Title of Dissertation: GRATING MAGNETO-OPTICAL TRAPS
FOR STRONTIUM
Peter Knox Elgee
Doctor of Philosophy, 2022
Dissertation Directed by: Dr. Gretchen Campbell
Department of Physics
This thesis describes the construction of a new laboratory and a new apparatus
for ultracold strontium experiments. It also describes the the creation of the first
strontium grating magneto-optical trap (MOT). The new strontium apparatus has
been improved to provide high, and controllable atomic flux, high magnetic field
coils, low vacuum pressure, plenty of optical access, and a streamlined computer
control system. This flexible apparatus is designed to operate several potential
experiments.
Our work with strontium grating MOTs will help to enable the creation of
compact, stable, and portable quantum devices that harness the unique capabilities
of alkaline earth atoms. We present the construction of the grating MOT apparatus,
the demonstration of the broad line strontium grating MOT, and the transfer to the
narrow line grating MOT. In addition, we show the effectiveness of sawtooth wave
adiabatic passage (SWAP) in the grating MOT geometry.
Lastly, we present work from the previous apparatus, measuring the isotope
shifts on the clock and intercombination transitions in strontium.
GRATING MAGNETO-OPTICAL TRAPS FOR STRONTIUM
by
Peter Knox Elgee
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
2022
Advisory Committee:
Dr. Gretchen K. Campbell, Co-chair/Advisor
Professor Steven L. Rolston, Co-chair
Professor Mario Dagenais, Dean?s Representative
Professor Luis A. Orozco
Dr. James (Trey) V. Porto
? Copyright by
Peter Knox Elgee
2022
Acknowledgments
This work and my time at the Joint Quantum Institute (JQI) would not have
been possible without the support of many people within and outside of the Physics
Department. Firstly, I would also like to thank the previous strontium graduate
students who worked on the project, Ben Reschovsky, and Neal Pisenti. Both of
them taught me a lot in the first half of my graduate school career and Neal in
particular set the foundation for the new lab. Hiro Miyake was helpful as our
postdoc as we finished up in the old laboratory. Nick Mennona and Alex Hesse
worked in the lab for a shorter time but made large impacts on the experiment, and
Nick has remained my friend even after leaving the lab. Ananya Sitaram has been
my peer in the lab throughout graduate school, we started in Gretchen?s group the
same semester and are now leaving the same semester, we have gone through the
ups and downs of graduate school together. Our two new students Sara Ahanchi,
and Tim Li have stepped up to the plate to take over the lab. Across the hall,
in the ErNa lab Avinash Kumar, Mo?nica Gutie?rrez Gala?n, Swarnav Banik, and
Madison Anderson were always patient with my many questions. Now a part of the
ErNa lab, Yanda Geng initially helped us in the lab and graciously offered his coding
expertise. I would also like to thank my fellow graduate students and postdocs in the
JQI, specifically: Francisco (Paco) Salces Ca?rcoba, Ana Valde?s Curiel, Dalia Ornelas
ii
Huerta, Sandy Craddock, Sarthak Subhankar, and Chris Billington for particularly
useful and friendly discussions.
I owe much to my advisor Gretchen Campbell, who brings a calm presence to
any issue in the lab, and is reassuring whenever things go wrong. Gretchen let me
join the lab at the start of my first year of grad school, and was supportive whenever
we wanted to take our investigations in a new direction. I would also like to thank
the other professors who helped me in the JQI, Trey Porto, Alicia Kolla?r, Steve
Rolston, and particularly Luis Orozco who was always invested in my success and
provided a friendly face to go to when things got tough. Alessandro Restelli worked
closely with us for all our electronics needs and is a constant source of energy and
ideas. Bookending my graduate school career is Daniel Barker who graduated from
our lab shortly before I joined and has now acted as a second advisor and mentor for
the last two years. Dan is endlessly patient and knowledgeable, and I am immensely
grateful for his help.
I would like to thank my friends for putting up with me during the stress of
graduate school, and Humberto for many lunches on campus away from the base-
ment of the physical sciences complex. Thank you to my family for their continual
love and care, my parents Sara and Steve, and my sister Lauren for many impromptu
calls for support. Finally, I couldn?t have done any of this without my partner Axan-
dre Oge. Oge has always been there for the good and the bad of the last six years,
I?m excited for what lies in the future for us.
iii
Table of Contents
Acknowledgements ii
Table of Contents iv
List of Tables vii
List of Figures viii
List of Abbreviations x
Chapter 1: Introduction 1
1.1 Strontium Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Isotope Shift Introduction . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 New Strontium Laboratory . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Grating MOT Introduction . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Chapter 2: Theory 10
2.1 Laser Cooling of Strontium . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.1 Basics of laser cooling and trapping . . . . . . . . . . . . . . . 10
2.1.2 Blue MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.3 Red MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.4 Dipole Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 Isotope Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Grating MOT Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 Sawtooth Wave Adiabatic Passage . . . . . . . . . . . . . . . . . . . 29
Chapter 3: Experimental Apparatus 33
3.1 Vacuum System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.1 Bakeout Chamber . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.2 AOSense Strontium Source . . . . . . . . . . . . . . . . . . . . 37
3.1.3 Main Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1.4 Grating Chamber . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.1 Bitter Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.2 Grating Coils and Magnets . . . . . . . . . . . . . . . . . . . . 48
3.2.3 Shim Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
iv
3.3 Laser Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3.1 Blue 461 nm System . . . . . . . . . . . . . . . . . . . . . . . 51
3.3.2 Repump Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.3 Red 689 nm System . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.4 Clock Laser System . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3.5 Dipole Trap Laser Systems . . . . . . . . . . . . . . . . . . . . 57
3.3.6 Miscellaneous Lasers . . . . . . . . . . . . . . . . . . . . . . . 57
3.4 Experiment Side Optics . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.4.1 Main Experiment Optics . . . . . . . . . . . . . . . . . . . . . 59
3.4.2 Imaging Stack . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4.3 Grating Optics . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.5 Computer Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Chapter 4: Validations 72
4.1 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2 Blue MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3 Red MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.4 Dipole Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.5 Blue Grating MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.6 Red Grating MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Chapter 5: Publications 94
5.1 Publication: Isotope shift spectroscopy of the 1S0 ? 3P1 and 1S0 ?
3P0 transitions in strontium . . . . . . . . . . . . . . . . . . . . . . . 94
5.1.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.1.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.1.3 Experimental procedure . . . . . . . . . . . . . . . . . . . . . 97
5.1.4 Measurement of the 1S0 ? 3P1 isotope shifts . . . . . . . . . . 99
5.1.5 Measurement of the 1S0 ? 3P0 isotope shifts . . . . . . . . . . 103
5.1.6 King plot analysis . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.1.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.1.8 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.1.9 Appendix: Modeling the inhomogeneous broadening of the
clock transition . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.2 Publication: Confinement of an alkaline-earth element in a grating
magneto-optical trap . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.2.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.2.3 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.2.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . 136
5.2.7 Data Availability . . . . . . . . . . . . . . . . . . . . . . . . . 136
Chapter 6: Future Ideas and Conclusion 137
v
6.1 Box Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.2 Rydberg Spin Squeezing . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.3 Sub-wavelength lattices . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.4 Grating MOT Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Bibliography 144
vi
List of Tables
1.1 Isotope Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Transition Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Isotope Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 g-Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.1 Repump Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.1 Intercombination Isotope Shift Systematics . . . . . . . . . . . . . . . 97
5.2 Measured Isotope Shifts . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.3 Clock Isotope Shift Systematics . . . . . . . . . . . . . . . . . . . . . 102
vii
List of Figures
1.1 Level Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 87Sr Red MOT Hyperfine Structure . . . . . . . . . . . . . . . . . . . 19
2.2 Grating Chip Photo . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Grating MOT Polarization Projection . . . . . . . . . . . . . . . . . . 26
2.4 Blue Grating MOT Force Profile . . . . . . . . . . . . . . . . . . . . . 27
2.5 Red Grating MOT Force Profile . . . . . . . . . . . . . . . . . . . . . 28
2.6 SWAP Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.7 Grating MOT SWAP Simulation . . . . . . . . . . . . . . . . . . . . 32
3.1 Bakeout Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2 Strontium Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Main Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4 Grating Vacuum Chamber . . . . . . . . . . . . . . . . . . . . . . . . 43
3.5 Bitter Coil Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.6 Bitter Coil Photo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.7 Blue Laser Block Diagram . . . . . . . . . . . . . . . . . . . . . . . . 52
3.8 Horizontal Main Chamber Optics . . . . . . . . . . . . . . . . . . . . 60
3.9 Vertical Main Chamber Optics . . . . . . . . . . . . . . . . . . . . . . 61
3.10 Source Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.11 Imaging Stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.12 Grating Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.1 Imaging Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2 Resonant Shells of Red MOT . . . . . . . . . . . . . . . . . . . . . . 79
4.3 Red MOT Experimental Sequence . . . . . . . . . . . . . . . . . . . . 81
4.4 Single Beam Dipole Trap Images . . . . . . . . . . . . . . . . . . . . 82
4.5 Cross Dipole Trap Image . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.6 Grating MOT Atomic Beam . . . . . . . . . . . . . . . . . . . . . . . 84
4.7 Resonant Shell of Red Grating MOT . . . . . . . . . . . . . . . . . . 87
4.8 Red Grating MOT Experimental Sequence . . . . . . . . . . . . . . . 89
4.9 Red Grating MOT Temperature Measurement . . . . . . . . . . . . . 90
4.10 Red Grating MOT Lifetime Measurement . . . . . . . . . . . . . . . 91
4.11 Red Grating MOT SWAP . . . . . . . . . . . . . . . . . . . . . . . . 92
4.12 SWAP Sweep Frequency . . . . . . . . . . . . . . . . . . . . . . . . . 93
viii
5.1 Spectroscopy Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.2 King Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.3 Detailed Lineshape Curves for 698 nm Clock Transition . . . . . . . . 117
5.4 Thermal Line Pulling for Clock Transition . . . . . . . . . . . . . . . 118
5.5 Grating MOT System Cut-Away . . . . . . . . . . . . . . . . . . . . 122
5.6 Loading Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.7 Atom Number as a Function of Detuning . . . . . . . . . . . . . . . . 128
5.8 MOT Parameters as a Function of Intensity . . . . . . . . . . . . . . 129
5.9 MOT Parameters as a Function of Source Current . . . . . . . . . . . 131
5.10 Temperature Measurement . . . . . . . . . . . . . . . . . . . . . . . . 132
ix
List of Abbreviations
AOM Acousto-Optic Modulator
AR Antireflection
CAVS Cold Atom Vacuum Standards
CG Center of Gravity
DDS Direct Digital Synthesizer
DMD Digital Mirror Display
ECDL Extended Cavity Diode Laser
EOM Electro-Optic Modulator
FSR Free Spectral Range
IGBT Insulated-Gate Bipolar Transistor
JQI Joint Quantum Institute
MOSFET Metal?Oxide?Semiconductor Field-Effect Transistor
MOT Magneto-Optical Trap
NEG Non-Evaporable Getter
NI National Instruments
NIST National Institute of Standards and Technology
OD Optical Depth
PBS Polarizing Beamsplitter
PDH Pound-Drever-Hall
PID Proportional Integral Diferential
PLL Phase Lock Loop
RF Radio Frequency
SRS Stanford Research Systems
SWAP Sawtooth Wave Adiabatic Passage
TA Tapered Amplifier
ULE Ultra Low Expansion glass
VCO Voltage Controlled Oscillator
x
Chapter 1: Introduction
1.1 Strontium Basics
Laser cooling and trapping was initially developed with alkali atoms, however,
strontium and other alkaline earth atoms have unique properties that make them
useful for a variety of applications, from quantum simulation to precision metrology.
Strontium has four stable isotopes, three bosons (84Sr, 86Sr, and 88Sr), and one
fermion (87Sr). The properties of these isotopes are summarized in Tab. 1.1. The
nuclear spin I is zero for all the bosonic isotopes of strontium and other alkaline
earth elements, and thus they lack any hyperfine structure.
The most fundamental change when considering alkaline earth atoms over al-
kali atoms is the inclusion of a second valence electron. These two electrons can
either be in a singlet or triplet configuration, creating more complicated level struc-
tures. The level structure of strontium applicable to our experiments is shown in Fig.
84Sr 86Sr 87Sr 88Sr
Abundance 0.56% 9.86% 7.00% 82.58%
Nuclear Spin 0 0 9/2 0
Scattering Length ?2a0 96a0 823a0 123a0
Table 1.1: Properties of the different stable isotopes of strontium. a0 = 5.29 ?
10?11 m is the Bohr radius.
1
1.1. To first order, electric dipole transitions don?t couple singlet and triplet states,
but the spin-orbit interaction mixes states with the same total angular momentum
J , allowing these transitions to happen [1]. For instance in strontium the ?inter-
combination? transition 1S 3 3 10 ? P1 is possible due to mixing between P1 and P1
[2]. This mixing effect also gives these transitions narrow linewidths, making them
useful for creating cold magneto-optical traps (MOTs) around a few ?K without the
need for sub-Doppler cooling.
The spin-orbit interaction does not allow for the transition 1S0 ?1P0, dubbed
the ?clock? transition, since J = 0 ? J ? = 0 is forbidden for dipole transitions.
However, The hyperfine interaction of the fermion, mixes states with the same F , so
that 3P state mixes with 3P , 1P , and 30 1 1 P2 to allow for the transition [2]. Since this
effect is much weaker than the spin-orbit interaction, the linewidth of this transition
is very narrow ? 1 mHz. While the clock transition is generally not allowed in the
bosonic isotopes, as they do not have hyperfine structure, applying a magnetic field
mixes 3P0 with
3P1, and allows for the transition [9].
In another contrast to alkalis, strontium and other alkaline earth atoms have
no electronic angular momentum (J = 0) in their ground states. Thus the ground
state is insensitive to magnetic fields which eliminates, or complicates, some useful
tools such as magnetic traps, Feshbach resonances [10], and sub-Doppler cooling.
While 87Sr has F = 9/2 in the ground state, it still has a very small g-factor due
to the small nuclear magneton. In addition, since the excited state of the clock
transition (3P0) also has J = 0, the transition is also magnetically insensitive. Even
the small g-factors of these states cancel when considering the differential magnetic
2
Figure 1.1: Level diagram for strontium. The blue MOT, red MOT / intercombina-
tion, and clock transitions from the ground state are shown with wavelengths and
linewidths [3?5]. The decay out of 1P1 state [6], and subsequent decays into the
3P states [7] (from context and alignment with other theses I assumed the 1D -12 P1
rate listed in [7], was in fact the 1D 32- P1 rate). The two repump transitions are also
shown with their wavelengths and linewidths [8].
3
sensitivity, and only the contribution coming from the state mixing remains [8]. The
narrow linewidth and magnetic insensitivity of the clock transition justify its name
as is used for some of the world?s best optical atomic clocks [11?13]. The clock
transition also makes strontium useful for different types of metrology and quantum
simulation discussed throughout this thesis.
While the bosonic isotopes of strontium have no nuclear spin, the fermion has
a relatively high nuclear spin (I = 9/2). In addition, the J = 0 nature of the ground
and clock states decouple the electronic angular momentum from the nuclear spin.
This decoupling also makes collisions and atomic interactions independent of the
sublevel mI , which creates an SU(N) spin symmetry in the nuclear spin sublevels,
where N can be chosen as long as it satisfies N ? 2I+1 = 10 for strontium [14, 15].
This large symmetry group makes strontium an attractive element for quantum
simulations of systems with this symmetry [16?22].
For reference, I have included a few tables with different properties of stron-
tium. Tab. 1.2 lists the wavelength, linewidth, saturation intensity, and fine struc-
ture g-factors of the blue MOT, red MOT / intercombination, and clock transitions.
Tab. 1.3 lists the different isotope shifts relative to 88Sr for the main transitions.
Tab. 1.4 lists the g-factors for the different excited hyperfine states of 87Sr.
1.2 Isotope Shift Introduction
The final experiment for the old strontium laboratory involved precision mea-
surements of isotope shifts on the intercombination and clock transitions in stron-
4
1S ?1P 10 1 S0 ?3P 1 31 S0 ? P0
? 460.9 nm 689.4 nm 698.4 nm
?/2? 30.2 MHz 7.4 kHz ? 1 mHz
I 2sat 40.4 mW/cm 3.0 ?W/cm
2 ? 10?10 mW/cm2
gJ ? 1 3/2
Table 1.2: Parameters for different transitions in strontium. The vacuum wave-
lengths (?), linewidths (?/2?), saturation intensities (Isat), and excited states fine
structure g-factors (gJ ?) are listed. Numbers from [3, 5, 8].
Isotope Shift (kHz) 1S 1 1 3 1 30 ? P1 S0 ? P1 S0 ? P0
84-88 -273 ?351.5 ?349.7
86-88 -126 ?163.8 ?162.9
87-88 (CG) -45 ?62.2 ?62.2
87-88 (F ? = 7/2) -8 1351.9
87-88 (F ? = 9/2) -68 221.7
87-88 (F ? = 11/2) -50 ?1241.5
Table 1.3: Isotope shifts for the different transitions in strontium relative to 88Sr.
For states with hyperfine structure the center of gravity (CG) shift, weighted by
number of sublevels, is shown along with individual hyperfine transitions. Isotope
shifts are from Refs. [23, 24].
g 1 3F P1 P1
F = 7/2 -2/9 -1/3
F = 9/2 4/99 2/33
F = 11/2 2/11 3/11
Table 1.4: Hyperfine g-factors for the excited states of 87Sr. This does not take into
account the nuclear magneton.
5
tium. Precision spectroscopy is a useful tool in general, as it provides the basis
for more precise experiments in the future. It can also provide insight into atomic
structure, and be used to search for new physics via comparison with theoretical
calculations. Isotope shifts in particular give insight into nuclear and atomic struc-
ture. In addition, by comparing the isotope shifts on two transitions we can extract
a linear relationship, known as King linearity [25].
The King linearity is a useful tool to track the change in the charge radius as
neutrons are added to the nucleus and the charge distribution changes. In addition,
this linear relationship has been proposed as a way to search for new physics, by
putting limits on new physics that would create deviations from this linear rela-
tionship [26, 27]. However, higher order effects within the standard model can also
cause nonlinearities, so the linearity of King plots only becomes a useful tool when
these effects are well accounted for.
Our lab measured the isotope shifts for both the intercombination and clock
transitions between four stable isotopes of strontium at the level of ? 10 kHz. With
the four isotopes of strontium we had just enough measurements to investigate King
linearity in this system, and found a significant deviation from linearity. This de-
viation is most likely due to higher order effects in the calculation of the center of
gravity of the intercombination line of 87Sr due to its hyperfine structure. Never-
theless, this work provided the first precision measurement of the clock transition
in 86Sr and 84Sr, and from our King plot analysis we were able to extract useful
information about the mass and field shift parameters for these transitions.
6
1.3 New Strontium Laboratory
Our work on isotope shifts was the last experiment performed in our old stron-
tium laboratory. While this project was being completed we were simulatneously
building a new laboratory space and apparatus and implementing a number of im-
provements. The new strontium laboratory aims to provide the tools required for a
variety of experiments that utilize the unique properties of strontium. We have in-
vestigated using Rydberg dressing to increase the precision of optical atomic clocks
[28, 29], generating sub-wavelength attractive potentials [30], and using a box trap
for quantum simulation [16, 17]. These experiments require a robust, stable, and
flexible apparatus.
During my time in grad school we have built up a new laboratory and appa-
ratus with the capabilities to execute these experiments. We implemented a new,
high flux, strontium source that has increased the atom number in our MOTs. We
designed and built new, high current magnetic coils for operating MOTs, or for other
high field investigations. We rebuilt some laser systems, and refined others. Finally,
we implemented laser cooling and trapping of strontium, from the 2D MOTs of our
source, to the broad blue MOT, to the narrow line red MOT, to dipole traps, which
prepare our atomic gases for future experiments.
7
1.4 Grating MOT Introduction
Many of the unique properties of strontium and other alkaline earth atoms
make them particularly useful for metrology and sensors. However many of these
applications only reach their full potential with stable, compact, and portable laser-
cooled quantum devices. For instance, portable quantum devices with strontium
could be used for gravimetry [31] or inertial navigation [32], devices capable of
being installed in satellites could be used as gravitational wave detectors [33?35],
and accurate dissemination of the SI second relies on the availability of simple and
stable atomic clocks [36]. Generally, experiments with laser-cooled atoms, and in
particular conventional 6-beam MOTs, require large laboratory spaces, large vacuum
chambers, and many laser beams, which makes them difficult to engineer into a
stable usable device.
In our laboratory, we worked on developing grating MOTs for strontium as one
solution to this issue. By using a single laser beam input and the diffracted beams
off of a grating to create the trapping forces, grating MOTs offer simplification,
stability, and miniaturization over traditional 6-beam MOTs. Only alkali atoms
had previously been trapped in a grating MOT before our work with strontium.
We created the first alkaline earth grating MOT with our strontium blue grating
MOT, implemented transfer to the narrow line red MOT, and showed sawtooth
wave adiabatic passage (SWAP) working in our narrow line grating MOT.
8
1.5 Outline of Thesis
The following chapters will go into detail on the work I have done over my time
in graduate school. Chapter 2 will give an overview of the theory behind the laser
cooling and trapping techniques for strontium, grating MOTs, and SWAP. Chapter
3 will give an overview of the new experimental apparatus, both the main apparatus
and the grating MOT apparatus. Chapter 4 will discuss the various cooling stages
performed on the main apparatus, and their optimizations and results, it will also
give a description of the imaging process and the results and process of running
the grating MOT. Chapter 5 includes two publications. One publication on our
measurements of Isotope shifts, which I supported at the start of the experiment,
and took over the for the second half of data collection. The other publication on
the blue grating MOT, which was a joint project with me and Ananya Sitaram,
and with the guidance of Daniel Barker. Chapter 6 will conclude the thesis with
the addition of forward-looking experiments that could be performed on our two
systems.
9
Chapter 2: Theory
2.1 Laser Cooling of Strontium
The experiments and applications that our laboratory aims to accomplish
require us to first to create a cold sample of atoms. In this chapter I will first
go through a relatively simple explanation of the laser cooling techniques used to
generate our cold atom gases. Much of the theory for laser cooling is presented with
more detail in Ref. [37]. I will then go through each cooling stage, and how they
apply to the specific case of strontium in our system.
