Contact Measurements on a Strongly Interacting Bose Gas
byRobert Johannes Wild
B.S. Physics, University of Arizona, 2004

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
2012This thesis entitled:
Contact Measurements on a Strongly Interacting Bose Gas
written by Robert Johannes Wild
has been approved for the Department of Physics

Dr. Eric A. Cornell

Dr. Deborah S. Jin

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
Wild, Robert Johannes (Ph.D., Physics)
Contact Measurements on a Strongly Interacting Bose Gas
Thesis directed by Dr. Eric A. Cornell

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 withdelta-function interactions. For bosons, the fact that contact spectroscopy can be used to probe thegas on short timescales is potentially useful given the decreasing stability of BECs with increasing
interactions. Successfully measuring the contact requires careful control of experimental parameterssuch as the magnetic ﬁeld and the RF probe pulse. In this thesis I report on measurements of thecontact, using RF spectroscopy, for an 85 Rb atomic Bose-Einstein condensate (BEC). The measured
contact exhibits beyond-mean-ﬁeld behavior, the degree of which is dependent on the rates of changeof the scattering length. A potential complication is the added possibility, for bosons, of three-bodyinteractions. In investigating this issue, we have located an Eﬁmov resonance for 85 Rb atoms withloss measurements and thus determined the three-body interaction parameter.

Acknowledgements

As with most academic undertakings, this thesis is a result of much collaboration. Even
though it is just my name on the cover, I could not have produced this work without all thescientiﬁc, technical, theoretical, emotional, and moral support that I received over the years. I
am deeply grateful for all of it, which has aﬀected not only this thesis but also the person I havebecome.
First and foremost I’d like to thank my two advisors, Eric Cornell and Debbie Jin. I am
constantly amazed by their physical insights and am grateful to have had the opportunity to workwith them. From them I learned how to be scientiﬁcally rigorous and to always question what I
think I know. My improved reasoning skills are the most valuable result of my work with them,and in spite of their scientiﬁc prowess, they always managed to be friendly and easy-going. I am
glad to have been able to build great personal relationships with both.
I would also like to thank my undergraduate advisor at the University of Arizona, Alex
Cronin, who showed me how to work in an “unstructured” lab and instilled conﬁdence in myabilities, without restricting my freedom to participate in my numerous extracurricular activities.
Thanks also to John Obrecht and Scott Papp, who taught me much about ultracold atom
experiments in my ﬁrst few years at CU. A special thanks goes to Juan Pino and Phil Makotyn,
with whom I spent the bulk of my time in the lab (in which I spent the bulk of my time in general).
Juan, Phil, and I worked together for several years, both in pairs and as a team of three, and wenow share that special bond that comes from sharing the frustrations and successes of experimental
physics and the many late nights in the lab. Our relationships consisted mainly of making fun of

veach other at every opportunity, often with two ganging up against the third, and I’m glad to have
had them to help make the work a bit more fun. Cathy Klauss joined just months before I left,but I am grateful for the overlap. She is not only intelligent, but quickly learned to hold her own
in the laboratory banter.
The tri/bi-group and all of its members over the years were also extremely helpful. They
were always available to answer those “dumb” questions you don’t want to ask your advisors, andI am glad to have become friends with so many of them.
I also want to thank the various undergrads with whom I worked. Rachel Umbel for beam
proﬁling work, Sophie Letournel for building a very good laser for the lab, Zach Newman for his
work on a crossed optical trap, and Chris Poulton for all the various excellent work he has alreadydone and continues to do in the lab.
Of course, JILA has been a fantastic environment to perform research, and I want to thankall the technical help I have received from the machine shop, the electronics shop (especially Carl
Sauer), the computing team, and the rest of the staﬀ that make the place run smoothly and allowed
me to focus on the experiment.
A special thanks to Brandon Peden, roommate for ﬁve years and great friend, with whom
I learned how to brew beer, make cheese, and enjoy scotch, and who philosophized with me overtopics ranging from math to women to everything in between, not being afraid to discuss absurd
hypothetical situations and all their ramiﬁcations.
I would also like to thank my family for all their support. Their unwavering belief in myabilities was an immense help throughout the years of my studies, especially when my own belief
was not quite as steadfast.
Finally, I would like to thank Colleen Cummings, who for the last ﬁve years has given meso much love and support, especially through those worst of times, that I am forever indebted. I
can only hope to amply return what she has given me, and am ever grateful to have her by myside.

Contents

Chapter
1 Introduction1

1.1

Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Strongly correlated systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3

Contact spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.4

Technical challenges and thesis contents . . . . . . . . . . . . . . . . . . . . . . . . .

72 Experimental Basics

92.1

85 Rb and its Feshbach resonance

2.2

The road to BEC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Magnetic Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

15

3.1

Gradient cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2

Magnetic-ﬁeld stabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.1

60 Hz AC line noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.2

Field monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.3

Magnetic-ﬁeld-induced atomic transitions . . . . . . . . . . . . . . . . . . . . 24

4 Radiofrequency pulses

25

4.1

Gaussian pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2

Shape checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.3

Rectiﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

vii
5 Detection
5.1

5.2

34

Condensate absorption imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.1.1

Imaging corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.1.2

BEC collapse and high-ﬁeld imaging . . . . . . . . . . . . . . . . . . . . . . . 38

5.1.3

High-intensity imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

In-trap imaging for contact measurements . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2.1

Transfer to the imaging state . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.2.2

Small cloud eﬀects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6 Two-body Contact

54

6.1

What is the contact? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.2

The contact signal and its characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.3

Extracting the contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.4

Contact measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

7 Three-body Contact

72

7.1

The Eﬁmov resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

7.2

Searching for C3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

8 Summary and Outlook

85

8.1

In Essence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

8.2

The next few weeks... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Bibliography

88

Appendix
A Eﬀective range contribution

92

viii
B Detailed procedure for the contact measurements

95

Figures

Figure1.1

Sketch of RF transition rate Γ. (a) Shows the ideal noninteracting spectrum, where
the whole population is resonant at a single frequency. (b) Correlations brought
about by interactions give rise to a ω −3/2 tail, shown by the red line, as well asa small shift of the resonance frequency. (c) The experimental broadening of thelineshape obscures this shift. Nearly all the atoms are within the area of the resonantlineshape, which gives us a measure of the single atom Rabi frequency. We use
diﬀerent methods to probe the resonant lineshape and the tail regimes, and the ﬁnal
contact measurement involves a ratio of these excitation rates. The magnitude of
the tail has been greatly exaggerated for illustrative purposes. . . . . . . . . . . . . .

2.1

6

The scattering length in units of the Bohr radius a0 as a function of magnetic ﬁeld.
We create our cold atom clouds on the high-ﬁeld side of the resonance to optimize
collisional properties. From there, we can easily tune the value of the scattering
length to either large positive or negative values. . . . . . . . . . . . . . . . . . . . . 11

2.2

A schematic of our evaporation. The optical trap ensures overlap between the 85 Rb(red points) and the 87 Rb (blue points), and is centered on the 85 Rb magnetic equilibrium position. Lowering its trap depth evaporates 87 Rb. The 87 Rb cools the 85 Rb
via collisions until all the 87 Rb has fallen out, and only 85 Rb remains in a purely
magnetic trap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

x
3.1

Our scheme to create a uniform ﬁeld. We start with a BEC in a purely magnetic
trap, whose center is shifted upward by a set of large shim coils. We then quickly
ramp down the coils, causing the minimum of the magnetic ﬁeld to move to thelocation of the atoms. A small shim coil ﬁne tunes this location. . . . . . . . . . . . 17

3.2

Measuring the magnetic-ﬁeld gradient. We create a large thermal cloud (pink circle
with dashed outline) and perform RF spectroscopy on a magnetically sensitive transition (in this case the |2, −2⟩ to |2, −1⟩ transition). As long as the spectral widthof the pulse is smaller than the cloud, diﬀerent parts of the cloud are resonant at
diﬀerent ﬁelds, and will outcouple at diﬀerent positions in space. The resulting slope
can be converted to a magnetic-ﬁeld gradient. . . . . . . . . . . . . . . . . . . . . . . 18

3.3

Correlating noise to the 60 Hz AC line. Part (a) shows how we correlated the noise
to the AC line. (a) Taking data on the side of a transition and using the slope allowed us to convert a scatter in signal to a scatter in B-ﬁeld. Correlating this scatterto the phase of the AC line at the time we applied the RF pulse shows a modulation in agreement with the 60 Hz line. A histogram (c) of the scatter reveals thecharacteristic double-peaked shape of sinusoidally distributed values. This structure
essentially disappears in (d), where we have synchronized the experiment to the ACline 1 second before the B-ﬁeld measurement. The bin size and axis range of (c) and(d) are identical. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4

The B-ﬁeld resonance giving rise to 2-photon transitions. Part (a) shows Fouriertransforms of oscilloscope traces taken with a hand held pickup coil close to the
trap. The blue curve shows the problem-causing 40 kHz feature. Changing the gainof the B-ﬁeld servos allowed us to eliminate the feature, as illustrated by the green
curve. A schematic of the 2-photon transition is shown in (b). . . . . . . . . . . . . . 24

xi
4.1

Some RF pulses in frequency space. In (a) we see the power spectrum of two standard
pulses (inset). A square pulse in time becomes a Sinc2 function, which looks much
like a Gaussian down to the ﬁrst minimum, but then exhibits signiﬁcant wings far
from the center (red curve). A Gaussian in time, however, becomes a Gaussian in
power (black curve). The curves have been scaled to give similar peaks and widths.
In (b) we see the same curves on a logarithmic scale, which makes the diﬀerence
clear. An ω −3/2 line has been added (blue), to show that the problematic wings ofa Sinc2 function persist with detuning. . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2

The Fourier transform of the RF power at system saturation. The labels correspondto the output power of the synthesizer before all ampliﬁcation. At 9 dBm output
we begin to risk damage to the ampliﬁers. The dashed black line shows a perfect
Gaussian for reference (a parabola on this scale). We see that saturation does not
aﬀect the nicely Gaussian shape of the pulse down to at least 60 dB below the peak
power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3

The Fourier transform of truncated Gaussians. The inset shows the shape of thetruncated Gaussians, at ±1 (blue), 2 (green), and 4 τ (black). The Fourier transformof the power shows the resulting frequency spectrum. The ±4τ truncation looks
unaﬀected, whereas the ±1τ truncation clearly begins to look similar to a squarepulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.4

Atoms spin-ﬂipped by truncated Gaussians. A truncated pulse is less eﬃcient attransferring atoms between magnetic sublevels. The pulse still exhibits lower efﬁciency at 2τ , but is fully Gaussian at 4τ . The theory line is scaled to the ﬁnal
value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

xii
4.5

Eﬀect of RF rectiﬁcation. (a) We send out a τ = 100 µs Gaussian pulse, and probe
the resulting ﬁeld with a 50 µs square pulse. The 100 µs delay gives the servo time
to respond and shift the ﬁeld. (b) At our maximum allowable power (8 dBm at thesynthesizer), we see rectiﬁcation amounting to 17 mG. At 8 dB lower power (c), therectiﬁcation has all but disappeared (d). Our contact measurements required powers
no more than -9 dBm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.1

A measurement of our eﬀective saturation intensity. We measure the OD vs Ii in
counts per pixel, and then invert the axes for plotting and ﬁtting via Equation
5.3. The red line shows this ﬁt. The measured OD is highly sensitive to the probe
intensity, especially at low intensities, where one might think that the correction due
to I/Isat is negligible.

5.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Comparison between measured and predicted BEC size. The diﬀerent shapes correspond to diﬀerent ARP eﬃciencies, giving rise to diﬀerent ODs. The red line is themodeled BEC size in expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.3

Eﬀect of photon re-absorption on OD. The black circles show the measured OD
of a BEC, calculated with Equation 5.6, as a function of I/I sat . For intensities
near I sat , the high scattering rate in conjunction with the high OD causes a large
fraction of atoms to re-absorb photons emitted from surrounding atoms. This makes
them unavailable to absorb from the probe beam, lowering the apparent OD. As theintensity increases, the scattering rate remains roughly constant, translating to asmaller fractional amount of rescattered light present in the cloud. The red squares
show the same experiment on thermal clouds with lower OD, where this eﬀect does
not occur.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

xiii
5.4

State diagram showing transfer to the imaging state. The atoms (shown in blue) startin the |2, −2⟩ state, and the RF pulse for contact spectroscopy moves a small fraction
to the |2, −1⟩ state. These are transferred via shaped ARP to the |3, −2⟩ state andRF π pulse to the |3, −3⟩ state (solid green arrows). Once there, the probe beam
cycles them to the F’=4 manifold (dotted green arrow). The entire procedure, fromthe |2, −1⟩ state to imaging, takes < 200 µs The transition of the cloud remainder
to the |3, −1⟩ state (red dashed arrow) is unwanted and would produce signiﬁcantfalse signal. The energy splittings are given in MHz, calculated for a ﬁeld of 160 G.

5.5

47

Shaped ARPs. A standard ARP is shown in part (a), where the RF power andtherefore the energies of the dressed states are constant in time. The frequency must
start far from resonance and end far from resonance for a well behaved ARP. A
shaped ARP (or ShARP) seen in (b) starts at low detuning and low power. The
power is adiabatically increased as the frequency ramps through resonance and thendecreased the same way. This allows for a quick, eﬃcient ARP, sweeping over a verysmall frequency range around the resonance.

5.6. . . . . . . . . . . . . . . . . . . . . . 48

Simulation of pixelation eﬀects. An absorption proﬁle is incident on a 2-dimensional
array of pixels, and the resulting pixelated OD, the sum over the pixles Σpx , is compared to the integral, IG , of the OD that gave rise to the absorption proﬁle. Part (a)shows the two conﬁgurations under which the simulation was run. In one conﬁguration, the Gaussian proﬁle was centered on the center of a pixel in both dimensions.
The corner conﬁguration formed the other “extreme”. Simulation results can beseen in part (b). When the Thomas-Fermi (TF) radius is roughly the size of a pixel,
the eﬀects can become very pronounced, even exhibiting unintuitive structure. The
error increases linearly with OD, but stays at a 3% or less correction for our normal
operating conditions of peak OD<1 and a TF radius of 2.4 pixels. . . . . . . . . . . 51

xiv
5.7

Schematic of the pixelsum technique. Shown is a standard image of a BEC with
30% thermal component, taken in the magnetic trap. For the pixelsum analysis, we
sum all the values of the inner square, and subtract the background as determinedby the “sidewalk” (hatched area). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.1

Schematic of the interparticle potential (a) and scattering wavefunction (b). The
potential V (r) looks like an attractive potential down to a distance given by the vander Waals potential. Outside of this eﬀective range re , which is ignored in the zerorange limit, the wavefunction can be universally described by the scattering lengtha. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.2

RF contact spectroscopy for a 85 Rb BEC. (a) Schematic of the contact transition. We
drive the interacting state into a lower magnetic sublevel, but the interaction-induced
energy requires less energy from the RF photon to eﬀect the spin-ﬂip. This results inthe transition frequency of the contact to be lower than the single atom transition.
∫∞(b) Example of spectroscopy signal S(ω), normalized so that −∞ S(ω)dω = 1 s−1 .
(c) The same signal as (b), shown on a smaller scale. The solid red line is a ﬁt to
the expected frequency dependence from Equation 6.6, while the dotted blue line
shows a ﬁt ignoring β(ω). The green line shows the expected Gaussian signal fromthe resonant lineshape. On the positive side of the transition the signal is consistent
with zero. (d) The tail signal multiplied by |ω|3/2 . This is how the ﬁts to the tailwere performed, and more clearly shows the ﬁnal-state eﬀects. The density hereis ⟨n⟩ = 5.8 × 1012 cm−3 . The data from the tail and main lineshape come from
diﬀerent experimental runs, but are shown together for illustration purposes. . . . . 59

xv
6.3

Expansion of the outcoupled atom cloud. (a) The size of the cloud of outcoupledatoms after the contact pulse with |ω| = 2π × 50 kHz detuning. The black line showsa sketch of a truncated Gaussian RF pulse for reference. The cloud expands with akinetic energy of 12 ~ω, as the excess energy is shared between two atoms. The redline is the predicted size due to the kinetic energy, added in quadrature with theresolution limit of our imaging system. (b) The peak OD of the clouds plotted inpart (a). The red line is a ﬁt to a w12 dependence (∼ t12 ), varying only an overallscaling factor. (c) The energy of the outcoupled cloud as a function of detuning,at 4.5 ms time-of-ﬂight expansion. We calculate the energy from the size of theoutcoupled cloud, accounting for the size of the cloud of non-spin-ﬂipped atoms at
ω = 0. The solid line is 21 |ω|
2π .

6.4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Deviation from signal linearity. On both the peak (a) and tail (b) of the transition,
we measure the outcoupled fraction fmeas of atoms vs RF pulse time and/or power,
and plot it vs the fraction expected fexp if the dependence were linear. We ﬁt this tothe simple saturation model of Equation 6.13 to extract a value for the asymptote
faysm and calculate the magnitude of the correction to the data. In (b) we combine
data from two diﬀerent scattering lengths for a larger range in signal. Individual ﬁts
to the two sets give the same result within the error bars. . . . . . . . . . . . . . . . 64

6.5α(a) C2
The contact vs a, measured at |ω| = 2π × 40 kHz. (a) Here I plot β(ω)
N0 , which is

directly proportional to the strength of the measured ω −3/2 RF tail. (b) The contact
C2
per particle N. The solid lines in (a) and (b) are the mean-ﬁeld predictions. The
0ﬁnal-state eﬀects shift what is a parabola centered about a = 0 in (b) to one centeredabout a′ = −565 a0 in (a), which enhances our signal at small a. . . . . . . . . . . . 66

xvi
6.6

Contact measurements as a function of the LHY energy ELHY . (a) The contact per
particle normalized by density1/3 . The red curve shows the mean-ﬁeld prediction forthe contact, and the blue curve shows the total contact including the LHY term.
In (b) the data are normalized to the mean-ﬁeld value of the contact to compare
to theory more clearly. In the limit of low interaction strength, the data match thetheory but cannot distinguish the LHY contribution. At higher values, the data
are systematically low. At values of ELHY approaching unity, we do not expect the
perturbative LHY theory to be valid. . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.7

Contact measurements as a function of ramp rates, expressed in terms of the “adiabaticity parameter” R. Higher values of this parameter correspond to faster ramps,
which show a value of the contact closer to the mean-ﬁeld prediction. Slower ramps
result in higher values closer to the LHY prediction. Diﬀerent shapes correspond to
diﬀerent days on which the data were taken. The small upper plot is a reproductionof Figure 6.6b, to illustrate the values of ELHY shown here. For lower interaction
strength, the LHY term is too small compared to the error bars to discern a ramp ratedependence. To calculate the mean-ﬁeld and LHY energies, we require knowledgeof the cloud density. The change in densities due to the ramp in a were calculatedusing a model for expansion that included only the expected mean-ﬁeld energy, andvary from 40% for the slowest ramps to 10% for the fastest. This reasoning is somewhat circular, but including the LHY term in this model causes at most an extra
5% decrease in density for the slowest ramps, and the resulting change is shown by
the open circles.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

xvii
7.1

√
A sketch of Eﬁmov states. The vertical scale gives the wave number K = ± m|E|/~2 ,and the horizontal gives 1/a, such that the origin corresponds to unitarity (a → ∞).
The green line shows the state corresponding to a bound dimer plus a free atom,and the blue lines show the bound trimers, for which there is an inﬁnite series approaching unitarity. Note that the scaling factor here is ∼ 2 to make the behavior
visible and to match the literature on Eﬁmov states. A realistic Eﬁmov state withscaling factor 22.7 would have an energy that is nearly indistinguishable from thedimer energy on the scale of this plot. . . . . . . . . . . . . . . . . . . . . . . . . . . 73

