ABSTRACT
Title of Dissertation: Analogue Cosmology Experiments
with Sodium Bose-Einstein Condensates
Swarnav Banik
Doctor of Philosophy, 2021
Dissertation Directed by: Dr. Gretchen Campbell
Department of Physics
Due to their high degree of controllability and precise measurement capabilities,
ultracold ensembles of neutral atoms are a leading platform for performing quantum
simulations. In this thesis, I will describe the design and construction of an analog
quantum simulator based on 23Na Bose-Einstein Condensates (BEC). Our system can
produce and trap BECs in arbitrary-shaped quasi two-dimensional optical dipole traps,
which can be dynamically altered during an experimental sequence. Such controlled
variation of the BEC?s spatial mode enables exploration of open questions in superfluidity,
atomtronics, and analogue cosmology. I will describe the implementation of our system
to study the inflationary dynamics of the early universe and report our recent results on the
simulation of cosmological Hubble friction. We expand and contract a toroidally shaped
BEC and analyze the time evolution of its collective phonon modes. These excitations are
analogous to fluctuating scalar fields in an expanding universe. The changing metric of the
expanding or contracting background BEC results in dilation of the phonon field through
a term dependent on the expansion speed, similar to Hubble friction in inflationary models
of the universe. We conclusively demonstrate the analogy by experimentally measuring
Hubble attenuation and amplification. Our measured strength of Hubble friction disagrees
with recent theoretical work [J. M. Gomez Llorente and J. Plata, Phys. Rev. A 100
043613 (2019) and S. Eckel and T. Jacobson, SciPost Phys. 10 64 (2021)], suggesting
inadequacies in the current model.
ANALOGUE COSMOLOGY EXPERIMENTS
WITH SODIUM BOSE EINSTEIN CONDENSATES
by
Swarnav Banik
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
2021
Advisory Committee:
Professor Steven Rolston, Chair
Dr. Gretchen Campbell, Co-chair/Advisor
Dr. Ian Spielman
Professor Rajarshi Roy
Professor Mohammad Hafezi, Dean?s Representative
? Copyright by
Swarnav Banik
2021
Acknowledgments
My experience in the ?Laser Cooling and Trapping? group at the Joint Quantum
Institute (JQI) has been enriching. The training and exposure received during the past
few years has not only had a transformational impact on my problem-solving approach
but also helped me appreciate the importance of collaborations in doing so. During my
Ph.D. I had the privilege to work with and learn from incredibly talented researchers.
The collaborative environment fostered by Bill Philips, Gretchen, Ian, Trey Porto, Steve
Rolston, Alicia Koller, and former graduate students Daniel Barker, Benjamin Rechovosky,
Neal Pisenti, Zach Smith, Varun Vaidya, Dan Campbell, Ana Valdes Curiel, Francisco
Salces Carcoba (Paco), Dalia Ornelas-Huerta, and Avinash Kumar made it a delight to
work in this group. As a new graduate student, I had a minimal experimental background
and was in absolute awe of the massive homemade multi-component quantum simulators
at JQI. A major part of my doctoral work involved the construction of one such setup.
This would not have been possible without the help and support that I received from my
colleagues. Here is my humble attempt to acknowledge and thank them for making this
experience as fulfilling as it was.
First of all, I would like to thank my advisors, Gretchen Campbell and Ian Spielman,
for being such great mentors. Throughout my Ph.D., their patience and encouragement
were instrumental in helping me develop my intellectual faculties and grow as an independent
ii
researcher. Gretchen?s ability to see through complex problems and bring our attention to
what matters most has helped us establish priorities for the project. Ian?s ability to adapt
relevant experimental techniques from different fields of research is impressive. It was
a privilege to learn from such great researchers. The construction of this lab would not
have been possible without the efforts of my fellow labmates Monica Gutierrez Galan,
Hector Sosa Martinez, and Madison Anderson. Hector?s presentation skills and Monica?s
workflow design and systematic approach to problem-solving are some of the many
invaluable skills I?ll take with me from grad school. When I was a young graduate student,
our former labmate Avinash Kumar patiently taught the basics of an AMO experiment,
such as optical alignment. I am very thankful for his eagerness to help.
I would also like to thank our electronics guru Alessandro Restelli for his help on
many electronic projects which now power our apparatus. Over the years, my interactions
with members of the Strontium lab, Neal Pisenti, Peter Elgee, and Ananya Sitaram, have
resulted in many productive collaborations. Insightful discussions with fellow JQI grad
student Sarthak Subhankar has helped me identify gaps in my understanding. Discussions
with our new post-doc Shouvik Mukherjee and my roommate Tamoghna Barik have
allowed me gain a perspective different from AMO physicists.
My friends Kumi Wickramanayaka, Umesh Kumar, Subhojit Dutta, Yidan Wang,
Nicholas Mennona, Sabyasachi Barik, and Nitish Mehta have been a constant source of
support. I am thankful to Kumi for cheering me up when things didn?t work in the lab. My
undergraduate classmate Umesh has always kept in touch, and I am genuinely grateful for
having him as a friend. Lastly, my family has been a great source of support throughout
these years. My parents have always gone out of their way to help me realize my dreams.
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I?m grateful for having such a caring family.
iv
Table of Contents
Acknowledgements ii
Table of Contents v
List of Tables vii
List of Figures viii
List of Abbreviations xv
Chapter 1: Introduction 1
1.1 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Chapter 2: Theory of Bose Gas 5
2.1 Ideal Non-interacting Bose Gas . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Mean Field Dynamics of Weakly Interacting BECs . . . . . . . . . . . . 6
2.2.1 Thomas Fermi Approximation . . . . . . . . . . . . . . . . . . . 9
2.2.2 Collective Excitations: Phonons . . . . . . . . . . . . . . . . . . 10
2.2.3 Damping of Low-energy Excitations . . . . . . . . . . . . . . . . 12
Chapter 3: Analogue Gravity 14
3.1 Previous Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 BEC Phonon Wave Equation and the Space-time Metric . . . . . . . . . . 17
3.3 Wave Equation in a Toroidal BEC . . . . . . . . . . . . . . . . . . . . . 20
Chapter 4: Experimental Apparatus 23
4.1 Vacuum Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2 Laser Frequency Stabilization . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3 Optical Dipole Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.3.1 Red-detuned Optical Dipole Trap . . . . . . . . . . . . . . . . . . 31
4.3.2 Blue-detuned Optical Dipole Trap . . . . . . . . . . . . . . . . . 33
4.4 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.4.1 Partial Transfer Absorption Imaging . . . . . . . . . . . . . . . . 39
4.4.2 Probe Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . 42
Chapter 5: Production of BEC 46
5.1 2D MOT Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
v
5.2 Laser Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.3 Forced Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.4 Transfer to Purely Optical Traps . . . . . . . . . . . . . . . . . . . . . . 59
Chapter 6: The Hubble Friction Experiment 62
6.1 Publication: Hubble Attenuation and Amplification in Expanding and
Contracting Cold-Atom Universes . . . . . . . . . . . . . . . . . . . . . 64
6.1.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.1.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.1.3 Experiment and Results . . . . . . . . . . . . . . . . . . . . . . . 69
6.2 Model for the Toroidal Potential and Scaling Laws . . . . . . . . . . . . . 75
6.3 Simulation: Impact of Phonon Phase on Hubble Friction Strength . . . . . 77
6.3.1 Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.3.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.4 Mode Purity of the Phonons . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.5 Fitting Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Chapter 7: Towards Erbium BEC: Scanning Transfer Cavity Lock 89
7.1 Laser Cooling of Erbium . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.2 Frequency Stabilization of 583 nm Laser . . . . . . . . . . . . . . . . . . 94
7.3 Fabry Perot Etalons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.4 Construction of the Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.4.1 Cavity Construction . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.4.2 Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.5 Results and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Chapter 8: Conclusions and Outlook 115
Appendix A: Characterizing Imaging Aberrations 117
Bibliography 124
vi
List of Tables
4.1 Detunings and intensities of the different laser cooling beams. For the
first five entries, detunings are from the cooling transition. For the last
three entries, detunings are from the repump transition. For reference, the
saturation intensity for the cooling transition is I 2sat = 6.26 mW/cm . . . 31
6.1 Best fit global parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.2 Fit results of the eight different techniques when applied to the expansion
data sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.3 Fit results of the eight different techniques when applied to the contraction
data sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.1 Erbium laser cooling transitions [1]. . . . . . . . . . . . . . . . . . . . . 94
7.2 Thermal expansion of cavity parts. . . . . . . . . . . . . . . . . . . . . . 104
vii
List of Figures
3.1 A moving fluid creating an acoustic horizon when supersonic flow avoids
upstream movement of sound waves. The colored circles represent time-
evolution of acoustic wavefronts in the fluid space-time. ex ? ey plane
represents spatial coordinates. Time is expressed in different colors according
to the color bar to the left. The vertical red lines denotes the sonic horizon.
(a) and (b) depict how the wavefront time-evolution changes with varying
fluid velocity v and sound speed cs, respectively. . . . . . . . . . . . . . 15
4.1 Schematic of the vacuum chamber. The source chamber, science chambers,
and the sodium oven are indicated. The blue circle points to the location
of the differential pumping tube, with the inset showing a schematic of the
tube. The yellow line indicates the push laser beam used for transferring
atoms from the source to the science chamber. . . . . . . . . . . . . . . . 24
4.2 Cross-section of the differential pumping tube. . . . . . . . . . . . . . . . 26
4.3 Energy levels for the 23Na D2 transition [2]. The yellow and green lines
indicate the cooling and repump transitions separated in frequency by 1.7
GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.4 Schematic of the optical setup inside the Toptica DL-pro laser [3]. The
inset shows a magnified view of the ECDL, indicating the piezo actuated
grating, which can be externally controlled via the voltage Vpiezo. . . . . . 28
4.5 Optical schematic of the SAS setup. The yellow lines indicate light from
a Toptica DL-pro laser which is first upshifted by 150 MHz and then
split into pump and probes. The signal from a balanced photodetector,
as shown in the rectangular box, is sent to a lock-in amplifier. . . . . . . . 29
4.6 (a) Lock-in scheme to obtain the derivative of the SAS absorption signal
VPD. The blue and purple lines represent VPD and the derivative signal
Vlockin respectively. The dashed box indicates the components of a lock-in
amplifier. The local oscillator is a Voltage Controlled Oscillator (VCO)
that drives the AOM. The servo generates the signal Vpiezo to control
the laser frequency. (b) VPD and Vlockin indicated in blue and purple,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
viii
4.7 Propagation of ODT beams in the science chamber. The main figure
shows the red-detuned ODT propagating along positive ex and being focused
down at the center of the chamber. The blue-detuned sheet ODT propagates
along positive ey. The inset shows beam propagation on the ey?ez plane.
The blue-detuned DMD ODT propagates along negative ez. Gravity acts
along negative ez. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.8 Schematic for the sheet trap optical setup. Blue-detuned laser light from
a single-mode fiber is collimated and propagated through lenses L1, L2,
L3 and a ? phase plate along the ey direction. (a) and (b) shows the beam
profile in the ez ? ey and ex ? ey planes, respectively. Figures (c) and (d)
show the simulated ez?ex beam intensity profiles at indicated Y positions,
with ez along the vertical axis and ex along the horizontal. The color bar
encodes the relative beam intensity for figures (c) and (d). The atoms, as
depicted by the yellow dots, lie close to the front focal plane of lens L3.
Gravity acts along negative ez. . . . . . . . . . . . . . . . . . . . . . . . 34
4.9 Simulations exploring slight vertical misalignment between laser spot and
? phase plate. The top panel is a schematic depicting the relative vertical
distance between the laser spot and the ? phase plate. The elongated
beam has a 1/e2 beam waist of 5 mm on a phase plate of clear aperture
radius 25 mm. The middle panel shows the ez ? ex intensity profile of
the beam at the front focal plane of L3. The color bar encodes the relative
beam intensity for the middle panel. The bottom panel is a vertical cut-out
showing the intensity profile along the ez axis from the middle panel. (a),
(b), and (c) correspond to 0, 3, and 5 mm misalignments, respectively. . . 35
4.10 (a) Schematic of the DMD optical setup. Po and Pi refer to the DMD and
atom planes, respectively. The purple lines are rays representing imaging
of the DMD pattern onto the plane Pi. The blue lines indicate the spatial
profile of the collimated laser beam as it travels through the optics. (b)
DMD being used as a binary amplitude grating. The top panel shows
binary patterns imprinted onto the DMD plane Po. The dark and bright
regions represent micromirrors in the OFF and ON positions, respectively.
The bottom panel shows the resultant atomic density distribution on the
atom plane Pi. (c) DMD being used as an analog amplitude grating using
halftoning. The right panel shows the overlap between the images of
adjacent DMD pixels when projected through the imaging system. The
right panel shows halftoning implemented in our system. . . . . . . . . . 37
4.11 Partial Transfer Absorption Imaging. (a) The atomic levels involved in
imaging the atomic sample. The inset shows a magnified view of the
transitions induced by microwaves. The splitting in the hyperfine levels of
the ground state is caused by a constant magnetic field applied throughout
the experiment. (b) Rabi oscillations for the transition |F = 1,mF = ?1?
to |F = 2,mF = ?2?. (c) Smith chart for the impedance matched microwave
antenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.12 (a) Schematic of absorption imaging optical setup. The dark field created
by the atoms cast a shadow on the probe. . . . . . . . . . . . . . . . . . . 43
ix
4.13 Optical densities obtained via absorption imaging without (a) and with (b)
PCA. The gray bar indicates optical densities for both (a) and (b). . . . . . 43
4.14 Principal Component Analysis. (a) The variance along each of the principal
components. (b) ?A?P as a function of the number of principal components
used for reconstructing the reference image P . . . . . . . . . . . . . . . . 44
5.1 Flux vs. sodium atomic velocity at different temperatures. The blue,
orange and green curves correspond to 150?C, 200?C, and 230?C, respectively.
As a point of reference, 3D MOT velocities are usually less than 100 m/s. 47
5.2 (a) Schematic of the old NIST experiment [4]. The main chamber is
separated from the sodium source by a long Zeeman slower tube wound
by magnetic field coils creating a space-varying magnetic field. (b) and
(c) are two orthographic views of the 2D MOT chamber. The cooling
arms cool atoms in the ex ? ey plane, as shown in (b). The push beam
pushes atoms into the main chamber along the ez direction, as shown in
(c). Sodium atoms effuse out of the oven in the positive ey direction. The
slower beam is propagated along negative ey. The black bars at the top
right and bottom center are scale bars for (a) and (b)/(c), respectively. . . 48
5.3 Cross-sectional views of the science chamber with the 3D MOT cooling
beams depicted in yellow and quadrupole coils in purple. . . . . . . . . . 50
5.4 3D MOT 1/e2 diameter D vs. time of flight tflight. The red curve is a
fit to the blue data points according to Eq. (5.1). The temperature of the
thermal atomic cloud is estimated to be T = 294(25) ?K by measuring
the rate of expansion, as described in the text. . . . . . . . . . . . . . . . 52
5.5 Critical laser cooling parameters vs. time t. Magnetic field gradient B?,
cooling laser beam detuning ?cool and power Pcool during 3D MOT and
PGC stages. The vertical dashed red lines separate the laser cooling steps
into three sections. The first 10 s long section is the 3D MOT loading.
PGC is performed for the remaining 2.25 ms and 1.5 ms sections. . . . . 53
5.6 Schematic of the transfer of atoms from a quadrupole magnetic trap to a
hybrid magnetic and optical trap. (a), (b), and (c) depict three snapshots
during the transfer process as the magnetic field gradient is reduced from
228 G/cm (a) to 120 G/cm (b) and finally dropped to 7.3 G/cm (c). Gravity
acts along the negative ez direction. The gravity compensation field gradient
is 8 G/cm. The top row is a schematic of the single IR beam dipole trap.
The origin marks the center of the magnetic trap. Atoms are adiabatically
transferred into the hybrid trap as the field gradient is lowered. The
bottom panel shows the change in potential energy landscape U along
ez at (x, y) = (0, 0), as the field gradient is reduced. . . . . . . . . . . . 54
5.7 Critical forced evaporation parameters vs. time t. Magnetic field gradient
B? (a), RF frequency ?RF (b), and IR ODT power Pir (c) during RF
evaporation, adiabatic transfer and hybrid trap evaporation stages. The
vertical dashed red lines indicate the separation of the three stages. . . . . 57
x
5.8 Transfer of BEC into purely optical traps. Critical parameters, magnetic
field gradient B? (a), IR ODT power Pir (b), sheet trap ODT power Pgs
(c), and DMD ODT power Pdmd (d) vs. time t. The vertical dashed lines
indicate the instant when the in-situ ODs (e)-(l) were imaged. (e)-(h) are
ODs on a vertical plane, with the horizontal axis pointing along the IR
beam (ex) and gravity acting along the vertical axis (ez). (i)-(l) are ODs
on a horizontal plane, with the horizontal axis pointing along the sheet
trap beam (ey) and the vertical axis pointing along the IR beam (ex). The
black bars in (e) and (i) correspond to 50 ?m, the length scales being
same for all horizontal and vertical images separately. The color bar at
the bottom depicts the OD scale for (e)-(l). . . . . . . . . . . . . . . . . 61
6.1 Ring-trapping potential and resulting atomic density. The green surface
schematically depicts the trapping potential; the orange lines mark the
typical chemical potential ?. The blue-dashed curve shows a power law
fit to the potential (up to ?) around ? = 0 giving exponent 2.02(3) for this
example. The measured 2D density n2D(?, ?) is shown in the ex?ey plane
(with peak density 165 ?m?2) and the white dashed arc marks the mean
radiusR. Because of the short 500 ?s TOF, the observed width of the ring
is slightly in excess of that anticipated from the in-situ T-F approximation. 66
6.2 Phonon evolution in a contracting toroidally-shaped BEC, averaged over
three measurements. (a) Density perturbations for a ring withRi = 38.4(6)?m
at 10 ms and 35 ms, and Rf = 11.9(2)?m at 45 ms and 53 ms. The
density scale of images before contraction is multiplied by 2. The horizontal
bar corresponds to 80 ?m. (b) Experimental measurements and (c) fit to
Eq. (6.1) of angular density perturbation ?n1D as a function of azimuthal
angle ? and time t, where the ring contraction occurs at ti. (d) Phonon
amplitude ?n as a function of time. The circles plot the phonon amplitude
obtained from fitting each time-slice of (b) to a sinusoid. The red curve
is the instantaneous amplitude from the fit in (c). The diamonds are the
measured mean radius of the BEC and the blue line is the programmed
radius of the trap. The grayscale bar encodes the value of |R?/R|, with
a maximum of 328(11) s?1 at tpeak = 41 ms. The arrow indicates ti =
38.2 ms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.3 Phonon amplitude ?n as a function of time t for (a) expanding and (b)
contracting tori. The symbols, curves, and grayscale bars are all as notated
in Fig. 6.2(d). The expansion data (a) used Ri = 11.9(2) ?m and Rf =
38.4(6) ?m, and vice versa for contraction (b). ti is varied from 6.5 ms to
23 ms for expansion and from 27 ms to 70 ms for contraction. Here, the
red curves show simultaneous fits to our complete data set, as discussed
in the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
xi
6.4 Phonon amplitude vs. phase. (a) Data (black circles), fit (red curve), and
oscillation envelope (blue curve) used to extract the amplitude Af at tpeak.
The grayscale bar is as notated in Fig. 6.2(d). (b) Ratio of amplitudes
Af/Ai vs. ?peak, the oscillation?s phase at tpeak. The black circles plot the
data. The gray dashed, blue solid, and gray dashed-dot curves show the
prediction of Eq. (6.1) for ?H = 0, 0.36, and 1, respectively, with ? =
0.52. The red line indicates the prediction for an adiabatic contraction. . . 73
6.5 Simulations to study the impact of phonon phase ?peak on Hubble friction
strength. (a) Phonon amplitude (solid black curve), and oscillation envelope
(dashed red curve) used to extract the amplitude Ai and Af at tpeak. The
blue curve denoted the radius R as a fucntion of time t. The grayscale bar
is as notated in Fig. 6.2 (d). tpeak is the time when |R?/R| attains maximum
value. (b) Ratio of amplitudes Af/Ai vs. ?peak, the oscillation?s phase at
tpeak. The solid red and dashed black curves correspond to a value of ?H
of 0.5 and 0, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.6 Relative contributions of different azimuthal modes in the imprinted phonon.
(a) Raw data n1D for phonons in a torus of radius R = 31.8 ?m. (b)
The black dots, red open circles and blue crosses show the time evolution
of =[Am] for modes with m = 1, 2, and 3, respectively. The colored
solid lines represent decaying sinusoidal fits according to Eq. (6.18). Each
colored fit curve corresponds to the data points of the same color. (c) The
amplitude Bm (see Eq. (6.18)) for the three modes evaluated from the fits
in (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.7 Phonon amplitude ?n as a function of time t for stationary tori of radii
11.9, 14.3, 16.7, 24.0, 31.2, 38.4 and 43.2 ?m from top to bottom. Here,
the red curves show simultaneous fits to the complete data set, as discussed
in Sec. 6.1.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
7.1 The Erbium ground state. . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.2 Laser cooling transitions in atomic Erbium [1]. The purple and green
arrows indicate the cooling transitions used for the 2D MOT and 3D MOT,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.3 Vapor pressure of Sodium and Erbium as function of temperature [5]. . . 95
7.4 Interference in a simple Fabry Perot cavity. The beam on entering the
cavity undergoes multiple reflections. These when constructively interfere
lead to a maximum in transmission intensity. . . . . . . . . . . . . . . . 97
7.5 Cavity transmission peaks with varying mirror reflectivity R , according
to Eq. (7.4). The red, green and purple curves correspond to a reflectivity
R of 0.6, 0.8 and 0.995, respectively. . . . . . . . . . . . . . . . . . . . 98
7.6 Schematic of a Fabry Perot with curved mirrors. The mirrors with radii
of curvature R1 and R2 are separated by a distance d. . . . . . . . . . . . 99
xii
7.7 Stability diagram for a two mirror cavity based on Eq. (7.5). The red
points (1), (2), and (3) indicate positions of plane parallel (R1 = R2 =
?), symmetric confocal (R1 = R2 = L/2), and concentric (R1 = R2 =
L) mirror configurations. The blue point (4) indicates the position of the
STC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.8 Schematic of the scanning Fabry Perot cavity. The curved mirror is attached
to the piezo which is epoxied to the invar spacer via a macor ring. The
plane mirror is attached on the other end of the spacer. . . . . . . . . . . 103
7.9 Linewidth measurement of the constructed STC. The blue points correspond
to measured transmitted intensity It as the cavity piezo is scanned. The
horizontal axis is the cavity resonance frequency ?cav relative to the master
laser frequency ?master which serves as a reference. The black line is a
Lorentzian fit to the data giving a linewidth of 1.14 MHz. . . . . . . . . . 105
7.10 Optical setup of the scanning transfer cavity. Light from the two lasers
are couple into a fiber and sent to the STC. Lenses with focal length f1 =
150mm and f2 = 100mm are used to mode match to the cavity. . . . . . 106
7.11 Mode matching optics for the cavity. The Gaussian beam from the fiber
is collimated using an asphere. Lenses f1 and f2 are placed appropriately
to achieve mode matching. di?s are the distances between the various
components. d0, d1, d2 and d3 are 127, 150, 120 and 317.5 mm respectively.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.12 Pulse Detection action by the analog board. A voltage pulse (a) detected
in the photo diode is sent through a number of electronic components
which ultimately drive the NAND gate signal (f) LOW. This digital pulse
is fed to the micro controller. . . . . . . . . . . . . . . . . . . . . . . . . 110
7.13 Non-uniform motion of the piezo leads to multiple kinks on the cavity
transmission peak. The yellow curve is the transmission peak and purple
is the digital signal sent to the microprocessor, signaling a transmission
peak?s occurrence. Each vertical division is 500 mV for yellow and 2V
for purple. Each division on the horizontal axis is 250 ?s. . . . . . . . . 111
7.14 Master and slave frequency drifts ?? as the master oscillates and drifts.
The master SatAbs lock was dis-engaged and the master laser frequency
was modulated by 5MHz in a period of 0.5 s. The top panel is data
acquired over long periods and shows the overall drift in the mean frequency.