2.1.1 Basics of laser cooling and trapping
In order to cool and trap atoms with light, we must find a way to apply a
force to the atoms. The first force we consider is the scattering force caused from
the absorption and emission of photons by atoms. First let us break down the inter-
action of atoms with photons into three processes: absorption, stimulated emission,
and spontaneous emission. A photon carries a momentum of p~ = h?~k, where ~k is the
wavevector, k = 2?/?. Thus absorption of a photon from a laser beam transfers
momentum to the atom in the beam?s propagation direction. Stimulated emission,
10
the time reversal of absorption, of a photon into the laser beam kicks the atom
with the same momentum but in the opposite direction of the laser beam. Thus
these two processes together, from the same laser beam, give no net force on the
atom, however, the inclusion of spontaneous emission changes this picture. When
an atom spontaneously emits a photon the emission is in a random direction, fol-
lowing a dipole distribution, this process provides no net momentum when averaged
over many emissions. So when we simply combine these three processes from one
laser beam, the only way to get net momentum transfer is absorption followed by
spontaneous emission. The rate for this process, for a two level atom, is simply the
rate of spontaneous emission: ??ee, where ?ee is the population in the excited state
from the density matrix, and ? is the decay rate. This gives a force of:
~
~ h?ks0?/2F = h?~k??ee = , (2.1)
1 + s0 + (2?/?)2
where s = 2?20 /?
2 is the on-resonance saturation parameter, ? is the Rabi fre-
quency and ? is the (angular) detuning. We now have a simple way to provide a
force on atoms in the propagation direction of a resonant laser beam. However,
while a constant force on an ensemble of atoms may be able to push it around, in
order to cool or trap the ensemble we require a velocity or spatially dependent force
respectively.
There are three main sources of line shifts that allow us to create these in-
homogeneous forces. Firstly the Doppler shift: here the frequency of light, in the
11
atom?s frame, shifts depending on the atomic velocity as
?? = ?~k ? ~v,
where ~v is the velocity of the atoms. With Doppler shifts alone you can cool atoms
in an optical molasses consisting of counter-propagating red detuned laser beams.
Secondly the Zeeman shift: here the energy of an atomic state is dependent on a
magnetic field, which for weak fields goes like
?E = gF?BmFB,
where gF is the g-factor, ?B is the Bohr magneton, B is the magnetic field, and mF
is the projection of the angular momentum along the magnetic field direction. With
the Zeeman shift alone you can create a magnetic trap where a spacial magnetic
field gradient creates a trapping potential for atoms in magnetic field sensitive states,
discussed briefly in Sec. 2.1.2. With the Zeeman and Doppler shifts together you can
create a Zeeman slower [38], where a spatially varying magnetic field is engineered
such that, as atoms are slowed with a laser beam, the Doppler and Zeeman shifts
cancel to maintain a constant resonance frequency, and thus a constant slowing
force. Additionally these two effects together form the foundation of a magneto-
optical trap (MOT) which will be discussed in Sec. 2.1.2, and 2.1.3. The last
shifts to consider are light shifts: here the presence of an oscillating electric field far
detuned from a resonance gives a shift, in a two level system, to the ground (g) and
12
excited (e) states of
h??2 h??2I
?Eg,e = ? = ? , (2.2)
4? 6Is?
where ? is the Rabi frequency, proportional to the magnitude of the electric field, I
is the intensity of the light, and Is is the saturation intensity. Thus for a laser beam
detuned below resonance, atoms in the ground state will be attracted to locations
of high intensity. In this way tightly focused laser beams can create a trap for
atoms called an optical dipole trap discussed in Sec. 2.1.4. All of these cooling
and trapping techniques depend on the specifics of the atomic element used. In the
following sections I will discuss how these techniques apply to strontium specifically.
2.1.2 Blue MOT
A standard six-beam MOT consists of a quadrupole magnetic field and six
counter-propagating, appropriately circularly polarized, red detuned, orthogonal,
laser beams. Our ?blue? MOT of strontium operates on the 1S 10 ? P1 transition
(see Fig. 1.1). We will first consider the bosonic isotopes where there is no hyperfine
structure, which simplifies the picture. Also, since the ground state has J = 0 we
only have to consider the Zeeman shift to the excited state sublevels. To consider the
force profile in the MOT we can use the scattering force from Eq. 2.1, and include
the relevant shifts to the transition in the detuning. If ? is the detuning from the
bare resonance, then including the Doppler and Zeeman shifts, the detuning from
13
the shifted resonance becomes:
?? = ? ? ~k ? ~v ? ?BmFB/h?,
where gF = 1 for the
1P1 state. We consider the scattering force in only the x
dimension of the MOT where the magnetic field has a linear gradient of B~ = dBxx?.
Here we combine the force from the two counter-propagating beams, one with ~k =
+x? addressing the mF = +1 excited state, and one with ~k = ?x? addressing the
mF = ?1 state to get
(
? 1
Fx = h?ks0
2 1 + s0 + 4(? ? kvx ? ?BdBxx)2/?2)
? 1 .
1 + s0 + 4(? + kvx + ?BdBxx)2/?2
Thus with red-detuned beams we get a spatial and velocity dependent force that
slows atoms, and traps them near x = 0. The force in equation 2.1 is in reality
an average over many photon kicks, so while an atom at the center of the MOT
with zero velocity may see no average force, there is still a non-zero variance to the
force, which causes heating in the MOT. The Doppler temperature is the lowest
temperature possible with Doppler theory due to this heating, and is defined as
h??
TD = ,
2kB
where kB is the Boltzmann constant, and ?/2? = 30.2 MHz. For the blue MOT this
temperature is 840 ?K. While there are many different complications that increase
14
the MOT temperature, or techniques to get below this limit, the Doppler limit gives
us a rough idea of how cold atoms in our blue MOT will be.
While the blue MOT is simple compared to the following red MOT cooling
stage, there is one extra complication coming from the level structure. The excited
state of the cycling transition, 1P1, can decay through the
1D2 state into the
3P1
and 3P2 states as shown in Fig. 1.1. The
3P1 state decays relatively quickly, with a
lifetime around 20 ?s, but the 3P2 state does not decay, requiring repump lasers to
excite atoms out of this state and keep them cycling, and therefore cooling. There
are many different repumping schemes for strontium [39?42], but the one we use
requires two lasers. The first at 707 nm pumps atoms out of the 3P2 state to the
3S1 state. From there they can decay to the
3P1 state or to the metastable
3P0
state. This second state is the excited state of the clock transition, with a very long
lifetime, during which atoms will be lost from the trap. To avoid this we use our
second repump laser at 679 nm to pump out of this state to the same 3S1 state.
Thus any atoms in the blue MOT that become dark to the cycling transition are
pumped through 3P1 back to the ground state.
While the metastable states complicate the blue MOT slightly, they also open
up an opportunity for magnetic trapping. While the ground 1S0 state cannot be
magnetically trapped due to the lack of angular momentum sublevels subject to a
Zeeman shift, the 3P2 state can be magnetically trapped. Even without repumps,
some of the atoms that decay to the 3P2 state land in sublevels that are magnetically
trapped by the quadrupole field of the MOT. This leads to techniques were, while
loading the blue MOT, you can leave the repump lasers off to load this magnetic
15
trap of 3P2 atoms, called the metastable reservoir, and pulse on the repumps at
the end of the blue MOT cooling stage to reintroduce these atoms to the system,
hopefully increasing the number of available atoms in total. We don?t use this
technique often, but it can be useful for the less abundant isotopes. Due to the long
lifetimes of the metastable states, running the magnetic trap allows for increased
accumulation these less abundant isotopes compared to the blue MOT. More details
can be found in Ref. [43].
So far we have been ignoring any hyperfine structure, which is appropriate for
the bosonic isotopes, but not for the fermion: 87Sr. Luckily, the blue MOT doesn?t
require much alteration, other than accounting for isotope shifts. The only other
change is to account for the additional hyperfine structure in the 707 nm repump
transition by broadening that laser in an attempt to address all of the new hyperfine
levels. More details on this broadening are described in Sec. 4.2.
The blue MOT allows us to cool and trap atoms out of our source, but the
cooling is not sufficient to load into a dipole trap or work with the clock transition,
so instead, we transfer to a second, colder MOT.
2.1.3 Red MOT
The next stage of cooling is a MOT operating on the narrow 1S0 ?3P1 ?inter-
combination? line. The linewidth of this transition is just 7.5 kHz, meaning that the
Doppler limit is about 1.8?10?7 K. While the narrow linewidth of the red transition
helps to achieve a colder temperature, it also has the side effect that light at a single
16
frequency is only resonant with a small shell of atoms in phase space. In addition,
the lower forces in the red MOT compared to the blue MOT cause gravity to become
significant, and the atoms fall to sit at the bottom of this resonant shell. Since the
blue MOT is relatively hot, it has a wide velocity distribution, and expands rapidly.
Thus the resonant shell of atoms that the red light can address is a very small per-
centage of the total blue MOT population. In order to achieve a sufficient transfer
efficiency between the MOTs we initially broaden the red cooling laser to address
as many velocity classes and positions as possible. This broadening is ramped down
over time to compress the MOT and cool to our final temperature. A more specific
description of this transfer with our experimental parameters is presented in Sec.
4.3.
Where the red MOT really becomes complex is with the addition of the hy-
perfine structure in 87Sr. Given the nuclear spin I = 9/2 our red MOT operates
on a F = 9/2 ? F ? = 11/2 transition. There are two properties of strontium that
complicate the fermionic red MOT: first is the negligible magnetic sensitivity of the
J = 0 ground state, and second is the narrow linewidth which fully resolves the
hyperfine structure. In the simple MOT description from the previous sections the
shift of ?+ transitions as a function of field have positive slopes, and vice versa for
??. This allows for selective excitation from light with the appropriate k-vector. As
long as gF and gF ? are close enough in value, adding in hyperfine structure will not
change this picture much, as shown for 87Rb in Fig. 2.1 (b). In particular, as long
17
as the ?+ transitions for all mF shift such that
gF ??B(mF + 1)? gF?BmF > 0,
the MOT can operate stably. Considering the stretched states mF = ?F this
requirement becomes:
F gF ? F
< <
F + 1 gF F ? 1
However in the 87Sr red MOT transition, shown in Fig. 2.1 (c),(d), due to the lack
of magnetic sensitivity in the ground state, these conditions are not met. The shift
in ?+ transitions from mF < 0 states have a negative slope with magnetic field.
Thus, depending on the mF state and position of an atom, it can be resonant with
the wrong laser beam and be kicked out of the trap. In addition for mF ? 3/2
(mF ? ?3/2), there is no resonant light at all on the +x(?x) side of the trap.
If we can keep atoms in the mF = ?F states on the side of the trap with ?
magnetic fields we can avoid these anti-trapping effects in two ways. First, an atom
in one of these appropriate states will be resonant with the correct laser beam at a
smaller displacement than the resonant position of the anti-trapping beam. Second,
the Clebsch-Gordon coefficients from these appropriate sublevels favor the trapping
transitions over the anti-trapping transitions as shown in Fig. 2.1 (d). Not only
does this mean the trapping transitions are stronger, but it also means that atoms
are pumped into the ideal stretched mF = ?9/2 state, where the trapping transition
is strongest. Unfortunately, this pumping becomes undesirable as soon as the atom
18
Figure 2.1: Effect of hyperfine structure on the red MOT performance. (a) For
comparison, the level structure of the F = 2 ? 3 transition in 87Rb, and its level
shifts from a positive magnetic field. (b) The level splittings from (a) in a field
gradient. For negative detuning you can see the ?+ transitions are grouped together
all in the same quadrant, and isolated from the ?? transitions, and vice versa. (c)
The level structure of the red MOT transition F = 9/2 ? 11/2 in 87Sr, and the
shifts from a positive magnetic field. The Clebsch-Gordan coefficients, normalized to
integer values, are shown on their respective transitions. For strontium, the ground
state field shift is negligible. (d) The level splitting from (c) in a field gradient.
Some ?+ transitions have a negative slope, which are anti-trapping, and vice versa
for ??. The slight offset between certain ?+ and ?? transitions are simply for clarity
and are not to scale.
19
crosses to the other side of the trap, where it will be in the wrong stretched state
and will not see any resonant laser beams at all.
While the arguments made above apply to the fermionic blue MOT as well,
that MOT works well enough because of the much larger linewidth. In the blue
MOT, the hyperfine structure is not resolved, so the MOT beams addressing the
F = 9/2 ? 11/2 cooling transition, are also addressing the F = 9/2 ? 9/2
transition. This added excitation randomizes the populations in the ground state
hyperfine manifold, negating the negative effect of the optical pumping, we call this
effect ?stirring?. For the red MOT transition the linewidth is much smaller than the
hyperfine splittings, so the hyperfine structure is fully resolved, and we do not see
the stirring effect from the cooling light. However, we can simply add in a stirring
laser resonant with the F = 9/2 ? 9/2 transition to replicate this effect from the
blue MOT and stabilize the trapping in the fermionic red MOT [44]. Details about
the implementation of this are discussed in Sec. 4.3.
2.1.4 Dipole Traps
The atoms in the red MOT are cold enough (2?5 ?K) to be captured directly
in a dipole trap. To analyze the potential from this trap we need to understand the
intensity profile of the beam used. First consider a single high power laser beam
focused to a Gaussian waist of w0 (typical values range from 10-300 ?m), and peak
intensity I0 detuned below resonance (
1S 10 ? P1). Then the transverse intensity
2 2
profile of the beam is I(r) = I0e
?2r /w0 . From the light shifts (Eqn. 2.2), this gives
20
us a transverse potential for atoms in the ground state of
h??2I e?2r
2/w2
0 0
V (r) = .
6Is?
Now consider the beam along the axial d?irection, with a waist of w0 at z = 0,
the beam diameter varies like w(z) = w0 1 + (
?z
2 )2 [45]. So the intensity, and?w0
potential along the axial direction at r = 0 become
I h??20 I0
I(z) = , V (z) = .
1 + ( ?z )22 6I ?(1 + (
?z 2
?w s ?w2
) )
0 0
Thus in both the axial and radial directions we have a trapping potential for the
ground state with negative detuning, although since the axial trapping is weak
often multiple crossed beams are used. In practice it is useful to approximate these
potentials as parabolas, so that atoms in the trap can be modeled as simple harmonic
oscillators, and we can write the potential as
1 1
V = m?2x2 + m?2y2
1
x y + m?
2
zz
2.
2 2 2
This treatment is appropriate for a two level atom, but in reality we have to consider
contributions to the light shift from many different transitions.
Dipole traps allow for the tight confinement and cold temperatures necessary to
perform many experiments, but one issue arises once you try to address transitions in
the trap, especially for the narrow transitions in strontium. The potential described
above depends on the state of the atom, specifically the AC polarizability, which is
21
a function of wavelength. So atoms in an excited state will see a different potential
than ground state atoms, and consequently the transition frequency will vary over
the spacial extent of the trap, causing inhomogeneous broadening. In particular,
while the ground state, and the 3P0,1 states of strontium are all trapped by a 1064 nm
dipole trap, the differential light shift causes the resonance frequency between them
to be dependent on the motional state of the atom. A detailed analysis on the effect
is described in the appendix of Ref. [24] (Sec. 5.1). In order to avoid this effect,
and homogeneously address the narrow transitions of strontium we can instead pick
a ?magic? wavelength for our dipole trap where the polarizabilities, and thus the
trapping potential, of the two states considered are equal. For the clock transition
this wavelength is 813 nm [8].
2.2 Isotope Shifts
The energy levels of a given element will shift between different isotopes due
to nuclear effects. This effect is important to take into account when laser cooling
and trapping multiple isotopes, and can also be used to probe the nuclear structure,
and put limits on new physics which motivated our work in [24] (Sec. 5.1).
The isotope shifts can be split into two parts. The first is the ?mass shift?
which comes from the change in the nuclear mass. The mass shift is further split into
two parts, the normal mass shift, from the change in the reduced mass of the electron,
and the specific mass shift which arises from electron correlations in multi-electron
atoms and is difficult to calculate. Put together, the mass shift between isotope A
22
and A? is proportional to 1/?AA? = 1/mA? ? 1/mA, where mA is the nuclear mass
of isotope A. The second part of the isotope shift is the ?field shift? which arises
from the change in the charge radius of the, and thus is proportional to ? ?r2?AA?
[25], the change in the mean square radius of the nuclear charge distribution. We
can then write the isotope shift of a transition (i) between two isotopes (A and A?)
as a combination of these shifts.
?
??AAi = Ki/?AA? + Fi? ?r2?AA? , (2.3)
where Ki is a constant associated with the mass shift, Fi is the field shift constant,
and ??AA
?
i is the change in the transition frequency. Here we have considered the
isotope shift of a transition which is simply the sum of the isotope shift of each state
involved.
You can extract a linear relationship between isotope shifts of different transi-
tions, which forms the basis for our work in [24] (Sec. 5.1). Consider that the mass
and field shift constants are independent of the isotope, and the factors they are
associated with (?AA? and ? ?r2?AA?) are independent of the transition. This allows
us to rewrite Eq. 2.3 in terms of the isotope shift of another transition j
( ) ( )
AA? ? Fi Fi AA??AA???i = Ki Kj + ?AA???F F jj j
This linear comparison between two transitions causes the nuclear charge radius,
which is difficult to calculate, to drop out, and allows us to extract information
23
about the field and mass shift constants. In addition, looking for deviations from
this linear relationship has been proposed as a method to search for new physics
beyond the standard model [26, 27].
2.3 Grating MOT Theory
While traditional six-beam MOTs are nicely symmetric and simple, they re-
quire good optical access and have many degrees of freedom in alignment and polar-
ization. One way to simplify the alignment and make the apparatus more compact is
to use a grating MOT instead. Grating MOTs use diffraction gratings to trap atoms
between a single input beam, and the diffracted beams off the gratings [46]. In this
section I will focus on the style of MOT we have in the laboratory: a tetrahedral
grating MOT, that has three gratings (or grating sections) oriented in a plane, and
angled 2?/3 relative to each-other, as shown in Fig. 2.2. The diffracted beams and
the input beam then combine to create a tetrahedral trapping region. The tetrahe-
dral geometry uses the minimum number of beams, and creates a relatively large
trapping region, but other groups use gratings with four sections [48], regardless, the
basic ideas of this section will still hold. The most significant difference between a
conventional six beam MOT, and a grating MOT relates to the polarizations of the
diffracted beams and the magnetic field vector. Since the k-vector of the diffracted
beams don?t align well with the magnetic field, the polarization of the diffracted
beams in the quantization basis of the atoms in no longer circularly polarized in the
correct direction, and instead the atom sees all polarizations. This decreases the
24
Figure 2.2: A photo of the grating chip used to make our MOTs. Photo from Ref.
[47]
effective power of the correct polarization, adds power to the anti-trapping polar-
ization, and adds ? polarization as well dependent on the angle of diffraction. For
instance, with ?+ input polarization, where the quantization axis is defined along
z?, the fraction of each polarization diffracted from the chip is shown in Fig. 2.3.
Here you can see that primarily the anti-trapping polarization, in this case ?+ is
diffracted from the grating. Additionally, the total power in the diffracted beams
is significantly lower than the input beam due to the efficiency of the grating. The
decreased power, and odd angles of the diffracted beams cause grating MOTs to
have anisotropic temperatures [49], and we see this effect when measuring the axial
vs transverse temperatures in our MOTs.
We will define our MOT such that the input beam is pointing in the ?z?
25
Figure 2.3: The fraction of each diffracted polarization as a function of diffraction
angle for a ?+ input polarization. This is for the quantization axis along z?, where
the angle is defined relative to the normal of the grating chip.
axial direction, and the grating is normal to that beam. Taking ? = ?1?, s0 = 1,
dBz = 55 G/cm for our blue MOT grating MOT, we can see the force profile in the
dz
axial and transverse directions in Fig. 2.4. These force profiles were created with
pylcp, a python package for simulating laser cooling physics [50]. The transverse
forces in x and y look similar to what we would expect for a traditional MOT,
although there is a slight asymmetry caused by the three-fold rotational symmetry
of the grating. The axial (z) direction, however looks significantly different from
a traditional MOT. The large dip on the +z side of the force profile is due to the
force from the input beam, but there isn?t a matching peak on the ?z side from
the diffracted beams. This is due to the lack of power in the correct polarization.
26
Instead of the traditional circular polarization, our trapping from the diffracted
beams comes from the ? polarized light which provides force in the the +z direction,
but is insensitive to magnetic field, so provides that force everywhere in the beam
path. However, the input beam is strong enough to overcome this constant force
and provide the force downwards. Consequently, this means that the zero point of
the force does not sit at the zero of the magnetic field, and its location depends
on a number of atomic, and experimental parameters. In particular, for 87Sr, due
to the hyperfine structure and the mismatch in magnetic sensitivity between the
ground and excited state, the force from the ? polarized light becomes magnetically
sensitive. This shifts the force profile so that the fermionic Sr blue MOT forms
closer to the field zero than the bosonic MOTs.
Figure 2.4: The force profiles in the blue grating MOT in x, y, and z, for ? = ?1?,
s = 1, dBz0 = 55 G/cm.dz
27
Now we can look at the red MOT force profile shown in Fig. 2.5, for ? = ?10?,
s = 90, dBz0 = 8 G/cm. Again, it?s the axial force that looks significantly different.dz
Here we do see the force from the correct polarization of the diffracted beams, but
the peak is much narrower. We see a peak because the saturation intensity of the red
MOT transition is much smaller than the blue MOT?s, so the power in the diffracted
beam is large enough to provide a traditional force. However, the difference in power
between the input and diffracted beams is still large, causing power broadening of
the input beam?s resonance compared to the diffracted beam?s resonance.