7.2

Example of a loss rate measurement to extract K3 . These data correspond to ascattering length of 730 a0 . Heating of the cloud causes its size to increase as seenin (a). The parameters extracted from a linear ﬁt to the volume go into a ﬁt to
the atom number via Equation 7.10, as seen in (b), resulting in a value of K3 =5.64 × 10−22 cm−6 /s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

7.3

A three-body loss resonance for 85 Rb. We plot the three-body event constant K3
vs the scattering length a, for clouds with a temperature of roughly 8 nK. Fromﬁtting Equation 7.11 to the black points, for which a < 1/kthermal , we extract a− =−759(6)a0 and η = 0.057(2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

7.4

Measurement of η vs temperature. (a) The loss resonances. For the data at 30 and
140 nK, we did not calibrate the scattering length as carefully as the data at 80 nK,
which is the same data as in Figure 7.3 shown over a smaller range of a. Moreover,the 30 nK clouds are likely not in thermal equilibrium, and the ensuing error in
the average density will cause a systematic error in the calculated value of K3 . (b)Measured values of eta vs temperature. We do not see a signiﬁcant change in η overa large range in temperature, suggesting that the experiments are performed in the
low-temperature limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

xviii
7.5

GRF as a function of ω, with κ∗ =39(1) µm−1 . We plot GRF for negative ω to ease
comparison with the measured RF tail, which occurs at negative detunings in our
experiment. GRF has a node at ω ≃ 2π × 27 kHz, suggesting that one should look
for a C3 contribution to the RF tail for smaller detunings. . . . . . . . . . . . . . . . 81

7.6

Changing the scattering length for increased density. We jump the scattering lengtha (green curve) to 50 a0 , causing an inward breathe with a period of roughly 50 ms.
Close to the turnaround point we ramp to 982 a0 and apply the RF pulse to measure
the contact (grey line). The blue curve shows the prediction for the relative densityof the PG model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7.7(a) The calculated frequency dependence of GRF (ω), shown again on a logarithmic
ω scale for reference. (b) The measured frequency dependence of the tail of the RF
spectrum for a = 982 ± 10 a0 . The solid red line is a ﬁt to the expected frequency
dependence of the two-body contact C2 /N0 including ﬁnal-state eﬀects. The dottedblue 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 dashedblack line. Our measurements are consistent instead with a C3 /N0 of zero. For thesedata the density is ⟨n⟩ = 1.0 × 1013 cm−3 . . . . . . . . . . . . . . . . . . . . . . . . . 82

7.8

Searching for a tail in the RF spectrum at -760 a0 . We detected no signal indicativeof a tail on the RF spectrum. The error bars give an estimate of our detection limit.
The green line shows the prediction for the contribution to the signal S from thetwo-body contact, which is highly suppressed due to ﬁnal-state eﬀects and expected
to be below our detection limit. A heuristic calculation from loss rates via Equation
7.13 predicts a three-body contribution shown by the blue line. The density here is⟨n⟩ ≃ 7 × 1011 cm−3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

xix
A.1 Eﬀects of the ﬁrst-order correction due to a ﬁnite eﬀective range. (a) and (b) show
the fractional corrections introduced by using α′ (a) and β ′ (ω), respectively. For the
calculation of α′ (a), a detuning of ω = 2π × 40 kHz was assumed. The magnitudeof the eﬀect roughly corresponds to the size of the error bars in our data, putting
it just below our detection limit. (c) The combined eﬀect on the total contact is on
the order of a few percent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
B.1 Timing schematic of a typical contact measurement. The vertical scale is arbitrary.
The widths of the grey RF pulses and the red probe pulse have been exaggerated toshow up at this scale. The relative timings and the width of the blue RF pulse areto scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Chapter 1

Introduction

1.1

Universality
Scientiﬁc advancement has historically been reductionist, wherein humanity has aspired to

explain things by simpler underlying principles. All the complex matter in the world was found to
consist of combinations of only a hundred or so diﬀerent elements. The elements were then found
not to be so diﬀerent after all, as they are merely made of diﬀerent combinations of the same threeparticles. And even some of these particles have been broken down to their constituents. However,
although this method of understanding the world has brought us extremely far, knowing everything
about a constituent part may tell us very little about the whole. For example, detailed knowledgeof a single water molecule does not readily lead to the prediction that a collection of water will
freeze at a certain temperature, much less that this solid form will actually be less dense than theliquid. Although determined by the constituent parts, such emergent behavior is too complex topredict only using knowledge of the parts. We cannot predict complex chemical behavior if weknow everything about the atoms, just as we cannot predict a person’s actions by knowing exactly
how a neuron works. Instead, we study the diﬀerent levels of complexity in their own right, withthe hope to sometime connect the emergent behavior with our knowledge of the constituent parts.
An example of this is liquid Helium. In 1908, Helium was ﬁnally cooled to such low temperatures as to liquefy it [1] (Helium has the lowest boiling point of all gases). The liquefaction
itself was expected, since all other gasses behaved similarly. But one can imagine the astonishmentthat emerged when its superﬂuid properties were discovered in 1938 [2, 3]. It behaved like no other2liquid, ﬂowing completely without friction and over the sides of any open container. Similarly, somemetals exhibit superconductivity when cooled enough, allowing currents to ﬂow through them without resistance (a superﬂuid of electrons, if you will). These quintessential examples of unexpected
emergent behavior have been subject to intense study, but trying to predict this behavior from detailed knowledge of the individual parts proves impractical at best. Instead, we attempt to explainthese systems using only a few “universal” properties.
The concept of universality is a powerful tool that allows one to study analogues of complex
and diverse systems in simpler, more controlled conditions, and to make theoretical predictions
based on the “most important” parameters. The basic idea behind universality is that many systems
can be completely described by a small number of parameters and do not depend strongly on details
speciﬁc to the constituents. For example, much of astronomical behavior can be explained using
the universal concept of gravity, where the universal parameter is the mass of an object. Knowing
only the masses of objects involved, we can very accurately predict the motions of heavenly bodies
regardless of their size, shape, and constitution. Of course, an object’s shape could aﬀect the pathof a very nearby mass, so one must make sure that predictions and experiments stay within relevant
limits of the universal regime. An atomic example of a universal parameter is the scattering length.
When two atoms interact via low energy s-wave collisions, the scattering length, a, describes
how strongly these atoms interact with each other (where a small magnitude scattering length
corresponds to weak interactions). The usefulness comes from the fact that details such as thetype of atom or particle, or the speciﬁc shape of the interatomic potential are irrelevant, and thesystem of atoms can be completely described by the scattering length. Liquid Helium is not fullyin this regime, since its density is so high that the details of the interparticle potential can notnecessarily be ignored completely. Nevertheless, much theory has been developed in the universal
framework to approach some of the universal physics involved. To realize these universal systems
and test the theories, we need low temperature dilute gasses, the diﬃculty of which has kept thescience on a mostly theoretical level through most of the twentieth century. In 1995, however,Bose-Einstein condensation was achieved in dilute gasses [4, 5, 6], making quantum mechanical3
many-body systems described by the universal parameter a experimentally accessible.

1.2

Strongly correlated systems
Weakly interacting dilute gases have been well understood in the framework of mean-ﬁeld

theory, where the interactions between particles are characterized simply by the mean-ﬁeld energy
gn, where n is the density of the atoms and g = 4π~a2m , m being the atomic mass [7]. This provides
a highly accurate description of the system in the dilute limit [8], which assumes na3 → 0. Forinteracting Bose gases, however, the interactions give rise to correlations that modify the energy.
Systems with strong quantum correlations, such as liquid Helium, represent a frontier in our understanding of the complex quantum systems found in nature. A ﬁrst step to understanding these
correlated systems was taken by Lee, Huang, and Yang in 1957, who found that the leading ordercorrection to the energy density of a BEC is given by [9, 10]
2π~2 an2E=mwhere the term proportional to()
128 √ 31+ √na + ... ,
15 π

(1.1)

√
na3 is often called the LHY correction.

However, it has proven diﬃcult to study strongly interacting atomic BEC, and only a few
groups have successfully measured beyond-mean-ﬁeld eﬀects in these system [11, 12, 13]. The maindiﬃculty is a fundamental one, as increasing a in order to increase the interaction strength brings
with it a rapidly increased rate of inelastic three-body collisions [14, 15]. In one of these collisions,
two atoms bind to form a molecule, and the third atom recoils with a kinetic energy equal to 2/3of the binding energy of the molecule for a single-species BEC. This results in the loss of all threeatoms from the BEC, as well as heating. The heating comes from collisions, as they leave the cloud,of the energetic particles with other BEC atoms, as well as the fact that loss is density dependent,
which causes more loss from colder parts of the cloud (anti-evaporation). Moreover, this three-bodyrecombination rate scales as a4 , so experiments on these systems either must stay at relatively low
interactions strength (done successfully by the groups around Salomon [13] and Hadzibabic [12]), ormust probe the system very quickly, before the losses become signiﬁcant. The latter is our strategy;

4
our measurement technique uses a short radio-frequency (RF) pulse to measure a quantity called
the contact. I will introduce the contact below, and a more thorough treatment will be given in
Chapter 6. Previously, our group has probed strongly interacting gasses using Bragg spectroscopy,
whereby an excitation is created with a two-photon transition [11]. The appeal of this method was
the same; it allowed for a fast probe of the system before three-body losses became signiﬁcant.
However, we were trying to resolve the extra energy provided by the LHY term, meaning that ourprobe pulses had to be long enough to provide the required energy resolution. The contact is not
an energy measurement, so this otherwise fundamental trade-oﬀ between probe time and energy
resolution is not an issue, as explained below.

1.3

Contact spectroscopy
Measuring the contact through RF spectroscopy is a way to quickly probe the BEC, and isbased on measuring short-range correlations in the gas. In 2005, Shina Tan introduced a parameter
termed the contact, which quantiﬁes the strength of short-range correlations in an ultracold gas,and showed how this quantity connects quantitatively to macroscopic thermodynamic propertiesof the many-body system via a set of universal relations [16, 17, 18]. Tan’s predictions have been
explored theoretically [19, 20, 21, 22, 23, 24] and experimentally [25, 26] for ultracold Fermi gases.
One of these relations pertains to its measurement technique. The contact manifests itself inthe tail of an RF spectrum, which can be thought of as due to RF “dissociation” of pairs of atoms
that are close to one another. The rate of transferring atoms to a non-interacting state in this tailis given by√Ω2 ~ α(a) C
lim Γ(ω) =
,
(1.2)
ω→∞
4π m β(ω) ω 3/2∫∞
where the integrated RF lineshape is −∞ Γ(ω)dω = πΩ2 N , ω is the detuning from the single-atom
resonance, and Ω is the single atom Rabi frequency [27]. Further discussion of this will follow in
Chapter 6, but the point here is that one can measure the outcoupled number of atoms on the tailof an RF lineshape as a measurement of the contact.

5
A sketch of how we actually measure the contact is shown in Figure 7.1. The frequency
spectrum of a single atom (or of a cloud of noninteracting atoms) looks like a delta function(Figure 7.1a), but interactions give rise to the tail in the frequency spectrum (Figure 7.1b), as
parameterized by the contact in Equation 1.2. The interactions also cause a slight shift in theresonant frequency, but to resolve this sub-kHz shift would require long interrogation times on theorder of milliseconds. In reality, of course, the single atom resonance is widened by things suchas inhomogeneous magnetic ﬁelds and ﬁnite probe times. This results in a spectrum more akin to
Figure 7.1c, where the tail has been greatly exaggerated compared to the resonant lineshape. We
use diﬀerent methods to probe the diﬀerent regimes of this spectrum. We use a short, spectrally
broad pulse to probe the resonant lineshape, which tells us the resonant frequency ω0 and gives us
a measure of the single atom Rabi frequency Ω. Longer, spectrally narrow pulses are used to probe
the tail of the transition, giving us the excitation rate Γ(ω).
Another powerful relation states that the contact, C, is connected to the derivative of thetotal energy of the system, E, with respect to a via the adiabatic sweep theorem [28, 29]:
~2
dE
=
C.
da
8πma2

(1.3)

With this, we can rewrite the LHY result in terms of the contact for a condensate as()
5 128 √ 3√
C = 16π na 1 +na + ... N0 ,2 15 π22(1.4)

where N0 is the number of atoms in the BEC. This enables us, for instance, to look for deviations
from the mean-ﬁeld prediction for the contact, where the lowest order correction comes from the
LHY term of Equation 1.1.
Since this method inherently measures an amplitude instead of a frequency, we can use very
short RF pulses to probe the system at timescales short compared to three-body loss rates, eventhough the spectral broadening of these pulses would be much too wide to measure the change inenergy (via Equation 1.1) directly. Of course we are not evading the uncertainty principle, and thismethod does not actually tell us what the energy is. Instead, it tells us how the energy changes

6

(a)

0

(b)

0

(c)
resonant lineshapetail0
Figure 1.1: Sketch of RF transition rate Γ. (a) Shows the ideal noninteracting spectrum, where thewhole population is resonant at a single frequency. (b) Correlations brought about by interactions
give rise to a ω −3/2 tail, shown by the red line, as well as a small shift of the resonance frequency. (c)The experimental broadening of the lineshape obscures this shift. Nearly all the atoms are within
the area of the resonant lineshape, which gives us a measure of the single atom Rabi frequency. We
use diﬀerent methods to probe the resonant lineshape and the tail regimes, and the ﬁnal contact
measurement involves a ratio of these excitation rates. The magnitude of the tail has been greatly
exaggerated for illustrative purposes.

7with scattering length (Equation 6.3). This provides a solution to overcome the major hurdle ofshort lifetimes caused by inelastic losses. Of course, that does not make the experiment easy, andthe technical requirements to make this approach work provide many smaller experimental hurdles.
Overcoming these hurdles required much work, and the resulting solutions form a large part of thisthesis.

1.4

Technical challenges and thesis contents
What are the experimental capabilities then, required to measure the contact? First, we need

to be able to create stable condensates, and we must be able to tune the interaction strength toaccess the mean-ﬁeld and LHY regimes. We want to do this quickly to minimize three-body losses
over the duration of the ramp. When we increase the scattering length on a timescale shorter than
the trap period, the BEC will be out of equilibrium. This starts a breathe mode and the density
initially drops very quickly along with our signal. Yet working at sub-millisecond times is diﬃcult
due to experimental constraints. For example, coil inductances and current-servo bandwidths oftencause magnetic ﬁelds to ring down before stabilizing. In order to alleviate that less fundamental
but still problematic issue, we need a weak spherically symmetric trap to maximize the density
oscillation period. These requirements were already met by our BEC machine before we started
thinking about the contact, as explained in detail in the thesis of Dr. Juan Pino [30]. I will merely
give an overview in Chapter 2, where I will also give a rundown of our cooling process.
To measure the contact via RF spectroscopy, we need the frequency width of our resonantlineshape to be smaller than the detuning at which we are measuring the tail. This requires us tominimize the large magnetic-ﬁeld gradients present in our magnetic trap. Also, since the magnitudeof the signal on the tail is a strong function of the RF detuning ω, we have strict requirements on
our trap magnetic-ﬁeld stability. These requirements and our solutions are discussed in detail in
Chapter 3.
The RF pulse must also meet certain requirements. The transition rate on the tail is muchsmaller than on the resonant lineshape, so great care must be taken to prevent any spectral “wings”

8
on the pulse from giving us a false signal. For this reason we require Gaussian shaped pulses, and
our RF system must in general be well behaved and not have adverse eﬀects on the rest of theexperiment. These issues are illuminated in Chapter 4.
Another important element is that of detection. Our weak, spherical trap has the eﬀect
that the condensate does not expand much when the trap is turned oﬀ, which causes two major
challenges for imaging. Even after 30 ms expansion, the optical depth (OD) is still too large to avoid
severe saturation problems with standard absorption imaging techniques. Also, the large, negative
background scattering length causes the BEC to collapse once the magnetic ﬁelds are turned oﬀ,
and the low expansion energy is not enough to overcome it. Aside from problems associated withimaging the condensate, we must also be careful about our detection schemes for the tail and forthe resonant lineshape, so that in the end we can claim an accuracy on the contact measurement towithin better than 10%. The imaging and detection techniques and systematics will be describedin Chapter 5.
With these technical issues properly delineated, I will talk about the contact in more detail.
I will illustrate its origins and limits, and will show data that exhibit beyond mean-ﬁeld behavior,as well as interesting time-dependent behavior in Chapter 6.
Recently, there has been made some theoretical advancement predicting a three-body contact
for bosons, related to Eﬁmov physics [31]. Chapter 7 will expand on this subject, show data on
Eﬁmov eﬀects in our system, and show recent developments on the three-body contact.

Chapter 2

Experimental Basics

In this chapter I will give an overview and general background of our experiment. This
experiment hinges mostly on the fact that we can produce a Bose-Einstein Condensate (BEC), andthat we can tune the interactions of the constituent atoms, which are characterized by the two-bodyscattering length a. I will describe the various steps and processes we utilize to give us that ability.
First, I will describe how we tune a via a Feshbach resonance. Then, I will describe our sympathetic
cooling techniques and various trapping schemes that result in a magnetically trapped, sphericallysymmetric BEC. This chapter is included mostly for the sake of completeness and many details
will be left out, as more complete descriptions can be found in the theses of Dr. Scott Papp [32]and Dr. Juan Pino [30].

2.1

85

Rb and its Feshbach resonance

Bose-Einstein Condensation is a hugely interesting phenomenon that has sparked countlessstudies of basic quantum mechanics (and since they’re so countless, I will refrain from ﬁlling thispage with references to back up my claim). One of the workhorses of these studies has been 87 Rb,
and by now its cooling and trapping properties are so well developed and documented that one
needs a good reason to work with a diﬀerent atom. This is especially the case for 85 Rb, which has
such unfortunate scattering properties that it is diﬃcult to cool to degeneracy [32]. Our reason,
then, for working with 85 Rb is that it has a Feshbach resonance at an experimentally accessible
magnetic-ﬁeld strength of 155 G.

10
A Feshbach resonance occurs when a two-body bound state is tuned to become degeneratewith the atomic scattering threshold. More in-depth explanations can be found here [33, 34], but
the point is that the two-body scattering length behaves as(a(B) = abg 1 −∆B − Bpeak)(2.1)

near the resonance, where for 85 Rb, abg = −443(3) a0 is the background scattering length inunits of the Bohr radius, ∆ = 10.71(2) G is the width of the resonance, and Bpeak = 155.041(18)G is its location in magnetic ﬁeld B, as measured in [35]. The shape of this resonance can be seenin Fig. 2.1. Thus, by tuning the magnetic ﬁeld at the position of the atom cloud, we can access ahuge range of interaction strengths, allowing us to study condensates beyond the mean-ﬁeld limit.