The bottom panel is a zoomed-in version of the top panel. Measurements
were made using the ?Angstorm WSU2? wavemeter via a fiber switch. . . 112
7.15 Histogram of the slave frequencies as it is locked (red) and unlocked
(blue). Measurements were made using the ?Angstorm WSU2? wavemeter
via a fiber switch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.16 Cavity transmission peaks (yellow curve) and corresponding pulses generated
by analog board (purple) as the cavity (a) or laser (b) is scanned in frequency.
Each vertical division is 500 mV for yellow and 2V for purple. Each
division on the horizontal axis is 250 s. The above data was taken by
scanning the cavity and laser at 3.4 GHz/s. . . . . . . . . . . . . . . . . 114
xiii
A.1 Comparison of length and spatial frequency scales for imaging BECs. (a)
and (b) correspond to imaging systems with spatial resolutions lmin andR.
The grid represents camera pixels. Thermal De-Broglie wavelength ?dB
and healing length ? are indicated. The dashed circles in (b) correspond
to (a)?s solid circles of the same color. (c) Length scales in (a) and (b) in
the Fourier domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
A.2 (a) Atomic density of weakly interacting disk-shaped BEC nexp(rl). (b)
Density variation of (a) ?nexp(rl) about the average 50 shots. The densities
in both (a) and (b) are in the same arbitrary units. . . . . . . . . . . . . . 119
A.3 (a) Experimentally evaluated density fluctuation power spectrum as a function
of the spatial frequency kl. (b) Result of fitting (a) to Eq. A.4. The dashed
circles enclose resolvable spatial frequencies for an imaging system with
NA = 0.28. The color bar holds for both (a) and (b). . . . . . . . . . . . 121
A.4 (a) Wavefront aberration determined by fitting Fig. A.4 (a) to Eq. A.4.
(b) Wavefront aberration reconstructed by projecting (a) onto a Zernike
polynomial basis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
A.5 Experiment to deterministically change the system defocus. (a) The imaging
system with image distance zi as indicated. (b) Variation of Zernike
defocus coefficient Z20 as a function of ?zi. . . . . . . . . . . . . . . . . 123
xiv
List of Abbreviations
AOM Acousto-Optical Modulator
BEC Bose-Einstein Condensate
CAD Computer Aided Design
CTE Coefficient of Thermal Expansion
DC Direct Current
DMD Digital Micromirror Device
FP Fabry Perot
FSR Free Spectral Range
GP Gross-Pitaevskii
IR Infra-Red
MOSFET metal-Oxide-Semiconductor Field-Effect Transistor
MOT Magneto-Optical Trap
NIST National Institute of Standards and Technology
ODT Optical Dipole Trap
PGC Polarization Gradient Cooling
PI Proportional Integral
PSF Point Spread Function
PTAI Partial Transfer Absorption Imaging
RF Radio Frequency
SPOT Spontaneous Force Optical Trap
STC Scanning Transfer Cavity
TEM Transverse Electro-Magnetic
T-F Thomas Fermi
UHV Ultra High Vacuum
ZS Zeeman Slower
xv
Chapter 1: Introduction
Since their first realization in 1995 [6, 7], Bose-Einstein Condensates (BEC) have
emerged as an ideal platform for performing quantum simulations. Early efforts in simulating
the Bose-Hubbard model in 3D optical lattices [8, 9] and tuning atomic interactions via
Feshbach resonances [10?13] have illustrated the ability of such systems to simulate
defect-free and fully configurable synthetic materials. Moreover, the coherent manipulation
of BEC matter waves has opened up the fields of atomtronics [14?19], atomic wave
interferometry [20], and laboratory simulations of cosmological phenomena [21?25], to
name a few.
Bose-Einstein condensation occurs due to the macroscopic occupation of the single-
particle ground state. The phenomenon was first predicted in 1925 by Albert Einstein [26]
using methods of quantum statistics developed by S.N. Bose [27]. At a temperature T ,
atoms of mass m can be regarded as wav?epackets with a spatial extent corresponding to
the thermal de Broglie wavelength ?dB = 2?h?2/mkBT . As these atoms are cooled, ?dB
increases, and at extremely low temperatures, it exceeds the inter-atomic spacing. These
individual wavepackets at ultracold temperatures coherently superimpose to form a single
collective matter wave. If the atoms are bosonic, all of them occupy the lowest quantum
state to form a BEC, thereby serving as a macroscopic lens into quantum phenomena.
1
Attaining quantum degeneracy requires cooling atoms to sub 100 nK temperatures, making
the production of BEC technically challenging. With the availability of tunable wavelength
lasers, significant progress in laser cooling [28?30] in the 1980?s and forced evaporation
techniques [31, 32] in the early 1990?s resulted in the laboratory realization of BECs, 70
years after the first prediction.
Depending on the system, quantum effects become apparent at different temperatures.
For example, metal electrons exhibit strong quantum phenomena below the Fermi temperature
(104 ? 105 K), liquid helium around 1K, and atomic nuclei around 1011K [33]. These
temperatures are significantly higher than those required for Bose condensing neutral
atoms. However, despite their technically challenging production, quantum-degenerate
gases are superior for performing quantum simulations due to the unprecedented control
and measurement capabilities they offer. Almost all atoms in a BEC occupy the same
quantum state, allowing the interactions between the component atoms to be described
by an effective mean-field interaction potential [33,34]. This approach reduces the many-
body problem into a single-body problem which is much easier to model. Furthermore,
the inter-atomic interactions can be tuned by orders of magnitude using Feshbach resonances [35].
In addition to the interaction potential, BEC systems offer tunability in terms of the
condensate fraction, temperature, and shape of the atomic cloud. Because of the dilute
nature of these atomic samples, they can be probed via optical means, thereby facilitating
an ease of measurement.
The process of Bose condensation is a purely statistical one and has nothing to
do with inter-atomic interactions. When gases are cooled, they liquify and eventually
solidify, at which stage the interactions become dominant. Avoiding this solidification is
2
a major challenge in producing quantum degenerate gases. Usually, atoms have a very
high molecular binding energy which energetically favors solidification [36]. However,
since it occurs at a rate proportional to the gas density, molecular binding is avoided by
reducing the atomic density.
In this thesis, I will describe the construction of our 23Na BEC apparatus, aimed
at performing quantum simulations. Historically, our group has used BECs to study
superfluidity [17,37?40], atomtronics circuits [18,19] and perform cosmological simulations [41].
In 2007, the old Sodium Rings lab at the National Institute of Standards and Technology
(NIST) observed superflow in the form of persistent currents in a toroidally shaped BEC [37].
Since then, many of our experiments have focused on exploring the persistent current?s
decay mechanism [17,39,40] and using them for atomtronic circuits [18,19]. Towards the
old apparatus?s end of life, the group performed some experiments to simulate cosmological
analogues in BECs [41]. In 2015, we started building a new apparatus to make a dual-
species BEC with Sodium and Erbium. The old apparatus had many limitations, which
were addressed in this new system. Additionally, we decided to add highly magnetic
Erbium with a magnetic moment 7?B, to exploit its long-range interactions for studying
strongly correlated many-body systems. In 2018, we Bose condensed 23Na, and after that
developed capabilities to confine the BEC in arbitrary-shaped two-dimensional Optical
Dipole Traps (ODT). Significant advances have been made on the Erbium front as well.
In this thesis, I will describe our apparatus, our approach to simulate analogue cosmology
using BECs, and a specific example where we simulate the cosmological phenomena of
Hubble friction. Since many references describe laser cooling, forced evaporation, and
production of BECs [42, 43], I will only briefly touch upon these topics. In the next two
3
chapters I will describe collective modes in a BEC, introduce the idea of analogue gravity
and explain how a BEC?s collective modes can be used to develop experimental systems
for analogue gravity.
1.1 Outline of the Thesis
In 2015, after decommissioning the NIST Sodium Rings apparatus, we embarked
on the journey of constructing the new Sodium-Erbium BEC laboratory. The rest of
the thesis describes the construction and working of this new apparatus and explains an
analogue cosmology experiment performed on it. It is structured as follows. In Chapter 2
I start with describing the theory of Bose-Einstein condensation and follow it with a
discussion on the origin and nature of collective excitations in BECs. Chapter 3 introduces
the idea of Analogue Gravity and explains how collective excitations in BECs could be
used for simulating cosmology. Chapter 4 describes the experimental apparatus, and
Chapter 5 explains the procedure employed for producing BEC and further transferring
them into arbitrary 2D Optical Dipole Traps (ODT). Chapter 6 reports the simulation of
Hubble attenuation and amplification in a toroidal BEC. Though these experiments were
performed with 23Na BEC, we have made significant progress towards trapping Erbium in
a Magneto-Optical trap (MOT). Chapter 7 describes a Scanning Transfer Cavity scheme
used for locking Erbium lasers in our laboratory. Finally, I end with with an outlook in
Chapter 8.
4
Chapter 2: Theory of Bose Gas
2.1 Ideal Non-interacting Bose Gas
Bose-Einstein condensation is a quantum-statistical phenomenon that occurs due to
the macroscopic occupation of the ground state. This section will discuss the behavior of
an ideal non-interacting ensemble of bosons and highlight the role of quantum statistics
in Bose condensation as the ensemble?s phase space densities approach unity. It follows
the discussion in Ref. [44].
Consider an system of N bosons with chemical potential ? and temperature T . The
occupancy of a state n() with energy  is described by the Bose-Einstein statistics, given
by
1
n() = , (2.1)
exp((? ?)/kBT )? 1
where kB is the Boltzmann constant. The density of states for a 3D homogeneous system
a() is given by
a() d = (2?V/h3)(2m)3/21/2d, (2.2)
where V is volume, m is the mass of the atoms, and h is Plank?s constant. Using Eq. (2.1)
5
and (2.2), it can be shown that the number of atoms in all the excited states Ne is given by
V
Ne = g3 3/2(z), (2.3)?dB
where g is the Bose-Einstein function1, and z = e?/kBT? lies between 0 and 1. Since
g3/2(z) increases monotonically with z, the total number of atoms in the excited state
satisfies the inequality
Ne ?
V
g (1). (2.4)
?3
3/2
dB
Since Eq. (2.4) sets an upper limit on the maximum number of atoms in the excited state,
the remaining atoms are in the ground state. If the phase space density (N/V )?3dB exceeds
g3/2(1), the only way to satisfy Eq. (2.4) is if all the excess atoms are in the ground
state. This results in a macroscopically large number of atoms occupying the ground
state, marking the onset of Bose-Einstein condensation. As evident from Eq. (2.4), the
necessary condition for Bose condensation is (N/V )?3dB > g3/2(1) ? 2.612. In terms
of the density n = N/V , and phase space density PSD = n?3dB, the condition for Bose
condensation is expressed as
PSD > 2.612. (2.5)
2.2 Mean Field Dynamics of Weakly Interacting BECs
Though the non-interacting picture of ideal Bose gases describes the phenomena
of Bose condens?ation, weak atom-atom interactions in a BEC lead to several interesting
1 1 ?g (z) = x
??1 z2
? ?(?) 0 z?1ex?1 dx = z + 2?+ z33? +...
6
phenomena. This section derives the time evolution of the BEC wavefunction. Starting
from the many-body hamiltonian in the second quantization, a mean-field approach is
applied to model the atom-atom interactions. The discussion here follows that in Ref. [33,
34, 45], where detailed derivations can be found.
The second quantization many-body Hamiltonian describing N interacting bosons
in an external potential V (r) is given by
? [ 2 ]
H? = dr ???
h?
(r) ? ?2? + V (r) ??(r)2m (2.6)
1
+ dr dr? ???(r) ???(r?) V ?int(r? r ) ??(r) ??(r?),
2
where m is the mass of the atom, and Vint is the two body interaction potential, and ???(r)
and ??(r) are the the boson field operators corresponding to the creation and annihilation
of particles at position r, respectively. The field operator ??(r, t), can be expressed as
??(r, t) = ?(r, t) + ???(r, t), (2.7)
where ?(r, t) is a complex function defined as the expectation of ??(r, t) such that
? ?
?(r, t) = ??(r, t) , (2.8)
and ???(r, t) represents a small perturbation. The complex function ?(r, t) is a classical
field representing the BEC wavefunction with |?(r, t)|2 equal to the condensate density.
This serves as the order parameter for Bose condensation. Since we wish to derive the
time-evolution of condensate wave function ?(r, t), we write the Heinsenberg equation
7
of motion for the operator ??(r, t)
?
ih? ??(r, t) = [[??, H?]?t ? ] (2.9)
? h?
2
= ?2 + V (r) + dr????(r?, t)V (r? ? r)??(r?int , t) ??(r, t).
2m
Due to its diluteness and low temperature, the atom-atom interactions in an alkali BEC
can be described by elastic two-body s-wave scattering processes [33, 45]. This allows
for the introduction of an effective mean field approach to model these interactions such
that the short wavelength degrees of freedom are integrated out. For dilute Bose gases, the
effective mean field interaction potential Vint = g?(r?r?) is a contact interaction between
two atoms at r and r?, where g = 4?h?2as/m, and as is the s-wave scattering length.
Applying this mean field interaction potential in Eq. (2.9) and neglecting the perturbing
operator ???(r, t) in Eq. (2.7), we get the time-dependent Gross-Pitaevskii (GP) equation
[ 2 ]
? h? ?2 ??(r, t)+ V (r) + g|?(r, t)|2 ?(r, t) = ih? , (2.10)
2m ?t
which describes the time evolution of the BEC wave function ?(r, t). It should be noted
that since the operator ??(r, t) has been approximated to the complex valued scalar field
?(r, t), Eq. (2.10) is simply the zeroth order approximation of Eq. (2.9). The first order
approximation developed by Bogoliubov, involves retaining terms in the interaction which
are at most quadratic in ???(r, t), and treating the resulting interaction as a perturbation on
the BEC wavefunction ?(r, t). This first order analysis reveals the existence of collective
excitations, which have a phonon-like linear dispersion relationship for long wavelengths.
Sec. 2.2.3 covers details of the first order analysis useful for understanding concepts in this
8
thesis. Higher order corrections, which involve retaining terms proportional to ???3(r, t)
and ???4(r, t), result in damping of the collective excitations. These are finite temperature
effects. Sec. 2.2.3 touches upon some of the commonly observed damping mechanisms
in BEC experiments.
The stationary solutions of ?(r, t) are obtained by considering the form ?(r, t) =
?(r) exp(i?t/h?), where ? depends only on the spatial coordinate r. This gives the time-
independent GP equation
[ 2 ]
? h? ?2 + V (r) + g|?(r)|2 ?(r) = ? ?(r) (2.11)
2m
The first term in Eq. (2.10) and Eq. (2.11) correspond to the kinetic energy, the second
corresponds to the external potential energy V , and the third term is the inter-atomic
interaction energy.
2.2.1 Thomas Fermi Approximation
Eq. (2.11) is not exactly solvable and requires the implementation of numerical
methods. However, for a BEC with large number of atoms and repulsive inter-atomic
2
interactions (as > 0), the kinetic energy term ? h? ?2?(r) can be neglected with respect2m
to the interaction and potential energy terms. This, known as the Thomas Fermi (T-F)
approximation modifies Eq. (2.11) to
V (r)?(r) + g|?(r)|2?(r) = ? ?(r), (2.12)
9
which is exactly solvable. The solutions for ? are given under the T-F approximation are
given by
???????V (r)? , if V (r) < ?? g|?(r)|2 = ?? (2.13)0, if V (r) > ?,
where |?(r)|2 = n(r) is the volume density.
2.2.2 Collective Excitations: Phonons
As mentioned in Sec. 2.2, a first-order correction to the hamiltonian in Eq. (2.9)
reveals the existence of collective excitations or phonons. In this section, starting from
the time-dependent GP equation, I will show how slight variations of ?(r, t) leads to
collective excitations, which appear as oscillations in the background condensate density
and phase.
The wavefunction ?(?r, t) can be expressed in terms of the atomic density n(r, t) and
phase ?(r, t) as ?(r, t) = n(r, t)ei?(r,t). Substituting this in Eq. (2.10) and comparing
the real and imaginary parts gives the continuity equation
?n
+? ? (nv) = 0, (2.14)
?t
and an equation analogous to Euler equation of fluid dynamics,
( )
?v 1
m +? mv2 + ?? = 0, (2.15)
?t 2
10
where the velocity of the condensate v of the condensate is defined as v = (h?/m)??, and
h?2 ?
?? = V + gn? ? ?2 n. (2.16)
2m n
2 ?
The term ? h?? ?2 n in Eq. (2.16) depends on spatial variations in the BEC density
2m n
and is known as the ?quantum pressure? [33]. It arises due to the kinetic energy term of
the GP equation. Neglecting this is equivalent to the T-F approximation, as described in
the previous section. Collective excitations can be expressed as small variations of the
density n, as
n(r, t) = n0(r, t) + ?n(r, t), (2.17)
where ?n are small fluctuations about the background condensate density n0. Linearizing
Eq. (2.14) and (2.15) with (2.17) and treating ?n and v as small, gives the following set
of coupled differential equations.
? ?n
+? ? (n0v) = 0 (2.18)
?t
?v
m +???? = 0. (2.19)
?t
where ??? is obtained by linearizing ??. Differentiating Eq. (2.18) and substituting the
value of ?v/?t from Eq. (2.19), we get
?2 ?n
m = ? ? (n0? ???) (2.20)
?t2
Under the T-F approximation, the quantum pressure term in Eq. (2.16) can be neglected.
11
Furthermore assuming |?n0| << |? ?n|, Eq. (2.21) can be expressed as
?2 ?n
= ? ? (c2? ?n) (2.21)
?t2
where c2 = gn0/m. Eq. (2.21) is a wave equation representing sound waves or phonons
propagating in the BEC with a local velocity c(r) which depends on the local density
n0(r).
2.2.3 Damping of Low-energy Excitations
Starting from the mean-field time-dependent GP equation (Eq. (2.10)), Sec. 2.2.2
derives the propagation of collective excitations as solutions to an undamped wave equation
(Eq. (2.21)). Due to absence of a damping term (? ? ?n/?t) in Eq. (2.21), one expects
the phonons to be long lived. However, collective excitation in a finite temperature BEC
always damp with time [39, 46, 47]. To understand the mechanisms of phonon decay, we
need to look beyond the mean-field GP approach and consider interactions between the
different collective and thermal excitations. Eq. (2.9) describes the time-evolution of the
condensate. A full solution to Eq. (2.9) would involve coupling between the different
excitations, which becomes apparent as one retains terms proportional to ???3(r, t) and
???4(r, t) in the hamiltonian H [33, 46]. The system could no longer be described as a
closed system and finite temperature effects need to be considered.
Two mechanisms for phonon decay in BECs are Landau [46] and Beliaev [48]
damping. While Beliaev damping is four-wave mixing process where an elementary
excitation decays into two or three lower energy excitation, Landau damping results from
12
the interaction of phonons with thermal excitations. Since the available final state phase-
space is very restricted for low-energy initial excitations, Beliaev damping is usually not
the dominant mechanism. Landau damping occurs due to scattering of a low energy
phonon with thermal excitations , which leads to phonon creation or anhilation. This
form of damping depends on the temperature and arise due to presence of thermal atoms
in the system.
In the next chapter, I will describe how collective excitations in a BEC can be
used to simulate analogous systems for studying cosmological phenomena. The concepts
developed in this chapter will be useful in establishing the analogy between fields in
cosmology and phonons in a moving BEC.
13
Chapter 3: Analogue Gravity
The broad goal of analogue gravity experiments is to mimic cosmological processes
involving general relativity and field theory in controllable systems [49, 50]. The genesis
of this field can be traced back to Unruh?s seminal paper [51] from 1981, where he
established an analogy between fields at a black hole event horizon and sound waves in a
moving fluid. His focus was to study mechanisms and assumptions involved in Hawking?s
prediction of black hole evaporation [52, 53]. Since the experimental investigation of
Hawking radiation is virtually impossible, developing analogous systems to study quantum
thermal radiation could improve our understanding of the evaporation process at trans-
Planckian length scales, where many of Hawking?s assumptions might not hold.
Unruh argued that analogous to light at a black hole event horizon, sound waves
with speed cs traveling in a fluid flowing at supersonic velocities v > cs will not be able
to travel upstream. Fig. 3.1 schematically illustrates this idea by displaying the time-
evolution of acoustic wavefronts on a 2D space as the fluid velocity v and sound speed
cs are varied. The colored circles represents acoustic wavefronts on the ex ? ey spatial
plane. The different colors correspond to different times, as indicated by the color bar
to the left. Fig. 3.1 (a) shows the time evolution for four different values of the fluid
velocity v with the same speed of sound cs across all four situations. The horizontal axis
14
Y Subsonic Supersonic
X
Supersonic Subsonic Y
X
Figure 3.1: A moving fluid creating an acoustic horizon when supersonic flow avoids
upstream movement of sound waves. The colored circles represent time-evolution of
acoustic wavefronts in the fluid space-time. ex ? ey plane represents spatial coordinates.
Time is expressed in different colors according to the color bar to the left. The vertical
red lines denotes the sonic horizon. (a) and (b) depict how the wavefront time-evolution
changes with varying fluid velocity v and sound speed cs, respectively.
represents the control parameter v. The leftmost ex ? ey panel in Fig. 3.1 (a) corresponds
to v = 0, where acoustic waves expand uniformly in all directions. However, as the
fluid attains a non-zero velocity, this is no longer true. Beyond the sonic horizon (vertical
red line), where the fluid velocity equals the speed of sound, the acoustic waves can no
15
Sonic horizon
Sonic horizon
longer travel upstream, creating causally disconnected regions in the fluid space-time.
