Experimental studies of a degenerate unitary Bose gas
byPhilip Makotyn
B.A., Elmhurst College, 2006
B.S., University of Illinois at Urbana-Champaign, 2006

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2014This thesis entitled:
Experimental studies of a degenerate unitary Bose gas
written by Philip Makotyn
has been approved for the Department of Physics

Dr. Deborah S. Jin

Dr. Eric A. Cornell

Date

The final copy of this thesis has been examined by the signatories, and we find that both thecontent and the form meet acceptable presentation standards of scholarly work in the above
mentioned discipline.

iii
Makotyn, Philip (Ph.D., Physics)
Experimental studies of a degenerate unitary Bose gas
Thesis directed by Dr. Deborah S. Jin

A dilute Bose-Einstein condensate (BEC) near a Feshbach resonance provides experimental
physics with a clean and controllable system to investigate strongly interacting many-body systems.
The ability to tune the scattering length allows BECs to be projected onto strong interactions froman initial weakly interacting state. However, historically, studying a bulk 3D strongly interacting
BEC has been difficult, as these systems are inherently unstable due to three-body inelastic collisions. Thus, 2D, 1D, lattice confined, and two component Fermi gases were used to explore thestrong interactions in an ultracold gas. In this thesis, I present the first measurement of a stronglyinteracting 3D 85 Rb BEC. I introduce our experimental system and the techniques we used to probe
the BEC. I first report on probes of a BEC in the weakly interacting regime. To probe a spherical
85 Rb BEC with strong interactions we implemented a novel experimental technique that allowed us

to quickly change the interactions, bypassing inelastic losses. We projected the BEC onto unitarity,
where the scattering length diverges and the interactions are infinite, in order to observe dynamics
and the unexpectedly long lifetime of the gas. Additionally, we observe a universality of the gas
with respect to the length scale set by the interparticle spacing.

Dedication

To my grandmother, babcia

Acknowledgements

Working at JILA these past years has been an incredible delight. The fast pace of research,ease of collaboration, and intellectualism make it truly unlike any other place. The specialness of
JILA comes from all the wonderful people that walk the halls.
First, and foremost, I would like to thank my advisors Debbie Jin and Eric Cornell. I’m
grateful for them as they took me in as a lowly electrical engineer and allowed me to dress up andplay as an experimental physicist. They are both paragon examples of physicists. Debbie continues
to amaze me with her ability to dive exactly to the root of any scientific issue. I know no one elsewith Eric’s ability to deeply understand a problem from multiple perspectives.
Much of the credit for the work in this thesis goes to my previous lab mates Juan Pino and
Rob Wild. Juan drilled into me the power of data driven decision making. Rob showed me how
think out of the box as he brought many creative solutions experimental problems. Recently, I’ve
had the pleasure to work with Catherine Klauss. Cathy has been instrumental in our studies of
the unitary Bose gas.
In addition, there are many people to thank for help in physics and distractions from physics.
The greater JILA bi-group including Ruth Bloom, Tara Drake, Rabin Paudel, Matt Grau havedone both. The JILA electronics shop and machine shop are a graduate students best friend. The
electronics shop, especially Carl Sauer and Terry Brown, have been a great help with any electronics
related issues, along with teaching me how little I actually know about electrical engineering. Hans
Green in the machine shop deserves lots of thanks for answering my many random questions.
Many friends have helped me along the journey of grad school including Sara Fairchild, Samvi
Berweger, Joanna Atkin, Craig Hogle, Will Kindel, Mike Foss-Feig, Jennifer Harlow, both the
Absolute Zeros and the Moosecats softball teams and many others.
Lastly, this thesis would never had been written without the writing help of Joanna Atkinand Laura Nash. I’m deeply in indebted to them both.

Contents

Chapter
1 Introduction to this thesis

1

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

2 Ultracold bosons

4

2.1

Two-body interactions and Feshbach resonances . . . . . . . . . . . . . . . . . . . . .

4

2.2

Bose-Einstein Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.3Interactions in a BEC and the LHY correction . . . . . . . . . . . . . . . . . . . . .

8

2.4

Three-body recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.5

Efimov resonance and trimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93 The Apparatus

11

3.1

How to make a 85 Rb BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2

Changing the magnetic field quickly . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2.1

Fast-B system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2.2

Pre-correcting for Faraday’s law . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 The story of probing strongly interacting systems

23

4.1

Liquid helium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2

Metastablity and local equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

viii
4.3

4.4

Studies in the perturbative regime of interactions . . . . . . . . . . . . . . . . . . . . 27
4.3.1

Bragg spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.3.2

Contact Spectroscopy and three-body contact . . . . . . . . . . . . . . . . . . 29

Infinite interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.4.1

Unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.4.2

Universality of physics at unitarity . . . . . . . . . . . . . . . . . . . . . . . . 31

5 Photon counting

33

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.2

Bragg Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.3

Heterodyne detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.4

Optical Layout and Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.5

Noise performance and Bragg dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.6

Conclusion

6 The contact. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
49

6.1

Introduction to the contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6.2

If at first you don’t measure C3 ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.3

RF contact spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.3.1

Experimental sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.3.2

Results of contact spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.4

C3 from loss measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.5

Discussion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7 Unitarity
7.1

71

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7.1.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

7.1.2

Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

ix
7.2

Experimental sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
7.2.1

7.3

Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Conclusions and arising questions from the results . . . . . . . . . . . . . . . . . . . 77

8 Future Directions

88

8.1

Looking back . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

8.2

One more thing: How we project onto unitarity . . . . . . . . . . . . . . . . . . . . . 88

8.3

8.2.1

The ramp onto unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

8.2.2

Loss of atoms, but no molecules. . . . . . . . . . . . . . . . . . . . . . . . . . 89

Low hanging fruit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Bibliography

94

Appendix
A DIY guide to measuring the contact

102

A.1 Steps to measure the contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
A.2 Calculate the contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

TablesTable
3.1

Fast-B coil parameters compared to bias coils. . . . . . . . . . . . . . . . . . . . . . . 14

3.2

The inductive current model constants.

6.1

Experimental timings for contact spectroscopy measuring the three-body contact at. . . . . . . . . . . . . . . . . . . . . . . . . 19

unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Figures

Figure2.1

85 Rb Feshbach resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.2

Feshbach molecule binding energy

7

3.1

A schematic of the fast-B system. The fast-B coils are part of a simple PID feedback. . . . . . . . . . . . . . . . . . . . . . . . . . . .

loop. We supply current through the coils from a power supply and control the flowusing a FET. We measure the current using a Hall effect current sensor. The set
point and measured current are subtracted to create an error signal. . . . . . . . . . 15
3.2

A scope trace of the step response of the fast-B coil. We plot the current set point(red) and measured current (blue). We observe the system response after the set
point is jumped from one value to another. The 10% - 90% step response time is 2.1µs. We characterize the bandwidth of the system using this step response time. . . . 16

3.3

The transient magnetic field and the contributions from the fast-B coils, the bias coils,
and the pinch coils when jumping the field from an initial value of approximately 163
G to the Feshbach resonance location at 155 G. The magnetic-field contribution of
the fast-B coil is plotted in part (a). The contribution to the magnetic from the bias
and pinch coils is shown in part (b). The steady-state values of the magnetic-field
contributions from the bias and pinch coils have been subtracted. Part (c) shows
the sum of the magnetic-field contributions from all the coils (blue) compared tomeasurements of the field (red). We attribute the disagreement between the dataand curve to eddy currents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

xii
3.4

The lumped-element circuit model of the inductive coupling. The fast-B coils inductively couple to eddy current, bias coil, and pinch coil inductances. The mutual
inductances are shown as arrows. The eddy currents are modeled as an LR circuit.
The transient bias and pinch currents are LRC circuits. . . . . . . . . . . . . . . . . 19

3.5

The magnetic-field contribution from the bias and pinch currents are shown in (a)and (b), respectively. The blue lines are measurements of the current through thecoils and the green lines are the models based on LRC components. The model andmeasured currents agree excellently. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.6

The full model compared to magnetic-field measurements. The magnetic-field contribution from the fast-B coils, calculated directly from the measured current in the
fast-B coils, is shown in blue. The red curve is the total magnetic field, after adding
the calculated contributions from the eddy currents, bias coils, and pinch coils. The
black points are measurements of the field using RF spectroscopy. The measurements differ from the model by less than 0.05 G, while differing from only the fast-B
magnetic field contribution by more than 0.5 G. . . . . . . . . . . . . . . . . . . . . . 21

3.7

A plot of the calculated magnetic-field contributions from the eddy currents, bias
coils, and pinch coils. We jump the fast-B coil approximately 10 G in 5 µs andcalculated the induced currents. The largest contribution is due to the eddy currents. 22

xiii
5.1

Bragg spectrum of a 85 Rb BEC of 5.5×104 atoms at |k| = 16 µm−1 , measured in two
different ways. The horizontal axis shows the frequency difference between the Bragg
beams. The vertical scale shows the number of excitations, which is measured using
traditional absorption imaging of ejected atoms (hollow circles), as well as with thephoton-counting technique presented here (black triangles). The error bars on black
points represent the shot noise in the photon counting measurements. The photon
counting measurements used three pulses of equal length. The first and the third
pulses used only a single weak beam to make an average background measurement,with no Bragg excitation. During the second pulse, both Bragg beams illuminate
the condensate to induce Bragg scattering and we subtract the averaged backgroundto count the number of photons gained or lost in the weak beam due to the Bragg
excitation.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

xiv
5.2

Diagram of the Bragg beams (a) and the heterodyne setup for Bragg scattering via
photon counting (b). The Bragg beam frequencies (ωlaser , ωlaser + ω) are offset byω, and create excitations at momentum ~k = 2~klaser sin(θ/2), where θ is the angle
between the Bragg beams and klaser = 2π/780 nm in our case. In (b) the three
beams (labeled weak, LO and strong) are derived from the same laser source andindividually fiber coupled. The optics for the LO beam are chosen to give the same
spatial mode as that of the weak beam. After being combined on a beamsplitter,
the weak and LO beams illuminate a photodiode and the beat signal is sent to a
demodulating mixer. The two quadrature outputs of the demodulator, I and Q, aresent to an oscilloscope for the measurement. Also illustrated in the figure is a servo
loop that functions to minimize the phase fluctuations of the RF input with respect
to the “LO” drive of the mixer. The servo minimizes the demodulator output, I, by
feeding back to a phase modulation input on the synthesizer that drives the “LO”
port of the mixer. This synthesizer is phase-locked to the synthesizer driving the
weak beam acousto-optic modulator (AOM), illustrated with a double-headed arrow.
Not shown in the schematic is the synthesizer driving the strong-beam AOM, whichis also phase referenced to the synthesizer driving the weak-beam AOM.

5.3

. . . . . . 41

Noise performance of heterodyne detection for a LO power of 250 µW. Vertical scale
is normalized to the shot-noise expected for the relevant pulse length and laser power.
The legend shows the different weak beam powers used. Low-frequency drifts makeour heterodyne scheme no longer shot-noise limited at long timescales. Inset has
the same normalized vertical units, with the data shown at different weak beam
intensities, corresponding to a constant 105 photons.

. . . . . . . . . . . . . . . . . 46

xv
5.4

Bragg excitations as a function of time, measured with photon counting. The numberRτof Bragg excitations is given by Nexc (τ ) = 0 (ṅ(t)− ṅavg )dt, where ṅavg is the averagerate of weak beam photons measured when no strong light is present. The Bragg
pulse begins at 0 ms. The deviation from expected τ 2 behavior (for small τ ) is
discussed in the text. These measurements were performed on resonance, with a
BEC of 400,000 87 Rb atoms.

6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Example of RF contact spectroscopy. (a) RF lineshape, S(ω), normalized so thatR∞
−1−∞ S(ω)dω = 1 s . The data at large detunings (circles) are multiplied by a factorof 300 to make the tail visible. Here the mean BEC density is hni = 4.9 x1012 cm−3 .
(b) Additional release energy of the outcoupled atom cloud. We calculate the energy
σ 2 −σ 2from the width of the expanded cloud, σ, using E = 23 m ∆t2 0 , where ∆t is the time
between the middle of the RF pulse and the absorption image (4.5 ms) and σ0 is thesize of the expanded cloud measured at ω = 0. The solid line is 21 |ω|
2π . . . . . . . . . . 53
6.2

C2
The contact vs a, measured at |ω| = 2π ×40 kHz. (a) The contact per BEC atom N.
0(b) The raw signal before final-state corrections. The solid lines in (a) and (b) show
the mean-field predictions. The dashed line includes the next-order LHY correction.
For this data, hni is typically 5.8 x1012 cm−3 , with (na3 )1/2 reaching a maximum of
0.043. We linearly scale the points to account for ∼10% variation in density. The
final-state effects shift the solid line from a parabola centered about a = 0 in (a) to
one centered about a0 = −565 a0 in (b), which enhances the raw signal at small a. . 55
6.3

A three-body loss resonance for 85 Rb. We plot the three-body event constant K3
vs a. From fitting Eqn. 6.7 to the solid points, for which a < 1/kthermal , we extracta− = −759(6)a0 and η = 0.057(2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

xvi
6.4

(a) The frequency dependence of GRF (ω), given our measured value for κ∗ . (b)Frequency dependence of the tail of the RF spectrum for a = 982 ± 10 a0 . The solidline is a fit of the data (•) to the expected frequency dependence of the two-bodycontact C2 /N0 including final-state effects. The dotted line corresponds to the samevalue of C2 /N0 , but ignores final-state effects. For comparison, the fit plus a trial
C3 /N0 term of 0.1 µm−2 is shown with the dashed line. Our measurements are
consistent instead with a C3 /N0 of zero. Here hni = 1.0 x1013 cm−3 . . . . . . . . . . 59

6.5

A schematic of the excitation rate, Γ, of the atoms as a function of frequency incontact spectroscopy. The black arrow indicates the location of the atomic transition.
The green curve is the Fourier broadened lineshape for a 40 µs gaussian pulse. The
black and blue curves show the tail at large detunings ω for C3 = 0 and for finite
C3 , respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.6

The magnetic-field ramp to unitarity. The field set point is changed at -2.4 ms withrespect to the trap turning off at 0 ms. The solid blue line is the calculated magnetic
field from the measured current in coils. The green points are measurements of themagnetic field using RF spectroscopy of Zeeman levels. The red filled guassian isthe power envelope of the RF pulse, which is proportional to the Rabi frequency
squared, Ω(t)2 . The ±4 σ gaussian envelope of the electric field starts at -2.31 ms
and ends at -1.99 ms. This magnetic-field ramp was done before the fast-B coils wereadded to the experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.7

The large detuning portion of a RF spectroscopy lineshape for unitarity interactions.
The x-axis corresponds to the detuning from the atomic transition. The y-axis is
proportional to the signal, which is the coupling rate to the new spin state. The
deviation from the C2 line (black) may be due to a C3 signal. . . . . . . . . . . . . . 67

xvii
6.8

An in-situ measurement of loss from a BEC at unitarity. The x-axis is the time after
ramping to unitarity and the y-axis is the number of 85 Rb atoms in the BEC. The
magnetic-field ramp started at approximately -0.1 ms and took 0.1 ms. The black
points are measurement of the number and the red line is linear fit to the data, which
has a slope of -19(2) µs−1 . Using this initial slope, we extract C3 =1.0(2) µm−2 . For
this data, the average density of the BEC is hni = 4.7 × 1012 cm−3 . . . . . . . . . . 69

7.1

The experimental sequence of projecting a gas onto unitarity interactions. The
magnetic field is ramped to and from unitarity in 5 µs. We turn the trap off andimage the atoms after a time-of-flight expansion. . . . . . . . . . . . . . . . . . . . . 75

7.2

Number of atoms measured using absorption imaging as a function of the time at
unitarity. The number measured without ramping to unitarity is shown at t = 0.
The solid line shows an exponential fit to the data (points), which gives a time
constant of 630 ± 30 µs.

7.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

The column-integrated momentum distribution ñ(k̃) versus the transverse momentum k̃ after evolving at unitarity for time t. The distribution measured without rampR
ing to unitarity is shown at t = 0. For each t, the integral ñ(k̃)2π k̃dk̃ = 8π 3 N (t).
For this data hni = 5.5(3) × 1012 cm−3 , which corresponds to kn = 6.9 µm−1 . Each
momentum distribution is obtained from several images for each of three expansion
times (7, 13, and 25 ms). The inset shows the same data plotted on log-linear axes.
The gray regions indicate the part of data that is contaminated by initial-size effects
and, therefore, does not accurately reflect the momentum distribution. We observe
the emergence of signal outside this region, and a saturation of ñ(k̃) for t >100 µs. . 79

xviii
7.4

The momentum distribution, n(κ), plotted versus the scaled momentum, κ. Datafor hni = 5.5 × 1012 cm−3 and hni = 1.6 × 1012 cm−3 are shown as the gray andcyan lines, respectively. The higher hni data is the average of measurements for 6
hold times t between 100 µs and 300 µs, while the lower hni data is the average of 4
measurements for t between 200 µs and 700 µs. The distributions are normalized soRthat n(κ)4πκ2 dκ = 8π 3 . The data for two different densities are consistent with asingle curve when plotted in scaled units. Inset: Plotting κ4 n(κ) for high hni, we do
not find clear evidence for a 1/κ4 tail at high κ.

7.5. . . . . . . . . . . . . . . . . . . . 80

The column-integrated momentum distribution ñ(k̃) versus the transverse momentum k̃ after evolving at unitarity for time t. The distribution measured without rampR
ing to unitarity is shown at t = 0. For each t, the integral ñ(k̃)2π k̃dk̃ = 8π 3 N (t).
For this data hni = 1.6(1) × 1012 cm−3 , which corresponds to kn = 4.5 µm−1 . Each
momentum distribution is obtained from several images for each of three expansion
times (7, 13, and 25 ms). The inset shows the same data plotted on log-linear axes.
The gray regions indicate the part of data that is contaminated by initial-size effects
and, therefore, does not accurately reflect the momentum distribution. We observe
the emergence of signal outside this region, and a saturation of ñ(k̃) for t >200 µs.

7.6

82

Number of atoms measured using absorption imaging as a function of the time at
unitarity for our low density data. For this data hni = 1.6(1) × 1012 cm−3 . The
number measured without ramping to unitarity is shown at t = 0. The solid line
shows an exponential fit to the data (points), which gives a time constant of 1200 ±100 µs.

7.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

The number of atoms in two momentum ranges vs. t. Data is for hni = 5.5 ×
1012 cm−3 . The black circles are the fraction of atoms with κ between 1.20 and 1.32.
The cyan trianges are the fraction of atoms with κ between 0.81 and 0.89. The lines
show fits of the data to ∆N0 (1 − exp−t/τ ), from which we extract the timescale forsaturation, τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

xix
7.8

The number of atoms in two momentum ranges vs. t. Data is for κ between 0.81
and 0.89. The black circles are the fraction of atoms with hni = 5.5 × 1012 cm−3 .
The cyan triangles are the fraction of atoms with hni = 1.6 × 1012 cm−3 . The lines
show fits of the data to ∆N0 (1 − exp−t/τ ), from which we extract the timescale forsaturation, τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

7.9

The time constant associated with the emergence of signal at high momentum plotted
as a function of scaled momentum, κ. Data for hni = 5.5 × 1012 cm−3 are shownwith solid circles, while data for hni = 1.6 × 1012 cm−3 are shown with open circles.

8.1

86

(a) The magnetic-field magnitude. The ramp is a linear ramp from 163 G to 155.04
G in 5 µs. (b) The adiabaticity parameter for ramp in (a). We plot time on theȧ Emol
x-axis and aȧ / Emol~ on the y-axis. When a / ~ , the change in a is adiabatic. Whenȧ Emola / ~  1 the change in a is diabatic.

