotope shifts: theory and practice
Atomic systems provide a truly exceptional testbed for theoretical physics.
Because they can be studied in near perfect isolation from external perturbations, it
is possible to make predictions that are built directly from first principles in quantum
theory. Simultaneously, there is an enormous bedrock of experimental techniques
that enable us to perform some of the most precise measurements in the history of
science on atomic systems. For example, recent results from the Yb lattice clock at
NIST report sensitivities approaching a few parts in 10−19 [109].2
This clean experimental system notwithstanding, complications in the math-
ematics of many-body theories make it difficult to do exact calculations; beyond
hydrogen, we lack analytical solutions, and numerical methods break down as the
computational complexity grows exponentially with the number of interacting par-
ticles. This leaves theorists trying to develop approximation methods which can still
capture relevant physics, and it again becomes valuable to have clean experimental
systems to test those approximations.
For our purposes here, we do not dig into the full complexities of ab initio
theoretical methods to interpret the experimental data. However, it is necessary to
2As one of my professors said at some point, the only thing we really know how to measure well
is frequency. Any precision measurement, at some level, is just a clever way of demodulating the
signal of interest in such a way that it becomes equivalent to a frequency measurement. In this Yb
lattice clock result, they are able to resolve sub-centimeter differences in the gravitational redshift
from earth’s gravitational field.
109
have a basic understanding of the underlying atomic theory. In the next few sections,
I present derivations of a few of the key results that will inform the subsequent
experimental analysis. I also point out places where further theoretical analysis
would add substantially to our understanding.
5.1.1 Mass shift
As with any problem in quantum mechanics, we begin with a Hamiltonian
describing our system. In the present case, we have the Hamiltonian of a multi-
electron atom,
P̂2 ∑[ ]p̂2 Ze2 ∑ e2
Ĥ0 = +
i −
2M 2m |r̂i|
+ | − | , (5.1)r̂ r̂
i i>j i j
where P̂ is the momentum of the nucleus with atomic number Z, M is the nuclear
mass, {p̂i, r̂i} are the momentum and position of the ith electron, m is the electron
mass, and e is the elementary charge. In the idealized case of a heavy atom,M →∞
and the first term of Eq. 5.1 is neglected. However, accounting for the non-infinite
mass of the nucleus gives rise to the mass shift electronic energy eigenstates. In
Sec. 5.1.2, we will discuss the so-called “field” (or “volume”) shift, which arises from
changes in the charge distribution of the nucleus as neutrons are added from one
isotope to the next.
Taking Eq. 5.1 and moving to the center of mass coordinate system, we can
write,
∑
P = − pi (5.2)
∑ i ∑
P2 = p2i + 2 pi · pj . (5.3)
i i>j
110
Substituting Eq. 5.3 into Eq. 5.1, we can write
∑[ ]p2 2 ∑ ∑
Ĥ = i
p
+ i
1
0 [ + + pi · pj + V (ri, rj)∑ 2M( 2mi ) ]M i>j i>j1 M +m 2 1 ∑ ∑= pi + pi · pj + V (ri, rj) , (5.4)2 Mm M
i i>j i>j
where we have dropped the hats and rewritten the potential energy as V (ri, rj). The
first term in square brackets is quickly recognized as the standard atomic Hamilto-
nian with reduced mass µ = Mm/(M +m), while the second term involves correla-
tions between electron momenta. The second term is often referred to as the mass
polarization term, and will give rise to the “specific mass shift” discussed below.
Let us now consider what happens when the mass M → M ′. We can write
the new Hamiltonian
∑[ ( ) ]1 M ′ +m 1 ∑ ∑
Ĥ = 2
2 M ′
pi + ′ pi · pj + V (ri, rj) (5.5)m M
i i>j i>j
= Ĥ0 + Ĥ1 ,
where Ĥ0 is the Hamiltonian from Eq. 5.4. We solve for Ĥ1 to find
Ĥ1 = (Ĥ − Ĥ0 )[ ]
1 − 1 1
∑ ∑
= ′ p
2
i + pi · pj . (5.6)M M 2
i i>j
At first order in perturbation theory, the energy shift associated with this change in
nuclear mass is simply
∆E(1)mass = 〈Ĥ1〉 . (5.7)
111
This energy shift has two terms. The first term,
( )
1 1 ∑ 〈p2− i 〉′ ,M M 2
i
is called the “normal mass shift”, while the second term,
( )
1 1 ∑
M ′
− 〈pi · pj〉 ,
M
i>j
is the “specific mass shift” and arises from electron-electron correlations. For sim-
plicity, one would like to neglect the specific mass shift because it is a difficult
many-body calculation to make. However, for multi-electron atoms, this shift can
be rather large and does not necessarily even act in the same direction as the nor-
mal mass shift. But, the important result here is that both con(tribution)s to (1)∆Emass
scale in the same way with changing mass, namely, with 1 = 1 ′ − 1 ; similarly,µ M M
when we take the difference in energy shifts between two energy levels to compute
the isotope shift of an atomic transition, that scales with µ as well. This will be
important later when we discuss King Linearity.
5.1.2 Field shift
The other isotopic shift that must be accounted for when calculating atomic
spectra arises from an additional term not included in Eq. 5.1. In writing down that
equation, we assumed the nucleus to be a point charge with an electrostatic energy
∼ −1V (r̂)
r̂
We know from E&M this is only technically true for an electric monopole, but it
reveals an avenue for probing nuclear structure by measuring properties of atomic
structure — deviations from the Hamiltonian in Eq. 5.1 will cause shifts which
112
Point charge
Nuclear radius r1
Nuclear radius r2
r1 r2
Distance
Figure 5.1: Schematic potential energy for a uniformly charged sphere of radius rnuc. In this simple
model, electrons with non-zero wavefunction overlap near r = 0 see a potential which deviates from
1/r. The size of the nucleus is approximately assumed to scale rnuc ∼ A1/3 with mass number A.
The field shift is proportional to the electron wave function |Ψ(0)|2 evaluated at the nucleus. See
Ref. [110].
depend on the nuclear charge distribution and on higher order electric and magnetic
multipoles of the nucleus.
Beyond accounting for higher order terms in the classical multipole expansion
of the nuclear charge, isotope shifts can also be sensitive to corrections from relativity
and quantum electrodynamics [107]. For example, neglecting these corrections would
predict field shift constants that have no J-dependence — that is, the field shift of
transitions to the same fine structure multiplet should be identical. If instead one
solves the Dirac equation for the electronic wavefunction, the solution is a bispinor
which differs in the overlap at r = 0 for fine structure levels with different J . This
will manifest as a “splitting isotope shift,” AA′ ′ ′δνSIS = δνAA − δνAAi j , between two
fine structure transitions i, j and isotopes with masses A,A′ [107, 111]. It will also
manifest as a field shift ratio which deviates from unity in the King plot, as discussed
in the next section.
113
Energy
5.1.3 King plots
Now that we have established some basic principles, we can discuss techniques
for analyzing isotope shifts.3 Perhaps the most common (and model-independent)
technique in this field is the King plot analysis. If we assume the mass and field
shifts factorize according to
′ ′ ′
δνA,Ai = δν
A,A A,A
i,mass + δνi,field , (5.8)
where δνA,A
′
i is the isotope shift for transition i between two isotopes with mass
numbers A,A′, then we can use the results from the previous discussion to rewrite
the isotope shift as
( )
′ 1 1 ′
δνA,A 2 A,Ai = Ki − + Fi δ〈rM M c〉 . (5.9)A′ A
Here, the coefficient Ki wraps up the physics of the mass shift (normal plus specific
mass shift), which are both proportional to
1 MA −MA′
= (5.10)
µA,A′ MA MA′
1
= − 1 .
MA′ MA
The inverse mass factor µA,A′ is introduced via the definition 5.10 to be consistent
with the literature [106,107,112]. The second term, F δ〈r2〉A,A′i c , represents the field
shift δνA,A
′
i,field, which is proportional to the change in the mean-square charge radius
of the nucleus, 〈 2 A,A′δ rc〉 .
3This discussion follows the well-presented discussion in Ref. [107].
114
Taking Eq. 5.9 and multiplying through by µA,A′ , we get
′ ′
µ δνA,AA,A′ i = Ki + Fi µA,A′δ〈r2〉A,Ac . (5.11)
Since the nuclear structure is independent of the electronic configuration, we can
′ ′
eliminate δ〈r2〉A,A′c if isotope shifts {δνA,A A,Ai , δνj } are measured on two different
electronic transitions {i, j}. With a little bit of algebra, we find
A,A′ − Fi Fiµ δν = K K + µ δνA,A′A,A′ i i j A,A′ j . (5.12)Fj Fj
′
Equation 5.12 shows a linear relationship between the modified isotope shifts µ A,AA,A′δνi
′
and µA,A′δνA,Aj on two different electronic transitions. This relationship is known as
the King Linearity, after the physicist who first noticed this scaling [103]. Plotting
the shifts of several (but at least three) isotopes scaled by the inverse mass factor
µA,A′ allows one to fit a line with slope Fi/Fj and intercept Ki − (Fi/Fj)Kj.
An alternative analysis is possible if we take previously measured changes in
the mean-square nuclear charge radius, δ〈r2〉A,A′c . Here, the field shift constant Fi
and mass shift constant Ki can be extracted directly on a single electronic transition
via Eq. 5.11. Precise knowledge of the normal mass shift, which is only limited by
knowledge of the atomic mass unit, then permits one to extract the much harder
to calculate specific mass shift and benchmark theoretical predictions. If data from
more transitions are available, one can do a multi-dimensional King analysis [107]
and tighten the uncertainties on parameters such as ′δ〈r2〉A,Ac .
5.1.4 Hyperfine effects
One difficulty with the King plot analysis discussed in the previous section
arises when considering isotopes with hyperfine structure. For a given fine-structure
transition, the nuclear angular momentum I combines with the electronic angular
115
momentum J to give total angular momentum F = I + J under the traditional L-S
coupling rules. For strontium, the bosonic isotopes have I = 0, which greatly simpli-
fies the electronic energy structure; however, the fermion 87Sr possesses I = 9/2. For
the two transitions considered here, this primarily complicates the 1S0 – 3P1 inter-
combination line, which splits into three hyperfine manifolds F = {11/2, 9/2, 7/2}
for the 3P1 fine structure level.
The first and most obvious complication arises in how one assigns a single fre-
quency to a transition which is split by several GHz (which is larger than the isotope
splitting), such that it can be plotted alongside the bosonic isotopes on a King plot.
Fortunately, there is a procedure for assigning a “center of gravity” to transitions
split by hyperfine interactions. Unfortunately, as we shall see, this procedure is
overconstrained, and does not account for shifts beyond the first-order perturbation
theory of the non-shifted level. Without a way to properly identify the line center,
we introduce a nonlinearity to the King plot and often studies of isotope shifts will
neglect fermionic isotopes due to such complications. Unfortunately in strontium,
there are only three stable bosonic isotopes, which means the King plot analysis
must either account for the hyperfine complications of 87Sr, or accept the limitation
that the fitted linear relationship is again overconstrained by insufficiently many
points. In the King plot analysis of our results discussed in Sec. 5.3, we shall see
that including 87Sr does indeed introduce a statistically significant nonlinearity that
requires further theoretical study. We will therefore use the over-constrained model
and propagate errors from the bosonic isotope shift measurements where noted in
calculating properties of atomic and nuclear structure. If one had access to the
radioactive 90Sr isotope, this complication would be alleviated, and deviations from
King linearity (or the lack thereof) might hint at beyond standard-model physics.
The high precision offered by narrow optical transitions makes this an intriguing
possibility for future investigation.
116
With that, we will now discuss the effects of hyperfine interactions on the
87Sr intercombination line transition.4 Given the fine-structure Hamiltonian Ĥ0,
which incorporates spin-orbit interactions but not the hyperfine interaction between
nuclear and electron spins, one can add a small interaction term ĤHFI which couples
the electronic and nuclear angular momentum and consider it in the framework of
perturbation theory. Specifically, our Hamiltonian becomes
Ĥ = Ĥ0 + ĤHFI , (5.13)
where the hyperfine interaction Hamiltonian is given by
∑
Ĥ = (−1)µ T e nHFI k,µTk,−µ . (5.14)
k,µ
Here, the operators T e nk,µ and Tk,µ are spherical tensors of rank k that act on the
electronic and nuclear subspaces, respectively.
