Thermometry and Cooling of Ultracold Atoms in
an Optical LatticebyPatrick Medley
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree ofDoctor of Philosophy
at theMASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2010
c Massachusetts Institute of Technology 2010. All rights reserved.

Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Physics
June 8, 2010
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Wolfgang Ketterle
John D. MacAurther Professor of Physics
Thesis Supervisor
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
David Pritchard
Cecil and Ida Green Professor of Physics
Thesis Supervisor
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Krishna Rajagopal
Professor of Physics, Associate Department Head for Education

2

Thermometry and Cooling of Ultracold Atoms in an Optical
LatticebyPatrick Medley
Submitted to the Department of Physics
on June 8, 2010, in partial fulfillment of therequirements for the degree ofDoctor of Philosophy

Abstract
Ultracold atoms of 87 Rb were prepared in a mixture of two hyperfine states, |F =1, mF = −1 > and |2, −2 >. This two-component system was then studied in thepresence of a magnetic field gradient and an optical lattice.
The presence of a magnetic field gradient separated the atoms into regions of
opposite spin, with a boundary region of mixed spin in the center. In the presence ofan optical lattice, the width of this region was found to be proportional to the system’s
temperature and inversely proportional to the strength of the magnetic field. This
allowed the measurement of the size of the boundary region to act as a thermometerfor the system, representing the first demonstration of spin gradient thermometry.
This thermometer represents the first practical method for thermometry in the Mott
insulator, and has features of high dynamic range and tunable sensitivity. Givensufficient optical resolution and control over the magnetic field gradient, the lower
limit of this thermometer is set by quantum magnetic ordering effects.
The dynamic response of this system to changes in magnetic field gradient wasstudied, both in the weak and strong lattice regimes. The result of these studies
was the development of spin gradient demagnetization cooling. By performing an
adiabatic drop in gradient strength while still in the superfluid, significant coolingof the entire system was observed. When the same process was performed in the
Mott insulator, the spin temperature was cooled dramatically, while remaining out
of equilibrium with the remaining degrees of freedom of the system. By reversing
the gradient direction, inverted spin populations with negative temperatures havebeen produced. Spin gradient demagnetization has produced the closest approach toabsolute zero yet recorded: 300 pK for the equilibrated system, and spin temperaturesof 75 pK as well as −75 pK. The ability to achieve these temperatures puts studies
of quantum magnetism in optical lattices within reach.
Thesis Supervisor: Wolfgang Ketterle3

Title: John D. MacAurther Professor of Physics
Thesis Supervisor: David Pritchard
Title: Cecil and Ida Green Professor of Physics

4

Acknowledgments
My time at MIT has been an interesting one. Sometimes frusterating, but more often
enjoyable, I will miss it greatly when I move on. There are many people who havehelped brighten my time here, and I hope to give them some of the thanks they
deserve.
First, I want to thank my thesis supervisors Wolfgang Ketterle and Dave Pritchard.
Both have given me excellent guidance and advice over the years here at MIT. I havealways been impressed with Wolfgang’s perception of physics, and the sharp discussions I have had with him in group meetings have been very enlightening. As for Dave,I have been impressed with his insight, not just into physics, but into all manner of
topics. When Dave spoke, it was rarely a dull topic.
The former members of the Rubidium lab that I worked with have also given
me excellent guidance, and helped me develop the skills I have today. Gretchen
Campbell and Erik Streed provided early guidance, and Gretchen especially playeda crucial role in maintaining lab and group cohesion. The Ketterle-Pritchard group
was never the same after she left. Micah Boyd, with whom I worked directly in myfirst couple years at MIT, was great fun to work with and introduced me to manythings that have greatly enlivened my life. All three of these people were involved
with the construction and early development of the Rubidium machine, and it is
only as a result of their efforts early on that any of the work I have done could beaccomplished.
After the three of them graduated, there was a vacuum in the Rubidium lab that
Jongchul Mun and I had to fill. The time I spent working with Jongchul was bothinteresting and fun, and he was a great person to work with in lab. I also enjoyedhaving the chance to discuss all manner of things with him, both physics-related andnot.
After Jongchul, in turn graduated, the Rubidium lab had four remaining members:
5myself, postdoc David Weld, and two other graduate students, David Hucul and Hiro
Miyake. David Hucul’s early departure was very saddening for me and the othermembers of the lab, but I am sure he will find success in the future at Maryland.
As for Hiro and David Weld, I could not imagine the latter half of my time at MIT
without them. They have greatly brightened my time here as I have spent countlesshours working alongside them. The time spent watching bits of television shows andenjoying internet silliness with them both while working late into the night runningthe machine is something without which my time would have been much less fun.
I am also greatful to the many members of BEC I, II, and III, and the other CUA
labs along the hallway, for lending me so many pieces of optics and other tools, as wellas having so much fun playing soccer and softball, and hanging out at conferences.
And of course, I’d like to thank my family for always being there for me. My
parents, Mike and Carol, have always supported me in whatever I did, and allowed
me to find my path in life. And my brother, Jonathan, who is currently also here
at MIT, pursuing his own research in chemistry, has been the source of many great
weekends playing games and having fun. And lastly, my fiancée Lara Rogers: thankyou for your constant love, and for your patience during the long periods of time we
had to spend apart. We shall be together soon.

6

Contents

1 Introduction

15

1.1

From BEC to Optical Lattices . . . . . . . . . . . . . . . . . . . . . .

15

1.2

Optical Lattices as a Simulation Tool . . . . . . . . . . . . . . . . . .

16

1.3

Why Temperature Matters . . . . . . . . . . . . . . . . . . . . . . . .

17

1.4

The Rubidium Lab . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

1.5

Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2 Theory of BEC and Optical Lattices

21

2.1

BEC in a Harmonic Trap without Interactions . . . . . . . . . . . . .

21

2.2

The Effect of Interactions . . . . . . . . . . . . . . . . . . . . . . . .

23

2.3

Atoms in Optical Potentials . . . . . . . . . . . . . . . . . . . . . . .

25

2.3.1

Scattering Forces . . . . . . . . . . . . . . . . . . . . . . . . .

25

2.3.2

Dipole Forces . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

Optical Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

2.4.1

Band Structure . . . . . . . . . . . . . . . . . . . . . . . . . .

27

2.4.2

Wannier Functions . . . . . . . . . . . . . . . . . . . . . . . .

28

The Bose-Hubbard Model . . . . . . . . . . . . . . . . . . . . . . . .

30

2.5.1

The Mott Insulator transition . . . . . . . . . . . . . . . . . .

31

2.5.2

Excitation Spectrum . . . . . . . . . . . . . . . . . . . . . . .

33

2.5.3

Two-Component Bose-Hubbard Model . . . . . . . . . . . . .

35

2.4

2.5

7

2.5.4Phase Diagram Of the Two-Component Bose Hubbard Model

3 Experimental Setup

36
41

3.1

Science Chamber Setup . . . . . . . . . . . . . . . . . . . . . . . . . .

41

3.2

State Preparation and Gradient Evaporation . . . . . . . . . . . . . .

44

3.3

Imaging the Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

3.4

Saturation Correction . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

3.5

Principal Component Analysis Correction . . . . . . . . . . . . . . .

48

3.5.1

Post-Atom Fringes . . . . . . . . . . . . . . . . . . . . . . . .

49

3.5.2

Pre-Atom Fringes . . . . . . . . . . . . . . . . . . . . . . . . .

50

Magnetic Field Gradient Calibration . . . . . . . . . . . . . . . . . .

52

3.6

4 Spin Gradient Thermometry for Ultracold Atoms in Optical Lattices

57

4.1

Model of Spin Gradient Thermometry. . . . . . . . . . . . . . . . .

58

4.2

Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . .

60

4.3

Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

4.4

Limits and Comparison to Other Techniques . . . . . . . . . . . . . .

64

4.5

Effects of Occupation Number ni > 1 . . . . . . . . . . . . . . . . . .

68

4.5.1

The Effect of Indistinguishability . . . . . . . . . . . . . . . .

68

4.5.2

Scattering Length Correction . . . . . . . . . . . . . . . . . .

70

4.6

Dependence of Temperature on Gradient . . . . . . . . . . . . . . . .

71

4.7

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

5 Spin Gradient Demagnetization Cooling
5.1

5.2

75

Basic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

5.1.1

Direct Cooling for Nonadiabatic Demagnetization . . . . . . .

76

5.1.2

Adiabatic Demagnetization as an Entropy Pump . . . . . . .

79

Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . .

81

8

5.3

Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

5.4

Modelling the Adiabatic Case . . . . . . . . . . . . . . . . . . . . . .

86

5.5

Equilibration and Adiabaticity . . . . . . . . . . . . . . . . . . . . . .

90

5.6

Isothermal Demagnetization . . . . . . . . . . . . . . . . . . . . . . .

91

5.7

Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

6 Conclusion

95

A Spin gradient thermometry for ultracold atoms in optical lattices

97

B Spin Gradient Demagnetization Cooling of Atoms in an Optical Lattice103

C Phase Diagram for a Bose-Einstein Condensate Moving in an Optical
Lattice

121

Bibliography

127

9

10

List of Figures
2-1 Bloch bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

2-2 Phase diagram for the superfluid-Mott insulator transition . . . . . .

33

2-3 Mott insulator excitations . . . . . . . . . . . . . . . . . . . . . . . .

35

2-4 Two-component phase diagram . . . . . . . . . . . . . . . . . . . . .

38

3-1 ODT and lattice diagram . . . . . . . . . . . . . . . . . . . . . . . . .

42

3-2 Magnetic field geometry . . . . . . . . . . . . . . . . . . . . . . . . .

43

3-3 Gradient evaporation procedure . . . . . . . . . . . . . . . . . . . . .

46

3-4 Fringe removal results . . . . . . . . . . . . . . . . . . . . . . . . . .

52

3-5 Stern-Gerlach calibration . . . . . . . . . . . . . . . . . . . . . . . . .

54

3-6 Magnetic field gradient zero measurement . . . . . . . . . . . . . . .

55

4-1 Images used for spin gradient thermometry . . . . . . . . . . . . . . .

61

4-2 Independence of temperature on gradient . . . . . . . . . . . . . . . .

63

4-3 Validation of spin gradient thermometry . . . . . . . . . . . . . . . .

65

4-4 Excitations in spin gradient thermometry . . . . . . . . . . . . . . . .

67

5-1 Demagnetization toy model . . . . . . . . . . . . . . . . . . . . . . .

78

5-2 Details of two experimental cooling protocols . . . . . . . . . . . . . .

82

5-3 Preparation of arbitrary spin temperatures . . . . . . . . . . . . . . .

85

5-4 Plot of width vs. field gradient. . . . . . . . . . . . . . . . . . . . .

86

5-5 Plot of temperature vs. field gradient . . . . . . . . . . . . . . . . . .

87

11

5-6 Particle-hole entropy distribution and number . . . . . . . . . . . . .

89

5-7 Spin entropy distribution . . . . . . . . . . . . . . . . . . . . . . . . .

89

12

List of Tables
4.1

87

Rb scattering lengths . . . . . . . . . . . . . . . . . . . . . . . . . .

13

70

14

Chapter 1
Introduction

1.1

From BEC to Optical Lattices

Fifteen years ago, Bose-Einstein Condensates (BEC) were first produced experimentally in atomic gasses [4, 19]. Their creation marked the opening of a new field of
atomic physics involving the study of ultracold atoms. Initially, the field focused on
measuring the properties of BEC themselves, but as time moved on the role of BEC
expanded. No longer was it solely the subject of direct experimental study: it increasingly has become a tool used to enable the creation of other cold atomic systems.
BEC have since been used in countless other roles. To count just a few: they havebeen used as a refrigerant to produce degenerate fermions, as a storage medium toslow and stop photons through electromagnetically induced transparency, and as asource of cold atoms to fill optical lattices.
It is this last use, to produce ultracold atoms in an optical lattice, that is thefocus of this thesis, and that represents an important area in the future of atomic
physics. While past studies of BEC, even many of those involving optical lattices, were
often concerned with the properties of the superfluid state, optical lattices also point
the way toward an area beyond superfluidity: static but highly-ordered, crystal-like15

structures where superfluidity is left behind. Traditionally the realm of condensedmatter physics, the advent of optical lattices puts the study of crystalline ordering
within the grasp of the atomic physicist as well. Already the superfluid to Mott
insulator transition has been the subject of thorough experimental study, and in the
future, many new and exotic phases will be open for experimental study in deep opticallattices. As this field matures, the study of phases exhibiting magnetic ordering will
be within reach, and the work presented in this thesis will represent a significant stepin making the study of these phases possible.

1.2

Optical Lattices as a Simulation Tool

Whereas atomic physics once dealt primarily with dilute gasses of free-moving atoms,
much of condensed matter physics deals with solids with essentially rigid crystal
structures. The new states made possible by optical lattices provide an important
link between these two fields. By subjecting a cloud of ultracold atoms to a preciselytailored optical lattice potential, a wide range of crystal structures can be simulatedusing the atoms loaded from the condensate as a proxy for particles in a crystal lattice.
This represents the possibility of a new frontier for the investigation of many-body
physics [7, 48].
An optical lattice is created by subjecting a cloud of atoms to a standing waveof light. Via the AC Stark effect, this creates a periodic potential for the atoms. As
atoms are attracted to the minima of this potential, they arrange themselves in aperiodic structure similar to the structure of a crystal.
Optical lattices can offer many advantages over the direct probing of condensed
matter systems. Optical lattices can be tailored to the specific problem you wish tosolve, providing flexibility in the states and identities of the constituent atoms, thestrength of their interactions, their dimensionality, and the geometry of the lattice16

that holds them. Many of these properties can be changed continuously and independently, even in the course of a single experimental run. This makes it possible fora single atomic system to model many different types of condensed matter systems
without the need to fabricate them individually. It also allows the creation of simple
model systems that may be difficult to fabricate in solids in order to study particular
phenomena of interest in a controlled manner. Already, many studies involving theMott insulator transition have been made, and as research in optical lattice simulation continues, interest is growing in producing states that exhibit more complicatedphysics, such as magnetically ordered states. Achieving these sorts of states, however, presents significant new problems to be overcome, not just in engineering thenecessary interactions, but also in producing systems cold enough to exhibit them.

1.3

Why Temperature Matters

To achieve a magnetically-ordered state, it is necessary to have at least two componentseither two separate atomic species, or two states of the same species. The moststraightforward such system to produce experimentally is a system of two spin statesof a single atom in a cubic optical lattice. This system is described by the two component Bose-Hubbard model, and can give rise, under the appropriate conditions,to two types of ferromagnetic ordering, as well as antiferromagnetic ordering [21].
However, this potential also highlights the difficulties in achieving such orderings.
One of the clearest such difficulties is that of temperature. The acievement oflower temperature scales has consistently led to new advancements in physics, andthe case of quantum magnetism is no exception. The strength of interactions between
atoms in an optical lattice is much weaker than those between their condensed mattercounterparts. As a result, extremely low temperatures, far below even the µK criticaltemperature for Bose condensation, are required before any magnetic effects could17reasonably be observed. For atoms of 87 Rb in a lattice deep enough to form a Mott
insulator, this temperature is in the range of tens to hundreds of pK.
The low temperatures demanded by such systems are themselves a significant
challenge, but of equal concern has been the lack of an effective method for measuring
temperature in this regime. Temperatures as low as 450pK have been measured in
atomic gasses before [46]; however, the method used to do so required the atoms
to be in an unperturbed harmonic trap. The same method is unable to measuretemperatures of atoms in the presence of a lattice. Other methods for measuring
temperatures of atoms in an optical lattice have been tried, but they are largely
unable to measure with precision temperatures as low as needed. [16, 29, 38, 51, 56]
This thesis presents a method that has been developed to solve this problem in asystem of two spin components. Through the use of a carefully-controlled magnetic
field gradient to induce low-energy modes of excitation in the distribution of spins,
it becomes possible to image the Boltzmann distribution of those spins directly. The
resulting spin gradient becomes a thermometer from which one can straightforwardly
read out the temperature. Furthermore, experiments will be described in which thisthermometer is transformed into a refrigerator, cooling the atoms to the lowest temperatures ever recorded.