2.2

The road to BEC
By now, the production of BEC in alkali atoms alone no longer makes the headlines, andmethods for trapping and cooling are omitted from papers to make room for more interestingscientiﬁc results. Nevertheless, there can be interesting variations in the standard methods, andthe painful truth is that graduate students often spend disproportionately large amounts of timeon the optimization of these methods; this warrants a section devoted to the various trapping andcooling stages in our experiment. Again, this is just a quick overview, and more information canbe found in [32, 30].
Most of the cooling on 85 Rb happens via collisions with colder 87 Rb (this process is referred
to as sympathetic cooling), since cooling 85 Rb by RF evaporation directly has proven diﬃcult due
to its collisional properties. Thus, the 85 Rb and 87 Rb clouds follow much of the same path throughthe experiment. We start out with a dual-species magneto-optical trap (MOT) to initially cool 85 Rb
and 87 Rb from a room temperature gas. After the MOT stage, we load the atoms into a magnetic
trap and transfer the atoms to our science chamber, where the signiﬁcantly higher vacuum gives us
long lifetimes of ∼ 500 seconds. This transfer uses pairs of coils that turn on and oﬀ in succession,

110

scattering length (a )

4000

2000

0

-2000

-4000
140

150

160

170

B-field (G)Figure 2.1: The scattering length in units of the Bohr radius a0 as a function of magnetic ﬁeld. We
create our cold atom clouds on the high-ﬁeld side of the resonance to optimize collisional properties.
From there, we can easily tune the value of the scattering length to either large positive or negativevalues.

12
as well as a moving coil mounted on a track. In the science chamber we load the atoms into a IoﬀePritchard (IP) style magnetic trap, and perform standard RF evaporation on 87 Rb, thereby alsocooling the 85 Rb sympathetically. However, since the 85 Rb and 87 Rb atom have diﬀerent magnetic
moments (in the |2, −2⟩ and |1, −1⟩ states, respectively), the cloud centers are spatially separated
due to gravitational sag, and the sympathetic cooling will only be eﬃcient until the sizes of theclouds are on the order of their spatial separation. This occurs at ∼ 10 µK and 1 million 85 Rb
atoms, which is still far from degeneracy for a trap where ωx = 2π × 13 and ωy = ωz = 2π × 210
Hz, and our cooling procedure requires a change in strategy. At this point things become moreinteresting, as we have developed a hybrid optical and magnetic trap to continue the sympathetic
cooling process and end with a BEC of 85 Rb in a 10 Hz spherical trap.
As mentioned in the Introduction, we require a weak, spherical trap to give us more time toprobe our strongly interacting BEC. This trap has the eﬀect of increasing the separation between
the species’ equilibrium positions (800 µm), as well as lowering the intraspecies collision rate bynearly a factor of ten (given constant phase-space density). That would increase our evaporation
time accordingly, and slow the experiments down to unacceptable rates. To avoid these issues,
we change the currents in our various IP and shim coils to create our 10 Hz magnetic trap, andquickly turn on a single-beam optical trap (OT) with a 230 Hz radial conﬁning potential. The
atoms are now conﬁned by the OT, ensuring that the two species are in the same location andthat collision rates remain high. The tricky part here is to shim the magnetic trap ﬁelds in away such that the 85 Rb magnetically trapped equilibrium position is aligned with the optical trap
location. A graphical depiction of this trapping scheme can be seen in Figure 2.2. Then we simply
decrease the OT intensity down to zero at a rate slow compared the relevant collision rates. The
87 Rb experiences normal evaporation, with high energy atoms falling out of the optical trap (weapply RF to then transfer them to an untrapped state), and continuously cools the 85 Rb cloud,which stays centered on the OT/magnetic trap throughout the whole procedure. When the OT iscompletely oﬀ, all of the 87 Rb atoms have fallen out of the trap, and the 85 Rb exhibits a signiﬁcantcondensate fraction, which we purify with some RF evaporation at the end. This evaporation

13
scheme regularly results in 85 Rb condensates of 60,000 atoms, although we have seen condensates
upwards of 100,000 atoms.
This method proved to be an eﬀective and reliable scheme to create 85 Rb condensates forseveral reasons. For one, we can be certain that the atoms in our cloud are only 85 Rb and onlyin the |2, −2⟩ state, as the 87 Rb or other spin states are trapped at very diﬀerent positions. The
same can not be said for optical dipole traps, for which contamination from other atomic species
or spin states can be an issue. Also, since evaporation is performed almost entirely on 87 Rb, weonly need to load relatively few 85 Rb atoms into the MOT at the very beginning. This means thatthe 85 Rb density remains low up until the end of the cooling cycle, suppressing losses of 85 Rb due
to three-body recombination. Loading fewer 85 Rb atoms also means that it presents a smaller heatload for the coolant (87 Rb), which means that the same amount of initial 87 Rb can reach lower ﬁnal
temperatures. As long as we load just enough 85 Rb into the MOT to suppress 87 Rb condensation
at the end of the cycle, the overlap between the two species is guaranteed and we reliably make
85 Rb condensates.

Of course, all methods have their drawbacks. The weakest link in the evaporation is ensuringthat the OT is centered on the 85 Rb equilibrium position. If the two traps are misaligned, thenthe 85 Rb will slosh out of the OT, causing heating and ending the sympathetic cooling process. Toalign the traps, we use external coils to shim the magnetic trap position to coincide with the OT
position, which roughly coincides with the center of the IP trap. For the vertical direction, we use
a pair of coils to apply a uniform vertical ﬁeld of about 35 G, shifting the magnetic ﬁeld minimum
upward. For the horizontal direction perpendicular to the OT, we use a set of coils that produce a
5 G ﬁeld at the atoms along the OT, with a gradient perpendicular to the OT, producing a force
on the atoms. We tune the magnetic trap rather that the OT position, since in our setup we cancontrol the currents in our shim coils much more precisely than the mirror mounts of our optical
trap. Due to experimental drift, this alignment generally has to be done every few weeks. However,with the traps aligned, we often do not need to optimize our system in the morning to achieve ourstandard BEC conditions, owing to the general reliability of this cooling scheme.

14

Total Potentialzgravityyymagnetic field minimum

85

optical trap

Rb eq. position

87

Rb eq. position

Not to Scale

Figure 2.2: A schematic of our evaporation. The optical trap ensures overlap between the 85 Rb(red points) and the 87 Rb (blue points), and is centered on the 85 Rb magnetic equilibrium position.
Lowering its trap depth evaporates 87 Rb. The 87 Rb cools the 85 Rb via collisions until all the 87 Rb
has fallen out, and only 85 Rb remains in a purely magnetic trap.

Chapter 3

Magnetic Field

As mentioned in the Introduction, measurement of the contact via RF spectroscopy involves
transferring atoms from one magnetic sublevel to another, at frequencies detuned from the centerof the resonant lineshape (the single atom resonant frequency) by 10-100 kHz. We require ourline width to be much smaller than this detuning, since the transition rate for the tail can beseveral hundred times smaller than for the resonant lineshape. This puts a signiﬁcant limit on thewidth caused by magnetic-ﬁeld gradients, which cause diﬀerent parts of the cloud to be resonant at
diﬀerent frequencies. Also, because the contact signal on the tail is a strong function of detuning(C ∼ ω −3/2 ), it is crucial to know the resonant frequency to within < 1 kHz (a 1 kHz uncertainty
at 20 kHz detuning results in an 8% uncertainty in the contact). This means that our absolute
magnetic-ﬁeld noise must be less than 2 mG at 155 G, or ∼ 12 ppm.
In this chapter I will discuss these problems in two parts. First I will talk about our methodsof magnetic gradient cancellation to decrease the RF line width. In the second section I will reveal
our techniques for characterizing and stabilizing our magnetic ﬁelds, which will be of interest to
anyone with magnetic traps and stringent stability requirements.

3.1

Gradient cancellation
One of the problems we were faced with from the beginning was making the width of ourresonant lineshape small enough for contact measurements. We needed to probe the tail at low
detuning without outcoupling a signiﬁcant number of atoms due to this width. This problem came16
up time and again, and represents one of the main diﬃculties of RF contact spectroscopy.
One cause of this problem is fundamental to our trap. The fact that we have our BEC in apurely magnetic trap means that we have a strong magnetic-ﬁeld gradient in the vertical direction
to support the atoms against gravity. Speciﬁcally, since the atoms are all in the |F = 2, mf = −2⟩2state, the gradient comes to 2mg
3µB = 22.4 G/cm, where m is the mass and g = 9.8 m/s . For 50,000
G µB
atoms in our trap, the 14 µm Thomas Fermi radius [36] results roughly in a 14 µm · 22.4 cm· 3h ≃15

kHz rms width on our |2, −2⟩ to |2, −1⟩ transition. We quickly found out that this would not benearly narrow enough to measure a decent contact signal, so before we even worried about RF
pulse widths and ﬁeld noise and such, we had to ﬁnd a way to circumvent this “trapping gradient”
issue.
Our solution to this problem is in no way unique or even necessarily the best solution.
It just turns out that with our system it represented the quickest and easiest way to solve theproblem satisfactorily. In Chapter 2 I explained that we have to shim the magnetic trap ﬁelds
so that the cloud’s sag position overlaps with the optical trap. The optical trap loads from amuch tighter 230 Hz magnetic trap, so its vertical position essentially coincides with the magnetic
minimum (geometric center) of the Ioﬀe-Pritchard trap. The exact center, of course, corresponds
to a vanishing vertical magnetic gradient. The simple solution, then, is to turn oﬀ the large verticalshim coils so that the position of the magnetic minimum once again corresponds to the position of
the atoms (see Figure 3.1).
This solution is slightly too simple (as most simple solutions simply are), since the positionof the atoms (i.e. the optical trap) does not exactly coincide with the magnetic minimum when thelarge shims are turned oﬀ, leaving us with a 5 G/cm vertical gradient. To cancel this gradient, wethen turn on another small vertical shim coil to ﬁne tune the position of the magnetic minimum.
One might ask why we do not just use the large coils to do the ﬁnal shimming, which would be aﬁne question (and our initial mode of operation). But it turns out that those coils are used to carryhigh currents and supply strong ﬁelds several times in the experiment, and the weak currents andﬁne adjustments necessary for the ﬁnal shim were in a regime where the high-current servo fails to17gravitymag. potential

mag. potentialyymagnetic min.
85

Rb position

magnetic min.

Large shims on

Large shims off

small shim off

small shim on

Figure 3.1: Our scheme to create a uniform ﬁeld. We start with a BEC in a purely magnetic trap,
whose center is shifted upward by a set of large shim coils. We then quickly ramp down the coils,
causing the minimum of the magnetic ﬁeld to move to the location of the atoms. A small shim coilﬁne tunes this location.

18
gravity

B-field
gradient

thermal cloud

Figure 3.2: Measuring the magnetic-ﬁeld gradient. We create a large thermal cloud (pink circle
with dashed outline) and perform RF spectroscopy on a magnetically sensitive transition (in thiscase the |2, −2⟩ to |2, −1⟩ transition). As long as the spectral width of the pulse is smaller than thecloud, diﬀerent parts of the cloud are resonant at diﬀerent ﬁelds, and will outcouple at diﬀerent
positions in space. The resulting slope can be converted to a magnetic-ﬁeld gradient.

reliably operate. Since we already had a small coil in place to handle smaller currents and produce
weaker ﬁelds (labeled as the “anti-gravity coil” in Scott Papp’s thesis [32]), it was a simpler solution
to use it instead.
This small coil is a 60 turn coil with a 7.6 cm diameter, about 5 cm below the atoms, throughwhich we put about 3 A of current just before we ramp down the large shim coils in 0.5 ms. We tune
this current by creating large, thermal clouds of 85 Rb and taking an RF spectrum of the atomic
resonance. In a gradient, diﬀerent parts of the cloud will be resonant at diﬀerent frequencies, soplotting the position of the outcoupled cloud vs frequency gives us a measure of the magnetic-ﬁeld
gradient (see Figure 3.2).
Of course, when we turn oﬀ the gradient several things happen. First of all, the cloud begins
to fall under gravity, but since the trap is still on and the curvature unchanged, it does not expand
like it would if we simply turned the trap oﬀ. The falling of the cloud does not pose a problem,since it occurs relatively slowly. In 5 ms of falling, the cloud moves only 120 µm (20 pixels on ourcamera), and the magnetic ﬁeld stays constant to within our detection sensitivities. As I explainin Appendix B, we perform all of our experiments within 2-3 ms of turning the gradient oﬀ.

19
Another issue is that we actually change the magnitude of the magnetic ﬁeld when we ramp
oﬀ the large shim coils. Those coils create a 26 G magnetic ﬁeld perpendicular to the trap bias√
ﬁeld, which corresponds to an overall decrease of (155 G)2 + (26 G)2 − 155 G = 2 G when we turnthem oﬀ. Since we make BEC near a Feshbach resonance (see Equation 2.1), this will cause thescattering length to jump from 100 a0 (where we generally make BEC) to 260 a0 . For our data in
Chapter 6 we simply worked with this, since we then continued ramping the ﬁeld to higher values
anyway. However, for experimental procedures where this ramp is unwanted, we have developed asimultaneous ramp in the bias ﬁeld that compensates the shim ramp and keeps the ﬁeld magnitude
constant.
This scheme to temporarily create a spatially uniform ﬁeld works well and rarely requires
optimization, but limits us to short times. Perhaps a better and more robust solution would be
to work with an optical trap in the ﬁrst place. To reach similar trapping frequencies and aspectratios, the obvious setup would be a crossed beam optical trap with a large waist and high power.
If this is achieved, then the magnetic ﬁelds can be made very uniform with few coils, for as long as
is required. For a time we investigated using this setup ourselves, but problems with optical accessconvinced us to pursue the magnetic route.

3.2

Magnetic-ﬁeld stabilities
For us to perform trustworthy measurements, the magnetic ﬁeld (i.e. the resonant transition

frequency) must be uniform not only in space but also in time. A typical measurement of thecontact (more on that in Chapter 6) involves measuring the resonant transition frequency, andthen taking enough data on the tail of the transition (the contact) and the peak of the resonantlineshape (for the Rabi frequency) to achieve fractional error bars no larger than 10%. Any drift in
the resonant frequency during the measurements would create a systematic error in the measured
contact. So we had two goals for stabilizing our ﬁelds. On the one hand, we needed to minimize
the shot-to-shot variation, so that we can ﬁnd the resonance and measure the contact with as few
points as possible. On the other hand, we needed to ensure that the ﬁeld does not drift signiﬁcantly

20
over the total time this requires. Our experimental cycle time is about 90 seconds, which sets theformer time scale, and we tended to take one half to one hour to complete the measurement, setting
the latter.

3.2.1

60 Hz AC line noise

Luckily, we did not have to start from scratch. Scott Papp designed the magnetic trap andcontrolling servos very well [32], and we already had a stability of about 4 mG shot-to-shot to workwith. To improve on this, we ﬁrst set out to measure the noise from the AC power line. As any
table top experimentalist knows, the 60 Hz (50 Hz in Europe) frequency noise is diﬃcult to shieldcompletely, and makes it onto most electronics if you look closely enough. To investigate how much
this moved our ﬁeld, we measured an RF spectrum and took statistics on the side of the gaussianline shape. Assuming the scatter was purely due to the ﬁeld moving, we could convert the scatteraround the expected line to a value of magnetic ﬁeld for every point. Taking a histogram of thesepoints clearly shows a double-peaked structure, which one would expect for a random sample of asine wave. We also measured the phase of the AC line for each point, and the oscillation roughly
agrees with the expected 60 Hz signal (see Figure 3.3a).
The brute force solution to this problem would be to try to track down the path by which the
60 Hz noise reaches the magnetic trap. However, given that we separately control 5 magnetic ﬁeld
coils with as many servos, and that it would be diﬃcult to suppress the noise without adversely
aﬀecting the servo bandwidth, we opted for a more elegant solution. We decided to synchronize
the experiment to the AC line such that the RF pulse would always occur at the same phaseof the AC line. Our experiment is controlled by a DIO-128 board by Viewpoint Systems with64 programmable digital output lines, which we load with our timings and output values at the
beginning of the experiment, and then trigger to run the experiment oﬀ an internal clock. Ourchange was to set the trigger to wait for the rising edge of the AC line to start its output cycle.
However, we found that simply synchronizing the beginning of the experiment to the AC line was
insuﬃcient because the phase randomizes too quickly. After just 10 seconds, we measured the phase21
noise to be 12% of a full 16.7 ms cycle. Therefore, we synchronize 1 second before the sensitive
measurements, where the phase jitter is at a mere 1% of a cycle. This resulted in a shot-to-shotstandard deviation of ∼ 2 mG with a more Gaussian distribution (Figure 3.3b).
However, there are important issues with this solution, which have to do with experiment
timing. Mainly, the extra wait for an AC line trigger introduces timing uncertainties between
the times before and after the synchronization. This includes up to 17 ms of hold time fromthe AC phase, and several hundred milliseconds due to the fact that the DIO board has to load
new values for the ﬁnal part of the experiment. The latter might be circumvented with creative
programming and diﬀerent hardware, but the former is fundamental to the synchronization. Thistiming uncertainty is a problem when measuring things like cloud oscillations in the trap, which
are aﬀected (and usually caused) by various trap changes 10-20 seconds earlier. For this reason we
only synchronize the experiment to the AC line when we perform ﬁeld-sensitive measurements.

3.2.2

Field monitoring

With the synchronization we reached our 2 mG ﬁeld stability requirement on the 90 second
timescale. However, as we were taking numerous frequency spectra throughout the day, we noticed
occasional random jumps in the ﬁeld on the order of 10 mG (5 kHz for the contact transition).
If such a jump occurred in the time between measuring the resonant transition frequency andmeasuring the contact signal on the tail, it would result in a large systematic error in the ﬁnal
value for the contact (20% error for 40 kHz detuning). Therefore we had to start monitoring theﬁelds in order to track down the cause of the jumps.
One monitoring system we use is a 6 12 digit multimeter (Agilent 34401A) to record the voltages
from the Danfysik current probes that we use to servo the ﬁelds. To achieve the necessary precision(40 µV on the probe correspond to 6 mG in the Bias ﬁeld), it was required to use the “slow 5
digit” multimeter setting, which averages for 167 ms. Thus, we could only take measurements ofthe steady-state B-ﬁeld before our ﬁeld ramps. However, we did notice occasional jumps in thevoltages on our Bias coils, which we could correlate with similar measurements of the servo control

22

(a)

(b)

(c)

(d)

Figure 3.3: Correlating noise to the 60 Hz AC line. Part (a) shows how we correlated the noise to
the AC line. (a) Taking data on the side of a transition and using the slope allowed us to convert ascatter in signal to a scatter in B-ﬁeld. Correlating this scatter to the phase of the AC line at thetime we applied the RF pulse shows a modulation in agreement with the 60 Hz line. A histogram(c) of the scatter reveals the characteristic double-peaked shape of sinusoidally distributed values.
This structure essentially disappears in (d), where we have synchronized the experiment to theAC line 1 second before the B-ﬁeld measurement. The bin size and axis range of (c) and (d) are
identical.

23
voltage, which corresponded in magnitude and direction to jumps that we saw in the magnetic ﬁeld.
It turns out that a 100 µV jump in the control voltage caused a 4 mG jump in ﬁeld. We found thatthese tiny voltage jumps were caused mostly by faulty BNC cable connections between our National
Instruments PCI-6733 analog output board and the servo control box. Reducing the number ofconnections, as well as cleaning them with oﬀ-the-shelf cleaner for electrical connections essentiallysolved this problem, and our current probe readings became stable to a B-ﬁeld equivalent 0.5 mG.
We also monitored external ﬁelds, with the worry that there might be changes in the magnetic
ﬁeld environment. To measure external ﬁelds, we set up a ﬂuxgate magnetometer roughly 1.5 metersfrom the position of the magnetic trap, and set it to take and record a measurement of the ambient
magnetic ﬁeld in between each cycle of the experiment, i.e. when all of our magnetic trap coils
were turned oﬀ. Surprisingly, we noticed and tracked down several eﬀects that could change themagnetic ﬁeld at the atoms.
The ﬁrst of these was the freight elevator of the physics building. The elevator is locatedroughly 13 meters from our experiment and is used numerous times per day mostly for janitorial
equipment. We noticed a total change of 1.5 mG, which happened to be parallel and therefore add
linearly to our bias ﬁeld, when we moved the elevator from the basement to the third ﬂoor (our lab
is on the second ﬂoor). It is not enough to cause us problems, but I include it for the interested
reader. If the experiment were adjacent to the elevator, the change in ﬁeld could have been large
enough to adversely aﬀect our experiments.
The more relevant ﬁndings were related to magnetized metal objects within our laboratory.
We found that several items were magnetized enough that moving them toward or away fromthe 2 meter vicinity of the science cell cause ﬁeld ﬂuctuations upwards of 5 mG. These included
metal stools, trash cans, stepladders, and carts that carried equipment such as oscilloscopes andspectrum analyzers. Also, small magnets attached to large pieces of metal (speciﬁcally equipment
racks) created signiﬁcant ﬁeld variations. Our solution is to avoid placing portable pieces of metalon the side of the room with our science cell, and to simply not move anything in the lab while wetake sensitive data.