Fig. 3.1 (b) shows a similar situation where the fluid velocity v is the same across all the
panels, but the speed of sound cs is varied. Here the panels are in reverse order because
now the region left of the red line represents supersonic behavior. In this chapter I will
introduce the idea of simulating cosmological phenomena in BECs and formally establish
the analogy in terms of a wave equation in curved space-time.
3.1 Previous Experiments
Due to their high degree of quantum coherence and ease of control and manipulation,
BECs were always considered a promising candidate for developing Unruh?s sonic analogues
of Hawking radiation. Early proposals for BEC analogues involved creating sonic horizons
in long and thin quasi 1D condensates by explosively expanding them or tuning interactions
via Feshbach resonances [54,55]. In addition, several proposals involved the implementation
of the de Laval nozzle in linear traps to obtain regions of supersonic and subsonic flows in
the same condensate, thereby creating a sonic horizon [21,56?58]. A detailed description
of the proposed de Laval nozzle implementation can be found Ref. [56]. Despite these
theoretical proposals, major experimental progress wasn?t made until 2010, when Jeff
Steinhauer?s group realized a sonic black hole for the first time in a BEC [59]. By
accelerating an elongated harmonically-trapped BEC across a step-like potential (also
called ?waterfall potential?), they achieved fluid velocities an order of magnitude greater
than the speed of sound. This resulted in the creation of black and white hole horizons in
the same condensate, which they exploited in later experiments to observe self-amplifying
16
Hawking radiation [23] and spontaneous Hawking radiation [24, 60].
Apart from simulating Hawking radiation, several BEC experiments have exploited
the sonic analogue for simulating other cosmological phenomena. Notable among them
is the observation of Sakharov oscillations, where synchronously generated sound waves
were created by quenching the atomic interactions via Feshbach resonances [22]. Another
interesting experiment was the observation of sonic analogue of dynamical Casimir effect,
which was performed by modulating the density of a BEC [61]. The goal of our experiments
is to simulate cosmological phenomena associated with the inflationary dynamics of the
early universe. In particular, we wish to study the evolution of scalar fields in expanding
and contracting universes. We use a toroidally shaped BEC, which serves as our analogous
universe. We expand and contract our BEC universe by dynamically varying the radius of
the toroid during an experiment. Such a toroidal BEC has azimuthally traveling collective
phonon modes, which serve as our fluctuating scalar fields. These phonon modes can
be created either by intentionally imprinting them or they could also be spontaneously
generated. Our experiments study the evolution of these phonon modes as the BEC
universe is expanded or contracted.
3.2 BEC Phonon Wave Equation and the Space-time Metric
Unruh?s analogy relies on the condition that acoustic scalar fields satisfy the wave
equation in curved space-time. Formally the geometric and causal structure of space-time
is expressed in the form of metric tensors. For Unruh?s analogy to hold, the metric tensor
associated with acoustic waves in fluids should algebraically depend on sound speed and
17
the medium?s velocity. Ref. [49] derives the metric related to the phonon field in a BEC.
This section follows that derivation and show that the phonon?s metric represents a curved
space-time.
The time evolution of a BEC wavefunction ?(r, t) is given by the time-dependent
GP equation, given by Eq. (2.10). We re-write the equation here.
h?2? ?2 ??(r, t)?(r, t) + V (r)?(r, t) + g|?(r, t)|2?(r, t) = ih? , (3.1)
2m ?t
m is the atomic mass, V (r) is the external potential and g is the GP interaction constant.
The wavefunc?tion can be expressed in terms of the atomic density n(r, t) and phase ?(r, t)
as ?(r, t) = n(r, t)ei?(r,t). Substituting this in Eq. (3.1) and comparing the real and
imaginary parts gives the continuity equation
? ?n h?= ? ? (n??), (3.2)
?t m
This equation is same as Eq. (2.14) with v replaced by (h?/m)??. We also obtain an
equation analogous to Euler equation of fluid dynamics,
?? h?2 ? 2? h? = ? ? ?2 h?n+ (??)2 + V + gn. (3.3)
?t 2m n 2m
Since we are dealing with only low-energy and long-wavelength collective excitation, the
?
relevant length scales are larger than the healing length ? = h?/ mgn. As a result the
first term in Eq. (3.3) can be neglected, resulting in the hydrodynamic approximation. We
18
express the condensate density and phase as
n(r, t) = n0(r, t) + ?n(r, t), and ?(r, t) = ?0(r, t) + ??(r, t), (3.4)
where ?n and ?? are small fluctuations about the background condensate density n0 and
phase ?0. Linearizing Eq. (3.2) and (3.3) with (3.4) gives the following set of coupled
differential equations.
? ?n
= ? h? ? ? [n0???+ ?n??0] (3.5)
?t m
2
? ? ?? h? ? ? ? ? h?
2
h? = ?0 ??+ g ?n D?2 ?n, (3.6)
?t m 2m
where D?2 represents a second order differential operator. By substituting the value of
?n from Eq. (3.6) into Eq. (3.5), we obtain a partial differential equation (PDE) for ??.
This PDE can be expressed in terms of the (3+1) dimensional space time coordinates
x? ? (t;xi) as
? (f?,?? ?? ??) = 0, (3.7)
where the tensors f?,? are given by
[ 2 ]?1
f 0,0[ = ?
h?
g ? D?2
2m
2 ]?1
f 0,j
h? h?
= ? g ? D? ?j[ 2 ]?02m m2 ? (3.8)1
f i,0
h?
= ?[ ?
i h??0 g ?] D?2m 2m
i,j 2 ?1
f i,j
n0 ? h? h? h?
= ? ?i ? g ? D? ?j0 2 ?0
m m 2m m
19
The acoustic compton wavele?ngth ?c = h?/mcs sets a lower limit on the length scale
of the problem, where cs = gn/m is the speed of sound. Since ?c corresponds to
the BEC healing length ?, the contribution of the second order differential operator D?2
can be neglected as is done in the Thomas-Fermi (T-F) approximation. The tensors f?,?
can now be represented by just numbers. By introducing a tensor g?,? which satisfies
?
?g g?,? = f?,? , Eq. (3.7) can be expressed as
?
?1 ??( ?g g?,? ?? ??) = 0, (3.9)?g
where g is the metric g?,??s determinant. This effective metric g?,? describes a curved
Lorentzian geometry. The effective wave equation (3.9) for the scalar field ? is analogous
to that observed by a massless scalar field over curved space-time. It is important to note
that even though the und(erlying fluid motion)is Newtonian and non-relativistic with a
space-time metric ? ? diag[?c2?,? light, 1, 1, 1] , the phonon scalar field ? evolves on a?,?
curved space-time given by the metric g?,? . This establishes an analogy between phonons
in a flowing BEC and scalar fields in the curved space-time of the universe.
3.3 Wave Equation in a Toroidal BEC
The previous section dealt with deriving the wave equation for phonons in a BEC
of arbitrary geometry. For our analog cosmology experiments, we use a toroidally shaped
BEC. This section focuses on that and derives the corresponding wave equation. It is
based on Ref. [62] which derives the wave equation starting from the action of the GP
equation and then applying Hamilton?s principle of least action. Ref. [62] represents the
20
action of the GP equation S for a condensate with density and phase variations given by
Eq. (3.4) in the T-F approximation. The resultant action can be expressed as
? [[ ] ]2
h?2 ?3 ?S = dt d x h +?? ? ??? ? c2 ij ? ?? ? ??h , (3.10)
2g ?t ?xi ?xj
where hij is the eucledian metric, such that hijxixj is the spatial line element and h =
det(hij). When applied to co-moving cylindrical coordinates (?, ?, z), such that ? =
r ?R(t), where R(t) is the time-varying mean radius, Eq. (3.10) becomes
2 ?h?
S [=[( dt d? dz d? (R(t) + ?)2g ) ] ]2 (3.11)
?
+?? ? ?z ? ? 2 ij ? ?? ? ??? + ? ?? c h .
?t ?? ?z ?xi ?xj
Since we are concerned only with the low-energy azimuthal excitations, Eq. (3.11) is
integrated along the radial (?) and azimuthal (?) directions. This gives
2 ? [( )2 ( )]h? V ? ?? 2? c? ? ??S = dt d? , (3.12)
4?g ?t R2 ??
where the volume of the condensate V and azimuthal speed of sound c? is given by
? ? 2
V 2?R c= 2? dz d? (R + ?), and c2? = dz d? . (3.13)V 1 + ?/R
Note that the term containing ? doesn?t contribute to V because of its odd symmetry.
Under a thin ring approximation (?/R << 1) it doesn?t contribute to c? as well. By
applying Hamilton?s principle of least action to the action in Eq. (3.12), we get the wave
21
equation ( )
2
?2
V? c
+ ? ? ? ?2t t ? ?? = 0. (3.14)V R
Eq. (3.14) describes the evolution of scalar fields in a toroidal BEC. The term proportional
to the change in volume V?V is analogous to the cosmological phenomenon of Hubble
friction which is responsible for exponentially attenuating scalar fields in an expanding
universe. In Chapter 6, I will discuss an experiment where we exploit this analogy and
simulate Hubble friction in a toroidally shaped BEC.
22
Chapter 4: Experimental Apparatus
Every successful run of our experiment involves cooling gaseous sodium atoms to
quantum degeneracy, shaping them using arbitrary two-dimensional (2D) Optical Dipole
Traps (ODT), and acquiring data by directly imaging the atomic sample. In this chapter,
I will describe the apparatus we built to perform the above steps. Compared to the
previous version of this experiment at NIST, we made significant modifications in this
setup. The NIST experiment used a glass cell, but we designed this experiment using
a stainless steel vacuum chamber. Fig. 4.1 shows a Computer Aided Design (CAD)
diagram of our vacuum chamber. It consists of a source and science chamber maintained
at high and ultra-high vacuum, respectively. Atoms are pre-cooled in the source chamber
in a 2D Magneto-Optical Trap (MOT). Design details of our 2D MOT are covered in
Chapter 5. As shown in Fig. 4.1, the pre-cooled atoms are then transferred from the source
to the science chamber using a push beam. I will first give an overview of the vacuum
system, then describe the laser cooling setup, followed by a detailed description of the
optical system design used for creating arbitrary shaped quasi 2D BECs. Finally, I will
examine some of the imaging challenges particular to our experiment and our techniques
to overcome them.
23
Differential 
Pumping Tube
Science Chamber
Sodium Oven
Push Beam
4 inches Source Chamber
(2D MOT)
Figure 4.1: Schematic of the vacuum chamber. The source chamber, science chambers,
and the sodium oven are indicated. The blue circle points to the location of the differential
pumping tube, with the inset showing a schematic of the tube. The yellow line indicates
the push laser beam used for transferring atoms from the source to the science chamber.
4.1 Vacuum Chamber
In the absence of vacuum, atoms collide with air molecules gaining enough energy
to escape their trapping potentials. This problem worsens if the traps are shallow, as is
often the case in ultracold atom experiments, our final ODTs being less than 1 ?K deep.
Moreover, in our experiments, we attain quantum degeneracy by forced evaporation in a
24
magnetic and optical dipole traps. The efficiency of this technique relies on having long
vacuum-limited trap lifetimes. Because of these reasons, quantum degenerate gases can
only be produced in Ultra High Vacuum (UHV) conditions.
We perform our experiments in a stainless steel UHV chamber pumped with getter,
ion, and titanium sublimation pumps. Since we plan on adding highly magnetic Erbium
(magnetic moment 7 ?B) to this system, the chamber is custom-made by Kimball Physics
with non-magnetic 316L stainless steel. The vacuum chamber is divided into two sections,
source and science chamber, as shown in Fig. 4.1. The source section contains the atomic
source, which is heated to about 250 ?C, to obtain a vapor of sodium atoms. Given
the metal outgassing at these temperatures, maintaining UHV conditions is not possible.
Therefore we separate the source section from the science chamber using a differential
pumping tube. Fig. 4.1 denotes the location and shows a schematic of the differential
pumping tube. Fig. 4.2 is a mechanical drawing of the tube?s cross section. The source
side is maintained at a pressure of about 10?8 torr while the science chamber is maintained
at 1.2 ?10?10 torr. Due to its small conductance, the differential pumping tube can hold
this pressure differential in steady state. We monitor these pressures with two Agilent
UHV 24P ion gauges, one on the source chamber and the other on the science chamber.
The science chamber is pumped using a SAES D-500 getter and a Gamma Vacuum 45S
ion pump in steady-state. In addition, we have a titanium sublimation pump which is
occasionally activated to counter a gradual increase in the science chamber pressure. On
the source side, pumping is achieved using a SAES D-100 getter pump.
Gases adsorbed in the metallic chamber limit the vacuum?s quality. Therefore,
while constructing this chamber, we baked it in two stages. First, we performed a week-
25
 inches
1.25 inches
1.54 inches
Figure 4.2: Cross-section of the differential pumping tube.
long high-temperature bake at 400 ?C without any vacuum viewports. This was performed
in a kiln while a turbo pump pumped out most of the adsorbed hydrogen. The second
stage was a month-long bake with the vacuum viewports at around 180 ?C to outgas water
vapor absorbed in the chamber walls. This brought the pressure down to low 10?8 torr.
After the two bakes, we activated the getter and ion pumps which decreased the pressure
to 2.8 ?10?10 torr, followed by titanium sublimation pump activation, which achieved
1.2 ?10?10 torr. The goal of maintaining UHV in the science chamber is to minimize
collisions of cold atoms with air molecules. Therefore, the vacuum limited lifetime of an
atomic trap serves as a metric to quantify experimental limitations. We measure the drop
in atom number in a magnetic trap as a function of the hold duration and measure the
lifetime to be 23 s.
4.2 Laser Frequency Stabilization
In our experiment, we perform laser cooling of 23Na atoms along the D2 line
of atomic sodium. Fig. 4.3 shows the various hyperfine levels of the sodium ground
26
Cooling
Transition Repump
Transition
Figure 4.3: Energy levels for the 23Na D2 transition [2]. The yellow and green lines
indicate the cooling and repump transitions separated in frequency by 1.7 GHz.
? ? ? ?
state ?32S1/2 and the excited state ?32P1/2 corresp?onding to the D? 2 tr?ansition [2]. W?e
perform Doppler and sub-Doppler cooling along the ?32S , F = 2 to ?32P , F ?1/2 1/2 = 3
transition with laser light at 589 nm, as shown by the yellow line in Fig. 4.3. To recycle the
atoms which spontaneously decay from th?e excited state?into?the F = 1 groun?d state, we
add a repump beam corresponding to the ?32S ? 2 ?1/2, F = 1 to 3 P1/2, F = 2 transition,
as shown by the green line in Fig. 4.3. This beam is roughly 1.7 GHz blue detuned to the
cooling line.
For laser cooling, we use two commercial Toptica DL-Pro 589 nm laser systems,
one at the cooling transition and the other at the repump transition. Fig. 4.4 shows
a schematic of these commercial lasers [3]. It consists of an External Cavity Diode
Laser (ECDL) followed by a Tapered Amplifier (TA) and a Second Harmonic Generation
(SHG) crystal in a ring cavity. The diode emits light in the infrared (IR), which upon
27
Energy
Steering 
Mirror
Diode
Grating
(Piezo Actuated)
Optical isolator DL pro
Folding
mirrors ECDL
Tapered
amplifier Mode matching optics
PD
Beam shaping
optics
Resonant doubling cavity
Figure 4.4: Schematic of the optical setup inside the Toptica DL-pro laser [3]. The inset
shows a magnified view of the ECDL, indicating the piezo actuated grating, which can be
externally controlled via the voltage Vpiezo.
amplification through the TA, is frequency-doubled to 589 nm. The ECDL has a piezo
actuated diffraction grating in the Littrow configuration, as shown in the inset of Fig. 4.4.
To control the laser frequency, we provide a feedback signal to this piezo and the diode
current. The overall feedback signal is represented by Vpiezo
The linewidths of the two Toptica DL-Pros lasers is about 2? 30 kHz, estimated
by a self heterodyne measurement with 2 km long optical fiber and an Acousto Optical
Modulator (AOM). Though these linewidths are much smaller than the 2? 9.8 MHz
natural linewidth of the sodium D2 line, the lasers have a long-term frequency drift of
0.8 MHz over 30 minutes, measured on an Angstrom WSU2 wavemeter. To correct
these drifts, we feedback the cooling transition Toptica laser and lock it to the D2 line
obtained via Saturation Absorption Spectroscopy (SAS). Fig. 4.5 is a schematic of the
28
from local oscillator
to Lock-in Amp
Probe Vapor Cell
Pump
Balanced PD
Figure 4.5: Optical schematic of the SAS setup. The yellow lines indicate light from a
Toptica DL-pro laser which is first upshifted by 150 MHz and then split into pump and
probes. The signal from a balanced photodetector, as shown in the rectangular box, is
sent to a lock-in amplifier.
(a) SAS lock feedback loop (b) SAS signals
Laser SAS
Local 
Servo ~ Oscillator
Amplifier Filter Mixer
Lock-in Amplifier
Figure 4.6: (a) Lock-in scheme to obtain the derivative of the SAS absorption signal VPD.
The blue and purple lines represent VPD and the derivative signal Vlockin respectively. The
dashed box indicates the components of a lock-in amplifier. The local oscillator is a
Voltage Controlled Oscillator (VCO) that drives the AOM. The servo generates the signal
Vpiezo to control the laser frequency. (b) VPD and Vlockin indicated in blue and purple,
respectively.
29
AOM
AOM
? ? ? ?
SAS?setup. We perform S
2 2 ?
? ? ?
AS on the crosso?ver of the ?3 S1/2, F = 2 to ?3 P1/2, F = 2
and 32S , F = 2 to ?1/2 32P ?1/2, F = 3 transitions. Light from the bare laser is up-
shifted by double passing through a 75 MHz AOM and then sent to the vapor cell heated
to 150 ?C. This beam is split into two low-intensity 40 ?W probe beams and a 1.1 mW
pump beam. One of the probe beams is used to obtain a Doppler-free signal. The other
probe is subtracted from the Doppler-free signal to eliminate the Doppler broadened dip
on a balanced photodetector. The Doppler-free spectroscopy signal VPD thus obtained is
shown in the inset of Fig. 4.5. This signal is symmetric about the transition, and therefore,
does not give any information about which direction the piezo voltage needs to be driven
to correct the drift. However, the phase of the electric field associated with the transmitted
probe has odd symmetry about the atomic resonance. To extract this phase information,
we perform frequency modulation of the SAS laser beam and use a lock-in amplifier
to get a signal proportional to the derivative of the SAS absorption signal Vlockin [63].
Fig. 4.6 shows a schematic of the frequency modulation and lock-in scheme used. For
the frequency modulation, we dither an AOM at a modulation frequency of 50 kHz. Once
the cooling laser is locked to the atomic transition, the repump laser is beatnote locked
to the cooling laser in a master-slave configuration. Light from the two locked lasers
is frequency shifted using AOMs and then sent to the experiment table via single-mode
optical fibers. Tab. 4.1 lists the frequency and intensities of the different beams used to
address the atoms.
30
Table 4.1: Detunings and intensities of the different laser cooling beams. For the first
five entries, detunings are from the cooling transition. For the last three entries, detunings
are from the repump transition. For reference, the saturation intensity for the cooling
transition is Isat = 6.26 mW/cm2
Laser Beam Detuning from transition Intensity (mW/cm2)
(MHz)
2D MOT cool -12 0.98
3D MOT cool -16 1.13
Zeeman Slower(ZS) -219 20
Push +9 9
Polarization Gradient -39 to -16 1.13
Cooling (PGC)
2D MOT rep. -8 30
3D MOT rep -8 3
Spin Polarization -26 0.005
4.3 Optical Dipole Traps
We use three far-detuned laser beams to generate ODTs for our experiment. Fig. 4.7
shows the propagation of the three ODT beams with respect to the science chamber. One
of them is red-detuned to the atomic transition and is used for evaporative cooling. The
other two are blue-detuned and are used in combination to trap atoms in arbitrary 2D
potentials. The subsections below discuss the optical setup for all three laser beams.
4.3.1 Red-detuned Optical Dipole Trap
We use a red-detuned ODT for evaporatively cooling atoms to quantum degeneracy.
The forced evaporation scheme will be discussed in Chpt. 5. For this ODT, we use a
1064 nm single beam from an IPG ?YLR30-1064-LP-SF? Ytterbium fiber laser. 10 W
of 1064 nm power is fiber-coupled into an NKT Photonics ?aeroGUIDE 15 PM SMA-
31
Blue-detuned
DMD ODT
Z
Y
Blue-detuned 
Sheet ODT
X
Red-detuned ODT Y
Figure 4.7: Propagation of ODT beams in the science chamber. The main figure shows
the red-detuned ODT propagating along positive ex and being focused down at the center
of the chamber. The blue-detuned sheet ODT propagates along positive ey. The inset
shows beam propagation on the ey ? ez plane. The blue-detuned DMD ODT propagates
along negative ez. Gravity acts along negative ez.
905? photonic crystal fiber and sent to the experiment table. The light out of the fiber is
collimated into a 800 ?m beam using a LMH-10X-1064 microscope objective with 10?
magnification and 0.25 numerical aperture. A 1:5 telescope expands the beam into a 4
mm waist, which is then focused by a 300 mm focal length spherical lens into a beam
32
waist of w0 = 25 ?m at the plane of the atoms.
4.3.2 Blue-detuned Optical Dipole Trap
Our experiments rely on creating arbitrary shaped 2-dimensional ODT. For this, we
use two blue-detuned lasers beams from a 532 nm IPG GLR-30 fiber laser. Blue-detuned
laser beams create repulsive potentials and can achieve smaller diffraction limited spot
sizes dspot. The diffraction limited spot size is given by dspot = ??/NA, where ? is
the wavelength and NA is the numerical aperture of the optical system. Due to their
smaller wavelength, blue-detuned ODTs allow us to achieve tighter confinements than
red-detuned light. One of the two 532 nm beams is used for vertical confinement (along
ez) and the other for confinement on the horizontal plane (ex ? ey). Fig. 4.8 shows
a schematic of the optical system used for vertical confinement. With the help of a
cylindrical lens, two spherical lenses, and a ? phase plate, we shape the TEM00 mode out
of a single-mode SM460 optical fiber into an elongated TEM01 mode, providing sheet-like
confinement for atoms in between the two lobes of the TEM01 mode. The vertical width of
this sheet trap is about 9 ?m at the focus of the final lens L3. For the planar confinement,
we project arbitrary patterns from a Digital Micrometer Device (DMD) onto the plane of
atoms through an objective lens stack [64], as shown in Fig. 4.10 (a). The two beams are
slightly shifted (? 10 MHz) in frequency to avoid any interference effects. Together, they
provide arbitrary shaped quasi 2D confinement for BECs.