1.4

The Rubidium Lab

This thesis describes experimental work performed in BEC IV, also known as Rubidium lab, as it is the only one of the machines in the Ketterle-Pritchard group to workwith 87 Rb. While all of the current research efforts in the Rubidium lab (and all the
results presented in this thesis) are focused on optical lattices, this was not the casewhen I first joined nearly six years ago. At that time, the lab had just publisheda paper on photon recoil in a dispersive medium [11], and still viewed itself largely18

as a machine optimised to produce high atom number BEC. Indeed, some of theearliest projects I worked on as a junior student were still focused on the superfluid:
an early experiment involving scattering of atom pairs from a condensate in a onedimensional optical lattice potential [13] and an experiment involving the quantum
Zeno effect [68].
However, as time passed, it became increasingly evident that optical lattices were
the way of the future. The experiments I worked on involved systems of increasingly lower dimension: a two-dimensional potential created by radio frequency (RF)dressing of a magnetic trap [9], a one-dimensional magnetic tube potential above a
hard disk platter [8], and ultimately, in “zero-dimensions” a measurement of the shellstructure of the Mott Insulator state via RF spectroscopy [12].
The final project I worked on before beginning the experiments detailed in thisthesis was a first step toward the ultimate goal of probing phase diagrams in opticallattices. In it, the transport properties of atoms in the presence of an optical latticewere studied using a moving lattice to simulate AC particle flow. The result wasa phase diagram showing stable versus unstable flow as a function of lattice depth
and speed [53]. This diagram was compared to theoretical predictions based on amean field model [1], and the results showed excellent agreement. Additionally, theseexperiments allowed a precision measurement of the lattice depth of the Mott insulatortransition.
While this result was a significant step forward, it also represented a kind of limit
in the interesting physics that was available in a simple, single component lattice. To
proceed further, into the realm of magnetic ordering, it was necessary to work with asecond component. Furthermore, the temperature scale required to probe this kindof ordering—the superexchange tempreature scale—was a scale far colder than what
had yet been achieved. The experimental work that this thesis comprises opens up
this scale, providing both a method to measure these temperatures in the presence of19

a lattice and a means by which these temperatures can be attained. The developmentof spin gradient thermometry and spin gradient demagnetization cooling represents animportant step towards bringing together the fields of atomic and condensed matterphysics. I can only hope that in the future this combined field will be as fruitful as
it appears today.

1.5

Outline

The remainder of this thesis will have the following structure. Chapter 2 will discuss
the basic theory underlying our experiments. It will begin with the theory of BEC, andthen discuss the theory of optical lattices, including the one and two component Bose
Hubbard models and the Mott insulator transition. Chapter 3 will cover the basic
experimental procedures and the setup of the machine used in the production of 87 Rb
BEC. The next two chapters will then describe two experiments performed to measure
and lower the temperature of a system comprising an optical lattice loaded with 87 Rb
atoms in two spin states. Chapter 4 describes how a thermometer was be created andused to measure temperatures in the presence of a lattice, resulting in the process of
spin gradient thermometry. Chapter 5, meanwhile, discusses how this thermometer
was used to cool atoms through the process of spin gradient demagnetization cooling,which has resulted in some of the coldest temperatures yet recorded.

20

Chapter 2Theory of BEC and Optical
Lattices

2.1

BEC in a Harmonic Trap without Interactions

Although all of the important experimental measurements described in this thesis
occur outside of the pure superfluid state, in optical lattice potentials, the atoms begin
as a BEC. Since the properties of BEC are important to the preparation of our atoms
and their transferral into the optical lattice, I will devote the first two sections of thischapter to a brief overview of BEC. For a more thorough discussion of BEC, I refer
you to previous theses from this group [9, 10, 52, 64], review articles [17, 18, 43, 44, 47],and books [32, 55, 60].
In this section, I will derive an exact expression for the critical temperature andcondensate fraction for the simplest model of trapped bosons: that of a gas of noninteracting bosons in a harmonic potential. For a harmonic potential with trap frequencies ωx , ωy , and ωz , the energy levels for noninteracting particles are given byequation 2.1. Recall that BEC occurred when the chemical potential µ became equal
the ground state energy, which for the noninteracting case was set to zero. In the21case of interactions, however, the ground state energy is no longer zero, because it
depends on interactions. However, we know that in a BEC, the ground state energy
is equal to the chemical potential. Thus

E(nx , ny , nz ) = h̄(ωx nx + ωy ny + ωz nz )(2.1)

If the gas is at temperature T and chemical potential µ, then the population in eachenergy state is given by the Bose Einstein distributionp(nx , ny , nz ) =

1

(2.2)

y ,nz )−µexp( E(nx ,n
)−1
kB T

and the total number is then

N=

1

X

(2.3)

E(nx ,ny ,nz )−µ)−1nx ,ny ,nz exp(
kB T

As the temperature of the system is lowered, the chemical potential (which is necessarily negative) must rise to maintain constant total number. Eventually, the chemical
potential will reach zero and the expression for the population in the ground state,p(0, 0, 0) will diverge. The temperature at which this will occur is the critical temperature for Bose-Einstein condensation, and can be calculated by solving equation
2.3 for T when µ is set to zero. In the limit of large N, so we can treat the above
sum as an integral over energy times a density of states (nx ,ny ,nz →

P

R

2

d 2h̄3 ωx ωy ωz ),the critical temperature is given bykB Tc = h̄ωho (N/ζ(3))1/3

(2.4)

3
where ωho
= ωx ωy ωz and ζ(3) ≈ 1.202 is the Riemann Zeta function. in our experi-ments with 87 Rb, this temperature will be typically be 300 − 400 nK. At and below

22this temperature, the distribution of particles changes so that the population of theground state and the excited states are described differently. The population of each
excited state is still given by equation 2.2, but the fraction of atoms in the groundstate—the condensate fraction—is now
N0 /N = (1 − (T /Tc )3 ).

2.2

(2.5)

The Effect of Interactions

The approximation of the trapping potential as harmonic is reasonably accurate—the
actual potential produced in the experiments described in following chapters is thesum of two gaussians. The approximation of the atoms as noninteracting, however, is
not so accurate. In reality, 87 Rb atoms do interact, albeit weakly. Under experimental
densities and temperatures, only two-body s-wave scattering collisions will contributeto the overall potential, although three-body collisions contribute to loss processes
and heating via recombination. The effective two-body potential generated by s-wave
scattering interactions can be written in terms of the scattering length as and the
distance between a pair of particles r12 as U (r) = 4πh̄2 as /mδ(r12 ). This adds anextra term to the original Hamiltonian, resulting in a corrected Hamiltonian given by

Ĥ =

Z

d~rψ̂ † (~r)(

4πh̄2 as †h̄2 2
∇ + Vtrap (~r) +ψ̂ (~r)ψ̂(~r))ψ̂(~r)
2m
m

(2.6)

where Vtrap is the trapping potential. For T  Tc so that the thermal fraction can beneglected, and neglecting also quantum fluctuations, we can then describe the BEC
as a single wavefunction ψ(~r) with the normalization condition d~r|ψ(~r)|2 = N . ThisR

wavefunction will then obey the Gross-Pitaevskii equation:ih̄∂ψ(~r, t)h̄2 2
4πh̄2 as=(∇ + Vtrap (~r) +|ψ(~r, t)|2 )ψ(~r, t).
∂t
2m
m
23

(2.7)

To solve the Gross-Pitaevskii equation, we should first write it in a time-independent
form, of the type Ĥψ = Eψ. The energy of the condensate is the ground state energy,
which can easily be derived by appealing to equation 2.3. Recall that BEC occurred
when the chemical potential µ became equal the ground state energy, which for thenoninteracting case was set to zero. In the case of interactions, however, the groundstate energy is no longer zero, because it depends on interactions. However, we knowthat in a BEC, the ground state energy is equal to the chemical potential. Thus,
we can insert it into the Gross-Pitaevskii equation to produce a time independent
version:µψ(~r) = (4πh̄2 ash̄2 2
∇ + Vtrap (~r) +|ψ(~r)|2 )ψ(~r).
2m
m

(2.8)

The final approximation to solve this equation is the Thomas-Fermi approximation:
we throw out the kinetic term from equation 2.8. The resulting density distribution
is then easy to calculate:n(~r) =

m(µ − Vtrap (~r)).
4πh̄2 as(2.9)

This approximaiton is valid as long as the ratio of kinetic energy to interaction energy
is small. Then, the kinetic term will make no important correction to the density
function. The length scale over which the kinetic term is important is given by thehealing length ξ—for changes in the wavefunction occuring over a distance shorter
than the healing length the kinetic term dominates over the interaction term. The
healing length is thus the length scale over which these two terms are equal, orh̄2
4πh̄2 as n
1
=
⇒ξ= √
2
2mξ
m
8πnas

(2.10). The function of the healing length is to take any sharp edges in the density distribution given by the Thomas-Fermi approximation and transform them into a more-24

or-less smooth transition over the width of the healing length. This will be importantlater on, as it will provide a limit to the resolution of the thermometry technique ofchapter 4 when applied by the superfluid. As we shall see, the presence of a reasonably deep lattice eliminates this limitation by lowering the condensate fraction to thepoint that the Gross-Pitaevskii equation is no longer valid.

2.3

Atoms in Optical Potentials

2.3.1

Scattering Forces

An atom illuminated by a laser beam is subject to two different types of forces:
scattering and dipole forces. In optical dipole traps and lattices, the dipole force isused to form potentials to trap atoms, while the scattering force leads to heating andis generally desired to be as small as possible. Scattering occurs as photons from thelaser transfer the atom into an excited state, then the atom undergoes spontaneousemission, sending out a photon of the same energy in a random direction. Assuming,
as is a good approximation for 87 Rb, that the atom has only one relevant transition
to be excited, the scattering rate can be written in terms of the laser intensity I,
the transition’s saturation intensity Isat , the detuning δ, and the transition’s naturallinewidth Γ:Γsc =I/IsatΓ.
2 1 + I/Isat + (2δ/Γ)(2.11)

Each photon scattered imparts a momentum kick on the atom of h̄k in the directionof the laser beam, where k is the wavevector of the light, as well as an additional kickof h̄k in a random direction. The kick along the laser beam is used in instruments
such as Zeeman slowers and magneto-optical traps to slow and cool atoms, but is not
put to any use in dipole traps. The main reason for this is that the random kick leads25

to heating, with a heating rateU̇ =h̄2 k 2Γsc .
2m

(2.12)For atoms at temperatures well below 1µK, as is typical in experiments involving
BEC, this heating usually dominates any cooling effect that can be harnessed, so thescattering rate should be made as small as possible.

2.3.2

Dipole Forces

The dipole force, on the other hand, has no dissipative component: it creates anordinary conservative potential. The dipole force arises because the electric field of
the light induces a dipole moment µ in the atom, which itself interacts with theelectric field with interaction energy Udip = −µ· E. The resulting shift in energy is
known as the AC Stark effect and in the limit of large detuning is equal to

Udip =h̄Γ2 I.
8δ Isat(2.13)The sign of the detuning determines the sign of the potential: blue-detuned light
repels atoms from positions of high intensity, while red-detuned light attracts to thosepositions. This allows one to produce an optical dipole trap (ODT) by focusing a reddetuned laser beam, as the focus of the beam will be a point of maximum intensity.
If a beam of wavelength λ is focused with a beam waist of w0 it will have a Rayleighrange of zR =πw02
λand will create a potential near the focus of

Vtrap (r, z) ≈ V0 [1 − (zr 2) − ( )2 ].
w0
zR(2.14)All optical traps used to hold atoms in the experiments described in this thesis areformed by one or more such traps, produced using light at 1064 nm, which is far to thered of the nearest optical transition—the D lines of 87 Rb at 780 nm and 795 nm. This26

large detuning is important to minimize scattering losses: note that while the dipole
force drops as 1/δ, the scattering rate drops as 1/δ 2 . Thus, a large detuning allows
deep optical traps with very low scattering rates. Typical spontansous scattering
rates even in very deep potentials are 0.01Hz, making them essentially negligible.

2.4

Optical Lattices

While an ODT is a simple and straightforward use of the dipole force, a more interesting potential can be made by taking advantage of optical interference to produce
a standing wave. If a laser beam is reflected back onto itself, it will create a standing
wave with a node at the reflecting mirror. The total potential this standing wave
creates is
Vlatt (r, z) = 4V0 cos2 (kz)[1 − (zr 2) − ( )2 ].
w0
zR(2.15)

In our experiments, a combination of ordinary ODT beams confines the atoms toa radius much smaller than the waist of the lattice beam, so the lattice potential
may safely be treated as a uniform sine wave with depth 4V0 . This depth is usually
2 2

measured in recoils, where Erec = h̄2mk for a laser with wavenumber k and an atom ofmass m. By creating an optical lattice in each of three orthogonal dimensions, one
creates a three-dimensional sine wave potential, where the wells are arranged like theatoms of a simple cubic crystal.

2.4.1

Band Structure

If we ignore the external potential, we can understand the behavior of particles in
the abscence of interactions by solving the Hamiltonian of a single particle in a sine
wave potential. In each dimension we can write:H=−h̄2 ∂ 2si Erec+(1 − cos(2kxi ))22m ∂xi227(2.16)

where si is the lattice depth, measured in recoils, for each dimension (usually all
dimensions have the same depth in our experiments, but this is not necessary). The
Schrödinger equation then admits solutions in the form of Mathieu functions, or
equivantly, by Bloch’s theorem, in the form of functions [50]
Faq [x] = eiµaq x Paq (x)(2.17)

where µ(a, q) is the Mathieu characteristic exponent, a = E/Erec − s/2, and q = s/4.
As µ appears in the leading exponent as though it were a momentum, and indeed isequal to momentum in the limit of zero lattice depth, it is called the quasimomentum. The function Paq is periodic with period π/k, and each function Faq [x] has asister function Faq [−x] which is also a solution with the same energy but negative
quasimomentum. These solutions are only normalizable if µ(a, q) is real, which is true
only within certain ranges. This implies the existence of energy bands where particles
below a certain threshold can only have energies within certain ranges. Figure 2.1
shows the appearance and growth of the first three energy bands as a function oflattice depth. It is often useful to rewrite Faq (x) in terms of the quasimomentum µ
and band level n, forming the Bloch function ψµn (x). Note that these functions also
depend on the lattice depth s, even though it has been left off as an index.

2.4.2

Wannier Functions

While Bloch functions are useful for describing atoms in a shallow lattice and in theabsence of interactions, as the lattice becomes deeper, and especially as interactions
are taken into account, a delocalized wavefunction such as the Bloch function becomesless and less useful as a basis to describe the atoms. Increasingly, atoms are confined
to a single site, and perhaps those adjacent to it, so an ideal basis would be one thatwas likewise localized. The Wannier functions provide just such a basis. For an atom
28

Energy (Erec)

Lattice Depth (Erec)

Figure 2-1: Bloch bands. Bloch bands begin to form with increasing lattice depth.
This plot shows allowed energies in light blue and forbidden energies in white, as afunction of lattice depth. The first three bands are labelled on the plot. Each band
appears at a progressively higher lattice depth.

29localized to the j th lattice site and in the nth band, the Wannier function is

wjn (x) =1Zdµe−iµj ψµn (x).
L

(2.18)

where L is the width of the band, and acts as a constant to normalize the wavefunctions. In experiments with ultracold atoms, nearly all of the atoms will findthemselves in the lowest band, with a large energy gap separating the other bands at
higher lattice depths, so we can generally take n to equal 1.
Using Wannier functions, we can finally begin to deal with interacting atoms ina lattice by producing a proper expression for tunneling and onsite interaction matrix elements. These matrix elements will be the key components in writing theBose-Hubbard Hamiltonian. The tunneling matrix element J is given by the overlap integral between the Wannier functions of two adjacent sites as coupled by the
Hamiltonian:
Jij =

Z

dxwi (x)(−h̄2 ∂ 2 x
+ Vlatt (x))wj (x).
2m ∂x2(2.19)The interaction matrix element, U , for a given site is the repulsion felt between two
particles located at the same site. Recalling that the interaction term is U (r) =4πh̄2 as /mδ(r12 ), we can easily write:
4πh̄2 as ZUi =dx|wi (x)|4 .
m(2.20)

With these ingredients, we are finally ready to tackle the Bose-Hubbard Hamlitonian.