(a)

(b)

24

<2,-2|
80 MHz<2,-1|
020

4060Frequency (kHz)80

40 kHz100

Figure 3.4: The B-ﬁeld resonance giving rise to 2-photon transitions. Part (a) shows Fouriertransforms of oscilloscope traces taken with a hand held pickup coil close to the trap. The blue
curve shows the problem-causing 40 kHz feature. Changing the gain of the B-ﬁeld servos allowed us
to eliminate the feature, as illustrated by the green curve. A schematic of the 2-photon transitionis shown in (b).

3.2.3

Magnetic-ﬁeld-induced atomic transitions

In this last section I will mention an admittedly peculiar problem we encountered in thehopes that it might be interesting (or perhaps even useful) to the reader. Contact spectroscopy
relies on detecting low RF transition rates on one side of an atomic resonance at detunings where
one would expect negligible rates on the other side. However, for a while we were detecting spurioustransitions on both sides, which the widths of our RF pulses could not account for (more on thatin Chapter 4). We ﬁnally tracked it down to a problem with our magnetic ﬁelds. It turns out thatthe trapping ﬁelds were ringing slightly at around 40 kHz, which we detected by placing a hand
held pickup coil next to the trap and taking the Fourier transform of the resulting scope trace,
seen in Figure 3.4a. The atoms could then absorb a 40 kHz photon from the trap along with an 80
MHz photon from our RF pulse to undergo a 2-photon transition to the ﬁnal state (Figure 3.4b).
Analogously, we can think of the oscillating ﬁeld as modulating the transition frequency, resulting
in sidebands at ±40 kHz with respect to resonance, and it was the blue sideband photon, alongwith the carrier photon, that was causing the spin ﬂips. Changing the servo parameters of thesystem eventually allowed us to eliminate the 40 kHz peak, along with the spurious transition rate.

Chapter 4Radiofrequency pulses

A recurring theme of this thesis is that the contact measurement on the tail via RF spectroscopy gives a small signal, constantly in danger of being swamped by the nearby resonant atomic
transition. To avoid this, we must be very careful to shape our RF probe pulse. This will be themain focus of this chapter, with a note on RF rectiﬁcation in our system at the end.

4.1

Gaussian pulses
Ideally, one would be able to perform RF spectroscopy using a perfectly narrow function in

frequency space. However, since that is experimentally not feasible, we are required to do the best
we can with a wider feature. From undergraduate physics courses we know that the shape of apulse in frequency space is simply the Fourier transform of that shape in time [37]:∫ ∞1
F (ω) = √f (t)e−iωt dt.
2π −∞

(4.1)

This applies to the ﬁelds, so squaring the result gives the pulse shape in power. The standard
and easiest RF pulse shape is a square pulse, usually created by a simple RF switch. However, its
power spectrum is a Sinc2 function, which has oscillatory wings with an appreciable amplitude out
to many times the full-width-half-max (FWHM) of the main line (see Figure 4.1a). This would
be unworkable, since the spectral component has to decrease much faster with detuning than theω −3/2 tail from the contact. The logical choice, then, is to use a Gaussian shaped pulse, because
its Fourier transform is also a Gaussian and falls oﬀ extremely rapidly with detuning, as seen in
Figure 4.1b.

26

Figure 4.1: Some RF pulses in frequency space. In (a) we see the power spectrum of two standard
pulses (inset). A square pulse in time becomes a Sinc2 function, which looks much like a Gaussian
down to the ﬁrst minimum, but then exhibits signiﬁcant wings far from the center (red curve).
A Gaussian in time, however, becomes a Gaussian in power (black curve). The curves have beenscaled to give similar peaks and widths. In (b) we see the same curves on a logarithmic scale, whichmakes the diﬀerence clear. An ω −3/2 line has been added (blue), to show that the problematic wingsof a Sinc2 function persist with detuning.

27
We create the actual Gaussian shape using an Agilent 33220A programmable arbitrary function generator with 800 points per Gaussian. Initially, we tried to create our pulses simply byconnecting the output of the function generator to the amplitude modulation (AM) port of ourAgilent E4420B frequency generator. However, we found that the frequency generator’s 10 kHz
bandwidth began to distort the shape of the Gaussian for τ < 25µs, where τ is the rms width.
Instead we opted for the LMH6503 linear variable gain ampliﬁer, which has a 100 MHz gain control
bandwidth and 70 dB adjustment range. It takes the unmodulated frequency from the synthesizer
and the Gaussian shaped voltage pulse from the function generator as inputs, and outputs a Gaussian RF pulse. The power and frequency is controlled by the synthesizer, and the rms width of thepulse by the function generator. It is important to note that the ampliﬁer linearly modulates thevoltage of the rf signal with the Gaussiant2

V (t) = e− 2τ 2 ,(4.2)

which is squared to get the power as a function of time. As already mentioned, to get the rms widthin frequency, one takes the Fourier transform of the magnetic field (proportional to the voltage),1
and then squares the result to get the power. This results in a ﬁnal frequency width of σω = 2τ.

4.2

Shape checks
Of course, this is all nice and tidy in theory, but just because we tell our pulse to be a well

behaved Gaussian does not mean that it is one. In fact, several issues could conspire to change theshape of our pulse and create non-Gaussian spectral components at nonzero detuning (wings) on
our spectral function.
One of these is the intrinsic nonlinearity of our RF system. The variable gain ampliﬁer will
begin to saturate above 0 dBm (1 mW) of peak output power. Later down the line we have another
5 W ampliﬁer (Minicircuits ZHL-5W-1), with a 1 dB compression rating at 5W output. The main
result of these nonlinearities are that the peak of the Gaussian is slightly suppressed. We measure
this suppression and ensure that the change in overall power is always below a few percent. But that

28
-40

3 dBm
0 dBm

power (dB arb)

-3 dBm

-60

-80

-100

-120

79.48

79.50

79.52

Frequency (MHz)Figure 4.2: The Fourier transform of the RF power at system saturation. The labels correspondto the output power of the synthesizer before all ampliﬁcation. At 9 dBm output we begin to risk
damage to the ampliﬁers. The dashed black line shows a perfect Gaussian for reference (a parabolaon this scale). We see that saturation does not aﬀect the nicely Gaussian shape of the pulse down
to at least 60 dB below the peak power.

does not answer the question of the wings. To measure the spectral function, we used a pickup coilto record the RF pulse on a fast oscilloscope (Tektronix DPO4034), and took its Fourier transform(similar to Section 3.2.3). The results can be seen for several diﬀerent powers in Figure 4.2. The
saturation corrections for the three powers were 2.6% for -3 dBm synthesizer power, 5% for 0dBm,
and 11% for 3 dBm (i.e. higher powers than we actually use in the experiment), yet we do not see
any broadening above the noise ﬂoor.
Another issue is the fact that a true Gaussian extends to positive and negative inﬁnity, yet
we have to choose a time to open and close our RF switch, which eﬀectively truncates the Gaussian.
We want to give the Gaussian enough of its shape to be well behaved, yet truncate it enough toallow for maximum experimental timing ﬂexibility. We can explore this by creating a Gaussian
pulse and truncating it with an RF switch, and then once again calculating the Fourier transform.
Figure 4.3 shows the eﬀects of truncating the Gaussian at diﬀerent multiples of its width. One

29

Figure 4.3: The Fourier transform of truncated Gaussians. The inset shows the shape of thetruncated Gaussians, at ±1 (blue), 2 (green), and 4 τ (black). The Fourier transform of the powershows the resulting frequency spectrum. The ±4τ truncation looks unaﬀected, whereas the ±1τ
truncation clearly begins to look similar to a square pulse.

30

Figure 4.4: Atoms spin-ﬂipped by truncated Gaussians. A truncated pulse is less eﬃcient attransferring atoms between magnetic sublevels. The pulse still exhibits lower eﬃciency at 2τ , but
is fully Gaussian at 4τ . The theory line is scaled to the ﬁnal value.

can clearly see how the spectral function begins to look more and more like that of a square pulse
as the truncations become tighter. At ±4τ , we no longer notice any wings above the noise ﬂoor,
which is at an appreciable 60 dB below the peak power.
Truncating the Gaussians also decreases the total power in the pulse. Calculations of the
Rabi frequency tend to be much easier if we can just assume an ideal Gaussian, so we need to ensurethat the integrated power in the truncated pulse is essentially equal to the full pulse. To test this,we resonantly drove our favorite 85 Rb atomic transition (|2, −2⟩ to |2, −1⟩), in the limit of low
excitation fraction, with various truncated Gaussians. The results can be seen in Figure 4.4. The
blue line is the integrated power of a truncated Gaussian, divided by the integral of a full Gaussian.
It is then scaled to the maximum transferred fraction. We see reasonable qualitative agreement
between the measured outcoupled fraction and the expected shape from our calculations, where wehave only adjusted an overall scaling factor.
Given the results from the Fourier transform and the power tests, we eventually decided on
truncating our Gaussians at 4τ (8τ total pulse length). A 3τ truncation should be usable as well,
but we decided to be conservative in that respect. It essentially guaranteed a highly Gaussian

31
shape, and timing issues turned out not to be a problem. Typical Gaussian pulses for the contact
have a τ = 100µs. We have found this RF system to be very robust and repeatable. For pulses
longer than τ = 25µs, the option of using the AM port on the frequency synthesizer seemed to also
work very well. Another option is to use a voltage variable attenuator such as ZX73-2500 from
Minicircuits, which has a much higher frequency range. However, its gain curve is highly nonlinear,which would have to be carefully accounted for in order to produce clean Gaussian pulses.

4.3

Rectiﬁcation
One ﬁnal important issue related to the RF system is that of RF rectiﬁcation. The contact,

as mentioned numerous times already, gives rise to a low transition rate on the tail of the resonance.
To be able to measure something, then, we must be able to either increase the pulse length or thepulse power enough for the signal to become detectable. What originally attracted us to measuring
the contact is the fact that the system can be probed quickly with respect to three-body loss rates,
so increasing the pulse length by many orders of magnitude is somewhat counterproductive. Thus,
it is generally the RF power that we increase to bring the contact up to detectable levels. However,
although it is relatively simple to buy very powerful RF ampliﬁers to keep throwing more power at
the problem, eventually one is limited by RF rectiﬁcation.
Rectiﬁcation, in the sense that we care about, means that the RF that is intended solely
for the atoms is transmitted throughout the room and makes it onto the servos that control ourmagnetic ﬁelds. Inside the servos are op-amps, which ideally would reject noise outside theirbandwidth (in the hundreds of kHz range). However, internal interferences actually cause the opamp components to produce a DC oﬀset on the “out” pin [38], which manifests itself in a change
in the magnetic ﬁeld. Great care can be taken with electronics design to avoid this problem, but it
is nearly impossible to eliminate completely.
To measure the eﬀect of rectiﬁcation, we produce two RF pulses: a strong Gaussian pulse
to modify the DC magnetic ﬁeld via rectiﬁcation, and a weak square pulse to probe the magnetic
ﬁeld using an atomic transition. The strong pulse is detuned from resonance by 1.0 MHz. We can

32then compare to the magnetic ﬁeld without the strong pulse to see if it caused a shift. The resultsof such a test can be seen in Figure 4.5. We see signiﬁcant rectiﬁcation for the strongest pulses our
system allows, as seen by the 17 mG magnetic ﬁeld shift. However, rectiﬁcation becomes negligible
at powers 9 dB higher than the maximum powers needed to probe the contact. We can therefore
rule out rectiﬁcation as a possible mechanism that would give us a systematic shift in the measured
contact.
The net results of all these systematic RF pulse checks are that we can be conﬁdently measure
the tail of an RF transition. The clean Gaussian spectral line allows us to rule out that we are
inadvertently exciting atoms via the resonant transition.

33

(a)

rectification pulse(c)probe pulse

(b)

(d)

Figure 4.5: Eﬀect of RF rectiﬁcation. (a) We send out a τ = 100 µs Gaussian pulse, and probe theresulting ﬁeld with a 50 µs square pulse. The 100 µs delay gives the servo time to respond and shift
the ﬁeld. (b) At our maximum allowable power (8 dBm at the synthesizer), we see rectiﬁcation
amounting to 17 mG. At 8 dB lower power (c), the rectiﬁcation has all but disappeared (d). Ourcontact measurements required powers no more than -9 dBm.

Chapter 5

Detection

Detection is one of the most important parts of any experiment. It connects us to the objectof our studies as an extension of our natural senses, and what it tells us forms the basis of everything
we can say about it. However, it is often seen as a black box that “just works” by anyone new inthe lab, and not understanding its complexities and subtleties can cause serious problems for theunaware. More speciﬁcally, it turns out that accurately calculating atom number from absorption
images is not at all straightforward, and many systematics can cause errors in the measured signal.
This applies to simply measuring condensate number in expansion, as well as to the in-trap imaging
we perform when making measurements of the contact.

5.1

Condensate absorption imaging

5.1.1

Imaging corrections

One of the eﬀects of our 10 Hz trap is a slowly expanding BEC when the trap is turnedoﬀ, which is a problem when it comes to imaging. In “standard” time-of-ﬂight (TOF) absorption
imaging, the trap is released, and the cloud expands until the optical density (OD) is low enoughto be imaged. However, the slow timescales our trap causes means that our BEC would have toexpand for 70 ms or more before the OD drops to measurable levels. Such long expansion times
are not feasible, since the cloud would hit the ﬂoor of our science cell in about 30 ms. Our initialsolution to this problem (see Section 5.1.3 for our current solution) was to transfer a small fraction(20-30%) of the atoms from our |F = 2, mf = −2⟩ trapping state to our |3, −3⟩ imaging state via

35
a deliberately ineﬃcient adiabatic rapid passage (ARP). This reduces the OD of our imaging state
by a factor of 3-5, which we then simply multiply up by a correction factor which we calibrate using
thermal clouds. But even for clouds with low OD, much care must be taken to correctly calculate
the atom number.
The reason that a low OD is required in the ﬁrst place is twofold. One problem is the limited
dynamic range of the camera, which will be addressed in Section 5.1.3. The other is that images
usually saturate at some value of OD with standard absorption imaging. A picture of our cloudis generated using three images: the ﬁrst image uses resonant light to probe the atoms, whichcasts a shadow onto the camera (Ishadow ). The second image contains the probe pulse again, but
without the shadow of the atoms (Ilight ). The ﬁnal image triggers the camera and opens all thesame shutters, but does not include the probe beam (Idark ). This dark frame is then subtracted oﬀ
the other two frames to account for room lights and a camera oﬀset. I will redeﬁne the resulting
frames using the convention in [39]: If = Ishadow − Idark and Ii = Ilight − Idark . This is done forevery pixel, and the simply measured ODmeas is then
ODmeas = ln

Ii
.
If

(5.1)

However, if any light in the probe beam cannot be absorbed by the atoms due to being oﬀ-resonant(what we call “bad light”) or simply by scattering around the atoms, then If cannot become
arbitrarily small, and the ODmeas will saturate to some value ODsat , regardless of how opticallythick the cloud actually is. Also, if the dark frame does not eﬀectively subtract light that is notresonant, the OD will saturate at lower values. For example, vibrations of the optics or nonrepeatable shutter timings could cause less ambient light to enter the dark frame than the shadow
frame. For our experiment, ODsat is typically around 3.5, and its ﬁnite value can be corrected forto get a modiﬁed OD using [40]ODmod = ln

1 − e−ODsat.
e−ODmeas − eODsat(5.2)

Also, the probe intensity can saturate the atomic transition, which causes a signiﬁcant change

36

60k
I

eff

sat

=

5200 ±400 cts/px

OD =0.72 ±0.03

0.4

0.6

40k

30k

i

I (cts/px)

50k

20k

10k

0
0

0.2

0.8

OD

Figure 5.1: A measurement of our eﬀective saturation intensity. We measure the OD vs Ii in counts
per pixel, and then invert the axes for plotting and ﬁtting via Equation 5.3. The red line shows
this ﬁt. The measured OD is highly sensitive to the probe intensity, especially at low intensities,
where one might think that the correction due to I/Isat is negligible.

37
in the measured OD even for low intensities. This eﬀect can be corrected for using
ODactual = ODmod + (1 − e−ODmod )

I
I sat

,

(5.3)

where I is the intensity of light at the atoms, and I sat is a constant, which is 1.16 mW/cm2 for
Rb on the cycling transition. To make this correction, then, one can try to carefully measure
the intensity of the probe beam at the atoms, taking into account losses at optical surfaces. It ismuch simpler, however, to measure an eﬀective saturation intensity I eﬀ
sat in the same units as one
measures Ii , in counts per camera pixel. One can then replace I/I sat with the equivalent Ii /I eﬀ
sat .
To do this, we create thermal clouds and image them using diﬀerent probe intensities. We can thenuse Equation 5.3 as a ﬁtting function with ODactual and I eﬀ
sat as the ﬁt parameters. An example ofthis can be seen in Figure 5.1.
The resulting corrections to our total BEC numbers come to about 5% for the ODsat correction and about 10% for the I/I sat correction, given a measured OD of 1 and an I/I sat = 0.1.
But since imaging can be so important, we wanted an independent conﬁrmation of our methods.
To get an independent measurement, we can look at the sizes of the clouds (which depend on
BEC number) and compare to expected sizes. The expected values come from a variational model
developed by Pérez-Garcia et al. [41] to simulate BEC dynamics using a Gaussian cloud ansatz.
This will be referred to as the PG model. Equations 10a-c in [41] reduce to~2ẅ + ν w = 2 3 +
m w
2

√

2 a~2 Nπ m2 w 4(5.4)√
for a spherical trap, where w/ 2 is the rms width of the Gaussian cloud, ν is the trap frequency,
m is the mass, a is the scattering length, and N the total BEC number. Setting ẅ to 0, we
ﬁrst solve for the equilibrium size w0 of this Gaussian condensate given our initial number andscattering length. We then plug in an a(t) that corresponds to the scattering length vs time in our
experiment, and numerically integrate Equation 5.4 to get the size as a function of time w(t). Thenwe must convert the size of the modeled cloud w to our ﬁt size σﬁt . The Thomas-Fermi radius RTF
of a BEC is calculated to be 1.78 times larger than the equilibrium width w0 independent of the38parameters. To go from the Thomas-Fermi radius RTF to our ﬁts, we simulate a three dimensional
spherical Thomas-Fermi distribution and integrate it through one dimension (as is the eﬀect ofimaging). We then ﬁt the resulting distribution to a two-dimensional Gaussian, resulting in a ratio
σﬁt /RTF = 0.44, which is also independent of the speciﬁc initial parameters. Thus, to convert the
PG results to the imaging ﬁts, we use σﬁt /w = 1.78 · 0.44 = 0.78. Using it, we predict the size of the
BEC vs number after 28 ms expansion and compare it to our measured sizes, seen in Figure 5.2.
For this test, we expanded at 500 a0 and subtracted our 8 µm resolution limit σres in quadrature√
2 − σ 2 . The good agreement between the data and the theory provides additional
via σ = σﬁtres
conﬁrmation of the validity of the OD corrections. Unfortunately, the dependence of BEC size on
the number is too weak to allow spatial extent to be our primary measure of condensate number.
This method of calculating OD via Equation 5.1 and then applying ARP, ODsat , and Isat
corrections works well enough, but can be quite tedious. Also, the corrections can change withexperimental drift. ODsat , for example, can change drastically with a small change in probe beamalignment or frequency components of the probe laser, and the ARP calibration has to be measured
every day. Carefully performing these necessary corrections is certainly manageable, but in the end
we decided to switch to a more robust high-intensity probing scheme described in Section 5.1.3.