Perturbation theory relates the shift of a particular energy eigenstate to matrix
elements of the interaction Hamiltonian ĤHFI; thus, we consider terms of the form
〈γ′IJ ′F ′m′F |ĤHFI|γIJFmF 〉
∑
(5.15)
F J I
= δ δ I+J+F ′ ′ e nF ′F m′ m (−1) 〈γ J ||T ||γJ〉〈I||T ||I〉 ,F F  k k
k k I J ′
(5.16)
where the matrix elements are expressed via the Wigner-Eckhart theorem as prod-
ucts of reduced matrix elements 〈|| · ||〉 and terms related to the angular momentum.
The state labels are the standard angular momentum quantum numbers for nuclear
4This discussion and derivation follows the discussion in Ref. [113], which is primarily concerned
with the hyperfine structure of 3P2.
117
spin (I), total electronic angular momentum (J = L + S), and the total angular
momentum (F = I + J), with γ encompassing all other quantum numbers. At first
order, the energy shift is the diagonal term 〈γIJFmf |ĤHFI|γIJFmF 〉, which can
be rewritten compactly as a sum over terms of increasing multipole order,
∆E(1) = ∑〈γIJFmf |ĤHFI|γIJFmF 〉 (5.17)
= Xk(IJF )〈T ek 〉J〈T nk 〉I (5.18)
k
with the function Xk(IJF ) wrapping all of the angular dependence
 F J IK I J
X I+J+Fk(IJF ) = (−1)    . (5.19) J k J I k I
−J 0 J −I 0 I
The matrix elements are related to the standard hyperfine coupling coefficients A,
B by
≡ 1A 〈T n1 〉I 〈T e1 〉J (5.20)IJ
B ≡ 4〈T n2 〉I 〈T e2 〉J (5.21)
For the 87Sr, 3P1 fine structure manifold, J = 1 and I = 9/2, so F ∈
{7/2, 9/2, 11/2}. In this case, the Wigner six-J symbol in the definition of Xk
is only non-zero for k ≤ 2; the spherical tensor structure coupling I = 9/2 and
J = 1 only supports up to a rank two operator. Thus, we do not have higher
order multipole coupling coefficients beyond the magnetic dipole term A and the
electric quadrupole term B. However, this framework could be extended for other
transitions with J > 1, which must include higher-rank tensor operators.
118
Practically speaking, measuring the hyperfine splittings in the 3P1 manifold
of 87Sr should allow us to calculate the energy shifts ∆E(1) from Eq. 5.17 and
back out the unperturbed energy level E(0), which would be the energy of some
hypothetical spin-zero fermionic isotope with mass number 87. However, an ob-
vious problem presents itself: in the derivation discussed thus far, we have only
treated the interaction Hamiltonian ĤHFI at first order in perturbation theory. But,
we are already discussing fitting a series of three frequency measurements for the
1S0(F = 9/2) – 3P1(F = 11/2, 9/2 , 7/2) transition to three unknowns — hyper-
fine coupling constants A and B, and the unperturbed energy level E(0). Thus, we
cannot experimentally disentangle the higher-order perturbation theory shifts from
what is available in the data, and the number we specify for E(0) is only accurate
assuming higher order shifts are negligible.
To see how reasonable this assumption is, we consider the second order per-
turbative term
∑ ∣∣∣ ∣2〈γ′IJ ′FmF |ĤHFI| ∣γIJFmF 〉∣
∆E(2) = , (5.22)
′ ′ EγJ − Eγ′J ′γ J
where the sum extends over all other fine structure levels |γ′J ′〉 6= |γJ〉. A similar
Wigner-Eckart decomposition yields an expression again proportional to reduced
matrix elements of ĤHFI between fine-structure energy levels, divided by the energy
difference EγJ − Eγ′J ′ .
The largest contribution to the shift ∆E(2) of the 3P1 state will come from
couplings to adjacent fine structure levels 3P0,2. Without calculating the reduced
matrix elements exactly, we estimate the magnitude of the second order contribution
to be smaller by roughly the first order hyperfine splitting in 3P1 divided by the fine
119
structure splitting Eγ,J=1 − Eγ′J ′=0,2. We find
( ) 1GHz · 1GHz∆E  THz ≈ 85 kHz for
3P
HFS  11.8 2∆EHFS · ∼  , (5.23)∆EFS 1GHz · 1GHz ≈ 180 kHz for 3P
5.6THz 0
which is non-negligible. A full theoretical treatment of the higher-order perturbation
theory shifts of the 87Sr fine structure levels is needed to unambiguously incorporate
that isotope alongside the bosons in the King plot analysis. Such a treatment is
beyond the scope of this thesis; we will instead rely on propagation of errors from
the measurement of isotope shifts between the three stable bosons in extracting
quantities from the King plot, but cannot say anything about deviations from the
predicted King linearity.
5.2 Precision measurement of strontium isotope shifts
We now turn our attention to the experimental measurement of isotope shifts
on the strontium “clock” (698nm) and “intercombination” (689nm) lines (see level
diagram in Fig. 1.1). We present the first reported spectroscopy of the clock tran-
sition in 86Sr and 84Sr, and the first systematic study of all isotope shifts on these
two lines. Our measurements of the intercombination line transitions are consistent
with previously reported values, in many cases with much smaller errors (see, for
example, Refs. [114–118]), and while certain experimental shortcomings discussed
below prohibited a measurement of the clock transition with state-of-the-art preci-
sion, this work paves the way for more detailed studies in the future. The results
discussed in the next few sections are in the process of being written up as a formal
publication, which will appear sometime next year.
120
5.2.1 Experimental setup
The experimental procedure is straight forward, and proceeds as follows. We
begin by laser cooling as described in Chapter 1.1. Briefly, we capture each isotope
from a Zeeman slower directly into a magneto-optical trap (the “bMOT”) using
the broad, 461nm 1S0 – 1P1 transition. Atoms are then transferred into a MOT
operating on the narrow, 689nm intercombination line (the “rMOT”), where we laser
cool to ≈ µK temperatures. At this point the rMOT is brought into spatial overlap
with the main 1064nm dipole beam via an adjustable magnetic shim field. Because
the isotope splitting is many tens to hundreds of MHz, the transfer from bMOT
to rMOT (which has a linewidth of ≈ 7.5 kHz) is isotope selective, and the atom
number can be varied by adjusting the time allowed for loading of the bMOT. We
typically load a few million to 107 atoms at temperatures ≈ 1 µK to 3 µK, depending
on the isotope.5
Measurement of the isotope shift then proceeds via direct spectroscopy, but
the details differ slightly for each line due to the technical reasons explained below.
In all cases, however, we measure the number of atoms remaining in the dipole
trap after performing spectroscopy as a function of the probe laser frequency by
on-resonant absorption imaging at 461nm, either in situ or after a short (few ms)
time-of-flight expansion to obtain sufficient signal-to-noise in the absorption image.
Atom number is extracted by numerically summing the optical density over a region
of interest [119], and subtracting a background count rate determined from a nearby
region of the image that contains no atoms.
5The atom number we can reliably load dropped precipitously towards the end of this work, as
discussed elsewhere — we speculate that we have completely coated the Zeeman slower viewport,
degrading the overall bMOT performance.
121
5.2.1.1 Spectroscopy of 689nm intercombination lines
Light for the intercombination line spectroscopy was derived from the “short
Steck” laser, which is referenced to the 689nmmaster laser via a dynamically tunable
beatnote lock (see Sec. 2.5.2 and Fig. 2.6). The light passes through a 100MHz AOM
and is fiber coupled to the experiment table. A photodiode on the opposite side of
the chamber monitors the power and servos the intensity by feeding back to the
RF drive amplitude of the AOM. The servo is based on the design from NIST [37],
and permits a digital sample and hold during the dithered spectroscopy described
below. The absolute frequencies quoted below are obtained by measuring an optical
beatnote between the 689nm master laser and a Menlo Systems frequency comb
(FC1500-250-ULN), which is referenced to a 10MHz Rubidium atomic clock (SRS,
FS725).
The spectroscopy beam is collimated at the atoms with a 1/e2 waist of ap-
proximately {1.25mm × 1.71mm}, and propagates at a 45◦ angle with respect to
the dipole trap. Power in this beam is typically 1 µW to 20 µW (I/Isat ≈ 10 at
1 µW), where the exact value depends on the transition and is chosen to maximize
the signal to noise on resonance without saturating the atom loss signal.
Bosonic intercombination line. Spectroscopy of the 689nm line in 88Sr, 86Sr, and
84Sr was completed over several days. After loading the dipole trap at high power
(≈ 3.2W in the main dipole trap beam, as described above), we perform a short
exponential ramp to the final power (≈ 1.2W, where the trap depth Udip ≈ 4.6 µK).
At this point, we perform spectroscopy in the absence of any AC stark shifts from
the dipole trap by dithering the spectroscopy laser out of phase with the dipole
laser, as in Ref. [120]. To maintain constant average power, we ramp the power
in the dipole beam to twice its nominal value in 50 µs and dither with a 50% duty
122
cycle and a 500 µs period (250 µs on followed by 250 µs in the dark, during which
the spectroscopy is performed). To ensure (near-)complete extinction of the dipole
light, we delay the spectroscopy pulse by 50 µs after the falling edge of the dipole
trap. We estimate the residual light shift due to incomplete optical extinction of the
dipole trap to be well below the other systematic uncertainties of our measurement.
The total duration is fixed by the integrated probe time of the spectroscopy pulses,
which is typically 1ms to 5ms at powers ranging from 1 µW to 2 µW depending on
the isotope. A small 500mG field is added to define a quantization axis and separate
transitions to 3P1 (m′J = ±1) such that we only address the mJ = 0 → m′J = 0
transition, which is first order field insensitive. After the spectroscopy is over, we
hold for several ms to allow lost atoms to fall out of the trap region, and measure
the remaining atom number by absorption imaging at 461nm.
Fermionic intercombination line. The procedure for spectroscopy of the 87Sr
intercombination lines is similar to that of the bosons with a few changes. Because
87Sr has ten mF levels which are roughly evenly populated, we cannot obtain suffi-
cient signal to noise if they are spectroscopically resolved by an external magnetic
field. Thus, we perform this spectroscopy at near zero magnetic field. We adjust
the probe power and duration to obtain sufficient contrast; because of the more
complex optical pumping dynamics, spectroscopy of the 87Sr F = 9/2 → F ′ = 7/2
required higher probe powers and durations. For this transition, we typically used
20 µW and up to 15ms probe time to achieve reasonable signal to noise. For the
F = 9/2→ F ′ = 9/2 and F = 9/2→ F ′ = 11/2 transitions, we typically used 5 µW
for 5ms to get similar results.
Data analysis and systematics. To extract useful information from the spectro-
scopic measurement, we must understand both its accuracy and its precision. The
123
precision is determined by the statistics of the measurement, while the accuracy
must take account of various systematic effects that shift the measured frequency
from the “true” frequency. Each scan of the resonance is time-tagged and fitted
to an inverted gaussian, which allows us to extract a line center and offset corre-
sponding to the average atom number during that scan. We simultaneously count
the frequency of the “master” 689nm laser with the frequency comb in a 1 s gated
window, and record the frequency of the beatnote lock reference (which is supplied
by a DDS). This allows us to reconstruct the absolute frequency of the spectroscopy
laser, flaser, averaged over the duration of the scan. We compute fabs, the absolute
frequency of the resonance position, as
fabs = flaser + fres (5.24)
We take many such measurements of fabs for each isotope, and tabulate them for
post-analysis.
To measure the systematic density shift (see Sec. 5.2.2.1), we keep everything
else constant but vary the atom number by varying the bMOT load time from ≈ 0.1 s
to 10 s depending on the isotope. This generates a scatterplot of {number, fabs}.
The slope of a line fit to this data gives the density shift of the resonance. We
compute this for each isotope, and apply the systematic offset based on the average
atom number across all measurements. The uncertainty of this systematic in the
final transition frequency is accounted for by the bootstrap analysis, whereby we
draw from the set of all density shift measurements (with replacement) to compute
the distribution of final transition frequencies. The density shift (and uncertainty)
quoted in Table 5.5 is generated from a fit to the complete set of experimental data,
and the uncertainty propagated from the fit uncertainty to the operating density of
the measurement.