33
(c) Collimated beam
(a) Sheet beam propagation along Z-Y 1
Z
Y
(c) (d)
Scale Z:X = 1:1
0
(d) Sheet trap
(b) Sheet beam propagation along X-Y
X
Y
(c) (d) Scale Z:X = 25:1 
Figure 4.8: Schematic for the sheet trap optical setup. Blue-detuned laser light from
a single-mode fiber is collimated and propagated through lenses L1, L2, L3 and a ?
phase plate along the ey direction. (a) and (b) shows the beam profile in the ez ? ey
and ex ? ey planes, respectively. Figures (c) and (d) show the simulated ez ? ex beam
intensity profiles at indicated Y positions, with ez along the vertical axis and ex along the
horizontal. The color bar encodes the relative beam intensity for figures (c) and (d). The
atoms, as depicted by the yellow dots, lie close to the front focal plane of lens L3. Gravity
acts along negative ez.
4.3.2.1 Vertical Confinement
The sheet trap is generated using a cylindrical (L1) and two spherical lenses (L2
and L3) and a ? phase plate placed at positions, as indicated in Fig. 4.8. The TEM00 beam
from a single-mode fiber is first collimated into a uniform beam with a 1/e2 diameter of
1360 ?m. This then passes first through the cylindrical lens L1 followed by the spherical
lens L2. L1 and L2 together form a telescope along the vertical direction ez thereby
magnifying the vertical beam waist by |f2/f1| = 7.5, where f1 =-40 mm is the focal
length of L1 and f2 =300 mm is the focal length of L2. As a result, the beam at L2?s front
focal plane has a vertical waist wz = 5.1 mm, while the horizontal waist gets focused
down to a small spot. This elongated beam passes through a ? phase plate, which adds
34
Rel. Intensity
(a) Perfectly aligned (b) 3 mm misalignment (c) 5 mm misalignment
Z
X
+30 +30 +30 1
0 0 0
-30 -30 -30
-750 0 +750 -750 0 +750 -750 0 +750
0
1 1 2
1
0 0 0
-30 0 +30 -30 0 +30 -30 0 +30
Figure 4.9: Simulations exploring slight vertical misalignment between laser spot and ?
phase plate. The top panel is a schematic depicting the relative vertical distance between
the laser spot and the ? phase plate. The elongated beam has a 1/e2 beam waist of
5 mm on a phase plate of clear aperture radius 25 mm. The middle panel shows the
ez ? ex intensity profile of the beam at the front focal plane of L3. The color bar encodes
the relative beam intensity for the middle panel. The bottom panel is a vertical cut-out
showing the intensity profile along the ez axis from the middle panel. (a), (b), and (c)
correspond to 0, 3, and 5 mm misalignments, respectively.
a ? phase difference between its top (Z > 0) and bottom (Z < 0) halves [40]. Finally,
the beam goes through the final lens L3 of focal length f3 =200 mm. L2 and L3 form a
telescope for the beam along ex, while it focuses the ez waist into a tight spot. Exploiting
the Fourier transform property of lenses, the beam?s intensity pattern is determined at the
front focal plane of L3, as shown in Fig. 4.8 (d) and Fig. 4.9 (a).
The bottom panel of Fig. 4.9 (a) shows that along Z, the intensity has a minimum
at z = 0, thereby providing vertical trapping. However, along the beam propagation
35
Rel. Intensity
Rel. Intensity
direction ey, the TEM01 intensity gradient introduces a not-so-intuitive antitrapping potential,
which pushes the atoms away from the focus of the imaging system. Even though the
dipole trap potential along the ey axis is zero, atoms trapped away from the ey axis
experience an intensity gradient along ey, pushing them away from L3?s front focal plane.
This problem is further exacerbated by slight misalignment in the beam shaping optics.
Fig. 4.9 is a simulation of how small misalignments in the ? phase plate relative to the
beam center can result in blue-detuned light leaking into the dark regions of the sheet trap.
As evident from these simulations, a 3 mm misalignment can cause significant distortions
to the trapping potential. In our experiment, we use a phase plate with a radius of 25 mm.
It is challenging to center this ? phase plate on the 5 mm (1/e2 radius) elongated beam to
better than 1 mm. Because of these reasons, our final system has some antitrapping, not
allowing us to place the BEC at the focus of the sheet trap. Instead, we place the BEC at
about a Rayleigh length away from the focus where the antitrapping force is not strong
enough to significantly distort the homogeneity of the sheet BEC. Since we do not sit at
the beam focus, the resulting trap frequencies are smaller for a given laser power. Taking
inspiration from [65], we plan to add a red detuned dipole trap to compensate for this
antitrapping potential in future designs.
4.3.2.2 Planar Confinement
For the 2D planar confinement, we use a Texas Instrument DLP7000 DMD. It has
a 1024 ? 768 array of individually addressable aluminum micromirrors arranged over a
rectangular grid. The DLP7000 has a micromirror pitch of 13.68 ?m and a maximum
36
(a) DMD optics (b) DMD as a binary amplitude grating
DMD
1
0
(c) DMD as an analog amplitude grating
NA = 0.27
Atoms 1
532 nm Laser beam
Image of the DMD pattern 0
Figure 4.10: (a) Schematic of the DMD optical setup. Po and Pi refer to the DMD and
atom planes, respectively. The purple lines are rays representing imaging of the DMD
pattern onto the plane Pi. The blue lines indicate the spatial profile of the collimated
laser beam as it travels through the optics. (b) DMD being used as a binary amplitude
grating. The top panel shows binary patterns imprinted onto the DMD plane Po. The dark
and bright regions represent micromirrors in the OFF and ON positions, respectively.
The bottom panel shows the resultant atomic density distribution on the atom plane Pi.
(c) DMD being used as an analog amplitude grating using halftoning. The right panel
shows the overlap between the images of adjacent DMD pixels when projected through
the imaging system. The right panel shows halftoning implemented in our system.
pattern projection rate of 32 kHz, allowing for a fast dynamic display of patterns essential
for our experiments. This is in contrast to phase Spatial Light Modulators (SLM) which
have a maximum update rate of a few hundred hertz, making them too slow for our
purposes. Fig. 4.10 (a) shows a schematic of the optical imaging system used for the
projection of DMD patterns. These micromirrors are bi-stable, limiting the DMD to a
binary amplitude grating, as shown in Fig. 4.10 (b). However, one can perform analog
37
Plane of atoms DMD pixels
Rel. density
Rel. density
modulation of the dipole trap beam?s spatial intensity profile using halftoning techniques.
Fig. 4.10 (c) shows how halftoning exploits the limited resolution of the projection optics
to create continuous gradients of intensity. Due to the finite size of the Point Spread
Function (PSF), the projection of the individual DMD pixels overlaps, creating a continuous
gradient in intensity. We use the Jarvis Halftoning algorithm [66] to choose the appropriate
binary DMD patterns to achieve this.
Since the DMD is a 2D array of micromirrors, it acts as a diffraction grating.
Therefore, care needs to be taken while aligning the DMD optics, to maximize the laser
power in only one of the diffraction orders. In our system, the blaze condition is achieved
for a 42? incidence angle with respect to the DMD surface. This reflects the first order at a
66? angle. Given our space constraints, such steep angles makes aligning optics difficult.
Since we are not starved for 532 nm laser power, we do not operate the DMD in the blaze
condition but at a 24? incidence angle which results in a reflection normal to the DMD
surface. The normal reflection additionally ensures that the DMD pattern is normal to the
optical axis of the imaging system, as shown in Fig. 4.10 (a).
4.4 Imaging
In ultracold atom experiments, observations are made by directly imaging the atomic
sample. Our experiments rely on the accurate estimation of the in-situ column density
n2D(x, y). These column densities are typically obtained via absorption imaging, where a
probe laser beam with detuning ? is made to pass through the atomic sample. The atomic
sample with a volume density n(x, y, z) absorbs the probe light, resulting in a decrease in
38
the transmitted intensity I(x, y, z) given by the Beer-Lampert?s law [42, 67]
dI
= ? In ?0 , (4.1)
dz 1 + I/Isat + (2?/?)
2
where Isat is the saturation intensity, ? is the natural linewidth, and ?0 is the absorption
?cross section. W? e pe?rform resonant a?bsorption imaging between the stretched states of?32S1/2, F = 2 and ?32P , F ?1/2 = 3 levels using a ?? polarized probe. As a result, ? =
0, and ?0 = 3?20/2?, where ?0 is the resonance frequency of the transition. Integrating
Eq. (4.1) along ez gives
( )
Ia Ip ? Ia
n2D(x, y) ?0 = ? ln + , (4.2)
Ip Isat
?
where n2D(x, y) = n(x, y, z) dz is the column density, Ip(x, y) and Ia(x, y) are the
cross-sectional probe intensities before and after absorption, respectively. We use Eq. (4.2)
to estimate the column density n2D. The optical density (OD) is defined as OD = n2D(x, y) ?0.
Typically, the intensities Ia and Ip are measured by capturing the probe intensities in the
presence and absence of atoms. Since Ia and Ib are measured in terms of the camera?s
photoelectron counts, Isat is calibrated in the same units as well.
4.4.1 Partial Transfer Absorption Imaging
As evident from Eq. (4.2), the transparency of the atomic cloud has a non-linear
dependence on the atomic density (n or n2D). Consider the situation of resonant (? =
0) absorption imaging when Ip << Isat. The transmitted probe intensity Ia decreases
39
(a) Imaging Levels (b) Rabi oscillations
Trapped atoms
(c) Impedance matching of antenna
Probe
Figure 4.11: Partial Transfer Absorption Imaging. (a) The atomic levels involved in
imaging the atomic sample. The inset shows a magnified view of the transitions induced
by microwaves. The splitting in the hyperfine levels of the ground state is caused by a
constant magnetic field applied throughout the experiment. (b) Rabi oscillations for the
transition |F = 1,mF = ?1? to |F = 2,mF = ?2?. (c) Smith chart for the impedance
matched microwave antenna.
exponentially with an increase in the column density n2D given by
Ia = I
??0 n2D
p e . (4.3)
BECs when imaged in-situ have large column densities (?0 n2D > 4). As a result,
small density variations lead to a negligible change in Ia. Due to the camera?s limited
dynamic range, it is often difficult to resolve small density variations in traditional in-
situ absorption imaging. For our experiments, we excite collective modes in a BEC and
monitor its time evolution. As such, the precision of our measurements relies heavily on
being able to resolve small density perturbations accurately. To this end, we image only
40
Energy
a fraction of our atomic sample, thus ensuring that we are far away f?rom the saturation ?
conditon of Eq. 4.3 [68]. While the BEC is prepared in the ground state ?32S1/2, F = 1,mF = ?1 ,
by applying microwaves pulses close to the sodium clock t?ransition of 1.77 GHz, w?e
transfer a small fraction (typically 5%) of the atoms to the ?32S1/2, F ?= 2,mF = ?2?
sta?te. These atoms?are then imaged with a probe beam resonant to the ?3
2S1/2, F = 2
to ?32P ?1/2, F = 3 transition. The energy levels and associated transitions are shown in
Fig. 4.11 (a). The microwave pulse duration provides a convenient handle on the fraction
of atoms imaged as shown in Fig. 4.11 (b). This method has the additio?nal advantage o?f
being minimally destructive, since the atoms which are not transferred to ?32S1/2, F = 2
do not interact with the probe laser.
For exciting these microwave transitions, the old NIST experiment used a half-wave
dipole antenna. Unlike our metallic vacuum chamber, the NIST BEC was produced in a
glass cell. Switching from a glass cell to a metallic chamber poses two problems. First,
metal surfaces alter the radiation pattern of an antenna. Second, a dipole antenna needs
to be placed farther from the atoms than in a glass cell due to the chamber?s geometry.
As a result, with the NIST half-wave dipole antenna and a 10W amplifier (ZHL-10W-
2G+), we obtained an order of magnitude smaller rabi frequency, therby making our
experiment sensitive to decoherence effects due to electronic noise on our bias magnetic
fields. The solution to this problem would be a more directional source of microwaves.
Commercial horn antennas have excellent directional properties. However, 1.7 GHz horn
antennas, such as the RF-Lambda RW430HORN15A, are much bigger than any of our
chamber viewports. With these limitations in mind, we decided to use a full-wave loop
antenna. This has three advantages. First, a loop antenna can be wound around a viewport,
41
thereby not restricting optical access. Second, such an antenna can be placed around our
recessed top or bottom viewports, closer to the atoms. Third, loop antennas are much
more directional than half-wave dipole antennas. Overall, these factors help increase
the radiation intensity at the location of the atoms. We measured a rabi frequency of 5
kHz with the loop antenna, which was five times larger than that with a half-wave dipole
antenna. Fig. 4.11 (b) shows the Rabi oscillations obtained in our experiment with a
full-wave loop antenna. To harness the full power of the microwave amplifier, the loop
antenna was impedance matched to the 50 ? output impedance of the amplifier, using a
triple stub tuner. Fig. 4.11 (c) is a smith chart of the frequency response of the antenna
after being impedance matched at 1.748 GHz with the stub tuner.
4.4.2 Probe Reconstruction
As described in the previous sub-sections, absorption imaging involves measuring
the cross sectional intensities Ia and Ip using a camera. Fig. 4.12 is a schematic describing
the process. As the probe moves through the atomic sample, atoms absorb light and
cast a shadow on it. The intensity pattern obtained from this image Ia is compared with
the reference intensity pattern Ip, to infer the column density information according to
Eq. (4.2). Therefore, conventional absorption imaging involves acquiring two images1,A
and P for estimating Ia and Ip, respectively.
Ideally, these two images should be identical except for the shadow formed by the
atoms. However, in practice, no two shots are the same. This happens due to many
1A third image is acquired to estimate the background noise, which is then subtracted from the two
primary images.
42
EMCCD
Camera
Dark Field
Bright Field
Probe
Figure 4.12: (a) Schematic of absorption imaging optical setup. The dark field created by
the atoms cast a shadow on the probe.
(a) (b) 
0.5
0
-0.5
Figure 4.13: Optical densities obtained via absorption imaging without (a) and with (b)
PCA. The gray bar indicates optical densities for both (a) and (b).
reasons, including mechanical vibrations. As a result, due to an inaccurate estimation
of the reference intensity Ip, traditional absorption imaging results in optical densities
with a noisy background, as shown in Fig. 4.13 (a). To get a more accurate estimate
43
(a)
Principal Component
(b)
Number of PCs used
Figure 4.14: Principal Component Analysis. (a) The variance along each of the principal
components. (b) ?A?P as a function of the number of principal components used for
reconstructing the reference image P .
of Ip, we reconstruct the reference image P from A by projecting it onto an eigenbasis
generated by many previously acquired reference images. This basis spans all possible
reference images. We use the orthogonalization technique Principal Component Analysis
(PCA) to convert the set of reference images into an eigenbasis. PCA has the advantage
of sorting eigenvectors in order of decreasing variances such that most of the shot-to-shot
variation is captured by the first few principal components, as shown in Fig. 4.14 (a).
This reduces the dimensionality of the image reconstruction process. For Fig. 4.13, we
have used 45 reference images to generate an eigenbasis. The optical densities obtained
via PCA are less noisy, as shown in Fig. 4.13 (b). For the difference image A ? P , we
evaluate the standard deviation across all pixels, as more and more principal components
are included in the reconstruction. This is plotted as ?A?P in Fig. 4.14 (b). As evident
44
Variance
(cam. counts) (cam. counts)
from this plot, the standard deviation saturates by the time seven principal components are
included, thus demonstrating the dimensionality reducing of the reconstruction process.
The higher-order principal components are just one-pixel wide, representing photon shot
noise.
45
Chapter 5: Production of BEC
In the previous chapter, I gave a detailed layout of our apparatus and a brief overview
of typical experimental techniques used. This chapter describes the procedure employed
to cool Sodium atoms to Bose-Condensation and further shape the BEC into arbitrary
2D patterns. Like most alkali atoms, Sodium is typically cooled by laser cooling and
forced evaporation techniques. Our experiment employs laser cooling followed by forced
evaporation in magnetic and hybrid magnetic and optical traps. Our new apparatus uses a
2D MOT as a high-flux source of cold sodium atoms, as opposed to the previous version
of this apparatus [4, 38, 39], which used a Zeeman slower. First, I describe our 2D MOT
pre-cooling procedure, followed by the various laser cooling steps in the science chamber.
Then, I detail our forced evaporation scheme, which combines cooling in magnetic and
optical trap to exploit the best of both traps. Finally, I end by describing the steps involved
in efficiently transferring the BEC into a purely optical trap. For each of these procedures,
wherever possible, I provide a time trace of experimental parameters that are critical to
that step.
46
Oven Temperature
0 200 400 600 800 1000
Velocity (m/s)
Figure 5.1: Flux vs. sodium atomic velocity at different temperatures. The blue, orange
and green curves correspond to 150?C, 200?C, and 230?C, respectively. As a point of
reference, 3D MOT velocities are usually less than 100 m/s.
5.1 2D MOT Source
To start the cooling process, a source capable of producing a high flux of cold
sodium atoms is needed. Since these atoms are eventually captured in a 3D MOT, the 3D
MOT capture velocity (typically less than 100 m/s) sets an upper limit on the temperature
of the captured atoms. To create sodium vapor, solid sodium metal is resistively heated.
The oven temperature determines the flux and velocity distribution of atoms in this vapor.
Fig. 5.1 depicts the flux of gaseous Sodium atoms at three different oven temperatures.
Here, the velocity distribution is determined using the Maxwell Boltzmann function, and
the temperature-dependent vapor pressure of sodium [2], which gives the overall flux.
Fig. 5.1 shows that only a tiny fraction of the atoms effusing out of the oven are within
the 3D MOT capture velocity. As a result, faster atoms need to be cooled before being
47
loaded into the 3D MOT.
(a)
~500 cm
(b) X (c) Z
To Main Chamber
Sodium Oven X
Slower and 
Y Repump
Slower and Repump Y
 
2D MOT
Cooling Beams ~10 cm Push Beam
Figure 5.2: (a) Schematic of the old NIST experiment [4]. The main chamber is separated
from the sodium source by a long Zeeman slower tube wound by magnetic field coils
creating a space-varying magnetic field. (b) and (c) are two orthographic views of the
2D MOT chamber. The cooling arms cool atoms in the ex ? ey plane, as shown in (b).
The push beam pushes atoms into the main chamber along the ez direction, as shown in
(c). Sodium atoms effuse out of the oven in the positive ey direction. The slower beam is
propagated along negative ey. The black bars at the top right and bottom center are scale
bars for (a) and (b)/(c), respectively.
To this end, the previous version of this experiment used a Zeeman Slower [4, 38].
Zeeman Slowers slow faster atoms into a much slower range of velocities by Doppler
cooling along a spatially varying magnetic field [28]. However, they have two significant
disadvantages. First, they are complex and bulky as they have carefully engineered
48
electromagnets that span about a meter. Fig. 5.2 (a) shows the Zeeman slower apparatus
used in the NIST lab. Second, the electromagnets require large currents with millisecond
turn-off times, limiting control and detection sensitivities for magnetic field-sensitive
experiments. Since we plan on making a dual-species BEC with Erbium which has
a relatively high magnetic moment (7?0), a Zeeman Slower is not ideal. Therefore,
in the current version, we have replaced the Zeeman Slower with a 2D MOT [69?71].
Fig. 5.2 (b) and (c) shows a schematic of our 2D MOT setup. The working principle of a
2D MOT is the same as a 3D MOT, with the difference that it cools and traps only along
two dimensions, as shown in Fig. 5.2 (b). Along the third dimension, a blue detuned laser
beam pushes the pre-cooled atoms into the 3D MOT, as shown in Fig. 5.2 (c). We based
our 2D MOT on the design in [71]. Permanent magnets generate a quadrupole magnetic
field with a 36 G/cm linear field at the center of the 2D MOT. A unique feature of the
design in [71] is the introduction of a Zeeman Slower beam which provides additional
Doppler cooling for high-velocity atoms effusing out of the oven. We observe a five-
fold increase in the 3D MOT atom number due to the introduction of this beam. This
beam enters the chamber through a vacuum viewport exactly opposite to the oven, as
shown in Fig. 5.2 (b) and (c). Thus, the window has a direct line of sight with the oven.
At oven temperatures above 170 ?C, it gets coated with a layer of shiny sodium metal.
Therefore, we heat the window to a temperature of 150 ?C by placing a 3? long and 1?
wide Thorlabs lens tube in front of the window and heat it using resistive heating tapes.
Due to space constraints, our sodium crucible is installed horizontally, thus increasing the
risk of molten sodium migrating to other parts of the chamber. We installed an additional
nozzle between the crucible and the rest of the 2D MOT chamber to avoid this migration.
49
5.2 Laser Cooling
Cooling Beams
Quadrupole Coils
Z
Y
X
Y
Figure 5.3: Cross-sectional views of the science chamber with the 3D MOT cooling
beams depicted in yellow and quadrupole coils in purple.
Once the atoms are pre-cooled in the 2D MOT, a push beam transfers these atoms
into the science chamber through the differential pumping tube. The push beam is 9 MHz
blue-detuned from the cooling transition to address atoms with positive velocities relative
50
to the beam?s wave vector. It is carefully aligned to the differential pumping tube to ensure
that the atoms reach the center of the science chamber, where they are trapped in a 3D
MOT. The push beam is not perfectly collimated but made to diverge slightly so that by
the time it reaches the center of the chamber, its intensity drops enough to not push atoms
out of the 3D MOT. The powers and detunings of the different 2D and 3D MOT beams
can be found in Tab. 4.1.
Fig. 5.3 shows a schematic of our 3D MOT setup. Six circularly polarized laser
beams intersect orthogonally at the center of the science chamber. These 3D MOT
cooling beams are 16 MHz red detuned to the cooling transition and just 1.13 mW/cm2 or
0.18 ?Isat in intensity. These intensities are significantly lower than most ultracold atom
experiments and can be attributed to an inefficient Polarization Gradient Cooling stage,
limiting the 3D MOT capture velocity. This also results in a larger mismatch between the
3D and 2D MOT capture velocities, leading to much slow loading rate of the 3D MOT.
Currently it takes around 10 s to load our 3D MOT to saturation, which is more than three
times the ? 3 s loading rate of the NIST experiment.