5.1.2

BEC collapse and high-ﬁeld imaging

Another imaging diﬃculty related to our system arises from the collisional properties of
85 Rb.

As explained in Chapter 2, 85 Rb has a background scattering length abg of -443 a0 , so thecondensate begins to collapse as soon as the magnetic ﬁelds are turned oﬀ (the low expansion energy
is not enough to overcome this). We allow the condensate to expand by simultaneously increasing
the bias ﬁeld as we turn the trap oﬀ, such that the magnitude of the magnetic ﬁeld stays constant.
The BEC then expands at a positive scattering length for ∼30 ms until the size has increased toabout twice the resolution limit. At this point, the original mode of operation was to turn oﬀ theﬁelds as fast as we can and image the BEC once the ﬁeld stabilizes. Even so, the condensate has
about 2 ms time at its background scattering length to ponder the merits of collapsing and act on

39

30

rms size( m)

OD < 2
OD < 1

20

10

0
0

20

40
N

BEC

60

80

3

(10 )

Figure 5.2: Comparison between measured and predicted BEC size. The diﬀerent shapes correspondto diﬀerent ARP eﬃciencies, giving rise to diﬀerent ODs. The red line is the modeled BEC size inexpansion.

40
the result. In order to predict the outcome we return once again to the Pérez-Garcia (PG) model.
If we take a 85 Rb BEC of 50,000 atoms at 100 a0 at rest in our trap and then instantly
turn oﬀ all the trapping ﬁelds, the PG model predicts that the BEC as a whole will collapse in
about 27 ms. But one of the limitations of this model is that the cloud shape is always conserved,
meaning that the cloud shrinks nicely as a whole. In reality however, smaller clumps of the cloud
can collapse locally on much faster timescales. To get an idea of this timescale, we use the factthat small condensates can be stable at negative scattering length. Assuming no trapping potential(ν = 0), we again solve for the stability condition by setting ẅ in Equation 5.4 to 0:√
wstable = −aN

2
π

(5.5)

(This illustrates another reason for making spherical clouds...to make the math easier). Here ais the background scattering length. If we constrain the peak density of this stable cloud to beequal to the initial peak density of our BEC, then we can solve for the width and atom number Npthat will form the smallest element of collapse in our BEC. Then we simply increase the number√
by Np , the statistical number ﬂuctuation in that packet, and use the PG model to predict thecollapse time of the packet, which comes out to be about 2 ms. This is certainly just a roughcalculation, as it does not take into account the expansion energy the cloud already has, but it tells
us that we cannot assume that 2 ms is too short to see losses from local implosions.
Our previous way to avoid collapse was simply to increase the scattering length during expansion to provide more kinetic energy and lower the density, but increasing a also greatly increases
the rate of three-body losses. And when varying the scattering length in expansion, we did not
see a regime that clearly avoids both collapse and three-body loss problems. Thus we decided toinstead image at high ﬁeld. We still need to transfer the atoms to the |3, −3⟩ imaging state, but
we can do that using a 7 µs RF π pulse, drastically reducing the time at negative a.
To image at high ﬁeld on the cycling transition, we need to shift the probe laser’s frequency by
252 MHz from the zero ﬁeld cycling transition. As explained in detail in [32], our lasers are controlled
via an oﬀset lock. The probe laser is overlapped with a master laser, and the resulting beat note41
mixed down with a voltage-controlled oscillator (VCO) to produce a signal that eventually feeds
back to control the frequency of the probe laser. The VCO frequency is changed by our computer
controller, but does not have enough tunability for both low and high-ﬁeld applications. To increase
its range, we ramp the VCO to its maximum detuning, and then switch over to a diﬀerent VCO thatworks at higher frequencies to continue the ramp. We have not had any problems with switching
between VCOs, and that method of increasing the laser tuning range has proven to be quite robust
for oﬀset lock lasers.
Unfortunately, our imaging axis does not coincide with the bias ﬁeld direction, meaning that
we cannot probe with pure σ− polarization. Instead, we change the polarization such that it islinear and perpendicular to the quantization axis. This is achieved by a waveplate mounted on
a ﬂipper, which we manually put into or out of the beam depending on which imaging scheme
we want to use. The polarization geometry results in a superposition of σ+ and σ− light (a nicetable of diﬀerent geometries and their resulting transitions can be found in [40]). Since we image
in a ∼160 Gauss ﬁeld, the σ+ and σ− transitions are separated by many linewidths, giving us apure cycling transition. In fact, the only result of the superposition is that it reduces the eﬀective
transition strength by a factor of 2.
The ability to image at high ﬁelds enables us much more ﬂexibility than before, as we do
not have to wait for ﬁelds to turn oﬀ to probe our system. The downside is that it is somewhat
of a nuisance to have to change the probe frequency every time we decide to image at a diﬀerent
ﬁeld. However, the 6 MHz natural linewidth of the transition is much larger than the magnetic
ﬁeld ﬂuctuations, so that we can just dial in the correct probe frequency once we know the ﬁeld
without recalibrating every time.

5.1.3

High-intensity imaging

In Section 5.1.1, I described a workaround to deal with the fact that our condensates have very
high OD, which involved performing ineﬃcient ARPs to transfer only a small fraction of the cloud tothe imaging state. However, the calibration of that fraction is cumbersome and possibly introduces

42
systematic errors. Also, the move to imaging at high ﬁelds means that the RF frequency for thattransfer depends on the ﬁeld, which increases the calibration requirements and the possibility oferror. But there is another solution. Fundamentally, a high OD leads to problems because of thelimited dynamic range of the camera CCD. Our camera has a 16-bit output, meaning a single
pixel can read 216 ≈ 65, 000 diﬀerent values. Thus, all else being perfect, the maximum OD thecamera can display is ln(216 ) = 11.1. However, as there is noise in the dark frame of the cameraand shot noise in the beam, a workable limit to the maximum OD is much lower. We measure
12 counts/pixel rms noise in the dark frame, so with I/I sat = 0.1, an OD signal-to-noise ratioof 2 will limit the maximum workable OD to 5.3. We numerically calculate this based on error
propagation through Equation 5.1. To beat this problem we clearly need more light to make itthrough the atom cloud. I mentioned in Section 5.1.1 that high probe intensity causes problems,
because saturation decreases the amount of light that gets absorbed. However, we understand how
this happens and can account for it, and if we signiﬁcantly increase the intensity that gets through,the noise becomes fractionally insigniﬁcant. Thus, imaging with intensities much higher than I sat
can allow for accurate quantitative measurements of optically thick clouds [39].
Instead of using the simple deﬁnition of OD (Equation 5.1) to create the image and thencorrecting for the saturation intensity, we use the full OD deﬁnition given by
ODnew = ln

Ii − If
Ii
+
,
If
I eﬀ
sat

(5.6)

which is just a rewritten form of Equation 5.3, assuming that the eﬀect from bad light is negligible.
Our imaging software uses this formula, with an empirically supplied I eﬀ
sat from data such as shownin Figure 5.1, to calculate the ODnew for each pixel. Using this method, we have been able to image
clouds with an OD on the order of 8 with negligible noise, and higher ODs should be possible.
But with all new and fancy techniques come new issues and systematics. One of these is
the problem of photon re-absorption. The calculations for OD assume that photons emitted fromthe atoms leave the system. However, there is a ﬁnite chance that another atom will absorb thisphoton, which will make it unavailable to absorb from the probe beam, and therefore decrease the43apparent OD. The higher the optical depth, the stronger this eﬀect. But for very high intensities,
the scattering rate eventually saturates to 1/2 the excited state decay rate [42]. At that point, theintensity from surrounding atoms stays constant as the probe intensity is increased, and an atom
will be increasingly likely to absorb from the probe beam. In other words, once you commit to highintensity imaging, you have to really let loose. An illustration of this eﬀect can be seen in Figure
5.3.
As we could not simply turn up the power of our probe laser to reach intensities of 15×I/I sat ,we mounted a 1 m focal length lens roughly 1 m upstream of the atoms to reach higher intensities
by decreasing the beam waist. The lens is also mounted on a ﬂipper mount, which allows us toeasily switch between imaging schemes. Such high intensities saturate the camera pixels, so we
also use a neutral density ﬁlter in front of the camera to decrease the intensity by a factor of 2.
One might naively think that this is counterproductive, but the important thing is to have a highintensity incident on the atoms. The neutral density ﬁlter does not change the fraction of lightabsorbed, and all we have to do is to measure our I eﬀ
sat with the ﬁlter in place to account for it.
But one must be careful with high-intensity beams. If too many photons are absorbed by
the atoms during the pulse, the resulting momentum transfer will result in a Doppler shift, andthe atoms are pushed out of resonance. This again decreases the apparent OD, leading to anothersystematic. To ensure this does not happen, the probe pulse times t must be kept very short. The
scattering rate Γ, the doppler shift δD , the photons absorbed Nph , and the velocity v of an atom
after absorbing photons are related via the coupled equations:
I γ
Isat 2Γ=I+
1 + Isat

(

4πδD
γ

)2

Nph = Γ · t

(5.7)

Nph h
v=
λm
)
(
c
1
δD =
,
1−
λ
1 + vcwhere γ = 2π × 5.98 MHz is the natural linewidth of the transition, λ = 780 nm is the probe44

peak OD

6

4

2

0
0

5

10

15

Probe Intensity (I/I

sat)

Figure 5.3: Eﬀect of photon re-absorption on OD. The black circles show the measured OD of
a BEC, calculated with Equation 5.6, as a function of I/I sat . For intensities near I sat , the highscattering rate in conjunction with the high OD causes a large fraction of atoms to re-absorb
photons emitted from surrounding atoms. This makes them unavailable to absorb from the probe
beam, lowering the apparent OD. As the intensity increases, the scattering rate remains roughly
constant, translating to a smaller fractional amount of rescattered light present in the cloud. The
red squares show the same experiment on thermal clouds with lower OD, where this eﬀect does notoccur.

45
wavelength, m is the mass of the atom, and c is the speed of light. We numerically solve theseequations vs time to calculate the expected error due to this eﬀect and ensure that it is small. We
use 5 µs probe times, which for an I/I sat = 10 and an OD of 6 corresponds to an error of 0.5%.
Finally, if the OD is low, imaging with high intensities results in unwanted noise. The lightand shadow frames will both have high intensity light, but the fractional diﬀerence between thetwo will be small. If we assume shot noise (a lower limit) on the beam and an I/I sat = 10, thenan OD of 0.1 results in a signal-to-noise ratio of 1.5 on the OD. Luckily, this is a per pixel noiseﬁgure, and low OD regions usually spread over many pixels. The spatial extent of thermal clouds
therefore help average down this noise to workable levels. However, because of this noise issue we
use high intensity imaging only when there is a high-OD feature we are interested in. For the lower
OD pure thermal clouds we switch back to low-intensity imaging.
Overall, the high-intensity imaging technique works very well for us. The value of I eﬀ
sat isdependent mostly on the reﬂectivity of the optics between the atoms and the camera, as well as
the camera’s quantum eﬃciency. Therefore it is generally not subject to experimental drift anddoes not have to be recalibrated unless the optics change (such as with the addition of the neutral
density ﬁlter). We have found that the best measurements of I eﬀ
sat (see Figure 5.1) are performed
with thermal clouds of low OD. Higher ODs can exhibit photon reabsorption, as well as cause
problems in the region of low intensity, where imperfections in the dark frame subtraction cancause signiﬁcant deviations from the saturation curve in Figure 5.1.

5.2

In-trap imaging for contact measurements

5.2.1

Transfer to the imaging state

We still use absorption imaging for our contact measurements, but we require several diﬀerent
additional techniques for our purposes. Mainly, when we perform RF spectroscopy to measure thecontact, we transfer 1-2% of the atom cloud from the |2, −2⟩ to the |2, −1⟩ state. The (diﬃcult)trick is to image and count that small fraction in the |2, −1⟩ state, without imaging any of the large

46
cloud left in the |2, −2⟩ state, which would cause signiﬁcant systematics. So the imaging process
must be highly eﬃcient and highly state-speciﬁc.
One important step toward enhancing imaging eﬃciency, or rather the signal-to-noise ratio,
is to wait for as little time as possible between the RF contact pulse and the probe pulse. The
atoms that we outcouple on the tail have high momentum, and their expansion energy is 12 ~ω,
where ω is the detuning from resonance (see Chapter 6). For a typical 2π×40 kHz detuning, theatoms move at a velocity of 14 µm/ms. Given that the starting size of our BEC is 14 µm, the OD
will drop signiﬁcantly if given more than a millisecond to expand. This is another reason that we
do not turn oﬀ the trap, but instead image at high ﬁeld.
To perform state-speciﬁc imaging, we need to eﬃciently transfer the |2, −1⟩ atoms to our|3, −3⟩ imaging state without putting any of the |2, −2⟩ atoms up there as well. We do this via
a two-step RF and microwave (µ-wave) process, as seen in Figure 5.4. A 2.8 GHz µ-wave photontransfers them to the |3, −2⟩ state, followed by a ∼ 85 MHz RF photon to get to the |3, −3⟩ imaging
state. Since we already have to ﬁnd the resonant frequency of the |2, −2⟩ to |2, −1⟩ transition forevery contact measurement, we can exactly calculate the other transition frequencies using theBreit-Rabi formula [43]. Initially, we used a short π pulse for both frequencies to transfer theatoms as quickly as possible (22 µs for the µ-wave and 16 µs for the RF), but we found that thiscaused noise in our signal. The µ-wave transition has a large ﬁeld sensitivity of 1.4 MHz/Gauss,
and the shot-to-shot ﬁeld ﬂuctuations were large enough to make this transfer highly unstable.
The obvious solution would be to sweep the frequency and transfer them using an adiabatic rapid
passage (ARP), which tends to be insensitive to magnetic-ﬁeld noise. However, the |2, −2⟩ to|3, −1⟩ transition (shown by the red dashed line in Figure 5.4) is only about 140 kHz away, and astandard ARP would transfer the 99% of atoms remaining in the |2, −2⟩ state into the |3, −1⟩ statewith very high eﬃciency. Unfortunately we found that, although we do not send out resonant RF
purposefully, some of the many atoms in the |3, −1⟩ state always make it into the |3, −2⟩ state andthen get pumped into the |3, −3⟩ state and get imaged. We solved this issue by shaping the ARPs
to avoid the unwanted transition.

47

...

F’=4

mf=0

F=3
-1
-2
84.85

-3
2837.00
2837.14

80.04

-2
-1

F=2

mf=0

...

Figure 5.4: State diagram showing transfer to the imaging state. The atoms (shown in blue) startin the |2, −2⟩ state, and the RF pulse for contact spectroscopy moves a small fraction to the |2, −1⟩
state. These are transferred via shaped ARP to the |3, −2⟩ state and RF π pulse to the |3, −3⟩ state(solid green arrows). Once there, the probe beam cycles them to the F’=4 manifold (dotted green
arrow). The entire procedure, from the |2, −1⟩ state to imaging, takes < 200 µs The transitionof the cloud remainder to the |3, −1⟩ state (red dashed arrow) is unwanted and would produce
signiﬁcant false signal. The energy splittings are given in MHz, calculated for a ﬁeld of 160 G.

48

(b)

Energy

(a)

Power

Power

RF Frequency

Time

Frequency

Frequency

Time

Time

Time

Figure 5.5: Shaped ARPs. A standard ARP is shown in part (a), where the RF power and therefore
the energies of the dressed states are constant in time. The frequency must start far from resonanceand end far from resonance for a well behaved ARP. A shaped ARP (or ShARP) seen in (b) starts at
low detuning and low power. The power is adiabatically increased as the frequency ramps throughresonance and then decreased the same way. This allows for a quick, eﬃcient ARP, sweeping overa very small frequency range around the resonance.

49
In the dressed-state picture, the resonant energy splitting between two states is given by ~times the Rabi frequency Ω. To complete a very eﬃcient ARP, one must start and end the frequency
sweep far away from resonance to be completely in a bare atom state at the beginning and endof the sweep. What constitutes “far away” is, not surprisingly, relative to Ω. A “standard” ARP,
with constant power and a linear frequency ramp, is shown in Figure 5.5a. To be able to perform
an eﬃcient ARP quickly (one of our requirements), one needs a high Rabi frequency. But the only
way to perform a standard eﬃcient ARP and not have to ramp through the nearby resonance would
be to use low power and sweep too slowly for our needs. We found a way around this by shaping
both the power and the ramp rate of the ARP as shown in Figure 5.5b. By creating a Gaussianenvelope on the power, we start at low detuning and low power to avoid the nearby unwanted
transition. As we ramp through the “good” resonance, the power adiabatically increases, which
also lets us increase the ramp rate. Using this method, we can ARP ∼80% of the atoms from the|2, −1⟩ to the |3, −2⟩ state in 100 µs, without transferring any detectable atoms from the |2, −2⟩
to the |3, −1⟩ state. The total sweep covers 100 kHz, making this ARP very insensitive to theshot-to-shot ﬂuctuations of the magnetic ﬁeld.
Once the atoms are in the |3, −2⟩ state, we pulse a 16 µs RF π pulse to transfer to the |3, −3⟩
imaging state with 97% eﬃciency. This transition only has 0.47 MHz/Gauss sensitivity (same as
the transition for the contact), so magnetic-ﬁeld noise is not an issue.
Unfortunately, as mentioned in Section 5.1.2, the fact that we image at high ﬁeld, with abeam perpendicular to the quantization axis, means that we automatically lose a factor 2 in OD
signal. But since we can image the outcoupled atoms <200 µs after the end of the contact pulse,
we probe much higher ODs than we would by waiting for the ﬁelds to turn oﬀ. When we ﬁrst
started contact measurements, we imaged the atoms at low ﬁeld about 4 ms after the RF pulse
for the contact. The downside of this was that the expanded clouds were not easily discernible on
the CCD image due to the low OD, and we had to trust the ﬁtting program to average out thenoise and ﬁt a cloud. But imaging after some expansion time also brings a great beneﬁt. Since theexpansion energy can be very large for the contact signal, the expanded size can serve as a clear

50
conﬁrmation that the atoms really are from the tail of the RF spectrum, rather than a systematic
unrelated to the contact.