8.2. . . . . . . . . . . . . . . . . . . . . . . . . 90

The column-integrated momentum distribution ñ(κ) versus the transverse momentum k/k̃. We ramp to unitarity in either 5 µs, 10 µs, or 20 µs. The cloud is held for100 µs and then ramped back to ∼ 163 G in 5 µs. The azimuthal averaged opticaldepth is after a time-of-flight expansion of 7 ms. We start with 6 × 104 atoms in the
BEC and an average density of, hni = 5.2 × 1012 cm−3 . . . . . . . . . . . . . . . . . . 91

8.3

The experimental sequence of projecting a BEC onto unitarity interactions andsearching for molecules. We ramp the magnetic field onto unitarity in 5 µs, holdfor 150 µs, ramp to a scattering length of 450 a0 in either 5 µs or 150 µs, and thenuse absorption imaging to measure the number of atoms. . . . . . . . . . . . . . . . . 92

xx
8.4

An RF spectrum of the cloud after ramps to unitarity and back. We use a 7 µs
square enveloped RF pulse on the F=2 mF =-2 to F=3 mF =-3 Zeeman transition,
which is the cycling transition. We perform absorption imaging directly after the
RF pulse. The magenta line is for a 5 µs ramp back while the black line is for a 150µs ramp back. The solid line is the expected transition probability for free atoms
exposed to a pulse with a square envelope. The dotted lines are predicted response
for a population of Feshbach molecules. Only the strength of the molecule response
is fit since the position of the curve is fixed relative to the single atom peak by theory. 93

Chapter 1

Introduction to this thesis

1.1

Introduction
Quantum mechanics provides a monumentally accurate description of the universe aroundus and is arguably the most successful theory of the physical world. Qualitatively, it describes
many concepts that were previously unexplained (e.g. atomic spectra, black-body radiation, theelectron work function, the double slit experiment, etc.). It is also hailed as the most quantitatively
correct physical theory because the calculation of the fine-structure using the anomalous magnetic
moment of the electron agrees with measurement to one part in 109 [1]. Recently, the 1995 creationof the first dilute gas Bose-Einstein condensation (BEC) [2] provided physicists with a new controllable macroscopic quantum mechanical system to study. This thesis describes studies of quantum
mechanics using 85 Rb BECs.
Quantum mechanics offers a complete description of the physics for two-body systems. However, the calculations needed to describe strongly interacting systems with more degrees of freedom
in the problem (e.g. more than two particles) quickly become impossible to do in a reasonableamount of time. And, even if one could calculate, for example, the ground state of an arbitrary
quantum system, it’s not clear how much insight into the physics of the system that gives. Thus, togain any an understanding of many-body systems, the problem must be simplified and described by
fewer parameters than the complete set of degrees of freedom [3]. Capturing a complete descriptionof system by fewer degrees of freedom typically works when the system has weak interactions: For
example, a weakly interacting BEC can be described using a single quantum mechanical wave-

2function [4]. For a BEC, this approach is only valid for a dilute system, where the length scale ofinteractions is much smaller than the interparticle spacing, na3  1, where n is the density and ais the scattering length.
So, Nature has played a cruel trick on us when making the most interesting systems exactly
the ones not understood in a mean-field approach.1 These are systems where the interactions arestrong, the systems aren’t dilute, and theoretical approaches are difficult. Since ab initio theory is
intractable, physicists use experimental studies to build an understanding of strongly interacting
many-body systems. Specifically, 85 Rb BEC has proved to be a particularly fruitful model systemfor exploring the transition from a weakly interacting mean-field description (na3  1) to a stronglyinteracting system (na3 > 1). The versatility of using 85 Rb BEC to study strongly interacting
systems is due to the ability to tune the interactions by changing the scattering length, a. We
typically start with a weakly interacting gas with na3  1, evaporate to BEC, and then increase
the interactions to the strongly interacting regime before probing the system. This thesis tells thestory of our lab’s studies of a BEC in the weakly interacting regime and of the first studies of astrongly interacting (na3 > 1) BEC.

1.2

Thesis outline
I begin with a brief review of ultracold boson physics in chapter 2. I describe various physicalphenomena of ultracold Bose gases relevant to our studies with 85 Rb BECs. I describe how wechange the interactions, which are parameterized by the scattering length, with a Feshbach resonance. I also give an account of the first-order perturbations to mean-field theory for the energy
density of a BEC. Lastly, I touch on three-body interactions in a BEC.
In chapter 3, I describe the apparatus and how we make a BEC. In our studies of strong
interactions we wanted to change the magnetic field much faster than the apparatus was previously
capable of. The approach we took was to add small additional coils that can change the field fasterthan the large trap coils. I describe these magnetic-field coils and the design challenges we solved
1

It’s possible physicists only find them interesting precisely because we don’t understand them.

3
implementing them.
After discussing the apparatus, I describe in chapter 4 how we transitioned from studying
weakly interacting to strongly interacting BECs. I include some background and context to thechallenges associated with strong interactions in a BEC. Our work started by probing BECs in
the weakly interacting regime, where we used two different probe techniques: Bragg spectroscopy
and contact spectroscopy. Chapter 5 describes the Bragg spectroscopy measurement and chapter
6 describes the measurement of the contact. Lastly, I discuss unitarity, where the scattering length
diverges. Our measurements of strongly interacting BECs are presented in chapters 6 and 7. I
finish with my thoughts on future directions for the experiment in chapter 8.

Chapter 2

Ultracold bosons

Before I discuss our studies of a BEC in the strongly interacting regime (na3 > 1), where
mean-field and perturbation theory are not applicable, I will discuss the BEC physics we do understand.
In the weakly interacting regime (na  1) a BEC is well described by mean-field theory. In
this regime, three-body losses are small and evaporating to BEC is relatively easy. The weaklyinteracting dilute gas is accurately described by averaging over two-body interactions. Therefore, I
discuss two-body scattering and our ability to tune the scattering length using a Feshbach resonance.
However, even for weak interactions, where the BEC is well described by mean-field theory,the interactions significantly affect the BEC. Last, even though three-body collisions are rare,
three-body physics can play an important role. This chapter provides a brief overview to weaklyinteracting BECs, but is not intended to be an in-depth review. I give citations to many excellent
review articles.

2.1

Two-body interactions and Feshbach resonances
Most of the time, atoms in an ultracold gas take turns interacting with each other; that is,they mostly undergo two-body collisions. When this is the case, we only have to understand thetwo-body scattering properties to describe the physics of the gas. We characterize the two-bodyinteractions with a physical quantity called the scattering length, a, and use a Feshbach resonance tochange the scattering length [5]. For 85 Rb, tuning the scattering length using a Feshbach resonance5
has allowed the exploration of a BEC in the strongly interacting regime [6].
The scattering length characterizes the two-body scattering in ultracold gases [7]. Generally,
the scattering cross section between atoms is dependent on short-range atomic potentials. To
understand the scattering properties of the gas, the complicated short-range potentials would haveto be considered; however, this calculation can be simplified by replacing the short-range atomic
physics with a delta-function pseudopotential with an associated length scale – the scattering
length. 1 The interactions are then completely characterized by the scattering length, rather thandescribed by the atomic potentials.
The remarkable thing about 85 Rb is our ability to conveniently change the scattering length
using a magnetic-field Feshbach resonance. The Feshbach resonance couples free atoms to a boundmolecular state at a specific magnetic field (we use the 85 Rb 155 G resonance [9] ). To change thescattering length, we tune the bound state energy relative to the free atom energy by changing themagnetic field. The scattering length is given bya = abg∆
1−
B − B0,(2.1)

where abg is the background scattering length, B is the magnetic field, B0 is resonance location,and ∆ is the width of the resonance. The binding energy of the Feshbach molecule, Emol is
Emol =

~2
,
ma2

(2.2)

were m is the mass of an atom. At the resonance location, B = B0 , the scattering length diverges|a| → ∞ and the molecule binding energy vanishes, Emol = 0.
The Feshbach resonance parameters for 85 Rb were precisely measured by Claussen et al. [10],allowing precise control over the interaction. The scattering length for 85 Rb is shown in figure 2.1
as a function of magnetic field. Also, we plot the energy of a Feshbach molecule as a function ofmagnetic field in blue in figure 2.2. The 155 G 85 Rb Feshbach resonance is convenient because
it is a wide resonance, with ∆ = 10.71 G [10]. The convenience of a wide Feshbach is countered1

Fermi was the first person to use a pseudo-potential when considering low energy electron scattering on groundstate atoms in 1934 [8].

6by the negative background scattering length abg = −443 a0 , which makes evaporating 85 Rb very
challenging. A large BEC is unstable for negative scattering length [11]. In order to evaporate
85 Rb to BEC, we must precisely control the magnetic field and set the scattering length to a small

positive value.

10000

0

(a )

5000

0

-5000

-10000
145

150

155

160

165

Magnetic Field (G)Figure 2.1: The 85 Rb 155 G Feshbach resonance. The x-axis is the magnetic field. The y-axis shows
the scattering length in units of the Bohr radius. By changing the magnetic field, the scattering
length can be tuned to any value. For this resonance abg = −443(3) a0 , ∆ = 10.71(2) G, and
B0 = 155.051(18) G [10].

2.2Bose-Einstein Condensation
An atomic Bose-Einstein condensation was first made in 1995 [2] and quickly became amodel macroscopic quantum system. Since it is a clean controllable system, many discussionsof macroscopic quantum systems now begin with BEC. The small length scale of interactions
compared to the interparticle spacing allows physicists to successfully describe a BEC with theGross-Pitaevskii (GP) equation. The GP equation was derived in 1965 (well before the experimental
observation of BEC) by Gross and Pitaevskii [12, 13] and describes the zero-temperature wave
function of bosons. However, the GP equation is only valid for a dilute BEC, where na3  1. The

Molecule Energy (MHz)

7

0.75

0.00

-0.75

-1.50

145

150

155

160

165

Magnetic Field (G)Figure 2.2: The binding energy of a Feshbach molecule. The x-axis is the magnetic field. The y-axis
is the energy of the Feshbach molecule. The black line is the energy of two free atoms. The blueline is the energy of a Feshbach molecule. At the Feshbach resonance, the binding energy vanishes.

time-dependent GP equation is∂i~ Φ(r, t) =
∂t

 2 2

~ ∇
2
−
+ Vext (r) + g|Φ(r, t)| Φ(r, t),
2m

(2.3)

where Φ(r, t) is the order parameter of the condensate, Vext (r) is the external potential, and g is
the coupling constant, which is given byg=

4π~2 a.
m(2.4)

The time-independent GP equation is
 2 2

~ ∇
2
µφ(r) = −+ Vext (r) + gφ(r) φ(r),2mwhere φ(r) is the order parameter of the BEC and µ is the chemical potential.

(2.5)

8

2.3

Interactions in a BEC and the LHY correction
Even for a weakly interacting dilute BEC, interactions modify the properties of the system.

Perhaps the most striking modification of a BEC is the shape of the condensate. With no interactions, the BEC wave function is the ground-state wave function of the trap. With finite interactions
and a large number of atoms in the BEC, even for na3  1, the interaction energy term is muchlarger than the kinetic energy term in the GP equation (equation 2.5). When neglecting the kinetic
energy term in equation 2.5 (this is called the Thomas-Fermi approximation), the BEC takes on
the shape of the external potential:n(r) = φ(r)2 = g −1 [µ − Vext (r)] .

(2.6)

If we increase na3 , the GP equation becomes less valid because the approximation of na3  1
was used to derive the GP equation. For a small but finite value na3 , a perturbation approach canbe used to describe the gas. The Lee, Huang, and Yang (LHY) correction [14] famously gives thefirst beyond-mean-field correction to the energy density of a BEC. The energy density of a BEC is
gn2
E=
2



128 √ 3na + ... .
1+ √
15 π

The first term in the equation is the mean-field contribution to the energy, while the

(2.7)
√

na3 term

is the LHY correction. There have been many beyond-mean-field studies of a BEC [15, 16, 17, 18,
19, 20, 21] and recently the LHY correction was precisely measured in a BEC [17].

2.4

Three-body recombination
In a Bose gas, the collisions mostly involve two bosons, but sometimes three collide. 2 When

three bosons collide, they have a chance to undergo a three-body recombination event. Three-bodyrecombination occurs when two free atoms form a molecule, with the binding energy converted
into kinetic energy. A third atom must be part of the collision to satisfy momentum and energy
2
In contrast, in a two-component Fermi gas three-body collisions are suppressed due to the Pauli exclusion
principle.

9conservation. The binding energy of a molecule is typically much larger than the energy per atom inthe BEC. The dimer and free atom then fly away from each other with much more energy than thechemical potential of the BEC. The fast moving atoms are typically lost from the trap, and if theycollide with other atoms on the way out the remaining cloud is heated. Additionally, three-bodylosses selectively removes more atoms from the coldest, most dense, part of the cloud, which overall
heats the cloud and is referred to as anti-evaporation.
An ultracold Bose cloud in a trap will experience three-body recombination events thatmanifest as loss of atoms from the cloud. The loss rate is described by the three-body loss coefficient
L3 , where the density and number of atoms in the cloud are given byṅ = −L3 n3

(2.8)

Ṅ
= −L3 hn2 i,
N

(2.9)

where N is the total number of atoms in the cloud and < n2 >=R

n3 d3 x/N . We obtain equation

2.9 by integrating equation 2.8 over all space. The three-body loss coefficient L3 scales as
L3 ∼ ~a4 /m,(2.10)

where a is the scattering length [22, 23]. The three-body losses increase as a4 in a cloud of ultracoldbosons [24, 25, 6]. The increase of three-body losses with scattering length has historically been
the limiting factor in studying strongly interacting Bose gases.

2.5

Efimov resonance and trimers
In addition to the Feshbach two-body bound state, shallow three-atom bound states exist for

ultracold bosons. In 1970, Efimov derived [26] an infinite series of trimer bound states for particles
with two-body interactions. These Efimov trimers result from the two-body interactions in thecontext of nuclear physics. Remarkably, even when all two-body interactions between atoms are
repulsive, a trimer bound state still exists. Recently, Efimov physics has been extensively exploredin the context of ultracold gases [27]. To observe Efimov physics in our ultracold gas system, we10measure the three-body loss rates of the cloud. Efimov physics modifies the a4 loss rate with a logperiodic structure. We discovered an Efimov resonance in 85 Rb by observing enhanced loss where
the Efimov trimer state is degenerate with the free atom energy; we describe our measurement ofEfimov trimers in chapter 6.

Chapter 3

The Apparatus

This chapter presents an overview of the apparatus and our procedure to make 85 Rb BECs,
along with details about my technical contributions to the apparatus. I am now the fourth graduate
student to do my Ph.D. work with the machine and the third student to make 85 Rb BECs in our
10 Hz spherical magnetic trap. Because the experiment has been excellently documented by mypredecessors in previous theses [28, 29, 30], I will give here a brief introduction to the experimentrather than many technical details. The majority of the chapter will focus on the latest improvement
we’ve made to the apparatus: the fast magnetic-field system.
The experiment was built to reliably make 87 Rb and 85 Rb BECs. In addition, we control theinteractions of 85 Rb atoms via a magnetic-field Feshbach resonance. BECs are made with a 90 s
cycle time. The machine has been quite reliable over the years, daily cooling to BEC for months
at a time. There have been with brief moments (of terror) when the experiment was out of order,
but most maintenance has involved minor trap alignments or tweaks of the laser system.1

3.1

How to make a 85 Rb BEC
Our experimental sequence is similar to many other ultracold atomic gas experiments [31,32, 33]. We initially collect as many atoms as we can in a magneto-optical trap (MOT), move theatoms to a higher vacuum region, then evaporatively cool the cloud. Our experimental sequenceis as follows: We start by trapping and laser cooling 85 Rb and 87 Rb from a room temperature
1

Examples of more challenging problems to fix include steam coming out of the cart coils, flaky transfer coil servo
grounding, and a broken motorized flipper spring.

12
vapor in a magneto-optical trap (MOT). We then use a moving magnetic potential to move theatoms from the MOT chamber, where the vacuum-limited lifetime (lifetime of the trapped gas)is only 5 s, to the science cell, which has a vacuum-limited lifetime of 500 s. Once in the science
cell, 85 Rb is cooled sympathetically while 87 Rb is evaporatively cooled. We evaporate 87 Rb first ina magnetic Ioffe-Pritchard trap [34], then a hybrid optical-magnetic trap, and finally in a 10 Hz
spherical Ioffe-Pritchard magnetic trap. We set a small (∼ 100 a0 ) scattering length in the 10 Hz
trap by tuning the magnetic-field near the Feshbach resonance location. We end with a pure 85 Rb
BEC by evaporating 87 Rb completely until no 87 Rb atoms remain trapped.
We use absorption imaging on a cycling transition to measure the number and temperatureof either the 87 Rb or 85 Rb cloud. First, we transfer the atoms to a state with a cycling transition(for 87 Rb |F = 2 mF = −2i and for 85 Rb |F = 3 mF = −3i) using either optical pumping or
RF techniques. Then, to measure the 2D optical density profile of the cloud, we pulse on a laser
resonant with the cycling transition and use absorption imaging techniques [35]. Because of thehigh optical density of the BEC, we use a high-intensity imaging pulse to increase the dynamic
range the imaging [36]. During my tenure in lab we regularly made 85 Rb BECs of 7 × 104 atoms.2

3.2

Changing the magnetic field quickly

3.2.1

Fast-B system

Since making BEC was a repeatable and reliable process, I was able to turn my attention
to developing new techniques to probe a BEC. We were motivated to change the interactions of
a BEC very quickly to study a strongly interacting gas, as described in chapter 2. The previous
method of changing the total magnetic-field magnitude was to tune the current flowing through thebias coils of the Ioffe-Pritchard trap. However, the bias coil step response time is ∼ 100 µs, and we
empirically found these ramps were too slow for our purposes. To change the field faster, we usedsmall low-inductance coils very close to the atoms, following the example of other labs [37, 31, 38].
2

The number of 85 Rb BEC atoms quoted in theses has increased from 4 × 104 in Juan’s thesis [29] to 6 × 104
in Rob’s thesis [30] to now 7 × 104 . I’m confident my successors will quote an even higher number after theirimprovements and optimizations to the apparatus.

13
To change the magnetic field, we developed a system we call the fast-B system. To be “quick”
with respect to the atoms, the magnetic-field ramp time has to be much faster than the fastest
timescale the atoms interact on. We can estimate a fastest timescale for any dynamics in the gas byconsidering the timescale set by the distance between atoms, which is 60 µs; t = 2m/(~(6π 2 n)2/3 ),where n = 5 × 1012 cm−3 is a typical density for our 85 Rb BEC. (A complete discussion of thetimescales of our experiment is given in chapter 4.) I next describe the coils, our motivations, andour design.
Our goal in designing the fast-B coils was to change the magnetic field as fast as possible(or at least much faster than 60 µs). The coils are coaxial with the bias magnetic-field direction
so that they generate a magnetic field that adds linearly to the magnitude of the larger field. The
coils are essentially inductors with series resistance, so, in order to change the magnetic field as
fast as possible, we need to maximize dI/dt. The current, I, inductance, L, and voltage, V , for an
inductor are related by
V =L

dI
.
dt

(3.1)

It’s clear from equation 3.1 that we should maximize V and minimize L to design the fastest coils.
Other labs in JILA have maximized V by charging capacitors to hundreds of volts (dangerous)
and minimized L by only using a single loop of wire for the coil. However, this approach is flawedsince it only considers the coils and ignores the rest of the system. With only one loop as the coil,
the wire leads from the power supply have the same (or more!) inductance than the coil itself;
therefore, this is clearly is not the ideal design strategy. And indeed, these systems are typicallylimited by the inductance through the relations of equation 3.1. However, the speed of change ofthe magnetic field isn’t maximized, even though the slew rate of the current is. The disadvantage
of a system limited by the slew rate is understood by considering a single turn coil whose leads
have more inductance than the coil. If we add one more loop in the coil the inductance stays about
the same, but the magnetic-field contribution and thus the magnetic-field ramp rate will double.
Instead, we designed the fast-B system by considering the coils as part of a larger feedback

14
system, shown in figure 3.1. We designed the coils to be limited by the feedback system rather
than by equation 3.1. The two coils have 10 turns, a radius of 0.5 cm (an inductance of ∼ 1 µH),
and are 2 cm apart. We require ∼ 10 G to change the interactions in the BEC from the mean fieldregime to unitarity. Assuming a 5 µs ramp time, the required voltage to drive these coils is 1.7 V,
which is small compared to the 30 V we use to power the system. The specifications for the fast-B
coils compared to the bias coils are listed in table 3.1. The 10 - 90 % step response of the coils
is tr = 2.1 µs, as shown in figure 3.2. This corresponds to a system bandwidth of 860 kHz (using
ωn ∼= 1.8/tr , equation 3.49 in [39]), which is approximately a two order of magnitude improvement
over our previous ramps. However, with these new and improved fast-B coils, we are too fastin changing the magnetic field! Induced currents in other coils and eddy currents in surrounding
conductors cause a transient magnetic field that opposes any fast change. These induced currents
limit our overall control of the atoms interactions and we have to correct for them.
Table 3.1: Fast-B coil parameters compared to bias coils.