To generate the 3D MOT magnetic field, a pair of coils in the anti-Helmholtz
configuration is used to produce a linear magnetic field gradient of 10.3 G/cm at the
center of the science chamber. The magnetic field and the six laser beams form the 3D
MOT providing a velocity and position-dependent force that cools and traps atoms close
to the center of the chamber. In addition to the gradient coils, we have three pairs of
Helmholtz coils along the three directions ex, ey and ez. These are capable producing
constant bias magnetic fields at the center of the chamber. We use them to correct for the
residual magnetic field or generate known magnetic fields whenever needed. We operate
51
12
10
8
6
6 8 10 12 14 16
Figure 5.4: 3D MOT 1/e2 diameter D vs. time of flight tflight. The red curve is a fit to the
blue data points according to Eq. (5.1). The temperature of the thermal atomic cloud is
estimated to be T = 294(25) ?K by measuring the rate of expansion, as described in the
text.
the experiment with a ?dark SPOT? MOT [72] which gives us atomic densities higher
than a traditional MOT. We place a glass plate with a black dot in the repump?s path. The
dark spot is imaged onto the center of the 3D MOT, creating a region with no repump. As
a result, atoms in the dark region decay into the F = 1 state and fall out of the cycling
cooling transition, thereby shielding them from the re-radiation forces that limit densities
in traditional MOTs. In our experiment, we image a black dot of radius ? 5 mm onto the
chamber?s center. The repump beam with the dark spot has 5 mW of power spread in a
ring around the dark region. We trap 370? 106 atoms cooled to a temperature of 294(25)
?K in the 3D MOT. The temperature T is measured by releasing the atomic cloud in time
of flight (TOF) and looking at its diameter D as a function of TOF, tflight, as shown in
Fig. 5.4. The sudden release of the atomic cloud converts all its energy kBT into kinetic
energy 1m dD, where kB is the Boltzmann constant, andm is the mass of a sodium atom.8 dt
52
As a result, fitting the data to the equation,
?
8kBT
D = t+D0, (5.1)
m
gives us an estimate of the atomic temperature. Here, D0 is the in-situ diameter of the
atomic cloud.
10
0
-15
-45
6
4
10 s 2.25 ms 1.5 ms
Figure 5.5: Critical laser cooling parameters vs. time t. Magnetic field gradient B?,
cooling laser beam detuning ?cool and power Pcool during 3D MOT and PGC stages. The
vertical dashed red lines separate the laser cooling steps into three sections. The first 10
s long section is the 3D MOT loading. PGC is performed for the remaining 2.25 ms and
1.5 ms sections.
After cooling atoms in the 3D MOT, the magnetic field gradient is turned off, and
the atoms are subjected to Polarization Gradient Cooling (PGC) [30,73]. The cooling light
is detuned linearly from -16 MHz to -39 MHz in 2.25 ms and then held at -39 MHz for 1.5
ms. Even though PGC reduces the temperature of the atoms to 178 ?K, it is significantly
higher than most Sodium BEC apparatuses [30]. Our current protocol to tune the cooling
53
light frequency is by varying an AOM?s frequency, whose operational bandwidth limits
the detunings we use for PGC. In the future, we plan to programmatically vary the laser
lock point instead, allowing for a frequency sweep of more than 200 MHz, hopefully
leading to a more efficient PGC. PGC marks the end of the laser cooling process. Fig. 5.5
summarizes the variation of different experimental parameters throughout the laser cooling
process.
5.3 Forced Evaporation
(a) Z (b) (c)Z Z
X X X
40 40 40
20 20 20
0 0 0
-20 -20 -20
-4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4
Figure 5.6: Schematic of the transfer of atoms from a quadrupole magnetic trap to a hybrid
magnetic and optical trap. (a), (b), and (c) depict three snapshots during the transfer
process as the magnetic field gradient is reduced from 228 G/cm (a) to 120 G/cm (b) and
finally dropped to 7.3 G/cm (c). Gravity acts along the negative ez direction. The gravity
compensation field gradient is 8 G/cm. The top row is a schematic of the single IR beam
dipole trap. The origin marks the center of the magnetic trap. Atoms are adiabatically
transferred into the hybrid trap as the field gradient is lowered. The bottom panel shows
the change in potential energy landscape U along ez at (x, y) = (0, 0), as the field gradient
is reduced.
Unlike laser cooling, forced evaporation doesn?t have a fundamental lower limit on
54
temperature, which makes it an ideal choice for attaining extremely cold atomic samples
[6,32]. At a given temperature, there are always some atoms in the high energy tail of the
Maxwell-Boltzmann distribution. Forced evaporation is a process where these energetic
atoms are ejected out of the system, thereby reducing the average energy of the ensemble
[74, 75]. The remaining atoms thermalize via inter-atomic collisions, resulting in net
cooling. Since the technique relies on collisional thermalization, atomic densities need to
be high enough such that thermalization times are shorter than the vacuum-limited sample
lifetime or other inelastic collision processes. Typically forced evaporation is performed
in a magnetic trap or an ODT. While magnetic traps are tunable to large volumes, ODTs
can confine atoms to tiny dimensions, limited only by a laser beam?s diffraction-limited
spot size. As a result, magnetic traps are ideal for mode matching low-density and high-
volume 3D MOTs, whereas ODTs help attain high densities for speedier evaporation.
Ref. [76] uses a hybrid magnetic and optical trap to exploit the best of both techniques
for Bose condensation of Rubidium atoms. In our experiment, we implement a similar
scheme to achieve a Sodium BEC.
For the magnetic trap, we use a pair of current-carrying coils in the anti-Helmholtz
configuration to generate a quadrupole trap. At the trap center, these coils produce a 1.30
G/cm linear field gradient for every ampere of current. A Lambda EMI 20-500 power
supply powers the coils in constant voltage mode. This power supply is voltage limited
at 20V. Since the coil circuit has a resistance of 103 m?, the power supply limits us to
a maximum current of 190 A or a field gradient of 252 G/cm. In addition to the high
field gradients, specific steps of the experiment require faster switching. The huge coils
make the circuit highly inductive, resulting in long switching times from one state to
55
another. Therefore we connected a bank of nine IXYS IXFN230N20T power MOSFETs,
in series to the coil, to quickly turn off the current. As a result, the turn-off time is limited
only by the 200 ns response time of the MOSFETs. These MOSFETs are connected
in parallel to each other to distribute the power dissipation equally, also water-cooled.
A proportional integral (PI) servo controls the MOSFET gate voltage to open or close
the circuit by monitoring the difference between the desired and the measured current
through the coils. To measure the coil current, we use an Ultrastab LEM-IT-500-S hall
probe sensor. For the ODT, we use a far red-detuned, 1064 nm single beam, focused to
a spot size of w0 = 25 ?m, as described in Sec. 4.3.1. Fig. 5.6 (c) shows a schematic of
the hybrid trap indicating the relative spacing between the IR beam and the center of the
magnetic trap. While the IR beam provides confinement along its radial direction, a weak
magnetic field gradient ensures confinement along the beam?s longitudinal axis ex.
Our forced evaporation scheme starts with radio-frequency (RF) evaporation in
the magnetic trap, followed by an adiabatic transfer into a hybrid magnetic and optical
dipole trap, and finally ends with evaporation in the hybrid trap. Of the three magnetic
sublevels of the F = 1 state, only the mF = ?1 is magnetically trappable. Therefore
to capture most of the atoms into the magnetic trap, atoms are spin-polarized into the
|F = 1, mF = ?1? ground state by first applying a DC magnetic field and then shining a
?? polarized laser beam propagating along the magnetic field. The magnetic field defines
a quantization axis and lifts the degeneracy between the various hyperfine magnetic sublevels.
On application of the ?? polarized light, atoms are transferred into the stretched state
|F = 1, mF = ?1?. This light is 26 MHz red detuned to the repump transition. After
spin-polarization, we quickly turn on the magnetic field gradient to a relatively low value
56
(a)
200
100
0
(b)
60
40
20
0
(c)
10
5
0
0 5 10 15 20 25
Figure 5.7: Critical forced evaporation parameters vs. time t. Magnetic field gradient B?
(a), RF frequency ?RF (b), and IR ODT power Pir (c) during RF evaporation, adiabatic
transfer and hybrid trap evaporation stages. The vertical dashed red lines indicate the
separation of the three stages.
of 68 G/cm in 2 ms. The smaller field gradient results in a larger trap volume, enabling
efficient atom transfer into the magnetic trap. We capture 250 M atoms of the 370 M
laser-cooled atoms (67 %) into the magnetic trap. Once captured, we compress the trap by
ramping the magnetic field gradient to 228 G/cm in 100 ms, resulting in higher densities
and a shorter re-thermalization time. RF evaporation is then performed by addressing
atoms in the high energy tail, providing enough energy to spin-flip to the magnetically
untrappable |F = 1, mF = 1? state. Fig. 5.7 plots the variation of critical parameters
during the forced evaporation process. We start with addressing higher energy atoms
with RF at 60 MHz and gradually sweep the frequency down to 10 MHz in 9.5 s, as
shown in Fig. 5.7 (b). The initial sweep from 60 to 25 MHz is performed in a 1.5 s linear
57
ramp, followed by three exponential ramps for 25 to 20 MHz, 20 to 13 MHz, and 13 to
10 MHz in 3, 3, and 2 s, respectively. The RF radiation is generated using an Agilent
33250A function generator that goes through an RF amplifier and is then applied to the
atoms using a loop antenna wound around a 2? lens tube placed near the bottom recessed
window. Due to Majorana spin flips near the center of the quadrupole field [77?79], we
see a significant loss of atoms if the RF frequency is swept below 10 MHz. Therefore we
stop here and perform the rest of the evaporation in the hybrid trap.
During the last 5 s of RF evaporation, we turn on the IR beam to full power. Once
the RF sweep is complete, we perform an adiabatic relaxation of the magnetic trap to
transfer atoms into the hybrid trap in 5 s. The vertical red dashed lines in Fig. 5.7 indicate
the start and end of the adiabatic expansion process where the field gradient is linearly
ramped from 228 G/cm to 7.3 G/cm in 5 s. This adiabatic expansion results in cooling
of the atomic ensemble, as shown in Fig. 5.6. As the field gradient is lowered, the more
energetic atoms are initially held in the quadrupole trap?s low-density tail away from the
dense cloud of colder atoms. Once the gradient falls below the gravity compensation
value of 8 G/cm, the hot atoms are ejected out of the trap resulting in a net cooling of
the ensemble. Lowering the gradient below gravity compensation lets us perform forced
evaporation by lowering the IR power since the magnetic trap will no longer trap the
ejected hot atoms. We achieve quantum degeneracy by ramping the IR power from 10 W
to 50 ?W in 12 s as shown in Fig. 5.7 (c). The IR power is dropped in two exponential
ramps: the first being 10 W to 2 W in 2 s, followed by 2 W to 50 ?W in the remaining 10 s.
Fig. 5.8 (e) and (i) shows an in-situ image of the atomic cloud at the end of IR evaporation.
The resulting cloud is cigar-shaped and expands anisotropically when released from the
58
hybrid trap, confirming that the evaporation results in Bose condensation.
5.4 Transfer to Purely Optical Traps
Once the sodium atoms are Bose condensed, we shape the BEC into arbitrary 2D
patterns using two blue detuned beams from a 532 nm IPG GLR-30 fiber laser. Sec. 4.3
describes the construction of these ODTs. Fig. 5.8 (a)-(d) plots the time variation of
experimental parameters critical to the transfer process. The sheet trap is turned on at
full power during the final 10 s of evaporation in the hybrid IR trap. This is followed
by linearly ramping the IR power to zero in the next 100 ms while the magnetic field
gradient is still maintained at 7.3 G/cm. I will refer to this sheet and weakly confining
field gradient as the ?hybrid sheet trap?. The region between the first two vertical dashed
lines depict the transfer from the hybrid IR to the hybrid sheet trap. Fig. 5.8 (e)-(l) are
in-situ ODs of the BEC at different stages of the transfer process. The hybrid sheet is
significantly shallower than the hybrid IR trap at full power. As a result, turning the sheet
beam on during IR evaporation doesn?t alter the BEC?s shape in the ex ? ey direction,
as shown in Fig. 5.8 (i). However, it helps efficiently transfer atoms from the hybrid IR
to the hybrid sheet trap. Fig. 5.8 (f) and (j) shows an in-situ image of the hybrid sheet
trap, where the weak field gradient provides the planar ex ? ey confinement. The radial
confinement of this disk can be altered by either changing the field gradient or the bias
magnetic field in the ez direction. Since the field gradient is a crucial parameter affecting
evaporation, we tune the ez magnetic field bias to alter the disk size.
Once in the hybrid sheet trap, we switch the DMD beam on and project a dark
59
circular disk with a radius similar to the BEC in the hybrid sheet trap. Fig. 5.8 (g) and
(k) show the in-situ ODs of the BEC at the end of this step. The third (from left) vertical
line also denotes this in Fig. 5.8 (a)-(d). We then linearly turn the field gradient to zero
in 500 ms. The BEC is now trapped in a purely optical trap and has an OD as shown
in Fig. 5.8 (h) and (l). Throughout the transfer of atoms across different magnetic and
ODTs, we maintained a small non-zero magnetic field bias. As a result, the trapped
atoms are maintained in the |F = 1, mF = ?1? state. Once we transfer atoms into this
purely optical disk trap, we can project different patterns on the DMD and obtain arbitrary
shaped quasi 2D BECs. Once the BEC is shaped in this purely optical trap, we are ready
to perform experiments on it. For example, Chapter 6 describes one such experiment,
where the shape of the BEC is altered dynamically during an experiment to simulate the
expansion of universe.
60
(e) & (i)
(f) & (j) (g) & (k) (h) & (l)
(a) 10
8
4
0
(b)
100
50
0
(c)
300
200
100
(d)
1000
500
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
(e) (f) (g) (h)
(i) (j) (k) (l)
OD
-0.5 0 0.5 1 1.5 2
Figure 5.8: Transfer of BEC into purely optical traps. Critical parameters, magnetic field
gradient B? (a), IR ODT power Pir (b), sheet trap ODT power Pgs (c), and DMD ODT
power Pdmd (d) vs. time t. The vertical dashed lines indicate the instant when the in-situ
ODs (e)-(l) were imaged. (e)-(h) are ODs on a vertical plane, with the horizontal axis
pointing along the IR beam (ex) and gravity acting along the vertical axis (ez). (i)-(l) are
ODs on a horizontal plane, with the horizontal axis pointing along the sheet trap beam
(ey) and the vertical axis pointing along the IR beam (ex). The black bars in (e) and (i)
correspond to 50 ?m, the length scales being same for all horizontal and vertical images
separately. The color bar at the bottom depicts the OD scale for (e)-(l).
61
Chapter 6: The Hubble Friction Experiment
The massive scale of the universe makes the experimental study of cosmological
phenomena difficult. Moreover, since performing controlled experiments is mostly impossible,
experimental verification of cosmological hypotheses relies on few naturally occurring
test cases. This has resulted in the emergence of the field of Analogue Gravity which
deals with developing analogous table-top experiments to simulate cosmological systems.
Unruh established one such analogy in 1981, where he demonstrated the similarity between
sound waves in a moving fluid and fields at a black hole event horizon, thereby hinting
at the possibility of simulating Hawking radiation in a moving fluid [51]. Since then,
analogous systems have been realized in various physical platforms ranging from classical
fluids to cold atomic systems. In this chapter, I will describe our approach towards
creating an analogous system for studying the inflationary dynamics of the early universe.
Our table-top setup comprises a toroidally shaped BEC of 23Na atoms, which we
use to simulate the universe?s expansion and contraction. The toroidal BEC is shaped
using the sheet and DMD traps, as described in Sec. 4.3, and serves as our analogous
universe. Since the DMD has a maximum frame update of 32 kHz, we can dynamically
vary the radius of this toroid during the experiment, hence simulating expansions and
contractions of our BEC universe. Finally, we wish to study the evolution of scalar fields
62
in the early universe. Analogous to fluctuating scalar fields, our quasi-2D BEC system
can have azimuthally traveling collective modes or phonons. In a typical experimental
sequence, we imprint phonons by creating density perturbations on the toroidal BEC by
halftoning binary patterns on the DMD, as described in Sec. 4.3. Because of the aspect
ratio of the toroid, the energy needed to excite non-azimuthal phonon modes is much
higher, thereby limiting our system to primarily have azimuthal phonons.
Exploiting the above analogy, in a previous work [41], our group simulated elements
of an expanding universe, including the redshifting of phonons in analogy to the redshifting
of photons. In a recent work [80], we have simulated and studied the cosmological
phenomenon of Hubble friction in detail. Hubble friction is the phenomenon responsible
for exponential decay of scalar fields in an expanding universe. This chapter starts with
a pre-print version of our recent work [80] in Sec. 6.1, followed by some supplemental
material that was instrumental in designing and understanding the experiment. All authors
significantly contributed towards the publication. The experimental setup was designed
and constructed by me, Monica Gutierrez Galan, Madison Anderson, and Hector Sosa
Martinez. The data acquisition was done by me, Monica, and Hector. I performed the
data analysis.
63
6.1 Publication: Hubble Attenuation and Amplification in Expanding
and Contracting Cold-Atom Universes
6.1.1 Abstract
In the expanding universe, relativistic scalar fields are thought to be attenuated by
?Hubble friction?, which results from the dilation of the underlying spacetime metric. By
contrast, in a contracting universe this pseudo-friction would lead to amplification. Here,
we experimentally measure both Hubble attenuation and amplification in expanding and
contracting toroidally-shaped Bose-Einstein condensates, in which phonons are analogous
to cosmological scalar fields. We find that the observed attenuation or amplification
depends on the temporal phase of the phonon field, which is only possible for non-
adiabatic dynamics, in contrast to the expanding universe in its current epoch, which
is adiabatic. The measured strength of the Hubble friction disagrees with recent theory
[J. M. Gomez Llorente and J. Plata, Phys. Rev. A 100 043613 (2019) and S. Eckel and
T. Jacobson, SciPost Phys. 10 64 (2021)], suggesting that our model does not yet capture
all relevant physics. While our current work focuses on coherent-state phonons, it can
be extended to regimes where quantum fluctuations in causally disconnected regions of
space become important, leading to spontaneous pair-production.
6.1.2 Introduction
During the early universe?s rapid expansion, primordially fluctuating scalar fields
are thought to have been exponentially redshifted and attenuated by the expanding spacetime
64
metric, where ?Hubble friction? contributes to the latter [81]. Unlike true friction, Hubble
friction is non-dissipative and therefore, while it attenuates scalar fields in an expanding
universe, it would amplify them in a contracting universe. In previous work [41], our
group showed that an atomic Bose-Einstein condensate (BEC) in an expanding toroidal
trap could simulate elements of an expanding universe, including the redshifting of phonons
in analogy to the redshifting of photons. Here, we build upon these studies by: including
contracting universes; measuring both Hubble attenuation and amplification with five-
fold increased precision; and showing that the magnitude of Hubble friction disagrees
with recent theoretical work [62, 82].
While the study of astrophysical systems is ordinarily limited to observations, the
development of well-controlled laboratory systems has enabled tabletop realizations of
general relativistic phenomena. Examples from a variety of physical platforms ranging
from classical fluids to cold atomic systems include: the realization of acoustic black
hole horizons [83?85]; stimulated and spontaneous Hawking radiation [24, 86, 87]; and
scattering processes around rotating black holes [88]. With their unprecedented control
and measurement capabilities, ultracold atoms are an emerging platform for realizing
minimal models relevant to high energy physics [89], astrophysics [23, 51, 90, 91], and
cosmology [21, 41, 54, 92].
In BECs, phonons are scalar fields that evolve approximately according to an effective
spacetime metric defined by the background BEC [21]. For toroidally-shaped BECs,
expanding or contracting 1D universes can be simulated by dynamically changing the
BEC?s radius and observing the evolution of azimuthal phonons. Unlike the expansion
observed in the photon-dominated epoch of the universe, we explore non-adiabatic expansions
65
and contractions where the rate of the metric change exceeds the oscillation frequency.
Figure 6.1: Ring-trapping potential and resulting atomic density. The green surface
schematically depicts the trapping potential; the orange lines mark the typical chemical
potential ?. The blue-dashed curve shows a power law fit to the potential (up to ?) around
? = 0 giving exponent 2.02(3) for this example. The measured 2D density n2D(?, ?) is
shown in the ex?e ?2y plane (with peak density 165 ?m ) and the white dashed arc marks
the mean radius R. Because of the short 500 ?s TOF, the observed width of the ring is
slightly in excess of that anticipated from the in-situ T-F approximation.
Phonons are predominately phase excitations with respect to the BEC?s order parameter.
For a toroidal BEC with radius R(t) (see Fig. 6.1), azimuthal phonons with mode number
m, have an approximate phase profile ??1D(?, t) ? ??(t) sin(m?) independent of r and z
and obey the wave equation [62] 1
{ [ ] }
V?(t)
?2t + 2? + ? + ?
2
t m(t) ??(t) = 0 (6.1)V(t)
at low energy (i.e., smallm). Here, the instantaneous angular frequency is ?m = mc?(t)/R(t),
for speed of sound c?(t). Because this manuscript focuses exclusively on them = 1 mode,
we omit the m subscript in what follows. The quantity in square brackets is reminiscent
1Eq. (3.14) derived in Chapter 3
66
of damping because it multiplies the first derivative of time. It includes two terms: a
phenomenological damping constant ? 2 and the non-dissipative ?Hubble friction? V?/V
arising from the changing metric defined by the background condensate. We model
the external potential (see Fig. 6.1) as quadratic in z and power law in ? = |r ? R|;
in the Thomas-Fermi (T-F) and thin-ring approximations, these lead to the BEC?s 3D
volume V ? R? and speed of sound c ? R??/2? , where the value of the constant ?
depends on the potential [62]. Rather than detecting ??1D, we measure the associated
density perturbation ?n1D(?, t) = ?n(t) sin(m?). The relationship between ?? and ?n
is ?t?? = ?(g/h?)(?n/R?), in terms of the Gross?Pitaevskii equation [45] interaction
constant g.
In our experiments, the potential V (?) is nominally fixed during expansion or
contraction, predicting V?/V = ?HR?/R with strength ?H = ?. In expanding systems
(R? > 0) the Hubble friction term attenuates phonons, while in contracting systems
it amplifies them. In the non-adiabatic regime R?/R?m, the timing of expansion or
contraction relative to the phonon?s temporal phase becomes important for subsequent
dynamics. We show this enhances or diminishes the impact of Hubble friction: because
the Hubble friction term includes the product of R?/R and ?n(t) ? ?t??(t), tuning
the timing of expansion or contraction relative to the oscillation changes the degree of
amplification or attenuation.
2This phenomenological damping term can account for Landau and Beliaev damping mechanisms [93]
as well as imperfections in the confining potential.