5.2.2

Small cloud eﬀects

With in-trap imaging also comes the problem of resolution. The Thomas-Fermi radius of our
BEC is generally 14 µm, corresponding to 2.4 pixels. With a cloud so small, one might wonder if
there are systematics associated with the pixelation. For example, the calculated OD is a nonlinear(logarithmic) function of the intensity, yet for each pixel we get simply a sum of the photons that hit
it. Thus, a signiﬁcant variation of the cloud OD over a length scale corresponding to a single pixel
can systematically lead to an error. To get an idea of the size of this error, we simulated a Gaussian
absorption proﬁle divided onto a 2-dimensional array of pixels, used the fraction of missing light ineach pixel to calculate that pixel’s OD value via the simple OD formula (Equation 5.1), and thensummed up all the OD values from the pixels. Comparing the integral of the Gaussian OD that
produced the shadow proﬁle IG to the summed pixels Σpx gives an idea of the error associated withthe pixelation. Some results of this simulation can be seen in Figure 5.6. This simulation does notaccount for the diﬀraction-limited resolution, which blurs the image and decreases the intensity
variation on a pixel, so the simulation shown in Figure 5.6 represents a worst-case scenario. Our
total resolution limit is about 1.4 pixels, or 8 µm (representing the rms width of the Gaussian thatﬁts the smallest clouds we can observe with our imaging system). Note that the eﬀect is dependent
on the position of the Gaussian with respect to the pixels, and increases with OD. For the ODs in
trap and a Thomas-Fermi radius of ∼2.4 pixels, we expect that this pixelation eﬀect adds an error< 3% to the atom number.
When measuring the contact, we alternate taking a measurement on the tail of the transition
and the peak of the transition, with the ﬁnal value for the contact involving a ratio of these two
measurements. Since the tail signal is dominated by the high-density BEC in the center, we use asingle, two-dimensional Gaussian ﬁt to ﬁnd the outcoupled number. However, on the peak of theresonant transition the rate is independent of density, and we outcouple equal fractions from the

51

(a)

Gaussian on pixel center

Gaussian on pixel corner(b)

Figure 5.6: Simulation of pixelation eﬀects. An absorption proﬁle is incident on a 2-dimensional
array of pixels, and the resulting pixelated OD, the sum over the pixles Σpx , is compared to theintegral, IG , of the OD that gave rise to the absorption proﬁle. Part (a) shows the two conﬁgurations
under which the simulation was run. In one conﬁguration, the Gaussian proﬁle was centered on
the center of a pixel in both dimensions. The corner conﬁguration formed the other “extreme”.
Simulation results can be seen in part (b). When the Thomas-Fermi (TF) radius is roughly the sizeof a pixel, the eﬀects can become very pronounced, even exhibiting unintuitive structure. The error
increases linearly with OD, but stays at a 3% or less correction for our normal operating conditionsof peak OD<1 and a TF radius of 2.4 pixels.

52
BEC and the thermal components. To measure the total number accurately, we require a doubleGaussian ﬁt to account for each component. But as both BEC and thermal are near the resolution
limit and similar in size, the ﬁts often cannot clearly distinguish the two components. Also, in
terms of general experimental wisdom, using diﬀerent techniques to measure the tail and the peakallows for unexpected measurement systematics. To ensure that this does not happen, we came up
with a technique based on summing over pixels. We choose a small region centered on the cloudand simply calculate the sum of all the pixel values. Then we create another region, a “sidewalk”
around the ﬁrst, and use it to calculate the average value of the background signal (Figure 5.7). We
subtract this average value from the inner region, leaving us with the signal from our atom cloud,independent of shape. To decrease the noise, we want the inner box to be as small as possible whilestill enclosing all of the signal. For the “sidewalk”, a smaller size allows less chance of a systematicoﬀset from diﬀerent parts of the CCD, but it must be large enough to average out local noise. We
analyzed numerous data sets, varying the sizes of the two regions, and ﬁnd the optimum sizes tobe 15 pixels on a side for the inner box and 35 for the outer (see Figure 5.7).
The main result from the analysis was that the method of summing over pixels agrees withthe ﬁtted Gaussian method as long as the subtracted background is small. This independently
veriﬁes that using the Gaussian ﬁts does not produce a systematic error due to the diﬃculty ofﬁtting very small clouds. We do not, however, use the pixelsum method for taking our data. Whenthe signal was relatively strong (as in Figure 5.7), we ﬁnd that the pixelsum method works very
well, often even giving smaller errors than the Gaussian ﬁts. However, for small peak ODs, when
the signal was comparable in magnitude to the subtracted background, we ﬁnd that this pixelsum
method results in very large ﬂuctuations compared to the Gaussian ﬁts.

53

Figure 5.7: Schematic of the pixelsum technique. Shown is a standard image of a BEC with 30%
thermal component, taken in the magnetic trap. For the pixelsum analysis, we sum all the valuesof the inner square, and subtract the background as determined by the “sidewalk” (hatched area).

Chapter 6

Two-body Contact

In this chapter I will explain what the contact is and try to elucidate its physical meaning.
I will also present data of contact measurements versus interaction strength, as well as some interesting dependences on timescales. In much of the literature up to the writing of this thesis, thecontact is presented for fermions. I will only deal with the contact for bosons which, for the mostpart, merely changes some prefactors in the equations by a factor of 2. However, the possibility of
three-body interactions among bosons gives rise to a three-body contact, which I will describe in
Chapter 7. For this reason, I will refer to the two-body contact as C2 and the three-body contact
as C3 . In this thesis, as well as in papers regarding fermions or ignoring three-body interactions,the contact refers to the two-body contact, C2 , unless speciﬁed otherwise.

6.1

What is the contact?
There are numerous working deﬁnitions of what the contact is, but I will begin with what

I think may be the most intuitive. The contact, C2 , is essentially the non-classical probability ofﬁnding two particles close to one another. Speciﬁcally, this probability due to the contact is higher
than in a classical, non-interacting gas. With atoms that interact only via s-wave collisions, thepresence of a two-body potential (Figure 6.1a) means that the scattering wavefunction will looksomething like Figure 6.1b. In the zero-range limit we ignore the complicated and species-dependent
short-range potential that gives rise to an eﬀective range re , and assume that the smooth, longerwavelength part of the wavefunction extends all the way to zero. Another way of representing the

55

(a)
V
r

(b)
zero-range limit

rY
re

a

r

Figure 6.1: Schematic of the interparticle potential (a) and scattering wavefunction (b). The potential V (r) looks like an attractive potential down to a distance given by the van der Waals potential.
Outside of this eﬀective range re , which is ignored in the zero-range limit, the wavefunction can beuniversally described by the scattering length a.

56
two-particle wavefunction Ψ(r) in the zero-range limit is to say(r)
A,lim rΨ(r) = 1 −
r→0
a

(6.1)

where r is the interparticle distance, a is the scattering length, and A is a constant that depends
on the problem [44]. Given this wavefunction, we can calculate the probability of ﬁnding a particle∫r′near another particle at r = 0, given by 4π Ψ(r)2 r2 dr, which is proportional to r′ in the limit of0′r ≪ a. The probability of ﬁnding the particle at 0 just goes as the volume 34 πr′3 , so the probabilityof ﬁnding a pair in a spherical volume of radius r′ goes as r′4 . Indeed, the coeﬃcient of the paircorrelation for small volume is given by the contact density, C2 (R), and the number of pairs of
atoms in a sphere of radius r′ is
Npair =r′4
C2 (R).
4(6.2)

The contact is obtained by integrating C2 (R) over all space. If the contact is zero, there is noenhanced correlation as r → 0, and you would expect the probability of ﬁnding pairs of particles
to go as r′6 (volume2 ) as in a non-interacting gas. This classical probability is unrelated to thecontact.
With only short-range interactions, the number of pairs close to one another, as parameterized
by the contact, has far-reaching consequences to the system as a whole. Indeed, if we change
the scattering length, the resulting energy change of the system depends on the contact. Thisdependence was ﬁrst derived by Shina Tan in 2005 (published in 2008) [16] and is referred to as
the adiabatic sweep theorem:
dE
~2
=−
C2 ,
d(1/a) s
8πm

(6.3)

where E is the total energy or the gas, and m is the mass of one atom. Combining the adiabatic
sweep theorem (Equation 6.3) with the LHY result (Equation 1.1) gives us)(
5 128 √ 3√
na + ... N0 ,C2 = 16π na 1 +2 15 π22(6.4)

where N0 is the number of atoms in the BEC. Not only can the LHY result be recast in terms ofthe contact, the fractional contribution of the LHY part of the contact has the added beneﬁt of a

57
factor 5/2 compared to the fractional contribution of the LHY energy, aiding its measurement.
Perhaps one of the best aspects of the contact is the way by which we can measure it. If
we look again to Figure 6.1b, we can ask ourselves what other eﬀects are due to this shape of the
wavefunction. To get the momentum distribution nk of the system, one takes the Fourier transformof the two-particle wavefunction. A noninteracting state looks just like a sine wave, giving a delta
function momentum distribution. However, the interacting wavefunction at low a results in a highmomentum tail in the momentum distribution that falls oﬀ as 1/k 4 and is also proportional to thecontact:
lim nk →

k→∞

C2
.
k4

(6.5)

Indeed, this relation is sometimes used as the deﬁnition of the contact. The momentum distribution
can be (and has been) used to measure the contact in fermions [26], but we need a much fasterprobe than time-of-ﬂight (TOF) imaging to avoid the timescale restrictions due to bosonic threebody losses. If we project the atoms into a noninteracting state via RF spectroscopy, the tailin the momentum distribution gives rise to a tail in the RF spectrum with an ω −3/2 dependence[45, 46]. This is a key result, since we can use very short RF pulses to probe the RF spectrum
before signiﬁcant three-body losses can occur. It turns out that atoms in the ﬁnal spin state dointeract with atoms in the initial state, modifying the scattering rate to [27]:Ω2lim Γ(ω) =
ω→∞
4π

√

~ α(a) C2,m β(ω) ω 3/2(6.6)

where the integrated RF lineshape is∫ ∞Γ(ω)dω = πΩ2 N

(6.7)

−∞

and Ω is the single atome Rabi frequency. In Equation 6.6, α(a)/β(ω) describes the ﬁnal-stateeﬀects; the a-dependent part is α(a) = (a′ /a − 1)2 , where a′ is the scattering length for interactions
between atoms in the ﬁnal spin state and atoms in the initial spin state, while the frequencydependent part is β(ω) = 1 + ~|ω|/E ′ , where E ′ = ~2 /ma′2 . For our system, the ﬁnal-state eﬀects
are characterized by a′ = −565 a0 [47] and E ′ /h =133 kHz.

58
Of course, there are limits of applicability of this analysis. In Equation 6.1, the wavefunction
was assumed to continue in the given form all the way to r = 0. However, once the distances arecomparable to the eﬀective range re , the wavefunction can no longer be described by the single
scattering parameter a, and the details of the interatomic potential must be taken into account.
Since these details vary between speciﬁc species of atoms, we no longer consider it the universal~
regime. Hence the above derivation is only valid for k ≪ 1/re , or ω ≪ 2mr
2 . The eﬀective range isegiven by [48]
re =

Γ

( )(
)
Γ 34
mC6 1/4( )~2
8 Γ 54

(1)

6π 2

4
√

(6.8)

where C6 is the coeﬃcient of the van der Waals potential, giving the limit ω ≪ 2π × 3.5 MHz. We
can include a ﬁrst-order correction due to the eﬀective range, which turns out to be small for theranges of a and ω that we probe. A derivation and the eﬀect of this correction can be found in
Appendix A. On the other side of the scale, the limit is set by the interparticle spacing. We only
consider the two-body wavefunction in the limit r → 0, and the above equations are valid only
for k ≫ n1/3 , or ω ≫ ~n2m , where n is the density, resulting in the limit ω ≫ 2π × 300 Hz for
2/3

n = 1013 cm−3 . Experimentally we stay within these limits by about a factor of 10 on each side,
so the contact formulation is valid for all our experiments.

6.2

The contact signal and its characteristics
For our RF contact spectroscopy, the RF drives a transition to a lower energy spin stateand one expects the |ω|−3/2 interaction-induced tail on the low frequency side of the lineshape.
This is schematically illustrated in Figure 6.2a. We drive the interacting |2, −2⟩ state into thefree-particle-like |2, −1⟩ state with extra kinetic energy. Since the absolute energy of the ﬁnal stateis lower, but with added kinetic energy, the transition frequency is lower than the bare transition.
Consistent with this expectation, we observe a tail for large negative detunings, while for similar
detunings on the positive side we ﬁnd that the signal is consistent with zero.
An example of RF contact spectroscopy at a = 1016 ± 10 a0 is shown in Figure 6.2b. We

(a)

59

(b)

(c)

(d)

Figure 6.2: RF contact spectroscopy for a 85 Rb BEC. (a) Schematic of the contact transition.
We drive the interacting state into a lower magnetic sublevel, but the interaction-induced energy
requires less energy from the RF photon to eﬀect the spin-ﬂip. This results in the transition
frequency of the contact to be lower∫ ∞ than the single atom transition. (b) Example of spectroscopy
signal S(ω), normalized so that −∞ S(ω)dω = 1 s−1 . (c) The same signal as (b), shown on asmaller scale. The solid red line is a ﬁt to the expected frequency dependence from Equation 6.6,
while the dotted blue line shows a ﬁt ignoring β(ω). The green line shows the expected Gaussian
signal from the resonant lineshape. On the positive side of the transition the signal is consistent
with zero. (d) The tail signal multiplied by |ω|3/2 . This is how the ﬁts to the tail were performed,
and more clearly shows the ﬁnal-state eﬀects. The density here is ⟨n⟩ = 5.8 × 1012 cm−3 . The datafrom the tail and main lineshape come from diﬀerent experimental runs, but are shown together
for illustration purposes.

60
deﬁne our signal, S(ω), as Γ(ω) normalized by the integrated lineshape. A detailed explanationof the experimental procedure and timings is found in Appendix B. The solid line in Figure 6.2bshows a ﬁt to the expected frequency dependence from Equation 6.6, while the dotted line shows
a ﬁt to |ω|−3/2 . One can clearly see the data deviate from the |ω|−3/2 line, owing to the ﬁnal-stateeﬀects manifested in β(ω). The corrections due to α(a) will be illustrated in Section 6.4.
The atoms outcoupled on the |ω|−3/2 tail have kinetic energy from the interactions, andthe kinetic energy released is given by the detuning from resonance. This excess energy is shared
between two pairwise interacting atoms, meaning that each atom will have a kinetic energy equal to1
2 ~ω. The absolute direction of this motion is random, so on average the energy is divided equally

into the three degrees of freedom. Thus we expect the radius w of the outcoupled cloud to expand√~ωin time t due to the kinetic energy as wKE (t) = 3m
t. Adding our imaging resolution wres = 7.5√
2 . We see preciselyµm in quadrature gives us the expected size versus time, w(t) = wKE (t)2 + wresthis expansion in Figure 6.3a. The Gaussian RF pulse, in this case cut oﬀ at ±2τ , is shown forreference. Here the cloud was still conﬁned in a trapping potential, but as the breathe period is
50 ms, we expect the initial behavior to resemble that of an untrapped cloud. Also shown is themeasured peak OD of the cloud in Figure 6.3b, which falls oﬀ as 1/w2 , and illustrates the beneﬁt
of imaging as soon as possible after the RF pulse to maximize the signal (as described in Chapter
5). Conversely, if we look at the energy of the expanded cloud versus detuning, we again see theexpected kinetic energy, shown by the line in Figure 6.3c. In this case the measurement was taken
in expansion at t = 4.5 ms. We calculate the energy from the width of the expanded cloud using
E = 32 m

6.3

w2 −w02, where w0 is the size of the expanded cloud measured at ω = 0.
t2

Extracting the contact
The derivations in the literature assume an inﬁnitely narrow probe in frequency that measures

the RF spectrum, along with an easily measured Rabi frequency. But in reality, we must limit
ourselves to ﬁnite probe times, which necessarily broadens the RF probe pulse in frequency. In
Chapter 4 I explained that we utilize Gaussian-shaped RF pulses for our contact measurements.

61

RF power

Figure 6.3: Expansion of the outcoupled atom cloud. (a) The size of the cloud of outcoupledatoms after the contact pulse with |ω| = 2π × 50 kHz detuning. The black line shows a sketch ofa truncated Gaussian RF pulse for reference. The cloud expands with a kinetic energy of 12 ~ω, as
the excess energy is shared between two atoms. The red line is the predicted size due to the kinetic
energy, added in quadrature with the resolution limit of our imaging system. (b) The peak OD of
the clouds plotted in part (a). The red line is a ﬁt to a w12 dependence (∼ t12 ), varying only an
overall scaling factor. (c) The energy of the outcoupled cloud as a function of detuning, at 4.5 mstime-of-ﬂight expansion. We calculate the energy from the size of the outcoupled cloud, accounting
for the size of the cloud of non-spin-ﬂipped atoms at ω = 0. The solid line is 12 |ω|
2π .

62
This section describes is greater detail how we rewrite the equations for the contact in terms ofﬁnite Gaussian pulses, and how we extract a value of the contact from our measured quantities.
In Equation 6.6, the single atom Rabi frequency Ω is really a Gaussian function of time given byt2Ω(t)2 = Ω2p e− τ 2 , where Ωp is the peak Rabi frequency of the Gaussian. If we integrate over all
time, we get a slightly modiﬁed version of Equation 6.6:
∫∞
−∞

τ Ω2p
lim Γ(ω, t)dt = Ntail = √
ω→∞
4 π

√

~ α(a) C2,m β(ω) ω 3/2(6.9)

where Ntail is the number of atoms we measure on the tail for a single pulse at ω. We do not directly
measure the Rabi frequency, but instead use the integral of the main lineshape to normalize thedata. This ensures that issues such as imaging eﬃciencies and eﬃciencies of the transfer to theimaging state are common-mode and cancel out. As above, the Rabi frequency in Equation 6.7 isalso a Gaussian in time, so

∫ ∞

t2

−∞

Γ(ω, t)dω = πΩ2p e− τ 2 N.

(6.10)

Integrating this over both time and frequency, and solving for Ω2p , we getΩ2p =√2 Npeak σLS,π Ntot τ(6.11)

where Npeak is the number of atoms we measure on the peak of the transition, Ntot is the total
number of atoms, and σLS is the Gaussian frequency width of the lineshape. If we assume that wehave a condensate fraction f and only BEC number N0 contributes to the contact, then we can
combine Equations 6.9 and 6.11 to solve for the contact per particle:
C2
4π 2
=
N0
I ·f
where I =√()mα(a) Ntail β(ω)|ω|3/2 ,~(6.12)√
2πNpeak σLS is the integrated lineshape. We use diﬀerent powers and pulse lengths

for measuring Npeak and Ntail to outcouple a small fraction of the cloud, so we have to scale thenumbers accordingly. We check that for both cases the outcoupled numbers are linear in time andpower, and correct for a small deviation from linearity. We have found that the deviation seems to
be dependent only on the outcoupled fraction, as long as we stay far below the saturation point of

63
our RF ampliﬁer. In Figure 6.4 we show the deviation from linearity of both the peak (a) and thetail (b) as a function of expected outcoupled fraction fexp , where the expected fraction is calculated
such that it asymptotes to the measured fraction fmeas for low values. The ﬁt is a simple saturation
model given byfmeas =

1
1 ,fexp + faysm
1(6.13)

fmeas
where faysm−fmeas is the deviation from linearity. We outcouple 1-2% of the atoms on the tail of thetransition, and 2-3% on the peak, resulting in a ∼ 5% correction for both. These mostly cancel, so
the ﬁnal correction due to nonlinearities is on the order of 1%.
We also know that non-condensed (thermal) atoms contribute to the contact signal as well, so
we have to account for them. In order to know what our thermal fraction is, we make sure to leave
enough thermal component in the cloud to be able to clearly measure two separate components,
while keeping the condensate fraction as high as possible. This results in a condensate fraction of∼ 70% for most of our data. We then model the contribution of the thermal atoms to the contact
based on the energy density and the spatial overlap of the two components. The total interactionenergy is given by (see Equation 13.15 in [36]),
)
]
∫ [ (
g
128 √
2
2
5
3
Eint =nB (r) + √nB (r) a + 2gnB (r)nT (r) + gnT (r) dr,
215 π(6.14)

where nB is the density of BEC atoms and nT the density of thermal (non-condensed) atoms,
and we have included the LHY correction to the energy in the ﬁrst term. We then use this result
in conjunction with the adiabatic sweep theorem (Equation 6.3) to calculate the total contact
including the thermal contribution. We repeat the calculation for the BEC atoms (just the ﬁrst
term in Equation 6.14), and compare the two to get a fractional contribution from the thermal
atoms, with which we adjust our data. The calculated correction due to thermal atoms is 5-12%
for our range of data. We check that the contact on a purely thermal cloud is consistent with thecalculated value of
Ctherm= 32π 2 na2 .
N(6.15)

64(a)

Peak saturation(b)

Tail saturation

Figure 6.4: Deviation from signal linearity. On both the peak (a) and tail (b) of the transition, we
measure the outcoupled fraction fmeas of atoms vs RF pulse time and/or power, and plot it vs thefraction expected fexp if the dependence were linear. We ﬁt this to the simple saturation modelof Equation 6.13 to extract a value for the asymptote faysm and calculate the magnitude of thecorrection to the data. In (b) we combine data from two diﬀerent scattering lengths for a larger
range in signal. Individual ﬁts to the two sets give the same result within the error bars.