2.5

The Bose-Hubbard Model

The Bose-Hubbard model is a model for the Hamiltonian for atoms in an opticallattice that assumes that the energy of an atom at a particular site is the sum of threeterms: the kinetic term, given by the tunneling matrix element to each adjacent site;30

the onsite interaction term, given by the interaction energy between that atom andany others at the same site; and the external potential term i = Vtrap (xi ), which isthe potential energy of the atom at that location given by whatever external trapping
potential is imposed on top of the lattice. For a lattice potential that is relatively flatcompared to the external trapping potential, so that Ui and Jij are the same for all
lattice sites, the full Bose-Hubbard Hamiltonian is then [42]:H = −JX<i,j>a†i aj + U/2

Xini (ni − 1) +Xi

(i − µ)ni

(2.21)

where a†i and ai are the creation and annihilation operator for an atom at site i andni = a†i ai is the number operator at site i, while < i, j > denotes the sum over allstates i 6= j.

2.5.1

The Mott Insulator transition

As U and J are the only two energy scales imposed by the lattice in the Bose-Hubbard
model, it is natural that the physics of this model depends on their ratio, given by
the dimensionless interaction energy u = U/J. As the lattice depth increases, U
rises linearly, while J falls exponentially. Thus, u rises exponentially with increasing
lattice depth. To understand what happens to the ground state wavefunction as u
rises, let us consider two extreme cases: the small u case, where J  U and the large
u case, where U  J.
The small u case corresponds to a vanishingly small lattice, and thus to a superfluid
state. This state would be best described in the basis of Bloch states, but one canwrite it also in the Bose-Hubbard basis. In the limit of no interactions, each atom
is independent, and is simply in the trap ground state. If the single particle ground31

state wavefunction is < x|φg >= φg (x), then the ground state for N particles is simply
X †|ψSF >∝ (iai φg (xi ))N |0 > .

(2.22)

If, as in the simplest case, φg (x) is a constant, we can easily see the expectedbehavior of a superfluid: phase coherence along with poissonian number fluctuations
for each site. As the lattice raises, however, and J begins to vanish in favor of U , thisphase coherence gives way to well defined values of atom number, and Fock states
form the new basis for the ground state. The transition between these two limits is
the Mott Insulator transition, and its ground state wavefunction
X †ni|ψM I >∝
wherei

ai |0 >

(2.23)

i ni = N . The actual value of ni is determined by the external potential andP

the chemical potential. We can think of filling each well with particles until the onsite
energy equals the chemical potential minus the trapping potential. In other words,
we can minimize the expression Ei = U/2ni (ni − 1) − (µ − i )ni . Since ni must be an
integer, however, the actual expression for ni moves in discrete steps:

ni = dµ − ie.
U(2.24)

This simple expression is only valid in the limit as J → 0—a more thorough analysis for nonzero J produces the phase diagram in figure 2.2. The actual point of theMott insulator transition, where certain sites first stop showing number fluctuations,occurs for sites at the top of the n = 1 peak. The precise location of the Mott insulator transition is not easy to measure, and has been the subject of many experimental
attempts [12, 24–26, 30, 42, 54, 66]. Nevertheless, a previous experiment performed inthis lab [53] measured the transition to be at u = 34.2±2.0 for 87 Rb in a 3-dimensional
32

0.03

J/U

0.02
Superfluid
MI n=1

0.01

n=2

n=3

n=4

n=5

0.00
0

1

2

3
(µ-ε i )/U

4

5

Figure 2-2: Phase diagram for the superfluid-Mott insulator transition. As J/Udecreases, the system will undergo a phase transition into the Mott insulator. The
system will form regions of superfluid and insulating regions with fixed atom number,determined by the value of (µ − i )/U at that lattice site. Higher chemical potentials
lead to higher atom numbers at each site.

lattice made with light at 1064µm, which occurs at a depth of 13.5 ± 0.2Erec . This
result is in good agreement with the prediction of mean field theory [42], although it islower than the predictions from quantum monte carlo [14,28]. The measurement was
precise, but may have had systematic errors affecting its accuracy, so the transition
point may be slightly under the measured value. Nonetheless, it seems safe to saythat any lattice above 14Erec is well into the Mott insulator regime.

2.5.2

Excitation Spectrum

An important difference between the superfluid state and the Mott insulator can beseen in their different excitation spectra. The superfluid state has a continuous excitation spectrum described by Bogoliubov theory. In terms of momentum k, interaction
33

parameter U0 = 4πh̄2 as /m and density n, the excitation energy is [65]sk =h̄2 k 2
+ U0 n)2 − U02 n2 .
(2m

(2.25)For small k this expression is approximately linear, implying excitations in the form ofphonons with speed of sound c =qnU0 /m while for high k we have free particle-like

2 2

excitations with k ≈ h̄2mk + nU0 .
This continuous excitation spectrum stands in sharp contrast to the Mott insulator, whose excitation spectrum is gapped. Since the only variable determining theenergy of a state in the deep Mott insulator is ni , the only form an excitation can takeis a change in distribution of ni . For example, in an n = 1 Mott insulator, one site
can be left vacant while another is occupied by two atoms. This can be thought ofas a particle-hole excitation, as a “hole” is created in the form of an n = 0 site, whilea “particle” is created in the form of an n = 2 site. Such an excitation is illustrated
in figure 2.3, and has energy U .
Of course, in a real system, the inhomogeneous trap potential will cause different
atoms to be at different effective chemical potentials. Looking in figure 2.2, any atoms
that happen to be in the superfluid region will have continuous excitation spectra. In
a harmonic trap, these atoms will be located around a particular radius determined by
the trap frequency, forming ellipsoids of superfluid commonly referred to as superfluid“shells.” However, for a reasonably deep lattice only a small fraction of the atoms will
be located in positions to form superfluid shells; the remainder will exhibit a gapped
excitation spectrum.
The effect of this gapped excitation spectrum is double-edged: on the one hand,for temperatures kB T  U there will be few excitations, giving a relatively puresystem, but on the other hand, this freezing out of excitations makes it very difficult
to read temperatures. Only the superfluid shell regions show significant responses to34

U

Figure 2-3: Mott insulator excitations. In the Mott insulator, the lowest energy
excitation occurs when an atom tunnels to an adjacent lattice site. This lowers theoriginal site’s occupation number by 1 while raising the new site’s by 1, at an energy
cost of U , the onsite interaction energy.

temperature changes in this range, but they can be very small and difficult to resolve.
As we shall see, using atoms in a different hyperfine state as a second component, astep necessary anyway for the observation of magnetic phenomena, provides a method
to overcome this difficulty.

2.5.3

Two-Component Bose-Hubbard Model

The presence of a second type of atom modifies the Bose-Hubbard Hamiltonian in thefollowing ways: for atoms numbered 1 and 2, we must introduce separate tunnelingmatrix elements J1 and J2 if the two components have different tunneling rates,
and we must introduce different interaction energies U11 , U22 , U12 to represent thedifferences in inter- and intra-species interactions. The resulting Hamiltonian is thenH=−X<ij>,σJσ a†σi aσj +

X Uσσ

(

i,σ

2

nσi (nσi − 1) + U12
35

Xi

n1i n2i +Xiσ(σi − µσ )nσi (2.26)

where sigma represents the index 1 or 2. In our experiments, the two species will
be two different hyperfine states of 87 Rb, the |F = 1, mF = −1 > state and the|2, −2 > state, where F and mF are the total spin and its projection along the axis of
the local magnetic field. The value J can be manipulated by the appropriate choiceof optical lattice: a spin dependent lattice can have different lattice depths for eachstate, and hence different J. Similarly, differences in U can be achieved either by using
Feschbach resonances to manipulate the relative scattering lengths of the two species(recalling that U ∝ as ), or again by the use of specially engineered spin dependentlattices [39, 49]. In the experiments described here, however, no use is made of spin
dependent lattices, so J is identical for the two species. Also, the scattering lengths ofthe two species are not manipulated, and the natural differences in scattering lengths
for 87 Rb are very small. The only remaining term is the different potentials, σi felt
by the species, and it is this term that is the key to allowing thermometry in thelattice. Nonetheless, it is instructive to briefly go over the resulting phase diagram
for lattices in the case of variable J and U , as we will see what magnetically orderedphases will arise and why they require the very precise temperature control that is
the subject of this thesis.

2.5.4

Phase Diagram Of the Two-Component Bose Hubbard
Model

In the Mott Insulator regime, J is much less than U , so it is appropriate to expand the
Hamiltonian perturbatively in J. Taking only terms out to order J 2 /U , and ignoring
the external potential, it is possible to use the Schrieffer-Wolf transformation [2,21,45]
to write the Bose-Hubbard Hamiltonian in the form

H=

X
<ij>

λz σiz σjz − λ⊥ (σix σjx + σiy σjy ) +36

Xi

hz σiz .

(2.27)

µµHere σiµ = a†k σkl
al for the Pauli matrices σkl, so σiz = n1i − n2i is the difference in

number at site i. Similarly, σix gives the component of the net spin angular momentum
pointing along the x-axis and σiy along the y. The term hz is a term proportional
to the applied magnetic field plus a constant which may be neglected as a being an
offset to the total field. Meanwhile λz and λ⊥ can be written in terms of U and J asλz =J12 + J22 J12 J22
−
−
2U12
U1 U2

(2.28)

J1 J2
.
U12

(2.29)

λ⊥ =Depending on the values of λz , λ⊥ , and hz , this Hamiltonian gives rise to a groundstate in one of three different magnetically ordered phases: a z-ferromagneticallyordered phase, where all spins align along the direction of the applied magnetic field;
an xy-ferromagnetically ordered phase, where the spins align orthogonal to the field;
and an antiferromagnetically ordered phase, where spins are alternately aligned andanti-aligned to the field in a checkerboard pattern. Roughly speaking, as U12 drops
relative to U1 and U2 , the xy-ferromagnet tends to be favored over the z-ferromagnet,
and as the tunneling difference β = J1 /J2 + J2 /J1 grows above its base value of 2, theantiferromagnet tends to be favored over the xy-ferromagnet. The resulting phase
diagram is shown in figure 2.4.
Because these phases arise as a perturbative correction to the J = 0 model of theMott insulator, the energy involved in creating them will be on the order of J 2 /U .
This scale is known as the superexchange energy, as it is the energy term involved in
an interaction where a particle tunnels to an adjacent site, interacts with whatever
particles are there, then returns. It is when the system is below this scale that it
is possible for it to exhibit the ordering effects of quantum magnetism [35]. Various
proposals [3,22] have focused on the realization of quantum spin Hamiltonians in thisregime. But the small value of this scale is a serious problem for atomic systems:
37

Figure 2-4: Two-component phase diagram. Two-component bosonic atoms in an
optical lattice obey this phase diagram very low temperatures. The three phases
depicted are labelled as follows: AF is the antiferromagnetic phase. X-FM is the
xy-ferromagnetic phase. Z-FM is the z-ferromagnetic phase. The variables λz and λ⊥used in the axes are defined in equations 2.28 and 2.29, respectively, while hz is themagnetic field. In the simplified case where J1 =J2 and U11 = U22 = Uσ , the verticalaxis is proportional to the ratio U12 /Uσ , plus a constant.

38

for 87 Rb in a lattice with average depth near the Mott insulator transition, recent
quantum Monte Carlo calculations have shown the Curie temperature for the xy
phase to be a mere 200 pK [15]. While the obvious answer might seem to be to raise
J or lower U to make J 2 /U larger, this would result in leaving the Mott insulator
regime and the failure of our perturbative assumption: the system would instead bea superfluid, so we could no longer probe the phase diagram we wish to. Thus, weare left with little alternative but to find a way to measure lattice temperatures in
the picoKelvin range, and find a way to produce such temperatures in experimental
conditions. Although some cooling schemes have been proposed [6, 36, 57, 58], theirimplementation has not been easy. Chapters 4 and 5 will describe how these twogoals have been achieved experimentally through the development of spin gradientthermometry and the use of adiabatic demagnetization cooling.

39

40

Chapter 3
Experimental Setup
In this chapter I will review the basic details involved in the preparation of ultracold
87

Rb atoms to be loaded into an optical lattice. The design of the experimental

apparatus has already been clearly described in Refs. [67], [52], and [69], so I will only
give a very brief description of the procedure for producing ultracold 87 Rb atoms. The
87

Rb machine is composed of an oven, Zeeman slower, and two chembers: the mainchamber and the science chamber. The main chamber contains a Magneto-Optical
Trap that catches and cools 87 Rb atoms exiting the Zeeman slower. The atoms arethen optically pumped into the in the |1, −1 > hyperfine state and transferred to amagnetic trap. The atoms are evaporatively cooled to a temperature of a few times
Tc before being loaded into an Optical Dipole Trap (ODT). This trap is produced byfocusing a 1064 nm wavelength laser through a lens on a translation stage. The stage
then moves the lens, translating the trapped atoms into the Science Chamber, where
they are then transferred into another ODT.

3.1

Science Chamber Setup

The Science Chamber is a vacuum chamber connected to the main chamber anddesigned specifically for use with optical lattices. The design of this chamber is
41

Y

Z

X

3“ OD

Figure 3-1: ODT and lattice diagram. This diagram shows the top view of the Science
Chamber. The configuration shows optical dipole traps in blue and optical lattices
in red. One lattice beam is oriented vertically through the atoms, coming out of thepage.

described in detail in Ref. [52]. Atoms are delivered into this chamber from the Main
Chamber using a translating ODT, then loaded into a crossed ODT formed by two
horizontal beams aligned 90 degrees apart from each other. The depth of the trap ofthese two beams is lowered to evaporatively cool the atoms, and the atoms form a
BEC. Aligned with the trap bottom created by these beams are three optical lattices,each formed by a retroreflected laser beam. One of the beams is vertical, while theother two are horizontal at 45 degrees from the beams of the crossed ODT. The
configuration of these five beams is shown in figure 3.1.
All of these beams originate from the same source, a 1064 nm laser. The frequencies of these beams are offset by at least 3 MHz from each of the others using
42

Y
B

B
Z

X
Bias
Coil

Bias
Coil

Z

Y

Gradient
Coil

X

B

Bias
Coil

Bias
Coil

Grad |B|

B
B

Gradient
Coil

Figure 3-2: Magnetic field geometry. These diagrams show the configuration of magnetic bias fields and gradients generated by the Science Chamber coils. The left shows
bias fields from a top view. The right shows a side view of ∇|B| near the atoms in
the absence of bias fields. With bias fields on, ∇|B| points along x axis only.

Acousto-Optical Modulators so that any interference effects are at high frequency andwill average out. The two optical dipole traps are given orthogonal polarizations, andeach of the lattice beams is also given a polarization orthoganal to each of the othertwo so as to further minimize interference effects.