67
(a) 50
0
-50
(b) 1 1
0 0
-1 -1
(c) 1 1
0 0
-1 -1
(d) 1 40
0 20
ti
-1 0
0 20 40 60 80
t (ms)
Figure 6.2: Phonon evolution in a contracting toroidally-shaped BEC, averaged over
three measurements. (a) Density perturbations for a ring with Ri = 38.4(6)?m at
10 ms and 35 ms, and Rf = 11.9(2)?m at 45 ms and 53 ms. The density scale of
images before contraction is multiplied by 2. The horizontal bar corresponds to 80 ?m.
(b) Experimental measurements and (c) fit to Eq. (6.1) of angular density perturbation
?n1D as a function of azimuthal angle ? and time t, where the ring contraction occurs
at ti. (d) Phonon amplitude ?n as a function of time. The circles plot the phonon
amplitude obtained from fitting each time-slice of (b) to a sinusoid. The red curve is
the instantaneous amplitude from the fit in (c). The diamonds are the measured mean
radius of the BEC and the blue line is the programmed radius of the trap. The grayscale
bar encodes the value of |R?/R|, with a maximum of 328(11) s?1 at tpeak = 41 ms. The
arrow indicates ti = 38.2 ms.
68
R (?m)
(a) Expansions (b) Contractions
Figure 6.3: Phonon amplitude ?n as a function of time t for (a) expanding and (b)
contracting tori. The symbols, curves, and grayscale bars are all as notated in Fig. 6.2(d).
The expansion data (a) used Ri = 11.9(2) ?m and Rf = 38.4(6) ?m, and vice versa for
contraction (b). ti is varied from 6.5 ms to 23 ms for expansion and from 27 ms to 70 ms
for contraction. Here, the red curves show simultaneous fits to our complete data set, as
discussed in the text.
6.1.3 Experiment and Results
Our experiments [39, 76] begin with quasi-2D BECs with N ? 1 ? 105 atoms
confined in a pair of blue-detuned (? = 532 nm) optical dipole traps. The chemical
potential is ? ? h ? 2.7 kHz. The harmonic vertical confinement, with frequency
?z/2? ? 1.2 kHz, is provided by a horizontally propagating Hermite-Gauss TEM01
beam. We generate nearly arbitrary space and time-dependent potentials in the r-? plane
by imaging ? = 532 nm laser light reflected by a digital micro-mirror device (DMD)
onto the BEC. We use these potentials to create toroidal traps with radius R (see Fig. 6.1)
ranging from 12 ?m to 39 ?m and radial width ? 5 ?m.
69
An azimuthal phonon excitation with mode numberm = 1 is generated by perturbing
the toroidal BEC with a potential V 3ph sin(m?) . This repulsive potential?generated by
the DMD?is applied for 2 ms, imprinting the phonon?s phase modulation onto the BEC.
After imprinting, the phonon evolves for an initial time ti from 6.5 ms to 70 ms, at which
point the torus is expanded or contracted using an error function profile [41], with 10 %-
90 % rise time 3.6 ms, and continues to evolve for up to ? 150 ms. For expansion, the
initial and final radii are Ri = 11.9(2) ?m and Rf = 38.4(6) ?m; these are reversed for
contraction 4. We detect the phonon at various points during the complete evolution using
partial transfer absorption imaging (PTAI [68]) after a short 500 ?s time of flight, giving
the 2D density n2D(?, ?) [see Fig. 6.1].
The phonon excitation?s density perturbation [see Fig. 6.2(a)] is ?n2D = n2D?n02D,
where n02D is the density with no phonon present. Integrating along r gives the azimuthal
density perturbation ?n1D(?, t). Figure 6.2(b) shows the time evolution of ?n1D, and
Fig. 6.2(c) shows the resulting fit to Eq. (6.1), from which we obtain both the red- or
blueshift (via c?) and the Hubble friction (from ?H). In our system, the phenomenological
damping ? is observed to depend on radius, and we parameterize ? in terms of the quality
factor Q = ?/2? = c?/2R?, which eliminates most of the radial dependence present in
?, (see [39, 47]).
Because the 3.6 ms expansion or contraction is a small fraction of the phonon
oscillation period, the overall fit is insensitive to how Q interpolates between Qi to Qf .
We therefore assume a simple linear dependence of Q on R. As shown in Fig. 6.2(b), our
3Vph is set to 0.8 times the overall potential depth.
4All uncertainties in this paper are the uncorrelated combination of 1-? statistical and systematic
uncertainties.
70
data typically has less than one oscillation before R changes; to reduce the uncertainty in
Qi and ?(Ri), we include fixed-radius rings in a simultaneous fit. These fits include as
free parameters ?H, Qi, Qf , ? as well as the initial amplitude ?ni, temporal phase ?0, and
speed of sound c?,i. c?(t) = c?,i (R(t)/R ??/2i) follows the expected scaling.
Figure 6.2(d) summarizes the outcome of this fit. The red curve is the time-dependent
density perturbation ?n obtained from the full fit, while the circles plot ?n from independent
fits to ?n sin(?) of each time-slice in Fig. 6.2(b). The blue curve displays the radius of the
DMD pattern while the diamonds plot R obtained from a 2D T-F fit to the observed
density distribution 5. The gray band plots R?/R during contraction, with maximum
R?/R ? 1.53? ?.
We study the hypothesized impact of the phonon phase on the Hubble friction
during expansion or contraction by changing ti [see Fig. 6.2(d)], thereby phase-shifting
the phonon by (c?,i/Ri)ti. We define tpeak as the time when the Hubble friction reaches
its peak ?strength, i.e., when |R?/R| is maximal. The phase of the phonon at tpeak is
? ? tpeakpeak dt ?(t) + ?0. Figure 6.3 shows example time-traces with multiple ti for0
both expansion and contraction, providing a complete picture to investigate the strength
of Hubble friction . The black circles show the time evolution of the phonon amplitude
?n(t) for a range of ti for both expansions (a) and contractions (b).
The red curves in Fig. 6.3 show the results of global fits of Eq. (6.1) to our complete
dataset, which includes 17 contractions and 11 expansions. The parameters ?H, ?, Qi, Qf ,
c?,i and ?ni are global, i.e., they are shared across all time traces. For each time trace, ?ni
5By contrast with Ref. [41], the BEC follows the contraction profile without overshoot or oscillation
because of tighter radial confinement.
71
is scaled by the atom number N(t) for that trace, and c?,i is correspondingly scaled by
? N(t)?/2 6; this accounts for both atom loss during and after expansion or contraction
and for overall drifts in atom number during data acquisition. Each global fit includes 7
additional time-traces, each with constant R, roughly from Ri to Rf . Because c?(R) ?
R?/2 in stationary rings, these additional datasets further constrain ?. We performed
separate global fits for expansion and contraction data, giving an independent measure of
their Hubble friction coefficients. Finally, we perform these global fits in eight different
ways, with the number of fit parameters varying between 32 and 101. Each fit yields
different best-fit values, but generally they agree within 2-?. These fitting methods differ
on whether the temporal and azimuthal phases are shared across the time traces and if
atom number varies within each time trace. Due to the large number of data points, the
degrees of freedom, in excess of 1 ? 104, do not vary significantly between the different
methods. We take the mean of the values obtained from the eight methods as the best
fit value. Their standard deviation is added in quadrature to the average 1-? uncertainty
from the fit to obtain the final uncertainty in the measurement.
Table 6.1: Best fit global parameters.
Qi Qf ? ?H c?,i ?ni
(mm/s) (rad?1)
Expansion 3.5(1) 4.4(2) 0.47(1) 0.28(4) 5.42(2) 7.47(13)
Contraction 7.8(3) 3.5(1) 0.52(3) 0.36(3) 4.36(4) 4.50(5)
Table 6.1 lists the best-fit values, with ?H different for contraction and expansion.
The values of ? are in agreement with each other and are about 1/2. For our power-law
6This can be derived from Eq.(4.8) and Eq.(4.20) of [62]
72
(a)
(b)
Figure 6.4: Phonon amplitude vs. phase. (a) Data (black circles), fit (red curve), and
oscillation envelope (blue curve) used to extract the amplitude Af at tpeak. The grayscale
bar is as notated in Fig. 6.2(d). (b) Ratio of amplitudes Af/Ai vs. ?peak, the oscillation?s
phase at tpeak. The black circles plot the data. The gray dashed, blue solid, and gray
dashed-dot curves show the prediction of Eq. (6.1) for ?H = 0, 0.36, and 1, respectively,
with ? = 0.52. The red line indicates the prediction for an adiabatic contraction.
potential model [62], ? ranges from 1/2 (for a harmonic potential) to 1 (for a hard-wall
potential). Our average value of ? ? 0.495 suggests that we have a harmonic potential in
both z and r.
Lastly, we confirm our expectation that the phonon phase ?peak has a marked impact
on the amplitude following expansion or contraction in the non-adiabatic limit. Our
experiments probe 1.3 < ?peak/? < 2.9. Fig. 6.4(a) illustrates our process for obtaining
the final amplitudes Af where we fit the oscillatory behavior to an exponentially decaying
sinusoid with the amplitude and temporal phase as free parameters (the remaining parameters
73
are drawn from the global fits). By contrast, the initial amplitude Ai is obtained from our
global fit, from the envelope of the decaying sinusoid evaluated at tpeak. Figure 6.4(b)
plots the fractional change in amplitude Af/Ai versus ?peak with black circles, and the
solid blue curve depicts Af obtained from our global fits 7. Our simulations (grey curves)
show that the significant oscillations for ?H = 0, give way to more uniform gain with
increasing ?H. The measured values ofAf/Ai are generally larger than would be expected
for ?H = 0, showing Hubble amplification due to contraction. Unlike Ref. [41], which
probed 1.8 < ?peak/? < 2.1, where Af/Ai has little dependence on Hubble friction,
our greater range of ?peak allows us to better constrain ?H . In addition to the overall
oscillation, there appears to be some additional dependence on ?peak not captured by our
model (data below ?peak/? < 2 generally lie above the ?H = 0.36 curve, and above for
?peak/? > 2). This additional dependence may indicate a more complicated damping
process for our phonons that could obscure our fitting for ?H .
The observed oscillatory dependence of Af on ?peak results from the rapid non-
adiabatic, i.e. superluminal, contraction in this experiment. The solid red curve emphasizes
this point by plotting the simulated behavior for a slow adiabatic contraction, computed
with ? = 0. No dependence on ?peak is present in this limit, as the phonon would undergo
many oscillations during expansion and therefore lose any dependence on initial phase.
The deviation from the adiabatic curve is associated with ?classical? stimulated emission
or absorption into or out of the phonon field, in much the same way that these processes
have been observed in acoustic black holes [23]. Direct observation of spontaneous
7The phenomenological damping term in Eq. (6.1) leads both Ai and Af to decrease in a common-mode
manner with increasing ?peak, but that decrease is absent in the ratio Af/Ai.
74
processes, i.e. pair production [25], would require an increase in our detection threshold.
While here we averaged three images per time point, the observation of spontaneous
Hawking radiation in Ref. [91] required a ? 104 image dataset.
Our data generally agrees with the predictions of Refs. [62, 82], with the notable
exception ?H 6= ?. Because we create large-amplitude phonons to maximize our detection
signal, it is possible that this leads to non-linear damping effects [48], compromising
our measurement of ?H, and potentially causing the additional dependence on ?peak seen
in Fig. 6.4(b). Likewise, the simple scaling of c? with R holds only in the thin-ring
approximation, and our smallest rings have thickness to mean radius ratio of 0.45. For
future experiments, our system is flexible enough to explore different metric scalings: to
date we focused on quasi-one-dimensional universes, we could also potentially simulate
two-dimensional (disc or square condensate) expansions or contractions where ?H > 1, as
suggested in Ref. [62]. Our experimental setup could also readily explore other analogue
gravity systems such as black hole horizons in 2D systems, where, for example the
acoustic metric resulting from quantized vortices could open new directions [94].
6.2 Model for the Toroidal Potential and Scaling Laws
The analysis in Sec. 6.1 assumes a power-law model for the trap potential. This
section defines the trap potential V and derives relevant scaling laws essential for data
analysis. The discussion here follows Ref. [62]. A cylindrically symmetric trap potential
75
V (?, z) can be described by the power-law
V (?, z) := ? [(?/? n??) + (z/z )
nz
? ] , (6.2)
where ? is the chemical potential of the BEC, n? and nz are the power-law exponents,
and ?? and z? are the T-F widths along e? and ez, respectively. Since the volume V is
independent of the chemical potential ?, the T-F widths scale according to
? ? ?1/n?? and z ? ?1/nz? (6.3)
The time evolution of a cylindrically symmetric BEC wavefunction ?(?, z, t) is
given by the time dependent GP equation
2
? h? ?2 ??? + V ? + g|?|2 ? = ih? . (6.4)
2m ?t
Under the T-F approximation, the spatial derivatives can be neglected. Assuming a
stationary time dependence ih? ?t?(r, t) = ? ?, Eq. (6.4) can be expressed as
? = g|?|2 + V, (6.5)
where |?|2 is the atomic density. Integrating Eq. (6.5) over the entire BEC volume gives
the expression ?
gN = d? dz (R + ?) [?? V (?, z)] , (6.6)
76
where N is the total atom number. Solving the integral on the LHS of Eq. (6.6), it can be
shown (see Ref. [62] for details) that
? ? (R??z ?1?) . (6.7)
Using Eq. (6.7) and (6.2),
? ? R??, (6.8)
where
1
? = . (6.9)
1 + 1/n? + 1/nz
This further gives rise to two other scaling laws for the volume V and azimuthal speed of
sound c?
V ? R??z? ? R? and c2? ? R?? (6.10)
These scaling laws were used for data analysis.
6.3 Simulation: Impact of Phonon Phase on Hubble Friction Strength
One of the goals of this project was the precise determination of Hubble friction
strength. We performed multiple simulations of the differential Eq. (6.1) to estimate the
impact that different experimentally controllable parameters would have on the strength
of Hubble friction. To better understand how these parameters affect the Hubble friction,
77
I will rewrite Eq. (6.1) by making the substitutions V?/V = ?H R?/R, and ?m = mc?/R.
The resulting differential equation is
{ [ ] }
?2 R?(t) ? mc?(t)
+ 2?(t) + ?H + ??(t) = 0, (6.11)
?t2 R(t) ?t R(t)
where the phase perturbations ?? are related to the density perturbations ?n by
? ?? g ?n
= ? . (6.12)
?t h? R?
As evident from Eq. (6.11), the Hubble friction strength depends on the value of
?H R?/R(t), and therefore its magnitude increases as the rate of change of radius R?
increases. Our radial trap frequency is about 500 Hz which sets an upper limit on |R?/R|.
Due to trap inhomogeneities, we cannot make a uniform ring larger than 40 ?m in radius.
Even though we can make rings with very small radii, doing so would violate the thin-ring
approximation assumed in the derivation of Eq. (6.1) and (6.11). Therefore, we limit our
ring radii between 12 and 44 ?m and perform expansions and contractions with |R?/R|
less than 328 s?1.
The other quantity that affects Hubble friction strength is the phase of ?n when
|R?/R| achieves its maximum value (?peak). From Eq. (6.11) and (6.12), the third term
can be shown to be proportional to R?/R(t) ?n. Hubble friction results in maximum
attenuation or amplification if the phase of ?n is such that it attains its maximum value
when |R?/R(t)| is maximum. To investigate this hypothesized impact we solved the
differential Eq. (6.11) for different phonon phases ?peak. The remainder of this section
78
discusses simulations that were performed to understand the impact of the phonon phase
?peak.
6.3.1 Simulation Model
To determine the time evolution of ?? or ?n, in addition to Eq. (6.11) and (6.12),
we need the time-evolution of R, c?, and ?. Both in the experiment and the simulations,
radius R is varied according to
???
???????Ri,?? [ ( ( ))]
if t < ti
R(t) = ??R + Rf?Ri?? i 1 + erf 5
t?ti ? 0.5 , if ti < t < t (6.13)2 t i + texp? exp??Rf , if t > ti + texp,
where ti denotes the start of ring dynamics and texp = 10 ms is the duration of ring
dynamics. Since, c ? R??/2? (see Eq. (6.10)), we assume
( )??/2
R(t)
c?(t) = c?i , (6.14)
Ri
where c?i is the azimuthal speed of sound in the initial ring of radius Ri. As described in
Sec. 6.1, ? is expressed in terms of the quality factor as
?(t) = m c?(t)/2 R Q(t), (6.15)
79
where we assume a linear dependence of Q on radius R given by
Qi ?Qf QfRi ?QiRf
Q(t) = R + , (6.16)
Ri ?Rf Ri ?Rf
where Qi and Qf are quality factors in the initial and final rings. We use equations (6.13),
(6.14), (6.15), and (6.16) along with the differential equation (6.11) and the density phase
relationship (6.12) to determine the time evolution of ?n.
6.3.2 Simulation Results
We solve the differential equation as described in the previous sub-section, to determine
the impact of ?peak on Hubble friction strength. Fig. 6.5 is one such simulation for
contraction experiments. In Fig. 6.5 (a), we start with a ring with initial radius Ri =
38 ?m and then contract it to a final radius Rf = 12 ?m according to Eq. (6.13). The
initial and final quality factors are assumed to be Qi = Qf = 10. The contraction starts
at time ti. The blue line indicates the ring radius as a function of time t. As the ring
contracts, phonons evolve according to Eq. (6.11). We plot the amplitude of the phonon?s
density perturbation ?n as the black curve in Fig. 6.5 (a). As is evident from the frequency
of oscillations, phonons experience a blue-shift in frequency upon contraction. The time
refers to the time when ?? ??tpeak ? ?R?/R? attains its maximum value. The phase of the phonon at
tpeak is given by
t
? ? peakpeak dt ?(t) + ? . We determine the amplitude of the decaying0 0
phonon envelope for both the initial and final phonons at the instant t = tpeak. These
amplitudes are referred to as Ai and Af respectively, as shown in Fig. 6.5 (a).
We study the hypothesized impact of the phonon phase ?peak on the Hubble friction
80
(a)
(b)
Figure 6.5: Simulations to study the impact of phonon phase ?peak on Hubble friction
strength. (a) Phonon amplitude (solid black curve), and oscillation envelope (dashed red
curve) used to extract the amplitude Ai and Af at tpeak. The blue curve denoted the radius
R as a fucntion of time t. The grayscale bar is as notated in Fig. 6.2 (d). tpeak is the
time when |R?/R| attains maximum value. (b) Ratio of amplitudes Af/Ai vs. ?peak, the
oscillation?s phase at tpeak. The solid red and dashed black curves correspond to a value
of ?H of 0.5 and 0, respectively.
by changing ti [see Fig. 6.5 (a)], thereby phase-shifting the phonon by (c?,i/Ri)ti. Fig. 6.5 (b)
summarizes this result where we plot the ratio Af/Ai as a function of the phonon phase
?peak. The dashed black and solid red lines are obtained by setting the value of ?H to 0
and 0.5, respectively. As a result, the black line represents the situation in the absence
of Hubble friction and contrasts it to the red curve which represents a non-zero Hubble
friction. As evident from Fig. 6.5 (b), there is a significant difference between the black
and red ratiosAf/Ai only for certain specific values of the phase ?peak. At all other phases,
the difference is small, indicating a strong dependence of the Hubble friction strength on
81
the phase ?peak. This simulation served as our motivation for varying the phase ?peak in
the experiment.
6.4 Mode Purity of the Phonons
The experiments in this chapter are performed with m = 1 azimuthal phonons
imprinted by halftoned binary patterns on a DMD. Though the patterns were selected to
imprint only m = 1 phonons, due to imperfections in the projected optical perturbation,
the excited phonons should be a linear combination of all possible phonon modes m.
These experiments follow the trap loading procedure described in Sec. 5.4. The final
ODT in Sec. 5.4 was a disk-shaped trap, as shown in Fig. 5.8 (h) and (l). Once the atoms
are transferred into this purely optical trap, the disk DMD pattern is adiabatically changed
to a ring of radius Ri. Atoms are then allowed to rest for 500 ms before the perturbation
is applied.
To project azimuthal phonons, we impart a perturbing potential Vph sin(?) using
a DMD [39]. While smaller Vph results in long-lived phonons with weaker density
perturbations ?n, large Vph results in stronger density perturbations that decay quickly.
For our experiments, we wish to detect ?n with large signal-to-noise ratio over multiple
oscillation periods. Therefore the value of Vph was a trade-off between the long decay
times and large signal-to-noise ratio. It was empirically determined to be 0.8 times
the overall potential depth. Sec. 2.2.3 discusses typical damping mechanisms for BEC
collective excitations. Since we mostly imprint the lowest order azimuthal modes, we
expect Landau damping to be the primary mechanism for decay. Landau damping is a
82
finite temperature effect which thermalizes BEC phonons by scattering with other thermal
phonons [46].
(a)
-0.6 0 0.6
(c)
1.2
1
(b) 1
0.8
0.5
0.6
0 0.4
-0.5 0.2
-1 0
0 50 100 150 1 2 3
Figure 6.6: Relative contributions of different azimuthal modes in the imprinted phonon.
(a) Raw data n1D for phonons in a torus of radius R = 31.8 ?m. (b) The black dots, red
open circles and blue crosses show the time evolution of =[Am] for modes with m = 1, 2,
and 3, respectively. The colored solid lines represent decaying sinusoidal fits according
to Eq. (6.18). Each colored fit curve corresponds to the data points of the same color. (c)
The amplitude Bm (see Eq. (6.18)) for the three modes evaluated from the fits in (b).
Since the DMD is a binary amplitude grating, we use Jarvis halftoning [66] to create
smooth intensity gradients for imprinting phonons with mode number m = 1. Fig. 6.6
demonstrates the purity of the modes generated by this process. Fig. 6.6 (a) is the raw
data corresponding to azimuthal density perturbation ?n1D(?, t) as a function of time t
and azimuthal angle ? for phonons in a torus with fixed radius R = 31.8 ?m. To estimate
the relative contribution of the different phonon modes, we evaluate the azimuthal spatial
Fourier coefficient Am(t) for the three lowest energy modes (m equal to 1, 2, and 3),
83
according to
?
?n
A 1m(t) = ?D(?, t) exp(i m ?) d? . (6.17)
?n1D(?, t) d?
Fig. 6.6 (b) plots the time-evolution of the imaginary part of Am for the three modes. We
fit =[Am] with a decaying sinusoidal function
Bm exp(??mt/2Q) sin(?mt+ ?), (6.18)
where ? is the temporal phase at t = 0, Q is the quality factor, and Bm is the initial
amplitude. The solid lines in Fig. 6.6(b) indicate these fits. Fig. 6.6(c) shows the initial
amplitude Bm for the three modes. The error bars correspond to the 95% confidence
interval in the fitted values of Bm. Bm of the desired m = 1 mode is 6.5(9) and 6.9(8)
times larger than the m = 2 and m = 3 modes, respectively, indicating that while our
oscillations are primarily m = 1, we may have a small admixture of other modes.