124
5.2.1.2 Spectroscopy of 698nm clock line
Light for spectroscopy of the clock transition is derived from the laser system
described in Chapter 3. Briefly, a home-built Littrow-configuration ECDL laser is
stabilized to a passive ULE cavity via PDH lock to the first phase-modulated side-
band generated by a broadband fiber EOM. Coarse tunability of the laser frequency
is provided by the RF drive of the fiber EOM, which provides a few-hundred MHz
offset between the atomic transition and the cavity resonance. Changing this drive
frequency enables us to span the full isotope shift of the transition. Fine frequency
tuning is provided by an 80MHz AOM, which also stabilizes the probe laser inten-
sity as described above for spectroscopy of the intercombination line. A schematic
of this setup was provided in Fig. 3.13. Again, the absolute frequency reference is
determined via optical beatnote with a Menlo Systems frequency comb.
Light is fiber coupled from the laser table to the experiment, and follows the
same beam path as the spectroscopy light for 689nm. However, we have removed
lens L1 in Fig. 2.2 such that the beam is focused through the center of the chamber
with a 1/e2 waist of approximately {332 µm × 461 µm}. This is necessary for the
bosonic isotopes in particular, because the line we are probing is a doubly-forbidden
J = 0 → J ′ = 0 transition. To make it allowed under E1 selection rules, we must
apply a magnetic field to mix in some 3P1 character [121]. The effective rabi rate Ω
then scales as
ΩLΩB
Ω = , (5.25)
∆
where ΩL is the Rabi rate coupling 1S ↔ 30 P1, ΩB is the magnetic field coupling
3P0 ↔ 3P1, and ∆ is the fine structure splitting between 3P0 and 3P1. For light
polarized at an angle θ with respect to the magnetic field, the resulting effective
125
rabi rate can be written
√
Ω = α I |B| cos θ , (5.26)
( √ )
where α = 19.8 mHz/ G mW/cm2 [121]. We operate far into the broadened
regime, where the linewidth is set by Ω; given the large inhomogeneous broadening
from stark shifts in the 1064nm dipole trap, this is necessary to achieve sufficent
signal to noise on our spectroscopy. To make an order of magnitude estimate, for
a field B ≈ 100G and 1mW of probe light, the rabi rate is Ω ∼ 40Hz. Given the
trap frequencies in our dipole trap are of a similar order of magnitude (ωx,y,z ≈
2π × {50, 4, 495} Hz), this is a very atypical regime to operate in. We discuss the
effect this has on the spectroscopy lineshape in Sec. 5.2.2.2.
The dithering technique used to probe the 689nm transition in the absence of
AC stark shifts does not work for spectroscopy of the clock transition, because it
causes too much heating during the 2 s probe time required to achieve sufficient signal
to noise in atom loss. Because of this, to reduce the sensitivity to long timescale
fluctuations in dipole trap power, we take an entire isotope shift measurement in a
single day between 88Sr and the target isotope. As discussed below in Sec. 5.2.2.1
with more detail, we account for this AC stark shift systematic.
Fermionic clock line. Spectroscopy of the clock transition in 87Sr is done at zero
magnetic field. We probe with typical powers ≈ 250 µW for 2 s, which is enough to
achieve reasonable signal to noise on the atom loss spectroscopy.
Bosonic clock line. Spectroscopy of the clock line in the bosonic isotopes requires
a large magnetic field. To do this, we reconfigure the MOT quadrupole field coils
from an anti-Helmholtz to a Helmholtz configuration; this is done by flipping the
126
0.00
0.05
Sr88,86
200 Sr84
150
100
50
0
10 8 6 4 2 0
Current setpoint (arb.)
Figure 5.2: Calibration data for the high-current magnetic field. Fit is to a quadratic model, where
the magnitude of the magnetic field is measured via spectroscopy of the Zeeman lines in 3P1 of
88Sr. Vertical green lines show the operation point for 88Sr, 86Sr (solid), and 84Sr (dash dotted).
The fit values with statistical uncertainties at the operating points are 109.57(2)G and 197.87(5)G,
respectively. A conservative estimate of the systematic error is . 100mG, which encapsulates a
small deviation from the model at high currents, as evidenced by the residuals plotted above. The
Landé g-factor for this transition is 2.101MHzG−1.
polarity of the lower coil with respect to the upper. Control of this is provided by an
interlocked H-bridge built out of high-current Contactor relays (LEV200A4NAF).
The fully debounced switching time is . 100ms, and is done during the hold after
loading of the dipole trap. The field is then ramped back up to 109.6(2)G for
88Sr and 86Sr, and 197.9(5)G for 84Sr. The field strength is calibrated by Zeeman
spectroscopy of the 3P1 (m′J = ±1) levels in 88Sr, using the known Landé g-factor
for this level (2.101MHzG−1).6 See Fig. 5.2 for more details, and Sec. 2.3 for a more
technical discussion of the coil electronics.
6Interestingly, at high enough fields, we start to spectroscopically resolve the magnetic field
gradient across our cloud. It is a fractionally small effect for spectroscopy of the 698 nm line, but
on the more sensitive Zeeman lines of 3P1 we can selectively blow away portions of our trapped
cloud.
127
Magnetic field (G) residuals
Data analysis and systematics. Each scan of the resonance is time-tagged and
fitted to an inverted gaussian, which allows us to extract a line center and offset
corresponding to the average atom number during that scan. We simultaneously
count the frequency of the 698nm clock laser with the frequency comb in a 1 s gated
window. As with the intercombination line, we calculate an absolute frequency,
fabs, for each scan of the transition by averaging the laser frequency over the scan
duration and adding the fitted line center. This is repeated for each isotope, with
a reference taken in 88Sr during the same day. The isotope shift is computed by
subtracting the measured frequency in each isotope from the reference frequency
measured in 88Sr.
To measure the systematic shifts (we explicitly measure density, AC stark,
probe power, and probe duration, see Sec. 5.2.2.1), we keep everything fixed but
vary the experimental parameter under consideration between a “low” and “high”
value. A linear fit to this data allows us to compute an offset, applied to the
average value of each parameter during the spectroscopy measurement. For the
recoil shift, we compute the systematic directly from the measured frequency and
known atomic mass; similarly for the second order Zeeman shift, we compute it
from the measured magnetic field strength and the known second-order constant
β = −23.8(3)MHzT−2 [122].
Unlike for the density shift measurement in 689nm, data presented for the
clock transition was obtained by interleaving scans with the spectroscopy laser on
and off. Points taken with the laser shuttered are used to normalize the atom number
during that scan, and apply a density shift correction based on an independently
measured density shift of the resonance position.
128
Sr88 Sr87
1.1
1.0 1.0
0.9
0.8
0.8
0.7 0.6
0.6
Integral fit 0.4
0.5 Gaussian fit
Raw data
0.4 0.2
100 75 50 25 0 25 50 75 100 100 75 50 25 0 25 50 75 100
Detuning (kHz) Detuning (kHz)
Sr84 Sr86
1.0 1.1
0.9 1.0
0.8 0.9
0.7
0.8
0.6
0.7
0.5
0.6
80 60 40 20 0 20 40 60 80 75 50 25 0 25 50 75
Detuning (kHz) Detuning (kHz)
Figure 5.3: Detailed lineshape curves for the 698 nm clock transition, with fits to a Gaussian model
and to the full lineshape (see Sec. 5.2.2.2). Given the atom number shot noise in our data, we
cannot resolve statistically significant differences between the two models. In both cases, the fit
error on the centroid is roughly 1 kHz, however the full lineshape model fits ω0 to be ≈10 kHz to
20 kHz red of the Gaussian line center. This is attributable to the thermal distribution of atoms
in a dipole trap operated far from the magic wavelength.
5.2.2 Results
A summary of the isotope shift measurement can be seen in Table 5.1, and the
absolute frequencies extracted from our spectroscopy are shown in Tables 5.2 and 5.3.
We also report the hyperfine A and B coefficients for 3P1 in Table 5.4. A breakdown
of the systematics for the intercombination and clock-line transitions can be found in
Tables 5.5 and 5.6, respectively. These will be discussed in more detail in Sec. 5.2.2.1.
We also show a detailed scan of each resonance for the clock transition in Fig. 5.3,
including the two newly measured lines in 86Sr and 84Sr. For a discussion of the fits
displayed in this figure, we defer to Sec. 5.2.2.2.
To generate confidence estimates in each parameter reported in the tables
129
Normalized Atom Number Normalized Atom Number
Normalized Atom Number Normalized Atom Number
below, we perform a bootstrap analysis. This is done by aggregating all data for
each measurement (including systematics) and resampling with replacement to cre-
ate an empirical bootstrap distribution. This sampled dataset is used to compute
the final parameter (e.g., fabs, the isotope shift, etc.). We repeat this 1000 times
to create a histogram of possible values for that parameter, which are shown in
Figures 5.4, 5.5 and 5.6. The 95% confidence interval (CI) is taken from this distri-
bution, and the mean is reported (along with the 95% confidence bounds) in each
table. It is worth noting that the 95% CI is a wider bound than the typical 1σ error
bar, and most of the uncertainty comes from incorporating the systematic shifts.
130
Sr88 Bootstrap distribution Sr86 Bootstrap distribution
(f0 = 434829121311.7 kHz) (f0 = 434828957492.5 kHz)
Lower  95% CI (-2.9 kHz) 0.16 Lower  95% CI (-10.8 kHz)
0.25 Middle 95% CI (-0.3 kHz) Middle 95% CI (-0.1 kHz)
Upper  95% CI (5.0 kHz) 0.14 Upper  95% CI (9.1 kHz)
0.20 0.12
0.10
0.15
0.08
0.10 0.06
0.04
0.05
0.02
0.00 0.00
4 2 0 2 4 6 8 40 30 20 10 0 10 20
Frequency (kHz) Frequency (kHz)
Sr84 Bootstrap distribution Sr87 (7/2) Bootstrap distribution
(f0 = 434828769816.4 kHz) (f0 = 434830473244.8 kHz)
0.035
Lower  95% CI (-8.1 kHz) Lower  95% CI (-24.3 kHz)
Middle 95% CI (-0.8 kHz) 0.030 Middle 95% CI (0.1 kHz)
0.08 Upper  95% CI (10.5 kHz) Upper  95% CI (26.4 kHz)
0.025
0.06
0.020
0.04 0.015
0.010
0.02
0.005
0.00 0.000
10 5 0 5 10 15 20 25 40 20 0 20 40
Frequency (kHz) Frequency (kHz)
Sr87 (9/2) Bootstrap distribution Sr87 (11/2) Bootstrap distribution
(f0 = 434829342988.3 kHz) (f0 = 434827879825.6 kHz)
Lower  95% CI (-35.1 kHz) Lower  95% CI (-30.4 kHz)
0.025 Middle 95% CI (-1.0 kHz) 0.030 Middle 95% CI (0.1 kHz)
Upper  95% CI (38.1 kHz) Upper  95% CI (30.0 kHz)
0.025
0.020
0.020
0.015
0.015
0.010
0.010
0.005 0.005
0.000 0.000
60 40 20 0 20 40 60 80 60 40 20 0 20 40
Frequency (kHz) Frequency (kHz)
Figure 5.4: Bootstrap distributions for the (absolute frequency) 689 nm intercombination line
transitions, including systematics. The 95% confidence interval (CI) is marked in each subfigure,
see text for more details.
131
Prob. Density Prob. Density Prob. Density
Prob. Density Prob. Density Prob. Density
87 isotope shift 86 isotope shift 84 isotope shift
(62186.8 kHz) (f0 = 163819.2 kHz) (f0 = 351495.3 kHz)
0.06 Lower  95% CI (-18.6 kHz) Lower  95% CI (-10.1 kHz) Lower  95% CI (-11.2 kHz)
Middle 95% CI (0.1 kHz) 0.12 Middle 95% CI (-0.1 kHz) Middle 95% CI (0.6 kHz)
0.05 Upper  95% CI (18.7 kHz) Upper  95% CI (12.5 kHz) 0.08 Upper  95% CI (9.5 kHz)
0.10
0.04
0.08 0.06
0.03
0.06 0.04
0.02 0.04
0.02
0.01 0.02
0.00 0.00 0.00
30 20 10 0 10 20 30 20 10 0 10 20 30 40 30 20 10 0 10
Frequency (kHz) Frequency (kHz) Frequency (kHz)
87 (7/2) isotope shift 87 (9/2) isotope shift 87 (11/2) isotope shift
(f0 = -1351933.1 kHz) (f0 = -221676.5 kHz) (f0 = 1241486.1 kHz)
0.035 Lower  95% CI (-26.2 kHz) 0.025 Lower  95% CI (-38.6 kHz) Lower  95% CI (-30.0 kHz)Middle 95% CI (0.3 kHz) Middle 95% CI (1.0 kHz) 0.025 Middle 95% CI (-0.3 kHz)
0.030 Upper  95% CI (23.8 kHz) Upper  95% CI (35.5 kHz) Upper  95% CI (30.7 kHz)0.020
0.025 0.020
0.015
0.020 0.015
0.015 0.010 0.010
0.010
0.005
0.005 0.005
0.000 0.000 0.000
40 20 0 20 40 80 60 40 20 0 20 40 60 40 20 0 20 40 60
Frequency (kHz) Frequency (kHz) Frequency (kHz)
87 Hyperfine A 87 Hyperfine B
(260085.0 kHz) (35667.4 kHz)
Lower  95% CI (-4.4 kHz) Lower  95% CI (-40.3 kHz)
0.20 Middle 95% CI (0.0 kHz) 0.020 Middle 95% CI (-0.9 kHz)
Upper  95% CI (3.7 kHz) Upper  95% CI (43.9 kHz)
0.15 0.015
0.10 0.010
0.05 0.005
0.00 0.000
7.5 5.0 2.5 0.0 2.5 5.0 7.5 10.0 60 40 20 0 20 40 60 80
Frequency (kHz) Frequency (kHz)
Figure 5.5: Bootstrap distributions for isotope shifts on the 689 nm intercombination line, including
systematics. The 95% confidence interval (CI) is marked in each subfigure. For 87Sr, the isotope
shift is taken relative to a hypothetical I = 0 isotope, and the bootstrap distributions for the
hyperfine A and B coefficients are shown in the lower panels, see text for more details.