Figure 2.5: The force profiles in the red grating MOT in x, y, and z, for ? = ?10?,
s dBz0 = 90, = 8 G/cm.dz
28
2.4 Sawtooth Wave Adiabatic Passage
Since the transition between the blue and red MOTs causes significant atom
loss, especially in our grating MOT where we lose around 85% of the blue MOT
atoms, we have investigated using Sawtooth Wave Adiabatic Passage (SWAP) [51,
52] in the grating MOT to increase the transfer efficiency. SWAP is a cooling
technique that works by providing a force against atomic motion from both the
absorption and stimulated emission of counter-propagating beams through adiabatic
passage. In SWAP two counter-propagating linear polarized beams are swept from
below atomic resonance to above it, following a sawtooth function as shown in Fig.
2.6. Consider an atom, starting in the ground state and moving along the axis of the
Figure 2.6: The frequency, ?, of the SWAP lasers as a function of time. The bare
resonance ?0 and the frequency shifted by the Doppler shift ?D are shown. The time
that the atom is in the excited state (te) and the ground state (tg) are shown below
the time axis. Below the plot is a diagram of an atom with a particular velocity v
interacting with the counter-propagating beams.
laser beams. From the Doppler shift the atom first becomes resonant with the beam
opposing its motion, which is blue shifted. As this beam sweeps over the resonance
29
it adiabatically transfers the atom to the excited state, and slows the atom down
by h?k/m. Then as the sawtooth wave continues above the unshifted resonance, the
laser co-propagating with the atom is Doppler shifted to the red onto resonance and
adiabatically transfers the atom back to the ground state, where the emitted photon
again slows the atom down. Then the sawtooth function of the sweep quickly falls
back to the start, repeating the process. Compared to traditional Doppler cooling
where stimulated emission works against the cooling, the time ordering in SWAP
ensures that both absorption and stimulated emission slow the atom.
The atom has to start in the ground state for the time ordering of SWAP to
work. If an atom starts in the excited state during a period of SWAP, the reverse
process will occur and the atom will be sped up. We can avoid this by making
spontaneous emission less likely in between the two transfers, and more likely at the
start or the end of the ramp. Thus the time between the transfers (te), where the
atom is in the excited state, should be less than the lifetime of the state, and less
than the total time in the sweep outside of the transfers (tg). The last requirement
[51]
?2 ? 1.
?
ensures that the frequency sweep rate ? (Hz/s) is slow enough compared to the Rabi
frequency of the beams ? to excite the atoms adiabatically.
Implementing SWAP in a MOT has additional complications. Since the coun-
terpropagating beams in a conventional MOT are ?+ and ?? they don?t address
the same sublevels. Thus, for a simple F = 0 ? 1 transition, atoms adiabatically
30
excited by one laser beam will not be adiabatically transferred to the ground state
by the other laser beam, reducing the efficiency of SWAP. With the right param-
eters, SWAP can still provide enhancement by suppressing stimulated emission by
the wrong laser beam [53]. When operating this way it becomes even more impor-
tant for the sweep period to be longer than the lifetime since atoms have to decay
through spontaneous emission every SWAP period to avoid anti-cooling. Alterna-
tively, one can switch the polarizations of the beams in between the absorption and
stimulated emission to maintain the full cooling efficiency of SWAP [54].
The complexities of polarization in a grating MOT ameliorate the issues with
performing SWAP in a grating MOT. Grating MOTs have mixed polarizations in
all directions (other than the main input beam), so even while operating the MOT
there is light at the right polarization to adiabatically transfer atoms to the ground
state and impart a cooling force. Fig. 2.7 shows the cooling force in the red grating
MOT as a function of atomic velocity for a traditional triangle wave modulation,
and sawtooth wave modulation.
31
a) b)
Figure 2.7: Modulated red MOT cooling forces. a) The force vs velocity along all
three axes for the red grating MOT with triangle wave modulation. b) The force vs
velocity for the MOT with SWAP. You can see the large enhancement in the cooling
forces along all directions when performing SWAP.
32
Chapter 3: Experimental Apparatus
The experimental apparatus involved in laser cooling and trapping experi-
ments are often complicated, interconnected, and expansive, and our laboratory is
no exception. Our apparatus is a combination of new and improved elements, and
old elements transferred directly from our old laboratory. In this chapter, I will
discuss the development of the new vacuum systems, our new magnetic field coils,
the variety of laser systems, the optics on the chamber side, and the implementation
of computer control throughout the apparatus. I will provide more detail on the
systems that we redesigned for the new laboratory; Refs. [5, 55, 56] provide details
on the pieces more directly taken from the old apparatus.
3.1 Vacuum System
Neal Pisenti completely redesigned the vacuum system for the new laboratory
to avoid the issues present in the design of the old vacuum system, including the
large size, lack of optical access, and issues with strontium coatings on windows. I
designed and built a high-temperature vacuum bakeout chamber to ensure we were
able to attain optimal pressure for this new chamber. In addition, we integrated a
high flux commercial source into the new design. The old system is described in
33
Refs. [5, 55, 56].
3.1.1 Bakeout Chamber
Working with ultra-cold gases requires a vacuum system in the 10?10?10?11 Torr
range. Lower pressure increases the lifetime of the trapped atoms and also improves
the atom number. However, the rate of desorption of water and hydrogen out of the
materials that make up your chamber, in our case polished stainless steel, limits the
ultimate pressure of the chamber. Thus it is necessary to bake your vacuum parts
at as high a temperature as possible to increase the rate of desorption and pump
out the water and hydrogen quickly. Our bakeout process consists of two stages.
First, before assembly, the parts are baked to a high temperature (over 400 ?C) in
a separate vacuum chamber to desorb the hydrogen from the bulk of the material.
Second, the whole system is baked at a lower temperature after it is assembled and
pumped down to desorb the water at the surface. Both bakes help lower the pressure
since the low temperature does not efficiently desorb hydrogen from the bulk of the
material, while parts of our vacuum apparatus, such as the viewports, cannot be
baked at the high temperature because of the glass to metal seal. The high tempera-
ture is particularly important because the rate of desorption approximately doubles
for each 10 ?C increase in temperature. Additionally, baking in a vacuum will not
create an oxide layer on the surface of the stainless steel parts, which would impede
the desorption. When the old vacuum chamber was assembled, the laboratory could
not perform a high-temperature vacuum bake. Although that chamber had a final
34
pressure of ? 5? 10?11 Torr, we wanted to do a high-temperature vacuum bake for
the new chamber to push the limits of the final pressure.
We designed and built a separate vacuum chamber large enough to accom-
modate each part and capable of reaching high temperatures. Fig. 3.1 shows the
bakeout chamber, it is made of a 14 in nipple with a valve to attach a turbo pump.
We use a chicken wire enclosure as a frame for heater tape and insulation. This
enclosure allows us to hold the temperature and pressure at around 425 ?C, and
1.5 ? 10?7 Torr respectively for weeks at a time, monitored by several external
thermocouples to control the heating and one vacuum thermocouple to monitor the
temperature at the parts that are being baked. We had a proportional integral dif-
ferential (PID) controller to actively stabilize the temperature of the bake, but due
to coupling between the different sections of heater tape, we found manual control to
be more effective. This bakeout chamber has become a useful resource throughout
the Joint Quantum Insitute (JQI) and has since been used by many groups.
The second bake after assembly brought our chamber to ? 150 ? C and was
monitored with a residual gas analyzer and an ion gauge. The pressure of our main
chamber is currently around 1?10?11 Torr, near the base pressure of our ion gauge,
where the reading is not reliable. For example, we discovered during the second
bake that the humidity around the ion gauge cable would affect the reading rather
significantly.
35
Figure 3.1: A photo of the full size assembled bakeout chamber with the insulation
frame.
36
3.1.2 AOSense Strontium Source
The strontium source and initial cooling stages on the old chamber had a few
issues that we wanted to address in the new laboratory. The old apparatus had a
large oven, a transverse cooling stage, and a long Zeeman slower. Together with the
main science chamber, the whole vacuum system took up almost an entire optical
table. In addition, the Zeeman slower was directly opposite a primary viewport
of the science chamber. Over time, strontium coated this viewport, reducing the
efficiency of the Zeeman slower and limiting the use of the viewport for other laser
beams. To address this issue, the old laboratory heated this viewport to reduce the
coating rate, but due to the size of the viewport and proximity to the main science
chamber, this solution was not optimal. The most substantial upgrade to the new
vacuum system is the commercial source, custom-made by AOSense, shown in Fig.
3.2. This source has a much smaller footprint and solves the issue of the strontium
coating.
The oven in the source is a small effusive oven, backed with an ion pump.
Since the oven is heated in vacuum, it can safely ramp up from room temperature
to operating temperature in under an hour, less than half the time of the previous
source. There is a short (< 7 cm) permanent magnet Zeeman slower after the oven.
This slower is a substantial improvement over the previous slower which was 35 cm
long and required additional water cooling. With our new source, the small Zeeman
slower window is heated to a constant temperature of 320 ? C with in-vacuum
heaters, and the window is not part of the science chamber itself (see Fig. 3.2).
37
Figure 3.2: A photo of the AOSense source before it was attached to the science
chamber. you can see the total size of the source including the minimal length of
the Zeeman slower. The progressive angle downwards is also visible, which allows
for the heated window the Zeeman slower beam goes through to be separate from
the science chamber.
38
This avoids the strontium viewport coating that occurred in the previous apparatus.
Following the slower are two, 2D MOTs that progressively angle the atomic beam
downward to 20? from horizontal, and into the vacuum chamber. These 2D MOTs
have completely internal optics and magnets, and require only a single input laser
beam each. Additionally, the angle means that there will be no flux of atoms into
the science chamber unless the 2D MOTs are on, allowing for an effective atomic
shutter, replacing our old pneumatic shutter. The tube connecting the source to the
main chamber is 10 cm long has an inner diameter of .6 cm, providing differential
pumping and maintaining the low pressure of the main chamber. The oven provides
an atomic flux of 3.4 ? 1012 s?1, at our current operating temperature of 420 ?C,
but can be operated at up to 540 ?C for a higher flux. Other groups who use
AOSense sources have had issues with clogs, they advised us to ramp our source
all the way down to room temperature at the end of the day, and that operating
at an intermediate temperature will increase the chance of a clog. Since the oven
reaches its operating temperature so quickly, operating at an intermediate value is
not necessary.
3.1.3 Main Chamber
The final issue with the old chamber was the lack of optical access. One main
viewport was coated with strontium, one main port to the chamber was used for
the input of the atomic beam, and the mini viewports were blocked by the bucket
windows on the top and bottom, rendering them useless. Our new science chamber,
39
shown in Fig. 3.3, is similar to our previous chamber, but with significantly more
optical access.
Figure 3.3: Photo of the chamber, before the source was attached or the breadboards
were assembled.
The science chamber is based off a modified spherical square base from Kimball
Physics. It has two recessed 2.69 in viewports on the top and bottom. These bucket
windows hold our MOT bitter coils, and a high-resolution imaging system, and they
40
were designed in combination with the chamber so the mini viewports have line of
sight to the atoms. There are three CF 4.5 in viewports, four CF 2.75 in viewports,
and fifteen mini CF 1.33 in viewports. Two of the mini viewports opposite each
other are specifically antireflection (AR) coated for the Rydberg laser wavelength
(317 nm), this laser is mentioned in Sec. 3.3.6. Only one mini flange doesn?t have a
viewport and is instead used for the input of the source. The source is supported by
2 in diameter posts, and clamped in place with added rubber to avoid torque on the
narrow connection to the main science chamber. There is a CF 6 in flange off of the
science chamber going to a cross that holds a final CF 6 in viewport, a Nextorr-1000
non-evaporable getter (NEG) pump, and a 75 L/s ion pump with a port for a turbo
pump. The turbo pump is only used for initially pumping down the chamber and
is disconnected when we are operating the experiment. Our vacuum chamber has a
pressure of ? 1?10?11 Torr, which is low enough for any science direction we might
take. Two custom breadboards are mounted around the chamber, one slightly below
the center of the chamber so that standard Thorlabs optics are centered, and one
breadboard above the chamber with cutouts for vertical optics.
3.1.4 Grating Chamber
The vacuum chamber for the grating MOT experiment was assembled rather
quickly and had a few iterations before we arrived at our current configuration. The
main part of the chamber is based on the design in Refs. [47, 57], and shown in
Fig. 3.4. Most of the apparatus is constructed within a nitrogen-filled glove bag
41
to avoid oxidation of the strontium. The strontium source consists of a 3D printed
titanium dispenser. The dispenser is a small cylinder, 5 mm in diameter, and 13 mm
long, one end of the cylinder has a cap that can be opened to load it with a few
mg of strontium, and closed tightly once fully assembled. The cylinder has a tab
on both ends to connect to the current feedthroughs and a small slit to generate
the atomic beam of strontium. This dispenser is generally operated at 12 A, and
gets up to the operating temperature in a few minutes, but we have pushed this up
to 14 A for short periods to increase the flux. Underneath the dispenser is a NEG
capaciTorr Z100 pump, which is reactivated every time we open the chamber to air.
Previous iterations of the apparatus had an atomic shutter, this was made from a 3D
printed titanium piece and controlled through an actuator on a ?micro? flange. The
actuator turned an axle, which would rotate the titanium piece above the dispenser
to block the atomic flux, or to the side to allow for flux. In practice, the shutter
would consistently snag on pieces within the chamber, often getting stuck fully or
halfway closed, and we would have to break vacuum to fix it. Additionally, the
micro-flange is difficult to seal and the hardware often strips. We decided to scrap
the shutter due to these issues, however, the flange remains as it is part of a custom
piece, and thus we have to work around this flange when designing coils or magnet
holders.
Above the dispenser is the differential pumping tube. The tube is 3 cm long
and has a 3 mm diameter, which corresponds to an N2 conductance of .11 L/s.
This isolates the trapping region from the source. In addition, since the magnetic
field extends into this region, the length of the tube acts as an effective Zeeman
42
a) b)
Heated Viewport
Grating Chip Atom Cloud
Current 
Feedthrough Dierential
Pumping Tube
Failed Shutter Flange Getter Pump
Dispenser
Figure 3.4: Renders of the final grating vacuum chamber. a) An external view of the
chamber, the getter pump sits below the dispenser which is supported by the current
feedthroughs. The vacuum tee, and connected ion and turbo pumps are not shown,
nor are the coils. b) A cutaway to show the internals of the vacuum chamber. The
dispenser sits below the differential pumping tube and grating mount. The atom
cloud sits above the grating at the center of the trap.
slower [58? ]. The machined part with the differential pumping tube also acts as a
mount for the grating chip. This piece was machined out of aluminum in the original
apparatus, however, we found that this created strong eddy currents, which made
the field switching between the blue and red MOTs impossible. We had the piece
re-machined out of titanium, which has a resistivity 15 times that of aluminum, and
this solved the eddy current issue.
The grating is held in place with a clamping piece (also remade out of tita-
nium). Details about the grating are described in detail in Ref. [58] and Sec. 3.4.3.
Three 2.75 in viewports provide optical access to the grating chamber.
The top viewport was placed directly on top of the cube in the original ap-
paratus, however, over time we noticed strontium build up on this viewport. We
43
replaced this viewport with a heated viewport (VHW-150-G), which is heated to
? 300 ?C. We have not seen strontium buildup on this new viewport, and it has not
increased the pressure in our chamber significantly; however, we had to add a 3 in
nipple to the top of the chamber to accommodate the larger radius.
One port of the grating chamber is connected to a vacuum tee which has a
75 L/s ion pump attached and a valve to connect a turbo pump. All together in the
final iteration of the chamber, after multiple leaks and redesigns, the apparatus has
a pressure of 1? 10?10 Torr.
3.2 Magnetic Fields
Both the main apparatus and the grating MOT apparatus use approximately
quadrupole fields to generate the magnetic field gradient for the trap, and shim coils
to make fine adjustments to the field zero. We have used a variety of techniques
to generate these magnetic fields in the laboratory, from simple coils, traditional
water-cooled coils, bitter coils, and permanent magnets. This section will expand
upon these different techniques.
3.2.1 Bitter Coils
One of the main changes to the new system is the introduction of bitter coils
to replace the wound coils of the previous setup. Our bitter coils are based off of
the design in Refs. [59, 60]. They consist of flat copper disks stacked on top of each
other and are then water-cooled in parallel between the plates that make up the coil.
44
This allows for more efficient water cooling than running the water serially through
the coil wire; thus bitter coils are often able to run at higher currents than wound
coils. We want the ability to operate at high fields for possible future experimental
directions, specifically implementing orbital Feshbach resonances, similar to Ref.
[61].
Fig. 3.5 shows a diagram of our bitter coil design. Between each disk is an
insulating polyester layer with channels cut into it to allow water to flow and a
small copper section to allow current to flow between the disks. Each of our coils
are formed by two concentric disk stacks, with a thicker rigid brass base to provide
structure. Current flows from the bottom of the outer stack to the top, where a
special copper layer connects the two stacks and allows the current to flow down
the inner stack (depending on the polarity). On top of each coil is a stainless steel
water manifold that distributes water through the various channels and defines the
direction of flow.
The construction of the bitter coils revealed many issues with the design.
Initially, the plan was to seal the coil closed using threaded rods, two of which would
be brass and used as current inputs. In practice, the brass rods would easily break
before the coil was sealed, so the final design includes copper bars as the current
connection. Getting the coil to seal watertight also posed problems. The Teflon
insulation alone was not enough to seal the coils, so we added rubber gaskets that
outlined the channels in the Teflon. With the tight tolerances of our design, these
gaskets could easily shift slightly out of place and spring a leak, making assembly
quite difficult. In the end, we also insulated the entire coil with silicone rubber
45
140 mm
Figure 3.5: Exploded diagram of the bitter coil. The black lines show how current
flows up and down the concentric coils. The blue lines show an example of how
water flows through an insulating layer between the plates.
sealant to make sure it would not leak. Lastly, since the water distribution manifold
is stainless steel there was a connection to ground through the optishield we put
in the water. While we were able to increase the resistance to ground by using a
non-conductive version of optishield for corrosion prevention, we also had to isolate
the control electronics to prevent current from flowing through ground when the
circuit was supposed to be closed. The coils have an inductance of ? 200 ?H and
give a field gradient per current of approximately 1 G/cmA axially when mounted
on the chamber. Fig. 3.6 shows a photo of a completed bitter coil.
The control electronics for this system were designed by Ananya Sitaram and
Alessandro Restelli. They use a metal?oxide?semiconductor field-effect transistor
(MOSFET) bank with feedback from a current sensor with varistors for protection.
46
Figure 3.6: Photo of a completed bitter coil before we insulated it with silicone
rubber sealant. Here you can see the total size of the coil and the water distribution
manifold, the top flange is 8 in in diameter. The rubber gaskets provide insulation
between the manifold and the sealing screws. The copper bars we use as current
inputs, and the final silicon sealant are not shown.
The first iteration had an insulated-gate bipolar transistor (IGBT) to further de-
crease the switching time, the electronics for that section would break from time to
time, and it made no difference in the MOT transfer efficiency, so we bypass it in the
final design. Currently, the coils only operate in an approximately anti-Helmholtz
configuration, but there is a break in the welding cables that supply the current for
future integration of an h-bridge to operate in a approximately Helmholtz configu-
ration.
47
3.2.2 Grating Coils and Magnets
We went through several configurations of coils and magnets for the grat-
ing MOT apparatus. The first iteration of the apparatus used purely permanent
magnets and is described in Ref. [58]. These magnets were arranged somewhat
asymmetrically to make room for the shutter flange but were still able to create a
quadrupole field with an axial gradient of 62 G/cm. Unfortunately, this configura-
tion is static and doesn?t allow us to switch to the red MOT gradient.
The second configuration incorporated both magnets and coils. Half of the
magnets were removed and 99 turn coils were wound above them. The coils supplied
approximately 30 G/cm axially when fully on and were controlled with a bipolar
MOSFET controller. This allowed us to operate the coils to add to the gradient of
the permanent magnets and reach the blue MOT gradient, or switch the currents
and operate them to cancel the permanent magnets and reach the low gradient of
the red MOT. The bipolar coils with separate control also allowed us to use the
same coils as the shim coils for the vertical direction. In practice, we ran into a
few issues with these coils. We were running at the maximum voltages of our power
supplies and were seemingly still not able to fully cancel the gradient of the magnets.
However, this was most likely due to the eddy current in the differential pumping
tube rather than an issue with the coils themselves. Furthermore, these coils were
not water-cooled, so they would overheat on the order of 30 s. While this could
be mediated by simply adding 1-2 s to the duty cycle of the experimental shots, it
made continuous alignment and optimization of the blue MOT difficult. We decided
48
to scrap these coils in favor of more traditional water-cooled coils, but I think this
configuration would work in the future given that we have solved the issue with the
eddy current.
The final configuration has two sets of coils: one set of large water-cooled coils
to supply the blue MOT gradient and one small set of coils to supply the red MOT
gradient. The large coils are very similar to the coils from the apparatus in the old
laboratory and supply approximately 0.55 G/cmA axially. We operate these coils
at 100 A for the blue MOT, to get an axial gradient of 55 G/cm. Unfortunately, our
bipolar controllers cannot operate at this current, so we use an IGBT to completely
and quickly turn these coils off and use a separate, more carefully controlled, set
of coils for the red MOT. This smaller set of coils is not water-cooled and gives us
0.2 G/cmA. We operate these coils with the bipolar controllers allowing us to use
them as the vertical shim as well. Near the max allowable current of the controllers,
they give us the 4 G/cm axial gradient we use for the red MOT. We would like
these coils to be a bit stronger, but cannot wind any more turns due to the current
geometric constraints of the chamber, a limited redesign would probably be able to
improve on this.
3.2.3 Shim Coils
Both the main and grating apparatus use shim coils to move the zero of the
MOT field gradient or zero the field in a dipole trap. There are shims along all three
axes on the main chamber, six coils in total. The vertical coils have 78 turns each,
49
and the horizontal coils have 60 each, except for the coil on the side of the chamber
that connects to the pump cross. Since this coil is wrapped around a larger flange
than its partner, it has 31 turns to compensate for the larger size and maintain a
flat magnetic field profile in the center of the chamber. All three sets of coils are
run with computer-controlled power supplies.
The grating MOT chamber only has shim coils in the transverse directions;
the axial direction is shimmed with the MOT coils. Three of these coils are wound
around 3D printed cage mounts and have 30 turns each. We have computer control
only in the y direction (along the imaging axis), while the x shim is manually tuned.