Expansion and contractions are performed by dynamically varying the ring radius in
subsequent DMD frames. Leveraging the DMD?s 32 kHz frame update rate, we smoothly
transition from the initial to final radii in 90 frames, such that the ring radius changes by
a distance corresponding to at most one DMD pixel in consecutive frames. To resolve
small density variations, we detect the phonons at different evolution times using PTAI,
where only 5-12% of the atoms are imaged. The transferred fraction is varied based on
the overall density of the sample to maintain similar levels of signal-to-noise ratio. Even
though we use the minimally destructive PTAI, only one image is acquired per repetition
84
of the experiment.
6.5 Fitting Method
In this section, we give a detailed description of our fitting methods. Our raw
data comprises of azimuthal density perturbations ?n1D(?, t) corresponding to expanding,
contracting, and fixed-radii tori. As mentioned in Sec. 6.1.3, we perform the global
fits in eight ways to mitigate potential systematic biases introduced by a fit parameter.
Performing fits in eight different ways allowed us to investigate any potential systematic
bias that 85 (i.e. 85 = 117 ? 32) of the 117 fit parameters might have caused. These
methods enumerated in roman numerals I-VIII vary in the number of fit parameters
?fit. Methods I and V assume the same temporal and azimuthal phases, methods II and
VI assume same azimuthal but different temporal phases, methods III and VII assume
same temporal but different azimuthal phases, and methods IV and VIII assume different
azimuthal and temporal phases across the datasets. Methods I to IV assume a constant
atom number throughout the ring dynamics. Since we observe an atom number loss
of up to 20 % during ring dynamics, we analyze the data with fitting methods V to
VIII, which fit the atom number N(t) to an exponentially varying function of time and
accordingly accounts for a change in c?(t) ? N(t)?/2. These differences in the fit
methodology are reflected in the number of fit parameters ?fit. Tab. 6.2 and Tab. 6.3 list the
fit results obtained from each of the eight fitting methods for expansions and contractions,
respectively. For both tables, the fit values corresponding to the eight methods agree
within 2-?.
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Table 6.2: Fit results of the eight different techniques when applied to the expansion data
sets.
Mtd. ?fit Qi Qf ? ?H c?,i ?ni
(mm/s) (rad?1)
I 32 3.5(1) 4.3(2) 0.48(1) 0.30(3) 5.44(1) 7.47(11)
II 43 3.5(1) 4.4(2) 0.47(1) 0.32(3) 5.44(1) 7.50(11)
III 43 3.5(1) 4.4(2) 0.47(1) 0.28(3) 5.42(1) 7.41(11)
IV 54 3.5(1) 4.5(2) 0.47(1) 0.29(3) 5.42(1) 7.41(11)
V 55 3.5(1) 4.3(2) 0.47(1) 0.26(3) 5.42(1) 7.52(11)
VI 66 3.5(1) 4.4(2) 0.46(1) 0.29(3) 5.41(1) 7.54(12)
VII 66 3.5(1) 4.5(2) 0.46(1) 0.24(3) 5.41(1) 7.44(11)
VIII 77 3.5(1) 4.5(2) 0.46(1) 0.27(3) 5.40(1) 7.45(11)
Table 6.3: Fit results of the eight different techniques when applied to the contraction
data sets.
Mtd. ?fit Qi Qf ? ?H c?,i ?ni
(mm/s) (rad?1)
I 32 7.7(2) 3.6(1) 0.51(1) 0.34(2) 4.38(1) 4.45(4)
II 49 7.6(2) 3.4(1) 0.49(1) 0.37(2) 4.40(1) 4.48(4)
III 49 7.7(2) 3.6(1) 0.51(1) 0.33(2) 4.38(1) 4.47(4)
IV 66 7.7(2) 3.5(1) 0.49(1) 0.37(2) 4.41(1) 4.49(4)
V 67 7.7(2) 3.6(1) 0.55(1) 0.37(2) 4.31(1) 4.53(4)
VI 84 7.9(2) 3.5(1) 0.53(1) 0.40(2) 4.33(1) 4.52(4)
VII 84 7.8(2) 3.7(1) 0.55(1) 0.35(2) 4.32(1) 4.55(4)
VIII 101 8.1(2) 3.5(1) 0.53(1) 0.39(2) 4.34(1) 4.54(4)
The large number of fit parameters result from including seven datasets corresponding
to fixed-radius tori in the simultaneous fits. As mentioned in Sec. 6.1.3, dynamic datasets
typically have less than one oscillation before the radius changes. Therefore we include
the fixed-radius datasets to reduce the uncertainty in Qi and ?(Ri). In addition to the
global fit parameters listed in Tab. 6.1, the fit includes the amplitude, quality factor,
azimuthal and temporal phases of the seven fixed-radius tori datasets as fit parameters.
This sets the minimum value of fit parameters ?fit to 32; 6 corresponding to the global
86
parameters listed in Tab. 6.1, 14 corresponding to the temporal and azimuthal phases of
the seven stationary tori, and 12 corresponding to the amplitude and quality factor of six
of the seven fixed-radius tori. The phonon in one of the seven fixed-radius tori is identical
to the initial phonon of each dynamic dataset. Therefore its amplitude and quality factor
are assumed to be the same as the global parameters ?ni and Qf , respectively. Since the
dynamic datasets typically have less than one oscillation before the radius changes, this
allows us to constrain the value of ?ni and Qf . Fig. 6.7 shows the phonon evolution in
the stationary ring. The black circles denote the measured value of ?n, and the red curve
corresponds to the simultaneous fits. Given the variation in amplitude, quality factor,
azimuthal and temporal phases across the seven fixed-radius datasets, these had to be
included as independent fit parameters.
87
10
0
-10
10
0
-10
10
0
-10
10
0
-10
10
0
-10
10
0
-10
10
0
-10
0 50 100 150 200 250
Figure 6.7: Phonon amplitude ?n as a function of time t for stationary tori of radii 11.9,
14.3, 16.7, 24.0, 31.2, 38.4 and 43.2 ?m from top to bottom. Here, the red curves show
simultaneous fits to the complete data set, as discussed in Sec. 6.1.3.
88
Chapter 7: Towards Erbium BEC: Scanning Transfer Cavity Lock
One of the goals when designing and building the new apparatus was to make a
dual-species degenerate gas mixture with Sodium and Erbium. Even though we haven?t
produced an Erbium BEC, significant progress has been made. For example, in 2017,
we successfully demonstrated an Erbium 2D MOT in the same vacuum chamber as
the sodium 2D MOT, thereby creating a single compact source of cold atoms for both
species. In addition, my colleague Madison Anderson has constructed an inductive oven
for producing a high flux of Erbium atoms. You can find more about that in her thesis,
expected shortly.
Erbium is a rare-earth metal from the lanthanide series with atomic number 68.
It has relatively high melting and boiling points of 1529? C and 2900? C, respectively.
Atomic Erbium has six stable isotopes 162Er, 164Er, 166Er, 167Er, and 168Er. 167Er is a
fermion while all others are bosons. Since the first experimental realizations of BECs [6,
7] and Fermi degenerate gases [95?97], the community of quantum gas researchers have
laid significant emphasis on tuning inter-atomic interactions to build fully controllable
quantum systems. Usually, for ultracold atoms, the interatomic interactions occur via
89
s-wave scattering, resulting in a mean-field contact potential Ucontact, given by
4?h?2a
Ucontact(r) = ? g ?(r), (7.1)
m
where a is the s-wave scattering length, m is the mass of the atom, g is the GP interaction
constant, and ?(r) is the kronecker delta function [98]. As evident from Eq. (7.1), Ucontact
is isotropic and short range. With the use of Feshbach resonances, the scattering length
a can be tuned by orders of magnitude [35], thereby giving us a convenient handle on
interactions. Since its first realization, Fesbach resonance have been used for many
applications [10?13, 99]. In addition to the above interactions, elements with high dipole
moment interact via a dipole-dipole potential given by
C 1? 3 cos2 ?
Udd(r
dd
) = , (7.2)
4? r3
where ? is the angle between the direction of polarization and the relative position of
the particles r, and Cdd is a constant proportional to the magnetic moment squared in
the case of magnetic dipole dipole interactions [98]. As evident from Eq. (7.2), unlike
Ucontact, Udd is anisotropic and long-range, making dipolar atoms like Erbium an exciting
candidate for studying quantum phases, otherwise not possible with alkali atoms. Erbium
has a relatively high permanent magnetic dipole moment of 7 ?B, where ?B is the Bohr
magneton. In contrast, alkali metals such as Sodium have a dipole moment of about 1
?B. In addition to the high magnetic moment, bosonic Erbium has multiple Feshbach
resonances densely packed into a small range of magnetic fields. Recent results from the
90
Ferliano group have revealed that 168Er and 166Er have 190 and 189 resonances over a 70
G range [100]. Compare this with 23Na, which is known to have just three resonances at
853, 907, and 1195 G [101, 102].
In 2005, the experimental realization 52Cr (dipole moment = 6 ?B) BEC, opened up
the field of ultracold mixtures of dipolar atoms [103]. This was soon followed by 164Dy
(dipole moment = 10 ?B) [104] in 2011 and 168Er [105] in 2012. Given its potential for
studying strongly correlated many-body systems, we wish to add an Erbium BEC to our
system and exploit these long-range and anisotropic interactions. The first step towards
Bose condensing Erbium atoms is to laser cool them. For this, we need a laser that is
frequency stabilized to the atomic transition of interest. The rest of this chapter will
first give a quick overview of the Erbium atomic structure and then describe the design
and implementation of a locking scheme to frequency-stabilize a 583 nm laser for laser
cooling Erbium in a 3D MOT.
7.1 Laser Cooling of Erbium
-3 -2 -1 0 +1 +2 +3
4f
6s 0
Figure 7.1: The Erbium ground state.
Early cold atom experiments focused on Bose-condensing alkali metals due to
91
their relatively simple energy level structure and high elastic to inelastic collision rate
constants. The energy diagram for Erbium is much more complex and has multiple
transitions that could be used for laser cooling [1]. The ground state of Erbium has
the electronic configuration [Xe] 4f12 6s2, where [Xe] represents the fully-filled atomic
Xenon electronic configuration. Fig. 7.1 shows how the 4f and 6s shells of atomic
Erbium are filled with electrons in the ground state. The two unpaired 4f electrons
have angular momentum magnetic quantum number ml = +2,+3, resulting in a large
magnetic moment with orbital angular momentum quantum number L = 5. The total
electronic spin S = 1 and the total angular momentum quantum number J = 6. As a result,
the Erbium ground state can be expressed as 3H6 in the term notation. In the case of
heavier elements, the spin-orbit interaction is no longer small compared to interactions
of spin and angular moments individually, thereby resulting in a breakdown of the L-S
coupling scheme. All inner and outer electrons independently couple via the L-S coupling
scheme, resulting in two total angular momentum quantum numbers J1 and J2. The
electrons in [Xe] and the 4f shell qualify as inner electrons, while all others are considered
outer. As a result, the atomic states are represented as coupling between these two J states
via jj coupling scheme. Fig. 7.2 is the energy level diagram of Bosonic Erbium. Since
Bosonic Erbium has zero nuclear spin, there is no hyperfine splitting.
Erbium has five possible laser cooling transitions [1], each with J = 7. Of these, we
use the broad 27 MHz, 401 nm line for 2D MOT and the narrow 583 nm transition for 3D
MOT. The excited state for the 2D and 3D MOT transitions are [Xe] 4f (3H6) 6s 6p (1Po1)
and [Xe] 4f (3H6) 6s 6p (3Po1), respectively. The terms in parenthesis indicate the L-S
coupling between the inner and the outer electrons, separately. The linewidth and other
92
Figure 7.2: Laser cooling transitions in atomic Erbium [1]. The purple and green arrows
indicate the cooling transitions used for the 2D MOT and 3D MOT, respectively.
parameters for these transitions are tabulated in Table 7.1. Below the 3D MOT excited
state [Xe] 4f (3H6) 6s 6p (3Po1), Erbium has two metastable states. However, [1] estimates
the transition rates to these states to be 0.017 s?1 and 0.0049 s?1, which is much smaller
than the fluorescence decay rate of the 3D MOT excited state to the ground state (106 s?1).
As a result, there is practically no leakage to dark states, and a re-pump is not needed. All
we need is a laser locked at the 583 nm atomic transition with a line width and frequency
stability better than 170 kHz. For the rest of the chapter, I will describe a simple and
robust method to achieve such a lock by utilizing available frequency stabilized lasers in
the lab.
93
583 nm 3D MOT
401nm 2D MOT
Table 7.1: Erbium laser cooling transitions [1].
? (nm) ? (s?1) ?? (MHz) I 2sat (mW/cm )
2D MOT 400.91 1.7 ?108 27 56
3D MOT 582.84 1.0 ?106 0.17 0.11
7.2 Frequency Stabilization of 583 nm Laser
We use a 583 nm, frequency-doubled semiconductor laser (Toptica SHG DL-Pro)
for the Erbium 3D MOT. The linewidth of this laser is supposed to be less than 2? 30
kHz when measured over a 1 s period. We verified it to be 2? 48(2) kHz by a delayed
self heterodyne measurement with a 2 km long optical fiber and an80 MHz AOM. Since
the Erbium 3D MOT transition has a 2? 170 kHz natural linewidth, the short-term laser
linewidth is good enough, and there is no need to narrow it further. However, over longer
durations, the laser frequency drifts at an approximate rate of 33 kHz/s (measured on an
Angstorm High Finesse WSU2 wavemeter), under typical laboratory conditions. This
corresponds to a drift equal to the Erbium 3D MOT transition linewidth in about 5 s.
Therefore, we need to frequency stabilize this laser with a locking scheme capable of
responding faster than 5 s or a bandwidth greater than 200 mHz.
Typical laser frequency stabilization schemes such as Saturation Absorption Spectroscopy
(SAS) lock the laser to the atomic transition by generating the corresponding signal in a
heated vapor cell (for example, see Sec. 4.2). However, these methods rely on significant
light absorption by the atomic gas. Erbium with a melting point of 1529?C has a very
low vapor pressure at temperatures typically used for heating glass vapor cells. Fig. 7.3
contrasts Erbium vapor pressures with Sodium, for which SAS is used as a locking
94
Figure 7.3: Vapor pressure of Sodium and Erbium as function of temperature [5].
scheme (see Sec. 4.2). At a relatively high temperature of 400?C, Erbium has a vapor
pressure less than 10?10 torr, while for Sodium, it exceeds 103 torr. As a result, very
few Erbium atoms are vaporized, resulting in a weak absorption and low signal-to-noise
ratio. A possible solution is using a heat pipe to heat Erbium to very high temperatures.
However, such systems are cumbersome and occupy significant optical table real estate.
An alternative could be using an ultra-stable reference such as an Ultra-Low Expansion
(ULE) cavity with frequency drifts in the order of tens of milli-hertz per second [106].
However, these are expensive, need special care such as vacuum engineering, and are
overkill for our problem.
Since we already have a 589 nm laser locked to the D transition of 232 Na, we
developed a scheme to use this laser to frequency stabilize the Erbium 583 nm laser in
a master-slave configuration. The locked master laser frequency drifts should be much
smaller than the Sodium natural linewidth of 2? 9.8 MHz. We measured this drift to be
95
about 100 kHz over a 30 min period (using an Angstorm High Finesse WSU2 wavemeter),
which could be further improved by simply increasing the bandwidth of the master laser
locking electronics. One way of transferring this stability is by locking the length of
an optical cavity to the master and then locking the slave to this cavity using a Pound
Drever Hall (PDH) scheme [107]. Such a scheme can provide sub 100 Hz stability but
involves setting up an entire locking apparatus, including cavity and electronics, for each
slave laser. We decided to implement a Scanning Transfer Cavity (STC) scheme, which
is bandwidth limited due to the piezo?s slow scan rate but offers a much simpler system
for locking multiple slave lasers to the same master. An STC is a Fabry Perot with piezo
actuated length tunability. A beam from the master and each slave laser is sent into
the cavity, while its length is scanned periodically. As the length scans, each of these
beams resonates with the cavity?s natural frequency and result in a transmission peak at
the output. The time difference between these peaks gives an estimate of the frequency
difference between the different lasers. This information is then used to generate a
feedback signal using a servo to stabilize the slave frequency to the master. The scan
rate of the STC piezo limits the bandwidth of such a locking scheme. Standard piezos are
adequate for achieving our modest bandwidth requirements (200 mHz). In the following
sections, I will first explain some of the basics of how Fabry Perot (FP) etalons work and
then describe the design and construction of our STC.
96
Transmitted  Beams
Mirror 1 Mirror 2
Figure 7.4: Interference in a simple Fabry Perot cavity. The beam on entering the cavity
undergoes multiple reflections. These when constructively interfere lead to a maximum
in transmission intensity.
7.3 Fabry Perot Etalons
In its simplest form, a Fabry Perot (FP) etalon consists of two parallel mirrors placed
at a separation L, as shown in Fig. 7.4. Light enters the cavity from one of the mirrors,
reflects multiple times, and eventually transmits through the other mirror. If the frequency
of this light is in resonance with the cavity, the multiple paths constructively interfere and
result in increased transmission intensity. Fig. 7.4 describes the amplification of light in a
plane mirror FP cavity. Consider a monochromatic beam travelling along ez with electric
field Ei ? ei?t?kz, incident on the front surface of the plane mirror cavity. Here t is time,
? is the frequency, and k ez = 2??/c ez is the propagation constant. The phase delay in
each round trip is given by ? = 4??L/c, where L is the length of the cavity. Transmitted
97
beams from the different reflections add up to give a total transmission Et, given by
T
E i? 2t = Eit1t2(1 + r1r2e + (r1r2) e
2i? + ..) = E , (7.3)
1? iRei?
where t1, t2 and r1, r2 are the transmission and reflection coefficients of the two mirrors,
respectively, and T andR are defined as T = t1t2 andR = r1r2. The intensity corresponding
to the transmitted electric field It is given by
? (1?R)2It = EtEt = Ii 2 . (7.4)(1?R)2 + 4R sin ?/2
From Eq. (7.4), we see that the transmitted intensity It achieves its maximum value when
1
0.8
0.6 R = 0.6
R = 0.8
0.4 R = 0.995
0.2
0
0.5 1 1.5
Figure 7.5: Cavity transmission peaks with varying mirror reflectivity R , according to
Eq. (7.4). The red, green and purple curves correspond to a reflectivity R of 0.6, 0.8 and
0.995, respectively.
the round-trip phase delay ? = 2q?, where q is an integer. This corresponds to frequencies
? = q ?fsr, where ?fsr = c/2L is the free spectral range of the cavity. Fig. 7.5 shows the
98
transmitted intensity as a function of laser frequency ? for different reflectivities R. As
R increases, the linewidth of the transmission peak ?? decreases, thereby improving the
frequency resolution of the cavity. The ratio of free spectral range ?fsr to linewidth ?? is
called finesse F . It can be shown that in the absence of other losses, the finesse is simply
?
a function of the reflectivity of the mirrors and is given by F = ? R/(1?R) [108,109].
Since the linewidth ?? is given by ?? = ?fsr/F , a high finesse helps achieve a narrow
linewidth for a fixed free spectral range.
The finesse of a cavity is degraded by factors such as diffraction and mirror surface
irregularities. Additionally, the finesse of a simple plane mirror FP is very sensitive to
mirror alignment as small angular misalignments are equivalent to corresponding surface
imperfections. In contrast, curved mirrors leave much more room for error as an angular
misalignment merely redefines the optical axis [110]. Given the many advantages of
resonators with curved mirrors, I will now formally explore the stability of modes in a
cavity with mirrors of different curvatures.
Mirror 1 Mirror 2
Figure 7.6: Schematic of a Fabry Perot with curved mirrors. The mirrors with radii of
curvature R1 and R2 are separated by a distance d.
Resonance in a FP is achieved by ensuring the same field distribution after each
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round trip. Consider a cavity of length L, made with mirrors of radius of curvatures R1
andR2, as shown in Fig. 7.6. It can be shown that for an electromagnetic wave to replicate
itself after each round trip, the geometry of the cavity should satisfy Eq. (7.5) [108, 109]
? L L0 (1 + )(1 + ) ? 1, (7.5)
R1 R2
Fig. 7.7 graphically depicts the stability condition (Eq. (7.5)) as a function of g1 and
g2, where g = 1 + L1 and g2 = 1 + L . The white region indicates the region ofR1 R2
stability where Eq. (7.5) is satisfied. As evident from Fig. 7.7, plane parallel (point 1)
and concentric (point 3) configurations lie at the edge of the stable region resulting in an
fairly unstable modes. A symmetric confocal configuration (point 2) lies within the stable
region and is very common for commercial FP cavities. Point 4 indicates the position of
our STC which is made using a plane and curved mirror.
To find the resonance condition, we consider eigenmodes of a FP cavity with mirrors
of arbitrary curvature. These eigenmodes can be described as Hermite Gauss modes of
order (l,m) given by
(? ) (? ) ( )
w0 2x 2y k
U(x, y, z)l,m = Hl Hm exp ?i(kz ? ?(l,m; z))? i (x2 + y2) ,
w(z) w(z) ? 2R
(7.6)
where Hn,m are Hermite polynomials, w(z) is the waist of the beam as a function of
distance z from the focus in the direction of propagation, and w0 is the minimum waist
at the focus. The phase term ?(l,m; z), called the guoy phase is given by ?(l,m; z) =
kz?(l+m+1) ?(z) where ? is given by ?(z) = tan?1(z/z0), z 20 = ?w0/? is the Rayleigh
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4
Unstable 3
 (1) Plane Parallel - (1,1)
Stable 2 (2) Symmetric Confocal - (0,0)(1) (3) Concentric - (-1,-1)
1 (2) (4) STC - (1, 0.58)
0
-1 (4)
-2 (3)
-3
-4
-4 -2 0 2 4
Figure 7.7: Stability diagram for a two mirror cavity based on Eq. (7.5). The red points
(1), (2), and (3) indicate positions of plane parallel (R1 = R2 =?), symmetric confocal
(R1 = R2 = L/2), and concentric (R1 = R2 = L) mirror configurations. The blue point
(4) indicates the position of the STC.
length, and l and m are integers. The phase accumulated in each round trip is given by
?
2kL ? 2(l + m + 1) ?? , where ?? can be approximated to ?? ? cos?1(? g1g2).
Resonance in such cavities occurs for frequencies ?lmq given by
( )
cos?1(??g1g2) c
?lmq = q + (l +m+ 1) . (7.7)
? 2L
?
As evident from Eq. (7.7), geometries with cos?1(? g1g2)/? equal to 0, 1/2 or 1 exhibit
a great deal of degeneracy between axial and transverse modes. These correspond to
plane-parallel, confocal and concentric arrangements, respectively. In cavities with degenerate
modes, slight geometric deviations cause broadening of the cavity linewidth. Though
such broadening eases coupling light into the cavity, it is not ideal for laser stabilization
101
purposes. For our STC, we use one plane and a R = 500 mm concave mirror separated
by 210.5 mm. Such a plane-concave mirror configuration not only satisfies the stability
criteria of Eq. (7.5) but also avoids degenerate modes. Point 4 in Fig. 7.7 depicts our STC.