132
Prob. Density Prob. Density Prob. Density
Prob. Density Prob. Density
Prob. Density
Prob. Density Prob. Density
Sr88 Sr87
(429228066386 kHz) (429228004212 kHz)
Lower  95% CI (-13 kHz) Lower  95% CI (-10 kHz)
0.06 Middle 95% CI (0 kHz) 0.07 Middle 95% CI (-0 kHz)
Upper  95% CI (13 kHz)
0.05 0.06
Upper  95% CI (11 kHz)
0.05
0.04
0.04
0.03
0.03
0.02
0.02
0.01 0.01
0.00 0.00
20 10 0 10 20 20 15 10 5 0 5 10 15
Frequency (kHz) Frequency (kHz)
Sr86 Sr84
(429227903444 kHz) (429227716731 kHz)
0.08 Lower  95% CI (-11 kHz) Lower  95% CI (-11 kHz)
0.07 Middle 95% CI (0 kHz)
0.07 Middle 95% CI (0 kHz)
Upper  95% CI (11 kHz) 0.06 Upper  95% CI (11 kHz)0.06
0.05 0.05
0.04 0.04
0.03 0.03
0.02 0.02
0.01 0.01
0.00 0.00
20 15 10 5 0 5 10 15 20 15 10 5 0 5 10 15 20
Frequency (kHz) Frequency (kHz)
Sr87 Isotope Shift Sr86 Isotope Shift Sr84 Isotope Shift
(62177 kHz) (162938 kHz) (349656 kHz)
0.06 Lower  95% CI (-14 kHz) Lower  95% CI (-17 kHz) 0.05 Lower  95% CI (-16 kHz)
Middle 95% CI (-0 kHz) 0.05 Middle 95% CI (0 kHz) Middle 95% CI (-0 kHz)
0.05 Upper  95% CI (14 kHz) Upper  95% CI (16 kHz) Upper  95% CI (17 kHz)0.04
0.04
0.04
0.03 0.03
0.03
0.02 0.020.02
0.01 0.01 0.01
0.00 0.00 0.00
30 20 10 0 10 20 20 10 0 10 20 30 20 10 0 10 20
Frequency (kHz) Frequency (kHz) Frequency (kHz)
Figure 5.6: Bootstrap distributions for isotope shifts on the 698 nm intercombination line, including
systematics. The 95% confidence interval (CI) is marked in each subfigure. Isotope shifts in the
lower panel are taken relative to 88Sr. See text for more details.
133
Prob. Density Prob. Density Prob. Density
Prob. Density
Prob. Density Prob. Density
Prob. Density
Isotope pair (kHz) 1S0 → 3P1 95% CI 1S 30 → P0 95% CI
88 – 87 62 187 [−19, 19] 62 177 [−14, 14 ]
88 – 86 163 819 [−10, 13] 162 938 [−17, 16 ]
88 – 84 351 495 [−11, 10] 349 656 [−16, 17 ]
Table 5.1: Summary of isotope shift measurements on each transition relative to 88Sr, with un-
certainties reported as a 95% confidence interval (CI) generated from the bootstrap distribution
shown in Figures 5.5 and 5.6. The isotope shift for the 87Sr intercombination line uses the weighted
line center from the three hyperfine manifolds in 3P1. See text for more details.
Isotope 1S0 → 3P0 frequency (kHz)
(this work) 95% CI (previous)
88 429 228 066 386 [−13 , 13 ] 429 228 066 418.0072(2) [a]
87 429 228 004 212 [−10 , 11 ] 429 228 004 229.8732(2) [a]
86 429 227 903 444 [−11 , 11 ]
84 429 227 716 731 [−11 , 11 ]
aRef. [122]
Table 5.2: Measured absolute frequencies for 1S → 30 P0. Uncertainties in this
work are reported as a 95% confidence interval (CI) generated from the bootstrap
distribution shown in Fig. 5.6. The discrepancy in absolute frequency for 87Sr and
88Sr points to an uncontrolled systematic, likely due to a convolution of thermal
line pulling with the measurement of the AC stark shift.
134
Isotope 1S → 30 P1 frequency (kHz)
(this work) 95% CI (previous)
88 434 829 121 311.7 [ −2.9, 5.0] 434 829 121 312.30(4) [a]
7/2 434 830 473 244.8 [−24.3, 26.4] 434 830 473 270(55) [b]
87 9/2 434 829 342 988.3 [−35.1, 38.1] 434 829 343 010(50) [b]
11/2 434 827 879 825.6 [−30.4, 30.0] 434 827 879 860(55) [b]
86 434 828 957 492.5 [−10.8, 9.1] 434 828 957 494(10) [c]
84 434 828 769 816.4 [ −8.1, 10.5] 434 828 769 718(111) [d]
aRef. [116] bRef. [115] cRef. [114] dRef. [117]
Table 5.3: Measured absolute frequencies for 1S0 → 3P1. Uncertainties in this
work are reported as a 95% confidence interval (CI) generated from a bootstrap
distribution, while for previous works the uncertainty corresponds to a 1σ error
bar. Given normally distributed errors, the 1σ error bar is a factor of two smaller
than the corresponding 95% confidence bound.
Hyperfine Parameter (kHz) (this work) 95% CI (previous)
A 260 085 [ −4, 4] 260 084(2)
B 35 667 [−40, 44] 35 658(6)
Table 5.4: Measured hyperfine A and B coefficients for 3P1. Uncertainties in this work are reported
as a 95% confidence interval (CI) generated from the bootstrap distribution shown in Fig. 5.5, while
the previous result (Ref. [123]) reports the uncertainty as a 1σ error bar.
135
5.2.2.1 Systematics
When performing a (quasi–) precision measurement, a full accounting of sta-
tistical and systematic errors is necessary to understand how well you know the
quantity you have measured. Statistical errors are easy, in the sense that they are
unambiguously calculated from the data.7 Systematic errors, on the other hand,
can be notoriously hard to calculate or estimate, and tabulating an exhaustive list
relies on understanding the physics of the measurement procedure.
We calculated or directly investigated the effect of several systematic shifts
to the measurements presented above. Because we are primarily interested in the
isotope shift, many systematics which have a large effect on the absolute transi-
tion frequency become (nearly) common-mode in the isotope shift measurement.
However, we will consider each in turn for a full accounting.
Recoil and Doppler shifts. The first and most straight forward systematic shift is
the photon recoil shift, which can be treated within the same theoretical framework
as the Doppler shift. Consider an atom in the electronic ground state |g〉 with some
initial momentum, |pi〉. Unless one is working in the Lamb-Dicke regime, where
the atomic motion is quantized and the photon recoil energy is much less than the
gap to the next excited motional energy level, the transition we are driving can be
written (in the lab frame) as
|g,pi〉+ ~ω → |e,pf〉 , (5.27)
where the atom has absorbed a photon with energy ~ω, and the electronic energy
level difference ∆E = Ee − Eg = ~ω0.
7This is perhaps an oversimplification, because there are many ways one can calculate and
report statistical errors, too! See discussion in the previous section, but as always, there is some
amount of judgement involved.
136
By conservation of energy and momentum, we require Ei = Ef , where
p2
E ii = + ~ω (5.28)
2m
p2f
Ef = ~ω0 + (5.29)
2m
and similarly,
pi + ~k = pf (5.30)
where ~k is the photon momentum. Substituting pf into the expressions above, we
arrive after a few lines of algebra at
~2k2
~(ω − ω0) = ~k · v + (5.31)
2m
where now v = p/m is the atomic velocity. The energy shift from the “true”
resonance, ~ω0, has two contributions as shown in Eq. 5.31. The first term, ~k ·v, is
the standard first-order Doppler shift, while the second term, ~2k2/2m is the recoil
shift.
In principle, one might consider higher order Doppler shift corrections, which
arise from taking into account relativistic expressions for the energy and momentum.
However, these corrections will arise in a Taylor expansion in the (small) quantity
 ≡ v/c, which for our µK-temperature samples, is ∼ 10−11. If we re-write the
Doppler term ~k · v in terms of the photon energy ~ω, we get
EDoppler = ~ω (5.32)
≈ 20 kHz× h . (5.33)
An additional factor of , to give the second order Doppler shift, will enter at 10−8 Hz,
137
which is far below any of our statistical uncertainties and anything we can detect.
Thus, they are neglected in the calculations here.
Density shift. The next systematic we consider is an interaction shift of the res-
onance, which scales with atomic density and is a consequence of the scattering
dynamics parametrized by the (temperature-dependent) scattering length. We do
not have an accurate measure of in situ density, but we do have a good measure of
atom number. At fixed trap frequencies, this becomes proportional to the atomic
density. We empirically measure a shift in the resonance for each isotope and tran-
sition while varying atom number by a factor of ≈ 2 to 4. The computed correction
is applied to the absolute frequency measure before taking the difference to compute
the isotope shift.
AC Stark shift. The AC stark shift arises from off-resonant coupling between our
levels of interest and other atomic energy levels. We do not tabulate the probe-
induced stark shifts because the shift is well below our other systematic uncertain-
ties, but for the clock line interrogation we must consider an AC stark shift arising
from the non-magic wavelength 1064nm dipole trap. We probe this shift empiri-
cally, measuring a change in the resonance frequency for each isotope as the trap
power is varied. A rough estimate of the differential AC stark shift, given the dipole
beam waists from Table 2.1, is 70.5 kHz which is very close to the measured shift in
Table 5.6.
2nd order Zeeman shift. Because we probe a mJ = 0 → m′J = 0 transition
(with the exception of 87Sr, which is done at zero magnetic field), there is no linear
Zeeman shift to contend with. However, the second order Zeeman shift (proportional
to the square of the magnetic field B), will in principle give rise to a systematic shift.
138
Because we probe at low fields for the 689nm spectroscopy, the second order Zeeman
shift is below our resolution for that transition. But the high magnetic fields required
to induce the clock transition in the bosonic isotopes requires proper accounting.
Here, we compute the systematic shift from the known magnetic field, calibrated
as discussed above in Fig. 5.2, and the known quadratic susceptibility to magnetic
field from the strontium clock literature. Specifically, there is a shift
∆ν = β |B|2 , (5.34)
where β = −23.8(3)MHzT−2 [122].
Probe power and duration shifts. We do not tabulate these for the intercom-
bination line transition because we probe for short times and low powers, and any
systematic shift is well below the resolution of our spectroscopy. However, for the
clock transition, we measure these explicitly again by varying both the power and
duration while holding every other parameter constant. These are tabulated in
Table 5.6.
139
140
88 87 86 84
Systematic Shift (kHz) F ′ = 7/2 F ′ = 9/2 F ′ = 11/2
Density 1.7± 1.7 −34.1± 14.5 −51.9± 26.8 −43.3± 15.6 5.1± 3.4 −1.4± 4.3
Recoil 4.8± (< 0.1) 4.8± (< 0.1) 4.8± (< 0.1) 4.8± (< 0.1) 4.9± (< 0.1) 5.0± (< 0.1)
Total 6.5± 1.7 −29.3± 14.5 −47.1± 26.8 −38.5± 15.6 10.0± 3.4 3.6± 4.3
Table 5.5: Summary of systematics for the bosonic 689 nm intercombination line. Here we report uncertainties as the uncertainty in the corresponding
systematic fit parameter propagated to the operating point during spectroscopy. This allows us to see which systematic errors are likely to dominate
the final bootstrapped confidence interval. Statistical uncertainties are all below 1 kHz and do not contribute substantially to the total error budget.