Coil separation
Coil radius
Coil inductance
Magnetic field per amp
Magnetic field curvature
Current required for 10 G
Voltage required for 10 G / 5 µs
Number of turns

3.2.2

Bias coil4.76 cm
3.8 cm
37 µH
1.609 G/A
0.087 G/A/cm2
6.2 A
45.6 V
8

Fast-b coil
2 cm
0.5 cm∼ 1 µH
1.109 G/A
7.83 G/A/cm2
9.1 A
1.7 V
10

Pre-correcting for Faraday’s law

The quick magnetic pulse from the fast-B coils creates a large change in magnetic flux close
to the atoms. Faraday’s law tells us that a changing magnetic field induces an orthogonal electricfield. If the induced electric field is in a conductor, it causes a current to flow, which generates a

15

Fuse

+
6800 F

-

Coils

Power
SupplyHall Sensor

- +

FET

Set
Point

Servo
Figure 3.1: A schematic of the fast-B system. The fast-B coils are part of a simple PID feedback
loop. We supply current through the coils from a power supply and control the flow using a FET.
We measure the current using a Hall effect current sensor. The set point and measured current are
subtracted to create an error signal.

magnetic field that opposes the original field. We found that two things happen when we ramp
the field quickly with the fast-B coils. First, the fast-B coils induce small transient currents in
two pairs of coils in the Ioffe-Pritchard trap: the bias coils and the pinch coils. Second, we see a
discrepancy between the measured currents in all the coils and the measured magnetic field. The
fast-B ramp and three sources of inductive coupling in our system are demonstrated in figure 3.3.
The fast-B magnetic-field ramp is shown in figure 3.3 (a). We ramp the field very quickly in order
to measure eddy currents and induce currents in the other coils. The induced currents in the bias
(green) and pinch (blue) coils are shown in figure 3.3 (b), where I have converted the currents into
magnetic field using measured calibrations of these coils. We subtract the bias and pinch current
offset values. Figure 3.3 (c) shows the sum of all the coils compared to measurements of the totalmagnetic field. The total magnetic field was measured using RF spectroscopy on atomic Zeeman
levels. We attribute the difference between the measured field and the expected total field fromthe currents due to eddy currents. So, although we control the current though the fast-B coils
exquisitely, we need to account and correct for the induced fields to control the magnetic field with

16

Figure 3.2: A scope trace of the step response of the fast-B coil. We plot the current set point (red)and measured current (blue). We observe the system response after the set point is jumped fromone value to another. The 10% - 90% step response time is 2.1 µs. We characterize the bandwidthof the system using this step response time.

the same precision.
To pre-correct for the eddy currents, we need to understand how they behave; we have made amathematical model to predict the induced currents. Our model is a simple circuit lumped-element
model, shown in figure 3.4. The coupling between the fast-B coils and other elements is treatedas mutual inductances between inductors. The fast-B coils are modeled as an ideal current source
flowing through an inductor. The eddy currents in surrounding conductors are modeled as an LR
circuit while the bias and pinch coils are modeled as LRC circuits. The eddy currents are physicallya current circulating in a conductor somewhere near the magnetic trap. The current path in the
inductor has an inductance and resistance. Thus, we model the eddy currents as a lumped-element

17
LR circuit. However, the bias and pinch coils are under an active feedback loop during the fast-B
jump. Empirically, a second order differential-equation LRC model fits the measured current well.
Also, any complicated many-order feedback system can be approximated by a second order system.
In feedback systems theory this is known as the dominant pole approximation [40]. The system ofcoupled differential equations then is
dIf
dIe
+ Le
+ Re Ie
=0
dt
dt
d2 If
d2 Ib
dIb
1
Mb 2 + Lb 2 + Rb+
Bb = 0
dt
dt
dt
Cb
d2 If
d2 Bp
dBp
1
+
Bp = 0,
Mp 2 + Lp 2 + Rp
dt
dt
dt
Cp
Me

(3.2)
(3.3)
(3.4)

where M is the mutual inductance, L is inductance, R is the resistance, and C is the capacitance.
The subscripts f , e, b, and p corresponds to the fast-B, eddy currents, bias coils, and pinch coils.
We can cast these equations into a more convenient form. Assuming B ∝ I the equations are
written as a function of magnetic field B, the coupling between the coils α, a characteristic decay
time τ , and a second-order parameter X. These equation are
dBf
dBe
1
=
+
Be
dt
dt
τe
d2 Bf
d2 Bb 1 dBb
1−αb 2 =
+
+
Bb
2
dt
dt
τb dt
Xb
d2 Bf
d2 Bp 1 dBp1−αp 2 =
+
+
Bp .
2
dt
dt
τp dt
Xp
−αe

(3.5)
(3.6)
(3.7)

To find the constants α, τ , and X we compare our measurements to the model. We change
the parameters till the model agrees with the measurements. The final constants are shown in table
3.2. The comparison between the bias and pinch coil currents and the model is shown in figure3.5. To calibrate the eddy current model we compare the currents through the coils and the totalmagnetic field. After finding the appropriate eddy current parameters, we can put all the pieces
together. Figure 3.6 shows the comparison between the measured magnetic field (black circles),the field contribution from the fast-B coils (blue), and the full model (red). After our calibrations,the model agrees excellently with the data. The total magnetic-field jump was about 10 G and weField (G)180Fast−B coil

−5

−10

0

0.1

0.2

0.3

Field (G)0.2
Bias and Pinch
0.1
0
−0.1

Field (G)

−0.2

0

0.1

0.2

0.3

155.5
Total Field
155.25

155
0

0.1

0.2

0.3

time (ms)Figure 3.3: The transient magnetic field and the contributions from the fast-B coils, the bias coils,
and the pinch coils when jumping the field from an initial value of approximately 163 G to the
Feshbach resonance location at 155 G. The magnetic-field contribution of the fast-B coil is plottedin part (a). The contribution to the magnetic from the bias and pinch coils is shown in part (b).
The steady-state values of the magnetic-field contributions from the bias and pinch coils have been
subtracted. Part (c) shows the sum of the magnetic-field contributions from all the coils (blue)
compared to measurements of the field (red). We attribute the disagreement between the data andcurve to eddy currents.

find the agreement better than 10 mG, except for one data point. The corrections we made for19

Eddy Currents

Me

Fast-b
Coil

Mb

Le
Re

Lb

Bias
Coil

Rb
Cb

Mp
Pinch
Coil
Lp

Rp
Cp

Figure 3.4: The lumped-element circuit model of the inductive coupling. The fast-B coils inductively
couple to eddy current, bias coil, and pinch coil inductances. The mutual inductances are shownas arrows. The eddy currents are modeled as an LR circuit. The transient bias and pinch currents
are LRC circuits.

this specific data were on the order of 500 mG, shown in figure 3.7. Further calibrations with moremeasured magnetic-field points would allow us to find better agreement.
Table 3.2: The inductive current model constants.

Eddy currents
Bias coil
Pinch coil

α
0.068
0.0269
0.016

τ
45 µs
13 µs
17.5 µs

X2.0787 × 10− 10 s−2
1.0731 × 10− 10 s−2

20Induced bias magnetic field

field (mG)

200
150
100
50
0
−50
−100

0

0.05

0.1

0.15

0.2

0.15

0.2

time (ms)Induced pinch magnetic field

field (mG)

150
100
50
0
−50
−100

0

0.05

0.1

time (ms)Figure 3.5: The magnetic-field contribution from the bias and pinch currents are shown in (a)and (b), respectively. The blue lines are measurements of the current through the coils and thegreen lines are the models based on LRC components. The model and measured currents agree
excellently.

At the end of the day, we are very happy with the performance of the fast-B magnetic-field
control system. We step the magnetic field in less than 5 µs. We can control the absolute magnetic
field within 10s of mG by pre-compensating for eddy currents. The fast-B system is reliable andrepeatable and doesn’t interfere with the rest of the apparatus.

21

155.4

Field (Gauss)

155.3
155.2
155.1
155
154.9
154.8
154.7
154.6

0

0.05

0.1

0.15

0.2

time (ms)Figure 3.6: The full model compared to magnetic-field measurements. The magnetic-field contribution from the fast-B coils, calculated directly from the measured current in the fast-B coils, isshown in blue. The red curve is the total magnetic field, after adding the calculated contributions
from the eddy currents, bias coils, and pinch coils. The black points are measurements of the fieldusing RF spectroscopy. The measurements differ from the model by less than 0.05 G, while differingfrom only the fast-B magnetic field contribution by more than 0.5 G.

22

600
eddy
bias
pinch

500

field (mG)

400

300

200

100

0

−100

0

0.05

0.1

0.15

0.2

time (ms)Figure 3.7: A plot of the calculated magnetic-field contributions from the eddy currents, bias coils,
and pinch coils. We jump the fast-B coil approximately 10 G in 5 µs and calculated the induced
currents. The largest contribution is due to the eddy currents.

Chapter 4

The story of probing strongly interacting systems

An open problem in physics is understanding how complex behavior emerges in systems madeup of many interacting constituents. The physical world is made up of large many-body quantum
systems and, although the microscopic behavior may be known for two particles, the many-bodysystem can be “more than the sum of its parts.” That is, many-body systems have complicatedbehavior that is not intuitive only from their two-body interactions. When the interactions are weak,
many-body systems can be understood by averaging over the contributions from other particles;
this averaging over all other particles is called a mean-field approach. However, this mean-fieldapproach fails when the interactions are strong. Thus, understanding these strongly interacting
many-body systems remains an exciting frontier of physics. Using a 85 Rb BEC we tune a manybody system from weak to strong interactions. We change the scattering length in a 85 Rb BEC andthen probe the system to build an understanding of how strongly interacting many-body systems
behave.
The behavior of two isolated interacting particles compared to two particles imbedded in a
many-body system can be fundamentally very different. A prototypical example of this difference isthe behaviour of electrons in a metal. Isolated electrons are charged particles that interact throughthe Coulomb interaction. When in a metal, surrounded by a sea of electrons, electrons are instead
described by Landau-Fermi liquid theory. This transition from few to many is a current hot topic
of research in ultracold gases [41, 42].
Dilute weakly interacting many-body systems can be understood in a mean-field approach,24
but mean-field theories break down for strongly interacting many-body systems, which lie at thefrontier of understanding in physics. To makes strides in understanding these systems, experimental
investigations of strongly interacting systems are essential.
One particular approach is to use ultracold gases to study strongly interacting systems.
Ultracold atomic gases are proving to be excellent model systems for the study of few and manybody physics. In particular, nearly all the aspects of an ultracold gas can be controlled: theparticle number, the density, the geometry of the system, the interactions, and the dimensionality.
Using this freedom of control, atomic physicists are beginning to create quantum systems using
both a bottom up or top down approach. For example, experimentalists are beginning to build
quantum systems from the bottom up, placing atoms one by one and coupling them together
arbitrarily [43, 44]. Also, efforts to make quantum computing gates, study quantum information,or to investigate quantum metrology take advantage of coupling trapped atomic ions [45, 46, 47].
Rather than building the systems from the bottom up, the top down approach is to tune theinteractions of bulk 3D, 2D, or 1D systems or to make artificial crystals of atoms in optical lattices[48]. Our particular experiment takes a top down approach by tuning the interactions in a bulk 3D
BEC from weak to strong interactions.
The use of Feshbach resonances to change the scattering length is an incredibly powerful tool
to tune interactions in a BEC. However, there is no free lunch for our BEC. Three-body inelastic
collisions begin to dominate the system as we increase the scattering length, since the three-bodyrecombination loss rate increases as a4 . As we try to access a strongly interacting regime (i.e.
where a is large), three-body recombination thwarts the study of an equilibrium 3D Bose gas. The
gas decays away during the ramp to large scattering length and an equilibrium is never reached.
Various experiments have side stepped this problem by putting bosons in a lattice and limiting thesite occupation to fewer than three atoms or reducing the dimensionality of the system to 2D or
1D systems, where inelastic losses are less prevalent. These approaches lead to interesting physics(Bose-Hubbard Hamiltonians, Tonk-Girardeau gases, etc.), but these systems give up degrees of
freedom in order to limit losses. Perhaps the most radical work around of inelastic losses is to25
study fermions instead of bosons.1

As we mentioned though, fermions don’t undergo three-bodyinelastic processes in a two-component Fermi gas because of the Pauli exclusion principle.
This chapter outlines how we transitioned from studying BECs with na3  1 to BECs withna3 > 1. We introduce our motivation for studying strongly interacting many-body systems. In ourstudies, we first probed a BEC with na3  1 using Bragg spectroscopy and contact spectroscopy.
Then, we increased the interactions in the BEC to na3 > 1 by tuning the magnetic field to the
Feshbach resonance so that the BEC is at unitarity where |a| → ∞. However, at unitarity the gas
is metastable and thus we needed to understand on what timescale it decays. We have performed
two measurements of this system: (1) measuring the evolution of the momentum distribution n(k)and (2) measuring the contact.

4.1

Liquid helium
The ubiquitous motivation for studying strongly interacting quantum liquids is superfluid

liquid Helium. In 1937, Kapitza and Allen discovered that liquid 4 He undergoes a phase transition
at 2.2 K [49, 50]. Below the phase transition temperature, 4 He becomes a superfluid, which is aliquid with zero viscosity. The desire to explain the properties of this quantum liquid has inspired
much of the theoretical work that was later used to describe Bose-Einstein condensation. However,
because 4 He is a stongly interacting quantum fluid, the theories were inadequate: Most theories
assume only pair-wise interactions between particles, but, for 4 He, the distance between particles
is roughly the same as the distance that characterizes the interactions. The assumption of only
two-body interactions breaks down for this system and therefore the theories don’t apply to liquid
4 He. The ability to tune 85 Rb BEC from dilute to strong interactions makes it a model system to

explore the physics of quantum liquids.
1

Not that fermions aren’t very interesting in their own right.

26

4.2

Metastablity and local equilibrium
Although a BEC is in the motional ground state of an external potential, it can decay to alower energy state (molecules and eventually a solid) and thus has a finite lifetime. If we are toexperimentally probe the BEC, we must probe faster than this lifetime of the gas. In addition to theloss mechanism above, there are technical reasons the BEC will have a finite lifetime. The dullest
reason for the BEC’s finite lifetime are collisions with background gas in the vacuum chamber. We
measured the vacuum-limited lifetime, or the lifetime due to background collisions, in our sciencecell to be 500 s, which is longer than our experiment cycle time. Thus, the vacuum-limited lifetime
is not a concern for the lifetime of the BEC, but three-body inelastic loss can occur on time scalethat is much faster. The process of three-body recombination converts atoms in the dilute gas into
tightly bound molecules. The lifetime due to three-body recombination during the evaporation to
BEC is long compared to the timescale of BEC measurements, but the three-body losses increaseas a4 . As a gets large, this lifetime can be shorter than the timescale of our measurements. In
addition, if we increase the scattering length quickly, interaction pressure excites a collective density
breathe mode of the cloud. This causes the density to begin decreasing and thus also reduces na3 .
This density breathe mode has a period of half the trap period. Since our trap has a 100 ms period,
we must probe the cloud faster than approximately 5 ms if we want the change in n to be small.
After we evaporate to degeneracy, the BEC is in thermal equilibrium (although not in thetrue ground state of 85 Rb). To probe larger na3 , we quickly increase the scattering length a using
the Feshbach resonance. Due to interaction pressure, the cloud will begin to “breathe” out, loweringn, and reducing na3 . After the quick change in a, the cloud is not in a global equilibrium; however,
the gas can be in equilibrium locally. That is, locally in the BEC, the atoms can behave as if they
are in equilibrium at the larger scattering length but with the local density unchanged from theinitial local density before the fast change in a. While locally the gas might be described with atemperature, this “temperature” varies across the trapped cloud. A minimum requirement for thesystem to stay locally in equilibrium is that the change in scattering length be slow compared the27
two-body timescale, which is characterized by the molecule binding energy.
The idea of a local equilibrium was important for our experiments that probed the BEC using
Bragg spectroscopy and contact spectroscopy. We changed the interactions to a region where the
LHY correction was small compared to one, but not insignificant. We changed a on a timescale thatwas much shorter than the trap period, so that the cloud was out of equilibrium globally; however,ȧ/a was much smaller than ~/Eb , where Eb is the binding energy of a two-body Feshbach molecule.
The breathe excitation and three-body losses caused na3 to decay by 50% within about 10 ms after
the change in a. Thus, it was important that our probes were faster than this decay time. However,at large enough scattering lengths the breathe and losses occurred during the ramps and the BEC
decayed before we reached the final scattering length. We found that three-body losses and breathe
limited us to na3 ≈ 0.006.

4.3

Studies in the perturbative regime of interactions
We developed two probes to study the interacting BEC in the perturbative regime: Bragg

spectroscopy (the results are described in chapter 5) and contact spectroscopy (the results aredescribed in chapter 6). For both experiments, we started with a dilute BEC where na3 < 10−6 ,then ramped a (we reached a max value of na3 ≈ 0.006), and probed the BEC before any significant
losses occurred. Here, we tried to ramp slowly enough that the gas remains in a local equilibrium.

4.3.1

Bragg spectroscopy

Bragg spectroscopy is a useful tool for probing ultracold quantum gases that has been applied
by many experimental groups to study BEC [51, 52, 53, 15]. A simple picture of Bragg spectroscopy
is the absorption of a photon from one laser beam and then stimulated emission into another. The
momentum difference of the two photons is imparted to the ultracold gas. For counter-propagating
beams, the Bragg scattering imparts twice the momentum of a single photon. For momentum
and energy conservation to hold, the energy and momentum difference between the two photons
must satisfy the dispersion relation for excitations of the BEC. By varying the energy, or frequency

28
difference, of the two laser beams and measuring the response of the atoms, Bragg scattering canbe used to measure the energy of excitations of the gas at a specific momentum. The Bragg signal
will have two peaks: one when the frequency difference of the lasers is positive and one where thedifference is negative. The positive and negative detunings correspond to a photon being either
added to or subtracted from a particular laser beam. To change the momentum imparted to thegas, we can change the angle between the lasers, which sets what fraction of a photon’s momentumis transferred to the cloud.
Papp et al. used Bragg spectroscopy to probe a 85 Rb BEC at various interaction strengths
using counter-propagating beams [15]. However, counter-propagating laser beams impart relatively
large-momentum excitations to the BEC. A Bragg excitation with twice the momentum of a photon
excites atoms in the BEC to free-particle states with E = ~2 k 2 /(2m), but much of the interesting
many-body physics (such as phonon excitations, or perhaps a roton, in the BEC) show up at low
momentum. So, to create low-momentum Bragg excitations, we reduced the angle between ourlaser beams from 180◦ to 12◦ . To measure high-momentum Bragg excitations, one can measureatoms kicked out of the cloud or, alternatively, if the interactions are strong enough that atoms
don’t escape the cloud, the total momentum transfer to the cloud. However, with smaller momentum kicks, atoms that undergo a Bragg excitation can no longer be differentiated in time-of-flight
expansion and the total momentum transferred to the cloud is much smaller and also very difficult
to measure.
Therefore, we needed another technique to count the number of Bragg excitations, and so
we looked at the light. The number of Bragg excitations is equal to the number of photons lostor gained by a Bragg beam. For our BEC of 50,000 atoms and for the small excitation regime,
this signal is at most 104 photons, which corresponds to an optical power on the order of a pW.
Measuring a small change in the power of an intense laser beam is technically challenging. Chapter
5 describes our shot-noise-limited photon counting technique for measuring the change in power ofa laser beam due to Bragg excitations.
The photon counting technique was successful as a proof of principle, but there were a few

29
issues when we applied it to measuring low-momentum Bragg spectroscopy of a strongly interacting
BEC. When performing Bragg spectroscopy, it is preferred to keep the number of excitations in
the BEC small compared to the total number of particles and the number of absorbed photons
from the beam small compared to the total number of photons. Neither of these conditions weretrue, though, if we maintained reasonable signal to noise in our measurement. Second, it wastechnically very difficult to control the pointing stability of the laser beam well enough. Lastly, amore fundamental problem was that the spectral width of the positive- and negative-going Bragg
peaks were comparable to the separation of the signals. Thus, the two peaks would destructively
interfere with each other, which stopped us from accurately measuring the peak position and width.
This problem, unlike the previous challenges, was a fundamental challenge to this measurement andwas going to be difficult to overcome. For these reasons we moved on to contact spectroscopy as adifferent probe of the interacting BEC.