In both systems the shims are not powerful enough to have much effect on the
magnetic field gradient of the blue MOT but can move the zero of the red MOT
field gradient significantly.
3.3 Laser Systems
Working with strontium requires many different laser systems. In the labora-
tory, we have one optical table for the blue lasers and dipole lasers, and another for
all the red lasers, including the repump lasers. The last optical table is reserved for
the vacuum systems and a few miscellaneous lasers that will only be discussed in
brief. This section will be a relatively brief overview of these laser systems, as many
of the lasers were transferred without alteration from the old lab and are described
and diagrammed in Refs. [5, 55, 56].
50
3.3.1 Blue 461 nm System
The first cooling stage of strontium, the blue MOT, uses light at 461 nm. Our
461 nm system consists of two lasers. The ?spectroscopy? laser is a Toptica DL pro,
which we lock to a hollow cathode lamp with polarization rotation spectroscopy [62]
on the 1S0 ?1P1 transition in 88Sr. This lock is sensitive to power variations and
magnetic field, so we stabilize the power of the spectroscopy beams, and cover the
hollow cathode lamp with mu-metal to provide magnetic shielding. The old setup
had an optical isolator (from a different laser system) nearby the cell, the field from
that isolator was written onto the mu-metal so when it was removed this field made
the lock unusable, and the mu-metal had to be replaced. The power lock is even
more important, as small shifts in power will significantly move the frequency, and
it only works well when the maximum power is significantly higher than the lock
point. Even with the magnetic shielding and power lock, the frequency of the lock
can shift by a few MHz over an hour or two. The laser is locked 100 MHz below
resonance using an acousto-optic modulator (AOM) that shifts the spectroscopy
laser beam up in frequency.
The main cooling laser is a new TA-SHG pro, which can output ? 900 mW of
light. Currently, this light is split into four main paths with polarizing beamsplitter
(PBS) cubes, light for the 2D MOTs, the Zeeman slower, the main MOT beams,
and the imaging probe beam with approximately 20 mW, 25 mW, 9 mW per beam,
and 300 ?W out of the respective fibers. These powers can easily be adjusted with
?/2 waveplates before each cube. The light for the 3D MOT is coupled into a 1 to
51
3 fiber splitter, and power locked using the attenuation of that path?s AOM, and a
photodiode on the experiment table. Fig. 3.7 is a diagram of this system.
199 MHz To Grating Probe
AOM ?/2 Shutter Laser
To Beatnote
Fiber PBS Beam Stop
Figure 3.7: Diagram of the blue laser system. The mirrors are omitted for simplicity.
Each arm of the blue laser system has its own AOM, and waveplate to control the
power. Optics components from Ref. [63].
The unshifted light from the zeroth order of the 2D MOT AOM is used to lock
the cooling laser below the spectroscopy laser with a beatnote lock. This beatnote
lock is somewhat noisy, and while it doesn?t seem to significantly effect the loading
of the blue MOT it can make fluorescence imaging less consistent. The beatnote
is ? 100 MHz for 88Sr and increases according to the isotope shifts for the other
isotopes, listed in Tab. 1.3, so that the unshifted laser is locked ? 200 MHz below
resonance. An individual AOM shifts each arm of the system to its respective
frequency relative to the lasing frequency: +160 MHz for the 2D MOT beam, -
388 MHz for the Zeeman slower beam, +155 MHz for the main MOT beams, and
52
To 2D MOT
173 MHz
To Zeeman Slow - 388 MHzer
To 3D MO 171 MHzT
To Main Probe
+200 MHz for the probe beam. This setup is changed and simplified from the old
apparatus, in particular, the frequencies have been shifted to remove a double pass
AOM, and decrease the amount of wasted power.
For the final iteration of the grating MOT system we use this laser for its high
power. When running the grating MOT the fiber splitter is replaced with a single
fiber going to the grating MOT beam input, and almost all the power is put into this
arm giving ? 200 mW after the fiber. Some power is still reserved for the beatnote
lock, and probe beam. We can switch between the main apparatus probe, and the
grating MOT probe using a waveplate and cube.
3.3.2 Repump Lasers
Our apparatus uses two different repump frequencies as mentioned in Sec.
2.1.2. We have two homemade, extended cavity diode lasers (ECDLs) in the Littman-
Metcalf configuration to provide light at these frequencies. Both these lasers are
locked through slow feedback using the reading from the wavemeter (High Finesse,
WS7). We have a Python program that reads the wavemeter and provides this feed-
back. Yanda Geng remade this program based on the old Labview code to operate
more quickly and fully separate our lab from Labview. The approximate locking
frequencies are shown in Tab. 3.1. The wavemeter has a precision down to 1 MHz,
but can drift by a few MHz over a day.
The 679 nm laser is sent through an AOM to provide more precise control of
timing and future imaging out of the 3P0 state. The atom number is more sensitive
53
Repump Frequencies (THz) 679 nm 707 nm
Bosons 441.3326 423.9136
Fermion 441.3301 423.9130
Table 3.1: The wavemeter values of the repump laser frequencies for the bosons and
the fermion. Note that the wavemeter drifts day to day, and the 679 nm laser is
shifted by 207 MHz above the wavemeter measurement by an AOM.
to the 707 nm repump which has to be broadened to cover the hyperfine transitions of
87Sr, as mentioned in Sec. 2.1.2. For both of these reasons, we need a larger amount
of power in this repumper. To achieve this we injection lock another diode laser
seeded with the homemade ECDL to provide sufficient power. We add sidebands to
the output of this laser with a fiber electro-optic modulator (EOM), which is driven
at many different frequencies, as discussed in Sec. 4.2, when we are trapping 87Sr.
3.3.3 Red 689 nm System
Operating the red MOT requires a laser at 689 nm with a narrow linewidth
to address the narrow transition. This narrow linewidth is achieved by locking a
laser to an Ultra Low Expansion (ULE) glass cavity from Advanced Thin Films
(ATF-6010-4) with a cavity linewidth of ? 6 kHz, I will refer to this laser as the
?cavity? laser. The cavity laser is sent through a double pass ? 1 GHz AOM. The
double passed diffracted order is sent through a fiber to a free-space EOM to lock
the laser to the cavity with a Pound-Drever-Hall (PDH) lock. This cavity and lock
are described in detail in Ref. [55]. The free spectral range (FSR) of the cavity is
approximately 1.5 GHz, so together with the double passed AOM, this allows us to
have full flexibility in the locking frequency of the cavity laser. The zeroth order
54
of the AOM is sent to a beatnote lock with an ECDL laser. We call this ECDL
laser the ?long Steck? since it is based on the design by Dan Steck [64] (and is a
bit longer than a similar laser in the laboratory we call the ?short Steck?). In the
previous system, this beatnote was measured with a large network of fiber splitters
that picked off light for the wavemeter and measured beatnotes for a variety of
lasers that are no longer in use. We replaced this setup with a single fiber splitter to
conserve power and simplify the system. Additionally, we added computer control
to the beatnote frequency, a new feature of the current apparatus.
The long Steck is used as the main cooling laser for the bosonic isotopes. The
beam path to the experiment has an AOM and shutter to shift, shutter, broaden,
and control the power of the beam. Then the light is sent through a 1 to 4 fiber
splitter to provide the light for the MOT beams. Typically we have ? 600 ?W per
beam for the red MOT.
As always, for the fermion, there are additional complications. In this case,
the light from the long Steck is used as the stirring laser, mentioned in Sec. 2.1.3,
as opposed to the cooling laser. Some of the light from the cavity laser is sent to
inject another diode laser which we then use for the cooling and can provide more
power, ? 2 mW per beam.
For the grating MOT, we replace the fiber splitter, which splits the main red
MOT beams, with a fiber that seeds a Toptica BoosTA tapered amplifier (TA).
The fiber-coupled output of this TA supplies ? 90 mW of power when operating
at its maximum current, and follows the broadening of the red laser well. While
searching for the fermionic red MOT in the grating system we have used a few
55
different configurations informed by our simulations. We tried seeding the TA with
the injection laser, and using it as the cooling laser, but with opposite circular
polarization. We also tried seeding the TA with the long Steck as the cooling laser
and operating the injection on the F = 9/2 ? 7/2 transition for stirring with the
same circular polarization for both. Lastly, we are trying the same configuration
but seeding the TA with both the injection, and long Steck combined onto a fiber
splitter.
3.3.4 Clock Laser System
We need another narrow laser, operating at 698 nm to address the clock tran-
sition. This laser is another long Steck style laser, which has one beam path with
an AOM going to the experiment and another which is sent through a fiber EOM
for a PDH lock to another ULE cavity. This second cavity is more carefully en-
gineered and sits on a vibration isolation stage and inside a sound-insulated box.
A very detailed description of the cavity is presented in Ref. [5], although a new
box had to be made for the new laboratory. The fiber EOM is driven with two
different frequencies one at 18 MHz for the PDH lock, the other at ? 800 MHz to
add sidebands to the laser. We lock one of these ? 800 MHz sidebands to the cavity,
rather than the carrier, This gives us the flexibility to lock the frequency of the laser
away from a cavity resonance. Previous iterations of the clock laser system had a
pre-stabilization cavity, to initially narrow the laser before the ULE cavity, but this
was scrapped as it didn?t improve the lock. The beatnote between the red cavity
56
laser and the clock laser is ? 200 Hz wide, putting an upper limit on the linewidth
of the clock laser.
3.3.5 Dipole Trap Laser Systems
We have two dipole trap lasers. The first is a 1064 nm, 30 W laser that was
transferred directly from the old laboratory. This laser system had four beam paths
for different dipole traps, but we have only used two of them in the new laboratory.
Each path has an AOM and mechanical shutter, for quick and complete extinction
respectively. We send ? 2 W to the science chamber for each dipole trap, but this
power can be lowered once the traps have been aligned.
The second dipole trap laser is an M squared SolsTiS ti-sapphire laser. This
laser has a wide frequency range and can provide ? 4 W at 813 nm, the magic
wavelength for the clock transition. We had a lot of issues with this laser in the
past. Previously we pumped the laser with a 532 nm IPG laser, but multiple of
these lasers failed, and eventually, the ti-sapphire laser failed as well. We now have
a new system with a pump from M squared which we hope will be more stable and
are working on getting the magic wavelength dipole traps working.
3.3.6 Miscellaneous Lasers
The laboratory has a few other lasers that I would like to mention here briefly.
First is the old blue laser system from the old apparatus. It is another TA-SHG
Pro from Toptica, similar to our current system, but is nearing the end of its life,
57
and can only put out ? 300? 400 mW. We previously used this system to run the
grating MOT, but after finding that we were power limited we switched to the new
laser system instead. The current plan is to use this system for blue detuned dipole
traps, specifically to make the walls of a box trap, discussed in Sec. 6.1.
The second laser is a Toptica TA-FHG Pro, this is a fourth harmonic genera-
tion laser at 317 nm laser to excite Rydberg levels off of the clock state, as discussed
in Sec. 6.2. This laser sits on the table with the vacuum chamber due to the dif-
ficulty of efficiently fiber coupling ultra-violet light. Currently, this experiment has
been put on hold, so the laser isn?t currently in use.
The final laser I would like to discuss is the frequency comb in the lab. The
comb is from Menlo Systems (FC1500-250-WG) and is a resource for the JQI as
a whole. The comb acts as an absolute frequency reference by translating op-
tical frequencies to a combination of three radio frequencies: the repetition rate
(? 250 MHz), the offset frequency (? 35 MHz), and the beatnote between the clos-
est mode of the comb and the laser being measured. The comb was a particularly
valuable resource for the work in Ref. [5], but is also useful as a way to get lasers
close to the narrow transitions in strontium. The comb has amplifiers to provide
power centered at 688 nm, 698 nm, 780 nm, 1112 nm, and 1166 nm, although the
688 nm and 698 nm amplifiers can?t be operated simultaneously. In addition, there
is a photonic crystal fiber to broaden the laser and provide power over a wide range
of wavelengths from 640-900 nm.
58
3.4 Experiment Side Optics
3.4.1 Main Experiment Optics
So far the main experiment requires optical access for blue and red MOT
beams, two dipole trap beams, and two imaging paths, with the future addition of a
blue detuned dipole trap. Figs. 3.8 and 3.9 show the optics on the vacuum chamber
that are used to shape and combine these beams.
The blue MOT beams are collimated and their polarization is set and cleaned
up with waveplates and PBS cubes. One of the blue MOT beams has a pickoff for
power stabilization. The red MOT beams have a similar setup and are combined
with the blue beams on a dichroic mirror. The horizontal blue and red MOT beams
enter the chamber at 45? relative to the imaging axis and are rotated to circular
polarization with dual-color ?/4 waveplates. These waveplates are actually ?/4 for
461 nm and 3?/4 for 689 nm, so the red beams have an extra ?/2 waveplate to
correctly account for this. There are mirrors to steer the blue and red MOT beams
separately, as well as to steer them together. The blue and red beams have 1/e2
power distribution radii of ?6 mm and ?3 mm respectively giving max I/Isat of
?.4 for the blue beams. We previously placed a telescope in these beams to further
expand them, but we found that we did better by taking it out, however, it?s likely
the atom numbers could be improved by further expanding these beams. Simple
retro-reflecting mirrors create the opposing beams, with waveplates to maintain the
correct circular polarization. The repump fibers are mounted to the chamber with
59
60
813 / 1064 nm
689 nm
461 nm
200 mm
Mirror Lens Picko Beam Stop ?/2 ?/4 PBS Dichoics 400 mm
Camera Fiber Flipper Mirror Photodiode
Figure 3.8: The horizontal optics for the main experiment. This diagram is not to scale. Optics components from Ref. [63].
Future Blue Dipole
Blue MOT
689 nm
461 nm
Future Dichoic Probe
Mirror Lens Picko ?/2 ?/4
Red MOT
Camera Fiber PBS Dichoics
Figure 3.9: The vertical optics for the main experiment. The vertical beam paths
have the high resolution imaging stack. The 50% opacity dichroic and pickoff are for
the future implementation of vertical red and blue detuned dipole traps respectively.
This diagram is not to scale. Optics components from Ref. [63].
Thorlabs cage mounts, and viewport washers and adapters machined at the JQI,
and enter on two of the mini viewports.
The x-direction dipole trap simply enters the chamber perpendicular to the
horizontal imaging, this beam is focused onto the atoms with a 400 mm focal length
lens, and the beam is shaped to provide a ? 30 ?m waist. The cross dipole trap
counter-propagates with the horizontal imaging probe beam. It is combined with
this beam on a dichroic mirror and is focused onto the atoms to create a ? 30 ?m
trap with a 200 mm focal length lens, which is also used as the first lens of the
imaging system. On the other side of the chamber, there is a flipper mirror to switch
between imaging and retro-reflection of the dipole beam to create a lattice. Both
dipole traps have pickoffs to photodiodes for power stabilization. The horizontal
61
imaging system is a simple two-lens system with a magnification of 1.4.
When the data in Ch. 4 was taken the vertical beam paths looked much like the
horizontal MOT paths. Since then we have added optics to allow for high-resolution
imaging and the box dipole traps. Parts of this system are still under construction.
The MOT beams are launched from below the chamber, along with an added probe
beam. On top of the chamber a high-resolution imaging stack, described in Sec.
3.4.2, sits inside the bitter coil. We plan to integrate a dichroic mirror for the
implementation of future 813 nm vertical dipole traps, however, we have had some
issues with fringing from this dichroic, so for now it is replaced with a simple mirror.
After a dual-color ?/4 waveplate a PBS beam splitter separates the MOT beams
from the probe beam. The MOT beams are collimated and retro-reflected, while the
probe beam is sent to the ?PIXIS? (PIXIS Excelon 1024B) camera from Princeton
Instruments. Future blue detuned dipole beams will be combined with the probe
beam with a simple beam splitter.
The AOSense source has beam shaping optics for the 2D MOT and Zeeman
slower laser beams that hang underneath the top breadboard. The 2D MOT beam
is shaped into an elliptical beam with cylindrical lenses, before being split between
the two 2D MOT inputs. The Zeeman slower beam is circularly polarized and sent
through the heated viewport. Fig. 3.10 shows these optics.
62
2D MOT
Atomic Beam
461 nm
Mirror Lens ?/2 ?/4 Zeeman Slower
Fiber PBS Polarizer
Figure 3.10: The optics for the AOSense source. The atomic beam is also shown
including the progressive angles of the 2D MOTs. The internal optics are not shown
as they are proprietary to AOSense. This diagram is not to scale. Optics components
from Ref. [63].
3.4.2 Imaging Stack
For the new strontium apparatus, we designed a high-resolution imaging lens
stack. The design for the stack was inspired by Ref. [65]. This design aims to
create a four lens stack out of stock Thorlabs lenses. We start by putting four lenses
and the viewport of the vacuum chamber into Zeemax and optimizing the imaging
resolution for 461 nm using the surfaces of the lenses. Then we replace one of the
lenses with the closest Thorlabs part and redo the optimization. This process is
iterated until all the lenses and the spacings are fixed and optimal. Fig. 3.11 shows
63
the lens stack and spacings.
The final lens stack is assembled with aluminum spacers that were precisely
machined to match the spacings. Since this lens stack is fitting inside one of the
bitter coils we cut slits into all the spacers, the lens tube, and even the retaining
rings, and filled the slits with Teflon to avoid eddy currents in these parts. The
stack has an NA of 0.29, and a resolution of ? 1 ?m at 461 nm measured on a test
setup.
3.4.3 Grating Optics
The optics on the grating MOT apparatus are less than ideal; space is limited
and they are supported by long posts rather than a proper breadboard. Fig. 3.12
shows a diagram of this setup. The blue MOT beam polarization is cleaned up
with a cube, this cube is useful for alignment as the back-reflected zeroth order of
the grating will be rejected by the cube. The beams are reflected by a dichroic
mirror before the circular polarization is set with a dual-color waveplate, so that
the dichroic doesn?t scramble the polarization. A telescope then expands the beam
before it enters the chamber and is diffracted by the grating. The red MOT beam
has a similar setup, with extra waveplates to provide separate control of the red
and blue polarization. The blue MOT beam has a ?9 mm radius and the red MOT
beam has a ?5 mm radius. In the future, it could be possible to use planar optics
to simplify and shrink the MOT beam shaping system[66].
The grating chip, shown in Fig. 2.2, has three grating sections arranged in
64
LC1611-A
19.23 mm
LB1607-A
33.50 mm
LA1417-A
17.14 mm
LE1418-A
Figure 3.11: The imaging stack we designed for the new strontium apparatus. The
Thorlabs part numbers for the lenses, and the spacings between them are shown.
65
Red MOT
Stirring
F = 9/2 ? 9/2
689 nm
Blue MOT 461 nm
707 / 679 nm
Mirror Lens ?/2 ?/4 Grating
707 nm
Fiber PBS Dichoic
679 nm
Figure 3.12: The optics on the grating MOT. The stirring laser for operating on
the F = 9/2 ? 9/2 transition has it?s own input. The repumps come in on the
viewport opposite the pump tee. The probe and imaging system are not shown and
have a simple linear setup. This diagram is not to scale. Optics components from
Ref. [63].
66
a circle with an inch diameter. The center of the chip has a triangular hole that
sits over the differential pumping tube. Each grating section diffracts 32% of the
blue light into each of the ?1 orders, at an angle of 27? relative to the normal, and
? 37% of the red light at an angle of 43?. With these angles, there is enough overlap
(43%) between the blue and red capture regions for the transfer between the MOTs
to work.
The repump beams are combined on a PBS and sent through the viewport
opposite the pump tee. The probe beam enters the chamber perpendicular to the
input MOT beams and the repumps, and is imaged with a simple two-lens imaging
system with a magnification of 0.25.
3.5 Computer Control
The old laboratory used SetList, a program created with labVIEW, to provide
computer control for the apparatus. Unfortunately, the user base of SetList is be-
coming less active, and the user experience was a bit clunky in the old laboratory.
In addition, we wanted to move away from labVIEW in general, so we decided to
change our computer control scheme for the new laboratory. We transitioned to
the labscript suite of software [67], which had already been in use by Ian Spiel-
man?s groups at the JQI. This software has a more active user base than setlist
and is python-based allowing for easy extensibility. The software suite consists of
four main programs: Runmanager defines the variables for experimental sequences,
Blacs communicates with the hardware and runs each shot, and also allows you to
67
manually control all the hardware when a shot isn?t running, Lyse allows you to
write custom scripts to analyze data from each experimental run as they come in,
and using Runviewer you can view the output of each experimental sequence. The
map of connections between all the devices in the lab is encoded in a python file
called the ?connection table?, and the logic of the experiment itself is written in
another python file using the labscript library. All the data for each experimental
shot is stored in an hdf5 file. This hdf5 file includes all the parameters of the shot,
the experimental logic, any results (including images), as well as the results of your
analysis scripts. The file is created with Runmanager, sent to Blacs where the shot
is run on all the devices, and the results are written to it, and finally, it is sent
Lyse for analysis. The organizational scheme of this new software keeps everything
organized and in one place, while simultaneously allowing us to split experimental
control over multiple computers. In addition, there are tools to automatically run
optimization algorithms using the M-Loop python package. While these algorithms
rely on a stable system, they have been useful in searching the large parameter space
of the red MOT.
Labscript is built off of the use of programmable ?pseudoclocks?, with one
?master pseudoclock? to provide the timing for each experimental shot. previously
we used a pulseblaster for our master pseudoclock, but Ananya Sitaram and Alessan-
dro Restelli designed an alternative, called Jane with more channels, and built-in
digital buffers [68]. This master pseudoclock is used directly for TTL signals to a
variety of devices, in addition to clocking ?intermediate devices? such as National
Instruments (now NI) cards, and Arduinos. The laboratory uses three NI cards (NI
68
PCI-6733, NI PCIe-6361) that have both digital and analog channels.
The AOMs are controlled in a variety of ways, their drivers, designed and
built at the JQI, have: a digital input that turns the radio frequency (RF) power on
and off, an analog input that controls the attenuation of the RF power, an analog
input to control the frequency of the voltage controlled oscillator (VCO) inside the
driver, and a TTL to control whether the frequency is determined by the VCO or
an external RF signal. Some of these inputs are unused by some AOM drivers, and
unfortunately, depending on the version of the driver, the RF TTL or attenuation
might have the opposite sign.