Magnetic field control is very important in our experiments, and it is performed
using six bias coils. Four small coils in the horizontal plane are configured to provide
a bias field of up to 15 G along the x-axis. Two larger coils, located above and below
the chamber are arranged in an anti-Helmholtz configuration to produce a magnetic
field gradient. In the presence of a strong bias field in the x direction, the gradient of
the absolute value of the field, ∇|B|, will point along the x-axis. In experiments, thestrength of this gradient can be varied from 2 G/cm to −1 G/cm. Figure 3.2 shows
the configuration of these coils and the fields they produce.
43

3.2

State Preparation and Gradient Evaporation

The Main Chamber produces atoms in the |1, −1 > hyperfine state, and the atoms
remain in this state as they are transferred into the Science Chamber and evaporated
to BEC. Since we wish to have atoms in a mixture of two states, we transfer a fractionof those atoms into the |2, −2 > hyperfine state. There are several reasons why the|2, −2 > state is an attractive choice for a second state. First, it is a stretched statein the same direction as the |1, −1 > state, so it cannot collide with a |1, −1 > atom,or with itself, and produce atoms in different hyperfine states. This means that the|1, −1 >/|2, −2 > mixture acts as a two-spin system with conserved magnetization.
Secondly, the transition from the |1, −1 > to the |2, −2 > state is a single photon
magnetic transition, so it is easy to drive with microwave radiation while at the sametime having an extremely long lifetime in the upper state. Finally, the two states
have opposite g-factors, so they are pulled in opposite directions by a magnetic field
gradient. This is important, because the interaction between spin and magnetic field
gradients is critical to our experiments.
The transition between |1, −1 > and |2, −2 > is driven with microwaves using a
rapid, nonadiabatic sweep of the magnetic field. A microwave signal at 6.844 GHz is
mixed with an RF signal at 36 MHz to produce microwave radiation at 6.808 GHz (aswell as a 6.880 GHz signal which is far off resonance and has no effect). The mixer
allows the use of RF function generators to provide easy control of the frequency andintensity of the microwave radiation, but the same effect could be achieved using asingle frequency microwave source at 6.808 GHz instead. The microwaves are fed
into a microwave horn, which then exposes the atoms to radiation at that frequency.
Meanwhile, the bias field is swept linearly from about 12 G to 13.5 G over the courseof 20 ms. This sweep causes the resonance frequency of the atoms to pass over thefrequency of the microwaves, transferring some, but not all, of the atoms to the|2, −2 > state. In the end, we want an equal mixture of the two states, but we44

initially prepare an excess number of |2, −2 > atoms. The reason for this is that in
the next step, gradient evaporation, the |2, −2 > atoms evaporate more rapidly due
to their high magnetic moment. Thus, we overpopulate the |2, −2 > state initially sothat after the evaporation the |2, −2 > atoms will be equal in number to the |1, −1 >atoms.

After preparing the two spin states, we then begin another round of evaporation in
the presence of a magnetic field gradient. The gradient is turned on and increased toa strength of 2 G/cm and held for between 1 and 4 seconds as the atoms evaporatively
cool. The reason for this extra evaporation step is that our state preparation adds
a great deal of entropy to the system, and evaporation in the presence of a gradientcan remove it [40]. The sweep is nonadiabatic, with the two states decohering within
milliseconds, so each atom can basically be thought of as having a random spin, either|1, −1 > or |2, −2 >. Thus, we have added entropy of around kB ln 2 per particle.
This entropy can be mostly removed through gradient evaporation. Because the twostates have opposite g-factors, they are pulled in opposite directions by the magnetic
field: the |1, −1 > atoms toward the region of weaker field and the |2, −2 > atoms
towards the region of stronger field.

As the spins segregate, their spin entropy decreases, changing into kinetic entropy
in the form of heat. This heat is then removed by evaporation. Because the |2, −2 >atoms have twice the magnetic moment of the |1, −1 > atoms, they are pulled morestrongly and evaporate more rapidly. The |2, −2 > population is initially made to be
more than half so that after this process they are equal in number to the |1, −1 >atoms. Ultimately, in a stong gradient, the two spin states are completely separated
on opposite sides of the cloud. Figure 3.3 shows a cartoon picture of this process. The
state that results from this evaporation is then ready to be used in our experiments.
45

Magnetic Field Gradient
2G/cm

0G/cm

Figure 3-3: Gradient evaporation procedure. Atoms are initially in the |1, −1 > state(black) and zero magnetic field gradient. Roughly half of the atoms are then swept
into the |2, −2 > state (white). The gradient is then ramped up to 2 G/cm, partiallyseparating the two spins. As the atoms evaporate, the entropy added from state
preparation is removed, and the spins segregate on opposite sides of the trap.

3.3

Imaging the Atoms

At the end of each experimental run, we image the atoms using resonant absorption
imaging. The atoms are illuminated with light at 780 nm, resonant with the F = 2
to F = 3 cycling transition of 87 Rb. Any atoms in the F = 2 hyperfine level casta shadow onto a camera—this image is the “probe with atoms” (PWA) frame. Two
additional images are taken immediately afterwards, one with the same light pulse
but no atoms, called the “probe without atoms” (PWOA) frame and one more frame
without the light, called the “dark field” (DF) frame. The absorption image is thenthe ratioABS =P W A − DF.
P W OA − DF

(3.1)

The number of atoms in a given pixel N (x, y) is then given by the equation

N (x, y) = −A ln ABS
,
σ0

(3.2)

where A is the area of a single pixel and σ0 is the resonant cross section. In-trap
images are taken with a magnification factor of 10, and our camera has 13µm pixels,
46

so the area A for in-trap images is (1.3µm)2 . This basic equation can be used togenerate a two dimensional image of atom number density; however, there are twocorrections we use in processing images that improve the accuracy of of our absorption
images.

3.4

Saturation Correction

The first correction involves saturation effects. Equation 3.2 is only valid in the limit
of unsaturated imaging. As the light intensity approaches or exceeds the saturation
intensity Isat , that equation no longer remains accurate. The obvious solution would
seem to be to always use light intensities well below saturation, but this is not always
possible. Sometimes, especially when imaging dense clouds of atoms in-trap, theparts of the cloud we are interested in measuring are optically dense. When this is
the case, unsaturated light will be almost entirely extinguished, so what signal does
get through will be only a few counts per pixel—this will result in high noise and poor
accuracy. In those cases, it is necessary to use much more intense light to retrieve
any signal at all, but of course, the signal will no longer give the correct density if
simply plugged into equation 3.2.
The solution is to use a correction term to cancel out the effects of saturation andretrieve the correct signal. To do this, we must know the properties of the atoms weare measuring and what the original intensity of the light was at each pixel. The firstpart is no problem, as the optical properties of 87 Rb are well understood. As for thesecond part, we can retrieve that information from the PWOA shot. We can thenuse an equation given by Ref. [59] to retrieve the real number of atoms:

N (x, y) =P W A − DFP W OA − P W AA[α ln(−].
σ0P W OA − DF
Isat

(3.3)

Here, Isat is the saturation intensity, in units of camera counts per pixel and α is a cor47

rection factor that depends on a variety of factors such as imaging beam polarizationand the structures of the upper and lower states. We determined α experimentallyby imaging a constant number of atoms with different intensities of light. The best
fit value was 2.1 for our primary imaging setup, and this value was used in all in trap
shots to determine the correct atom number. For many of our experimental runs,
specifically those performed with unsaturated light, the correction factor is basicallynegligible and equation 3.3 provides no more accurate informaiton than equation 3.2.
Nonetheless, when imaging clouds with large atom number, the flexibility given byequation 3.3’s correction helps significantly.

3.5

Principal Component Analysis Correction

While the saturation correction described above is a correction based on the interaction between 87 Rb atoms and light, there is a second correction we also make thatinvolves only the laser. Specifically, the problem is the appearance of fringes arisingfrom the division of the PWA by the PWOA images to obtain an absorption image.
We normally assume that the only difference between the PWA and PWOA images isthat the atoms are gone—in other words, that the intensity distribution of the laser isthe same. But this is not necessarily true: the shots are taken about 1 second apart,and in this time the shape of the light may change. Imaging artifacts can arise due tovibrations in the different optics components, changes in intensity, and other effects.
These give rise to fringes in the absorption image.
One way of thinking about this problem is that the PWOA shot is equal to what
the PWA shot would have been in the absence of the atoms, plus some fringes thatarise due to vibrations and other imperfections in the apparatus. Basically, what wewant to know is not what the PWOA was, but rather what it should have been. We
can make a good guess at what the PWOA should have been by looking at correlations
48

between different parts of an image. By masking off the part of the frame containing
the atoms in the PWA, we can infer what that area would have been be comparing therest of the image to a set of many PWOAs from different shots. We can then think ofreconstructing the “true” PWOA as a weighted sum over the set of several different
PWOAs, where the weighting coefficients are based on a comparison between themasked PWA and each PWOA in the set. The mathematical method of determining
these coefficients is principal component analysis.
Principal component analysis can be used on a set of vectors to generate a principal
component matrix [62, 63, 72]. In this case, the original set of vectors are the masked
PWA shots and the (unmasked) PWOA shots. Let the principal component matrix be
P , whose column vectors pi are called the principal component vectors. The vectors
are two-dimensional arrays of the same size as our images, and P has an additional
dimension equal in length to the number of images used to generate it. Similarly,
let us call the PWA image IP W A and the PWOA image IP W OA , where again each
vector is a two-dimensional array. These images are used to generate the principal
component matrix. Finally, we choose a mask around the atoms, and so can break0
0
the PWA image into two parts, IP W A = I−
P W A + IP W A where IP W A is the masked off

region—the part with the atoms. The correction prescribed by principal component
analysis then depends on whether the fringes in an image arise due to elements in the
beam path before or after the atoms.

3.5.1

Post-Atom Fringes

If the fringes arise from elements in the beam path after the atoms, then the correction
is effectively one to the dark field of each of the probe beams. Each frame, the PWA
and PWOA, must then be corrected. The corrected frames can be straightforwardly
written as
I0P W A = IP W A −

X

49

i

(I−
P W A · P )i pi(3.4)

I0P W OA = IP W OA −Xi

(IP W OA · P )i pi(3.5)

and the normalized absorption image is
I0ABS =

I0P W A − IDF
I0P W OA − IDF(3.6)

However, in our experiments, we find that this type of correction, being an additive
correction of the same type as the dark field, does not significantly diminish thevisible fringes. This is reasonable, as the path length between the atoms and camera
is shorter, and contains fewer optical elements, than the path length leading up to
the camera. Thus, the multiplicative form of correction, which accounts for pre-atom
fringes, will be the one we use in the experiment.

3.5.2

Pre-Atom Fringes

When the fringes arise from elements in the beam path prior to the atoms, as wefind to be the case in our experiment, then a multiplicative correction should beapplied. This correction is only to the PWOA shot, and essentially involves first
subtracting the fringes of the PWOA, then adding in the fringes of the PWA, so thatin the end these fringes will divide out cleanly and give an accurate absorption image.
The procedure is more complicated than the additive correction, proceeding in thefollowing manner:
First, we must decompose the basis vectors pi in the same manner as the PWA00
shots, writing pi = p−
i + pi where pi is the part of the basis vector under the mask.

Next, we renormalize the remaining part of the basis vector by writing
p0i = q1−p−
i · pi

50

p−
i .

(3.7)

Now we can extract the principal component coefficients ci from the PWA:
ci = IP W A · p0i .

(3.8)

Now, we are ready to correct the PWOA image in two steps.
In the first step, we subtract the fringes of the PWOA in the same way as we didfor post-atom fringes:
I0P W OA = IP W OA −Xi

(IP W OA · P )i pi .

(3.9)

In the second step, we then add to the PWOA the fringes that the PWA frame has,so that we may properly divide them out later:
I00P W OA = I0P W OA +X

ci pi .

(3.10)

i

Finally, the absorption image is given by
I0ABS =IP W A − IDF.
I00P W OA − IDF(3.11)The effect of using this image processing routine is to significantly reduce visible
fringes in images. This allows a more accurate in-trap image of the atom cloud
to be resolved. Figure 3.4 shows an example of a pair of frames before and after
fringe removal. Principal component analysis requires many frames to be useful insubtracting fringes, so we use all the shots in a data set to produce the principal
component matrix. One problem to keep in mind is that bad frames—those caused
by camera malfunctions, for example—can cause the fringe subtraction routine toworsen, rather than improve, the images. We occasionally had such bad frames,
and it was important to remove them prior to computing the principal component
51

Figure 3-4: Fringe removal results. The left frames show absorption images of ouratoms without fringe removal, while the right frames show the results of fringe removalusing principal component analysis. Imaging artifacts are clearly removed (note, forexample, the uneven “blotches” on the left part of the images which are gone in thecorrected images), while the image of the atoms themselves is preserved.

matrix. Removing the bad frames afterwards is not sufficent, as they will contiminate
the remaining frames through the erroneous matrix.

3.6

Magnetic Field Gradient Calibration

A final topic important to the experiments presented in the following chapters isthe calibration of the magnetic field gradient. Precise control over the strength anddirection of the magnetic field gradient, as well as accurate knowledge of the point
at which the gradient strength is zero, are very important both for spin gradientthermometry and spin gradient demagnetization cooling. This section will describe
the procedures used to calibrate all three of these.
52

The direction of the gradient must remain constant, in the x direction, over thefull range of gradients used. To assure this, a strong bias field, of approximately 15
G, is maintained in the x direction. As long as this field is maintained, the directionof the gradient will stay constant, as any gradients in orthogonal directions will bequadratically suppressed. To see why, consider that we can expand the magnetic field
to first order in ~x as~ = B0 x̂ +BX

Bi0 x~i(3.12)ii
is the gradient due to our antihelmholtz coils (other terms give rise
where Bi0 = ∂B∂xi

to curvature and can be neglected for well-aligned coils). Then the magnitude of themagnetic field |B| can be written
|B| =

√

B2 =

s

B02 + 2B0 Bx0 x +X(Bi0 )2 x2i .

(3.13)

i

Finally, to get ∇|B|, we expand the square root in the limit of large B0 to get∇|B| = Bx0 x̂ + O(B 0 /B0 )(3.14). Thus, we can treat ∇|B| as a single number value B 0 , representing the strength of
the gradient along the x axis, ∇|B|· x̂.
To be sure that as we vary the strength of the gradient, we do not introduce
magnetic field components in the ŷ and ẑ directions, we make a Stern-Gerlach measurement at various gradient strengths. The atoms are prepared in a mixture of|2, −2 > and |1, −1 > as described in section 3.2. The gradient is set to a variablevalue, then the trap holding the atoms is turned off, and the two atomic species separate in time of flight. As long as the atoms are separated along the same axis for all
values of gradient applied, we know the gradient is along that axis alone. If they do
not remain along the x axis, coils are adjusted and shimming currents are added until
53

140

Separation (arb)

120
100
80
60
40
20
0

2

4
Gradient setting (V)68Figure 3-5: Stern-Gerlach calibration. Atoms in the |2, −2 > and |1, −1 > states
are dropped in time of flight in varying magnetic fields. The x axis is voltage of thecontrol apparatus, and is proportional to the magnetic field gradient. These data
can be fit to an absolute value function to extract the strength of the gradient in
G/cm/V. These data fit to a gradient strength of approximately 0.336 G/cm/V. The
vertical offset from zero is due to the finite trap size.

they do. This same measurement gives us the gradient strength, as the separationof the atoms for a given time of flight should be proportional to the strength of thegradient. Figure 3.5 shows a plot of such a set of measurements; the separation of theatoms is an absolute value function because we do not differentiate between species
in this measurement.
The most important measurment, however, is the location of the magnetic fieldzero. This value can be estimated from a Stern-Gerlach experiment, but for our
purposes we need to be much more precise in our zero determination, at a level ofabout 1 mG/cm. To achieve this level of precision, we make an in-trap measurement of
the atoms. After gradient evaporation, the two atom species will be on opposite sidesof the trap. If the gradient is then lowered, the atoms will remain on opposite sides
until the gradient passes through zero and reverses, whereupon the atoms will switch
54

Delta X (arb)

2

0

-2

-4

-6
0

2

4
Gradient (arb)

68

Figure 3-6: Magnetic field gradient zero measurement. Atoms in the |2, −2 > and|1, −1 > states are evaporated in a strong magnetic field gradient. The gradientis then lowered to a new value, and the positions of the |2, −2 > atoms and theatom cloud as a whole are measured. The x axis is gradient in arbitrary units withan unknown offset, and the y axis is the difference in positions, also in arbitrary
units. The fit curve is to guide the eye; the gradient zero value is where the positiondifference equals zero.

sides. To determine the zero point, we make a series of measurements comparing thecenter position of the |2, −2 > atoms to the center of the atoms as a whole. The zerois where the difference between the two goes to zero. Figure 3.5 shows an example ofthe data from such a calibration run. This procedure can measure the gradient zero
position accurate to a few mG/cm. To achieve a precision of 1 mG/cm, the sameprocedure is used, but the gradient is lowered more rapidly, in a manner similar tothe process of spin gradient demagnetization cooling described in chapter 5, and theatoms are imaged in the presence of the lattice. This produces much colder atoms,
which have a correspondingly higher sensitivity to the gradient zero.