88-87 88-86 88-84
Systematic Shift (kHz) 88 87 88 86 88 84
Density 0.8± 1.6 −3.8± 1.2 0.2± 0.3 −0.9± 0.8 0.4± 0.9 −2.3± 0.9
Recoil 4.7± (< 0.1) 4.7± (< 0.1) 4.7± (< 0.1) 4.8± (< 0.1) 4.7± (< 0.1) 4.9± (< 0.1)
AC Stark 73.7± 2.2 71.2± 3.4 76.3± 2.2 73.9± 2.1 71.9± 2.1 73.0± 2.1
2nd Order Zeeman −2.8± (< 0.1) 0.0± (< 0.1) −2.8± (< 0.1) −2.8± (< 0.1) −2.8± (< 0.1) −9.1± (< 0.1)
Probe Power 3.5± 1.6 1.3± 0.3 3.5± 1.6 3.5± 1.6 3.6± 1.6 3.6± 1.6
Probe Duration 3.4± 3.3 3.3± 1.3 3.4± 3.3 3.4± 3.3 3.4± 3.3 3.4± 3.3
Thermal line pulling −22± 4 −17± 4 −22± 4 −8± 4 −22± 4 −21± 4
Total 61.3± 6.1 59.7± 5.6 63.2± 5.9 73.9± 5.9 59.2± 5.9 52.4± 5.9
Table 5.6: Systematic frequency shifts and uncertainties for the 1S 3 880 → P0 transition. The three columns for Sr correspond to the three independent
isotope shift measurements. Here we report uncertainties as the uncertainty in the corresponding systematic fit parameter propagated to the operating
point during spectroscopy. This allows us to see which systematic errors are likely to dominate the final bootstrapped confidence interval. The thermal
line pulling, discussed in Sec. 5.2.2.2, is not included in the bootstrapping procedure but added as a separate systematic to calculate the final frequencies
and isotope shifts. Statistical uncertainties on the absolute frequency are . 1 kHz for all transitions and do not contribute substantially to the total
error budget.
5.2.2.2 Detailed analysis of the clock transition lineshape
As discussed previously in Sec. 3.4, because our dipole trap was not tun-
able to the magic wavelength for either transition (λmagic,698 = 813 nm, λmagic,689 =
914 nm [59]), we were unable to perform recoil-free spectroscopy in a deep optical
lattice, as is done for true precision clock experiments. Given this limitation, it
is important to understand the inhomogeneous broadening mechanisms that might
affect our spectroscopic signal.
We begin with a simplified, semi-classical treatment of atom loss from the
trap, following the line of argumentation from Ref. [124]. Since our signal is a laser-
induced one-body loss after some probe time tp, we start by calculating a loss rate
coefficient K(δω, I, T, Utrap). The observable, namely, atom number, is governed by
the differential equation
dN
= −K(δω, I, T, Utrap)N . (5.35)
dt
The loss rate coefficient is a function of the laser intensity I and detuning δω =
ωlaser − ω0 (relative to the bare atomic resonance ω0), and and is assumed to be
proportional to an ensemble average of the scattering rate over all atoms in the
trap. It carries a dependence on theromdynamic quantities such as the temperature
T , and is parametrized by the confining potential Utrap.
From Eq. 1.1, the scattering rate is
( )
Γ s0
Γscat = . (5.36)
2 1 + s0 + (2∆/Γ)
2
We rearrange this expression, pulling out constant terms to write
Γscat ∝ ( 1′ )2 (5.37)Γ + ∆2
2
141
√
where Γ′ = Γ 1 + s0 is the saturation-broadened linewidth. For an atom in the
trap with a given phase space coordinate (r,p), the effective detuning ∆ can be
written
− p · k∆ = δω −∆Utrap(r) . (5.38)
m
The term δω − p · k/m is the Doppler-shifted laser frequency, and ∆Utrap(r) is the
differential stark shift which arises from operating the dipole trap away from the
magic wavelength. Specifically, we can write
1 ( )
∆Utrap(r) = m ω̄
2
e −(ω̄2 2g r ) (5.39)2
1 2
= mω̄2
ω̄
gr
2 e − 1 , (5.40)
2 ω̄2g
where ωg(ωe) is the ground (excited) state trap frequency, and ω̄ is the geometric
√
mean of the trap frequencies in all three dimensions. Since ωi ∝ αi, where αi is
the AC polarizability of state i ∈ {g, e}, we find
( )
αe
∆Utrap(r) = Utrap(r) − 1 . (5.41)
αg
At 1064 nm, assuming g is the ground 1S0 state and e is the clock state 3P0, we can
compute αe/αg ≈ 0.7.8 This has the effect of re-scaling the trap potential, such that
∆Utrap(r) = αUtrap(r) (5.42)
8We calculate the polarizability by taking a sum-over-states [125], with dipole matrix elements
computed predominantly from values in the NIST database. Where better line strength data is
available, we have substituted those instead. Finally, we add a small ad-hoc offset to make the
static polarizability match the value from Ref. [15], which is calculated with much more complicated
theoretical machinery. We justify this by noting it makes the predicted magic wavelengths match
much more closely to their empirical values, and can be understood as compensating for incorrect
or incomplete transition dipole matrix data.
142
with α = (αe/αg − 1) ≈ −0.295. In the limit of operating at the magic wavelength,
αe = αg such that α = 0 and the spatial dependence drops out of the equation.
We now turn our attention to solving for the loss rate coefficient K, by taking
an ensemble average over the scattering rate expressed in Eq. 5.37. Because we are
interested in deriving a shape function which can be fit to real atom loss data, we
ignore normalizations and overall constants which will be brought into a final fit
parameter. The ensemble average of Eq. 5.37 is proportional to
∫ ∫ [ ]
d3r e−Utrap(r)/kBT 3
2 1
d p e−p /2mkBT ( ,
Γ′
)2 ( )
+ δω − p·k − 2αU
2 m trap(r)
(5.43)
where we have taken an integral over phase space {r,p} weighted by the appropriate
Boltzmann factor. Here, kB is the Boltzmann constant, T is the temperature, and
m is the atomic mass.
We wish to make this dimensionless to easily work in a numerical fitting routine
with experimental data. Focusing on the integral d3p = dpxdpydpz first, we can
choose p̂z to point along k. Thus, p · k = pzk, and the Boltzmann factor must be
rewritten
e−p
2/2mkBT = e−(p
2 2
x+py)/2mkBT e−p
2
z/2mkBT (5.44)
The integral over px and py now factors, and can be brought into an overall scale
√ √
factor. We define the dimensionless v√ariable y ≡ pz/ 2mkBT , dpz = 2mkBTdy,
where now pzk/m = βy, with β ≡ k 2kBT/m. In convenient units, for 88Sr and
√
2π/k = 698 nm, β/2π = 19.7 kHz T with T measured in µK. This parameteriza-
tion of y serves to scale the momentum pz to the most probable momentum at a
given temperature.
143
Putting it all together, the integral from Eq. 5.43 becomes
∫ ∫ [ ]
d3r e−Utrap(r)/kBT dy e−y
2 ( 1′ )2 (5.45)Γ + (δω − βy − αUtrap(r))22
where we’ve neglected overall constants. Turning back to the integral over r,
we know Utrap(r) = f(r2). Thus, we can pull the angular integral from d3r ≡
r2 sin θdr dθ dφ into an overall constant, leaving just the integral in r given by
∫ ∫ [ ]
r2dr e−r
2mω̄2/2kBT dy e−y
2 ( 1′ )2 ( )2 , (5.46)Γ + δω − βy − α 1mω̄2r2
2 2
where we have also replaced Utrap(r) with its exp√licit form 1mω̄2r2.2
Defining the dimensionless variable x ≡ r mω̄2/2kBT = r (ω̄k/β) (which
scales r by the ratio of the trap potential energy to the thermal energy kBT ), we
can rewrite the integral as
∫ ∫ [
2 2
dx x2e−x dy e−y ( ′ )2 ( ( ) )
]
1
2 . (5.47)Γ + δω − βy − αm β2x2
2 2k2
The term αm/2k2 can be evaluated exactly now (note, there is a factor of ~ to
convert energy to angular frequency units, which I’ve dropped previously by setting
it to unity). Again, for m = 88 amu and k = 2π/698 nm, we find
αm ≈ −2.52× 10−6 s . (5.48)
2k2
What is left is to put the angular frequencies δω, Γ′ into units commensurate with β,
such that the numerical integration is more friendly. Recall β/2π ∼ 104 at T = 1 µK,
and our empirical linewidth is of a similar order. So let us express frequency in units
144
of 10 kHz, such that
ω̃ ≡ 2π × 10 kHz (5.49)
δω = ω̃ δ̃ (5.50)
Γ′ = ω̃ Γ̃ (5.51)
( β) = ω̃ β̃ (5.52)αm
ω̃ ≡ α̃ ≈ −0.158 . (5.53)
2k2
We re-write the integral finally as
∫ ∫  ∞ ∞
K ∝ dx x2e−x2 2 dy e−y ( ) ( 1 ) 2 2 , (5.54)
0 −∞ Γ̃ + δ̃ − β̃y − α̃β̃2x2
2
having pulled 1/ω̃2 out into the overall normalization constant.
Returning to Eq. 5.35, we use our expression Eq. 5.54 for K to solve for atom
number as a function of the renormalized detuning δ̃ and intensity Γ̃. We get a
simple exponential,
N(τ) ∝ e−aK(δ̃,Γ̃,β̃)τ , (5.55)
( )
which can be used as an integral fit function to vary the parameters {a, δ̃0 ≡ ω̃laser − δ̃ , Γ̃, β̃}
in a least-squares minimization routine, where τ is wrapped in to the overall nor-
malization of K, given by the fit parameter a in Eq. 5.55.
We perform a fit using Eq. 5.55 to the loss spectra shown in Fig. 5.3. The
large atom number shot noise makes it difficult to differentiate the quality of the
fit between the full integral lineshape and a simple Gaussian model, but there is
significant thermal line pulling from a full accounting of the lineshape. To account
for this systematic, we numerically simulate the systematic Gaussian fit offset as a
145
1.0
10
0.8
15
0.6
20
0.4 25
0.2 30
100 50 0 50 100 1 2 3 4
Detuning (kHz) Temperature (uK)
Figure 5.7: Effects of thermal line pulling on the clock transition. We simulate the lineshape model
discussed in the text as a function of temperature (left panel), and extract a systematic offset to
the Gaussian fitted center (right panel). This systematic shift is −7.6(3) kHz µK−1.
function of temperature, and find it to be −7.6(3) kHzµK−1, as shown in Fig. 5.7.
Typical temperatures for each isotope are {2.9 µK, 2.2 µK, 1.1 µK, 2.7 µK} for
{88Sr, 87Sr, 86Sr, 84Sr}, with uncertainties ≈ ±500 nK. Given this, we add an addi-
tional systematic to Table 5.6. This is used to seed a parametric bootstrap distri-
bution of the systematic shift due to thermal line pulling, which is included in the
final bootstrap distributions discussed above.
5.3 King plot analysis
Armed with the results from Sec. 5.2.2, we conduct a King plot analysis to
extract properties of strontium nuclear structure. As we can see from Fig. 5.8,
including 87Sr introduces a strong non-linearity to the King plot. The likely origin
of this was discussed previously in Sec. 5.1.4 — from the experimental data, it is
ambiguous what the proper “spin-zero” 87Sr intercombination line center should be.
If we add a 400 kHz offset to that 3P1 isotope shift, we can bring all three isotopes
into alignment. This is roughly the order of magnitude we expected second-order
perturbative shifts to appear from couplings between the 3P levels. Clearly, however,
a more careful theoretical analysis of these shifts is necessary to include 87Sr on the
146
Normalized number
Fit offset (kHz)
0.25
0.00
0.25
650
625
600
575
550
525
500
475
475 500 525 550 575 600 625 650 0.25 0.00 0.25
Intercombination transition (GHz-amu) residuals
Figure 5.8: King plot, fit to all isotopes. Error bars, where not visible, are smaller than the
point. The line does not pass through any of the points, indicating a clear non-linearity likely due
to misidentification of the hyperfine triplet linecenter in 87Sr 3P1. A shift of roughly 400 kHz is
sufficient to bring that isotope into line with the bosonic isotope shifts. Clearly a more detailed
theoretical accounting of higher order corrections is necessary to properly include 87Sr in the King
plot.