The laboratory also has a variety of direct digital synthesizers (DDSs) and
phase locked loops (PLLs) that are controlled using Arduinos. These are used for
beatnote locks, and to control the broadening of the 707 nm repump laser. Each
Arduino takes a clock TTL and a serial connection that sends it commands. The
code on these Arduinos was originally written to work with SetList, but we were
able to get them to work nicely with labscript with a bit of modification and a new
labscript device script. They are capable of controlling multiple DDSs by setting
the frequency and doing frequency ramps that are clocked internally by the DDS
chip to avoid any lag from the Arduino. Often when controlling beatnote locks
the frequency that the DDS is set to is multiplied by a PPL before the beatnote
comparison happens, so this multiplication has to be factored in when programming
an experimental shot. There are still a few DDSs which we have not yet implemented
into labscript, such as the DDSs that control the clock fiber EOM sidebands, or the
?1 GHz AOM for the red cavity laser, but future implementation of computer
69
control for these DDSs could be useful.
There are three cameras we use in the lab that are controlled by labscript. The
?Flea? (FL3-U3-13Y3M) currently images the grating MOT, and the ?Grasshopper?
(GS3-U3-28S4M) currently provides the horizontal imaging, both of these cameras
are from Point Grey, and work with NI drivers. The last camera is the PIXIS, which
we are implementing into the vertical imaging system, and has its own software and
driver. Labscript has a very nice camera class for cameras that integrate with NI
software. The class has a widget in blacs that allows for manual continuous imaging,
which has been a valuable resource for our optimizations of the blue MOT. The
connection table defines separate attributes for the cameras in manual and buffered
(during a shot) operation. Unfortunately, we had to write a new labscript class for
the PIXIS, and so far have not been able to get continuous imaging to work.
To broaden the red cooling and stirring lasers, as well as perform SWAP, we
use a Stanford Research Systems (SRS) function generator (DS345). The function
generator has a TTL to turn the output on and off, and an analog input to control
the amplitude of the oscillation. We switch between sawtooth and triangle waves
manually. This signal is added to analog signals for the VCOs of the cooling and
stirring laser?s AOM drivers in order to broaden the lasers.
The remaining digital channels control: laser shutters, camera triggers, the
IGBT in the grating MOT coil circuit, turning on and off the grating red MOT coils
current lock, and turning on and off the bitter coil current lock. The remaining
analog channels control: shim coil currents, the grating blue MOT coil current, the
set point for the top and bottom current of the grating red MOT coils, the set point
70
for the bitter coil current, and the set point for the blue laser and dipole beam power
locks.
In conclusion, we have built a new and improved apparatus for ultracold stron-
tium experiments. Our new apparatus has a high flux source, good optical access,
and coils capable of providing high magnetic fields. We have eliminated the issue
of strontium coating viewport windows from the old apparatus, and significantly
decreased the size. Our old and new laser systems are fully integrated into the ex-
periment and everything is managed through a more streamlined, and extendable
computer control system.
71
Chapter 4: Validations
This chapter walks through the process of trapping strontium atoms in the
various traps, the techniques we used to initially find appropriate trap parameters,
and the result of each cooling and trapping stage. These traps include the blue and
red MOTs of both the main apparatus and the grating system, as well as the dipole
traps of the main apparatus. In addition I will discuss the imaging techniques we use
to measure the trap properties. The experimental parameters and results presented
here are typical values found empirically, but they can change significantly day to
day, or depending on what we are specifically doing in the laboratory. I will use
?field? in this chapter to refer to the magnetic field unless otherwise stated.
4.1 Imaging
We extract all the information we need from our trapped atoms through imag-
ing. From measuring atom number, temperature, and peak optical depth for opti-
mization, to techniques for finding initial parameters that will trap any atoms at
all, imaging is fundamental. In our laboratory we use two simple forms of imaging:
fluorescence imaging, and absorption imaging.
Fluorescence imaging is the simpler of the two; a probe beam of resonant light
72
interacts with the atoms, and the resulting fluorescence is captured, off the probe
beam axis, by an imaging system, and focused onto a camera. In practice it is
difficult to extract accurate atom numbers from fluorescence imaging, as the light
collected depends on the solid angle of the optical system, and how that solid angle
overlaps with the dipole radiation pattern. We mostly use fluorescence imaging for
initial alignment and optimization of our blue MOTs, as fluorescence images can be
taken continuously while the MOT is active. However, in our system, the blue MOT
beams are acting as our probe for fluorescence imaging, as we do not have a dedicated
fluorescence probe beam. This is not ideal as the MOT beams are detuned from
the resonance. Thus, optimization of the MOT detuning using fluorescence imaging
will not be accurate, as detuning closer to resonance will increase the fluorescence
even with the same number of atoms. Despite this, in the grating MOT system
fluorescence imaging has been a useful tool, as it has a better signal to noise than
the absorption imaging in our apparatus. More details on how we image our clouds
in the grating MOT system are described in Sec. 4.5 and 4.6.
Absorption imaging is the more accurate and versatile tool. To take an absorp-
tion image a low power resonant probe beam is sent along the axis of the imagining
system where it is absorbed by the atoms. The light that is not absorbed is imaged
onto a camera. This first image (Iatoms) is proportional to the transmitted intensity.
Then a second image (Iprobe) of the probe beam is taken once the atoms have fully
left the interaction region. This allows you to account for the probe beam power,
and spatial variations in the beam profile. We also take a background image (Ibg)
with the probe beam off. Consider the amount of light dI that will be absorbed by
73
Figure 4.1: An example of the three images we take when doing absorption imaging,
and the optical depth they combine to create. The probe image is in the upper left
which is taken once the atoms have left the frame. The atoms image is in the upper
right, and you can see the shadow of the atomic cloud. The background image is in
the lower left. Finally the optical depth, the combination of these three images, is
in the lower right.
a beam propagating a distance dz through a cloud of atoms with density n. If we
take I  Is this drop in intensity becomes [1]:
I
dI = ?n(x, y, z)?0 dZ
1 + (2?/?)2
where ?0 is the on resonance cross section, and assuming a two level system ?0 =
74
3?2/2?. If we take ? = 0 and integrate over z we are left with
( ) ( )
? If ? Iatoms ? Ibgln = ln = n
? 2D
(x, y)?0,
Ii Iprobe Ibg
Where n2D is the density of atoms integrated along z. We then define the Optical
Depth (OD) as n2D?0. Adjustments to this analysis have to be considered when
probing with higher powers, with detuned beams, and when dealing with something
other than a simple two-level system. For the bosonic isotopes of strontium the
extension beyond a two-level system does not change much, but the fermion requires
a more complicated analysis of the absorption cross section, described in detail in
Ref. [56]. This analysis shows that the highest cross section, and best absorption,
occurs with a circularly polarized probe beam for 87Sr. Once the OD is calculated
getting the atom number is a simple matter of integrating the OD over the extent
of the cloud in x and y.
If we assume the atoms in the trap are in thermal equilibrium, then the velocity
distribution in one dimension is a Maxwell-Boltzmann distribution:
( )1/2
m ?mv2x
f(vx) = e 2kbT ,
2?kbT
where kb is the Boltzmann constant. As the cloud of atoms expands in time of
flight the spacial distribution will be a convolution of this distribution with the
initial spacial distribution of atoms in the trap. Thus we generally fit the images
to a Gaussian function, which becomes a better assumption as the time of flight
75
increases. Ideally we would fit to a 2D Gaussian function, but this is generally slow,
and we have found fitting to 1D slices in both dimensions of the image to be efficient
and accurate. We do an iterative fit, where the slice of the image we fit to along
one axis is determined by the center found from the previous fit to the orthogonal
axis. This ensures the fits find the true center of the cloud in the 2D image.
Once we are able to take images in time of flight we can calibrate the mag-
nification of the imaging system by dropping the MOT, and watching it fall. By
measuring the acceleration of the MOT downwards, and matching it to gravity we
can convert between pixels on the camera and meters. Currently, the horizontal
imaging system on the main apparatus has a conversion of 3.4?10?6 m/pixel (mag-
nification of 1.4), and the imaging system on the grating MOT apparatus has a
conversion of 19.4 ? 10?6 m/pixel (magnification of 0.25). Looking at the MOT
after a time of flight also gives us a measure of the temperature. After a time t the
width of the cloud becomes
w2
kbT
= w20 + t
2, (4.1)
m
where w0 is the initial width of the cloud. Thus fitting the MOT expansion allows
us to extract the temperature of the cloud in both axes imaged by the camera.
4.2 Blue MOT
The blue MOT of the main apparatus is relatively simple. We operate at a field
gradient around 55 G/cm, and with about 9 mW of power in the MOT beams, giving
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a peak I/Isat of ? 0.4. After loading the trap for 2 to 4 s we trap ? 3? 107 atoms
of 88Sr, and for the other isotopes we trap atoms approximately proportional to the
atomic abundances. For 87Sr, the 707 nm repump EOM is modulated at a variety
of frequencies to account for the hyperfine structure. In the main apparatus we
modulate at 571 MHz, 1240 MHz, and 480 MHz. These frequencies are empirically
found informed by the values from the old laboratory [5]. Due to the magnification
of our imaging system and relatively high temperature of the blue MOT, in time of
flight the atomic cloud quickly expands past the field of view. This makes it difficult
to take a good measurement of the temperature of the atoms in the blue MOT, but
we expect that it is somewhere around a few mK.
After loading the blue MOT we ramp the power in the MOT beams down,
while ramping the field gradient up. We call this section of the blue MOT exper-
imental sequence the ?Doppler cooling? stage. Fig. 4.3 shows the laser power and
field ramp during this stage. Lowering the power should allow us to reach a tem-
perature that is closer to the Doppler limit, which is only achievable in the limit
of low saturation, while ramping the field up compresses the MOT, counteracting
the expansion of the MOT that occurs when the power is lowered. Lowering the
temperature facilitates better transfer between the red and blue MOT. In practice,
we currently get the most efficient transfer when the blue power is dropped to zero,
and the field barely increases, although the transfer is not very sensitive to the field
values, so they can change depending on the specific optimization.
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4.3 Red MOT
Finding the red MOT is quite a bit more complicated than the blue MOT.
Firstly, since the red MOT requires first loading the blue MOT, and the red MOT is
not visible by eye, we cannot continuously load and optimize the MOT at the same
time, and we have to use destructive imaging to analyze the red MOT. Second,
there must be significant spatial overlap between the red and blue MOT in order
for there to be sufficient transfer. Furthermore, the lower field gradient of the red
MOT causes any stray field to considerably move the field zero. This means we can
move the red MOT around in space easily with the shim fields, but if the fields are
not correct the red MOT field zero can easily be in a completely different location
than the blue MOT, and transfer will be impossible.
We have found a relatively consistent way to find the 88Sr red MOT with
our main apparatus. First, there is a visible dip in fluorescence and a change in the
shape of the blue MOT from the red MOT beams when they are on resonance. Since
this is visible even while continuously imaging the fluorescence of the blue MOT,
it allows us to easily confirm the laser is close to resonance. Once we are close, we
leave the red light on while looking at absorption images of the blue MOT in time
of flight with the field gradient set to an appropriate red MOT value (between 3
and 5 G/cm). After a few ms only a diffuse background gas from the blue MOT
is visible. Atoms in the background gas see a force towards the field zero, and
are excited to the 3P1 level wherever the light is resonant with the Zeeman shifted
transition. Since the linewidth of the red MOT transition is so narrow this condition
78
is only satisfied for a thin shell of atoms where the field is appropriate, and these
shells appear in the images as ellipses. We can look at the center of these shells and
how they move with changing frequency to adjust the shim fields so that the center
of the shells is overlapped with the blue MOT. This also tells us where our eventual
red MOT will be, so we can tune other parameters until the MOT appears at that
spot. Additionally, the radii of these ellipses tell us the approximate detuning of the
red light. Fig. 4.2 show example images of these shells.
Figure 4.2: Absorption image of the 689 nm light interacting with the blue MOT
background. The resonant shells are marked with a dashed line. In (a) we can see
that the center of the resonant shell is well below the blue MOT, so we have to use
the shim fields to bring it up. Image (b) shows the shell better aligned with the blue
MOT. From here we are likely to get a red MOT with just a few adjustments of
other parameters.
We start loading the red MOT with an artificially broadened laser, about
2 MHz wide, and slowly ramp it down to a single frequency. Broadening the laser
enables us to initially capture more velocity classes from the blue MOT and greatly
increases the final atom number. While we ramp down the broadening we also ramp
down the power in the red MOT beams to further cool the MOT using the RF power
to the AOM before the red fiber splitter. Fig. 4.3 shows example parameters for the
79
blue to red MOT transfer. We have captured ? 1?107 88Sr atoms in the red MOT,
at a temperature of ? 5 ?K. We have also achieved a red MOT of all other stable
isotopes, with atom numbers consistent with the relative abundances. For 87Sr the
stirring laser, mentioned in Sec. 2.1.3, is broadened in the same way, and in phase
with the cooling laser, and the power is ramped down similarly as well, this keeps
the stirring continuously working while the broadening is narrowed.
4.4 Dipole Traps
The main difficulty with transitioning from the red MOT to the dipole trap
is aligning the two in space. We want the in situ red MOT to sit right where the
focus of the dipole beam or beams are. To get the initial alignment we first set up
a temporary imaging system, along the axis of the dipole trap, but on the other
side of the chamber from the input dipole beam. Then we add in a flipper mirror
so we are able to switch between putting the dipole trap and a probe beam through
this imaging system. Then we take in situ images of the red MOT which allows
us to focus the imagining system, and note the location of the red MOT in the
images. We then swap in the dipole beam for the probe beam, and use the same
imaging system to align the image of the dipole beam up with the previous image
of the MOT to give approximate transverse alignment. The trap can be aligned
axially by adjusting the final lens of the dipole trap to focus it on the camera. Since
we previously focused the imaging system with the red MOT, this process aligns
the focus of the dipole trap onto the atoms. Throughout this process we lower the
80
Figure 4.3: Example red MOT sequences for the MOT in the main apparatus. a)
The broadening of the laser detuning starts out large and decreases to zero over
time. b) The power in the red MOT beam decreases as well to cool the atoms in
the MOT to the final temperature. We ramp this power with the AOM RF power
linearly which leads to the exponential shape of the optical power ramp. c) The blue
power decreases during the Doppler cooling stage using the power lock, and d) the
field gradient increases slightly. After the Doppler cooling stage the field gradient
drops to the inital red MOT value where it is slowly ramped up.
81
Figure 4.4: (a) One of the first images we took of our dipole trap 10 ms after the
red MOT was dropped. You can see the red MOT atoms that were not captured
by the dipole trap continuing to fall. (b) The dipole trap after some optimization
and after holding the dipole trap for 5 s. You can see the weak trapping in the axial
direction, and the slight angle to the trap that still needs to be leveled.
power in the dipole beam significantly, and add a neutral density filter to avoid
any damage to the camera. While this process gives us rough alignment, chromatic
shifts in the imaging system between the probe wavelength (461 nm) and the dipole
trap wavelength (1064 nm in this example) can give a significant offset from perfect
alignment. We make up for this final misalignment by randomly sampling a large
space for the final red MOT shim coil values. This moves the red MOT around in
space with a very good chance of overlapping the dipole beam at some point. You
can see the result of this alignment in Fig. 4.4. After this process the red MOT
shim fields might not be at optimal values for the blue to red MOT transfer, but
once the initial trap is found it is simple to slowly walk the dipole beam alignment
and shim values back to optimal values.
For many applications a single beam dipole trap is not sufficient, and two (or
more) beams, intersecting at a point, must be used. The process for aligning this
second, cross dipole beam is the same as the first, we do this alignment without
82
Figure 4.5: An image of the atoms in the cross dipole trap with two dipole trap
beams. Note that the magnification of our imaging system changed between this
image and those in Fig 4.4.
operating the main dipole trap. Once we move the red MOT so that some atoms
were trapped by the cross dipole beam we have two different sets shim values for the
two dipole beams. All that is left is to walk the main dipole beam?s shim values to
the values for the cross beam so that both beams and the red MOT all overlap. Our
cross beam comes in on the same axis as our permanent horizontal imaging system,
so a permanent dichroic is used to combine the probe and dipole beams. However,
since the main probe beam and cross dipole beam counter-propagate through the
chamber, a temporary imaging system is still required on the opposite side of the
chamber from our permanent imaging system.
The traps discussed here are 1064 nm dipole traps, but for most of our appli-
cations we want magic wavelength 813 nm attractive traps. This initial alignment
83
should be sufficient to easily swap out the light for the magic wavelength and still
see the trap, but there will be some chromatic shifts which will impact the quality
of the trap. Thus, we did not invest time in fully optimizing the 1064 nm dipole
traps while we did not have access to 813 nm light.
4.5 Blue Grating MOT
Figure 4.6: An image of the atomic beam out of the dispenser source with the
magnetic fields off and the MOT beams on resonance.
Just like the blue MOT of the main apparatus, the blue grating MOT is
relatively simple to find. The atomic fluorescence of the atomic beam is visible
when the magnetic field is off, and the MOT beam is on resonance, as seen in Fig.
84
4.6. This allows for the resonance to be found independent of alignment and the
magnetic field. Once the field is on, the easiest way to align the MOT beam is to
look at the 0th order diffraction of the grating which should retro-reflect back onto
the main beam. In practice however, just like most MOTs, perfect retro-reflection
is not optimal. The physical z position of the coils or magnets can also be adjusted
to optimize the MOT.
While the MOT is loading we take fluorescence images approximately every
200 ms. This allows us to see the MOT loading rate. Unfortunately, due to some
noise in the blue MOT laser lock, the fluorescence sometimes instantaneously drops,
but in general there are enough usable images. Fig. 5.6 shows examples of these
loading curves. From the fits to these curves we can calculate the final relative atom
number, unfortunately, it is difficult to get an accurate absolute atom number using
fluorescence imaging. Additionally, in the first iteration of the blue grating MOT,
absorption imaging was only possible when the atom number was optimal, which
restricted the data that could be taken with this method, and added significant noise
to the data we could take. To combat this issue, we normalized the fluorescence
imaging to the absorption atom number using a constant averaged normalization
factor. This allowed us to get a stable and accurate atom number from the loading
curves. Since the failure of the atomic shutter prohibits us from shutting off the
atom beam and looking at the decay of the MOT, we instead use the loading curves
to extract the lifetime of the MOT. Sec. 5.2.4 discusses more details on these
loading curves and their interpretation. Since the fluorescence in the loading images
is from the blue MOT beam, in order to create Fig. 5.7, we had to account for the
85
dependence of fluorescence on detuning. Luckily this is simple using the equation
for the scattering rate (Eq. 2.1).
Since the imaging system on the grating MOT apparatus demagnifies, com-
pared to the magnification of the main apparatus imaging system, we get a larger
field of view. This allows us to take images with varying time of flight, and measure
the temperature of the MOT. This process is described in Sec. 5.2.4.
After the publication of Ref. [58] (Sec. 5.2), which did not include any results
from the fermion 87Sr, we found the fermionic blue grating MOT. By adjusting the
magnets to account for the differential offset in position between the fermion and the
bosons as discussed in Sec. 2.3 we were able to trap the fermion. For the fermionic
grating MOT, the 707 nm repump is modulated at 1315 MHz and 499 MHz, these
values are informed by Ref. [69]. Additionally, in the second iteration of the grating
MOT apparatus, the atom number in the blue MOT was improved by an order
of magnitude, up to 3 ? 107. This was likely due to using the same blue laser as
the main apparatus, which has more available power, for the MOT beam, and an
improvement in the vacuum pressure down to approximately 1 ? 10?10 Torr. This
increase in the atom number makes absorption imaging a much more consistent tool.
To transfer atoms from the blue grating MOT to the red grating MOT we
operate the same Doppler cooling stage as on the main apparatus. However, since
we are currently operating the blue grating MOT coils at the maximum current of
the power supply, we cannot ramp the field gradient up during this stage. In the
grating MOT the length of this stage that gives an optimal transfer is much shorter
than in the main apparatus, as seen in figure 4.8, this could be because we cannot
86
compress the MOT with a field gradient.
4.6 Red Grating MOT
The red grating MOT has difficulties that are not present in the conventional
red MOT. The capture region is smaller and does not overlap completely with the
blue capture region due to the difference in diffraction angle. The high transverse
temperature of the blue grating MOT means that atoms can quickly leave this
capture region. Additionally, before the apparatus was refined, the slow rate of field
switching further exasperated this issue as it left more time for the atoms to expand
before the red MOT field value was reached.
a) b)
Figure 4.7: a) An image of the resonant shell from the red MOT beams in the
background of the blue grating MOT. b) The same image with the shell marked
with a dashed white line.
The comparatively low atom number in the blue grating MOT also adds dif-
ficulty by decreasing our signal to noise ratio across the board. For example, the
signal of the resonant shells discussed in Sec. 4.3, is greatly reduced, especially at
low field gradient, and they are less useful as a diagnostic tool. An example image
87
of a resonant ring in the blue grating MOT is shown in Fig. 4.7. Despite this, the
rings were useful in diagnosing the issues we had with the field switching. From
the relative insensitivity of the radii of the rings to the red laser detuning, we could
determine that our field gradient was still high even 2 ms after commanding a low
field. Additionally, we could tell by how the rings shifted in the +z direction over
time that we had an eddy current from the grating mount.
Once the issues with field switching were resolved, as discussed in Sec. 3.1.4,
the main method we used to search for the red MOT was a large random sampling
of the appropriate parameter space to search for any initial signal. In this way, we
found a very faint signal of diffuse atoms at 10 ms time of flight (longer than blue
MOT atoms remain visible), and from here a blind adjustment of alignment got us
our first red Grating MOT.
Example parameters for the transfer between the blue and red grating MOTs
are shown in Fig. 4.8. Here you can see the short Doppler time where the blue MOT
power decreases to zero, the frequency broadening and compression, the decrease in
the red MOT power to compensate for this compression, and the field switching. The
broadening in the red grating MOT is done using a sawtooth function to facilitate
SWAP cooling. In addition to perform SWAP we increase the broadening, which
pushes the top edge of the broadening above resonance.