7.4 Construction of the Setup
The setup for the STC lock includes a homemade Fabry Perot cavity, an electronic
board for converting the analog cavity transmission peaks into digital pulses, and a microprocessor
to generate the feedback to control the laser frequency. This section describes the construction
of all three parts.
7.4.1 Cavity Construction
The FP cavity acts as a tunable, narrow bandpass filter capable of detecting the
master-slave frequency difference with a precision better than the 583 nm Erbium transition
natural linewidth. The time difference between the resonance peaks gives an estimate of
the frequency difference. As a result, key considerations in its design are [110]
? mechanical stability and isolation from temperature and pressure fluctuations
? attainment of high finesse to obtain a narrow linewidth
? and an optimal optical layout to achieve mode matching.
We will touch upon these aspects in the following sections.
102
Figure 7.8: Schematic of the scanning Fabry Perot cavity. The curved mirror is attached
to the piezo which is epoxied to the invar spacer via a macor ring. The plane mirror is
attached on the other end of the spacer.
7.4.1.1 Mechanical and Thermal Stability
Fig. 7.9 is a schematic of the STC. It comprises a ?Invar36? metallic spacer, a piezo
to scan the separation between the two resonator mirrors, and a macor ring between the
piezo and spacer surfaces. Macor being an insulator avoids short-circuiting the inner and
outer electrodes of the piezo. The two mirrors are glued to the ends of this assembly
using ?Torr-Seal? epoxy. An important consideration for the cavity design is the drift in
its free spectral range with ambient temperature changes. The length L in the expression
?fsr = c/2L is the distance between the mirrors. In our case, it is the sum of the lengths
of the invar spacer (Lin), macor ring (Lm) and piezo (Lp). For a 1?C rise in temperature,
103
the change in FSR, ??fsr is given by
c ?Lin + ?Lm + ?Lp
?? (7.8)fsr = ,
2L L
where ?Lin/m/p are the corresponding change in lengths. Since, for a 1? C rise in
temperature, the fractional change in lengths are given by coefficients of thermal expansion
(CTE), the above expression can be re written as
Lm Lp
??fsr = ?fsr (?in + ?m + ?p ), (7.9)
Lin Lin
where ?in/m/p are the corresponding CTEs. Here, the length of the invar spacer is
assumed to be very large compared to the macor and piezo. The CTE and dimensions
of the different parts that make up the cavity are tabulated in Table 7.2.
Table 7.2: Thermal expansion of cavity parts.
Part Material CTE Length ? Lxx Lin
(10?6/?C) (mm) (10?6/0C)
Spacer (in) Invar 36 1.2 190.5 1.2
Piezo (p) APC Material- II ? 20 ?
Insulating Ring (m) Macor 5.2 2.54 0.069
It is evident from Table 7.2 that despite macor having a higher CTE, the invar
spacer?s expansion limits the thermal stability of cavity free spectral range. Invar was
chosen for its low CTE. For a cavity with ?FSR = 712 MHz, ??fsr ? 903 Hz/?C, much
smaller than the linewidth of the Erbium 583 transition. The piezo used is a standard, off-
the-shelf ring piezo from ?APC International?. It is 20 mm long and has outer and inner
104
diameters 19 mm and 16 mm, respectively. The piezoelectric coefficient d31 = -175,
resulting in an axial elongation of 0.467?m for every 200 V applied across the inner and
outer electrodes. For a 200 mm long cavity, this corresponds to a frequency scan greater
than 1.5 ? ?FSR.
7.4.1.2 Finesse and Linewidth
1
0.8
Data
0.6 Lorentzian fit
0.4
0.2
0
-10 0 10
Figure 7.9: Linewidth measurement of the constructed STC. The blue points correspond
to measured transmitted intensity It as the cavity piezo is scanned. The horizontal axis
is the cavity resonance frequency ?cav relative to the master laser frequency ?master which
serves as a reference. The black line is a Lorentzian fit to the data giving a linewidth of
1.14 MHz.
Our goal is to make a FP with the smallest possible linewidth. This can be achieved
by either using mirrors with high reflectivity or by increasing the length of the FP. While
higher reflectivities lead to larger finesse F , large cavity lengths decrease the free spectral
range ?fsr. We use mirrors with 99.5% reflectivity, with an expectation of an ideal finesse
of 625. Mirrors with reflectivities higher than this are very expensive. We decided to
go with a cavity length of 210.5 mm; any longer would make aligning the laser beam
105
through the cavity challenging. This gives us a ?fsr = 712 MHz. Based on the ideal
finesse of 625, we expected to get a cavity linewidth of 1.14 MHz. Fig. 7.9 shows one of
589 peaks as the cavity?s natural frequency ?c is varied. By fitting the transmission peak
to a Lorentzian line-shape and assuming a 712 MHz FSR, we obtain a linewidth of 1.5(2)
MHz, corresponding to a finesse F = 469.
7.4.1.3 Mode matching
Figure 7.10: Optical setup of the scanning transfer cavity. Light from the two lasers
are couple into a fiber and sent to the STC. Lenses with focal length f1 = 150mm and
f2 = 100mm are used to mode match to the cavity.
Fig. 7.10 shows a schematic of the STC and associated optics. To couple maximum
light into the (0,0) Hermite gauss cavity mode, the beam?s wavefront curvature should
match that of the mirrors at their respective locations [111, 112]. This corresponds to a
500 mm curvature at the curved mirror and minimum waist at the flat mirror. Since the
length of the cavity (L) is 210.5 mm, the required waist size at the plane mirror can be
106
Figure 7.11: Mode matching optics for the cavity. The Gaussian beam from the fiber is
collimated using an asphere. Lenses f1 and f2 are placed appropriately to achieve mode
matching. di?s are the distances between the various components. d0, d1, d2 and d3 are
127, 150, 120 and 317.5 mm respectively.
estimated using Eq. 7.10.
?
zR R
R(L) = L(1 + ( )2) =? zR?= L ? 1L L (7.10)
?zR
z 2R = ?w0/? =? w0 = ?
where zR is the Rayleigh length. This turns out to be 215.133 ?m. We achieve this using
two lenses of focal lengths f1 and f2. We perform a ray matrix analysis to determine
appropriate placement of these lenses [109, 111, 112]. The optical setup is as shown in
Fig. 7.11 where capital letters indicate the ray transfer matrix of each optical element.
Matrices for the two lenses with focal length f1 and f2 are given by Eq. 7.11. C,D,E
and F are the matrices for propagation in free space.
107
?? ?? ?? ?1 0 1 0?
A = ?? ??B = ?? ??
? ? ?? ? ??
1 1
f1
?? ?
? 1 1
f2 ? ? ? (7.11)
?? 1 0?? ?? 1 0?? ??? 1 0??? ??? 1 0?C = D = E = F = ??
d0 1 d1 1 d2 1 d3 1
When solved we obtain the following results.
? 2f1wAwB =
4d20 ? 8d0f1 + 4f 21 + k2w4A
f1(?4d20 + 4d0f 2 41 ? k wA)d1 = ?
4d20 ??8d 2 2 40f1 + 4f1 + k wA (7.12)
2f w2 + 4f 2w2 22 0 2 0wB ? k2w4w4d2 = ? 0 B2w20
2f w2 + 4f 2w2w2 2 4 42
d = B 2 0 B
? k w0wB
3
2w2B
In our setup, light out of the fiber is collimated into a beam of 430 ?m 1/e2 radius, using
an aspheric lens. The first spherical lens is placed 127 mm (d0) away from the asphere.
The two lenses have focal lengths f1 = 150 and f2 = 100 mm. This gives a value of
150, 120 and 317.5 mm for d1, d2 and d3 respectively. Once mode matched, we perform
a two-mirror walk with mirrors M1 and M2 (see Fig. 7.10) to maximize coupling. This
step is tricky as the cavity is single mode. Two things make this process slightly easier.
First, we try to align the beam reflected off the first cavity mirror with the input beam.
Second, we look at the transmitted cavity mode on a camera while walking the mirrors
and scanning the cavity. When roughly aligned, we see multiple modes on the camera
as the cavity scans. A photodetector will capture these different modes as multiple peaks
for a single free spectral range scan. The rest of the alignment is done with the goal of
108
eliminating all but one mode.
7.4.2 Electronics
7.4.2.1 Pulse Detection Board
The photodetector at the output of the cavity (see Fig. 7.10) detects the transmitted
light. We designed an electronics board to detect peaks in transmission intensity and
generate digital pulses whenever they occur. Fig. 7.12 is a schematic of this circuit. The
photodetector signal is sent through two channels. One detects voltage values greater than
a certain threshold using a commercial comparator. The other detects a local maximum
in voltage using a differentiator and comparator in series. The differentiator generates a
derivative of the signal, which is compared against a reference. A NAND gate takes the
output of these channels and goes to a digital LOW when both conditions are satisfied.
Fig. 7.12 depicts how the different components in the circuit work. The LOW pulses
generated by the NAND gate serve as the digital input to the microprocessor.
7.4.2.2 Microprocessor
The microprocessor used is a Teensey 3.2 board. An Arduino IDE code evaluates
the time difference between two consecutive peaks as the cavity scans. This code also
eliminates spurious peaks detected by the photo-diode at the exit of the cavity. We
generate an analog feedback signal using the Audrino PI library, which is fed to the
slave laser?s ECDL grating piezo. While implementing this code, we realized that each
cavity transmission peak generates multiple digital pulses, as shown in Fig. 7.13. The
109
Figure 7.12: Pulse Detection action by the analog board. A voltage pulse (a) detected in
the photo diode is sent through a number of electronic components which ultimately drive
the NAND gate signal (f) LOW. This digital pulse is fed to the micro controller.
yellow curve in Fig. 7.13 is a transmission peak, and purple pulses are digital inputs
generated by the analog pulse detection board. Due to non-uniform motion of the cavity
110
Figure 7.13: Non-uniform motion of the piezo leads to multiple kinks on the cavity
transmission peak. The yellow curve is the transmission peak and purple is the digital
signal sent to the microprocessor, signaling a transmission peak?s occurrence. Each
vertical division is 500 mV for yellow and 2V for purple. Each division on the horizontal
axis is 250 ?s.
piezo, the transmission peaks have multiple kinks, resultingin multiple digital pulses. The
analog board interprets these kinks as separate peaks and sends out false signals to the
microprocessor. We get around this problem by rejecting inputs that are spaced closely in
time.
7.5 Results and Outlook
The microprocessor?s Proportional Integral (PI) output controls the slave?s ECDL
grating piezo voltage as the cavity scans. PI gains and set-points are digitally controlled
by turning rotary encoders on the analog electronics board. We were successful in locking
the 583 nm slave to the locked master. Also, we were able to scan the slave by scanning
the master laser frequency by 5 MHz in 0.5 s, indicating qualitative stability of the lock
for slow drifts, as shown in Fig. 7.14. The frequency stability of the locked laser was
111
20
0
-20
0 50 100 150 200 250
0
-5
-10
22 23 24 25 26 27 28 29 30 31
Figure 7.14: Master and slave frequency drifts ?? as the master oscillates and drifts. The
master SatAbs lock was dis-engaged and the master laser frequency was modulated by
5MHz in a period of 0.5 s. The top panel is data acquired over long periods and shows
the overall drift in the mean frequency. The bottom panel is a zoomed-in version of the
top panel. Measurements were made using the ?Angstorm WSU2? wavemeter via a fiber
switch.
measured using an ?Angstorm high finesse WSU2? wavemeter. Fig. 7.15 is a histogram
of the slave frequency measured over 1 minute of it being locked and unlocked. The
locked laser clearly has a smaller spread in frequencies than when it?s unlocked. However,
the frequency spread is roughly 1.5 MHz, similar in magnitude to the cavity linewidth.
Ideally, we expect the cavity to resolve frequencies much smaller than its linewidth. In
this case, the sub-optimal resolution can be attributed to the non-uniform motion of the
cavity piezo, as shown in Fig. 7.13. The analog board reads these kinks as individual
peaks and sends the information to the microprocessor. Though the microprocessor code
112
Master Laser
Slave Laser
0.04
0.03
Unlocked
0.02
Locked
0.01
0
-5 0 5
Figure 7.15: Histogram of the slave frequencies as it is locked (red) and unlocked (blue).
Measurements were made using the ?Angstorm WSU2? wavemeter via a fiber switch.
picks only one of the many digital pulses, this could correspond to any position on the
cavity peak. As a result, we should expect uncertainty in frequency estimation equal to
the cavity linewidth. To investigate this hypothesis, we look at the cavity transmission
peaks as we scan the laser frequency but hold the piezo stationary. Fig 7.16 (a) shows
the cavity transmission peaks when the cavity is scanned, but the laser frequency is held
constant. Fig 7.16 (b) corresponds to the opposite case where the laser is scanned, but
the cavity is held stationary. The kinks in the transmission peaks appear only when the
cavity is scanned, thereby indicating a non-uniform piezo motion. Even though this STC
setup has failed to stabilize the 583 nm slave laser within the Erbium atomic transition
natural linewidth (2? 170 MHz), we have successfully identified a major shortcoming in
the scanning motion of the piezo. A possible reason for the non-uniform motion could
be the excessive weight of the piezo. By replacing this with a lighter piezo and stacking
the cavity vertically we might be able to relieve some of the strain in its free movement,
113
Probability of Occurance
thereby eliminating the kinks in the cavity transmission peaks.
(a) (b)
Figure 7.16: Cavity transmission peaks (yellow curve) and corresponding pulses
generated by analog board (purple) as the cavity (a) or laser (b) is scanned in frequency.
Each vertical division is 500 mV for yellow and 2V for purple. Each division on the
horizontal axis is 250 s. The above data was taken by scanning the cavity and laser at 3.4
GHz/s.
114
Chapter 8: Conclusions and Outlook
While building this new apparatus, we made several improvements to alleviate
many of the problems faced in the previous setup. For example, the imaging system
in the previous experiment had a numerical aperture of 0.09 and the response was heavily
aberrated [38]. The current experiment has an improved imaging system with a numerical
aperture of 0.28 and is designed for minimal aberrations [64]. One can gauge the degree
of improvement by comparing the typical radial trap frequencies of the toroidal traps. In
this apparatus, we make rings with radial trap frequencies ?r = 2? 500 Hz, a factor of five
better than the NIST experiment. Moreover, for the NIST experiment, it was impossible
to create uniform rings without making significant corrections to the spatial mode of the
DMD ODT beam. For the analogue cosmology experiment, tighter radial confinement has
allowed us to explore rapid expansion and contractions without exciting collective modes
in the radial direction, thereby making our system more 1D than the NIST version. In
future, we plan to exploit this improved resolution for a better phonon detection threshold
and attempt to observe spontaneous pair production. Some progress has been made in this
direction (see Appendix A). Additionally, the improved resolution will help create weak
links of much smaller dimensions, enabling the study of tunneling effects in a superfluid
ring and building upon our previous work in atomtronic circuits [17?19].
115
Another significant improvement was switching the Zeeman slower source with a
more compact 2D MOT. This source is at least three times smaller in volume and doesn?t
have large electromagnets with long tun-off times. Since we plan on making a dual-
species BEC with highly magnetic Erbium, this is particularly useful. The long-range and
anisotropic nature of dipole-dipole interactions in Erbium makes it an exciting candidate
for strongly correlated many-body physics. Erbium has many Feshbach resonances which
are densely packed. Its bosonic isotopes 168Er and 166Er, have respectively 190 and 189
resonances over 70G [100]. This makes Erbium a promising candidate for applications
involving tuning interactions, not limited to but including analogue cosmology.
116
Appendix A: Characterizing Imaging Aberrations
To extend the cosmological analogy to the quantum domain, we plan on exploring
the phenomena of pair production in the early universe [21, 92, 113?116]. As illustrated
by our previous work [41], this involves the precise detection of spontaneously generated
excitations, which relies on the ability to resolve spatial correlations caused by them.
The healing length of condensates ? determines the length scale of these correlations.
Eq. (A.1) defines the healing length [33] as
?
h?2
? = , (A.1)
2m?
where m is the mass of an atom, and ? is the chemical potential of the BEC. Fig. A.1
helps in visualizing these length scales in the context of imaging systems with different
resolutions. Fig. A.1 (a) corresponds to a system whose spatial resolution is limited by
the pixel size lmin, as indicated by the grid. Fig. A.1 (b) corresponds to a system whose
resolution spot size is significantly larger than the camera pixel size. The solid and dashed
circles represent atoms as imaged by the two imaging systems. The thermal De-Broglie
wavelength ?dB represents the spatial extent of atoms. These appear as objects of size R
when imaged through an imaging system with a finite resolution, as shown in Fig. A.1 (b).
Fig. A.1 (c) depicts these length scales in the Fourier domain, indicating the spatial
117
(a) Ideal Imaging System (b) Real Imaging System
Very High Resolution Finite Resolution
Resolution Spot, R
CCD Pixels CCD Pixels
(c)
Resolvable Spatial Frequencies for (a)
Resolvable Spatial 
Frequencies for (b)
Figure A.1: Comparison of length and spatial frequency scales for imaging BECs. (a)
and (b) correspond to imaging systems with spatial resolutions lmin and R. The grid
represents camera pixels. Thermal De-Broglie wavelength ?dB and healing length ? are
indicated. The dashed circles in (b) correspond to (a)?s solid circles of the same color. (c)
Length scales in (a) and (b) in the Fourier domain.
118
frequencies resolved by the two imaging systems. Ideally, we wish to have an imaging
resolution much smaller than ? so that resolving spatial correlations for frequencies near
and smaller than ??1 is straightforward. However, since our typical chemical potentials
are of the order of 1 kHz, ? < 1 ?m, which is of the same order as our imaging system?s
diffraction-limited resolution. As a result, imaging systematics such as diffraction and
aberrations obscure the correlation information. Ref. [117, 118] derives a method to
determine these systematic effects in an uncorrelated atomic sample experimentally. We
employ the same technique to characterize the modulation transfer function (MTF) of our
imaging system.
(a) (b)
0 1.5 -0.75 0.75
-100 -100
0 0
+100 +100
-100 0 +100 -100 0 +100
Figure A.2: (a) Atomic density of weakly interacting disk-shaped BEC nexp(rl). (b)
Density variation of (a) ?nexp(rl) about the average 50 shots. The densities in both (a)
and (b) are in the same arbitrary units.
We start by acquiring images of 50 similar atomic samples of weakly interacting
BECs. The 2D column densities corresponding to each of the 50 samples are similar
but not identical. Fig. A.2 (a) is the density corresponding to one of these samples
119
nexp(rl), and Fig. A.2 (b) is its density fluctuation about the mean density evaluated
across the 50 samples ?nexp(rl) = nexp(rl) ? ?nexp(rl)?, where ?..? denotes averaging
across the 50 samples. These fluctuations in the atomic density ?nexp are a result of
shot-noise. There?fore for a?n ideal imaging system, we expect the density fluctuation2
power spectrum ?? ???nexp(kl)? to be spectrally flat, where .?. represents spatial Fourier
transform. However, due to the finite resolution of the imaging system, diffraction and
imaging aberrations result in stronger density-density correlations for spatial frequencies
less thanR?1, conveniently giving a measure of the MTF. According to [117], the Fourier
transform?o?f the M?? ?
T?F, M(kl) is related to the average of the density fluctuation power??2spectrum ??n(kl) by the equation
??
M2 k ?( l) = ? ?? ?2??n(kl)? . (A.2)
? Fig.?A.3 (a) is the experimentally obtained RHS of Eq. (A.2) estimated by averaging?? 2??nexp(k ?l)? across the 50 images. The strong DC component arises due to the finite extent
of the atomic cloud and needs to be removed for accurate determination of the MTF. For
this, we refer to the relationship between the Fourier transformed MTF M(kl) and the
resonant absorption imaging pupil function p, derived in [117]. The relationship is given
by
???? ??? ?2 ?M2 ?= ??n(kl) = ? ?R? ?[p?]?2 , (A.3)
where the pupil function can be represented in the form of amplitude A and wavefront ?
120
(a) (b)
-0.75 -0.75
0 0
+0.75 +0.75
-0.75 0 +0.75 -0.75 0 +0.75
0 1E-4
Figure A.3: (a) Experimentally evaluated density fluctuation power spectrum as a function
of the spatial frequency kl. (b) Result of fitting (a) to Eq. A.4. The dashed circles enclose
resolvable spatial frequencies for an imaging system with NA = 0.28. The color bar holds
for both (a) and (b).
aberrations as
p(r) = A(r) ei2??(r). (A.4)
?? ? ?2
We fit the experimentally obtained ????nexp(k ?l)? (see Fig. A.3 (a)) to the pupil function
(A.4) with the wavefront aber?ra?tion rep?re?sented by an even polynomial of order six.
Fig. A.3 (b) shows the fitted ?? ?2??n(kl)? , which according to Eq. (A.2) is equal to
M2(kl). This fitted MTF doesn?t have the spurious DC component. Fig. A.4 (a) shows the
wavefront aberration ? obtained from the fit. As a sanity check, we go a step further and
estimate the types of aberrations present in our imaging system, which involves projecting
the wavefront aberration ? onto a Zernike polynomial basis [119], to obtain the first 10
121
Zernike coefficients. Fig. A.4 (b) is the wavefront aberration reconstructed from these
Zernike coefficients.
(a) Polynomial fit (b) Zernike fit
1 1
0 0
-1 -1
-1 0 1 -1 0 1
-2 2
Figure A.4: (a) Wavefront aberration determined by fitting Fig. A.4 (a) to Eq. A.4. (b)
Wavefront aberration reconstructed by projecting (a) onto a Zernike polynomial basis.
Our next goal is to intentionally introduce an aberration in our system and then
look for the desired effect, serving as a sanity check for the above scheme. Since we have
complete control over the system defocus, we now concentrate on the Zernike defocus
coefficient Z20 . Fig. A.5 (a) is a schematic of the imaging system where zi is the distance
between the exit pupil and the camera. We intentionally defocus the system by varying
zi and record the Zernike defocus coefficient Z20 . Fig. A.5 (b) plots Z
2
0 as a function of
?zi. ?zi = 0 represents the optimal focal position. As zi is varied on either side of this
optimal value, Z20 increases in magnitude, thereby confirming optimal focus at ?zi = 0.
122
(a) (b)
Y
X
Object Plane Image Plane
-2 -1 0 1 2
Figure A.5: Experiment to deterministically change the system defocus. (a) The imaging
system with image distance zi as indicated. (b) Variation of Zernike defocus coefficient
Z20 as a function of ?zi.
123
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