King plot.
Instead, we proceed with our bootstrapped isotope shift distributions for 86Sr
and 84Sr, and compute the field shift ratio F698/F689 as well as the intercept term
K698 − (F698/F689) K689 from these shifts alone. This is shown in Fig. 5.9, and
tabulated in Table 5.7.
Parameter Experiment 95% CI
F698/F689 0.9979 [ 0.9952, 1.0008 ]
K698 − (F698/F689) K689 −2.0 [−3.9 ,−0.3 ]
Table 5.7: Summary of atomic and nuclear structure parameters extracted from the King plot
analysis. The field shift ratio F698/F689 and the difference K698 − (F698/F689) K689 (GHz amu)
are taken from the mean of the bootstrap distribution calculated over 86Sr and 84Sr only.
147
Clock transition (GHz-amu) residuals
300 Lower  95% CI (0.9952) 0.5 Lower  95% CI (-3.9089)
Middle 95% CI (0.9978) Middle 95% CI (-1.9852)
250 Upper  95% CI (1.0008) 0.4 Upper  95% CI (-0.3265)
200
0.3
150
0.2
100
50 0.1
0 0.0
0.994 0.996 0.998 1.000 1.002 1.004 6 5 4 3 2 1 0 1
Field shift ratio [F698/F689] Intercept [K698 - (F698/F689)*K689] (GHz-amu)
Figure 5.9: Bootstrap distributions of the field shift ratio F698/F689 (left) and the intercept term
K 86 84698 − (F698/F689) K689 (right). These are generated from the Sr and Sr isotope shifts only.
5.4 Discussion and future directions
Here, we have presented the first measurement of the 698nm clock transition
in 86Sr and 84Sr, and the first systematic measurement of strontium isotope shifts
between all isotopes on the narrow 689nm intercombination and 698nm clock tran-
sitions. While our measurement achieves a fractional uncertainty ≈ 10−11, there are
several aspects which could be markedly improved to reduce both systematic and
statistical errors.
The first and most substantial improvement would be to operate both transi-
tions in a magic-wavelength lattice. The lack of a magic-wavelength trap manifests
itself directly in the lineshape one measures for these nominally ultra-narrow optical
transitions, and in the inability to perform coherent spectroscopy with reasonable
signal-to-noise — the dominant linewidth is set by inhomogeneous broadening in-
stead of the laser or (ultimately) the natural width of the atomic transition. Truly
narrow features would allow one to integrate down noise much more quickly, and
a magic-wavelength lattice would remove much of the uncertainty due to the AC
stark shift. It would also allow us to take advantage of our narrow laser to address
the entire ensemble coherently, reducing the currently necessary long probe time,
larger magnetic fields, and higher probe intensities used in the 698nm spectroscopy.
148
Prob. Density
Prob. Density
Our lab recently purchased a Ti:Sapphire laser, which can be tuned between
813nm and 914nm (the magic wavelengths for the clock and intercombination lines,
respectively). This laser will be incorporated in the next generation strontium sys-
tem described in Chapter 6. We briefly considered bringing magic-wavelength light
from this system into the current apparatus, but due to legacy optics there is no
simple way to combine dipole beams at these wavelengths with the existing beam
paths.
149
Chapter 6: SrII and future directions
When I joined the strontium team in 2012, the system described in Chapter 2
was already under vacuum and ready to start trapping. One of the first things I did
as a graduate student was help wrap shim coils onto the chamber, and a few weeks
later we achieved our first bMOT. However, the current experiment was scheduled
to ultimately move to the new Physical Sciences Complex (PSC) building, which
was nearing the end of construction.1 The timing and details of that move have been
rather fluid, but when we had money in the budget to re-work the vacuum system
and buy new lasers, the decision was made to build up “SrII” in tandem with the
operation of the “SrI” apparatus in the Computer & Space Sciences (CSS) building.
Lessons learned building the first apparatus produced a laundry list of ideas
to improve the operation and performance of some future system. In this chapter, I
describe the design and construction of the new strontium machine. As I write this,
the vacuum system is fully assembled and holding a pressure . 1× 10−11 torr, and
is ready for the installation of lasers.
6.1 Vacuum system
The SrI machine was plagued by several issues related back to vacuum design
choices. First, the high optical-access “spherical square” vacuum chamber from Kim-
ball Physics, which formed the main science chamber, in reality ended up with very
restricted optical access. This was a result of two factors: first, the 1.33" mini-CF
1Of course, ahead of schedule and under budget!
150
viewports had no line of sight to the center of the chamber due to a miscommunica-
tion with the manufacturers of the recessed bucket windows, which are installed on
the 8" flanges on the top and bottom of the chamber. Second, the Zeeman slower
enters the chamber on one of the large 4.5" CF ports, which precluded using that
axis for more optically demanding laser beam paths. The limited optical access
necessitated some creative optical contortions. In addition, the Zeeman slower win-
dow was not heated adequately or placed on a short bellows as is done in other
experiments, and so over time we have coated that viewport with strontium and
drastically degraded the performance of our Zeeman slower. We are nearing the
point in early 2018 where we’ll either need to pull the plug and move to the new
lab, or open vacuum to replace that viewport.
For some reason, the atomic flux of our experiment has always suffered slightly
relative to other strontium groups. We speculate this is related to the aggressive
differential pumping between our oven and main chamber. The first iteration of our
oven used a complex, slightly-focusing array of microcapliaries manufactured out of
a vacuum-compatible epoxy. However, it was later discovered to not be rated to
the 650 ◦C temperatures we heat it to, and we suspect outgassing from the epoxy
was responsible for poisoning the nearest ion pump twice in roughly five years of
operation. The upgraded oven and nozzle, which was installed over winter break in
2015, is based on the design in Ref. [24] and involves a clamped array of stainless
steel hypodermic needles.2 The new design roughly doubled our steady-state atom
number, and allowed quick progress towards quantum degeneracy in 87Sr. However,
as mentioned previously, the increased atomic flux also increased the rate at which
the Zeeman slower viewport was being coated, and the atom number has drifted
down continuously since the initial oven upgrade. This has a very real impact
on productivity — load times for the bMOT have increased, which increases the
2For details of this design, see Ben Reschovsky’s thesis [20] or the discussion in Sec. 2.1.
151
0.008 20
Initial velocity dist.
0.007 (extracted T = 408C) 18
Final velocity dist. 16
0.006
14
0.005
12
0.004
10
0.003
8
0.002
6 5 cm dia.
0.001 7 cm dia.4
8 cm dia.
0.000 2
0 50 100 150 200 250 300 350 400 450 0 50 100 150 200 250
Velocity (m/s) Beam power (mW)
(a) (b)
Figure 6.1: Zeeman slower simulation data from AOSense. The figure on the left is an example
simulation of initial and final atomic velocity distributions, assuming a Zeeman beam with a 1/e2
beam diameter of 8mm and 120mW of power. On the right, we show a summary of the simulations.
For each beam diameter and power, the detuning was optimized to give the highest flux below
60ms−1. From the simulations, we see that at a 5mm diameter there is reduced spatial overlap
between the atomic beam and the Zeeman slower light, which degrades performance. There is
little difference between 7mm and 8mm beam diameters, although the optimal detuning will vary
slightly (see Table 6.1).
duty cycle for an experimental shot; we are strained for power, particularly in the
Zeeman slower beam, which necessitates daily peakup of fiber and AO alignments
and the use of the rather touchy injection-locked 461nm diode to provide light for
the Zeeman slower. And lower atom number makes each stage of optimization from
bMOT through evaporation more crucial to maintain high enough numbers to reach
quantum degeneracy.
I mention some of this history as a motivation for design choices made on SrII.
First, to avoid the headache of poor atomic flux, we chose to purchase a strontium
beam source from AOSense (see Sec. 6.1.1). These commercial atomic beam sources
have been used successfully in a few labs at JILA, and our decision to buy rather
than design our own drove much of the remaining vacuum design. The final system
described in the next section attempts to remedy many of the design flaws in the
current system.
152
Probability density
Trappable frac. of total flux (%)
6.1.1 AOSense atomic beam source
The AOSense “Beam II” is designed to be a compact, high-flux atomic beam
source ready to install on a standard cold-atom apparatus3. At its core, it is a short,
permanent-magnet Zeeman slower attached to a dual 2D MOT which provides trans-
verse beam collimation and 20◦ deflection from the Zeeman slower axis. The oven
is controlled by an in-vacuum heating element, which reduces power consumption
and eliminates the need for water cooling or careful thermal load management be-
tween the oven and the rest of the chamber. The Zeeman slower viewport has a
secondary in-vacuum hot window which is kept at a high enough temperature to
avoid strontium buildup. All other optics for beam shaping and splitting between
axes on the 2D MOT stages are also housed in-vacuum, making the entire package
easy to integrate with another system. One simply has to provide a beam of 461nm
light at the appropriate detuning and cold atoms emerge out the other end.
6.1.2 Science chamber
The science chamber itself is a modified spherical square from Kimball Physics,
made from 316 stainless steel. We had one of the 4.5 in conflat ports enlarged to
accept a 6 in minimum-length nipple, which connects the science chamber to a large
pump body. The pump body is effectively a shrunken reducer cross, which adapts
the 6 in nipple from the chamber into an 8 in flange. The pump body contains two
pumps — first, a Nextorr-1000 non-evaporable getter (NEG)/ion pump from SAES
provides the bulk of the pumping speed on hydrogen and nitrogen. We originally
planned to just use this single pump, but the NaEr group at the JQI found their base
pressure was argon-limited. Thus, we place a second 75L s−1 ion pump from Gamma
Vacuum on the other port of the pump body. The ion pump has a secondary 2.75 in
3“Just add lasers!”
153
1/e2 Diameter (mm) Power (mW) Detuning (MHz) Captured frac. (%)
20 −580 5.2
40 −585 9.3
60 −595 11.2
80 −595 12.7
5
100 −605 12.9
120 −610 12.6
150 −620 12
180 −620 11.9
20 −560 3.5
40 −575 9.9
60 −585 13.5
80 −585 14.1
7
100 −590 15.3
120 −590 17
150 −595 17.7
180 −610 17.7
20 −565 2.1
40 −565 9.4
60 −575 12.9
70 −585 14.2
80 −590 15.1
90 −595 16.1
8
120 −595 17.1
150 −605 18
180 −605 18.7
200 −605 19
220 −605 19.4
250 −615 19.5
Table 6.1: Summary of Zeeman slower simulations from AOSense, including optimal detunings.
There is little difference in performance between 7mm and 8mm beam diameters, but depending
on the power available, the detuning should be adjusted for peak performance. See Figure 6.1 for
a convenient graphical representation.
154
CF Size (in.) Thickness (cm) 5 Time Constants (days)
350 ◦C 400 ◦C 425 ◦C
1.33 1.9 48.6 33.9 24.2
2.75 1.3 21.6 15.1 10.8
4.5 1.7 40.0 27.9 19.9
6 2.0 52.6 36.7 26.2
8 2.2 67.0 46.7 33.4
10 2.5 81.3 56.7 40.5
Table 6.2: Summary of H2 outgassing rates for various flange sizes. The time constant scales
quadratically with the flange thickness. Data from [126].
port which can be used to attach a turbo-molecular pump for the initial pump-down
and bakeout. There is also a small half-nipple on the pump body which contains
an ion pump (Agilent UHV-24p) for pressure monitoring if needed. The AOSense
beam source attaches to the science chamber via one of the 1.33" ports next to the
pump body, and care was taken to avoid any obstruction of optical access to the
center of the chamber.
Before assembly, the science chamber and pump body were both baked under
vacuum at 400 ◦C for several days to reduce the outgassing rate of hydrogen from
the stainless steel bulk. The rationale for this procedure over other similar bakeout
schemes comes from the vacuum sciences division at NIST [127]. This paper found
that a high-temperature bake (& 400 ◦C) under vacuum is responsible for the bulk
of gain in H2 outgassing. They also found that a second high-temperature air bake
could further reduce outgassing by a factor of two, likely due to the formation of
an oxide layer that acts as a diffusion barrier. However, we decided not to risk that
second bake due to risk of contamination.4 The bakeout chamber itself was built
from a stock 14 in nipple, with a custom 14 in to 2.75 in reducer on one flange to
4Interestingly, they’ve also found (although perhaps anecdotally) that the benefits of a high
temperature bakeout don’t accrue until you reach above 400 ◦C, and in fact might increase the
base pressure for bakeouts between 250 ◦C and 400 ◦C. The thinking here is that an intermediate-
temperature bake is enough to bring much of the hydrogen to the surface, but not enough to
increase the desorption rate from the steel.