In the red grating MOT we can capture around 3 ? 106 atoms, achieve a
temperature of around 3 ?K, with a trap lifetime of 0.8 s. The measurements of
temperature and lifetime are shown in Fig. 4.9 and Fig. 4.10 respectively.
We also investigate the effect of SWAP, discussed in Sec. 2.4, on the grat-
88
Figure 4.8: Example red MOT sequences for the grating MOT. a) In the grating
MOT the broadening is much larger at first to account for the higher temperature of
the blue grating MOT, and to facilitate SWAP. In addition, for SWAP the frequency
of the upper edge of this broadening decreases slightly over time, and starts above
the resonance, and the final broadening is left at 100 kHz rather than zero. b)
The corresponding decrease in red MOT beam power is shown. c) The blue power
decreases during the Doppler cooling stage (linearly in AOM RF power), but in
the grating MOT the length of this stage is much shorter. d) The field gradient
switches from the blue MOT value to the red MOT value, unfortunately these are
the maximum operating values for the respective coils, so we cannot ramp the field in
either the Doppler stage or broad red MOT stage like we do on the main apparatus.
89
Figure 4.9: A measurement of the red grating MOT temperature. We measure the
temperature by looking at how the width of the cloud increases in time of flight. The
fits to this expansion, using Eq. 4.1, are shown as well. These fits yield temperatures
of 4.5(6) ?K in the transverse (x) direction, and 2.9(1) ?K in the axial (z) direction.
ing MOT by switching between sawtooth and triangle sweeps, as well as sawtooth
sweeps with the wrong slope (anti-SWAP). Fig. 4.11 shows how the atom number
varies between these three different methods of modulation, and you can see the
enhancement in atom number that SWAP provides. SWAP at 20 kHz modulation
almost doubles our atom number when compared to the optimal triangle wave mod-
ulation at 20 kHz. We can match the sweep rate of the triangle and sawtooth waves
by halving the frequency of the triangle sweep to 10 kHz, but this simply drops
the atom number from the triangle modulation further. Lastly, modulating with a
sawtooth wave with the wrong slope pushes atoms out of the trap, and drastically
90
Figure 4.10: A measurement of the red grating MOT lifetime. A fit to the expo-
nential decay of atoms in the trap as a function of hold time is shown in red. This
yields a lifetime of 0.8 s.
drops the atom number. In addition Fig 4.12 shows how the atom number varies as
a function of the sweep frequency.
In conclusion, on the main apparatus we got our new strontium source up and
running, and transferred atoms between the progressive cooling stages of strontium
from the source, to the blue MOT, to the red MOT, to dipole traps. We have
improved the atom number in the red MOT compared to the old apparatus. The
next step for the main apparatus involves implementing magic wavelength dipole
traps and preparing for future experiments discussed in Ch. 6. On our grating MOT
apparatus we implemented the first alkaline earth grating MOT, performed transfer
to the red grating MOT, and showed enhancement of the transfer efficiency with
91
Figure 4.11: Atom number vs detuning for SWAP vs a triangle wave sweep. The
detuning shown is just an estimation of the final detuning of the laser beam after the
broadening is decreased down to 100 kHz. The data for our optimal SWAP MOT
is shown in blue with a sawtooth wave at 20 kHz. The standard swept MOT with
a triangle wave is shown in red, and a sawtooth wave MOT where the sawtooth
has the wrong slope (anti-SWAP) is shown in green, both with a 20 kHz sweep
frequency. We also show the triangle wave MOT with a 10 kHz sweep frequency,
shown in purple, to match the sweep rate of the SWAP MOT (note that these data
were taken a different day so they were normalized to 20 kHz SWAP MOT).
92
Figure 4.12: The atom number we can achieve through SWAP at a variety of dif-
ferent sweep frequencies. Our atom number is optimal when the sweep frequency is
20 kHz.
SWAP in the grating geometry. The immediate next step for the grating apparatus
is to find a red MOT of 87Sr.
93
Chapter 5: Publications
5.1 Publication: Isotope shift spectroscopy of the 1S 30 ? P1 and
1S 30 ? P0 transitions in strontium
This work was completed in the old strontium laboratory, published as Ref.
[24] and led by Neal Pisenti. Isotope shifts were a straightforward avenue to in-
vestigate given the laboratory?s capability to operate with all the stable isotopes in
strontium, and the lack of precision spectroscopy of the narrow transitions in the less
used isotopes. In addition with four stable isotopes, we had enough data points to
investigate the linearity of the results, which became more interesting given recent
proposals to use this linearity as a way to probe new physics. The precision spec-
troscopy was further motivated by the recent purchase of an optical frequency comb
by the JQI. Halfway through the data collection, Neal moved on to other projects
and I took over the work in the laboratory for the second half of the experiment,
specifically the spectroscopy and systematics of the clock transitions. Much of the
text was written by our postdoc Hiro Miyake, with significant input from the rest
of the group. This work found a significant deviation from linearity in the king plot
due to the hyperfine structure of 87Sr, and work since has provided more possible
94
explanations for this effect [70?72].
5.1.1 Abstract
Isotope shift spectroscopy with narrow optical transitions provides a bench-
mark for atomic structure calculations and has also been proposed as a way to
constrain theories predicting physics beyond the Standard Model. Here, we have
measured frequency shifts of the 1S 3 10 ? P1 and S ? 30 P0 transitions between
all stable isotopes of strontium relative to 88Sr. This includes the first reported
measurements of the 1S ? 3P isotope shift of 88Sr-86Sr and 88Sr-840 0 Sr. Using the
isotope shift measurements of the two transitions a King plot analysis is performed,
revealing a non-linearity in the measured values.
5.1.2 Introduction
Isotope shifts of atomic transition frequencies arise due to the difference in
neutron numbers for different isotopes with the same atomic number. For a given
element, these shifts can be systematically analyzed using a King plot, which eluci-
dates the contributions of the field and mass shifts [25]. The King plot is typically
expected to be linear, and the experimentally determined value of the slope provides
a good benchmark for theoretical predictions [27]. Any deviations from linearity as
was observed in Sm [73] and Ba [74], or between predicted and experimentally mea-
sured values of the slope as was observed in Ca+ [75], are important for refining
atomic structure calculations [76]. Furthermore, recent theoretical proposals have
95
suggested that linearity in King plots could be used to put constraints on higher-
order effects on isotope shifts or on physics beyond the Standard Model [26, 77].
Strontium has many favorable properties for studying isotope shifts, including an
abundance of stable isotopes and very narrow optical transitions [40]. In addition,
prior theoretical work has proposed the measurement of strontium isotope shifts as
a promising probe of new physics [26, 77].
Strontium has four stable isotopes: three bosons (88Sr, 86Sr, and 84Sr), and one
fermion (87Sr). Mixing between the singlet and triplet fine structure manifolds leads
to narrow-linewidth optical transitions, and these transitions have found use in both
strontium and other alkaline-earth-(like) atom experiments [78, 79]. In particular
for strontium, the 1S0 ? 3P1 intercombination-line transition at 689 nm (linewidth
?/2? = 7.4 kHz) is used during laser cooling to operate a narrow-line magneto-
optical trap (MOT) [40, 80], and the even narrower 1S ? 30 P0 clock transition at
698 nm (?/2? ? mHz) is the foundation for state-of-the-art optical clocks operating
at a precision at the 10?18 level [11, 12, 81]. The clock transition is strictly forbidden
by angular momentum considerations, but becomes weakly allowed via hyperfine
mixing in 87Sr or by application of an external field for the bosonic isotopes [9].
While the 1S 30 ? P0 clock transition has been extensively studied in 87Sr and
88Sr [9, 11, 12, 81?85], there have been no previous measurements of the transition
in either 86Sr or 84Sr as far as we know [86]. Here we report the first isotope shift
spectroscopy measurements of the clock transition for both 84Sr and 86Sr relative to
the most abundant isotope 88Sr. Furthermore, we measure all isotope shifts of the
1S0 ? 3P1 intercombination-line transition relative to 88Sr, permitting the first King
96
plot analysis of strontium for these two transitions. Given the very narrow linewidths
involved, extensions of this work could place stringent experimental constraints on
the King linearity, ruling out candidate theories for physics beyond the Standard
Model or benchmarking state-of-the-art atomic structure calculations.
5.1.3 Experimental procedure
All of the isotope shift spectroscopy was performed using laser-cooled stron-
tium atoms at temperatures of a few ?K, held in an optical dipole trap (ODT). After
applying the spectroscopy light, we monitored atom loss by performing absorption
imaging.
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 5.1: Measured systematic frequency shifts and uncertainties for the 1S 30 ? P1
transition. Uncertainties indicate one standard deviation.
The laser lights used for spectroscopy of both the 1S0 ? 3P1 and 1S0 ?
3P0 lines were generated using two home-built external-cavity diode lasers based
on the design in Ref. [64]. The frequency of the 689-nm laser was stabilized via
an optical phase-locked loop [87] to the master laser of the 689-nm narrow-line
MOT system. The master 689-nm laser was locked using the Pound-Drever-Hall
(PDH) method [88, 89] to a cavity constructed from ultra-low expansion (ULE)
glass and housed in a temperature-stabilized vacuum chamber. To stabilize the
frequency of the 698-nm laser, we passed a few percent of the light through a wide
bandwidth electro-optic modulator and locked the first phase-modulated sideband
97
via PDH to a second, independent ULE cavity [90]. This cavity was housed in
a separate temperature-stabilized, acoustically-isolated vacuum chamber. These
locking schemes for both lasers allowed us the flexibility to shift the frequency of
either laser to span the isotope shifts of its respective transition. For both the
689-nm and 698-nm lasers, the light was referenced to a frequency comb (Menlo
System FC1500-250-ULN [91]) to account for long-term drift and provide a frequency
reference. Fine frequency control of each laser beam was achieved by adjusting the
drive of an acousto-optic modulator, which was also used to stabilize the intensity
of the spectroscopy pulse. The spectroscopy laser linewidth was characterized by
locking independent 689-nm lasers to each ULE cavity. A heterodyne beatnote at
689 nm between the two separate lasers was measured to be approximately 200 Hz
wide, which bounds the expected spectral performance of both systems.
The remainder of the apparatus used for the spectroscopy has been described
in detail previously [43]. Laser cooling of all isotopes proceeds according to well-
established techniques [40], with a MOT first operating on the broad 1S 10 ? P1
transition at 461 nm, followed by a narrow-line MOT operating on the 689-nm
intercombination-line transition 1S0 ? 3P1. For all isotopes, temperatures in the
narrow-line MOT are typically a few ?K, low enough to efficiently transfer the
atoms into a single beam, far-detuned ODT at 1064 nm. Typical temperatures
in the ODT are {2.9 ?K, 2.2 ?K, 1.1 ?K 2.7 ?K} and typical atom numbers are
{1, 0.1, 0.5, 0.2}?106 for {88Sr, 87Sr, 86Sr, 84Sr} respectively, with trap frequencies
{?x, ?y, ?z}/(2?) = {50, 4, 495} Hz in the horizontal, axial, and vertical directions
respectively. For the bosons, the variation in atom number is mostly due to the
98
difference in the natural abundance of each isotope, whereas for the fermionic isotope
the atom number is also limited by the additional complexity of the narrow-line
MOT [44].
5.1.4 Measurement of the 1S 30 ? P1 isotope shifts
After loading the atoms into the ODT, the magnetic field was set to 0.05 mT
(0.5 G) to resolve the 3P1(m = 0) state for the even (bosonic) isotopes. For the
odd (fermionic) isotope, which has hyperfine structure, the magnetic field was set
to zero, meaning that the Zeeman splitting was not detectable within the lineshape.
The strength of the magnetic field was calibrated by addressing the 1S0(m = 0)?
3P (m?1 = 1) transition of
88Sr, for which the Zeeman shift is known [80]. To eliminate
the effect of AC Stark shifts from the ODT we implemented a stroboscopic procedure
where, with a typical period of 500 ?s, the ODT was turned on and off with a duty
cycle of 50% (duration of 250 ?s), and applied the 689-nm probe laser when the
ODT was off, similar to the procedure used in Refs. [92, 93]. The spectroscopy light
was used to induce atom loss from the trap through light scattering and subsequent
recoil, which is primarily an incoherent process where there is absence of coherence
between the pulses of the stroboscopic method with the switching of the ODT.
The 689-nm spectroscopy beam was aligned at an angle of approximately 45? with
respect to the ODT, both in the horizontal plane. The spectroscopy beam was
collimated with a 1/e2 beam waist of 1.25 mm in the horizontal direction and 1.71
mm in the vertical direction at the position of the atoms. The polarization of the
99
spectroscopy beam was set to be linear along the direction of the magnetic field.
The total illumination duration used for spectroscopy was set to between 1 ms and
15 ms, corresponding to multiple stroboscopic pulses, and the peak optical intensity
was at most 0.1 mW/cm2 (Isat = 3 ?W/cm
2). These values were chosen to ensure
atom loss of approximately 50%. After the spectroscopy pulse was completed, the
atoms were released from the ODT and we performed absorption imaging on the
1S0 ? 1P1 transition to measure atom loss as a function of the spectroscopy laser
frequency.
For all four isotopes, data was taken across several days and referenced to the
frequency comb. Then the frequencies were averaged to obtain a single line center for
each isotope. A final isotope shift was found by subtracting the measured absolute
frequencies relative to 88Sr, and the total errors were added in quadrature. For
the 87Sr isotope shift, we weight the measurements of each excited-state hyperfine
manifold F ? ? {11/2, 9/2, 7/2} to find the nominally unshifted line center in the
absence of the hyperfine interaction [94]. However, it is important to note that this
model fits three parameters (hyperfine A and B coefficients and an unshifted line
center) from three isotope shifts, and thus is completely determined by the available
data1. A more accurate theory of higher order shifts from other fine structure levels
will be necessary to assign a more accurate isotope shift for 87Sr. This is currently
an area of ongoing theoretical research [96].
To calculate the final value for the isotope shift, we also evaluated system-
1We determined |A| = 260085? 2 kHz and |B| = 35667 ? 21 kHz, consistent with previous
results [95].
100
atic effects, as summarized in Table 5.1. Since many of the systematic effects are
common to both isotopes, and the isotope shift is found from a difference in those
frequencies, many potential systematic effects are common mode and cancel to a
high degree. This is particularly true for the even isotopes, where there is no hy-
perfine structure. For example, even though a magnetic field is applied during the
spectroscopy pulse for the even isotopes, the Zeeman shift is identical to within our
experimental uncertainties, and does not lead to a correction to the final isotope
shift. Therefore, as shown in Table 5.1, the remaining systematic effects are those
that are not common mode: the density shift and recoil shift.
The density shift arises due to the different scattering lengths and atom num-
bers between different isotopes in our experiment. The cumulative effect is a non-
zero differential density shift to the final isotope shift value. We experimentally
determined this density shift for each isotope by measuring the line center at differ-
ent atom numbers while keeping all other parameters the same. A linear fit allowed
us to extrapolate from our operating atom number to a nominal ?zero-density? fre-
quency, yielding the systematic density shift shown in Table 5.1. The photon recoil
shift [97] was also accounted for and was calculated from known physical quantities.
To first order, the 1S ? 30 P1 transition is magnetic field insensitive, and
our measurements were performed at a low magnetic field of 0.05 mT (0.5 G) for
the bosons and zero magnetic field for the fermion. Therefore both the first and
second order Zeeman shifts were negligible at our level of accuracy. The stroboscopic
procedure described above removed any AC Stark shifts due to the 1064-nm trapping
beam. Finally, since the intensity in the 689-nm spectrosopy pulse was low (at most
101
0.1 mW/cm2) and the probe times were short (a few ms), systematic shifts from the
probe pulse were below our experimental uncertainty.
Isotope Shift (kHz) 1S ? 3P 10 1 S 30 ? P0
88-84 351495.8? 0.3? 4.6 349656? 1? 10
88-86 163818.7? 0.3? 3.8 162939? 2? 11
88-87 62186.5? 0.6? 11.7 62171? 1? 23
88-87 (F ? = 7/2) ?1351933.1? 2.1? 14.6
88-87 (F ? = 9/2) ?221676.6? 0.4? 26.9
88-87 (F ? = 11/2) 1241485.8? 0.3? 15.7
Table 5.2: Measured isotope shifts relative to 88Sr. For 87Sr (1S 30 ? P1), contri-
butions from the three excited-state hyperfine manifolds are weighted to establish
the fine-structure line center. Uncertainties are one standard deviation and indicate
statistical and systematic uncertainties.
After applying corrections for the systematic effects, the final values for 1S0 ?
3P1 isotope shifts are shown in Table 5.2. The total systematic uncertainties are
determined by adding the individual systematic uncertainties for each isotope in
Table 5.1 in quadrature. Our results are consistent with a previous measurement of
the 88Sr-86Sr isotope shift, which reported a value of 163817.4? 0.2 kHz [98].
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 51? 5 42? 22 53? 5 51? 5 50? 5 51? 5
Thermal ?22? 4 ?17? 4 ?22? 4 ?8? 4 ?22? 4 ?21? 4
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
Total 39? 8 31? 22 40? 7 51? 7 37? 7 31? 7
Table 5.3: Systematic frequency shifts and one standard deviation uncertainties for
the 1S0 ? 3P0 transition. The three columns for 88Sr correspond to three inde-
pendent isotope shift measurements. Uncertainties indicate one standard deviation.
102
5.1.5 Measurement of the 1S ? 30 P0 isotope shifts
The procedure for measuring the 698-nm transition differed from the measure-
ment of the 1S 30 ? P1 intercombination-line transition in several key ways. Since
the clock transition is strictly forbidden by angular momentum considerations for
the bosonic isotopes, a much larger field was necessary to induce a transition in
these isotopes. For 88Sr and 86Sr a magnetic field of 10.96?0.02 mT (109.6?0.2 G)
was used, and 19.79?0.05 mT (197.9?0.5 G) was used for 84Sr. For measurements
of 87Sr, which is weakly allowed due to hyperfine mixing, we applied zero magnetic
field. For all isotopes, the 698-nm spectroscopy pulse was applied for 2 s with typical
peak intensities of 0.87 W/cm2 for the even isotopes and 0.12 W/cm2 for the odd
isotope (Isat ? 0.4 pW/cm2). These values were chosen to ensure approximately
50% atom loss. Atom loss was induced by the light scattering and subsequent recoil
of the 698-nm light, which ejected the atoms out of the trap. Representative line
shapes for the 1S0 ? 3P0 transitions are show in Fig. 5.1 for each isotope.
The spectroscopy beam was aligned in the horizontal plane at an angle of ap-
proximately 45? with respect to the ODT, and was focused onto the atoms with a
1/e2 waist of 330 ?m in the horizontal direction and 460 ?m in the vertical direction.
The beam was linearly polarized parallel to the magnetic field. Finally, because of
the long interrogation time needed for sufficient atom loss (and therefore sufficient
signal to noise), we were unable to apply the stroboscopic procedure used to mea-
sure the 1S ? 30 P1 transitions, resulting in large AC Stark shifts from the trapping
beam. Due to the modified experimental procedure for the clock transition, addi-
103
tional systematic shifts included: thermal shifts, second order Zeeman shifts, and
spectroscopy pulse shifts.
Figure 5.1: Spectroscopy of the 1S0 ? 3P0 transition for each strontium isotope.
The normalized atom number is shown as a function of the laser detuning. The
solid line is a Gaussian fit to the data.
For the 1S 30 ? P0 transition, the dominant systematic effects were the AC
Stark shift and what we call the thermal shift. The AC Stark shift arises from
the differential polarizability of the 1S0 and
3P0 states at 1064 nm. The thermal
shift arises from the inhomogeneous broadening and shift from the thermal motion
in the intensity distribution of the ODT. Experimentally, the AC Stark shift was
determined by measuring the resonance frequency as a function of the ODT inten-
sity. However, varying the intensity of the ODT also varied the trap depth, which
in turn varied the temperature of the atomic cloud. This led to additional shifts
104
in the resonance frequency due to both the Doppler shift and the inhomogeneous
differential AC Stark shift. To distinguish the effects of the AC Stark shifts from
the thermal shifts we took the thermal average using the Maxwell-Boltzmann distri-
bution and modeled the scattering process from the spectroscopy pulse [92, 99] (see
the Appendix 5.1.9). As shown in Table 5.3, the experimentally determined values
for the AC Stark shift for each isotope agree with each other within the uncertainty.
Therefore the AC stark shift is common mode and cancels to a high degree.
For even isotopes, the systematic shift for the first order Zeeman effect is zero
since we probe a J = 0? J ? = 0 transition with no hyperfine structure. To deter-
mine the second order Zeeman shifts for the 1S0 ? 3P0 transitions we used our cali-
brated magnetic field measurements and the known second order Zeeman shifts [9],
which are identical for all even isotopes. Spectroscopy of 87Sr was performed at zero
magnetic field, and so the Zeeman shift was well below other systematic effects [2].
The last systematic effects evaluated for the clock transition were related to
the spectroscopy laser, occurring due to the relatively long probe time (2 s) and
high peak intensities (0.87 W/cm2). To measure these systematics, the transition
frequency was measured as a function of both pulse power and duration, and the shift
was extrapolated to zero. Finally, the density shift and recoil shift were obtained
using the same procedure as described for the 1S0 ? 3P1 transition.
The final values for the isotope shift of the clock transition, including system-
atic corrections, are shown in Table 5.2. The systematic shifts are summarized in
Table 5.3. The total systematic uncertainties for the clock transitions in Table 5.2
are determined by adding the individual systematic uncertainties for each isotope in
105
Table 5.3 in quadrature. Comparing to prior measurements of the 88Sr-87Sr isotope
shift which were all approximately 62188 ? (< 1) kHz [82, 83, 85], our result of
62171? 24 kHz is consistent to well within one standard deviation.
5.1.6 King plot analysis
We performed a King plot analysis using our measured values of the isotope
shifts, including the first measurements of the 88Sr-86Sr and 88Sr-84Sr isotope shifts
for the clock transition. A King plot analysis is a systematic approach to quantita-
tively and visually analyze isotope shifts of different atomic transitions referenced
to the same isotope by relating the isotope shifts between different transitions [25].