55

56

Chapter 4Spin Gradient Thermometry for
Ultracold Atoms in Optical
Lattices
As atoms in optical lattices are increasingly used to simulate condensed matter systems, the temperature of those atoms becomes more and more important to know.
New realms of physics open themselves up at ever decreasing temperatures. Mott
insulator shells first form at temperatures T ∼ 0.2U , and the layers between thoseshells become superfluid at T ∼ zJ where z is the coordination number. For twocomponent systems, magnetic ordering first arises at temperatures T ∼ J 2 /U . All ofthese temperatures are well below the condensation temperature Tc , and the lowest
are difficult to achieve at all, even in the absence of a lattice. Before one can eventhink about achieving these temperatures, however, it is important to have a method
to measure them. While methods exist to measure temperatures in this range in the
superfluid state, the development of similarly effective methods that can be appliedin the Mott insulator has proven more difficult [16, 29, 38, 51, 56]. The spin gradientthermometry method described in this chapter provides such a method applicable in57

a two-component system. These experimental results have reported in [74], a reprintof which is included in Appendix A.

4.1

Model of Spin Gradient Thermometry

Spin gradient thermometry admits a very simple theoretical treatment. The system involved consists of a collection of 87 Rb atoms in two different hyperfine states,|2, −2 > and |1, −1 >. These atoms are put in the presence of a three-dimensional
cubic optical lattice made from light at 1064 nm. The lattice is typically raised toa depth of 15Erec , putting the system above the Mott insulator transition. Finally,
the atoms are subjected to a magnetic field gradient. Since the two atoms have different magnetic moments, they are pulled in different directions by the gradient: the|2, −2 > atoms are pulled towards the region of higher field while the |1, −1 > atoms
are pulled towards the region of lower field. The ground state of the system, assuming a gradient high enough that we can neglect superexchange interactions, is for theparticles to be completely segregated on opposite ends of the trap. Conversely, athigh temperatures and low gradients, we should see a completely random mixture,with each particle equally likely to be on either side of the trap. The behavior ofparticles in between these two extreme cases is what we must understand.
Assume we have a lattice with total occupation number n = 1, composed of an
equal mix of |2, −2 > and |1, −1 > atoms. The magnetic moments of the |2, −2 >and |1, −1 > atoms are −µB and µB /2, respectively for a difference of ∆µ = 3µB /2.
First, we can treat the force on these atoms due to a magnetic field gradient as aconstant force on both plus a symmetric force separating two particles of pseudospin+1 and −1, with effective magnetic moments of ∆µ/2. The constant force will justshift the trap, so it can be neglected, leaving only the symmetric force. Then, as
long as we are in a regime where spin coherence effects are negligible, we can treat
58each lattice site as being isolated, with a partition function factorizable as Z = Zσ Z0 .
Here, Z0 is the partition function corresponding to the atom number ni , and Zσ isthe partition function for the spin. The latter is then given byβZσ = cosh( ∆µ· B(x))
2

(4.1)

for magnetic field B(x) and β = 1/kB T .
This expression is generally valid when the occupation number ni = 1, while
for occupation numbers other than 1, there are two correction factors. One is due
22
to a difference in scattering lengths, and is proportional to ∆U = U12 − U11 +U.
2

This difference, due to different scattering lengths, is very small for 87 Rb and cangenerally be neglected. The other correction is due to the indistinguishibility of theparticles, and acts as a multiplier on the temperature, usually less than a factor of
two unless atom numbers are very large. The experimental data presented in thischapter ignore these corrections and treat atoms as being in the n = 1 case, while
section 4.5 examines the corrections more closely and quantifies their size.
Setting these corrections aside, however, we can use this partition function toln Z
to get the energy ateasily find the expected spin. Using the formula < E >= − ∂ ∂β

a site, we then divide by ∆µ· B(x)/2, half the energy difference between the spin up
and down states. We can then write < s > as a function of position:β< s(x) >= tanh(− ∆µ· B(x))
2

(4.2)

By measuring the actual spin distribution in a known magnetic field and fitting itto the above equation, one can then extract the temperature. Roughly speaking,as the temperature rises, the area near the boundary between the two spin regions
changes from a sharp step to a gradual slope. The width of this boundary region
is proportional to the temperature, and inversely proportional to the gradient of the59magnetic field.
Because the magnetic moment of the atoms will always follow the direction of
the magnetic field, only the magnitude of the field is physically relevant. So we
can write the field term as a constant field plus a gradient B 0 = ∂|B|
, in the form∂xµ· B(x) = µB 0 (x − x0 ). The actual value of the constant magnetic field term isirrelevant to the physics of the problem: the position offset x0 actually acts as a fit
parameter to determine the ratio of spin up to spin down. If x0 is in the center of thecloud, for example, then the spin ratio is 1 : 1.

4.2

Experimental Procedure

A mixture of 87 Rb atoms in the |2, −2 > and |1, −1 > states is prepared in the presenceof a magnetic field gradient as described in section 3-2. Initially, the strength of themagnetic field gradient is 2 G/cm, and in the same direction as the weakest axis of
the trap. This gradient is always kept in the same direction, but its magnitude can bechanged to any of a variety of other values. In equation 4.2, the argument of the tanhfunction is proportional to the magnetic field gradient, while inversely proportional
to the temperature. Thus, applying a lower field gradient makes this thermometry
technique sensitive to lower temperatures.
After lowering the gradient to the desired value, a three dimensional optical lattice
is raised, typically to a depth of 15Erec . Since this depth is past the Mott insulatortransition for 87 Rb, which occurs at 13.5Erec , the spin distribution should be decoupled from the particle-hole distribution, meaning equation 4.2 is a good approximationof the spin distribution. This distribution can then be measured and fit to extract
the temperature.
The measurement of the spin distribution is done via a pair of absorption images.
Although it is possible to extract the spin distribution from just a single image, using
60

Figure 4-1: Images used for spin gradient thermometry. Data on the left were taken
at a lower optical trap power than data on the right. Panels a and b are images ofthe spin distribution. Panels c and d show the mean spin versus x position. The fit
to c gives a temperature of 52 nK; the fit to d gives a temperature of 296 nK. The
inset of a shows the axes referred to in the text. The bar in b) is a size scale.

two images simplifies the fitting procedure and guards against systematic errors. On
two consecutive runs, an absorption image is taken: the first run images the atoms in
the |2, −2 > state, while in the second run the |1, −1 > atoms are optically pumped
into the F = 2 manifold, and then an image of all the atoms is taken. The difference
between twice the first image and the second image shows the mean spin times thecolumn density of atoms at each pixel, so the difference divided by the image of all
atoms shows the mean spin as a function of x and y. Since the mean spin should
only depend on x (the direction parallel to the gradient), integrating over the y axis
gives a simple curve < s(x) >, which can then be fit to a hyperbolic tangent to
extract the temperature. Figure 4.1 shows a pair of composite images at two different
temperatures, along with the mean spin < s(x) >.
If the density of |2, −2 > atoms is ρ2 (x) and the density of all atoms is ρ(x), then61the fitting function used is
3
B0
2ρ2 (x) − ρ(x) = ρ(x) tanh[ µB(x − x0 )].
4 kB T(4.3)

The only free parameters used to fit are the position offset x0 , which measures thespin ratio, a vertical offset to cancel imaging errors, and the temperature, T .

4.3

Experimental Results

Figure 4.2 shows the results of this thermometry performed on 87 Rb atoms at two
different temperatures, each at a variety of magnetic field values. As predicted byequation 4.2, the inverse width of the spin distribution is proportional to the strengthof the magnetic field gradient for each constant temperature curve. However, the lowtemperature curve disagrees with this prediction for high gradients. This is a resultof finite optical resolution of our apparatus–the real width is too small to resolve athigh gradients and low temperatures. This effect was approximated by applying a 4µm Gaussian blur to the predicted curve, shown as a dash-dotted line. This curve
agrees with both the low and high gradient measurements. As a rule, however, thiseffect will limit our ability to resolve temperatures using an arbitrary gradient: tomeasure colder temperatures, it is necessary to lower the gradient to such a pointthat the width is no longer less than our optical resolution. If one leaves the gradient
too high, the result is to overestimate the temperature.
In order to gauge the accuracy of spin gradient thermometry, it is necessary tocompare it to other thermometric techniques. While there currently exist no othertechniques capable of precisely extracting temperatures of atoms in optical lattices atthe coldest ranges measured in our experiments, there do exist other reliable methods
at higher temperatures. Figure 4.3 shows a comparison of one such technique to spin
gradient thermometry. The other method estimates the temperature by a comparison
62

Figure 4-2: Independence of temperature on gradient. The inverse of the width of
the spin profile is plotted as a function of magnetic field gradient for two data sets
at two different temperatures. For constant temperature, a linear curve is expected.
The width is defined as the distance from the center to the position where the mean
spin is 1/2. The solid (dashed) line assumes a temperature of 123 nK (7 nK) andperfect imaging. The measured width of the colder data set saturates at high gradientbecause of finite imaging resolution. The dotted line assumes a temperature of 7 nK
and an imaging resolution of 4 µm.

63

to the width of the cloud in the lattice, using the relation σ 2 = kB T /mω 2 , where σis the 1/e2 half-width of the cloud along one axis and ω is the trap frequency alongthat same axis [51]. The x-axis is the strength of the optical dipole trap in arbitrary
units. Because the depth of the optical dipole trap determines the rate of evaporative
cooling, higher trap depths should correspond to higher temperatures.
The width is extracted from the wings of the atomic cloud, and the simple relation
to the temperature depends on the approximation that the atoms are noninteracting.
For high temperatures, this is a reasonable approximation as there will be many empty
sites in the lattice, so atom-atom interactions are less important. However, as temperatures fall and the atoms become more closely packed, the atomic cloud becomesincompressible and this approximation breaks down. Figure 4.3 agrees with this expectation: at high temperatures, the two techniques give similar results, while at low
temperatures width thermometry gives consistently higher estimated temperatures
than spin gradient thermometry. The good agreement between these thermometry
techniques lends credence to the claim that spin gradient thermometry also produces
accurate results in temperature ranges where no other technique can currently be
used.

4.4

Limits and Comparison to Other Techniques

One of the most useful aspects of spin gradient thermometry is its wide range of applicability. As the data in Figure 4.3 indicate, spin gradient thermometry can provide
accurate temperature values at high temperatures, even those higher then the criticaltemperature for Bose condensation. Moreover, this same technique can be used tomeasure ever lower temperatures by setting the magnetic field gradient to lower values. The lowest temperature we measured in the lattice using ordinary evaporative
cooling techniques was approximately 1 nK, but even this temperature is by no means
64Figure 4-3: Validation of spin gradient thermometry. Comparison of two measured
temperatures versus final power in one of the optical trapping beams. Squares represent the results of in-trap cloud width thermometry, and circles represent the resultsof spin gradient thermometry. Error bars represent estimated uncertainties. The
dashed line is a linear fit to the spin gradient thermometry data. The closeness of
this fit suggests that the temperature reached is proportional to the trap depth.

the lower limit of the resolution of this thermometry technique. The ultimate lower
temperature limit of spin gradient thermometry arises from a comparison of the optical resolution available and the precision of control over magnetic field gradient. For
our apparatus, for example, control over the gradient on the order of 1 mG/cm will
give a lower limit of approximately 50 pK. Of course, depending on the lattice depth
and atom-atom interactions, superexchange ordering may arise, in which case thatwill instead determine the lower limit. In chapter 5, we will see how a new coolingtechnique developed in this lab has measured spin temperatures down to this limit.
To further shed light on the utility of this technique, in is helpful to compare spin
gradient thermometry to other thermometric techniques available for use, both in andout of the lattice. Because temperatures measured in the condensate do not always
correspond to temperatures in the lattice, it is most helpful to compare the range
65

of entropies per particle, S/N kB , in which a thermometer can function. Condensate
fraction thermometry, for example, has difficulty measuring condensate fractions below 10%, and as a result its effective range is from .35 < S/N kB < 3.5. The cloud size
thermometry discussed above and used in figure 4.3 has a similar lower limit, although
its upper limit is unbounded. Theromometry based on interference peak visibility iscomputationally intensive, requiring advanced Quantum Monte Carlo calculations,but has been used to measure temperatures as low as 0.08U [70]. However, this technique is not applicable in the deep Mott insulator [56]. Measurement of the widthof the conducting layers between Mott shells is the only other technique that can beused in the Mott regime [27, 29, 35], but the range is narrow: 0.4 < S/N kB < ln(2).
Here the upper limit is the entropy at which the Mott shells melt, while the lower
limit is determined depends on the optical resolution and trap shape, but will notvary too much from this estimated value for typical experiments.
By way of comparison, spin gradient thermometry is applicable in the range 0.1 <Sσ /N kB < ln(2), where Sσ is the total spin entropy. By adjusting the strength ofthe magnetic field gradient, entropy can be freely shifted between spin and particlehole degrees of freedom to keep Sσ /N kB within this range. In essence, spin gradientthermometry measures the penetration of a boundary region similar to Mott shellwidth thermometry. However, whereas Mott shell width thermometry’s boundary
is of a strength fixed by the lattice geometry, spin gradient thermometry involves
a boundary whose strength can be freely tuned by changing the gradient. This isrelated to the fact that the energy of lowest energy excitations of the Mott insulator
is determined by U , as shown earlier in figure 2.3, which is intimately tied to thelattice depth and not easy to change. For temperatures much less than U , thisfreezes out excitations everywhere but the narrow superfluid shells, making resolutionof the temperature very difficult. By comparison, the lowest energy excitations inspin gradient thermometry are given by B 0 ∆µ∆x, where ∆x is the distance between
66

B‘∆µ∆x2

Figure 4-4: Excitations in spin gradient thermometry. In contrast to the Mott insulator, which has a fixed energy U as its lowest excitation, spin gradient thermometry
introduces an excitation of tunable strength. The lowest energy excitation is a spin
exchange over a distance ∆x between adjacent lattice sites, and has total energy
B 0 ∆µ∆x, or B 0 ∆µ∆x/2 per particle. In this graph, the gradient is oriented to pushspin up particles to the left, and the ground state energy for each site, including theenergy due to the magnetic field gradient, has been set to zero.

lattice sites. These excitations are shown in figure 4.4. Because B 0 is a freely settableparameter, one can make these excitations as large or small as needed in order toresolve them.
Furthermore, because of the decoupling between the spin and particle-hole degrees
of freedom, the bulk porperties of the Mott insulator are not significantly changedby the existence of a second spin component or the gradient. Rather, they behave
as an extra thermometer attached to the cloud as a whole. It is only in the regimeof superexchange-driven magnetic ordering, when even spin gradient thermometry
breaks down, that the presence of a gradient and a second component begin to modify
the bulk properties of the system.
67

4.5

Effects of Occupation Number ni > 1

The technique of spin gradient thermometry is most easily applicable in a latticewith occupation n = 1 at all sites. At higher densities, where ni > 1, there are twocorrections for spin gradient thermometry, one to the partition function as a whole,due to the indistinguishibility of the bosons, and one to the potential as a resultof unequal interaction strength between species. This section will quantify these two
effects, the first of which takes the form of a multiplier to temperature, and the secondof which acts approximately as an added term.