4.3.2

Contact Spectroscopy and three-body contact

After developing the photon-counting Bragg-spectroscopy technique, we turned our attention
to a new probe of interacting BECs and measured the contact (our measurement of the contact
is in presented chapter 6). In 2008, Shina Tan published a number of seminal papers [54, 55, 56]
describing a quantity called the contact. Tan showed that the contact is an extensive thermodynamic quantity of a Fermi gas that relates microscopic physics to thermodynamic quantities of thegas. The contact is related to a number of thermodynamic quantities, such as the pressure, thetotal energy, the momentum distribution, and the virial theorem, and to a number of microscopic
quantities, such as the probability of two particles to be within a certain volume. Braaten et al.
later extended the contact to bosons and to considering three particles instead of two [57]. To
measure the contact in our lab, we used the fact that the contact is related to the amplitude of along tail on one side of a RF lineshape [58, 59, 60].
We measured the contact for atomic bosons using RF spectroscopy and related it back to
the total energy of the gas [19]. Our measurement of the contact is the first for atomic bosons

30
and we showed how the contact changes with scattering length. We explored the contact from themean-field regime to the regime of beyond-mean-field interactions. Measuring the contact uses RF
spectroscopy and allows us to quickly probe the BEC before losses occur. We apply an RF pulse
for the contact measurement as soon as the magnetic fields are stable after changing a. We canpulse the RF field on for very short durations since we have large RF power and good coupling to
the atoms. For very fast measurements, we are limited by the spectral Fourier width of the shortpulse; however, the contact comes from the relative amplitude of the signal in the tail, which is lesssensitive to broadening than a frequency measurement.
We made successful measurements of the contact due to two-body interactions. Braaten
also predicted a contact signal proportional to three-body interactions in a BEC [57]. This threebody contact signal would be more apparent at smaller detunings on the excitation tail. So, tomeasure the three-body contact, we repeated the RF spectroscopy measurements, but with smaller
detunings. Although we increased the interactions to as large as na3 = 0.002, we saw no evidenceof a three-body contact [19]. This was the largest na3 we were able to ramp to before inelastic
losses and density increases limited us. However, we can tune the scattering length to unitarity,
where we saw preliminary evidence for a finite three-body contact.

4.4

Infinite interactions

4.4.1

Unitarity

Our Bragg spectroscopy and contact spectroscopy measurements were performed on a interacting BEC in the beyond-mean-field regime, but with na3 still much less than one. The goalof studying strongly interacting BECs was stymied, since three-body losses caused the BEC todecay during the ramp from a weakly interacting BEC to a na3 > 1 regime. Experiments studying
interacting BECs could only reach “modestly strong” interactions [17, 15, 18, 61] with na3 < 0.008.
Alternatively, experimentalists used thermal Bose gases to explore na3 > 1 with measurements ofthe three-body losses at unitarity [62, 63]. However, thermal gases aren’t macroscopic quantum

31
systems where quantum mechanics and interactions determine the behavior of the gas.
We took a different approach to probe a strongly interacting BEC with na3 > 1. Instead of
ramping slowly compared to molecule binding energy, which is a minimum requirement for local
equilibrium in the gas, we forsake equilibrium and quickly “jump” the scattering length. Our quick
jump bypasses losses and projects the BEC on strong interactions. We measured the loss timescale
and probed the dynamics of the system for short times compared to the timescale of the losses. To
jump the interaction strength, we installed two small coils that quickly change the magnetic field.
Chapter 3 described these fast-B coils.
Projecting a BEC onto unitarity raises many questions. If we consider the local physics, isan equilibrium of the many-body system reached before there is significant number loss? Does thesystem evolve coherently? Do losses dominate the system? A priori we don’t know either the loss
or many-body dynamics timescales. It becomes an experimental question to answer. We describe
our measurements after fast projection of a BEC onto a strongly interacting state in chapter 6 andchapter 7.

4.4.2

Universality of physics at unitarity

As we increase the scattering length to where a diverges (a → ∞), it can no longer be aphysically relevant length scale. In the mean-field regime, the scattering length is an essential
parameter in describing the gas. When a → ∞, another physical length scale must effectively
replace a. For a thermal gas, this could be the thermal de Broglie wavelength, but the de Broglie
wavelength is not a relevant length scale for a BEC at zero temperature. The only length scalethat remains is the distance between particles, which can be written in terms of the density as
n−1/3 . The size of the cloud or the trap parameters can also provide a length scale; however, these
are not intrinsic to a BEC. The length scale that remains is the distance between particles, so thedegenerate Bose gas is universal in the sense all the physics is captured by the interparticle spacing.
This means that energies scale as n2/3 , momenta as n1/3 , rates as n2/3 , and timescales as n−2/3 . Wenote that here we are ignoring any length scales due to three-body interactions (for example, the32size of an Efimov trimer could provide another length scale). Deviations from universal dependenceof n due to three-body physics would be very interesting to observe.
The expected universality of the gas gives us reason to hope the losses may not dominate the system at unitarity. Naively, if the loss and equilibration rates scale as Γloss ∝ a4 andΓequilibrium ∝ a2 , respectively, one would expect that the losses always dominate as a → ∞. Instead, when we replace a with the interparticle spacing, all rates must scale as Γ ∝ n2/3 . It is thenan experimental question as to whether loss or equilibration is faster. For 85 Rb atoms, the previous
experimental observation (figure 6.3 in chapter 6) of a relatively narrow, and therefore long-lived,
Efimov resonance (characterized by a dimensionless width, η ∼ .06  1) [19] is indicative that
atoms close together do not decay instantaneously to deeply bound molecular states. As we seein chapter 7, equilibrium (or at least saturation of the dynamics) indeed appears to be faster by afactor of ten compared to the loss rate at unitarity.

Chapter 5
Photon counting for Bragg spectroscopy of quantum gases1

In this chapter we demonstrate a photon-counting technique for detecting Bragg excitationof an ultracold gas of atoms. As we explained in chapter 4, we use Bragg spectroscopy to probe aweakly interacting BEC. By measuring the response of the light field to the atoms, we derive a signal
independent of traditional time-of-flight atom-imaging techniques. With heterodyne detection weachieve photon shot-noise limited detection of the amplification or depletion of one of the Bragg laser
beams. Photon counting for Bragg spectroscopy is useful for strongly interacting gases where atomimaging detection fails. In addition, this technique provides the ability to resolve the evolution of
excitations as a function of pulse duration. However, we found probing low momentum excitations
posed a challenge due to the positive and negative branches of the Bragg spectrum interfering witheach other.

5.1

Introduction
Ultracold quantum gases provide clean, controllable model systems for exploring quantum

many-body physics [3], and a powerful probe of these interacting quantum systems is the spectroscopy of low-energy excitations. For example, intriguing behaviors such as superfluidity, fermion
pairing, and exotic weakly bound molecules can be revealed in their characteristic excitation spectra. In many cases, measured spectra can be directly compared with many-body theory to test our
understanding of these rich systems.
1This chapter is an expanded version of a previously published paper [64]34
Any experiment designed to probe excitations requires the same essential components: a way
to excite the sample and a way to measure the response. To drive the excitations one applies afield to perturb the gas, and spectra are obtained by measuring the response of the ultracold gas asa function of the driving-field frequency. In ultracold atom experiments, this response is measured
by imaging the atom cloud after the perturbation, and observing the response as some change inthe density distribution of the imaged cloud.
There exists in any excitation spectroscopy, however, an alternative way to measure theresponse of the quantum gas to a driving field. Just as the quantum gas has responded to the probe
field, the probe field must have an equivalent response to the quantum gas. In this chapter, we
present a technique to measure this complementary information in Bragg spectroscopy by measuring
the change in the number of photons in one of the laser fields used to drive the Bragg excitations.
While in this work we use Bragg spectroscopy, this technique may more generally be applicableto other excitation spectroscopies as well [65]. For ultracold atoms where the number of atoms istypically small (∼ 105 ), the challenge, of course, is to have adequate signal-to-noise in measuring
photon number to detect the Bragg response. For smallish atom samples, this will typically requirea photon shot-noise limited measurement. To that end, we detail the experimental setup andthe techniques of a heterodyne-based detection scheme used to realize shot-noise-limited photon
counting. We also explore the advantages and limitations of this technique.
To motivate tackling the challenge of photon counting, we point out that this technique
avoids some of the pitfalls inherent to spectroscopy measurements where the cloud is imaged. For
example, strong interactions, which complicate the response seen in the density distribution of a
Bose-Einstein condensate (BEC) [15], provide no additional complications for the photon counting
approach. Indeed, a primary motivation for this work is our desire to extend Bragg spectroscopy of
BEC to the case where we have both strong interactions and low momentum excitations. Finally,
we present a powerful new feature of our technique, which is that we can directly probe the time
evolution of the excitation process, even during the course of a single laser pulse.

35

5.2

Bragg Spectroscopy
Conceptually, a Bragg excitation can be thought of as a coherent scattering process involving

absorption of a photon from one of the Bragg beams and stimulated emission into the other: a twophoton transition. It is this process of either absorption or emission that we aim to measure with
our photon-counting technique. The process leaves the excited atoms in the same internal state,
but with a new momentum, k, determined by the geometry and wavelength of the Bragg beams.
The two lasers have slightly different frequencies, to account for the energy of the excitation, andwe vary this frequency difference, ω, for our spectroscopy.
Typically, the Bragg response is measured by looking at the cloud using time-of-flight atom
imaging, where the gas is suddenly released from the trap and allowed to expand before imaging.
Bragg excitations in a weakly interacting BEC then appear as atoms in a distinct cloud outside of themain cloud. The position of the new cloud, which is seen after the expansion from the trap, reflects
the momentum of the excitation. This new cloud’s density reflects the strength of the excitation.
Pioneering studies of weakly interacting BECs have been done with such measurements [66, 67, 68],
but, there are limitations. In particular, in the case of strong interparticle interactions [15, 69] or
low-momentum excitations [68], the excitations no longer appear as a clearly distinguishable second
cloud in the momentum-space image and the response becomes more difficult to quantify.
We show a measured Bragg lineshape for a weakly interacting 85 Rb BEC in figure 5.1. On
the horizontal axis, we have the frequency difference between the two Bragg beams, which sets theenergy of the Bragg excitation. On the vertical axis, we have the Bragg signal, namely, the numberof excitations due to the Bragg process. We define this signal such that it can be either positiveor negative, reflecting the direction of the momentum transfer. The number of excitations due tothe Bragg pulse are counted in two different ways, and one can see that the photon counting andthe time-of-flight imaging signals agree well with each other. The two sets of data were acquired
simultaneously, with each cycle of the experiment providing both a photon-counting and an atomimaging data point. This demonstrates the complementary nature of the two techniques. The lines

36
in figure 5.1 are individual fits of the Bragg spectrum to two Gaussian functions. These fits can beused to extract a center frequency and an RMS width. In the rest of this chapter we describe indetail the photon-counting technique demonstrated here.

4

1x10

photons
atoms

3

5x10

0
3

-5x10

4

-1x10

-20

-16

-12

12

16

20

Figure 5.1: Bragg spectrum of a 85 Rb BEC of 5.5 × 104 atoms at |k| = 16 µm−1 , measured in
two different ways. The horizontal axis shows the frequency difference between the Bragg beams.
The vertical scale shows the number of excitations, which is measured using traditional absorption
imaging of ejected atoms (hollow circles), as well as with the photon-counting technique presented
here (black triangles). The error bars on black points represent the shot noise in the photon counting
measurements. The photon counting measurements used three pulses of equal length. The first
and the third pulses used only a single weak beam to make an average background measurement,with no Bragg excitation. During the second pulse, both Bragg beams illuminate the condensate toinduce Bragg scattering and we subtract the averaged background to count the number of photons
gained or lost in the weak beam due to the Bragg excitation.

A two-photon Rabi frequency characterizes the Bragg response and gives us an understandingof how the excitation rate will scale with the various experimental parameters. The two-photon

37
Rabi frequency is given byΩBragg =Ωweak Ωstrong
2∆int

(5.1)

where Ωweak and Ωstrong refer to the individual Rabi frequencies associated with each laser field,
labeled here as weak and strong (the Bragg beam to be measured is the weak beam, for reasonsthat will be clear later). The detuning, ∆int , in rad/s, refers to the detuning of the beams fromthe atomic transition, and is much larger than ω, the small frequency difference between the two
beams. Using the Rabi frequency, we can define an excitation rate, Γ(ω) whereZΓ(ω)dω =π
S(k)Ω2Bragg
2

(5.2)

here S(k) is the static structure factor, and gives the strength of the Bragg resonance for a particulark. [70, 71].
In this paper, we describe photon counting detection for Bragg spectroscopy of BECs of Rb
atoms. In our experiment, we can create nearly pure BECs of either 85 Rb or 87 Rb atoms, witha final number of roughly 4 × 104 or 4 × 105 , respectively, in a roughly 10 Hz spherical magnetic
trap. Thanks to a magnetic-field Feshbach resonance at 155 G, we can study strongly interacting
condensates using 85 Rb atoms. The 87 Rb condensates, on the other hand, have far greater atom
number and give us a robust tool for characterizing the photon-counting technique with an increased
signal-to-noise ratio.
Regardless of the method used to measure a Bragg response, adequate signal-to-noise will
always be a prerequisite. The signal is set by the number of Bragg excitations, which is typicallyless than 10% of the total atoms in the sample in order to probe the linear response. For our 85 Rb
condensates, that would be some 4,000 excitations. If we allow for 105 photons in the weak beam,
this corresponds to a signal-to-noise ratio (SNR) of only 9 on resonance. This assumes a detector
with perfect quantum efficiency as well as a shot-noise limited detection scheme.
For the best signal-to-noise ratio, assuming that a shot-noise limited measurement is available,
we look to minimize the total number of photons in the weak beam. Keeping the excitation rate (andhence the signal) constant when decreasing the intensity of the weak beam necessitates increasing

38
the intensity of the strong beam (see equation 5.2). However, single-photon processes, namely
off-resonant scattering of the stronger beam, limit the maximum permissible intensity in the strong
beam, and this, in turn, limits the minimum intensity in the weak beam for a desired excitation
rate. Note that having a large mismatch between the two Bragg beam intensities is a distinct
requirement for photon counting compared to the usual atom-response detection, where there is no
reason not to have equal intensities in the two beams.
Another limit to the minimum photon number in the weak beam is set by the spatial profileof the weak beam at the atoms. We tailor this spatial profile by focussing the weak beam onto theatoms, attempting to match the transverse spatial profile of the weak beam to that of the condensate
in order to minimize the number of photons that would never interact with the condensate, and only
add to the shot-noise of the measurement. In minimizing the number of weak beam photons, one
could easily enter a regime where the number of Bragg photons scattered is a significant fraction of
the weak beam photons themselves. In this regime, the transverse spatial profile of the weak beam
intensity would be modified due to the scattering of Bragg photons out of (or into) the weak beam;
this could, in effect, burn a “hole” in the probe. We describe these as propagation effects and,to simplify the interpretation of the Bragg response, we keep the total number of photons in theregion of overlap between the weak beam and the BEC large compared to the number of excitations.
This provides another limit on the minimum number of weak beam photons and, therefore, themaximum signal-to-noise ratio for the measurement.
In order to then make shot-noise limited measurements of the weak light, one could consider
using detectors with integrated cascading amplification, such as avalanche photodiodes and photo
multiplier tubes. However, in considering this option for our purposes, we found that avalanche
photodetectors suffer from pulse-pileup for the powers we use, while photo multiplier tubes sufferfrom low quantum efficiencies in the IR, which at 5% or so are too low to be useful. Therefore we
consider the alternative of using detectors in their linear mode of detection. Silicon photodetectors
have quantum efficiencies as high as 90% and capacitances on the order of 3 pF. However, for thepulse lengths we seek (∼ 1 ms) the Johnson noise associated with the necessary feedback resistor

39
corresponds to the shot noise of a 1 µW beam at 780 nm and therefore makes direct shot-noiselimited measurements impossible for the powers used in these experiments. Our solution to thesechallenges is one as old as the AM radio: heterodyning.

5.3

Heterodyne detection
To motivate heterodyne detection, we consider the following scheme. We direct the weak

Bragg beam onto a photodiode, and after some stages of amplification, directly measure a voltage,
Sdirect , proportional to the number of photons per second incident on the detector. To illustrate therole of shot-noise, we consider our signal to be proportional to the number of photons, Nweak , that
hit the detector in some time τ . In terms of an average voltage, the signal is Sdirect = eRηNweak /τ ,where η is the quantum efficiency of the detector, e is the charge of an electron, and R is thetransimpedance gain of the amplifier.
The signal-to-noise ratio, SNRdirect , on this direct weak beam detection is given by
SNRdirect =eRηNweak /τqδS2 + δe2

(5.3)√
where the shot-noise associated with the photocurrent is δS = (eR/τ ) ηNweak . Other sources ofnoise (dark noise from the detector, Johnson noise introduced in the amplification stages, noiseon background light incident on the detector, etc.) will be referred to as electronic noise, and are
represented by δe . Shot-noise limited detection is defined as the regime where δe is an insignificant
contribution to the total noise (δe  δS ). In this case,
SNRshot-noise =pηNweak .