This is a function of the mass and field shift constants, which are independent of the
isotopes and depend only on the transitions under consideration [100]. Specifically,
the isotope shifts between isotopes of mass numbers A and A? on two transitions i
and j can be written
A,A? Fi Fi ??A,A???i = Ki ? K + ?
A,A
j A,A???j , (5.1)Fj Fj
where 1/?A,A? = 1/mA? ? 1/mA is the inverse mass constant, mA is the mass of
isotope A [101], Ki is a constant associated with the mass shift of transition i, Fi
? ?
is the field shift constant for transition i, and ??A,Ai = ?
A A
i ? ?i is the isotope shift
between isotopes A and A? on transition i [25, 75]. For our particular analysis, we
have A = 88, and A? ? {87, 86, 84}, i ? 1S ? 30 P0 at 698 nm, and j ? 1S0 ? 3P1 at
689 nm. An important point to note is that Eq. 5.1 describes a linear relationship
106
between isotope shifts of different transitions.
The King plot for our measured isotope shifts is shown in Fig. 5.2. A linear
fit to all three points weighted by their uncertainties leads to a field shift constant
ratio of F698/F
F698
689 = 0.987 ? 0.008 and K698 ? K689 = 5.20 ? 5.31 GHz?amu,F689
where the statistical and systematic uncertainties are added in quadrature. We
have also performed a linear fit by replacing our measurement of the 88Sr-87Sr 698-
nm transition isotope shift with the more precise value from Ref. [85]. This leads
to values of F698/F689 = 0.981? 0.005 and K ? F698698 K689 = 8.56? 3.45 GHz?amu,F689
which are consistent with values obtained using our measurement of the 88Sr-87Sr
698-nm transition isotope shift.
Since there is some uncertainty in deriving the frequency for 87Sr due to the
hyperfine structure, we also fit the data after excluding this point to obtain a field
shift constant ratio of F698/F689 = 0.998?0.002 and K ? F698698 K689 = ?1.87?1.03F689
GHz?amu where the uncertainties are propagated from the uncertainties of each
point for both axes. Compared to this two-point linear fit, the 88Sr-87Sr 689-nm iso-
tope shift we determined would have to increase by 136.2 kHz to become consistent
with a linear King plot. Given that our data points with their uncertainties lie well
outside of the straight line fit to all three points, the results in Fig. 5.2 suggest a
possible nonlinear contribution to Eq. 5.1, or may indicate significant uncertainties
in the determination of the center-of-mass of the 87Sr 3P1 hyperfine structure. In
particular, our data indicates a nonlinearity using the nonlinearity measure defined
in Ref. [26]. Future theoretical and experimental studies should help to explain our
observations, including better calculations of the hyperfine mixing within the 3P
107
(a) (b)
(c) (d)
Figure 5.2: King plot of the measured strontium isotope shifts. (a) Linear fit to
the three points derived from the six isotope shift measurements. Solid black line
is a fit using all six of our measured isotope shifts, and the dashed gray line is a fit
by replacing our measured 88Sr-87Sr 698-nm transition isotope shift with the value
from Ref. [85], which is more precise than our measurement. The black points are
derived from our measurements, and the gray point is using the 88Sr-87Sr 698-nm
transition isotope shift from Ref. [85]. The fits are weighted by the uncertainties of
each point. Error bars and the difference between the 87Sr points derived from our
measurement and from Ref. [85] are not visible at this scale. (b)-(d) Close up of
each point in (a) with error bars shown.
108
states and a prediction of the King plot slope.
5.1.7 Conclusions
In summary, we have presented the first spectroscopy of the 1S ? 30 P0 clock
transition in 86Sr and 84Sr, and reported their isotope shifts relative to 88Sr. In
conjunction with improved measurements of the intercombination line isotope shifts,
we performed a King plot analysis and extracted constants related to the field and
mass shifts. Hyperfine effects in 87Sr complicate this analysis, but the experimental
precision permitted by these two narrow optical transitions make it a rich testbed
to benchmark state-of-the-art theory. Furthermore, it has been suggested that a
comparison of isotope shifts between neutral and ionic strontium could set stringent
limits on new physics [26, 77]. However, an improved theory, accounting for our
observed nonlinearity would be essential. Alternatively, one could also perform this
measurement with the radioactive bosonic isotope 90Sr (half-life of approximately
29 years [102]) to avoid complications due to the hyperfine structure.
Future improvements on the measured frequencies will be possible by ap-
plying techniques successfully used with state-of-the-art strontium optical clocks,
such as the use of magic-wavelength dipole traps to minimize the differential AC
stark shift [103, 104] and optical lattices to suppress motional broadening and recoil
shifts [97]. These advances should further suppress statistical and systematic errors
on both transitions, allowing measurements with fractional uncertainties down to
the level of 10?18 [11, 12, 81]. Our results, combined with other recent measurements
109
of isotope shifts in Ca+ [105] and Sr+ [106], will further help to refine refine atomic
structure calculations and constrain new physics.
5.1.8 Acknowledgments
We thank Luis Orozco, Marianna Safronova, and Charles Clark for fruitful dis-
cussions and Nicholas Mennona for experimental assistance. This work was partially
supported by the U.S. Office of Naval Research and the NSF through the Physics
Frontier Center at the Joint Quantum Institute.
5.1.9 Appendix: Modeling the inhomogeneous broadening of the
clock transition
In general, the AC Stark shift is different for different atomic states due to
state-dependent polarizabilities. The exception to this is if one operates the dipole
trap at specific laser wavelengths typically referred to as the ?magic wavelength?
where the ground and excited states experience the same AC Stark shifts. For
strontium atoms, the magic wavelength is 813 nm for the 698-nm clock transition
and 914 nm for the 689-nm intercombination transition [8, 11]. In our experiment,
the optical dipole trap uses 1064-nm laser light, a wavelength where the two states,
1S and 30 P0, have different polarizabilities. This leads to inhomogeneous broadening
which must be accounted for. The resulting lineshape is further complicated by the
temperature of our atomic samples. Here we describe our method for modeling and
accounting for this inhomogeneous broadening due to both the differential AC Stark
110
shift and the thermal shift.
We model the inhomogeneous broadening process using a semi-classical treat-
ment of atom loss from the trap due to the spectroscopy pulse [99]. We can model
the atom loss from the spectroscopy pulse after some probe time, by calculating the
loss rate coefficient K. The time-dependent atom number in the presence of the
spectroscopy pulse is governed by the differential equation
dN
= ?K(??, I, T, Utrap)N, (5.2)
dt
where the loss rate coefficient K is a function of the laser detuning ?? = ?laser ? ?0
(?laser is the frequency of the probe laser and ?0 is the bare atomic resonance fre-
quency), the probe laser intensity I, the atomic cloud temperature T , and the dipole
trap potential Utrap. The loss rate is modeled to be proportional to an ensemble av-
erage of the scattering rate over all atoms in the trap.
The scattering rate can be written [37]
( )
? s0
?scat = 2 , (5.3)2 1 + s0 + (2?/?)
where ? is the transition linewidth, ? is the effective detuning from resonance,
s0 ? I/Isat is the on-resonance saturation parameter, I is the excitation laser inten-
sity, and Isat is the saturation intensity. We rearrange this expression, pulling out
111
constant terms to write
1
?scat ? 2 , (5.4)(??/2) + ?2
?
where ?? = ? 1 + s0 is the saturation-broadened linewidth. For a thermal atom in
a far-detuned optical dipole trap with a given phase space coordinate (r,p), ? can
be written
p ? k
? = ?? ? ? (Ue(r)? Ug(r)) , (5.5)
m
where the term ??? p ? k/m is the Doppler-shifted laser frequency, p is the atomic
momentum vector, k is the probe laser wavevector, m is the atomic mass, and Ue(r)?
Ug(r) is the differential AC Stark shift which arises from different polarizabilities
between the states e and g. Note that in the treatment here we neglect all other
systematic frequency offsets which do not depend on position, since these appear
simply as frequency offsets and do not cause any inhomegeneous effects. We also
neglect gravity in our model since the atoms are tightly confined in this direction.
We can approximate the trapping potential for the far-detuned optical trap as a
112
parabola and write
1 1
Ue(r)? Ug(r) = Ue,0 + m??2r2e ?(Ug,0 ? m
2 2
2 2 ) ??gr (5.6)
= U ? 1U + m ??2 ?(??2 2e,0 g,0 e g r ) (5.7)2
1 ??2
= U 2 2 ee,0 ? Ug,0 + m??gr ? 1 (5.8)2 ??2g
= ?U0 + ?Utrap(r), (5.9)
where ??g(??e) is the geometric mean of the ground (excited) state trap frequencies in
all three dimensions and the trap is effectively spherical in these coordinates. Since
?
??i ? ?i, where ?i is the AC polarizability of state i ? {g, e}, we find
( )
?e
?Utrap(r) = Utrap(r) ? 1 , (5.10)
?g
where U 2 2 1 3trap = m??gr /2. For 1064-nm light, with g the S0 state and e the P0
state, we compute ?e/?g ? 0.7. This can also be written as a re-scaling of the trap
potential, such that
?Utrap(r) = ?Utrap(r), (5.11)
with ? = (?e/?g ? 1) ? ?0.295. Note that operating the dipole trap at the magic
wavelength would lead to ?e = ?g, which means ? = 0, and therefore the spatial
dependence would drop out of Eq. 5.5.
We now turn our attention to solving for the loss rate coefficient K by taking
113
an ensemble average over the scattering rate expressed in Eq. 5.4 using the detuning
defined in Eq. 5.5. Because we are interested in deriving a lineshape function which
can be fit to experimentally measured atom loss data, we ignore normalization and
overall constant terms which can be condensed into a single fit parameter. Taking
the ensemble average of Eq. 5.4 leads to
? ? [ ]
K ? d3r e?Utrap(r)/(kBT ) d3 ?p2p e /(2mkBT ) 1
(??
,
/2)2+(????U0?p?k/m??Utrap(r))2
(5.12)
where we have taken an integral over phase space (r,p) weighted by the Boltzmann
factor. Here, kB is the Boltzmann constant.
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 p?z to point along k. Thus, p ? k = pzk, and the Boltzmann factor can be
rewritten
?p2e /(2mkBT ) = e?(p
2
x+p
2
y)/(2mk
2
BT )e?pz/(2mkBT ) (5.13)
The integral over px and py now factors out, and can be brought into an overall scale
?
fa?ctor. We define the dimensionless variable y ? pz/ 2mkBT . After defining ? ?
k 2kBT/m, this becomes pzk/m = ?y. In convenient units, for
88Sr and 2?/k =
?
698 nm, we get ?/2? = 19.7 kHz? T with T measured in ?K. This parameterization
of y serves to scale the momentum pz to the most probable momentum at a given
114
temperature.
Putting it all together, the integral from Eq. 5.12 becomes
? ? [ ]
K ? d3r e?U 2 1trap(r)/(kBT ) dy e?y .
(??/2)2 + (?? ??U0 ? ?y ? ?U 2trap(r))
(5.14)
With regards to the integral over r, since we have scaled the trap to be effectively
spherical, we can write U (r) = f(r2trap ). Thus, we can pull the angular integral
from d3r ? r2 sin ?dr d? d? into an overall constant, leaving just the integral in r
given by
? ? [ ]
K ? 2 ?r2m??2r dr e g/(2kBT ) dy e?y2 1 ,
(??
2
/2)2+(????U0??y??m??2r2g /2)
(5.15)
where we replaced Utrap(r) with its explicit form?m??2r2g /2.
Defining the dimensionless variable x ? r m??2g/(2kBT ) = r (??gk/?), which
scales r by the ratio of the trap potential energy to the thermal energy kBT , we can
rewrite the integral as
? ? [ ]
2 2 1
K ? dx x2e?x dy e?y ( ( ) )2 ,
(??/2)2 + ?? ??U ? ?y ? ?m 20 2 ? x22k
(5.16)
where for our system, ?m/(2k2) ? ?2.52? 10?6 s.
Returning to Eq. 5.2, we use our expression Eq. 5.16 for K to solve for atom
115
number and obtain
N(?)
= e?K? , (5.17)
N(0)
which can be used as an integral function to fit the four parameters
{a, (? ?0 + ?U0(Itrap)) ,? , ?}, where a is an overall normalization factor for K, in
a least-squares minimization routine. We keep the ?U0(Itrap) term explicit and
highlight its dependence on the optical dipole trap laser intensity Itrap. We use this
expression to extract the AC Stark shift systematic correction.
Note that in theory, the integral in Eq. 5.16 ranges over the entire real line.
In our numerical implementation, we truncate these integrals at finite values. In
our experiment, we typically have Utrap ? 160 kHz, and so Utrap/(kBT ) ? 8 and
we take the position integral out to five times the thermal energy scale. Since the
integrand is convolved by a gaussian, continuing the integration further in the wings
contributes only marginally to the final value, and the truncation does not change
the result above other uncertainties. The ratio Utrap/(kBT ) ? 8 also allows us to
approximate the trap as harmonic.
As an example, we perform a fit using Eq. 5.17 to the loss spectra shown in
Fig. 5.3. It is difficult to visually differentiate the quality of the fit between the full
integral lineshape and a simple Gaussian model, but there is a non-negligible thermal
line shift from a full accounting of the lineshape as is evident in the fit parameters. To
account for this systematic shift, we numerically simulate the systematic Gaussian
fit offset as a function of temperature, and find it to be ?7.6?0.3 kHz/?K, as shown
116
1.1
1.0 1.0
0.9
0.8
0.8
0.7 0.6
0.6 0.4
0.5 88Sr 87
0.4 0.2
Sr
100 75 50 25 0 25 50 75 100 100 75 50 25 0 25 50 75 100
Detuning (kHz) Detuning (kHz)
1.1 1.0
1.0 0.9
0.9 0.8
0.7
0.8
0.6
0.7
86Sr 0.5 84Sr
0.6
75 50 25 0 25 50 75 80 60 40 20 0 20 40 60 80
Detuning (kHz) Detuning (kHz)
Figure 5.3: Lineshape curves for the 698-nm clock transition. The curves include
a Gaussian model (dashed line) and a full lineshape model (solid line) fit to the
averaged data points (circle points). In both cases, the fit error on the centroid
is roughly 1 kHz, however the full lineshape model fits a different ?0 which varies
as a function of temperature and is red of the Gaussian line center by up to 20
kHz. This is attributable to the thermal distribution of atoms in a dipole trap with
inhomogeneous AC Stark shifts.
117
Normalized Atom Number Normalized Atom Number
Normalized Atom Number Normalized Atom Number
(a) (b) 
Figure 5.4: Effects of thermal line shift on the clock transition. (a) Lineshape
simulations as a function of temperature with 0.79 ?K (solid line), 1.6 ?K (dashed
line), 2.7 ?K (dash-dotted line), and 4.1 ?K (dotted line). (b) Systematic offset to
the Gaussian fitted center as a function of temperature and a linear fit to the data
extracted from the simulation in (a).
in Fig. 5.4. With this result and measured temperatures of {2.9 ?K, 2.2 ?K, 1.1 ?K
2.7 ?K}, we obtain a systematic frequency shifts of {?22 ? 4 kHz, ?17 ? 4 kHz,
?8? 4 kHz, ?21? 4 kHz} for {88Sr, 87Sr, 86Sr, 84Sr} respectively.
5.2 Publication: Confinement of an alkaline-earth element in a grat-
ing magneto-optical trap
Our work on the strontium grating MOT was completed in collaboration with
the cold atom vacuum standards (CAVS) group at the National Institute of Stan-
dards and Technology (NIST), especially Daniel Barker, who suggested we try creat-
ing a strontium grating MOT. Our laboratory had the infrastructure at that point
to run strontium laser cooling experiments, and the group at NIST had grating
118
chips, custom vacuum pieces, and experience with grating MOTs. Ananya Sitaram
and I were the senior graduate students on the project together, and we were able
to get our blue MOT up and running relatively quickly. The following publication,
written up during the COVID-19 pandemic and published as Ref. [58], is the result
of that work. We were able to create and characterize the first grating MOT for
alkaline earth atoms.
5.2.1 Abstract
We demonstrate a compact magneto-optical trap (MOT) of alkaline-earth
atoms using a nanofabricated diffraction grating chip. A single input laser beam,
resonant with the broad 1S 10 ? P1 transition of strontium, forms the MOT in
combination with three diffracted beams from the grating chip and a magnetic field
produced by permanent magnets. A differential pumping tube limits the effect of
the heated, effusive source on the background pressure in the trapping region. The
system has a total volume of around 2.4 L. With our setup, we have trapped up
to 5 ? 106 88Sr atoms, at a temperature of approximately 6 mK, and with a trap
lifetime of approximately 1 s. Our results will aid the effort to miniaturize quantum
technologies based on alkaline-earth atoms.
5.2.2 Introduction
Laser-cooled alkaline-earth atoms have applications in a wide range of quantum
devices, including atomic clocks [84, 107, 108], gravimeters [31], and spaceborne
119
gravitational wave detectors [33, 34]. The transition from a laboratory to field-
based applications will require a drastic reduction in the size and complexity of laser-
cooling systems. For example, proposals to detect gravitational waves using alkaline-
earth atoms require atom interferometers capable of being installed in satellites [34].
Compact versions of these laser-cooled systems are also necessary in order make the
unprecedented accuracy of alkaline-earth atomic clocks widely accessible [36].
Laser-cooling experiments typically use a magneto-optical trap (MOT) to cap-
ture, cool, and confine the atoms. Conventional MOTs use three orthogonal pairs
of well-balanced, counterpropagating laser beams to confine atoms at the center of
a quadrupole magnetic field. As such, MOTs require large vacuum chambers with
optical access along all axes, and have many degrees of freedom in alignment and
polarization. Compound optics can generate all necessary beams from a single in-
put beam, reducing the complexity of the optical setup. For example, pyramidal
retro-reflectors maintain the beam geometry of conventional MOTs [109]. However,
the MOT forms inside the retro-reflecting optic, limiting optical access [110, 111].
Tetrahedral reflectors form the MOT above the optic, maintaining optical access
but breaking the geometry of a conventional trap by using only four beams [112].
Tetrahedral MOTs can also be planarized by using diffraction gratings [46, 113, 114].
Thus far, only experiments with alkali atoms have been successfully miniaturized
using such grating MOTs [47, 113, 114]. Here, we demonstrate a compact, grating
MOT system for alkaline-earth atoms.
Alkaline-earth atoms pose unique challenges to miniaturization. First, sources
for alkaline-earth atoms must be heated to high temperatures (over 350 ?C) to
120
create sufficient flux of atoms to load a MOT. Outgassing from the hot source
can increase the background pressure, decreasing the trap lifetime, and equilibrium
atom number. Second, alkaline-earth atoms require large magnetic field gradients
(on the order of 5 mT/cm), often created with large water cooled coils [115]. Third,
with the high Doppler temperature of the broad 1S ? 10 P1 transition, and lack
of sub-Doppler cooling, strontium and other alkaline-earth systems usually operate
a second, subsequent MOT on the narrow 1S0 ? 3P1 transition to achieve lower
temperatures. The ideal compact system must have the capability to operate at the
two different cooling wavelengths.
Our system, designed around a diffraction grating chip, mitigates the above
issues associated with miniaturizing a MOT for alkaline-earth atoms (see Fig. 5.5).
First, a 3 cm long differential pumping tube separates the vacuum chamber into two
regions: the source chamber and the science chamber. The source chamber con-
tains a vacuum pump and a low-outgassing dispenser [116] that vaporizes strontium
atoms. The atoms then travel through the differential pumping tube before entering
the science chamber. Second, we create the magnetic field gradient for the MOT
using permanent magnets, which are less complex than typical, water-cooled coils.
The magnetic field gradient extends into the differential pumping tube, forming an
effective Zeeman slower when combined with the input laser beam. Lastly, the first
order diffraction efficiency of the grating we use is optimal at a wavelength of 600 nm,
a middle ground between the two laser-cooling wavelengths (461 nm and 689 nm)
for strontium. Our compact alkaline-earth grating MOT system also maintains the
optical access and achieves the atom number necessary for future quantum devices.
121
Figure 5.5: A cut-away model of the grating MOT system, with coordinates speci-
fied in the bottom right. The diffraction grating in the middle of the science cham-
ber diffracts light to form the MOT beams. The input laser beam (not shown)
propagates along the +z? direction. 3D-printed magnet holders position permanent
magnets around the top and bottom of the chamber. The top magnet holder is
translucent to show the configuration of the magnets in the holders. The polarity of
the magnets is shown with red and blue coloring. The dispenser source sits below
the differential pumping tube and is pumped by a non-evaporable getter (NEG)
pump.
122
5.2.3 Apparatus
Our apparatus is shown in Fig. 5.5. The vacuum system is comprised of two
chambers, separated by a 3 cm long, 3 mm diameter differential pumping tube with
an N2 conductance of 0.11 L/s. The MOT is located in a science chamber with
four CF275 [91] viewports, and pumped with a 75 L/s ion pump (not shown). The
source chamber is located below the differential pumping tube and is pumped with
a 40 L/s non-evaporable getter (NEG) pump. Our source of Sr atoms is a 3D-
printed titanium dispenser, described in Ref. [116]. We run a current between 12 A
and 14 A through the dispenser, effusing strontium towards the differential pumping
tube. Together, the source and science chambers are approximately 2.4 L in volume,
although this estimate does not include the ion pump or the magnet holders. In
typical strontium experiments, the sources alone are often at least 2 L in volume.
The base vacuum pressure of 2? 10?7 Pa in the science chamber could be improved
by replacing the large ion pump with a small hybrid NEG/ion pump, which would
also reduce the size of the apparatus.
The grating chip is located above the differential pumping tube, and has a
triangular hole through its center, allowing atoms to enter the science chamber. The
grating chip was fabricated at the National Institute of Standards and Technology,
and consists of three linear gratings arranged in a triangle. The parameters of the
chip are the same as those in Ref. [47], except with a trench depth of 150(2) nm.