155
attach the turbo pump. A chicken-wire frame wrapped with heater tape and many
layers of fiber glass insulation drops over the entire assembly to provide even heating,
and several thermocouples monitor the temperatures and adjust power to the heater
tape accordingly. After the chamber was assembled, we performed a much gentler
bake at slightly above 100 ◦C to remove any water and residual contaminants from
inside the vacuum chamber.
The supports for the chamber are made from thick-walled 1.5 in OD stainless
steel pipe, which are filled with lead shot. While not as good as the dynamically-
damped posts you can buy from Thorlabs or Newport, they have the advantage of
not being nickel-plated and thus are non-magnetic. However, the damping is slightly
improved from a standard stainless steel post. We also purchased large, 3/4" custom
breadboards from BaseLab Tools, which feature cutouts at convenient locations for
optical access to the chamber. One breadboard is supported at mid-level, such that
the beam height for standard Thorlabs optomechanical components is level with
the center of the chamber, and a second breadboard is mounted above the chamber
for beam paths that must enter vertically. Each viewport on the main chamber
is attached with a custom adapter washer that allows cage components or other
mechanical assemblies to be mounted directly to the chamber. A CAD rendering of
the whole system can be seen in Fig. 6.2. Table 6.3 also lists each port with optical
access to the atoms, the corresponding distance, and the highest-NA stock lens that
can be used along that path.
6.2 Laser systems
Here, I describe the planned laser systems, beam paths, and proposed power
budgets for the SrII experiment. This should serve as a reference for future students,
based on some of our experiences in SrI, but may change with the facts on the
ground. The main cooling light at 461nm is generated from a Toptica SHG system,
156
Breadboards w/
custom cutouts
Science chamber
Lead-damped posts
NEG/ion pump
2D MOT
Zeeman window
Ion gauge
Bitter coil
Connection for
turbo pump
Pump body
Ion pump (75 L/s)
Figure 6.2: CAD rendering of new SrII apparatus. The main chamber is a modified spherical
square from Kimball Physics, with one 4.5 in CF port replaced by a 6 in minimum-length nipple
to attach to the pump body. The top and bottom ports accept custom recessed viewports from
MDC/Insulator Seal, which have a 2.69 in clear aperture. About the axis of symmetry, there are
three 4.5 in viewports, four 2.75 in viewports, fifteen 1.33 in mini-viewports, and one 6 in viewport
which attaches on the opposite side of the pump body. All viewports except for two, which are
reserved for the ultra-violet (UV) Rydberg dressing laser, have a broadband anti-reflection (AR)
coating covering visible through ∼ 1.2 µm. The pump body is a 8 in to 6 in four-way reducer cross
with a custom 2.75 in port welded on for an ion gauge (Agilent UHV-24p). We employ both a
75L s−1 ion pump (Gamma Vacuum 75S) and an integrated non-evaporable getter (NEG)/ion
pump from SAES (NexTorr-1000). The NEG provides 1000L s−1 pumping on N2, which we ex-
pect to be conductance-limited. The integrated ion pump is a 20L s−1 pump, which enhances the
pumping speed on Noble gasses and other non-reactive species. The Gamma Vacuum ion pump
has a secondary 2.75 in CF port, which we attach to an all-metal angle valve (VAT, 54132-GE02)
and use for initial pumping with a turbo-molecular pump. All vacuum parts were baked for several
weeks at & 400 ◦C in an evacuated bakeout chamber (see text) to reduce the outgassing rate of
hydrogen. A secondary in situ bake at slightly above 100 ◦C was performed while attached to the
turbo pump, and pressures . 1× 10−8 torr were achieved before cooling down and valving off the
all-metal angle valve. Final pressures are at the detection floor of our ion gauge, . 1× 10−11 torr,
although we see a high degree of correlation between pressure reading and humidity in the labora-
tory. We suspect this is due to hydroscopic plastic in the ion gauge connector, which causes small
changes in the leakage current used to indicate pressure.
157
Viewport Distance (washer face to atoms) Lens / max N.A.
4.5" 132mm f = 150mm, (2" dia.)NAmax = 0.17
f = 150mm, (1" dia.)
2.75" 134mm NAmax = 0.084
" mm f = 150mm, (0.5" dia.)1.33 135 NAmax = 0.042
" mm f = 500mm, (2" dia.)6.0 438 NAmax = 0.025
Recessed bucket f = 66mm, (2" dia.)134mm eff NAmax = 0.29
Table 6.3: Summary of viewports available on SrII science chamber, along with the working
distance to the center of the chamber and maximum clear aperture/numerical aperture (NA) using
catalog lenses. The numbers quoted for the recessed bucket window assumes a custom-designed
objective made from catalog lenses but aberration-corrected at 461 nm. Each viewport is mounted
to the vacuum chamber with a custom stainless-steel washer which adapts to 15, 30, or 60mm cage
systems, depending on the viewport. Alternatively, 4-40 threaded holes allow for easy mounting
of custom assemblies to the vacuum viewports.
and is split into several arms (see Fig. 6.3). Each arm represents a beam path to the
experiment, and is equipped with an AOM for frequency and power stabilization,
and a shutter where high optical extinction is desired. Unused light from the zeroth
order of the Zeeman slower AOM can be used to monitor on the wavemeter and
generate a beatnote to lock the SHG laser to a secondary spectroscopy laser. As
described previously, this beatnote locking scheme allows flexibility to adjust the
frequency quickly to address different isotopes within a single experimental shot. In
the SrI system, the optomechanics of this laser system were very unstable; care was
taken in the sketched layout to keep beam paths short and direct; since we do not
require large frequency sweeps from any of our AOMs, we avoid using them in a
double-passed configuration.
The 689nm laser systems sketched in Fig. 6.4 follow a similar design philoso-
phy, privileging compact and simple layouts over infinite flexibility. A single master
laser is locked to a stable Fabry-Pérot cavity via the Pound-Drever-Hall method.
158
Zeeman slower 
(-595 MHz, 90 mW at table)
Probe B
0th order
to fiber splitter
(WM + beatnote)
AOMs
AOM (+250 MHz)
(-345 MHz) 250 mW 25 mW
-250 MHz 300 mW 50 mW 25 mW
Probe A
500 mW
200 mW
75 mW 2D MOT
(-40 MHz, 25+ mW at table)
125 mW AOM
AOM (+210 MHz) Key:
(+205 MHz)
Optical shutter
AOM
3D MOT
(-45 MHz, into three-way fiber splitter, PBS
15+ mW/arm)
Figure 6.3: Planned layout for the SrII 461 nm laser system. Note, better AOM diffraction efficiency
might be achieved (especially for the Zeeman arm) if the light coming out of the laser is detuned
further and AOM frequencies are adjusted accordingly. All AOMs are 200MHz models from
IntraAction (ATM-2001A1), except for the Zeeman arm AOM which is a 350MHz model (ATM-
3501A1). Half-wave plates are omitted for clarity, but should be installed before each beam cube
to appropriately set the power splitting between each arm. Polarization-maintaining (PM) fiber
carries light from the laser table to the apparatus. Light for the 3D MOT is split in a 3-1 fiber
coupler array (Evanescent Optics) for each arm, which is retro-reflected on the apparatus. The 2D
MOT AOM can be used as an optical beam shutter for atoms emerging from the AOSense source;
shutters are required on the 3D MOT arms and on the probe arms to achieve full extinction of
461 nm light during the rMOT and dipole trapping stages of the experiment.
159
Light for this lock is intensity stabilized by noise-eating on the zeroth order of an
AOM; the rejected first order can be used as a wavemeter monitor. Remaining light
is split four ways (or more, as needed) on polarizing beam cubes, and fiber coupled
to a beatnote lock with a slave laser. All fibers and fiber splitter/combiners in the
beatnote network should be FC/APC connectorized; in the past, back-reflections
from the front facet of a fiber have destabilized the lasing mode and degraded laser
performance in strange ways. Note here we do not require any high-frequency AOMs
to offset the laser light to the cavity resonance, since light for the experiment is gen-
erated from the system of beatnote slave lasers.
Each slave laser is similarly simple – light from the laser passes an optical
isolator, then the rejected port of a polarizing beam cube is fiber coupled to the
beatnote lock. The remaining light passes an AOM, where the zeroth order is fiber
coupled for a wavemeter monitor, and the deflected order is fiber coupled to the
experiment. In the case of light for the red MOT, we forego the AOM and shutter
and instead couple light directly to a tapered amplifier (TA).5 The TA current should
be interlocked on the optical power from the slave seed laser, and amplified light
should at this point be coupled through an AOM/shutter and into a four-way fiber
splitter (Evanescent Optics) to the apparatus.
6.3 Magnetic field coils and high-NA imaging
Beyond basic laser and vacuum systems, the final necessary pieces for a work-
ing neutral-atom quantum gas experiment include a way to generate magnetic fields,
a way to image your gasses (preferably with high resolution), and a computer con-
trol system to orchestrate experimental sequences. The first two of these will be
addressed in this section, and we will reserve the next Section 6.4 to discuss plans
for an upgraded computer control architecture.
5This is the hilariously named “boosTA”, from Toptica.
160
BN1
Master laser
BN2
BN3
First order
To Wavemeter
Beatnote slave schematic
Zeroth order BN4 1st order
intensity locked & to experiment
sent to ULE cavity 
PDH
0th order to 
Wavemeter
Reference light
from master
for Beatnote Lock To Hamamatsu PD
(G4176-03)
Figure 6.4: Planned layout for the SrII 689 nm laser system. The “Master” laser is locked to a
convenient mode of a stable Fabry-Pérot cavity via the PDH technique. The remaining light is split
into four (or more) arms, which reference beatnote locks for slaved lasers. The beatnote circuit
is described in more detail in Sec. 2.5.2. For the beatnote laser used to reference the red MOT,
the final AOM is omitted and instead light is first passed to a Tapered Amplifier (TA). Remaining
beatnote slaves include stirring laser for 87Sr, optical pumping, and optical Stern-Gerlach (OSG).
Halfwave plates are omitted for clarity.
161
Most neutral-atom experiments at the JQI have built high-current magnetic
field coils using hollow square copper tubing wound onto a form. To remove heat
dissipated in the coil by the high current, a closed-loop cooling system runs chilled
water through the hollowed center. During design discussions for the SrII system,
we became aware of the Bitter-type electromagnet. Instead of running chilled water
serially through the whole coil, a Bitter coil is constructed from interlocking copper
disks with water cooling channels between each layer. In this design, the cooling
occurs in parallel, drastically increasing the coil’s ability to dissipate heat.
Recently, there has been interest in so-called “orbital” Feshbach resonances in
alkaline-earth(-like) atoms [128–130]. Because the ground state of alkaline-earth
species is an electron spin singlet, the magnetic moment is (nearly) zero and thus
affords no magnetically-tunable Feshbach interaction like there is in alkali species.
However, given spin exchange interactions and a slight differential Landé g-factor
between 1S0 and 3P0, one can generate a Feshbach-type resonance using the clock
transition [128].
To do this, one operates in a mixture of ground |g〉 and excited |e〉 orbital
states. Because of the differential Landé g-factor between the two orbital levels,
there is a small energy difference between an atom in |g ↑〉 and |e ↑〉 (with the spin
state designated by ↑↓), which is tunable over some range with a magnetic field.
Consider the (s-wave) scattering interaction of two such atoms. We can write two
scattering channels, designated |o〉 for “open” (i.e., energetically allowed) and |c〉 for
“closed” (energetically forbidden), which in properly symmetrized form are
|o〉 = √1 [|g ↑〉1|e ↓〉2 − |e ↓〉1|g ↑〉2] (6.1)
2
|c〉 = √1 [|e ↑〉1|g ↓〉2 − |g ↓〉1|e ↑〉2] (6.2)
2
At infinite separation, |g ↑〉1|e ↓〉2 and |g ↓〉1|e ↑〉2 have an energy offset δ propor-
162
tional to ∆m ↓ ↑F = mF −mF and given by
δ = (ge − gg)∆mFµBB (6.3)
where the differential Landé g-factor δgF = ge − gg arises from hyperfine mixing
between 3P0 and 3P1. This energy difference now provides a handle for tuning the
energy of bound states in the open and closed channels by changing the magnetic
field; given some spin exchange interaction, this now behaves analogously to a tra-
ditional magnetic Feshbach resonance.