(5.4)

This is difficult to achieve in direct detection, where for typical experimental values of τ = 1 ms,
Nweak = 105 , η = 85 %, we might expect δe = 1000 × δS for a silicon photodetectors with a highbandwidth, low-noise transimpedance preamplifier.
We use a heterodyne scheme to overcome these difficulties. The idea of heterodyne detection
is to amplify the signal optically before detection, so that electronic noise is of no consequence. We

40
do this by measuring the beat of the weak beam against another, more intense beam, which wewill refer to as the local oscillator (LO). Because the beat signal, Sbeat , goes as the square root of
the product of the intensities of the weak and the LO beam, we are left with a signal that is muchstronger than that of the original, weak beam signal.
The advantage heterodyne offers is that one may arbitrarily increase the intensity of the LO
beam, so that the shot-noise from the LO photons dominates the electronic noise, and the totalnoise is given byqpδS2 + δe2 ' δS = (eR/τ ) ηNLO

(5.5)

where NLO is the number of photons from the LO beam during a time τ . Because increasing the LO
beam intensity also increases the optical gain, we are left with a SNR on Nweak using heterodyne
detection as
SNRhetero = CpηNweak

(5.6)

where the contrast, C, is a number between zero and one that describes the quality of the modematching between the two heterodyne beams. One notes that for perfect contrast (C = 1), SNRheterois our stated goal for a shot-noise limited measurement as in equation 5.4. These principles of optical
heterodyning are well established, and we refer the reader to Refs. [72, 73, 74] for a more thoroughdiscussion.
Measurements of the LO shot-noise, δs , also serves to calibrate the overall gain of our system,
R. Provided one knows η (which is readily available from the photodiode’s datasheet) and has acalibration for NLO (which is straightforward for the relatively high power of the LO beam), one
can rewrite equation 5.5 as(τ /e)
R = δS √
.
ηNLO

(5.7)

41

5.4

Optical Layout and Electronics
While the principle behind shot-noise limited heterodyne detection is straightforward, theimplementation of such a technique has a number of subtleties, which we lay out here. We illustrate
both the optical and the RF design in figure 5.2. We begin our discussion on the optical side ofthings, and then consider the RF.
strong(a)wlaser + w

BEC

(b)

beamsplitter

BEC

AOM

q

weak

photodiode
mixer

LO

RF

wlaser

“LO”

AOM

PM

I Q

Servo
to oscilloscope

Figure 5.2: Diagram of the Bragg beams (a) and the heterodyne setup for Bragg scattering via
photon counting (b). The Bragg beam frequencies (ωlaser , ωlaser + ω) are offset by ω, and create
excitations at momentum ~k = 2~klaser sin(θ/2), where θ is the angle between the Bragg beams andklaser = 2π/780 nm in our case. In (b) the three beams (labeled weak, LO and strong) are derived
from the same laser source and individually fiber coupled. The optics for the LO beam are chosen
to give the same spatial mode as that of the weak beam. After being combined on a beamsplitter,
the weak and LO beams illuminate a photodiode and the beat signal is sent to a demodulating
mixer. The two quadrature outputs of the demodulator, I and Q, are sent to an oscilloscope for themeasurement. Also illustrated in the figure is a servo loop that functions to minimize the phasefluctuations of the RF input with respect to the “LO” drive of the mixer. The servo minimizes thedemodulator output, I, by feeding back to a phase modulation input on the synthesizer that drives
the “LO” port of the mixer. This synthesizer is phase-locked to the synthesizer driving the weak
beam acousto-optic modulator (AOM), illustrated with a double-headed arrow. Not shown in theschematic is the synthesizer driving the strong-beam AOM, which is also phase referenced to thesynthesizer driving the weak-beam AOM.

In figure 5.2(a), we show the two Bragg beams intersecting at the location of the BEC. The
third, much more intense LO beam, shown in figure 5.2(b), functions only to amplify the weak
beam for our heterodyne detection, and avoids the atoms altogether. All three beams are derived
from the same laser, and are offset in frequency after passing through acousto-optical modulators(AOM). For these experiments we use a diode laser at 780 nm with a coherence length on order of42
100 m. Each AOM has a dedicated synthesizer, with an additional synthesizer driving the mixer
as well. All three of these synthesizers are phase-locked to each other to ensure coherence between
the Bragg beams.
There are two challenges in optimizing this heterodyne interferometer for shot-noise limited
detection. The first is achieving a contrast, C, as close to unity as possible. In equation 5.6, we seethat the signal-to-noise ratio scales linearly with the contrast. Therefore, we would like the best
possible mode-matching between the two beams, both spatially as well as in terms of polarization.
In light of these two requirements, we launch the two beams into polarization-maintaining fibers
identical in both core sizes and output couplers. The weak and the LO beams subsequently passthrough different sets of lenses, which serve to match the focussed size of the weak beam to the sizeof the BEC, as well as to match the spatial mode of the LO beam to that of the weak when they
recombine on the beamsplitter. To avoid interactions between the intense LO light and the atoms,
the LO beam has its own optical path that avoids the vacuum chamber and the atoms. In addition,
we have an optical isolator (not shown in figure) in between the beamsplitter and the photodiodeto extinguish LO light backreflections off of the photodiode that interact with the atoms.
The second challenge is to minimize any jitter in the relative phase (φ) between the two
beams. As we will see later, this phase jitter introduces noise that limits us from achieving shotnoise limited detection for longer Bragg pulses. We find that the biggest source of instability in therelative phase is due to the optical fibers. These fibers are sensitive to mechanical vibrations andacoustical noise. We thus use relatively short fibers (1 m) to minimize phase jitter between the two
beams, while still providing the necessary filtering of the spatial mode.
Combining the two beams with a beamsplitter provides good spatial overlap, at the costof weak beam attenuation from light lost at the unused beamsplitter port. This can be includedin equation 5.6, by simply replacing η with a lower effective quantum efficiency for the detection
system. For the best signal-to-noise ratio, we would like to measure as much of the weak beam as
possible. This must be balanced, however, with our need for large enough LO power to overwhelm
the electronic noise. Typically, we work with 90% reflection of the weak beam and 10% transmission

43of the LO beam, corresponding to 250 µW of LO power on the photodiode. This 90/10 ratio isachieved with a partially reflective, gold-coated piece of glass.
To measure and maximize the contrast, we replace the light launched into the weak beamfiber with light at the same frequency and intensity as the LO. With this homodyne interferometer,
one can easily assess the quality of the contrast by eye, by looking at the pattern of the interference
fringes at the photodiode. Our procedure is to first align the beam such that the fringes (seenusing a temporary lens to expand the beam) are a circular pattern of rings. We then optimize thecollimation of the LO beam by minimizing the total number of rings seen. We found this to bean essential step in getting good contrast. Once the beams are closely mode matched by eye, we
perform the final steps of alignment by directly monitoring the DC output of the photodetector.
We found it useful to modulate the phase of one of the beams (achieved via a moving plate ofglass in one of the beam paths), enough to wrap around 2π, so that the DC output swings between
fully constructive (DCmax ) and fully destructive (DCmin ) interference. We are able to quantify thecontrast throughC=

DCmax − DCmin,
DCmax + DCmin(5.8)

true when the intensities of the two beams are the same.
We also take steps to minimize the electronic noise (δe ) that the LO’s shot-noise must overcome. The relevant noise for the heterodyne measurement is that at the frequency of the beatnote, ∆LO . Our photodiode circuit is designed to minimize the effects of inherent voltage noise at
the op-amp input at this frequency (2π × 70 MHz) by way of a standard “tank” circuit. The tankcircuit consists of an inductor between the op-amp input and ground. The inductance is chosen
so that when the photodiodes’s internal capacitance is also considered the two make a resonant
LC circuit at ∆LO . We are able to reduce the receiver (photodiode plus amplifier [75]) dark noise√
to 2 pA/ Hz which is typical for most commercially available transimpedance amplifiers. The
subsequent stages of amplification are chosen so that the noise they add is small compared to that
introduced in this first stage of amplification.

44
The voltage from the photodetector then goes to an RF mixer. As shown in figure 5.2(b), themixer is a four-terminal device, called an I,Q-Demodulator; this is essentially two mixers in one,with the RF input split between the two. Port I is the same as the IF (intermediate frequency)
output of a standard mixer, while the Q port is the output of a second mixer whose “LO” drive hasa phase offset 90 degrees with respect to the phase of the I port’s “LO” drive. In our application,
the outputs of the demodulator have the formp
SI = (2CeRη/τ ) NLO Nweak cos φ

(5.9)

p
NLO Nweak sin φ(5.10)

SQ = (2CeRη/τ )By summing the squares of SI and SQ , we are able to measure the amplitude of the beat signal,
regardless of the relative phase φ. For a known LO beam power, ∝ NLO /τ , our signal is directly
proportional to the rate of weak beam photons, ṅ = Nweak /τ . We measure SI and SQ with adigitizing oscilloscope and perform the sum in subsequent software analysis.
While we now have a signal that in principle is insensitive to the relative phase φ between
the weak beam and the LO beam, the demodulator is imperfect, due to non-linearities and voltage
offsets in the I and Q outputs as well as deviations from perfect 90 degree phase offset. We therefore
servo the phase of the RF driving the “LO” port of the demodulator, as illustrated in figure 5.2(b).
The servo minimizes SI by actively feeding back to the phase-modulation input of the relevant
synthesizer, and this reduces the sensitivity of the quadrature sum to demodulation imperfections.

5.5

Noise performance and Bragg dynamics
In figure 5.3, we present data illustrating the shot-noise limited performance of our heterodyne

detection. We calculate the noise by measuring the standard deviation, σ∆ , of the difference inphoton number, ∆i , for M consecutive Bragg pulses of equal length,v
uM −1
u X (∆i − ∆)¯ 2σ∆ = tM −1i=1(5.11)

45
where ∆i = Nweaki+1 − Nweaki , and Nweaki is the measured number of photons in the weak beam
¯ = 0), and we know that the standard
for pulse i. Since on average the difference should be zero (∆√deviation in photon number, σ = σ∆ / 2, we are left with
v
uM −1

X Nweaki+1 − Nweaki 2
1 u
σ=√ t
.
(5.12)
M −1
2i=1

We plot the noise on our measurements of weak beam photon number, normalized to the expected√
shot-noise (δshot = ηNweak ) for that particular pulse duration and laser power, as a function of theduration of a fixed intensity pulse. In the inset of the figure, we plot the same noise measurement
for the case where the weak beam intensity is varied to keep the total number of photons fixed at
a constant 105 .
For a large range of pulse durations, the measurement is at, or within a factor of two of,the shot-noise limit. For pulses shorter than 1 µs the noise is artificially low due to an inline,
low-pass filter. The increased noise at longer timescales sets an upper limit to the time available
for our Bragg measurements and is probably caused by residual phase drift in our system. Servoing
the laser power provided no significant improvement in the noise performance of our heterodyne
detection.
A potentially useful feature of a photon-counting measurement is the ability to measure thedynamics of Bragg excitations during a single laser pulse. We demonstrate this capability in figure
5.4, where we plot the number of excitations, Nexc , as a function of time, τ . The data were takenusing a condensate of 400,000 87 Rb atoms, with the Bragg detuning set to be on resonance with themeasured Bragg transition at ω = 2π × 250 Hz for a momentum transfer given by k = 1.5 µm−1 .
For Fourier-limited pulses we expect Nexc (τ ) to go initially as τ 2 , however an interesting featureillustrated in figure 5.4 is the suppression of Nexc (τ ) for pulses short compared to the inverse Bragg
resonance, τ < ω −1 = 0.3 ms. For these short pulses, the associated energy uncertainty makes it
impossible to resolve a +k excitation from a −k excitation. Photon emission from the one processcancels photon absorption from the other process. In our experiments, we did not observe Rabi
flopping in the time-dependent data, which may be due to dephasing. For the measurements here,

46

noise (

noise

10

/

shot

)

3
2

1
100µs

1ms

10ms

pulse length

11000 pW
100 pW
10 pW
2 pW

100ns 1µs 10µs 100µs 1ms 10ms 100ms
pulse length

Figure 5.3: Noise performance of heterodyne detection for a LO power of 250 µW. Vertical scale
is normalized to the shot-noise expected for the relevant pulse length and laser power. The legendshows the different weak beam powers used. Low-frequency drifts make our heterodyne scheme nolonger shot-noise limited at long timescales. Inset has the same normalized vertical units, with thedata shown at different weak beam intensities, corresponding to a constant 105 photons.

the weak beam profile used was much smaller (7 µm 1/e2 waist) than that of the BEC (22 µm
Thomas-Fermi radius), which complicates the Bragg response. We present this data, however, toillustrate a promising feature only available with a photon-counting approach.

5.6

Conclusion
We have demonstrated photon counting as a viable technique for Bragg spectroscopy inultracold atoms. By measuring the response of the driving field to the atoms, one has a measurement
independent of, and complementary to, the traditional atom cloud time-of-flight imaging. Our

47

6

10

3

4

2

0
0.0

0.2

0.4

0.6

0.8

1.0

Figure 5.4: Bragg excitations as a functionR of time, measured with photon counting. The numberτof Bragg excitations is given by Nexc (τ ) = 0 (ṅ(t) − ṅavg )dt, where ṅavg is the average rate of weak
beam photons measured when no strong light is present. The Bragg pulse begins at 0 ms. The
deviation from expected τ 2 behavior (for small τ ) is discussed in the text. These measurements
were performed on resonance, with a BEC of 400,000 87 Rb atoms.

heterodyne scheme approaches shot-noise limited detection of a weak Bragg beam, which thenallows us to measure the number of photons added to, or depleted from, that beam. We haveshown that this measurement technique can be used simultaneously with time-of-flight imaging,and that it can probe the time dependence of the excitation process.
There are several issues to consider in applying photon-counting for Bragg spectroscopy of a
BEC. Because the weak Bragg beam waist needs to be roughly matched to the transverse profileof the condensate, careful alignment must be maintained for the photon counting approach. In

48
addition, Bragg spectroscopy for small momentum transfers can be complicated by the fact thatthe upward and downward portions of the lineshapes (see in figure 5.1) can begin to merge for
Bragg frequencies near zero. Determining a resonant frequency from fits to these slightly mergedlineshapes is nontrivial (in particular the fitting parameters, line widths and line center, are nolonger orthogonal), and, as a result, greater signal-to-noise is needed. We have also observed
asymmetries between the upward and downward portions of Bragg lineshapes taken with photon
counting, which we attribute to the relatively small number of Nweak photons necessitated by thephoton counting approach. We speculate that these asymmetries are related to the propagation
effects (for instance difficult-to-model effects associated with spatial variations in the Rabi frequency
as the beam propagates through the sample), or nonlinearities, inherent in a weak beam that has
been significantly enhanced or depleted by the Bragg process. These effects will be strongest when
the number of Bragg photons is a significant fraction of Nweak , or of the number of atoms in thecondensate. These issues may limit the usefulness of this technique to systems whose atom number
is larger than the 40,000 atoms in our 85 Rb BECs.
Measurements of the scattered Bragg photons could open the doors to new investigations of
the temperature-dependent structure factor, as proposed by Stamper-Kurn [76]. More generally,
this technique of measuring the probe in order to detect atom-light interactions could be applied toother types of ultracold atom spectroscopy as well, and it seems that the two technologies, photon
counting and time-of-flight imaging, marry nicely.

Chapter 6
The contact 1

A powerful set of universal relations, centered on a quantity called the contact, connects thestrength of short-range two-body correlations to the thermodynamics of a many-body system withzero-range interactions. In this chapter we discuss measurements of the contact for an 85 Rb atomic
Bose-Einstein condensate (BEC) using RF spectroscopy and loss measurements of the BEC. For
bosons, the fact that contact spectroscopy can be used to probe the gas on short timescales isuseful given the decreasing stability of BECs with increasing interactions. A complication is theadded possibility, for bosons, of three-body interactions. The different functional form predicted
for the RF lineshape due to two-body interactions compared to three-body interactions allowed us
to differentiate between the two. In our investigations, we also located an Efimov resonance for
85 Rb atoms with loss measurements and thus determined the three-body interaction parameter.

Systems with strong quantum correlations represent a frontier in our understanding of thecomplex quantum systems found in nature. Atomic Bose-Einstein condensates (BEC) provide a versatile system in which to explore beyond mean-field physics. Ultracold atoms experience two-body,
short-range interactions that are well described theoretically by a delta-function pseudopotential
characterized by an s-wave scattering length a. In the simplest BEC experiments the values of aand of the density n are such that interactions are too weak, compared to the kinetic energy costof correlations, to take the gas out of the mean-field regime. In many experiments an additionallattice potential is applied to suppress the kinetic costs and thus turn on interesting correlations.
1

Much of this chapter appeared in a previously published paper[19]50
The external lattice, however, imposes an artificial orderliness not found in bosons in the wild. To
explore strong interactions in a more naturalistic bulk three-dimensional gas, one can increase a bymeans of a magnetic-field-tunable Feshbach scattering resonance [5]. Such efforts are motivated for
instance by a desire to make better conceptual connections to the iconic strongly correlated fluid,
liquid helium.
In practice it has proven difficult to study atomic BEC with increasing a and only a few
experiments have measured beyond-mean-field interaction effects [15, 17, 18]. The problem is thathigher a is accompanied by a dramatic increase in the rate of inelastic three-body processes [23, 22].
This leads to large losses and significant heating of the trapped gas on a timescale similar to thatfor global equilibrium of the trapped cloud. Probes of the gas that require global equilibrium, suchas measurements of the density distribution or the amplitude or frequency of collective density
oscillations in a trap, are therefore limited to systems that are only modestly out of the meanfield regime. Our strategy for exploring BEC with larger interaction strengths is to start from anequilibrated weakly interacting gas, change the interaction strength relatively quickly, forsaking
global equilibrium, and then use a fast probing technique to look at local many-body equilibrium(as described in chapter 4) in the trapped gas [15]. We show how RF contact spectroscopy is a fastprobe of short-range correlations in a weakly interacting BEC, before presenting measurements ofthe contact for a strongly interacting BEC at unitarity.

6.1

Introduction to the contact
A central challenge in many-body physics lies in elucidating the dependence of an interacting

many-body system on the strength of the few-body interactions. For ultracold Fermi gases, it
has been shown that the dependence of the energy on a can be connected to the strength of twoparticle short-range correlations through a set of universal relations that were introduced by Shina
Tan [54, 55, 56]. These universal relations, which involve a quantity termed the “contact” holdtrue for any locally equilibrated gas regardless of the temperature, interaction strength, or numberof particles. Tan’s predictions have been explored theoretically [77, 78, 79, 80, 81, 82] and verified

51
experimentally [83, 84] for strongly interacting Fermi gasses. Recently, a number of theoretical
papers have discussed extending the ideas of the “contact” to bosons [85, 57, 86, 87]. We now
address experimentally whether contact spectroscopy can be used to probe interacting bosons.
The derivation of Tan’s universal relations does not depend directly on the quantum statisticsof the particles, however, it does assume that the interactions are fully described by a single
parameter, a. While this is true for an ultracold two-component (spin-up and spin-down) Fermi
gas, it is in general not true for a Bose gas, where three-body interactions give rise to Efimov
resonances [26]. A number of recent experiments probing few-body physics in ultracold Bose gases
have observed Efimov resonances [88, 89, 90, 16, 91], however, many-body effects of the three-bodyinteractions have not been observed. To explore contact spectroscopy for bosons, we begin byexamining RF spectroscopy assuming that three-body interactions do not significantly affect thismeasurement. Following this, we present a measurement of the three-body parameter for 85 Rb
using trap loss rates for a non-condensed gas, and then look for many-body effects manifested in athree-body contact for a BEC, C3 [57, 92].
The two-body contact, C2 , is an extensive thermodynamic variable that is connected to thederivative of the total energy of the system, E, with respect to a [55, 85, 86].
dE
~2
=
C2 .
da s 8πma2(6.1)

Combining this with the energy density of a BEC (equation 2.7) in the perturbative beyond-meanfield regime the predicted contact for a condensate is
5 128 √ 3√
C2 = 16π na 1 +na + ... N0 ,2 15 π22(6.2)

where n is the atom number density, m is the atomic mass, and N0 is the number of atoms in the
BEC.
To measure C2 using RF spectroscopy [58, 59], an RF pulse drives a Zeeman transition andtransfers a small fraction of spin-polarized bosonic atoms into another spin state. Interactions
give rise to an asymmetric tail in the RF spectrum, which can be thought as RF “dissociation”

52
of pairs of atoms that happen to be very close to each other. Ignoring C3 , and assuming that themeasurement is done in the linear regime, the rate for transferring atoms to the final spin state inthis tail is given by [60]Ω2lim Γ(ω) =
ω→∞
4π
where

r

~ α(a) C2,m β(ω) ω 3/2

(6.3)

R∞

2
−∞ Γ(ω)dω = πΩ N , Ω is the Rabi frequency, and N is the total number of atoms. In Eqn.