4.5.1

The Effect of Indistinguishability

The partition function Zσ described in section 4.1 is valid for sites with n = 1 filling,
and also for distinguishable particles at any occupation number. It is not, however,valid for indistinguishable bosons with n > 1. This is because the degeneracy of thedifferent levels is miscounted. For example, in the case of n = 3, there are four possible
total spin values for the site: 3, 1, −1, and −3. In the classical (indistinguishable)case, these values have degeneracy 1, 3, 3, 1, and these degeneracies are reflected in thesimple model where Z3 = Z13 . This partition function would imply that the spin perparticle for an n = 3 site is the same for an n = 1 site. However the same spin levels in
the indestinguishable case all have degeneracies of 1. Thus, Z3 6= Z13 . Instead, we canwrite the general n particle partition function as Zn =Pn/2

βj=−n/2 exp(− 2 ∆µ· B(x)· 2n),or in a simpler form,
Zn =sinh[(n + 1)β/2∆µ· B(x)].
sinh( β2 ∆µ· B(x))(4.4)

This partition function can then be used to predict the mean spin per particle ata site. This value is easily calculated as< s >=n+1
ββcoth[(n + 1) ∆µB 0 x] − 1/n coth( ∆µB 0 x).
n
2
2
68

(4.5)

It can easily be checked that this simplifies to equation 4.1 in the n = 1 case, but for
n 6= 1 the form of the equation is not so easy to work with. It is especially not simple
to fit to when, as is common in experiments with higher filling factors, one must
integrate over a range of sites with different values of n. Nonetheless, with a goodknowledge of how many atoms are in each given n state, a good approximation canbe made. As n increases, the spin function changes from a hyperbolic tangent to asimilar function with a steeper slope. This makes it appear as though the temperature
has decreased. Thus, a site with n particles will look like it is a n=1 site, but withtemperature of approximately

Tapparent =3
Treal .
n+2

(4.6)

So, by fitting the temperature as though it the lattice were at n = 1 filling,
then multiplying by n+2, we get an approximately correct temperature. If we are3
integrating over a range of values for n, the correction to the temperature can thenbe approximated as
X

Treal = (

n

fn

n+2
)Tapparent
3

(4.7)

where fn is the fraction of atoms in sites with occupation number ni = n. The
correction to the temperature caused by this is independent of the temperature itself,
and is a relatively small multiplier: for a lattice of pure n = 2 sites, for exemple,
the temperature appears 25 percent colder than it really is. Because sites with threeatoms or more decay rapidly through 3-body recombination [12], most experiments
will not have many lattice sites with three or more atoms, especially when trying toachieve lower temperatures.
69

Property
SymbolValue|1, −1 > s-wave scattering lengtha11100.44a0|1, −1 > /|2, −2 > s-wave scattering lengtha12
98.09a0|2, −2 > s-wave scattering lengtha22
95.47a0
Table 4.1: 87 Rb scattering lengths. Values are taken from [34]. The uncertainties ofthese values are not given, but a measurement of a22 − a11 in the same reference has
uncertainty 0.31a0 .

4.5.2

Scattering Length Correction

In addition to the above correction to the partition function, there is also a correction
to the potential the atoms feel in sites where ni > 1. This arises when the interactionenergy U depends on spin. Although atoms in both states experience the samelattice potential in our experiment due to our lattice’s large detuning, there is a slightdifference in their interaction energies due to the states’ different scattering lengths.
In 87 Rb, the scattering lengths of atoms in the F = 2 and F = 1 manifolds are nearly
identical, there is nevertheless a small difference: the best known values [34, 73] aregiven in terms of the bohr radius a0 in table 4.1. While the variation between thevalues a11 , a12 , and a22 is small on its own, ranging over only a few percent in value,22= 0.135a0 , that matters.
it is actually the much smaller difference ∆a = a12 − a11 +a2

To see why, imagine two sites, separated by distance ∆x, one with n atoms in the|2, −2 > state and one with the same number in the |1, −1 > state. The energy cost
per particle of switching one of the atoms from one well with one from the otherwould normally be ∆µB 0 ∆x/2. However, the interaction energy difference adds an
22
additional cost equal to (n − 1)∆U where ∆U = U12 − U11 +U, the difference between
2

the interspecies interaction energy and the mean of the two intraspecies interactionenergies. Since Uij ∝ aij , we can write ∆U ≈ U ∆a/a, which in a 15Erec lattice isabout kB · 40 pK. This should be treated as a rough value, as the scattering lengthsof 87 Rb very nearly cancel, so this remainder is similar in size to the precision of
measurement.
70

Clearly, this energy scale is much smaller than any of the temperatures measured
above, but in chapter 5, temperatures are reached that are sufficiently low that thiscorrection becomes significant—comparable in size to the partition function correction. Thus, it is worth writing out the full form of the correction to the potential now.
It can be written in a self-consistent manner as a function of the spin distributions(x) in the following manner:

E(x) =∆µ 0
B x + s(x)(n − 1)∆U
2

(4.8)

Because ∆U is positive, meaning these two states are naturally immiscible, the effectis to make the sample appear colder than it really is. Of course, given the size of theerror bars on ∆U compared to its size, it is even possible that the sign of the effect is
opposite, favoring a mixture of spins rather than a segregation at zero temperature.
Nonetheless, the best estimate is that a small positive correction term must be added
to temperatures when they drop below 100 pK or so.

4.6

Dependence of Temperature on GradientAs a final point of consideration, on careful examination of figure 4.2, one questionthat may arise in the reader’s mind is this: Why do the data points in a given set all
fit to the same temperature? At first glance, the answer may appear obvious: eachshot was prepared in the same manner, with the same trap depth and lattice depth, soof course they should be the same temperature. However, this is not quite true. One
thing that does vary between the shots is the final strength of the gradient. Whileeach shot is prepared by evaporatively cooling the two components in a gradient of
2 G/cm, that gradient is changed to the various values on the x-axis before turningon the lattice. Might this changing gradient not affect the temperature? After all,section 4.4 describes how changing the gradient moves entropy between spin entropy
71and bulk entropy, so it is natural to question whether this transfer of entropy would
heat or cool the system. Indeed, one would expect that if the gradient is lowered,more entropy would go into spin, leaving less in the bulk and thus cooling the system
as a whole. Since we do not see this in the data in figure 4.2, we must explain why.
That is, why is the process of lowering the gradient isothermal in our experimentrather than, say, adiabatic?
The answer to this question lies in the exact manner in which the gradient was
lowered. In each experimental run, the gradient was lowered from 2 G/cm to its
final value over a very long time: the gradient was changed linearly over the courseof 400ms. Then, the atoms were held an additional 400ms before ramping up thelattice over the course of about 150ms. Thus, the atoms had nearly a second to
spend “adjusting” to the new lattice depth. The reason this kept the system at its
initial temperature is the same reason it was at that temperature to begin with: thebalance between heating and evaporative cooling. The trap depth determines thetemperature of the atoms by determining the rate of evaporative cooling. As theatoms cool, the rate of evaporation drops. But this evaporative cooling is balancedby heating due to three body losses, collisions with background gas particles, andother mechanisms. The final temperature is the point at which this heating and theevaporative cooling rate balance. If the atoms are suddenly cooled when the gradientchanges, the rate of evaporative cooling will drop, and on net, the system will begin
to heat up. Given enough time, this heating will continue until the atoms are backto their initial temperature. Thus, a very slow change in gradient, such as was donein our experiment, will be approximately isothermal.
But what about a more rapid change. one that did not allow time for heatingfrom collisions and other mechanisms to warm the atoms back up? Naturally, theexpectation is that we really would see cooling as entropy was shifted between spin
and bulk excitations. A fast enough ramp time could be almost perfectly adiabatic,72

allowing for a maximum cooling effect. And indeed, it is precisely this phenomenon,
analogous to the condensed matter technique of adiabatic demagnetization refrigeration, that is used in the experiments of chapter 5 to achieve temperatures well into
the picoKelivn range. Also in that chapter, data will be presented that explicitly
check this transition between isothermal and adiabatic changes in gradient.

4.7

Conclusion

Spin gradient thermometry represents an important new technique for measuring thetemperature of ultracold atoms in optical lattices. We have demonstrated its wide
dynamic range, and its compatibility with other techniques within their own dynamic
ranges. It is well suited for measuring lattice temperatures for two-component systems, making it a valuable tool for use in quantum simulation of magnetically-orderedsystems. The limitations of this technique are very few, arising only at especially highdensities or at very low temperatures, as superexchange physics begins to play a role.
With future experiments, such as those described in the next chapter, it is hoped thatthese limits will be put to the test, leading to a new frontier in atomic physics.

73

74

Chapter 5
Spin Gradient Demagnetization
Cooling
Spin gradient thermometry provides the ability to measure very cold temperaturesin the presence of an optical lattice, but it does not, on its own, provide a means
to achieve them. For 87 Rb, magnetically ordered states in the Mott insulator are
expected to arise at temperatures around 100 − 200 pK [21], which is significantly
below the temperatures usually accesible in ultracold atoms, even outside of a lattice.
However, the same physics that allows spin gradient thermometry to measure thetemperature of a two-component system can be harnessed to cool that same system.
By changing the strength of the gradient separating the two spin components, it ispossible to substantially cool the system. This process, which we call spin gradientdemagnetization cooling, is analogous to the condensed matter process of adiabatic
demagnetization refrigeration. By using a time-varying magnetic field gradient tochange the spin energy scale, it is possible either to cool the spins as an isolated system, or to pump entropy into the spin degrees of freedom, thereby cooling the system
as a whole. Using this technique, we have prepared states with spin temperaturesof 75 pK and −75 pK. We have also produced bulk temperatures in an apparent
75equilibrium at 300 pK, although we cannot rule out long-lived metastable excitations
that may fail to couple to our thermometer. These temperatures represent the lowest
temperatures which have been achieved in any system [33, 46, 71]. The experimental
results described in this chapter were reported in [5] and is included in Appendix B.

5.1

Basic Theory

To understand the basic theory of this cooling technique we need only look at thetheory behind spin gradient thermometry. In the presence of a magnetic field gradient,a two-component Mott insulator will separate into to regions of opposite spin, witha boundary region of mixed spin in the center. The width of this region will be
proportional to the temperature of the spins and inversely proportional to the strengthof the magnetic field gradient. If the strength of the magnetic field gradient changes,
then one or both of the other two properties—the temperature or the width—mustchange to maintain this equality. The way these two change depends on the manner inwhich the gradient is changed, so it is useful to look at two different models to explain
it. The first will be most useful if the gradient is changed nonadiabatically and thespin system is held out of equilibrium with the particle-hole degrees of freedom, while
the second covers the case where it is changed adiabatically and in thermal contact
with the particle-hole degrees of freedom.

5.1.1

Direct Cooling for Nonadiabatic Demagnetization

In the first, simplest case, we treat the nonadiabatic case in which the spin degreeof freedom is isolated from the particle-hole degrees of freedom. This case can beaccomplished experimentally by placing the atoms in a deep Mott insulator. In a
Mott insulator, the rate of spin relaxation is proportional to J 2 /U , a quantity which
can become very small at high lattice depths. Then, if the strength of the gradient76

is changed on a timescale much shorter than the spin relaxation time, we can treat
the spin as an isolated system and ignore the particle-hole degrees of freedom (which
remain at their original distribution and temperature).
Because the spin system is “frozen” in place, the width of the domain wall will
not change, and the total spin entropy is conserved. In this case the temperature of
the spin system will scale proportional to the strength of the magnetic field. To see
why this is, recall the thermodynamic definition of temperature

T =

∂E(S)
|N,V .
∂S(5.1)

If E(S) is simply scaled by a constant without changing the configuration of thesystem, then obviously T will be scaled by the same constant.
A good toy model to illustrate this effect is that of a system of spin-1/2 particles
with magnetic moment µ in a uniform magnetic field of initial strength B1 . Figure
5.1 illustrates this system. The lower state has energy −B1 µ/2 while the upper hasenergy of B1 µ/2. If the population in the upper state is p, then one can easily workout from the Boltzmann distribution that the temperature is

T1 =

µB1
.
kB ln 1−pp

(5.2)

If the magnetic field is changed to B2 while the magnetization is held fixed then
the new temperature must be T2 = T B2 /B1 . Furthermore, if the field is reversed
in direction, then the temperature will be negative as a result, indicating that theswapping of the two energy states has led to an inverted population.
The toy model illustrated in figure 5.1 is actually a reasonable approximation of
the spin physics behind the process of adiabatic demagnetization refrigeration usedin condensed matter systems. However, the full process of adiabatic demagnetization
refrigeration is more closely analogous to the adiabatic case, which is discussed in the77Magnetic
Field:

B1

B2

-B2

p
1-p

E
p
1-p
Temperature:B
T 2= T 1 2
B1

T1

-T 2

Figure 5-1: Demagnetization toy model. This model illustrates the physics behing
the spin part of adiabatic demagnetization cooling. If the particles cooled by demagnetization are decoupled from other degrees of freedom, then their spin can be cooledin exactly this manner. In real atomic systems, the difficulty of controlling magnetic
fields to the necessary µK level limits the practicality of using this simple scheme ofdemagnetization.

78

next subsection. Regardless, magnetic field demagnetization of this sort is impracticalfor use in atomic systems due to difficulties in magnetic field control [23]. However,a magnetic field gradient can be used to generate a similar effect in a much easiermanner. As discussed in chapter 4, a two-component Mott insulator in the presenceof magnetic field gradient will have a spin distribution given by< s(x) >= tanh(−∆µB 0 x
).
2kB T

(5.3)

Clearly, if the spin distribution is held fixed and the gradient B 0 is changed, thetemperature must shift by the same factor. Magnetic field gradients are much easier
to control than magnetic fields, so that cooling that would require an impracticalµG magnetic field control can be achieved with gradient control at the relatively
easy mG/cm level. Also, magnetic field gradients can easily be reversed in direction without inducing spin flips, allowing the straightforward production of negative
temperature systems.

5.1.2

Adiabatic Demagnetization as an Entropy Pump

The nonadiabatic case described above is simple because the spin distribution is held
fixed. As a result, it is only the spin temperature that is cooled—the particle-holedegrees of freedom are thermally isolated and remain at their starting temperature.
This is fine if we only wish to cool the spin degree of freedom, but if we wish tocool the system as a whole, it becomes more complicated. As this process requires
the system to equilibrate, it is best used as a model when the gradient is changed
in a shallow lattice, or none at all, so the system remains in equilibrium throughout
the demagnetization process. If this condition holds, we can model the process asadiabatic.
In contrast to the nonadiabatic case, this process will be almost directly analogous
79to adiabatic demagnetization refrigeration [20, 31]. In adiabatic demagnetization refrigeration, a paramagnetic material is prepared in a strong magnetic field. The
magnetic field strength is then decreased, and entropy is absorbed into spin-flip excitations, cooling the system. In our system, the total magnetization is conserved, so
there are no spin flips; however, the same role is played by spin exchanges across theboundary region of randomly oriented spins in the center of the system. This willallow the elimination of particle-hole excitations in favor of spin-exchange excitations.

In adiabatic spin gradient demagnetization, the gradient is lowered, causing theenergy scale of the spin degrees of freedom to fall while the energy of the particle-holedegrees of freedom remain fixed. But since these two parts of the system are now
in thermal contact, entropy will flow out of particle-hole excitations to create spin
exctations. As a result, the width of the central “domain wall” of random spin will
widen. However, it will not widen proportionally to the change in magnetic field:
since entropy is leaving the particle-hole excitations, they are cooled, and therefore
the entire system must be colder. Roughly speaking, if the gradient changes from B10
to B20 , then the final temperature will be
T2 =B20 S0 + ∆S
T1
B10
S0

(5.4)

where S0 is the initial spin entropy and ∆S is the entropy absorbed by the spins. The
size of ∆S will depend on the relative heat capacities of the spins and the rest of thesystem, which will depend on various factors including the density of atoms and thelattice depth. In a sense, the spins are used as a “sponge” to soak up the entropy in
the rest of the system. In the end, at very low gradients, nearly all the entropy canbe transferred into the spins, and the final temperature of the system will be muchcolder. Indeed, if the starting entropy is low enough (below about kB ln 2 per latticesite), then in the limit as the gradient strength approaches zero, all of the entropy
80could be absorbed into the domain wall.
This gives us a lower limit for the temperature that can be achieved by adiabatic
spin gradient demagnetization. If the total starting entropy of the system is Stotal ,of which the spins initially contain Sspin , then applying equation 5.4 gives us anexpression for the maximum cooling factor:
B 0 Sspin
T1
= 10
.
T2
B2 Stotal(5.5)

For our experiment, the ratio of magnetic field gradients available is around 1000. If
we have a starting temperature of about 5 nK, and about half the total entropy were
initially in the spins, with half initially in the particle-hole degrees of freedom, thiswould suggest a lower limit of somewhere around 10 pK, assuming total transfer ofentropy and no additional heating. This equation also applies for the nonequilibrium
case, except that in that case, ∆S = 0, so Sspin /Stotal should be set to 1.
Another limit on the lowest achievable temperature stems from the effects of spin
coherence, which will start to become important near the Curie temperature of thespin-ordered phases. Spin correlations reduce the heat capacity of the spin degrees of
freedom, because a magnetic domain containing multiple sites can only hold as much
entropy as a single site could in the absence of correlations. These practical andtheoretical limits on spin gradient demagnetization cooling do not appear to preclude
cooling below the Curie temperature of the spin-ordered phases. This technique thus
provides a specific and realistic method of realizing magnetic phase transitions inlattice-trapped ultracold atoms.