S-wave orbital Feshbach resonances have been observed in Ytterbium [129,130],
and should exist in strontium as well but only at very large fields. To see why this is,
we consider the pseudo-potential model from Ref. [128], which predicts a scattering
length as given by
√
−as0 + √mδ/~2(a2 2s0 − as1)as = (6.4)
a mδ/~2s0 − 1
where a + −s0 = (as +as )/2 is the average of s-wave scattering lengths in the symmetric
(a+s ) and anti-symmetric (a−s ) spin channels, a = (a−s1 s −a+s )/2,m is the mass (in this
case 87 amu), and δ is the magnetically-tunable energy difference between the open
and closed channels. The resonance position predicted by Eq. 6.4 occurs when the
scattering length as diverges, that is, when δ → ~2/ma2s0. Crucially, the magnetic
field at which this occurs is a function of the differential Landé g-factor, but also on
the scattering lengths for the symmetric and anti-symmetric spin channels, contained
in as0.
Using scattering numbers from Ref. [1], the known differential Zeeman shift
mF · 109Hz/G [59], and assuming a spin mixture of mF = ±9/2, we find the scatter-
ing resonance should occur near 3000G. This field is likely to be unreachable. The
reason this technique works in ytterbium is the spin-symmetric scattering channel
163
Figure 6.5: CAD renderings of the bitter coil and a photo of everything fully assembled. Because
the current-carrying brass rods snapped easily, we instead welded a rectangular-profile copper bar
to the brass bottom layer, which can be seen in (c) attached to the low-gauge welding cable. The
entire coil is potted in epoxy to reduce the risk of water leaks.
has a very large s-wave scattering length a+s = 3300 a0, much larger than strontium,
for which both a+s and a−s are near 100 a0. Stated another way, in ytterbium there
is a near-threshold bound state supported by the closed channel scattering potential
which can be pushed around slightly to yield a scattering resonance.
That all said, in discussions with theory collaborators, there may exist a near-
threshold p-wave resonance in strontium.6 If this indeed the case, we want the
ability to still apply moderately high magnetic fields to access this resonance. This
motivated the decision to build Bitter coil electromagnets for SrII.
The design for our coils was based loosely on the design in [131,132], with
additional tips from [133] and discussions with the CAVS group at NIST. A CAD
rendering is shown in Fig. 6.5. The coil consists of two concentric stacks of copper
disks with insulating polyester spacers sandwiched between each layer. Cooling
water passes through eight vertical cutouts passing through all layers, and then
horizontally between each layer in space cutout from the polyester spacer. The final
layer at the bottom of the coil is slightly thicker and machined from brass, which
provides better mechanical rigidity than copper for clamping the coil together, but
has a higher conductivity than steel. The high-current input and return happens on
6Paul Julienne, private comm.
164
vertical copper bars welded to the bottom of the brass clamp. The original design
followed ideas from Refs. [131,132], where current was brought on a threaded brass
rod passing through one of the vertical water cooling channels. However, we had
difficulty tightening the coils to the point where they would seal without snapping
the brass rod.7
Our primary modification from the design in Refs. [131,132] is to add a second
inner coil, and simplify the water distribution manifold. We constructed a dual-
chamber stainless steel reservoir which attaches to the copper coils; water enters the
bottom chamber, passes through the coil, and returns to the top chamber before
returning to the chilled water closed loop. The inner structure of this can be seen
in the exploded view of Fig. 6.5(c).
Imaging system. In an imaging system, aberrations arise when light deviates
significantly from the paraxial approximation, as is the case for high-NA optics.
They are classified into different categories depending on the physical origin (defo-
cus, spherical aberration, coma, etc.), and the goal of a good optical design is to
engineer these aberrations to cancel to a high degree. If this can be accomplished,
the resulting image will be limited by diffractive effects only.
The high-resolution imaging system designed for SrII is based off the principles
in [134]. Using Zemax or other ray-tracing design software, one creates a “best-form”
lens for diffraction-limited imaging at a particular wavelength. An iterative process
follows: replace one lens with the closest stock-catalog lens available, and re-optimize
the design with the remaining surfaces. Repeat this process until you are left with an
7Our Bitter coil design was made by Alex Hesse, a visiting student from Heidelberg, and the
painstaking construction was taken over by Peter after Alex’s departure. Getting these coils to seal
has been an ongoing nightmare, as we first thought to use teflon gaskets to both seal and provide
electrical insulation. This plan did not work, and we eventually reverted back to the polyester
spacers with laser-cut rubber gaskets as in [131]. We also found that machining a clamp plate, and
using the 1/4-20 through-hole mounts in the water manifold to pre-tension the coil and check for
leaks, has been crucial to preventing the threaded rods from snapping.
165
LC1611 LB1607 LA1417 LE1418
(plano-convex) (bi-convex) (plano-convex) (meniscus) Vacuum Window
f = -150 f = 175 f = 150 f=150
Atoms
Figure 6.6: High-NA imaging objective for SrII, corrected at 461 nm for aberrations introduced
by the vacuum viewport. It has an image-space NA of 0.29, and a diffraction-limited spot size of
1.9 µm. It is constructed out of catalog 2" lenses, and optimized with Zemax as described in the
text.
objective made from stock lenses. Surprisingly, this seems to converge to the same
set of lenses for a variety of wavelengths, with the precise lens spacing set by details
of the imaging wavelength and the optical thickness of the final vacuum viewport.
The final design for our high-resolution, infinity-corrected objective can be found
in Fig. 6.6. It has an effective focal length of 66mm, and an image space NA of
0.29. At 461nm, this corresponds to a diffraction-limited spotsize of ≈ 1.9 µm.8
The entire objective is set in a 2" lens tube, with cylindrical spacers machined from
aluminum.
6.4 Computer Control
Historically, labs that are a part of the laser cooling and trapping group in the
JQI have used an amalgamation of LabView VIs to visualize and program experi-
8There are, of course, many metrics one can use to define resolution. Here, we are taking the
characteristic spotsize given by the first null of the airy disk, the so-called Raleigh criteria. The
airy radius, then is half of this.
166
mental sequences. Collectively known as SetList, this computer control system has
been previously described in Sec. 2.4 and in Ref. [19].The original SetList was rewrit-
ten circa 2014 to utilize some of LabView’s object-oriented programming structures,
and the newer version was much improved. However, we find that we have pushed
SetList far beyond the intended capabilities of LabView. The code has grown un-
wieldy and undocumented, the GUI is frustratingly slow to respond to user input,
and there are intermittent bugs which can be time-consuming to address and solve.
In recent years, with the growing popularity of Python, there has been movement
in the AMO community to adopt control and analysis software written in Python.
On the analysis side, there is a wealth of well-developed fitting, plotting, and im-
age analysis packages. On the experimental control front, the community seems to
have coalesced around three candidates: the “Sandia” control system9, LabScript
Suite [135], and ARTIQ [136].
The Sandia system was first developed at Sandia by Peter Maunz, and later
extended by ion trapping groups at the University of Maryland. The core of the
system is based on an Opall Kelly FPGA; a GUI written in Python/Qt allows the
user to compose “pulse programs,” which utilize a subset of the Python language,
and instruct the FPGA on deterministic timing. The pulse programs support data
acquisition and conditional branching, which makes the construction of complex,
real-time behaviors possible. The FPGA is directly connected to several “fast” pe-
ripherals such as DACs, ADCs, and DDSs, as well as digital I/O. All data is reported
back to the computer and plotted in real-time for easy analysis. For a more detailed
description, see the thesis by Kenneth Wright [137].
LabScript Suite is another Python-based system developed at Monash Univer-
sity. It utilizes a very similar timing architecture to the one employed by SetList,10
9To my knowledge, this system has not appeared in published form, but has been adopted by
several ion trapping groups at the University of Maryland
10More specifically, it uses a PulseBlaster or its generic cousin the “PineBlaster” to deliver pseudo-
167
but uses a Pythonic scripting language instead of LabView for programming of se-
quences. LabScript also includes an image analysis pipeline, which allows the user to
write custom analysis scripts which interact seamlessly with images acquired during
the course of an experiment. Files are stored in the popular HDF5 format, which
can then be interfaced with other analysis programs if desired. LabScript Suite has
been adopted by a few neutral atom labs at the JQI and NIST, and is being actively
developed by those groups.
Finally, ARTIQ is a “next generation control system for quantum information
experiments.”11 It is being actively and heavily developed by a professional software
company, and the accompanying hardware is being developed by a group at War-
saw University. Like the two previous solutions, ARTIQ is Python-based but aims
to be more agnostic to particular hardware toolchains. For comparison, the San-
dia system is tied to the Opall Kelly hardware and frontpanel software drivers for
communication, and LabScript Suite assumes a particular control architecture. The
stated goals of the ARTIQ ecosystem are to serve as a backbone “operating system”
for quantum information and other disciplines with stringent timing requirements.
Careful thought and planning has gone into making it robust, well-tested, and ex-
tensible. And the fact that it is developed by professional software and hardware
engineers means we physicists can leverage their expertise in design architectures
while focusing on the end physics applications.
While each system has its upsides and pitfalls, we have decided to deploy
Labscript Suite in the SrII laboratory.12 We have also been working to develop a
clock triggers to a heterogeneous array of peripherals. These peripherals are programmed at the
top of a given experimental cycle, and update their outputs to the next state when triggered to do
so by the PulseBlaster.
11See the project GitHub page for more information: https://github.com/m-labs/artiq
12Our original intention was to deploy ARTIQ, but I seemed to be the only person pushing this
path. Given the collective experience with Labscript Suite at the JQI, and the relative newness
of ARTIQ, keeping with a known solution seemed prudent. ARTIQ certainly has a vastly more
powerful real-time control infrastructure, but as it stands we are without a compelling use case.
168
replacement for the PulseBlaster, using the PYNQ development platform. Even-
tually, this might be extended into a larger ecosystem of custom instrumentation
electronics.
6.5 Outlook
As the experiment transitions fully to the new lab space, we have begun to
build the capability for incorporating Rydberg-dressed interactions in the clock
state. Based on the proposal in Ref. [138], the idea is to generate spin squeez-
ing on the clock transition to reduce the quantum projection noise, which will soon
limit atomic clock performance.
Another idea which we would like to pursue is to generate uniform 1D poten-
tials with a blue-detuned dipole trap imaged onto the atoms with a digital micro-
mirror array (DMD). There are a number of interesting proposals to study spin
physics in a 1D box potential [6, 7]. These proposals all rely on the highly group-
symmetric Hamiltonian accessible in a quantum simulation with strontium. Because
the momentum eigenstates of a particle in a box all have identical overlap integrals,
the interaction energy does not depend on the exact levels that are occupied. This
adds an additional permutation symmetry between momentum eigenstates in a uni-
form square-well potential.
If one thinks of the external energy eigenstates of the box potential as a syn-
thetic lattice dimension, we have changed the spin-spin interaction energy scale from
nearest-neighbor J2/U (with J the hopping energy and U the onsite interaction en-
ergy) to all-to-all coupling with constant strength. This model is exactly solvable
and can be implemented with thermal atoms, which is interesting from the point of
And because the development thus far has targeted ion trapping groups (who have supplied the
majority of the funding for both ARTIQ hardware and software), concise programming examples
tailored to the way neutral-atom experiments are structured have not been created. Without some
level of institutional experience, or a very high level of python-fu, the learning curve was going to
be steep.
169
view of benchmarking a strontium quantum simulator.
One other proposal for the 1D box potential enables efficient estimation of the
eigenspectrum of the nuclear spin density matrix ρ̂. The experiment is remarkably
simple — a Ramsey pulse sequence is performed on the clock transition, and the
dynamics depend on the state of the nuclear spin subspace. Careful analysis of the
Ramsey dynamics allows one to extract this information.
Finally, we might revisit the isotope shift measurements discussed in Chap-
ter 5, this time with a magic-wavelength lattice. Norbert Linke is building a Sr+
experiment at the JQI, and there might be the possibility for combining precision
measurements in both neutral and ion systems for a more complete test of the King
linearity. We have had some discussions with theory collaborators about how to
handle the nonlinearities in 87Sr, which will be necessary before moving forward
with this research program.
In all, some very exciting times lie ahead for the new strontium experiment.
Improvements in the design of the apparatus will enable a richer exploration of the
physics accessible with ultracold strontium, and I look forward to seeing what is
possible.
170
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