6.3, α(a)/β(ω) describe final-state effects; the a-dependent part is α(a) = (a0 /a − 1)2 , where a0 is
the scattering length for interactions between atoms in the final spin state and atoms in the initial
spin state, while the frequency-dependent part is β(ω) = 1 + ~|ω|/E 0 , where E 0 = ~2 /ma02 .
Our experiments probed 4-8×104 Bose-condensed 85 Rb atoms in a gas with a 60% condensate
fraction. The atoms are in the |F = 2, mF = −2i state, where F is the total atomic spin and mF
is the spin projection. They are confined magnetically in a 10 Hz spherical harmonic trap witha variable magnetic bias field. We work at magnetic-field values near a Feshbach resonance at
155.04 G [10], and during the final stages of evaporation, the field is set to give a ∼100 a0 . Afterevaporation, we ramp the bias field in order to change a on a timescale that is fast compared tothe trap period, but adiabatic with respect to two-body timescales, with ȧ/a never reaching morethan 0.01~/(ma2 ) (ȧ being the time derivative of a) [93].
An example of RF contact spectroscopy at a = 497 ± 5 a0 , where a0 is the Bohr radius, isshown in figure 6.1a. Roughly 1 ms after the magnetic-field ramps, we probe the BEC using a
Gaussian envelope RF pulse to drive the |2, −2i to |2, −1i transition. We determine Γ(ω) from thenumber of atoms transferred to the |2, −1i spin state divided by the RF pulse duration. We thendefine our signal, S(ω), as Γ(ω) normalized by the integrated lineshape. We fit S(ω) to a Gaussianlineshape (dashed black line in figure 6.1a) and take the center to be the single-particle transition
frequency ω0 . The center of the RF lineshape will be shifted due to mean-field interactions by anamount typically less than 2π × 0.5 kHz. For the main lineshape, we use short RF pulses with aGaussian rms width for the field amplitude, τ , of 5 µs; this sets the observed width of the lineshape.
At larger detunings, we use longer pulses, with an rms width of 25 to 200 µs, and an increased RF

53

12x10

-6

x 300

S

8x10

4x10

(a)

-6

-6

0
E/h (kHz)

40

(b)

20
0
-60

-40

-20
/2

0

20

(kHz)

Figure6.1: Example of RF contact spectroscopy. (a) RF lineshape, S(ω), normalized so thatR∞
−1−∞ S(ω)dω = 1 s . The data at large detunings (circles) are multiplied by a factor of 300 to
make the tail visible. Here the mean BEC density is hni = 4.9 x1012 cm−3 . (b) Additional releaseenergy of the outcoupled atom cloud. We calculate the energy from the width of the expanded
σ 2 −σ 2
cloud, σ, using E = 23 m ∆t2 0 , where ∆t is the time between the middle of the RF pulse and theabsorption image (4.5 ms) and σ0 is the size of the expanded cloud measured at ω = 0. The solidline is 12 |ω|
2π .

54
power, Ω2 , such that we outcouple 1-2% of the gas. We normalize the signal for the different τ andΩ2 , making small (5%) corrections for measured nonlinearity in Ω2 τ .
For our experiment, the RF drives a transition to a lower energy spin state and one expects
the 1/|ω|3/2 interaction-induced tail on the low frequency side of the lineshape. Consistent withthis expectation, we observe a tail for large negative detunings, while for similar detunings on thepositive side, we find that the signal is consistent with zero. The solid line in figure 6.1a shows a fit
to the expected frequency dependence from Eqn. 6.3, while the dotted line shows a fit to 1/|ω|3/2 .
For our system, the final-state effects are characterized by a0 = −565 a0 [94] and E 0 /h =133 kHz.
The 1/|ω|3/2 tail, due to the contact, corresponds to an expected 1/k 4 tail in the momentum
distribution n(k) [60, 84]. In figure 6.1b, we show the expansion energy of the outcoupled atoms,
measured by releasing the gas from the trap and imaging the cloud after 3 ms of expansion. In
the region of the observed tail our data show good agreement with the prediction (line in figure6.1b) that the additional release energy should be 21 ~|ω|, where the factor of 12 comes from theassumption that the excess energy of the RF photon is shared between two pairwise interacting
atoms [95].
The strength of the RF tail is shown as a function of a in figure 6.2. As expected, we
see the strength of the RF tail increase as a increases. In comparison with theory, our contact
measurements are larger than the mean-field prediction (solid line in figure 6.2), but not as large
as the prediction including the next order LHY term given in Eqn. 6.2 (dashed line in figure 6.2).
While beyond-mean-field physics is evident in the contact data shown here, we see evidence thatthe measured strength of the RF tail depends on the speed of the magnetic-field ramp to increasea, with C2 increasing for slower ramps. We plan to carefully explore this intriguing dependence on
ramp speed in order to probe experimentally local microscopic dynamics in the beyond mean-fieldregime.
We now turn our attention to C3 , which is connected to the derivative of E with respect to

55

8

(a)

6
4

2

-1

C /N0 ( m )

10

2

2

-1

C /N0 ( m )

0
8

(b)

6
4
2
0

-500

0

500

scattering length1000

a (a )
0

1500

a

C2
Figure 6.2: The contact vs a, measured at |ω| = 2π × 40 kHz. (a) The contact per BEC atom N.
0(b) The raw signal before final-state corrections. The solid lines in (a) and (b) show the mean-field
predictions. The dashed line includes the next-order LHY correction. For this data, hni is typically
5.8 x1012 cm−3 , with (na3 )1/2 reaching a maximum of 0.043. We linearly scale the points to accountfor ∼10% variation in density. The final-state effects shift the solid line from a parabola centeredabout a = 0 in (a) to one centered about a0 = −565 a0 in (b), which enhances the raw signal at
small a.

56
a three-body interaction parameter κ∗ [57, 92]
dE
2~2
=−
C3 .
dκ∗
mκ∗

(6.4)

Three-body short-range correlations contribute a predicted additional term to the RF tail at large
detunings that should be added to the right-hand side of Eqn. 6.3 [57]:~Ω2 GRF (ω)
C3 .
2m
ω2

(6.5)

Here, GRF (ω) is a log-periodic function rooted in Efimov physics:
GRF (ω) = 9.23 − 13.6 sin[s0 ln(m|ω|/~κ2∗ ) + 2.66].

(6.6)

Efimov physics predicts an infinite series of successively more weakly bound trimers whose binding
2 2

energies at unitarity (a → ∞) are given by ~ mκ∗ (e−2π/s0 )l , where l is an integer and s0 is 1.00624
for identical bosons [96]. We note that there is as yet no prediction for final-state effects on the C3
contribution to the RF tail.
In order to determine κ∗ for 85 Rb atoms, we performed measurements of loss rates as afunction of a. With these measurements, we locate an Efimov resonance, which is shown in figure
6.3. For these measurements, we make non-condensed clouds of 1.5 × 105 atoms at a temperature
T = 80 nK. After ramping the magnetic field to realize the desired a on the a < 0 side of the
Feshbach resonance, we use absorption imaging to measure the number of atoms and cloud size as
a function of hold time. We then extract the three-body event rate constant K3 , which is defineddby dt
N = −3K3 hn2 iN when all three atoms are lost per event. In extracting K3 , we assume that

all of the measured loss is due to three-body processes and we account for the observed heating of
the gas, which causes additional decrease in n in time. We fit the measured K3 vs a to the expected
form for an Efimov resonance for non-condensed atoms [96],
K3 =

~a4
4590 sinh(2η),
sin2 [s0 ln(a/a− )] + sinh2 η m(6.7)

where η parameterizes the decay rate into deeply bound molecules and the resonance location,a− , is related to κ∗ through a− = 1.56(5)κ−1∗ [96]. Because this expression comes from a T = 0

57

cm /s)

10

0.1

3

K (10

-22

6

1

a = 1/kthermal

0.01
-10

3

scattering length-104a (a )
0

Figure 6.3: A three-body loss resonance for 85 Rb. We plot the three-body event constant K3 vs a.
From fitting Eqn. 6.7 to the solid points, for which a < 1/kthermal , we extract a− = −759(6)a0 andη = 0.057(2).

58
theory, we only fit the data for a < 1/kthermal , where kthermal =√2mkB T /~ and kB is Boltzmann’sconstant. From the fit we find κ∗ =39(1) µm−1 .
To see how the three-body parameter might impact the many-body physics, we plot the
expected frequency dependence of GRF (ω) in figure 6.4a. Note that GRF (ω) has a node at |ω| ∼2π × 27 kHz and a smaller magnitude at larger |ω|. Eqn. 6.5 has a frequency dependence given
by GRF (ω)/ω 2 , which suggests that the largest contribution from C3 will be for smaller |ω|. The
prediction for the C3 term (Eqn. 6.5), like the C2 term (Eqn. 6.3), is valid for ω → ∞. For the caseof the C2 term, the RF tail arises from two-body short-range correlations at distances that are small
compared to the interparticle spacing, which requires |ω|  ~n2/3 /m. For our typical experimental
parameters, ~n2/3 /m ∼ 2π × 1 kHz and this requirement is always satisfied. However, for the case~of C3 , the prediction for the C3 tail contribution to the RF tail may only be applicable for |ω| > ma2[97], where the frequency dependence makes it more difficult to observe this contribution to the RF
tail.
The results of our search for C3 can be seen in figure 6.4b, where we examine the frequency
dependence of the RF tail for a BEC at a = 982 ± 10 a0 . Residual magnetic-field gradients broaden
the central feature in the RF spectrum, and this limits our data for the tail to |ω| ≥ 2π × 10 kHz.
We fit the data to the predicted frequency dependence of the C2 contribution, shown by the solidline. We can see that our data fit very well to the expected frequency-dependence for the two-bodycontact with final-state effects, and we do not observe any deviation consistent with a three-bodyterm. Fitting the data to both contributions gives an upper limit for C3 /N0 of 0.07 µm−2 .
In the regime of perturbative interactions, such as assumed in the LHY calculation, one would
expect that the short-range correlations in the BEC are dominated by two-body effects. This isconsistent with our measurements, where no clear signature of three-body effects is seen in thefrequency dependence of the interaction-induced tail in RF spectroscopy. In general, this paves
the way for using RF spectroscopy to measure the two-body contact for BECs and thus measure
beyond-mean-field physics.
A further result of our studies is the location of the 85 Rb Efimov resonance. When a− is

59

G

20

RF

0

-6

10

S

(a)

(b)

-7

10

-8

10

-100

-10
/2

(kHz)

Figure 6.4: (a) The frequency dependence of GRF (ω), given our measured value for κ∗ . (b) Frequency dependence of the tail of the RF spectrum for a = 982 ± 10 a0 . The solid line is a fit of thedata (•) to the expected frequency dependence of the two-body contact C2 /N0 including final-state
effects. The dotted line corresponds to the same value of C2 /N0 , but ignores final-state effects.
For comparison, the fit plus a trial C3 /N0 term of 0.1 µm−2 is shown with the dashed line. Our
measurements are consistent instead with a C3 /N0 of zero. Here hni = 1.0 x1013 cm−3 .

60expressed in units of the mean scattering length of the van der Waals potential [98] for 85 Rb (78.5
a0 ), we find a value of -9.67(7) [99], which is very similar to reported results for 133 Cs (for multiple
Feshbach resonances) [100] and for 7 Li [91]. This adds to the empirical evidence suggesting that thethree-body parameter depends only on the coefficient of the 1/r6 part of the two-body potential
and not on the details of a three-body potential at short range [100]. In the many-body physics ofan interacting BEC, three-body correlations may yet play a significant role. For example, it will
be interesting to look for three-body effects on BECs with strong interactions (at unitarity in thenext section), or at a = a− . We discuss in the next section measurements of the contact for a BEC
at unitarity using the RF techniques developed here.

6.2

If at first you don’t measure C3 ...
In the previous section we presented measurements at finite na3 of the value of the two-bodycontact, but only set an upper limit for the three-body contact. The idea then was to increase theinteractions even further to look for a three-body contact. Rather than do so in incremental steps,
we tuned the magnetic field to unitarity, where a → ∞ 2 . This section describes our preliminary
results in measuring the three-body contact for a BEC at unitarity. The contact can be thoughtof as a measure of the chance that either two or three particles are very close to each other. The
two-body contact, C2 , is proportional to the probability that two atoms are near each other andC3 is proportional to the probability of three atoms being close. Intuitively, the BEC must havea non-zero three-body contact. Since atoms in the BEC undergo inelastic losses, we know C3 isnon-zero, since three atoms must be close to each other during an inelastic loss process. Thus, one
way to probe the contact is by measuring inelastic collisions [57]. Moreover, since three-body losses
generally increase with increasing scattering length [23, 22], it stands to reason C3 also increases
with scattering length. One then might expect the largest C3 at unitarity, which is presumably thelocation of maximum loss.
One caveat to the data presented here is that these measurements were taken before we
2

These measurements inspired our later work characterizing the degenerate unitarity Bose gas.

61characterized the unitarity Bose gas (chapter 7) and before we developed faster ramps to unitarity(chapter 3). The magnetic-field ramps to unitarity for this work were much slower (taking approximately 100 µs) than the ramps we used to investigate the unitarity Bose gas (which took 5µs). For a BEC at unitarity, we observed dynamics of the unitarity Bose gas on a time scale of
10s of µs, which is shorter than the magnetic-field ramps used for the measurements in this chapter. Therefore, the results of this section may be dependent on the specifics of the magnetic-fieldramps. Nevertheless, our preliminary measurement of the three-body contact is interesting and any
perturbation to this result due to the specifics of the ramps to unitarity would be an interestingmeasurement of non-equilibrium many-body dynamics of a BEC.
Like for C2 , we first consider the adiabatic sweep theorem (shown previously as equation 6.4)
to predict C3 , which is
dE
2~2
=−
C3 ,
dκ∗ s,amκ∗(6.8)

where E is the total energy of the gas and κ∗ is an Efimov three-body parameter [57]. Equation
6.8 relates C3 to how much the total energy of the BEC changes when one adiabatically changes
κ∗ . However, unlike the scattering length, κ∗ is not a tunable parameter in our system. Instead,
2 2

the parameter κ∗ is related to the binding energy of the Efimov trimer at a = ∞ through Et = ~ mκ∗[96]. Also, the dependence of the energy density of a BEC on κ∗ has not been predicted and thus
we are unable to use it to predict C3 by applying the adiabatic sweep theorem. Thus, even if we
could tune κ∗ it is unclear how the energy of a BEC would change. Nonetheless, even without aprediction, we can measure C3 using two different methods: RF spectroscopy (which we call contact
spectroscopy) and by measurements of three-body rates.
Before discussing our measurements, we can apply the adiabatic sweep theorem to is a gasof dilute Efimov trimers at a = ∞ to provide context for values of the contact. We can use theadiabatic sweep theorem (equation 6.8 and equation 6.1) to calculate both the two-body and threebody contact, since the trimer energy is a function of both the scattering length and κ∗ . The energy

62of an Efimov trimer near unitarity is [96]~2 κ2∗
Et ≈ −
m



2.11
1+
κ∗ a

(6.9)

Applying the adiabatic sweep theorem (equation 6.8 and 6.4) we calculate C2 /N and C3 /N to beC2
2.11 × 8π
= κ∗
N
3
2
C3
κ
= ∗
N
3

(6.10)(6.11)

We can calculate the expected two- and three-body contact for a for a gas of 85 Rb Efimov
trimers. For the lowest energy Efimov trimer κ∗ = 39 µm−1 , which then gives C2 /N = 689 µm−1
and C3 /N = 507 µm−2 . Presumably, the three-body contact in a gas of Efimov trimers, where all
the atoms are bound into triplets, close to each other, gives a sort of upper limit of the three-body(1)

contact in a Bose gas. The next higher energy trimer has a κ∗ = κ∗ /22.7 = 1.7 µm−1 , which thengives C2 /N = 30 µm−1 and C3 /N = 1 µm−2 .

6.3

RF contact spectroscopy
The technique for measuring the contact at unitarity using contact spectroscopy is essentiallyunchanged from that for the lower na3 measurements: we take an RF lineshape, driving a transition
between two atomic Zeeman levels, which in our case are split by approximately 80 MHz. In theabsence of a contact, the lineshape should be symmetric. However, including the contact leads toan asymmetric lineshape, with a tail extending to large detunings on one side, as shown in figure6.5. When we add at C3 term to equation 6.3 it becomes [57]" r#
1
~ 1 α(a)GRF (ω)~2lim Γ(ω) = Ω
C2 +
C3 ,
ω→∞
4π m ω 3/2 β(ω)
2mω 2(6.12)

where, as before, ω is the detuning from the Zeeman transition frequency, Ω is Rabi frequency, andR∞Γ is normalized by the sum rule −∞ Γ(ω)dω = πΩ2 N .
The three-body contact, C3 , then gives rise to a deviation from the |ω|−3/2 two-body contact
tail. We use the G(w)|ω|−2 deviation to search for a three-body contact signal. In general, the63signal at smaller detunings of ω is where the C3 signal will be the largest compared to C2 . Figure
6.5 illustrates this by showing the bare RF transition (as a delta function at ω = 0), the broadenedlineshape due to Fourier width of the RF pulse (green), and the difference between a C2 signal(black) and a signal that includes C2 and C3 (dashed blue).

6.3.1

Experimental sequence

As was the case for measuring the contact at finite scatting length, we started with a weaklyinteracting BEC and ramped the magnetic field to change the scattering length. The magnetic fieldmust be stable during the RF pulse since both the scattering length and the transition frequency aremagnetic-field dependent. In addition, we needed to cancel any magnetic-field gradients (primarily
the vertical gradient that supports the cloud against gravity) that would otherwise cause broadeningof the RF lineshape. Also, any time variations of the magnetic field during the RF pulse broaden
the lineshape. Minimizing these technical effects is important for looking for a C3 contribution at
relatively small ω. We plot the magnetic-field ramp used to measure C3 in figure 6.6, which shows
the transient magnetic-field contribution from the trap bias coils (blue line) and the measured
magnetic field (green circles), following a sweep that started from an initial value of ∼ 164 G. The
set point for the bias coil is changed at -2.4 ms and the current through the magnetic-field coils
settles in approximately 100 µs (this measurement was performed before we had implemented coils
capable of driving 5 µs ramps). The discrepancy between the field contributed from the coils (line)
and the measured field (points) is due to eddy currents that decay on a 100 µs timescale. We show
the power envelope of the truncated gaussian RF pulse in red.
In addition to the magnetic-field stability, we also have to consider and account for density
and number losses during the magnetic-field ramp and during the RF pulse. The time we have toprobe the BEC at unitarity is much shorter than when we probed the BEC at weak interactions.
Three-body losses and the global expansion of the cloud (“breathe”) causes the number and density
to decay on a timescale similar to our ramp time. The curve in figure 6.6 shows the fastest magneticfield change our system allowed at the time; this ramp was chosen to avoid, as much as possible,

64

100

(Hz)

10

1

C

+

C

3

C

0.1

0.01
-50

2

-40

-30

-20

2

-10

-0

(kHz)
Figure 6.5: A schematic of the excitation rate, Γ, of the atoms as a function of frequency in contact
spectroscopy. The black arrow indicates the location of the atomic transition. The green curve isthe Fourier broadened lineshape for a 40 µs gaussian pulse. The black and blue curves show thetail at large detunings ω for C3 = 0 and for finite C3 , respectively.