65
The very low signal (due to the low densities) currently prevents us from measuring it with highprecision, but future technical improvements should be able to allow the experimenters to further
explore the thermal contact.
Finally, to compare the contact to the theory prediction given by Equation 6.4, we need toknow both the scattering length and the density of the cloud during the RF probe pulse. Precise
measurements of the Feshbach resonance, which are used to determine the scattering length as afunction of magnetic ﬁeld, were performed in [35]. Since we determine the resonant RF frequency
for every contact measurement, we can exactly calculate the magnetic ﬁeld, which gives us thescattering length a. The error bars from the Feshbach resonance measurements, along with theuncertainty in the magnetic ﬁeld, result in ﬁnal errors on the scattering length of roughly 1% in
the range that we probe (300 to 3500 a0 ). We determine the density of the BEC using a numericalmodel based on mean-ﬁeld interactions (the PG model described in Section 5.1.1), calculating thespace- and time-averaged density of the cloud after ramping to various values of a. Since we havea weak spherical trap and a fast probing technique, our average density does not drop more than
20% for fast ramps in a over our full range of scattering lengths.

6.4

Contact measurements
To investigate LHY physics represented by Equation 6.4, we measure the contact for diﬀerent

values of the scattering length. Here is where the ﬁnal-state eﬀects due to α(a) manifest themselves.
The contact signal comes from the fact that we project the wavefunction of the interacting particles
onto a state with diﬀerent interactions. If the interactions of that ﬁnal state were exactly the sameas the initial state, we would not expect to outcouple any atoms. Thus the size of the signal ishighly dependent on the diﬀerence between the scattering lengths of the ﬁnal and initial state. The
eﬀect of α(a) then, is to shift what would be a parabola centered about a = 0 to one centered abouta′ = −565 a0 , which enhances our signal at small a. This eﬀect can be seen in Figure 6.5. The
solid line is the mean-ﬁeld prediction including ﬁnal-state eﬀects, whereas the dotted line ignores
them. For these data, the space-averaged density has been averaged over the diﬀerent points to66

α(a) C2
Figure 6.5: The contact vs a, measured at |ω| = 2π × 40 kHz. (a) Here I plot β(ω)
N0 , which is

directly proportional to the strength of the measured ω −3/2 RF tail. (b) The contact per particle
C2
N0 . The solid lines in (a) and (b) are the mean-ﬁeld predictions. The ﬁnal-state eﬀects shift what isa parabola centered about a = 0 in (b) to one centered about a′ = −565 a0 in (a), which enhances
our signal at small a.

67

(a)

(b)

Figure 6.6: Contact measurements as a function of the LHY energy ELHY . (a) The contact per
particle normalized by density1/3 . The red curve shows the mean-ﬁeld prediction for the contact,
and the blue curve shows the total contact including the LHY term. In (b) the data are normalized
to the mean-ﬁeld value of the contact to compare to theory more clearly. In the limit of low
interaction strength, the data match the theory but cannot distinguish the LHY contribution. At
higher values, the data are systematically low. At values of ELHY approaching unity, we do not
expect the perturbative LHY theory to be valid.

68
be ⟨n⟩ = 5.8 × 1012 cm−3 . Since the data generally have varying densities, depending on the
scattering length that we ramp to, we have to normalize out the density. If we normalize C2 to theinterparticle spacing n1/3 and BEC number N0 , we can rewrite Equation 6.4 to give the contact inits dimensionless form C2′ :C2′ =))4/3 ((√
C2
5 128 √ 3
2
3
√
na
1+
na + ... .
= 16π
2 15 πN0 n1/3(6.16)√√In Figure 6.6a we plot C2′ versus ELHY , where ELHY = 15128
na3 is the value of the fractionalπ
LHY correction to the energy. The red line is the mean-ﬁeld prediction, and the blue line includes
the contribution from the LHY term. The advantage of our measurement method is clearly evident
in the range of the x-axis. We can measure the contact up to such high interaction strengths that
ELHY can no longer be said to be a perturbation on the mean-ﬁeld energy, and we no longer expect
the LHY result to hold.
In order to see the data more clearly, we then normalize the contact per particle by themean-ﬁeld prediction to get
C2=
N0 16π 2 na2)(5
1 + ELHY + ... ,2(6.17)

shown in Figure 6.6b. The mean-ﬁeld value is now 1, and the LHY term is a line with a slope of
5/2. For the lowest values of ELHY , where the separation between the mean-ﬁeld and the LHY
prediction is on the order of the error bars, the data show excellent agreement with the predictedvalue. At values of ELHY approaching unity, we would not expect the perturbative LHY theory to
be valid, so the mismatch between data and theory comes as no great surprise. However, we do
expect the LHY theory to be valid in the intermediate regimes, so we must take a closer look at
our methods.
When we change the scattering length, we ramp a as quickly as possible to avoid the destructive three-body loss rates. The resulting increase of interaction energy brings the system out ofglobal equilibrium and starts a breathe oscillation of the cloud. However, an ongoing assumption
is that the system is in equilibrium locally, because we change a on a timescale that is adiabatic

69
with respect to the binding energy of a dimer, with ȧ/a never reaching more than 0.01~/(ma2 ) (ȧbeing the time derivative of a). One might then wonder if we are possibly ramping too quickly
for the physics underlying the LHY correction to evolve. Going one step further, we can ask what
determines the relevant timescale. One (possibly naive) guess would be to say that the relevant
2time scale is given by the mean-ﬁeld chemical potential µ = 4π~m na. We then look at how large thefractional rate of change µ̇/µ of this energy is, compared to the characteristic time given by thatenergy, µ/~. This gives us a unitless “adiabaticity paramater” (R) for our ramps,
R=

µ̇/µ
.
µ/~

(6.18)For the data in Figure 6.6, R ranges ranges between 2 and 3. When we ramp to our ﬁnal scattering
lengths, we ramp in such a way to keep R roughly constant throughout the ramp.
In Figure 6.7 we show the results of changing our ramp rates at two values of ELHY . Herethe data are normalized to represent where, fractionally, they lie between the mean-ﬁeld and LHY
predictions. The red line at 0 represents the mean-ﬁeld contact, and the blue line at 1 is the fullcontact with the LHY term. We can clearly see a dependence on the ramp rate, suggesting that
we are indeed ramping too quickly for the LHY energy to completely manifest itself. Also, thefact that the value of the contact is in between the mean-ﬁeld and the LHY values right aroundwhen our R is equal to 1, suggests that the chemical potential is the relevant energy scale. For
lower values of ELHY , the LHY term is too small compared to the error bars to discern a ramp ratedependence.
A more convincing argument would be to show that the contact actually saturates to the
LHY and mean-ﬁeld values for very slow and very fast ramps, respectively. Unfortunately, achieving
these rates present a nontrivial experimental challenge. The fastest ramps we are currently able toachieve are limited by the gain of our servos and the inductance of our magnetic-ﬁeld coils. This
limit could be surpassed by installing a separate low-inductance coil pair driven by a capacitor
bank, which could change the ﬁelds by the necessary amount as fast as ∼ 5 µs [49], giving us
another factor of 20 in ramp rate. This is quite feasible, and it is likely that the experiment will

70

slowerm& m
m h

fasterslowerm& m
m h

fasterFigure 6.7: Contact measurements as a function of ramp rates, expressed in terms of the “adiabaticity parameter” R. Higher values of this parameter correspond to faster ramps, which show avalue of the contact closer to the mean-ﬁeld prediction. Slower ramps result in higher values closer
to the LHY prediction. Diﬀerent shapes correspond to diﬀerent days on which the data were taken.
The small upper plot is a reproduction of Figure 6.6b, to illustrate the values of ELHY shown here.
For lower interaction strength, the LHY term is too small compared to the error bars to discerna ramp rate dependence. To calculate the mean-ﬁeld and LHY energies, we require knowledge ofthe cloud density. The change in densities due to the ramp in a were calculated using a model for
expansion that included only the expected mean-ﬁeld energy, and vary from 40% for the slowest
ramps to 10% for the fastest. This reasoning is somewhat circular, but including the LHY term inthis model causes at most an extra 5% decrease in density for the slowest ramps, and the resulting
change is shown by the open circles.

71
move in this direction in the not-too-distant future. The other extreme for the ramp rate is more
problematic.
Achieving very slow ramp rates is technologically trivial, but two major issues prevent us
from easily obtaining trustworthy measurements. One of these is the change in density. When wechange a, the resulting change in interaction energy causes the cloud to expand for the start of abreathe mode. We rely on the PG model to calculate the change in density of our cloud from theinteraction-induced breathe and obtain the value of ELHY . But the results of that model depends
on the energy of the gas, and whether or not we must include the LHY corrections to the energy.
For fast ramps the density drop is on the order of 10%, and the inclusion (or exclusion) of the
LHY energy in the PG model will not incur a signiﬁcant total error. For slow ramps the density
will drop more like 30% from the initial density, and exact knowledge of the energy is much more
important. For the slowest ramps shown in Figure 6.7, using the LHY rather than the mean-ﬁeld
energy causes a density change of ∼ 5%, which moves it closer to the LHY prediction as shown by
the open circles. Because of this, measuring an LHY term tends to be circular when large density
corrections are made. The other major issue is the ever-recurring problem of three-body losses. RF
contact spectroscopy is an appealing measurement technique because it can be performed quickly
to avoid the loss issues. Taking a long time to ramp a means we once again have to account forsigniﬁcant losses in the condensate for every measurement. Again, this is a fundamental issue
inherent in bosons.
To summarize, we have used RF contact spectroscopy to measure beyond-mean-ﬁeld eﬀects instrongly interacting BEC. At low interaction strengths, where the LHY term is a < 5% perturbationon the energy, the measured contact shows excellent agreement with theory. At higher interaction
strengths, we see clear deviation from the mean-ﬁeld contribution to the contact. These beyondmean-ﬁeld eﬀects are dependent on ramp rate, and evidence suggests that the relevant time scale isdetermined by the chemical potential. Nonetheless, RF contact spectroscopy is a powerful tool for
probing strongly interacting Bose gasses before three-body losses become signiﬁcant, even letting
us reach a regime where the LHY contribution to the energy is calculated to be of order 1.

Chapter 7

Three-body Contact

Beyond some prefactors in the formulas due to particle statistics, the two-body contact
applies equally to bosons and fermions. However, the possibility of three-body interactions inbosonic systems means that one has to take into account the possibility of a three-body contact.
The three-body contact, similar to its counterpart, measures the probability for triples of identicalbosons to be close to one another [31]. Continuing the analogy, just as the two-body universal
parameter is the scattering length, a, the universal parameter describing the three-body physics isthe Eﬁmov parameter κ∗ . An important thing to keep in mind is that the three-body contact also
connects a few-body interaction parameter with many-body eﬀects. One fundamental property ofEﬁmov physics is that it predicts an inﬁnite series of successively more weakly bound trimers withdiscrete scaling factor eπ/s0 ≈ 22.7, where s0 ≈ 1.00624 for identical bosons [50]. A sketch of thisbehavior can be seen in Figure 7.1. The parameter κ∗ can be deﬁned by the energies of trimers at
unitarity, where a → ∞, as
Etrimer =~2 κ2∗ ( −2π/s0 )l
e
,
m

(7.1)

where l is an integer, and m is the mass of one atom. In this way, we can understand κ∗ to be thecharacteristic momentum of the most tightly bound trimer, and its inverse gives the approximate
size of the trimer molecule [50].
The three-body contact C3 has an “adiabatic sweep theorem” like Equation 6.3, which deﬁnes

73

K

√
Figure 7.1: A sketch of Eﬁmov states. The vertical scale gives the wave number K = ± m|E|/~2 ,and the horizontal gives 1/a, such that the origin corresponds to unitarity (a → ∞). The green
line shows the state corresponding to a bound dimer plus a free atom, and the blue lines show thebound trimers, for which there is an inﬁnite series approaching unitarity. Note that the scaling
factor here is ∼ 2 to make the behavior visible and to match the literature on Eﬁmov states. A
realistic Eﬁmov state with scaling factor 22.7 would have an energy that is nearly indistinguishable
from the dimer energy on the scale of this plot.

C3 in terms of the derivative of the energy
(
)
∂E
2~2
κ∗
=−
C3 .
∂κ∗ a
m

(7.2)

The three-body contact also manifests itself in an additional contribution to the tail of the momentum distribution that goes as
lim n(k) =

k→∞

F (k)
C3 ,
k5

(7.3)

where the log-periodicity of Eﬁmov physics manifests itself in the function
F (k) = A sin[2s0 ln(k/κ∗ ) + 2ϕ](7.4)

with numerical constants A = 89.2626 and ϕ = −0.669064 [31]. This results in an additional term
to the RF tail at large detunings that should be added to the right-hand side of Equation 6.6:~Ω2 GRF (ω)
C3 .
2m
ω2

(7.5)

Here, GRF (ω) is a log-periodic function given by[
GRF (ω) = 9.23 − 13.6 sin s0 ln

(

mω
~κ2∗

)

]
+ 2.66 .

(7.6)

74
As of the writing of this thesis, there is no prediction for ﬁnal-state eﬀects on the C3 contribution
to the RF tail.
Our aim, then, is to see if we can ﬁnd a contribution to the interaction-induced RF tailfrom three-body eﬀects, or to show that possible three-body eﬀects are dominated by two-bodycorrelations and can be ignored in measurements of the contact. Although four-body interactions(or greater) can occur, it is expected that they do not require extra universal parameters, whichmeans that there is no need to deﬁne a four-body (or higher) contact. The total contact in theuniversal regime can be completely described by C2 and C3 [51].

7.1

The Eﬁmov resonance
To measure a three-body contact via Equation 7.5, we need to know the value of κ∗ . This

will tell us the frequencies for which GRF = 0 and the ones for which GRF is high, which can inturn aid us in maximizing the chances of detecting C3 . One way to measure κ∗ , which is a value
unique to each atomic species, is to ﬁnd the value of the scattering length a− on the negative sideof the Feshbach resonance where the energy of a trimer becomes degenerate with the energy ofthree free atoms (see Figure 7.1). At this value of a we expect that the probability of three atoms
close together is resonantly enhanced, manifesting itself in a peak in the three-body recombination
rate. Similar measurements of Eﬁmov resonances have been reported for several other ultracold
atom systems [52, 53, 54, 55, 56]. For identical bosons, the value of a− is related to the three-bodyparameter through κ∗ = −1.56(5)/a− [57].
We measure the 85 Rb Eﬁmov resonance using non-condensed clouds of 1.5×105 atoms at atemperature T = 80 nK. We ramp the magnetic ﬁeld to realize the desired scattering length on thea < 0 side of the Feshbach resonance and wait for a variable time t. We then turn oﬀ the magnetic
ﬁelds (except for a small quantization ﬁeld), ARP all the atoms to the |3, −3⟩ imaging state, anduse high-intensity imaging (see Section 5.1.3) to probe the clouds at 3 ms expansion time. This
time is small compared to the trap frequency, so the clouds still exhibit position information (weexpect the cloud size to have increased by 2% during this time). We measure the number and size

75
vs time, and extract the three-body event rate constant K3 for various values of the scattering
length.
The three-body event rate constant for a homogeneous gas with a number density n is deﬁnedbydn = −3K3 n3 ,
dt

(7.7)

where the factor of 3 in front comes from the fact that, in our case, three atoms are lost per
event. But since losses also cause heating, which contributes to a decrease in density, we integrate
Equation 7.7 over all space for the number loss rated
N = −3K3 ⟨n2 ⟩N,
dt

(7.8)
∫

n3 dr

where ⟨n2 ⟩ is the density-weighted mean square density ⟨n2 ⟩ = ∫ ndr . Our 500 s vacuum-limited
lifetime and previous experiments on 85 Rb suggest that one- and two-body losses can be ignored
for the range of magnetic ﬁelds we probe [58]. For a thermal distribution in a harmonic trap, we
can rewrite this as
8 N3d
N = −3K3 √,dt
27 V 2(7.9)

where the volume V = 8π 3/2 w3 and w is the rms width of the spherical cloud. We perform a linear
ﬁt to the volume of the cloud given by V (t) = A + Bt. Plugging this into Equation 7.9 and solving
for N (t) gives us our ﬁtting function for the rate:√
N (t) = N0

A(A + Bt).
√48 N 2 K3 t + A(A + Bt)
027(7.10)

Figure 7.2 shows an example of this procedure. The volume vs time is ﬁt in Figure 7.2a,setting A and B for the ﬁt to the number data in 7.2b via Equation 7.10. Using this procedure,
we extract K3 values for a range of a, plotted in Figure 7.3. We ﬁt the measured K3 vs a to the
expected form for an Eﬁmov resonance for non-condensed atoms [57],K3 =4590 sinh(2η)~a4.
sin2 [s0 ln(a/a− )] + sinh2 η m(7.11)

76

(a)

3

Volume ( m )

200
150
100
50
0

1.5

5

Number (10 )

(b)

1.0

0.5

0.0
0

1

2

3

4

time (s)Figure 7.2: Example of a loss rate measurement to extract K3 . These data correspond to ascattering length of 730 a0 . Heating of the cloud causes its size to increase as seen in (a). The
parameters extracted from a linear ﬁt to the volume go into a ﬁt to the atom number via Equation
7.10, as seen in (b), resulting in a value of K3 = 5.64 × 10−22 cm−6 /s.

77

6cm /s)

10

0.1

3

K (10

-22

1

a = 1/k
0.01

3

-10

scattering length

thermal4-10a (a )
0

Figure 7.3: A three-body loss resonance for 85 Rb. We plot the three-body event constant K3 vsthe scattering length a, for clouds with a temperature of roughly 8 nK. From ﬁtting Equation 7.11
to the black points, for which a < 1/kthermal , we extract a− = −759(6)a0 and η = 0.057(2).