5.2

Experimental Procedure

The initial state of the system is a two-component BEC at zero lattice depth in astrong gradient. The two components are the |2, −2 > and |1, −1 > hyperfine states
81Figure 5-2: Details of two experimental cooling protocols. a: Experimental phase
diagram of lattice depth vs. applied gradient. Dashed lines show two different paths
along which one can move between the high-gradient superfluid state and the lowgradient Mott insulating state. b: Lattice depth (solid line) and gradient strength(dashed line) versus time for the lower path in panel a. c: Lattice depth (solid line)
and gradient strength (dashed line) versus time for the upper path in panel a. The
shape of the lattice rampup is designed to ensure maximum equilibration.

of 87 Rb, and the gradient strength is initially 2 G/cm. The final state will be one ofhigh lattice depth, beyond the Mott insulator transition, and low gradient. The way
this state is reached depends on what type of experiment is to be performed. The two
experimental pathways used differ in which is done first: the lowering of the gradientor the raising of the lattice. Figure 5.2 depicts these two orderings as different paths
in an experimental “phase space” of gradient vs lattice depth.
In the lower path, as shown in part b of figure 5.2, the gradient is first lowered withthe lattice still at zero depth. Because this occurs in the superfluid, we can expect thechange to be adiabatic so long as the rate of change of the gradient compares favorably
to the trap frequency. The typical demagnetization time is 100ms, which should be
sufficiently slow when compared to the lowest trap frequency of 36Hz. Next, thelattice is raised to a depth of 15Erec , which is above the Mott insulator transition
point of 13.5Erec . The lattice is raised in three steps: a cubic spline to an intermediate82

lattice depth (typically 6 or 7Erec ), held for a brief time, then raised again in a cubic
spline to its final depth. The process takes between 150 and 200ms in total. The
reason for the intermediate hold is an attempt to maximize equilibration in the lattice.
At higher lattice depths, tunneling slows down exponentially, while low lattice depths
are sensitive to vibrational heating, so spending more time at intermediate depths
helps ensure maximum equilibration without introducing too much extra heating.
This produces a system that has been cooled similar to the adiabatic case described in the theory section. Although the particle-hole and spin degrees of freedom
may not be perfectly equilibrated—certainly they are frozen in at some point as thelattice crosses the Mott insulator transition—they are nevertheless in equilibrium tothe greatest extent possible. As a result, the domain wall width will be widened
significantly from its initial width, and will provide an estimate of the temperatureof the system as a whole.
The upper path, on the other hand, is used to produce a nonequilibrium coolingof the spin degrees of freedom alone. This path is depicted in part c of figure 5.2.
Here, the gradient is kept at a strength of 2 G/cm while the lattice is raised in thesame, three-step manner as in the adiabatic case. At this point, the lattice is thenheld in the Mott insulating state, so the tunneling rate will very small, proportional
to the superexchange term J 2 /U , and the atoms can be treated as frozen in place. In
this case, we can say that the particle-hole and spin degrees of freedom are isolated
from each other, and will be essentially separate systems with fixed entropy andindependent temperatures. The spin temperature can now be directly cooled bylowering the gradient. The gradient is lowered over the course of 100ms, which inthis case is much faster than the tunneling rate, supporting the assertion that weare creating a spin temperature isolated from the rest of the system. Since thespins are isolated, they can even be brought to negative temperature distributions byreversing the direction of the gradient. These distributions should remain at a stable83temperature on timescales on the order of the superexchange time.

5.3

Experimental Results

Each of the two experimental procedures described above was performed over a variety
of different values of final magnetic field gradient. The results of the nonadiabatic path
are shown in figure 5.3. The simple nature of this experiment allows one to essentially
set the temperature at whim by selecting a sufficiently low final field gradient value.
For negative field gradients, this produces negative temperatures. The minimum
temperature limit is ultimately set by one’s optical resolution and ability to measure
field gradient, so this procedure allowed the production of spin temperatures of 75 pKand −75 pK. Since the total energy is monotonic in 1/T , these temperatures represent
the most extreme (coldest and hottest) thermodynamic states ever measured in spin
systems [33, 71]. Because these spin temperatures are well isolated from the bulk
temperature, they display long lifetimes. The inset of figure 5.3 shows temperature
versus hold time for both positive and negative spin distributions. If a negative
temperature distribution is held for several seconds in the lattice, its temperature
becomes more negative, as expected.
The results of the adiabatic path, on the other hand, are more difficult to model
theoretically, and reach lower temperatures, but they are also more interesting inthat they produce a system closer to true equilibrium. Figures 5.4 and 5.5 show
the results of initial spin gradient demagnetization cooling experiments. As the field
gradient is reduced, the width of the domain wall is observed to increase (see figure5.4), indicating the transfer of entropy from the kinetic degrees of freedom to thespins. The width increases much less steeply than would be expected for an isothermal
sample; this demonstrates cooling. The observed domain wall width can be converted
to a temperature using spin gradient thermometry. The measured temperature falls
84

Figure 5-3: Preparation of arbitrary spin temperatures. Main Plot: measured spintemperature versus final gradient, for the case of rapid gradient change in the Mott
insulating state. Inset: Temperature versus hold time in the lattice of spin distributions at negative (dashed line) and positive (solid line) initial temperature. Note that,as expected, the temperature of the negative distribution becomes more negative as
it heats. Error bars represent the standard deviation of the data at each point.

85

Figure 5-4: Plot of width vs. field gradient. Demonstration of spin gradient demagnetization cooling. Measured domain wall width vs. final magnetic field gradient, for thecase of adiabatic gradient lowering in the superfluid state followed by lattice rampup.
The dashed line is an isotherm, meaning that it represents the expected widening behavior assuming no cooling. The dotted line shows the minimum measurable width,
given our optical resolution. The blue curve is the theoretical prediction, assuming
adiabatic demagnetization from an initial temperature of 6.2 nK and including theeffects of finite optical resolution.

rapidly as the gradient is lowered (see figure 5.5). The lowest measured temperature ison the order of 300 pK, which is within a factor of 2 of the expected Curie temperatureof the XY ferromagnet [15]. Theoretical curves in the two figures show good agreementwith the data and were fit with only one free parameter: the initial temperature prior
to demagnetization.

5.4

Modelling the Adiabatic Case

Figures 5.4 and 5.5 include a theoretical fit to the cooling data, assuming an initial
temperature of 6.2 nK, which is derived from a 1-paramater fit to the data. This section will explain the method used to model this case. The theory for the spin degrees
86Figure 5-5: Plot of temperature vs. field gradient. Demonstration of spin gradient demagnetization cooling. Theoretical temperature and measured temperature vs. final
magnetic field gradient, for the case of adiabatic gradient lowering in the superfluid
state followed by lattice rampup. These measurements are the same as those plottedin Fig. 3. The dashed line follows the isothermal trajectory and the dotted line shows
the optical resolution limit. The blue curve is the theoretical prediction, assuming an
initial temperature of 6.2 nK and not including the effects of finite optical resolution.

87

of freedom is straightforward, as already discussed, but the particle-hole degrees of
freedom are more complicated. To deal with them, we will follow the theoretical treatment from Refs. [29] and [35]. To calculate the partition function for the particle-holedegrees of freedom, we make two approximations: first, that J = 0, so that each site
can be treated separately, and second, that the number of particles at a site is no morethan one different than the T = 0 number, n0 = dµ/U e. This second approximation
is called the particle-hole approximation, and gives a partition function

z0 = 1 + exp(−β(U n0 − µ)) + exp(β(U n0 − U + µ)).

(5.6)

The particle-hole approximation is valid for kB T < 0.2U , which is certainly true
for the temperatures measured in our experiments. From this we can get the meanoccupation numbern ' n0 + (exp(−β(U n0 − µ)) − exp(β(U n0 − U + µ)))/z0 .

(5.7)

Here we use the position-dependent chemical potential µ(r) = µ − Vtrap (r).

Now that we have both the partition function for the spin degrees of freedom (fromequation 4.1) and the partition function for the particle-hole degrees of freedom, we
can directly calculate the entropy of each as a function of temperature at varying
gradients. Figure 5.6 shows calculated values for particle number and particle-hole
entropy per site, while figure 5.7 shows a calculation of spin entropy per site andtotal entropy per site at a variety of gradients and temperatures. We can then plot atemperature vs. gradient curve given an initial temperature (which we treat as a fit
parameter), a measurement of the total atom number and trap frequencies from ouroriginal data, and the assumption of fixed total entropy. The result is the fit curve
shown in figures 5.4 and 5.5.
88Figure 5-6: Particle-hole entropy distribution and number. The upper two plots show
atom number as a function of position for two different temperatures, given our known
trap frequencies and atom number. Each graph is a slice through the center of the
trap. The lower right shows entropy per site for a the same slice. Finally, the lower
right shows both atom number and entropy per site along a cut through the center.

Figure 5-7: Spin entropy distribution. Calculated distributions of spin entropy per
particle at various values of magnetic field gradient and temperature. The upper left
diagram shows spin entropy alone, while the other three show total entropy, including
both particle hole and spin entropies.

89

5.5

Equilibration and Adiabaticity

In the nonadiabatic path, lack of equilibration is a feature: we are interested indirectly cooling the spins, so minimizing the thermal contact with the remainder ofthe system can only help. However, on the adiabatic path, we are interested in using
the spins to cool the remainder of the system, so a lack of equilibration can be aserious obstacle. Hence, the degree to which the system remains in equilibrium along
the adiabatic path is an important thing to assess. Since there are two steps along
the path, we can estimate how well the atoms equilibrate along each of these steps
by testing their reversibility. If a given step is adiabatic, it is reversible, and should
also be in equilibrium at all times.
We have tested the reversibility of the gradient ramps used in spin gradient demagnetization cooling by decreasing, then increasing, then decreasing the gradient,and we observe no detectable difference between the resulting data and data produced by a single gradient ramp. This indicates that the gradient ramps used in spin
gradient demagnetization cooling in the superfluid state are adiabatic, as is expectedbased on the trap frequencies (36, 141, and 156 Hz) and total demagnetization times(100 ms).
However, equilibration in the Mott insulator is more difficult to demonstrate because equilibration timescales get very long as the system approaches the Mott insulator transition. This makes it nearly certain that the system leaves equilibrium atsome point during lattice rampup, so the question may really be whether this disequilibrium is significant. Still, the previously demonstrated agreement between spin
gradient thermometry and trap width thermometry at high temperatures (as shownin chapter 4) indicates that the kinetic and spin degrees of freedom are reasonably
well equilibrated in that regime. The strong fits in this colder regime to theoreticalpredictions is a further indication of good equilibration—both the hyperbolic tangent shape of the spin distribution and the good one-parameter fit to our model of90

demagnetization (which assumes adiabaticity).
Nonetheless, when we perform the same test of reversibility on the lattice raising
step as we did on the demagnetization step, heating is generally observed. Thus,
we cannot rule out the existence of long-lived metastable excitations in the Mott
insulating state which do not couple to the spin degrees of freedom and thus do
not influence our temperature measurement. This result is consistent with otherexperiments which have also seen evidence of long equilibration times in the Mott
insulator [41, 61]. On the bright side, the long lifetime of such excitations may mean
they are so decoupled from the spin degrees of freedom as to be irrelevant to the bulk
magnetic properties of the spin system. So if we are interested in using this coolingtechnique to observe spin ordering, these types of excitations may not interfere.
The reason for the poor equilibration of some of these excitations is the verysmall second-order tunneling rate, J 2 /U , which for our system is on the order of
1Hz at the deepest lattice depths. How much of a problem this is depends on the
degree of equilibration that happens at lower lattice depths, and how much remains toequilibrate above the Mott insulator transition. The same second-order tunneling rate
governs the formation of spin-ordered domains, so the same problems would be faced
if we were to try to achieve magnetic ordering in this system. Other, similar systems
may not have so hard a time: for example, equilibration time of a lighter atomic
species (e.g. 7 Li) would be faster by up to an order of magnitude due to the higher
recoil energy, making such species ideal candidates for spin gradient demagnetization
cooling.

5.6

Isothermal Demagnetization

An additional question closely related to the question of equilibration is the one posedin section 6 of chapter 4: why did the thermometry performed in those experiments
91not show evidence of demagnetization cooling? In that section it was suggested that,
although the experimental procedures of the thermometry were similar to those of
demagnetization cooling, the very long time spent demagnetizing (about 1s) may
have allowed other heating processes to cancel it out.

To test this explanation, we set up our system in a way similar to our thermometry runs, and with higher optical trap depths to create similar starting temperatures(warmer than our best demagnetization runs). We then performed demagnetization
runs alternately with fast demagnetization, as we normally do, and slow demagnetization, as we did in the original thermometry experiments. These showed a marked
difference: the fast demagnetization runs showed cooling similar to what we havedescribed above, while the slow demagnetization runs showed little to no cooling,
producing an isothermal demagnetization just as we observed in our thermometry experiments. This indicates that our explanation in section 4-6 was probably accurate:
demagnetization must be done rapidly, lest various uncontrolled heating mechanisms
cancel it out and reduce it to an isothermal demagnetization. However, it is also
worth noting that more aggressive demagnetization starting from the lowest possible
initial temperature still showed some deviation from the isothermal line, even withthe longer wait time. This suggests that some of our coldest results in the previous
chapter, for example our quoted lowest temperature of 1 nK, may have inadvertently
been the result of demagnetization cooling. The necessity of measuring these temperatures at very low gradients (to be able to resolve them optically) means also thatthey were subjuct to a large demagnetization, which may not all have been eliminatedby heating before we imaged the atoms.
92

5.7

Future Prospects

Spin gradient demagnetization cooling represents a significant step forward in the fieldof quantum simulation. The lack of a method to cool atoms in an optical lattice has
been an important stumbling block in the effort to probe Hamiltonians exhibiting
quantum magnetism. Many cooling mechanisms have been proposed [6, 36, 37, 58],
but await experimental realization. By contrast, the method presented here canbe immediately implemented in existing experiments. We have demonstrated theability to produce and measure spin distributions at arbitrary temperature by rapidlychanging the magnetic field gradient applied to a two-component Mott insulator. In
the same system, we have proposed and demonstrated a new cooling technique for theMott insulator, based on adiabatic demagnetization of a domain wall prior to lattice
rampup. This procedure has produced final temperatures of 300 pK, although wecannot rule out the presence of long-lived metastable excitations which do not couple
to our thermometer. Ultimately, both of these techniques are capable of cooling a
two-component Mott insulator below the superexchange temperature. It remains tobe seen which technique will be of most use: the answer to this question depends onwhether low spin temperature alone, or low temperature for the entire system is mostimportant. However, the choice between these two techniques allows for the possibilityof tailoring spin and particle-hole entropies to suit the needs of a given experiment.
This opens a realistic path towards experimental observation of superexchange-driven
phase transitions in optical lattices.