65
density and number loss.
Then, to measure the contact, we apply a ±4σ truncated-gaussian RF pulse with an rms
width of σ = 40 µs. We apply the pulse as soon as the field was somewhat stable. The magnetic
field has for the most part stopped changing where the power envelope of the gaussian pulse is in
figure 6.6. The RF pulse spin flips atoms from the F=2, mf =-2 state to the F=2, mf =-1 state. We
selectively image the spin-flipped atoms after further transferring them to the F=3, mf =-3 imaging
spin state. For reference, the experimental timings are listed in table 6.1.
Table 6.1: Experimental timings for contact spectroscopy measuring the three-body contact at
unitarity
experimental step
ramp to cancel vertical magnetic-field gradient
jump the magnetic field
gaussian RF pulse (±4σ)
Adiabatic rapid passage to |3 − 2i state
RF pulse to |3 − 3i state
absorption imaging
trap off

6.3.2

start time
-6.7 ms-2.4 ms-2.31 ms
-1.95 ms
-1.8 ms-1.7 ms
0 ms

end time-4.8 ms≈ -2.3 ms-1.99 ms
-1.85 ms
-1.72 ms
-1.675 ms
0 ms

Results of contact spectroscopy

Figure 6.7 shows the contact spectroscopy measurement at various detunings. We plot a
curve with a C2 contribution and a curve including both a C2 and a C3 contribution. Clearly, thedata is not consistent with a single C2 /|ω|3/2 curve. The additional signal at smaller detunings
could be due to a finite C3 term. Fitting the data to only a C2 curve we find C2 = 51(10) µm−1 .
If we include a C3 term we find C2 = 53(10) µm−1 and C3 /N = 6(1) µm−2 .
However, there are a number of potential problems with our contact spectroscopy measurement. First, as shown in figure 6.6, the magnetic field is still changing when we start the RF pulse,which is 90 µs after the jump to unitarity. Before doing contact spectroscopy we first find the66

Magnetic Field (G)

155.2
155.1
155
154.9
154.8
154.7
154.6
−2.4

−2.3

−2.2

−2.1

−2

−1.9

−1.8

t (ms)
Figure 6.6: The magnetic-field ramp to unitarity. The field set point is changed at -2.4 ms withrespect to the trap turning off at 0 ms. The solid blue line is the calculated magnetic field fromthe measured current in coils. The green points are measurements of the magnetic field using RF
spectroscopy of Zeeman levels. The red filled guassian is the power envelope of the RF pulse, which
is proportional to the Rabi frequency squared, Ω(t)2 . The ±4 σ gaussian envelope of the electricfield starts at -2.31 ms and ends at -1.99 ms. This magnetic-field ramp was done before the fast-B
coils were added to the experiment.

67

2

/N (

s)

10

1

C /

3/2

2

C /

+C /

2

3

3/2

2

0.1
-50

-40

-30

-20

-10

0

(kHz)
Figure 6.7: The large detuning portion of a RF spectroscopy lineshape for unitarity interactions.
The x-axis corresponds to the detuning from the atomic transition. The y-axis is proportional to
the signal, which is the coupling rate to the new spin state. The deviation from the C2 line (black)may be due to a C3 signal.

68transition frequency by taking an RF lineshape. We use a ± 4σ truncated-gaussian RF pulse with
σ = 40 µs to find the center frequency. We don’t observe any broadening of this initial RF lineshape. However a shifting magnetic field, that may not necessarily broaden the gaussian lineshape,
could cause an asymmetries in the wings of the lineshape, which could appear as a C3 signal. Second, the detuning from the transition frequency is much smaller for measuring C3 compared to theC2 measurement. At these closer detunings the contact signal may be significantly contaminatedby the broadened atomic lineshape. We can measure approximately how much the contact signal
is contaminated by measuring the excitation rate on the opposite side of the contact tail. For thecontact spectroscopy data with detuning ω = 40 kHz the amount of signal due to contaminationfrom the Zeeman lineshape is 10−2 times the contact signal. For the data detuned by ω ≈ 10
kHz, the additional signal from Zeeman transition is 0.3 times the measured signal. Further careful
investigation needs to be done to rule these systematics out of the C3 signal we measure.

6.4

C3 from loss measurement
A complementary method to using RF spectroscopy to measure C3 is using the three-bodyinelastic loss rate. As we stated before, three-body loss is evidence that C3 must have a finite value.
The relationship between C3 and Ṅ is [101]
Ṅ =

−12η~
C3 ,
s0 m

(6.13)

where η is the same unitless inelastic three-body Efimov parameter from before.
To measure the loss rate, we take in situ images of the cloud after a certain hold time at
unitarity. Like for contact spectroscopy, we ramped the magnetic field to unitarity in approximately
100 µs when measuring the loss rate. Figure 6.8 shows the number of BEC atoms as a functionof time at unitarity. Fitting the loss to a line we extract C3 /N = 1.0(2) µm−2 . Described in thenext chapter, after implementing our faster ramps, we measured the exponential decay timescaleof a BEC (figure 7.2), which for a hni = 5.5 × 1012 cm−3 BEC was 630 µs. This corresponds to
C3 /N = 3.1(6) µm−2

69

40

N/10

3

30

20

10

0
0.00

0.25

0.50

0.75

time (ms)Figure 6.8: An in-situ measurement of loss from a BEC at unitarity. The x-axis is the time after
ramping to unitarity and the y-axis is the number of 85 Rb atoms in the BEC. The magnetic-fieldramp started at approximately -0.1 ms and took 0.1 ms. The black points are measurement of
the number and the red line is linear fit to the data, which has a slope of -19(2) µs−1 . Using
this initial slope, we extract C3 =1.0(2) µm−2 . For this data, the average density of the BEC ishni = 4.7 × 1012 cm−3

6.5

Discussion and Outlook
We attempted to measure C3 using two different methods. The measured three-body contact,

C3 /N = 6(1) µm−2 , using contact spectroscopy is larger than the contact measured using threebody loss, C3 /N = 1.0(2) µm−2 and C3 /N = 3.2(6) µm−2 . These values are close to the expected
three-body contact for a gas of the Efimov trimers in the second energy branch, C3 /N = 1 µm−2 .
The value of C2 /N derived from RF spectroscopy, 52 µm−1 , may be compared to the prediction forthe shallow Efimov trimer of 30 µm−1 These preliminary results suggests that further measurements

70of the three-body contact would be fruitful and perhaps connect the unitarity Bose gas to Efimov
physics.3
Perhaps most interestingly, since the duration of the magnetic-field ramps was approximately
the same timescale as the dynamics of the unitarity Bose (chapter 7) it is unclear how the measured
contact is affected by ramp speed. Measuring the contact as a function of magnetic-field ramp speed
would shed light on this question. In addition, our ability to now change the magnetic field much
faster could allows us to measure the contact at various times. Observing a time-varying C2 and C3
would illuminate how two-body and three-body correlations evolve in the BEC at unitarity. Future
experimental investigations of C2 and C3 seem very promising for the capable hands of future 85 Rb
experimenters.

3

It’s low hanging fruit!

Chapter 7
What happens to a BEC at unitarity? 1

This chapter describes our recent results of projecting a BEC onto unitarity and observing theensuing dynamics of the gas. As chapter 4 discussed, our work with 85 Rb has pushed towards larger
and larger interaction strengths. However we’ve been foiled in the pursuit by the metastability ofa strongly interacting Bose gas. Overcoming these challenges, this work is the first study of adegenerate Bose gas with na3 >> 1, where n is the atom number density and a is the two-bodyscattering length. Our experiment was technically possible after implementing the fast-B magnetic
field control system that I describe in chapter 3. I will describe the context of the unitarity Bosegas, our experimental sequence, and our results.

7.1

Introduction

7.1.1

Background

Understanding the rich behavior that emerges from systems of interacting quantum particles,
such as electrons in materials, nucleons in nuclei or neutron stars, the quark-gluon plasma, andsuperfluid liquid helium, requires investigation of systems that are clean, accessible, and havetunable parameters. Ultracold quantum gases offer tremendous promise for this application largely
due to an unprecedented control over interactions [5]. Specifically, a, the two-body scattering lengththat characterizes the interaction strength, can be tuned to any value. This offers prospects forexperimental access to regimes where the behavior is not well understood because interactions are
1

Much of this chapter has been submitted for publication[102]72
strong, atom-atom correlations are important, mean-field theory is inadequate, and equilibrium
may not be reached or perhaps does not even exist. Of particular interest is the unitary gas, wherea is infinite, and where many aspects of the system are universal in that they depend only on theparticle density and quantum statistics. While the unitary Fermi gas has been the subject of intense
experimental and theoretical investigation [103, 104, 105, 106, 107, 108, 109], the degenerate unitary
Bose gas [110, 111, 112, 113, 20, 21, 114, 115] has generally been deemed experimentally inaccessible
because of three-body loss rates that increase dramatically with increasing a. Here, we investigate
dynamics of a unitary Bose gas for timescales that are short compared to the loss. We find that the
momentum distribution of the unitary Bose gas evolves on timescales fast compared to losses, andthat both the timescale for this evolution and the limiting shape of the momentum distribution are
consistent with universal scaling with density. This work demonstrates that a unitary Bose gas canbe created and probed dynamically, and thus opens the door for further exploration of this novel
strongly interacting quantum liquid.
As I showed in chapter 2, a powerful feature of atom gas experiments that provides access tothese new regimes is the ability to change the interaction strength using a magnetic-field Feshbachresonance. In particular, at the resonance location, a is infinite. For atomic Fermi gases, accessing
this regime by adiabatically changing a led to the achievement of superfluids of paired fermions andenabled investigation of the crossover from superfluidity of weakly bound pairs, analogous to theBardeen-Cooper-Schrieffer (BCS) theory of superconductors, to Bose-Einstein condensation (BEC)of tightly bound molecules [105, 106]. For bosonic atoms, however, this route to strong interactions
is stymied by the fact that three-body inelastic collisions increase as a4 [23, 22]. This circumstancehas limited experimental investigation of Bose gases with increasing interaction strength to studying
either non-quantum degenerate-thermal gases [62, 63] or BECs with modest interaction strengths(na3 < 0.008) [15, 17, 19, 61].
The problem is that the loss rate scales as n2 a4 while the equilibration rate scales as na2 v,
where v is the average velocity. Thus, it would seem that the losses will always dominate as |a|is increased to ∞. Even if we were to forsake equilibrium and suddenly change a in order to73
project a weakly interacting BEC onto strong interactions [116, 117, 61, 118], one might expectthat three-body losses would still dominate the ensuing dynamics for large a. In this work, however,
we use this approach to project a BEC “instantaneously” onto the unitary gas regime. We observe
dynamics that in fact saturate on a timescale shorter than that set by three-body losses and thatexhibit universal scaling with density.

7.1.2

Universality

One of the intriguing aspects of the unitary gas is that since a diverges, it can no longer
be a physically relevant scale for describing the system and its behavior. For a gas near zerotemperature, such as a BEC, the only physical scale that remains at unitarity is the interparticle
spacing. (In principle, the size of the cloud, or, equivalently the trap parameters, can provide alength scale, although one that is not intrinsic to the system. In addition, we are ignoring here any
explicit three-body interactions, which could provide an additional length scale.) The gas behavior
should then be universal in the sense that it is characterized only by the density n. This means thatenergies scale as n2/3 , momenta as n1/3 , and times as n−2/3 , which we parameterize respectively
by En ≡ ~2 (6π 2 n)2/3 /2m, kn ≡ (6π 2 n)1/3 , and tn ≡ ~/En .
The universality that makes the unitary gas so remarkable also provides a reason to hope that
rapid three-body loss will not necessarily be an insurmountable barrier to experimental explorationof bulk (as opposed to lattice-confined) degenerate Bose gases with unitarity-limited interactions.
For the degenerate unitary Bose gas, both the loss rate and the equilibration rate must scale as n2/3 .
The comparison of the two rates then hinges on unknown numerical prefactors, and it becomes anexperimental question whether losses dominate or a local equilibrium can be reached. In addition,
we note that on resonance, the shallow bound state that exists for finite positive a disappears,
shown in figure 2.2, so that loss requires atoms to decay to deeply bound molecular states [119].
For 85 Rb atoms, the previous experimental observation (figure 6.3) of a relatively narrow, andtherefore long-lived, Efimov resonance (characterized by a dimensionless width, η ∼ .06  1) [19]is indicative that atoms close together do not decay instantaneously to deeply bound molecular

74
states.

7.2

Experimental sequence
Our experiments begin with a 85 Rb BEC with 6 × 104 atoms confined in a 10 Hz sphericalmagnetic trap. The magnetic field, B, is set 8 G above the 85 Rb Feshbach resonance at B0 = 155.04
G [10]. This sets the initial a to 150 a0 , which gives the BEC a Thomas-Fermi density distribution
with an average density hni = 5.5 × 1012 cm−3 . With a typical initial temperature < 10 nK, thethermal deBroglie wavelength is large compared to hni−1/3 and is not a relevant length scale in
the physics of the ensuing experiment. Starting with this BEC in the extremely dilute limit, withhnia3 < 10−5 , we then decrease B to B0 in 5 µs. During the final 3 µs of the ramp of B, hnia3
goes from an essentially dilute value of 10−4 to hnia3  1.
After allowing the cloud to evolve at unitarity for a time t, we measure the momentum
distribution of atoms by ramping, equally rapidly, back to small a and allowing the gas to expand
ballistically before imaging the cloud using resonant, high-intensity absorption imaging [36]. Weshow the experimental sequence in figure 7.1 From an azimuthal average of the image, we extract
a momentum-space column density ñ as a function of the component of momentum perpendicular
to the line of sight, k̃. By imaging at various times of flight (7, 13, 25 ms), we increase the dynamic
range of our data and reduce the region of k̃ that is obscured by initial-size effects. We repeat thisexperimental procedure for various t to explore the evolution of the momentum distribution as afunction of time at unitarity.

7.2.1

Methods

Magnetic-field control: To rapidly change the magnetic field, we use an additional pair ofcoils, each with 10 turns and a diameter of 1.0 cm, 2.8 cm apart. The step response of the system
has a 10–90% rise time of 2.1 µs; thus, the 5 µs magnetic-field sweep used in the measurements is
below the maximum bandwidth of the system. We characterize and pre-correct for induced currents
from mutual inductances between these coils and the magnetic trap coils as well as eddy currents

75

t
time

Figure 7.1: The experimental sequence of projecting a gas onto unitarity interactions. The magnetic
field is ramped to and from unitarity in 5 µs. We turn the trap off and image the atoms after atime-of-flight expansion.

in surrounding conductors. Taking into account roughly equal contributions from uncertainty in
our magnetic field and the uncertainty in the Feshbach resonance location B0 [10], we estimate that
our experiments are within ± 50 mG of the Feshbach resonance, which corresponds to |a| > 95, 000
a0 .
Loss rate at unitarity: Using the initial loss rate implied by the exponential fit to the dataRshown in figure 1, and using dN/dt = −L3 n(r)3 d3 r, we extract L3 = 5(1) × 10−23 cm6 /s.
Unitarity-limited three-body loss rates for a non-degenerate Bose gas have been recently investigated by Rem et al. [62]. Using Eqn. 5 from Rem et al. [62] and the Efimov resonance width, η,from Wild et al. [19] , the predicted L3 for 85 Rb atoms at a temperature of 10 nK is 3×10−20 cm6 /s,
which is two and a half orders of magnitude larger than what we measure. On the other hand, after
the jump to unitarity, universality suggests that we should use an effective temperature determinedby the interparticle spacing. Using En /kB = 135 nK, where kB is Boltzmann’s constant, gives an

76

N

/10

4

6
4
2
0

0

100
t (

200

300

s)

Figure 7.2: Number of atoms measured using absorption imaging as a function of the time at
unitarity. The number measured without ramping to unitarity is shown at t = 0. The solid line
shows an exponential fit to the data (points), which gives a time constant of 630 ± 30 µs.

estimate for L3 of 1.7 × 10−22 cm6 /s, which is within a factor of 4 of our measurement. For the
low hni data, L3 is a factor of 6.2(5) larger than for the high hni data. Using the unitarity limit
for Tn , we would expect this ratio to be 5.2(6).
Momentum distributions: For the time-of-flight expansion, the 10 Hz spherical magnetic trap
is turned off over 2 ms, while keeping the magnitude of the total magnetic field constant. Because
the trap turns off in a time that is much shorter than the trap period, it has a negligible effecton the momenta of the atoms. We image the atoms using a 5 µs imaging pulse. The directionof the imaging beam and the magnetic-field direction are shown in figure 5. For each hold timeat unitarity, we repeat the experiment four times for each of three different times of flight, texp :25 ms, 13 ms, and 7 ms. Each image is azimuthally averaged, and the curves for the same TOF

77
are averaged together. We then combine the averaged curves into a single momentum distribution,ñ(k̃), using the largest texp data at the smallest k̃ and the smallest texp data at the largest k̃. This
minimizes the initial-size effects at small k̃, while improving the signal-to-noise ratio at larger k̃.
In combining the curves, we enforce agreement in the overlap regions by applying a multiplicative
factor to the data for shorter texp . This additional scaling factor, which ranges from 1.07 to 1.26 forthe texp =13 ms data and from 1.5 to 2.1 for the texp =7 ms data, reflects systematic uncertainties
that become increasingly important as ñ(k̃) decreases by orders of magnitudes.
At small k̃, the measured ñ(k̃) is distorted by the initial size of the BEC (the Thomas-Fermi
radius is 16 µm for the higher hni data and 22 µm for the lower hni data) and by our imaging
resolution (characterized by a gaussian width of approximately 6 µm). The gray regions in figure
2, and the corresponding regions where the data are shown as dashed lines in figure 3, are bounded
by a radius of 58 µm in the expanded cloud image. In the absence of the jump to unitarity, theBEC with hni = 5.5(3) × 1012 cm−3 has 97% of the atoms within this radius after an expansion
time of 25 ms. We note that all the effects discussed here cause low-momentum atoms to appear atlarger radii than one would expect from the product of velocity and texp . Therefore, integrating thesignal up to a particular momentum gives a lower limit to the number of atoms that have momenta
below that value. We use this fact in extracting a lower bound for the density in phase space.

7.3

Conclusions and arising questions from the results
From images of the expanded cloud, we also obtain the number of atoms, N , which we showin figure 7.2 as a function of t. Fitting an exponential decay to this early time data yields a time
constant of 630 ± 30 µs. A fact that is immediately clear from this data is that the number loss
at unitarity occurs on a timescale that is much longer than the few µs duration of our ramps onto
and away from the Feshbach resonance.
For understanding the dynamics at unitarity, it is also important to know the timescale
for changes in the spatial extent of the condensate. Specifically, exploring the possibility of local
equilibration only makes sense while the density of the gas remains largely unchanged. The spherical

78
aspect ratio of the trap was chosen to optimize the duration of inertial confinement. A priori, it
is not obvious whether the BEC will expand or collapse after the jump to unitarity. With in-situ
images of the gas at unitarity, we find the cloud size remains unchanged to within our measurement
precision for ∼500 µs and then slowly increases. The measured change in the spatial volume of thecondensate is (6 ± 9)% during the first 500 µs at unitarity.
Equipped with this information regarding the timescales for number loss and for expansionof the trapped gas at unitarity, we now consider the measured momentum distributions. These
are shown in figure 7.3 for various t, with the inset showing the same data on a log-linear plot.
Given the finite times of flight before imaging, the data at small k̃ are strongly affected by the
initial size of the BEC and do not accurately reflect ñ(k̃); the gray regions in figure 7.3 indicate
where initial-size effects are non-negligible, and we see that a significant fraction of the signal lies
within this region. Nevertheless, the data clearly show the emergence of signal at high k̃, outside
the gray regions. The signal at high k̃ grows as a function of t before saturating in approximately
100 µs. In this time, the gas has not yet lost a significant number of atoms or significantly reduced
its density. The fact that the timescale for the shape of ñ(k̃) to stop evolving is very different thanthe loss timescale clearly points to a mechanism for this dynamics that is distinct from three-bodyloss. Furthermore, the much shorter timescale for saturation of ñ(k̃) suggests the existence of a“quasi-equilibrium” metastable state of a degenerate Bose gas at unitarity.
To look for evidence of universality, we repeated the measurements for a lower initial densityof the BEC. Here, we begin the experimental cycle by increasing a to 400 a0 for 20 ms, increasing
the cloud radius and reducing hni to 1.6 × 1012 cm−3 . We measure a loss timescale for the lower
density data shown in figure 7.6. Following the jump to unitarity, the measured ñ(k̃) for lower initial
hni also shows the emergence of signal at high k̃ at unitarity, see figure 7.5. The distributions are
similar to those measured for the higher hni (figure 7.3), except that the dynamics occur over alonger time scale, with ñ(k̃) saturating in approximately 200 µs. We extract the three-dimensionaln(k) using an inverse Abel transform. To compare the two saturated curves, in figure 7.4, we show
the saturated momentum distributions as a function of the scaled momentum, κ = k/kn , where kn

794

t = 0 µst = 15 µs
t = 35 µs
t = 100 µs
t = 170 µs

6ñ(k̃)(106 µm2 )

10
3

5

10

4

10
2

3

10
1

0

0
0

5

5

10

10

15

15

20

k̃ (µm−1 )Figure 7.3: The column-integrated momentum distribution ñ(k̃) versus the transverse momentumk̃ after evolving at unitarity for time t. The
R distribution measured without ramping to unitarity
is shown at t = 0. For each t, the integral ñ(k̃)2π k̃dk̃ = 8π 3 N (t). For this data hni = 5.5(3) ×1012 cm−3 , which corresponds to kn = 6.9 µm−1 . Each momentum distribution is obtained fromseveral images for each of three expansion times (7, 13, and 25 ms). The inset shows the samedata plotted on log-linear axes. The gray regions indicate the part of data that is contaminatedby initial-size effects and, therefore, does not accurately reflect the momentum distribution. We
observe the emergence of signal outside this region, and a saturation of ñ(k̃) for t >100 µs.