78
Because this expression comes from a T = 0 theory, we only ﬁt the data for a < 1/kthermal ,√where kthermal = 2mkB T /~ and kB is Boltzmann’s constant. From the ﬁt, we extract a− =−759(6) a0 and η = 0.057(2). The error in a− is relatively small because of the fact that the
Feshbach resonance for 85 Rb was measured with very high precision in [35], and the uncertainty
from those measurements dominate our errors. Uncertainties of our measurement of the magnetic
ﬁeld or from the ﬁt to Equation 7.11 are negligible. In ﬁtting our observed loss rates to Equation
7.11, we initially included an overall multiplicative scaling factor, to account for systematics in our
measurements, but found it to be consistent with one. We do not expect to be able to measureanother resonance a factor of 22.7 more negative in scattering length (-17,000 a0 ) due to the ﬁnite
temperature of the gas. A factor of 22.7 lower (-33 a0 ), is smaller in magnitude than the van der
Waals length, where universality no longer applies.
Since the η parameter essentially determines the width of the resonance, we can think of itas proportional to the inverse lifetime of the trimers. The trimers can decay into a deeply bounddimer and a free atom. The width can conceivably increase if the temperature of the cloud is too
hot, due to the increased thermal de Broglie wavelength moving the system closer to unitarity. To
check that we are in the regime where η is independent of temperature, we repeat the measurementof η for 30 and 140 nK. The results are plotted in Figure 7.4. We can see that over a factor
of four in temperature, η does not show signiﬁcant change, suggesting that we are indeed in the
low-temperature limit.
The value of a− is not expected to be universal, since it comes from the details of the shortrange potential [15, 59, 57]. However, it gets interesting when expressed in units of the mean
scattering length of the van der Waals potential [60]( )()
mC6 1/4
1 Γ 34)(,ā = √~2
8 Γ 54(7.12)

where C6 is the coeﬃcient of the van der Waals potential. For 85 Rb, ā = 78.5 a0 , so we ﬁnd avalue for aā− of -9.67(7). Note that in other works, a− is sometimes normalized instead by the van( 6 )1/4[34]. It turns out that other experiments that have measured
der Waals range RvdW = 12 mC
~2

79

(a)

(b)

Figure 7.4: Measurement of η vs temperature. (a) The loss resonances. For the data at 30 and 140
nK, we did not calibrate the scattering length as carefully as the data at 80 nK, which is the samedata as in Figure 7.3 shown over a smaller range of a. Moreover, the 30 nK clouds are likely not inthermal equilibrium, and the ensuing error in the average density will cause a systematic error in
the calculated value of K3 . (b) Measured values of eta vs temperature. We do not see a signiﬁcantchange in η over a large range in temperature, suggesting that the experiments are performed in
the low-temperature limit.

80
Eﬁmov resonances have found very similar result in 133 Cs [61], 6 Li [62], and 7 Li [55, 56, 63],
with values for aā− ranging 8-10. This empirical evidence suggests that the three-body parameter
depends only on the coeﬃcient of the 1/r6 part of the two-body potential and not on the detailsof a three-body potential at short range [61].

7.2

Searching for C3
From the measured location of the Eﬁmov resonance, we can extract κ∗ =39(1) µm−1 . This

gives us the expected shape of GRF (ω), which is plotted in Figure 7.5. Note that GRF (ω) has anode at ω ≃ 2π × 27 kHz and a smaller magnitude at larger ω. Since the C3 term in Equation 7.5
goes as GRF (ω)/ω 2 , this suggests that the largest contribution from C3 will be for ω < 2π × 27 kHz.
Like the C2 term, the prediction for the C3 term is valid for ω → ∞. For the case of the C2 term,
the RF tail arises from two-body short-range correlations at distances that are small compared tothe interparticle spacing, which is always satisﬁed. However, for the case of C3 , the prediction for
the three-body contribution to the RF tail may have a more limited range of applicability [64]. In~
particular, the C3 theory may only be applicable for ω > ma
2 [65], where the frequency dependence

makes it less likely to contribute signiﬁcantly to the RF tail.
The results of our search for C3 can be seen in Figure 7.7, where we examine the frequency
dependence of the RF tail for a BEC at a = 982 ± 10 a0 . To maximize our chances of measuring
a C3 contribution, we increase the density of the clouds by 65% by jumping the scattering lengthfrom 100 a0 to 50 a0 and waiting 1/4 of the trap period before ramping to the ﬁnal value of 982
a0 . This creates a density oscillation as seen in Figure 7.6, which we predict using the PG model(see Section 5.1.1). We ﬁt the data to the predicted frequency dependence of the C2 contribution,
shown by the solid line. The dotted line is the same ﬁt but shown without including the ﬁnal-statecorrection 1/β(ω). We can see that our data ﬁt very well to the expected frequency-dependence
for the two-body contact with ﬁnal-state eﬀects, and we do not observe any deviation consistent
with a three-body term. Fitting the data to both contributions gives an upper limit for C3 /N0 of
0.07 µm−2 . Also shown is a trial C3 /N0 term of 0.1 µm−2 with the dashed line. For reference,

81
20

G

10

RF

0

-10
-150

-100

-50

/2

(kHz)

-0

Figure 7.5: GRF as a function of ω, with κ∗ =39(1) µm−1 . We plot GRF for negative ω to ease
comparison with the measured RF tail, which occurs at negative detunings in our experiment. GRF
has a node at ω ≃ 2π × 27 kHz, suggesting that one should look for a C3 contribution to the RF
tail for smaller detunings.

RF pulse

1000

2.0

800

1.5

600

1.0

400

0.5

200

0.0
-30

-25

-20

-15

-10

-5

0

0

(

a a
scattering length

relative density

0

)

2.5

time (ms)Figure 7.6: Changing the scattering length for increased density. We jump the scattering lengtha (green curve) to 50 a0 , causing an inward breathe with a period of roughly 50 ms. Close to theturnaround point we ramp to 982 a0 and apply the RF pulse to measure the contact (grey line).
The blue curve shows the prediction for the relative density of the PG model.

82

G

20

RF

0

-6

10

S

(a)

(b)

-7

10

-8

10

-100

-10

/2

(kHz)

Figure 7.7: (a) The calculated frequency dependence of GRF (ω), shown again on a logarithmic ω
scale for reference. (b) The measured frequency dependence of the tail of the RF spectrum for
a = 982 ± 10 a0 . The solid red line is a ﬁt to the expected frequency dependence of the two-bodycontact C2 /N0 including ﬁnal-state eﬀects. The dotted blue 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 black line. Our measurements are consistent instead with a C3 /N0 of
zero. For these data the density is ⟨n⟩ = 1.0 × 1013 cm−3 .

83~= 277 kHz for this scattering length.
ma2

Another idea was to look for a C3 contribution on the peak of the Eﬁmov resonance. The
thought was that the ﬁnal-state eﬀects suppress the two-body contribution to the contact, while
the presence of the Eﬁmov resonance might enhance the contribution from the three-body contact.
Since a BEC begins to implode at negative scattering length, we ﬁrst create condensates at 100
a0 , then we decrease the density by a factor of 5 by jumping a to 400 a0 and waiting 1/4 of a
trap period, before jumping the magnetic ﬁeld to the Eﬁmov resonance. This should give us ∼4
ms to measure the RF spectrum before the condensate begins to collapse locally (see Section 5.1.2
for an explanation of this prediction). The results of that test are shown in Figure 7.8, where allmeasurements of the tail are consistent with zero. The error bars roughly represent our detection
limit. The two-body contact prediction is shown by the green line, which is below our detection
limit due to the aforementioned ﬁnal-state suppression. The blue line shows a heuristic prediction
for the three-body contact from the loss rate, given by [65]~dN = −4η
C3 .
dt
s0 m

(7.13)

Given that this prediction lies signiﬁcantly above our detection limit, the data suggest that the~
ω > ma2 limit does deﬁne the range of applicability for the C3 theory, and/or that ﬁnal-state eﬀects

for the three-body contribution also cause signiﬁcant suppression of the signal, or that Equation
7.13 is incorrect. Of interest is the prediction of Equation 7.5, that if the scattering length werechosen such that the C2 term is zero, the transition rate goes negative, indicating that the theory
is at this point still incomplete.
In short, our investigations for a three-body contribution to the contact have shown thatthe short-range correlations in the BEC are dominated by two-body eﬀects, as we see no clear
signature of three-body eﬀects in the frequency dependence of the interaction-induced tail in RF
spectroscopy.

84

-4

10

-6

10

S
-8

10

-10

10

-30

-20

/2

-10

0

(kHz)

Figure 7.8: Searching for a tail in the RF spectrum at -760 a0 . We detected no signal indicativeof a tail on the RF spectrum. The error bars give an estimate of our detection limit. The green
line shows the prediction for the contribution to the signal S from the two-body contact, which
is highly suppressed due to ﬁnal-state eﬀects and expected to be below our detection limit. A
heuristic calculation from loss rates via Equation 7.13 predicts a three-body contribution shown by
the blue line. The density here is ⟨n⟩ ≃ 7 × 1011 cm−3 .

Chapter 8

Summary and Outlook

8.1

In Essence
In this thesis I presented an experiment on 85 Rb BEC near a Feshbach resonance, for which wemeasured the contact via RF spectroscopy. To fulﬁll the technical requirements for this experiment
we stabilized our trapping magnetic ﬁelds to ∼ 2 mG rms ﬂuctuations, and minimized the magnetic
ﬁeld variation across the cloud to a ∼ 4 kHz rms width on a 0.47 MHz/G transition. We also created
Gaussian-shaped RF probing pulses without stray spectral components down to 60 dB below thepeak power. To probe our BECs we developed imaging technology that easily spans imaging inmagnetic ﬁelds of zero to upwards of 200 G, giving us the ability to probe clouds with optical depths
greater than 6. We also developed a technique to eﬃciently measure only the atoms outcoupledfrom the tail of an RF lineshape, without letting the 99% of atoms in the original state contaminate
our images.
With these technical requirements resolved, we measured the two-body contact, experimentally probing the tail of the RF spectrum to high precision. We clearly see a deviation from the
1/|ω|3/2 tail consistent with predictions for the ﬁnal-state eﬀects. From the ratio of the transition
rate on the tail to the resonant transition rate, we extracted a value for the two-body contact. We
then measured the contact for varying values of the scattering length, ranging from a regime that iswell-described by the mean-ﬁeld approach, through a regime where we expect the correction to theenergy to be described by the Lee-Huang-Yang (LHY) formalism, all the way to a regime where na3
is so high that we no longer expect the LHY theory to be valid. In the low a regime our data match

86
the mean-ﬁeld prediction, and at higher interaction strengths we see beyond-mean-ﬁeld behavior.
The data do not match the LHY prediction, likely due to the physics underlying the LHY energy
not evolving adiabatically with fast changes in the interaction strength. When we vary the ramp
rate, we see a clear trend towards the mean-ﬁeld prediction for the contact for faster ramps andtoward the LHY prediction for slower ramps. This opens up an exciting area of research regarding
the dynamics of strongly interacting bose gasses.
We also investigated the possibility of a three-body contact, related to Eﬁmov physics. In
order to predict the shape of the three-body contribution to the RF tail of the frequency spectrum,
we measured the location of an Eﬁmov resonance on the a < 0 side of the Feshbach resonance. We
measured loss rates on non-condensed clouds to extract the three-body event rate constant K3 as afunction of the scattering length. This gives us a resonance located at a− = −759(6) a0 and a width
given by η = 0.057(2). Evidence suggests this data was taken in the low-temperature regime. Fromthe location of the Eﬁmov resonance we calculate the three-body universality parameter κ∗ = 39(1)µm−1 . Our subsequent measurements for a three-body contribution to the RF tail are consistent
with zero. 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. 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 and probe non-equilibrium many-body dynamics.

8.2

The next few weeks...
Strongly interacting systems have long been a challenge for theorists and experimentalists

alike, and the 85 Rb experiment is poised to investigate much that has not been explored by eithergroup. The dynamics of the LHY correction that we have already measured lack a theoreticalfoundation, and the experiments on those dynamics will likely be reﬁned and extended to give
clear results in the future. The two-body contact in general has proven such a useful tool thatthe experiment will probably continue to use it to study strongly-interacting BECs. Speciﬁcally,work is currently in progress to measure the contact at unitarity, where the scattering length is no87longer a relevant length scale. Instead, the interparticle spacing becomes the relevant parameter,changing the density dependence of three-body losses. Also, I imagine that the book on the threebody contact away from unitarity is not yet completely closed, as the current team will come up
with new ideas and techniques to search for it in diﬀerent regimes. It will be exciting to see whatsorts of new and interesting science the 85 Rb experiment (through its capable keepers) will reveal
in the future.

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Appendix A

Eﬀective range contribution

We calculate the transition rate for a dimer for a ﬁrst-order approximation to include eﬀects
from the short-range potential in the two-body contact. We start with Equation 7 of [66] for theFrank-Condon factor between the initial state with scattering length a and the ﬁnal state with a′ ,F (K) =

2m
(1 + k 2 a2 )−2 (sin δ ′ + ka cos δ ′ )2 ,
π~2 k

(A.1)

1
r′
k cot δ ′ = − ′ + e k 2 + ...,
a
2

(A.2)

where δ ′ is given by

and the eﬀective range re′ is deﬁned in the limit |a′ | ≫ ā by
( )

4
Γ 14
′
ā.
re =
2

6π

(A.3)

The mean scattering length of the van der Waals potential ā is given by( )()
1 Γ 34mC6 1/4
(
)
√
ā =
.
~2
8 Γ 54(A.4)

This results in an eﬀective range for 85 Rb of 229 a0 . Following the derivation in [66], we express
k 2 a2 = K/Eb and k 2 a′2 = K/Eb′ where, K = ~2 k 2 /m and Eb = ~2 /ma2 is the binding energy, andcombine Equations A.1 and A.2. After quite a bit of algebra, this comes out to be
2
F (K) =
π
′

( ′
)2
1/2
K 1/2 Eb Eb′a− (1 − ρ),a(K + eb )2 (K + Eb′ (ρ − 1)2 )

(A.5)

2 ′

where ρ = re k2 a . If we used only the ﬁrst term in Equation A.2, then ρ = 0 and Equation A.5
reduces to Equation 10 in [66]. Now we deﬁne the RF oﬀset energy E = Eb + K, and take the limit

93of E ≫ Eb (large detuning) to get
( ′
)2
a
1/2
−
(1
−
ρ)
a
2 Eb
(
).
F (K) =
3/2
πE
(1 − ρ)2 + EE′
b

(A.6)

Now we plug in E = ~ω and Eb = ~2 /ma2 to put things into more familiar units:( ′F (K) =)2aa − (1 − ρ)

2
1
(
).
√
3/2
π aω
~m (1 − ρ)2 + ma′2 ω

(A.7)

~

We can also substitute in the two-body contact for the dimer, C2 = 8π
a , using the adiabatic sweep2

theorem (Equation 6.3). Finally, we use Γ(ω) = hΩ
2 F (ω) from Equation 3 of [66] to getΩ2Γ(ω) =
4π√~ α′ (a) C2,m β ′ (ω) ω 3/2(A.8)

where the only diﬀerence between this result and Equation 6.6 is a modiﬁed deﬁnition for the terms
that include ﬁnal-state eﬀects:( ′)2aα (a) =− (1 − ρ)a)(ma′2 ω
′2.
β (ω) = (1 − ρ) +
~
′

(A.9)
(A.10)The eﬀect of including this eﬀective range contribution is seen in Figure A.1. It may bemeasurable as a slight deviation from the ω−dependence, but the expected deviation is on theorder of our error bars and as of yet too small to detect. Since the total contribution of ﬁnal stateeﬀects to the contact go as β(ω)/α(a), the corrections largely cancel, as seen in Figure A.1c.

94

(a)

(b)

(c)

Figure A.1: Eﬀects of the ﬁrst-order correction due to a ﬁnite eﬀective range. (a) and (b) show
the fractional corrections introduced by using α′ (a) and β ′ (ω), respectively. For the calculation ofα′ (a), a detuning of ω = 2π×40 kHz was assumed. The magnitude of the eﬀect roughly corresponds
to the size of the error bars in our data, putting it just below our detection limit. (c) The combined
eﬀect on the total contact is on the order of a few percent.

Appendix B

Detailed procedure for the contact measurements

The procedure described below is typical for our experiment, but may not be the exact
procedure used for any speciﬁc set of data. Before and after a contact measurement, we takemeasurements of BEC number using high-ﬁeld and high-intensity imaging as described in Chapter
5. We check that those measurements do not vary greatly, and average them to determine thecondensate number during the contact measurements. The contact measurement itself consists ofthree parts, for which the magnetic-ﬁeld ramps are identical. For mostly historical reasons, I willdeﬁne 0 ms to be the time when we turn oﬀ the trapping ﬁelds. We ramp oﬀ the large shim coils
to minimize the magnetic-ﬁeld gradient at -4 ms in 0.5 ms, while simultaneously ramping the bias
ﬁeld to compensate and keep the scattering length constant. At -3.5 ms, we begin a 1 ms ramp in
the bias ﬁeld to reach our ﬁnal value of a. The Gaussian RF pulse is always centered on -1.5 ms.
We ﬁrst determine the center frequency of the single-atom transition. For this measurement
we make non-condensed clouds, and probe the |2, −2⟩ to |2, −1⟩ transition with Gaussian pulsesof an rms width τ = 100 µs, outcoupling 10-20% of the atoms with a Gaussian line width of ∼ 4
kHz. We also include the RF pulses that normally transfer the |2, −1⟩ atoms to the |3, −3⟩ imaging
state, but detune them by 5 MHz to avoid this transfer. This ensures that any RF rectiﬁcation
will be common-mode between the measurement of the center frequency and the measurement of
the RF tail. We then turn oﬀ the magnetic ﬁelds at 0 ms, and expand for 15 ms, applying a
vertical magnetic gradient for 5 ms to physically separate the magnetic sublevels. This imaging is
performed like standard absorption imaging, at low ﬁeld and with low probe intensities. We ﬁnd

96that ﬁtting both clouds and looking at the outcoupled fraction signiﬁcantly reduces noise due tonumber ﬂuctuations, enabling us to reach < 1 kHz uncertainties in the center frequency with 5-6
well-picked points on the lineshape.
Once we know the single-atom frequency, we calculate and set the resonant frequencies for the
RF transfer to the imaging state, as well as the probe laser frequency. To measure the integratedlineshape of the BEC, we tune the |2, −2⟩ to |2, −1⟩ RF to resonance and probe with a short, τ = 5µs Gaussian pulse, outcoupling on the order of 2-3% of the atoms. Such a short pulse is always
pulse-width limited, and we only need the peak of the lineshape to determine its integral to withina small correction described in Section 6.3. The RF transfer pulses occur at -1 ms and -0.9 ms topump the atoms to the imaging state. We then probe at high-ﬁeld (in-trap) but with low intensity,using a 25 µs probe pulse.
For the measurement on the tail of the transition, we change only the initial RF pulse lengthto τ = 100 µs, and increase the power, always outcoupling ∼ 1 − 2% of the atoms. Since we“truncate” our Gaussians at ±4τ , this pulse “begins” at -1.9 ms (600 µs after the end of themagnetic ﬁeld ramp) and ends at -1.1 ms (100 µs before the ﬁrst RF transfer pulse). In order toavoid systematics caused by experimental drift, we alternate between measuring the peak and thetail of the transition. A sketch of the timings can be seen in Figure B.1.

scattering length a (arb)RF
R to tF ra
in to t ns
-tr ra fe
ap ns r t
im fer o |3
ag to ,-2
i n |3 >g ,pu 3>
ls
e

ct
on
ta
R

F

ra
m

p

fo

rc

en
d

rt
st
a
p
ra
m

intensity of RF or probe (arb)

be
gi

n

gr
ad
ie
nt

of
f

97

ail
ent
for t surem
mea

of
eak
for p hape
s
line

0

0
-4

-3.5

-2.5

-1.5

-0.8

-1
-0.9

0

time (ms)Figure B.1: Timing schematic of a typical contact measurement. The vertical scale is arbitrary.
The widths of the grey RF pulses and the red probe pulse have been exaggerated to show up at
this scale. The relative timings and the width of the blue RF pulse are to scale.