This trench depth minimizes the 0th order diffraction at 600 nm, which is between
the 461 nm and 689 nm cooling transition wavelengths for strontium. Each linear
123
grating diffracts 32(1) % of the normally-incident 461 nm light into each of the ?1
diffraction orders with an angle of 27.0(5)?. 4 % of the light is diffracted into each of
the ?2 orders, and 11 % is diffracted into the 0 order, with the remainder lost due
to the aluminum coating [117]. The 0 order light does not disrupt the MOT because
of the hole in the grating directly beneath it. The diffraction angle is a trade-off
between confinement of the 461 nm MOT and overlap with the second stage 689 nm
MOT beams. For normally incident, circularly polarized light, the stokes parameters
of the grating chip at 461 nm are Q = ?0.23(1), U = ?0.13(1), V = 0.96(1), where
Q = 1 (Q = ?1) corresponds to s (p) polarization defined relative to the plane of
reflection for each linear grating.
Two sets of grade N52 NdFeB magnets, arranged roughly in a dodecagon,
create the magnetic field for the MOT. Within each set, the poles of the magnets
are aligned. The magnets are housed in 3D-printed magnet holders made of poly-
lactic acid (PLA) that are designed to produce a compact setup with high magnetic
field gradients, as shown in Fig 5.5. Due to the geometric constraints of the vac-
uum chamber, the configuration of magnets is asymmetric, and the principal axes
are rotated from those Fig 5.5. We achieve maximum gradients of {3.5 mT/cm,
2.7 mT/cm, 6.2 mT/cm} along the {x??, y??, z?} axes, respectively, where x?? and y?? are
rotated by ??/6 from x? and y?. By removing magnets from the holders, we can lower
the gradient to {1.9 mT/cm, 1.9 mT/cm, 3.8 mT/cm} along the {x??, y??, z?} axes, re-
spectively. The field gradient extends to z ? 50 mm, where z = 0 corresponds to
the B = 0 and z ? 40 mm corresponds to the position of the source.
A single laser beam, red-detuned from the 1S ? 10 P1 transition at 461 nm,
124
enters through the top viewport along the +z? axis and is normally incident upon
the diffraction grating chip. The input MOT beam has a 1/e2 radius of 12 mm and
a maximum power of 92 mW. For the 1S 10 ? P1 transition with natural linewidth
?/2? = 30.5 MHz, Isat = 40.3 mW/cm
2, giving a maximum peak I/Isat ? 1.
Intensities I/Isat reported herein always refer to the peak intensity of the input
beam. The central portion of the beam continues through the hole in the diffraction
grating and through the differential pumping tube. This beam, combined with the
magnetic field gradient, allows for a small amount of initial slowing of the atoms,
similar to a Zeeman slower. Atoms can be lost from the MOT because the excited
1P1 state decays at a rate of 610 s
?1 to the 1D2 state, which in turn decays to the
3P
manifold. To mitigate the atom loss, two repump lasers, with wavelengths 679 nm
and 707 nm, address the 3P0 ? 3S1 and 3P ? 32 S1 transitions, respectively. More
information on the repump scheme can be found in Ref. [42]. The repump beams
are combined together on a 50/50 beam splitter, and then combined with the input
MOT beam using a polarizing beam splitter.
We use absorption and fluorescence imaging along x? to characterize the MOT.
Absorption images are taken after the MOT atom number equilibrates using a probe
beam resonant with the 1S 10? P1 transition with I/Isat ? 0.01. We use the atom
number from the absorption images to calibrate the atom number extracted from
fluorescence images taken during loading. The Labscript suite software [67] controls
the experiment and data collection. More detailed information on the laser systems
can be found in Ref. [5].
125
Figure 5.6: MOT loading curves with a source current of 13 A, I/Isat = 1, axial
magnetic field gradient of 6.2 mT/cm, and detuning ?/? = ?1. The blue dots
show the MOT atom number N as a function of time t. The black curve is a fit to
Eq. 5.18. The green triangles show the MOT loading without repump lasers, and
subsequent recapture from the metastable reservoir. The dashed line indicates when
the repump lasers were turned on.
5.2.4 Results
We measure atom number, loading rate, lifetime, and temperature to charac-
terize the MOT. During each experimental shot we take a sequence of fluorescence
images while the MOT loads and construct a loading curve. Fig. 5.6 shows typical
loading curves at an axial magnetic field gradient of 6.2 mT/cm. For a MOT with
no light assisted collisions, the loading rate R, MOT lifetime ? , and equilibrium
126
atom number N0 = R? , are extracted by fitting each loading curve to the single
exponential
N(t) = R?(1? e?t/? ). (5.18)
An example fit is shown with a solid black curve in Fig. 5.6. The quality of the fit to
Eq. (5.18) indicates light assisted collisions and secondary scattering are negligible.
At the higher gradient of 6.2 mT/cm, we observe typical loading rates of 4?106 s?1
and a vacuum-limited lifetime of 1 s. We observe a similar loading curve at the
lower gradient of 3.8 mT/cm.
Fig. 5.7 and Fig. 5.8 show the MOT parameters as a function of detuning
from resonance and intensity, respectively. We find the maximum atom number
of approximately 4 ? 106 at a source current of 13 A and ?/? ? ?1, a typical
detuning for a conventional 6-beam Sr MOT [118?120]. As shown in Fig. 5.8(a),
the atom number continues to increase with I/Isat, even at our maximum intensity,
indicating that more laser power would be beneficial. The increase in N0 = R? is
only partially due to the increase in the loading rate R, shown in Fig. 5.8(b). Part of
the atom number increase is due to an increase in the lifetime with intensity, shown
in Fig. 5.8(c). The lifetime increase suggests that the trap depth is increasing
with laser power, which in turn increases the escape velocity for a Sr atom that
undergoes a background gas collision [121, 122]. However, the interplay between
MOT temperature and tighter radial confinement with increasing intensity may
also play a role.
We can also use the MOT to continuously load a magnetic trap, which con-
127
Figure 5.7: The equilibrium atom number N0 as a function of MOT beam detuning
?/?, with a source current of 13 A, axial magnetic field gradient of 6.2 mT/cm, and
I/Isat = 1. The optimal detuning is ?/? = ?1. Most of the error bars are smaller
than the data points, and represent the standard error about the mean.
128
Figure 5.8: MOT loading parameters as a function of I/Isat: (a) equilibrium atom
number N0, (b) loading rate R, and (c) lifetime (?). Here, the source current is
13 A, the detuning ?/? = ?1, and axial field gradient is 6.2 mT/cm. The error
bars on the points are comparable to the marker size, and represent the standard
error about the mean.
129
sists of atoms that are trapped in the metastable 3P2 state. With strontium, the
metastable magnetic trap is often used to increase the capture of rare isotopes and
was key to the realization of quantum degeneracy [119, 123]. By operating the MOT
without repump light, atoms are shelved in the 3P2 state where they are trapped
by the MOT magnetic field. When the repump light is turned on after atoms have
accumulated in the magnetic trap, we see a sharp increase in the MOT atom num-
ber as shown in Fig. 5.6. The recovery confirms that atoms are being caught and
held in the magnetic trap, however we do not see a transient enhancement above
the equilibrium atom number as demonstrated elsewhere [124]. Given our densities,
and vacuum-limited atom number, we would not expect enhancement from mag-
netic trap loading. Adding a depumping laser could enhance the loading rate of the
magnetic trap and increase the atom number [43].
We investigate the effect of the source current on the atom number, loading
rate, and lifetime, shown in Fig. 5.9. The source current sets the temperature of the
source, which in turn determines both the vapor pressure and the average velocity of
atoms leaving the source. At our highest achievable source current of 14 A, limited
by the ampacity of our electrical feedthroughs, we trap 5 ? 106 atoms, but have
still not saturated the atom number. Based on the fit presented in Ref. [116], we
estimate the source temperature at 13 A to be over 600 ?C. When the source current
is increased from 0 A to 13 A, the vacuum pressure in the science chamber increases
by 3? 10?8 Pa, suggesting that the differential pumping is sufficient. The increase
in pressure is consistent with the small lifetime decrease shown in Fig. 5.9(c).
To determine the temperature of the MOT, we measure the width of the atomic
130
Figure 5.9: MOT loading parameters as a function of source current: (a) equilib-
rium atom number N0, (b) loading rate R, and (c) lifetime (?). Here, I/Isat ? 1,
?/? = ?1, and axial field gradient is 6.2 mT/cm. The error bars on most of the
points are comparable to the marker size, and represent the standard error about
the mean.
131
Figure 5.10: Temperature measurement of the atomic cloud. The rms width of the
atomic cloud in the y? direction (blue circles) and in the z? direction (green triangles)
are plotted against time of flight and fitted to the expansion function discussed in the
text (black curves). The calculated temperatures based on the fits are 7.8(9) mK
and 4.6(4) mK for y? and z?, respectively, where the errors in parentheses are one
standard deviation. This data was taken with a source current of 13 A, I/Isat = 1,
axial magnetic field gradient of 6.2 mT/cm, and detuning ?/? = ?1. The error
bars represent the standard error about the mean.
132
cloud as it expands in time of flight, shown in Fig. 5.10. A Gaussian fit extracts
the root-mean-square (rms) width, w, of the cloud in both the y? and z? directions.
The extracted widths are binned by time of flight, and the error bars are calculated
from the standard error about the mean. We fit the da?ta to w(t)
2 = w0(t)
2 +v2 2rmst ,
where w0 is the initial rms width of the cloud, vrms = kBT/m is the rms velocity,
kB is Boltzmann?s constant, m is the atomic mass, and T is the temperature of the
atomic cloud. The temperature is 7.8(9) mK and 4.6(4) mK for y? and z?, respectively,
where the errors in parentheses are one standard deviation, which is consistent
with a conventional six-beam MOT [118]. The temperature is not equal in the
two dimensions because the diffusion coefficient and velocity damping constant are
different in the axial and radial directions in a grating MOT. For our MOT, the
ratio of the temperatures along y? and z? is 1.7(2), consistent with the ratio of 1.9
from the theory in Ref. [49].
While the discussion has focused on trapping 88Sr, strontium has a number
of stable isotopes. The isotope abundances for strontium are 82.58 %, 7.00 %,
9.86 %, and 0.56 % for 88Sr, 87Sr, 86Sr, and 84Sr, respectively. Our setup can also
trap around 7 ? 105 atoms of 86Sr at a source current of 13 A, consistent with the
abundances above. Likewise, we would also expect to trap around 5 ? 105 atoms
of 87Sr, but were unable to realize a MOT of 87Sr. The hyperfine structure of
87Sr poses at least two complications. First, we might not have sufficient repump
power to adequately address all necessary hyperfine transitions [8]. Second, the
hyperfine structure combined with the non-trivial geometry and polarizations of
the grating MOT may significantly weaken the already limited transverse confining
133
forces [44, 113]. The theoretical details of the latter are beyond the scope of this
work and will be presented in a future publication.
5.2.5 Discussion
We have realized a grating MOT of alkaline-earth atoms in a compact 2.4 L
apparatus. Our permanent magnet design supplies the necessary field gradients for
the MOT and allows for a degree of tunability, while the differential pumping tube
limits outgassing from the hot source. The MOT traps up to 5?106 atoms of 88Sr at
a loading rate of 4?106 s?1, with a lifetime of approximately 1 s. This performance is
comparable to vapor loaded, six-beam strontium MOTs [118, 125]. We also observe
MOTs of 86Sr with 7? 105 atoms, consistent with the relative isotopic abundance.
In the future, upgrades to our apparatus could be made to improve the per-
formance of the MOT and decrease the size of the system. We could improve the
quality of the vacuum and reduce the size of the apparatus by replacing the ion
pump with a hybrid NEG/ion pump. Improving the quality of the vacuum would
increase atom number and lifetime, potentially allowing us to observe a MOT of
84Sr. The system could be further miniaturized by using a fiber-coupled and pho-
tonically integrated chip to expand the MOT beam to the appropriate size without
additional optics [126, 127].
With additional upgrades to our apparatus, we would be able to transfer atoms
to a second stage MOT operating on the narrow 1S 30 ? P1 transition at 689 nm.
The grating has good diffraction efficiency at both the 461 nm and 689 nm cool-
134
ing wavelengths. In addition, 43% of the capture volume created by the 689 nm
diffracted beams overlaps with the 461 nm capture volume, facilitating transfer be-
tween the MOTs. As discussed in Ref. [113] and Ref. [128], the diffracted beams
of a grating MOT have a complicated polarization projection onto the z? axis. The
mixed polarization projection of the diffracted beams reduces the confining force,
which might limit the second stage MOT because the force due to gravity is no
longer negligible. The second stage MOT also requires a low magnetic field gradi-
ent to operate, which we could achieve by incorporating electromagnets with our
permanent magnet assembly. Due to its low field gradient, the second stage MOT
is highly sensitive to stray magnetic fields, thus we would also need to incorporate
shim coils to ensure proper positioning. The electromagnets and shim coils would
also allow us to null the magnetic field to allow operation of optical clocks. We plan
to make these upgrades to the apparatus and attempt second stage cooling in future
experiments.
The implementation of field-deployable quantum devices relies on compact sys-
tems. Alkaline-earth-based quantum sensors have been proposed as platforms for
atom interferometers and atomic clocks. Compact interferometers could be used for
inertial navigation [32], and gravitational wave detection in space [34]. Deployable
networks of optical clocks will be important for improved time and frequency metrol-
ogy [36], and tests of fundamental physics [35]. Our results show that alkaline-earth
grating MOTs are a promising step towards the development of compact optical
clocks and other quantum devices.
135
5.2.6 Acknowledgements
Ananya Sitaram and Peter Elgee contributed equally to this work. We thank
Francisco Salces Carcoba and Hector Sosa Martinez, for their careful reading of the
manuscript. We also thank the NIST Center for Nanoscale Science and Technology
NanoFab staff for allowing us to use the facility to fabricate grating chips. This
work was partially supported by the NSF through the Physics Frontier Center at
the Joint Quantum Institute.
5.2.7 Data Availability
The data that support the findings of this study are available from the corre-
sponding author upon reasonable request.
136
Chapter 6: Future Ideas and Conclusion
Our new strontium apparatus opens the way for more experiments in the lab.
Our new high flux source provides high atom numbers, our chamber has a large
amount of optical access for additional beams, and our new imaging system allows
for the creation of high resolution optical potentials, and imaging. With these
improvements, the lab is ready to move forward with the next set of experiments.
We have a variety of ideas for future directions to take the laboratory, including
quantum simulation in a 1D box trap with SU(N) ? Sn symmetry, Rydberg spin
squeezing of the optical clock transition, the creation of sub-wavelength attractive
potentials with three-level systems, and a variety of new direction for the grating
MOT experiment. These possible experiments are outlined in this chapter.
6.1 Box Trap
Quantum simulation relies on the creation of hamiltonians that match that of
a system to be studied. Hamiltonians with SU(N) ? Sn symmetry can be used to
study spin models with infinite range interactions [16] or estimate the spectrum of
an N dimensional density matrix that is encoded in the nuclear spin of each atom in
the box trap [17]. As mentioned in the Sec. 1.1, the nuclear spin of 87Sr creates an
137
SU(N) symmetry, where N ? 10, in the ground and clock states. This symmetry
can be expanded to SU(N) ? Sn, where n is the number of atoms, by putting the
atoms in a 1D box trap potential.
We would create this system by loading ultracold atoms into a tight dipole
trap, such that the atoms only occupy the lowest energy level of the harmonic
oscillator potential in the transverse x and y directions. In the z direction we would
add the walls of the 1D box trap with a repulsive dipole trap, and allow the atoms
to occupy any motional level of the box trap. By initially spin polarizing the gas,
we could ensure that no level is doubly occupied. The wavefunctions for the levels
of the box trap are ?
?j(z) = 2/L sin(j?z/L)
We can model the interactions between atoms in the box trap with a pseudo-
potential V (~r) ? ?(~r), where ~r is the separation between the atoms. Then the
interaction energy between atoms initially in states ?i(z)?j(z
?) scattering to states
?i?(z)?j?(z
?) becomes
? L
U ? ?i?(z)?j?(z)?i(z)?j(z)dz
0
where we have integrated over the transverse directions and integrated the delta
function over z?. From energy conservation considerations and this integral, the
result is that U is a constant when (i, j) = (i?, j?) or (j?, i?), and zero otherwise.
Thus this interaction can swap states of atoms between two box trap levels, and the
138
energy is independent of the levels involved, this is associated with a permutation
symmetry Sn for the n levels of the box trap that are occupied. For the ground
and clock state where the nuclear spin is decoupled from the electronic angular
momentum, this implements an SU(N)? Sn symmetry.
We have started to implement the box trap in the laboratory. Initially, we
planned on using a digital mirror display (DMD) to generate the walls of the box
trap, but the low efficiency at 461 nm prompted us to switch to a printed mask.
We have gotten the mask printed with a variety of sizes to generate an ? 10 ?m
length trap with our current imaging system. We plan on using the old 461 nm
system to generate the blue-detuned repulsive dipole trap that will create the walls
of the trap, and using the high-resolution imaging system to image this potential
onto the atoms. For these experiments to work, the walls of the box trap must be
high enough, and the atoms cold enough, that the occupied box trap levels maintain
the Sn symmetry. In addition, the transverse dipole trap must be deep enough, that
at the given temperature only the lowest motional level of this trap is excited.
6.2 Rydberg Spin Squeezing
One of the limits on strontium optical clocks is the quantum projection noise
related to measuring the clock transition on independent atoms [129]. To beat this
limit and decrease the noise on the measurement entanglement has to be added to the
system so the measurements are no longer independent. By adding interactions, and
generating ?spin-sqeezing? in the ensemble of atoms, the noise in the measurement
139
of the proportion of atoms in the ground |g? or clock |c? states is improved at the
expense of the noise in the phase (when operating on the equator of the Bloch
sphere).
One way to add interactions and implement this spin squeezing is by dressing
the clock state with a Ryberg state |r? so that the Rydberg dressed clock state
is |c?? ? |c? +  |r? [28, 29]. Due to the large electric dipole moments of Rydberg
atoms, they interact strongly through the Van der Waals force. This interaction can
be used to implement entanglement and spin squeezing in an ensemble of atoms.
Our laboratory has a laser operating at ? 317 nm, mentioned in Sec. 3.3.6 to address
Rydberg levels off the clock transition and implement this dressing for investigations
of spin squeezing in strontium optical clocks.
6.3 Sub-wavelength lattices
Most often potentials for quantum simulation in cold atom systems are created
with optical lattices and dipole traps. In general, the spatial scale of these potentials
is limited by the wavelength of the light used. As an alternative, the Rb/Yb mixtures
group at the JQI has used a three-state system to create effective potentials with
structure smaller than the wavelength of the coupling lasers [130]. The basic idea
of this scheme uses a homogeneous probe field ?p that couples the ground state
to the intermediate state, and a spatially varying control field ?c that couples the
intermediate state to the third state. This system then has a dark state that varies
in space, when ?p  ?c the dark state is the ground state, and when ?c  ?p the
140
dark state is the third state. In between these two extremes the dark state (and
energy) varies quickly in space creating the sub-wavelength structure of the lattice.
So far this scheme has only been used to create repulsive sub-wavelength structure,
but it can create attractive potentials with the right parameters [30]. Based on my
discussions with Sarthak Subhankar, it seems that attractive potentials might be
easier to achieve with strontium than with ytterbium, motivating this as a potential
experiment for the new strontium laboratory. Sub-wavelength attractive potentials
have a variety of applications from expanding the parameter space of quantum
simulators, to improving quantum information processing.
6.4 Grating MOT Ideas
Our strontium grating MOT started as a side project, but it has proven to be
an interesting avenue to explore. There are several improvements and new direc-
tions the project could take. The most immediate goal is to find a red MOT of 87Sr
in this system. As mentioned in Sec. 2.3, the interaction of the narrow linewidth,
hyperfine structure, and complex polarizations in this system make it hard to in-
tuitively understand which conditions will provide stable operation. Despite this,
our simulations have indicated that it should be possible to operate a red grating
MOT of 87Sr by stirring on the F = 9/2? 7/2 hyperfine transition. So far we have
not found a signal, but are hopeful that by increasing the power, narrowing the
parameter space to search, and using the higher signal to noise ratio of fluorescence
imaging we will find something soon.
141
The next step to further prove the usefulness of our strontium grating MOT
system, could be to build an optical clock or atom interferometer. These systems
would require a redesign of the chamber to allow for more optical access and the
inclusion of an optical lattice or in the case of the interferometer a tower to increase
the path length. There are also upgrades that could be made to the apparatus
regardless of intent including improving the size and efficiency of the pumping, re-
implementing an atomic shutter that will not jam, and redesigning the coils and
coil electronics to produce both the red and blue MOT gradients with a single set
of coils.
Another direction could be combining the strontium grating MOT and lithium
grating MOT from the CAVS laboratory at NIST. The lithium grating MOT is used
as a pressure standard by looking at atom loss out of a magnetic trap, with the
ability to measure pressure more accurately than ion gauges [131]. By varying the
background pressure in the chamber, and measuring it with the lithium trap as a
pressure standard, we could measure the loss of strontium atoms and extract the
scattering cross section. Background gas collisions are estimated to cause shifts to
optical clocks at the 10?19 level [12, 13], so by doing this measurement we would be
showing a proof of principle for the CAVS system, and aid the next generation of
optical clocks.
142
6.5 Conclusion
During my time in graduate school, I was fortunate to help complete an ex-
periment on a mature apparatus. We precisely measured the isotope shifts of the
clock and intercombination transitions in strontium, finding a deviation from King
linearity for 87Sr. Additionally, we were able to build a new and improved strontium
laboratory for a new generation of experiments. We were able to work through all
the complexities of laser cooling strontium, ending with ultracold strontium atoms
loaded in optical dipole traps, and set up the apparatus for the future of the group.
While building up this apparatus, we were lucky to have Dan Barker suggest we
make a strontium grating MOT. We developed the first alkaline earth grating MOT,
a step towards realizing practical compact quantum devices that take advantage of
the unique properties of strontium. This project gave us a lot to work on while
the COVID-19 pandemic slowed down the speed of progress on the main apparatus.
From our continued work on this system, we were able to complete the transfer
between the broad and narrow line grating MOTs and show that SWAP works
efficiently in the complex polarizations of a grating MOT.
143
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