80

4

n( )

10

3

10

4

10

5

n( )

2

10

0
0

1

2

1

10

0

10

-1

10

0

1

2

3

Figure 7.4: The momentum distribution, n(κ), plotted versus the scaled momentum, κ. Data for
hni = 5.5 × 1012 cm−3 and hni = 1.6 × 1012 cm−3 are shown as the gray and cyan lines, respectively.
The higher hni data is the average of measurements for 6 hold times t between 100 µs and 300 µs,
while the lower hni data is the averagefor t between 200 µs and 700 µs. The
R of 4 measurements
2
3
distributions are normalized so that n(κ)4πκ dκ = 8π . The data for two different densities are
consistent with a single curve when plotted in scaled units. Inset: Plotting κ4 n(κ) for high hni, we
do not find clear evidence for a 1/κ4 tail at high κ.

81
is calculated at the average density hni. We find that the shape of the distributions for the two hni
are very similar.
Given that our data are consistent with a universal shape for the saturated n(κ) at high κ,
we now discuss aspects of this distribution. First, we note that although much of the signal remains
at small κ where our data are affected by initial-size effects, the population with κ > 0.5 for thesaturated n(κ) is nearly 50% of the initial N . Particularly, it is suggestive that saturated density
distribution appears to have a low momentum peak, much like a BEC. Second, for short-range
interactions, such as those that give rise to the s-wave scattering length for atoms, one expects
a 1/κ4 tail at high momentum for an equilibrium gas, where the amplitude of this tail is thethermodynamic parameter known as the contact [55]. We do not find evidence for a 1/κ4 tail athigh momentum, which would appear as a flat line for large κ in figure 7.3(inset); however, a 1/κ4
tail may exist below our detection limit at large κ where the signal-to-noise ratio is poor.
Finally, we consider the low-κ part of the momentum distribution and the question of whether
or not the gas remains degenerate after the rapid sweep to unitarity. At low κ, initial-size effects
can play a non-negligible role. However, this effect is such that we can obtain a lower limit on thefraction of atoms that have κ < κmax by integrating our n(κ) data up to κmax . This allows us to
extract a lower limit for the density of atoms in phase space. Specifically, we calculate the averageoccupancy per state at low κ, which is given by the number of atoms divided by the number ofstates in phase space:Nhρocc i =  3
8π

κZmax


2

n(κ)4πκ dκ /




V 4π
3 3
(~kn ) κmax ,
h3 3

(7.1)

0
3
where we conservatively use for the effective coordinate-space volume, V = 4π
3 RTF , where RTF is

the Thomas-Fermi radius of the initial weakly interacting BEC. For the higher hni data, where theeffects of the initial size are smaller, choosing for example κmax = 0.26 gives 23% of the atoms andhρocc i = 7.1 for t = 170 µs = 3 tn . The fact that this lower limit for the density in phase space ismuch larger than 1 for a significant fraction of the atoms indicates that the gas is degenerate.
In addition to considering the saturated n(κ), we present the observed timescale for the82

4

t = 0 µst = 33 µs
t = 51 µs
t = 221 µs
t = 310 µs

6ñ(k̃)(106 µm2 )

10
3

5

10

4

10
2

3

10
1

0

0
0

5

5

10

10

15

15

20

k̃ (µm−1 )Figure 7.5: The column-integrated momentum distribution ñ(k̃) versus the transverse momentumk̃ after evolving at unitarity for time t. The
R distribution measured without ramping to unitarity
is shown at t = 0. For each t, the integral ñ(k̃)2π k̃dk̃ = 8π 3 N (t). For this data hni = 1.6(1) ×1012 cm−3 , which corresponds to kn = 4.5 µm−1 . Each momentum distribution is obtained fromseveral images for each of three expansion times (7, 13, and 25 ms). The inset shows the samedata plotted on log-linear axes. The gray regions indicate the part of data that is contaminatedby initial-size effects and, therefore, does not accurately reflect the momentum distribution. We
observe the emergence of signal outside this region, and a saturation of ñ(k̃) for t >200 µs.

83

N

/10

4

6
4
2
0

0

200

400
t (

600

s)

Figure 7.6: Number of atoms measured using absorption imaging as a function of the time at
unitarity for our low density data. For this data hni = 1.6(1) × 1012 cm−3 . The number measured
without ramping to unitarity is shown at t = 0. The solid line shows an exponential fit to the data
(points), which gives a time constant of 1200 ± 100 µs.

dynamics in figures 7.7, 7.8 and 7.9, we examine the timescales for the dynamics that give rise to
the saturated n(κ). As can be seen in figure 7.3, the evolution of the momentum distribution is
not uniform, with the higher momentum population saturating earlier. In figure 7.7, we plot the
number of atoms, ∆N , within a specific momentum range as a function of t, for two different rangesof momentum κ. We also plot the timescales for the same κ and different hni in figure 7.8. In
each case, we find that the number of atoms within the specific momentum range grows and thensaturates. We find that the timescale for this saturation increases for smaller κ, and for smallerhni. We fit the data for each different κ range to an exponential and extract a time constant τ .
To look for universality in the timescales, we normalize τ by the characteristic time set

84

2500
2000

N

1500
1000
500
0
0

k = 1.2 - 1.32
k = 0.81 - 0.89

100

200

300

t ( s)
Figure 7.7: The number of atoms in two momentum ranges vs. t. Data is for hni = 5.5×1012 cm−3 .
The black circles are the fraction of atoms with κ between 1.20 and 1.32. The cyan trianges are thefraction of atoms with κ between 0.81 and 0.89. The lines show fits of the data to ∆N0 (1−exp−t/τ ),from which we extract the timescale for saturation, τ .

by the interparticle spacing, tn , where tn is 57 µs and 130 µs for the data at higher and lower
hni, respectively. Plotting the normalized τ /tn vs. κ, we find that the momentum-dependent
dynamics at our two different densities are consistent (figure 7.9). We conclude that the timescale
for n(κ) dynamics is universal in that it depends only on the density, or interparticle spacing. The
momentum dependence of the timescales remains to be understood, although it is perhaps not
unexpected that higher momenta saturate faster.
In conclusion, we have projected initially weakly interacting BECs onto unitarity-limited
interactions and measured the resulting momentum-space dynamics. Key findings of this work are
as follows: (1) The momentum distribution of the unitary gas evolves and then saturates on a

85

2500
2000

N

1500
1000
500
0
0

<n> = 5.5 10<n> = 1.6 10

100

200

12

12

cm
cm

-3

-3

300

t ( s)
Figure 7.8: The number of atoms in two momentum ranges vs. t. Data is for κ between 0.81
and 0.89. The black circles are the fraction of atoms with hni = 5.5 × 1012 cm−3 . The cyan
triangles are the fraction of atoms with hni = 1.6 × 1012 cm−3 . The lines show fits of the data to∆N0 (1 − exp−t/τ ), from which we extract the timescale for saturation, τ .

timescale that is significantly shorter that the timescale for three-body loss. (2) Both the shape ofthe saturated momentum distribution and the timescale for the dynamics appear to be universal. (3)Intriguingly, the saturated momentum distribution fits well to the distribution expected for an ideal
degenerate Bose-Einstein gas with fugacity z = 1. (4) The low-momentum part of the momentum
distribution indicates that the density of atoms in phase space exceeds 1, and, hence, the gas
is degenerate. These findings support the conclusion that the gas reaches a locally equilibrated,metastable state and open the door for experimental investigation of a degenerate unitary Bose gas– something that was previously considered inaccessible.
This work raises some interesting questions: To what extent can the gas locally be described

86

3

/ t

n

2
1
0
0

1

2

3

Figure 7.9: The time constant associated with the emergence of signal at high momentum plotted
as a function of scaled momentum, κ. Data for hni = 5.5 × 1012 cm−3 are shown with solid circles,
while data for hni = 1.6 × 1012 cm−3 are shown with open circles.

87
by a temperature, and is this temperature below the critical temperature for a Bose gas withunitarity-limited short-range interactions? What is this critical temperature in units of the criticaltemperature for an ideal Bose gas? At high momentum, what is expected for the contact, and doesa high-momentum tail whose amplitude corresponds to the contact exist beyond the range of ourdata, or below our detection limit? Finally, what does the observed momentum dependence of the
dynamics tell us about the evolution of the system at unitarity?

Chapter 8

Future Directions

8.1

Looking backThis thesis is my story of how our experimental studies of a 85 Rb BEC progressed fromstudying the mean-field regime na3 ≈ 10−6 , to measuring first-order beyond-mean-field physics(na3 ≈ 0.002), to the first studies of a BEC in a truly non-perturbative strongly interacting regime(na3 > 1). Our studies of mean-field and beyond-mean-field effects used Bragg spectroscopy andcontact spectroscopy to probe the gas. At unitarity, we tried to answer a few basic questions:
What is the lifetime of the gas? Does an equilibrium exists? Is the gas degenerate? In addition to
tackling these basic questions, we measured the contact to probe the gas. The results of this work
have proven to be very interesting.
Our studies of a BEC at unitarity have produced a number of surprising results. Most
surprising is that a degenerate gas at unitarity exists at all. Prior to these studies, an equilibrium
degenerate Bose gas was thought to be impossible due to three-body loss. Instead, the gas appears
to come to an equilibrium, or steady state, faster than atoms are lost. This surprising fact has
inspired a number of theoretical works describing a degenerate unitarity Bose gas [120, 121, 122].

8.2

One more thing: How we project onto unitarity
Studying a gas at unitarity has opened up a number of interesting questions for our lab topursue. The fast-B coils (described in chapter 3) allow us to control the interactions in a BEC fasterand more precisely than we could before. We used the coils to project the BEC onto unitarity and89back to small a, but slower ramps are also possible. I will present “one more thing” regarding ourstudies of a BEC and few open experimental results that don’t fit well into the previous chapters.
These studies shed light onto future directions.

8.2.1

The ramp onto unitarity

As discussed in chapter 4, when we project the BEC onto unitarity the 5 µs ramp time ismuch shorter compared to any BEC evolution timescale. However, the 5 µs change in magnetic
field is not faster than the 16 ns timescale associated with molecule binding energy at the startof the ramp. The change in interactions is diabatic at the end of the ramp since the moleculebinding energy vanishes (in a two-body zero-density picture). Thus, at some point, the change ininteractions changes from adiabatic to diabatic with respect to the two-body energy scale set by
the Feshbach molecule binding energy. To find this transition point, we consider an adiabaticity
parameter aȧ / Emol
~ . This adiabaticity parameter for a 5 µs ramp in shown in figure 8.1.
For various ramp speeds, the point at which the change in interactions becomes diabatic isdifferent. Faster ramps become diabatic at smaller scattering lengths than slower ramps. If thistransition point affects how the BEC at unitarity projects onto unitarity interactions, then thesaturated momentum distributions could be a function of ramp’s speed. The question is: does the
speed of the ramp to unitarity affect the momentum distribution? Figure 8.2 shows the saturated
momentum distribution for ramps of 5 µs, 10 µs, and 20 µs. We observe little if any effect of thedifferent ramp speeds. This suggests that the saturated momentum distributions are universal as
we concluded in chapter 7. Additional studies of the effect of varying the ramp speeds to and fromunitarity is an exciting future direction of the experiment.

8.2.2

Loss of atoms, but no molecules.

While exploring the ramps to and from unitarity, we observed another interesting phenomena: a loss of atoms with slower ramps away from unitarity after the momentum distribution has
saturated. This is not a total surprise, as many experiments observe losses when they slowly ramp

90

HaL

Field HGL

162

160

158

156

-2

0

2

4

6

time HΜsL

HbLa Emol

a Ñ

10

4

100

1

-2

0

2

4

6

time HΜsL
Figure 8.1: (a) The magnetic-field magnitude. The ramp is a linear ramp from 163 G to 155.04 G
in 5 µs. (b) The adiabaticity parameter for ramp in (a). We plot time on the x-axis and ȧa / Emol~ȧ Emolon the y-axis. When aȧ / Emol,thechangeinaisadiabatic.
When
/

1
the
change
in
a
is
~
a
~
diabatic.

across the Feshbach resonance [5]. Sweeping the magnetic field slowly across the resonance tracks
free atoms into Feshbach dimers, which why this is a common technique used to make dimers.
Therefore, the lost atoms should appear as Feshbach molecules. We measured the population of91

OD x 1000

104

5

s

10

s

20

s

103

102

101

0

1

2

~

3

k k

n

Figure 8.2: The column-integrated momentum distribution ñ(κ) versus the transverse momentum
k/k̃. We ramp to unitarity in either 5 µs, 10 µs, or 20 µs. The cloud is held for 100 µs andthen ramped back to ∼ 163 G in 5 µs. The azimuthal averaged optical depth is after a time-offlight expansion of 7 ms. We start with 6 × 104 atoms in the BEC and an average density of,
hni = 5.2 × 1012 cm−3 .

Feshbach dimers using RF spectroscopy. Figure 8.3 shows the magnetic-field ramps to and fromunitarity, which we ramp to unitarity from 163 G in 5 µs, hold for 150 µs, and ramp to 160.35 G ineither 5 or 150 µs. We ramp back to 160.35 G so any Feshbach molecules that exist in our sample
have 1 ms lifetime [123], much longer than time taken to image the sample. We apply an RF pulse
to selectively spin flip either the atoms or Feshbach molecules into imaging state. Figure 8.4 showsan RF spectroscopy measurement for the 5 µs ramp to unitarity and a 150 µs return ramp fromunitarity. The large peak in the lineshape is the signal from unbound atoms; the broader signal
with a larger frequency is due to Feshbach molecules. While the number of free atoms decreases

92
after a slow ramp from unitarity, curiously, the signal in the molecule tail doesn’t grow with thisloss of atoms. One possible explanation is that the lost atoms are converted into Efimov trimers.
A macroscopic population of Efimov trimers in the unitarity BEC would suggest that three-bodyinteractions are very important at unitarity. However, further studies are needed to verify thispotential explanation.

150 s
time

Figure 8.3: The experimental sequence of projecting a BEC onto unitarity interactions and searching for molecules. We ramp the magnetic field onto unitarity in 5 µs, hold for 150 µs, ramp to ascattering length of 450 a0 in either 5 µs or 150 µs, and then use absorption imaging to measure
the number of atoms.

8.3

Low hanging fruit
The previous ideas are two out of many different possibilities for the experiment to explore.

The quasi-equilibrium of the na3 > 1 degenerate Bose gas has opened a new regime to probe stronglyinteracting many-body physics. Further probes of the unitarity degenerate Bose gas would shedlight on the system. In addition to this new regime, the fast-B system allows us to control theinteractions with unprecedented speed and precision. Thus, we are excellently poised to study thetime dynamics of interactions in BECs. Measuring the time-resolved contact is one possibility of

93

35k
30k

N85

25k
20k
15k
10k
5k
0
0.0

0.5

1.0

1.5

2.0

RF Freq (+2671 MHz)
Figure 8.4: An RF spectrum of the cloud after ramps to unitarity and back. We use a 7 µs squareenveloped RF pulse on the F=2 mF =-2 to F=3 mF =-3 Zeeman transition, which is the cyclingtransition. We perform absorption imaging directly after the RF pulse. The magenta line is fora 5 µs ramp back while the black line is for a 150 µs ramp back. The solid line is the expectedtransition probability for free atoms exposed to a pulse with a square envelope. The dotted lines
are predicted response for a population of Feshbach molecules. Only the strength of the molecule
response is fit since the position of the curve is fixed relative to the single atom peak by theory.

time dynamics in a BEC.

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Appendix A

DIY guide to measuring the contact

A.1

Steps to measure the contact
During the experiment we measure the number of atoms after an RF pulse. However, thecontact is defined as√lim Γ(ω) = Ω2

ω→∞

~1/2
GRF (ω)~C2 + Ω 2
C3
√
3/2
2mω 2
4π mω

(A.1)

To calculate the contact from the number of atoms we measure using a specific RF power, detuning
ω, and pulse duration. From equation A.1 it’s clear to calculate the C2 and C3 we need Γ(ω)/Ω2 .
We measure the contact in a number of steps, shown below.

Step 1) - Find the center frequency
We take an RF lineshape to find the center transition frequency, which corresponds to a detuningω = 0. Additionally, we check that the lineshape is narrow in frequency space so it doesn’t contaminate the contact signal, where specifically for our experiment the lineshapes would have a gaussian
width less than 10 kHz.

Step 2) - Take data on the peak
We lower the RF power and measure the number on the peak of the lineshape. The data points on
the peak of the lineshape, where ω = 0, is a measure of the Rabi frequency. We historically useda truncated ± 4σ (in electric field) with a τpeak = 5 µs long Gaussian pulse, which gives a Fourier

103
limited 22.5 kHz wide gaussian lineshape. The RF pulse should outcouple 1-2 % of the total number in the gas, which we denote as Npeak , since we observe non-linearities if we transfer more atoms.

Step 3) - Take data on the tailWe then measure the number of atoms outcoupled on the tail of the lineshape. The data points
on the tail of the lineshape, detuned by a frequency ω, is a measure of the excitation rate for thecontact, Γ. We measure Ntail by outcoupling 1-2% of the gas. Historically we used a gaussian pulse
with τtail is 100 µs and 40 µs pulse for contact at finite na3 and contact at unitarity, respectively.

Step 4)
Lastly, we calculate the contact using the formula below.
1
Ntail
√
dB
Γ(ω)
τtail π 10 tail /10
=
2
Npeak
1
1
Ωtail
2
2
dB
Ntotal 2π τpeak 10 tail /10
Ntail√
τtail πNpeak
1Ω2peak =2
2
Ntotal 2π τpeakΓ(ω) =Ω2tail = Ω2peak × 10(dBtail −dBpeak )/10

A.2

(A.2)

(A.3)
(A.4)
(A.5)

Calculate the contact

To calculate the contact we start with two formulas given by Braaten[57]
#
" r
1
~
1
α(a)
G
(ω)~
RF
lim Γ(ω) = Ω2
C2 +
C3
ω→∞
4π m ω 3/2 β(ω)
2mω 2
Z ∞
Γ(ω)dω = πΩ2 N

(A.6)
(A.7)

−∞

We calculate C2 and C3 from Γ(ω)/Ω2
Note that the number of outcoupled atoms for a square pulse is
Nout = Γ(ω) × ∆t

(A.8)

104
When we use a gaussian pulses the outcoupled number of atoms is
Z ∞

−t2

e τ 2 Γ(ω)

Nout =

(A.9)

−∞

√
=τ πΓ(ω)(A.10)√
Ntail = Γ(ω)τtail π(A.11)

Then the number on the tail is thenNow, to calculate the contact all need is Ω2tail is. We use the sum ruleZ ∞Γ(ω)dω = πΩ2 Ntotal(A.12)−∞

Since we’re Fourier limited we know the functional form of Γ(ω), which is
−ω 2
2σω2
2π
σω = √
2 2πτtailΓ(ω) = Γ0 e−(A.13)(A.14)and like before
Γ0 =

Npeak
√
τ pi

(A.15)

when we take the integral of equation A.12 we findNpeak √√
2πσω = πNtotal Ω2peakτ π(A.16)

Then, the Rabi frequency isΩ2peak =Ntail
1
2
Ntotal 2π 2 τpeak

No we have all the pieces to calculate Γ/Ω2 and can calculate the contact.

(A.17)