93

94

Chapter 6
Conclusion
It seems clear that quantum simulation will be a field of significant scientific interest
and advancement in the coming years. The desire to understand quantum magnetic
phenomena, especially those leading to exotic but incompletely understood behavours
such as superconductivity, has attracted substantial interest in the field of ultracold
atoms. The work presented in this thesis clears a significant hurdle in the path towardachieving these goals.
The development of spin gradient thermometry and spin gradient demagnetization
cooling represents a significant step forward in the field of ultracold atoms. For
the first time, we have the ability to measure temperatures in an optical latticeat whatever scale and precision we need. We also have the ability to cool atoms
colder than ever before, with cooling of spin temperatures by a factor of 1000 within
relatively easy reach. Moreover, the experimental technique is simple to implement
and generalizable to nearly any system with at least two components with different
magnetic moments. The cooling technique could even be used as a refrigerator for
sympatetic cooling of a third species, if needed. The existence of these techniques,
and the new realm of temperatures they allow, opens the door to new fields of study
in quantum magnetism and simulation.
95

The next goal of the Rubidium lab will be to look for these orderings, beginningwith the search for an xy-ferromagnetically ordered phase in the Mott insulator. To
achieve this objective, a spin-dependent lattice will be implemented to tune the interactions between the two different spin states. In conjunction with the technique ofspin gradient demagnetization cooling, advancements in this direction may be stunningly fast. However, other problems already touched on in this thesis, especially thelong equilibration times observed in the Mott insulator, may hinder the observationof quantum magnetic phenomena in 87 Rb.
As an alternative approach, driven largely by the discoveries made while developing these techniques, a new lab is being founded in the Ketterle-Pritchard group to
focus on using 7 Li in place of 87 Rb in an optical lattice. It is hoped that this substitution will be able to overcome the problem of equilibration we faced in 87 Rb. The
high mass of 87 Rb made equilibration in the Mott insulator slow and uncertain; 7 Li’s
lower mass and correspondingly higher tunneling rate may help resolve this problem.
All other things being equal, the tunneling rate near the Mott insulator transition
is inversely proportional to the mass, so lighter elements have a natural advantage
in this regard. Furthermore, the presence of usable Feschbach resonances in 7 Li, afeature lacking in 87 Rb, may allow the tuning of interactions without resorting tospin-dependent lattices. Still, regardless of which lab is best able to advance into thisnew regime, the outlook of the field is bright, and I have every expectation of successfor both labs in the future.

96

Appendix A
Spin gradient thermometry for
ultracold atoms in optical lattices
This appendix contains a reprint of Ref. [74]: David M. Weld, Patrick Medley,
Hirokazu Miyake, David Hucul, David E. Pritchard, and Wolfgang Ketterle, Spingradient thermometry for ultracold atoms in optical lattices. Phys. Rev. Lett.,
103(24):245301, (2009).

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PRL 103, 245301 (2009)

Selected for a Viewpoint in Physics
PHYSICAL REVIEW LETTERS

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Spin Gradient Thermometry for Ultracold Atoms in Optical Lattices
David M. Weld, Patrick Medley, Hirokazu Miyake, David Hucul, David E. Pritchard, and Wolfgang Ketterle
MIT-Harvard Center for Ultracold Atoms, Research Laboratory of Electronics, and Department of Physics,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA(Received 20 August 2009; revised manuscript received 10 October 2009; published 7 December 2009)
We demonstrate spin gradient thermometry, a new general method of measuring the temperature ofultracold atoms in optical lattices. We realize a mixture of spins separated by a magnetic field gradient.
Measurement of the width of the transition layer between the two spin domains serves as a new method ofthermometry which is observed to work over a broad range of lattice depths and temperatures, including in
the Mott insulator regime. We demonstrate the thermometry using ultracold rubidium atoms, and suggestthat interesting spin physics can be realized in this system. The lowest measured temperature is 1 nK,
indicating that the system has reached the quantum regime, where insulating shells are separated bysuperfluid layers.
DOI: 10.1103/PhysRevLett.103.245301

PACS numbers: 67.85.d, 03.75.Mn, 05.30.Jp, 75.10.Jm

Ultracold atoms trapped in optical lattices represent a
new frontier for the investigation of many-body physics[1,2]. The existence of novel physics at decreasing energy
scales drives the quest for lower temperatures in the atomic
Mott insulator. Insulating Mott shells form at a temperature
T  0:2U, where U is the interaction energy. At the lower
temperature T  zJ, where J is the tunneling amplitude
and z is the number of nearest neighbors, the conducting
layers become superfluid and the system enters a quantum
insulator state [3]. At the even colder temperature scale
T  J 2 =U, superexchange-stabilized phases can exist in
the two-component Mott insulator; this is the regime ofquantum magnetism [4]. Various proposals [5,6] have focused on the realization of quantum spin Hamiltonians inthis regime. Detection of superexchange-driven phase transitions in these systems remains a major goal of ultracold
atomic physics. Perhaps the most important barrier toexperimental detection of such a phase transition is therequirement of temperatures well below 1 nK [4].
Additional cooling methods [7–10] will be needed to reachthis very interesting temperature scale. However, it is clearthat to assess current methods and to validate future cooling techniques, low-temperature thermometry of the Mott
insulator is needed.
Thermometry of systems in the Mott insulating state hasremained a challenge [3,11–14]. In this Letter, we discussand demonstrate a simple and direct method of thermometry using a magnetic field gradient which works in thetwo-component Mott insulator.
The theory behind this method of thermometry is
straightforward. The system under consideration is an
ensemble of atoms in a mixture of two hyperfine statesloaded into a three-dimensional optical lattice in the presence of a weak magnetic field gradient. The two states havedifferent magnetic moments, and are thus pulled towards
opposite sides of the trapped sample by the gradient. At
0031-9007=09=103(24)=245301(4)zero temperature, the spins will segregate completely, anda sharp domain wall will exist between the two spin
domains (a small width due to superexchange coupling istypically negligible). This system has the same bulk physics as the single-component Mott insulator, but includes
additional degrees of freedom in the form of spin excitations in the domain wall. At finite temperature, spin excitations will increase the width of the domain wall. Thiswidth will depend in a simple way on the field gradient, thedifferential Zeeman shift, and the temperature, and canthus be used as a thermometer.
For an incoherent mixture of two spins, the partition
function for an individual lattice site can be
P approximately
factorized as Z ¼ Z Z0 , where Z ¼  expð BðxÞÞ,  is 1=kB T,  is the magnetic moment of thespin , BðxÞ is the spatially varying magnetic field, and
Z0 is the partition function of the particle-hole degrees of
freedom (for which see [3]). This approximation is generally valid for the case of one atom per lattice site; foroccupation number n > 1, it is valid when the mean of theintraspin interaction energies U  is equal to the interspin
interaction energy U"# , which is a good approximation in
87 Rb [15]. Since the total magnetization is fixed, the average value of the magnetic field is canceled by the corresponding Lagrange multiplier; we include this in thedefinition of BðxÞ. We are free to treat the two states as
having pseudospin þ1 and 1; making that identification,
the mean spin hsi as a function of position, gradientstrength, and temperature has the simple form
hsi ¼ tanhð    BðxÞ=2Þ;(1)where  is the difference between the magnetic momentsof the two states. A fit of the measured spin distribution
with a function of this form will give the temperature of thesystem. When the Zeeman shift due to the magnetic field
gradient is a linear function of position, imaging of the spin

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PRL 103, 245301 (2009)PHYSICAL REVIEW LETTERS

distribution essentially corresponds to direct imaging of
the Boltzmann distribution.
The apparatus used to produce ultracold 87 Rb atoms is
described in Ref. [16]. After cooling, approximately 105
atoms are held in a far-red-detuned crossed optical dipole
trap with trap frequencies between 100 and 200 Hz. A
three-dimensional cubic optical lattice, formed by three
retroreflected beams each of radius 150 m, overlaps
the trapping region. Since spin gradient thermometry does
not depend on the number of atoms per lattice site n, we
perform measurements at a range of n values between 1
and 4. The trapping and lattice beams are all derived fromone fiber laser, with a wavelength  of 1064 nm. Magnetic
field gradients up to a few G=cm can be applied withexternal coils, and calibrated using Stern-Gerlach separation of the different spin states after release from the trap.
The gradient is applied along the x direction, which is theweakest axis of the crossed dipole trap. Absorptive imaging of the atoms is performed with a camera pointing downalong the vertical z axis.
The sequence of steps used to measure temperature
is as follows. First, a sample of 87 Rb atoms in thejF ¼ 1; mF ¼ 1i state is prepared by evaporation in the
optical trap. Here F and mF are the quantum numbers forthe total spin and its projection on the z axis, respectively.
The atoms are then placed into a mixture of the j1; 1i andj2; 2i states by a nonadiabatic magnetic field sweep
through the microwave transition between the two states.
This pair of states was chosen in order to avoid spinexchange collisions. A magnetic field gradient of
2 G=cm is applied along the weak axis of the trap andresults in additional evaporation, which is intended toremove the entropy created by the state preparation [17].
At this point, the field gradient is changed to the value to be
used for measurement; lower gradients are used for lowertemperature measurements to keep the domain wall width
larger than the imaging resolution. The optical lattice isthen adiabatically ramped up, typically to a depth of
14:5ER , where ER ¼ h2 =2m2 is the recoil energy and mis the atomic mass. The transition to the Mott insulator
occurs at 13:5ER . At this point, the spin structure depends
on the temperature as discussed above.
There are several ways to measure the resulting spin
distribution. One way is to first take an image of the F ¼ 2
atoms in the 14:5ER lattice, then in a second run to illuminate the atoms with an optical repumper beam resonantwith the F ¼ 1 to F0 ¼ 2 transition for a few s prior to
imaging. This method gives an image of all atoms and an
image of just the F ¼ 2 atoms; appropriate subtraction can
provide the spin distribution. It is possible to determine thetemperature from a single image of one spin, but the data inthis Letter were all taken using pairs of images to guard
against systematic errors.
The temperature can then be measured by fitting the spin
distribution to the hyperbolic tangent form. The resulting

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thermometer has high dynamic range and variable sensitivity, works at all accessible temperatures of interest, andrequires only the simplest fitting procedures.
Figure 1 shows data of the type used for spin gradientthermometry. An image of the total atom density and an
image of the spin density are obtained as discussed above.
Both images are then integrated along the y direction,
which is transverse to the gradient. The spin distribution
is then fit by a function of the form ðxÞ tanhð34 B djBj
dx xÞ,
where ðxÞ is the total density distribution. The only freeparameters in this fit are a horizontal and vertical offset andthe temperature T ¼ 1=kB .
Figures 2 and 3 show the results of this thermometry on
ultracold 87 Rb atoms in an optical lattice. Figure 2 shows
the linear scaling of the inverse width of the domain wall as
the magnetic field gradient is varied while holding thetemperature constant. For widths larger than the opticalresolution, the scaling is as predicted by Eq. (1). The twodata sets plotted in Fig. 2 were taken at two different
temperatures: 7 and 123 nK, according to the best-fit
theoretical lines. Finite optical resolution or motion of
the atoms during imaging will blur the measured spin
profile and result in an overestimate of the domain wall
width at high gradients. This effect was modeled by applying a Gaussian blur of radius 4 m to the theoretical 7 nK
spin profile at various gradients. The resulting curve, plotted as a dash-dotted line in Fig. 2, reproduces the saturationof measured width observed in the experimental data. The
effect of finite resolution is always to overestimate thetemperature.
Figure 3 shows the measured temperature plotted as afunction of the power in the dipole trapping beam which
confines the atoms in the direction of the magnetic field
gradient (the x direction). Higher powers in this beam lead
to less effective evaporation, and thus higher final tempera-

FIG. 1. Images used for spin gradient thermometry. Data on
the left were taken at a lower optical trap power than data on theright. Panels (a) and (b) are images of the spin distribution.
Panels (c) and (d) show the mean spin versus x position. The fit
to (c) gives a temperature of 52 nK; the fit to (d) gives atemperature of 296 nK. The inset of (a) shows the axes referred
to in the text. The bar in (b) is a size scale.

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FIG. 2. Independence of the measured temperature on theapplied field gradient. The inverse of the width of the spin profileis plotted as a function of magnetic field gradient for two data
sets at two different temperatures. For constant temperature, alinear curve is expected. The width is defined as the distancefrom the center to the position where the mean spin is 1=2. The
solid (dashed) line assumes a temperature of 123 nK (7 nK) andperfect imaging. The measured width of the colder data set
saturates at high gradient because of finite imaging resolution.
The dotted line assumes a temperature of 7 nK and an imaging
resolution of 4 m.

tures. As a check of the new method of thermometry, Fig. 3
also presents an analysis of the same data using an existing
method of thermometry, based on measurement of the intrap width of the atomic cloud along the direction perpendicular to the gradient. This second method is based on thewell-known relation 2 ¼ kB T=m!2 , where  is the 1=e2
half-width of the atomic cloud and ! is the trap frequency

FIG. 3. Validation of spin gradient thermometry. Comparisonof two measured temperatures versus final power in one of the
optical trapping beams. Squares represent the results of in-trap
cloud width thermometry, and circles represent the results ofspin gradient thermometry (see text for details). Error bars
represent estimated uncertainties. The dashed line is a linear fit
to the spin gradient thermometry data. The closeness of this fit
suggests that the temperature reached is proportional to the trap
depth.

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in the direction along which the width is measured [12].
The width is determined by a fit to the wings of the trapped
cloud. Trap width thermometry is based on a noninteracting approximation, and will fail at temperatures less than U
when the system starts to become incompressible. As in
Ref. [12], all points on this plot are in the high-temperaturesingle-band regime (T is less than the band gap but greaterthan the bandwidth). For the temperatures plotted in Fig. 3,
the agreement between the two methods is reasonably
good, and gives confidence in the use of spin gradientthermometry in regions of parameter space where no otherthermometer exists.
The large dynamic range of spin gradient thermometryis evident in Fig. 3. Thermometry can be performed attemperatures so high that no condensate exists before
lattice ramp-up. The lowest temperature we have measured
was achieved by using the new thermometry as a feedback
signal, enabling adjustment of experimental parameters for
optimization of the final temperature in the Mott insulator.
This method allowed us to achieve a measured temperature
as low as 1 nK. At the lattice depth used here, U is 37 nK,
and zJ is 6 nK. The measured temperature is thus well
below Tc ¼ zJ, the predicted critical temperature for thesuperfluid layer between the n ¼ 1 and n ¼ 0 Mott domains. According to the treatment of Ref. [3], at 1 nK thesystem should be well inside the quantum regime, withconcentric quantum insulator shells separated by superfluid layers. This represents the first direct demonstrationthat this temperature regime has been achieved in the Mott
insulator.
At a given value of the magnetic field gradient, very low
temperatures will result in a width of the transition region
smaller than the imaging optics can resolve (see Fig. 2).
However, the width can be increased by decreasing themagnetic field gradient. The lowest measurable temperature will then depend on the minimum achievable gradient
as well as the optical resolution, which are technical rather
than fundamental limitations. In our apparatus, background gradients with all coils turned off are of order
103 G=cm, which, given our imaging resolution of afew m, would in principle allow measurement of temperatures down to 50 pK or the superexchange scale,
whichever is higher.
It is instructive to compare the useful range of this new
method of thermometry with that of existing methods. To
facilitate meaningful comparison with non-lattice-based
methods, we discuss the range of entropy per particle
S=NkB at which a given thermometer works, rather than
the range of temperature. Condensate fraction thermometry works for 0:35 < S=NkB < 3:5, where the lower limit isset by the difficulty of detecting a thermal fraction less than
10%, and the upper limit is set by disappearance of thecondensate. Thermometry based on the thermal cloud size
has a similar lower bound, but extends to arbitrarily highvalues of S=NkB . Quantitative thermometry based on the245301-3

PRL 103, 245301 (2009)PHYSICAL REVIEW LETTERS

visibility of interference peaks upon release from the lattice requires state-of-the-art quantum Monte Carlo calculations fitted to the data. This technique was recently used
to measure temperatures as low as 0:08U in the superfluid
phase near the Mott insulator transition [18]. This method
cannot be applied deep in the Mott insulating state [11].
Measurement of the width of the conducting layers between the Mott shells is the only previously proposed
method which works directly in the Mott insulating state[3,4,19]. However, this method requires tomographic techniques, and the useful range of entropy is rather narrow:
0:4 < S=NkB < lnð2Þ, where the upper limit is set by themelting of the Mott shells, and the lower limit is an
estimate based on typical trapping parameters and opticalresolution. Counting only spin excitations, the range ofspin entropy per particle at which spin gradient thermometry works in our system is 0:1 < S =NkB < lnð2Þ, where
the lower limit is a function of optical resolution andsample size and the upper limit corresponds to the point
at which the domain wall becomes as wide as the sample. It
is important to note that spin gradient thermometry canwork even if the entropy of the particle-hole excitations
lies outside of this range in either direction. For example,
spin gradient thermometry can work at arbitrarily highvalues of the total entropy per particle S=NkB , assuming
the field gradient is increased to the point where S =NkB <lnð2Þ.
The method of thermometry presented here works because the two-component Mott insulator in a field gradient
has a spectrum of soft and easily measurable spin excitations. The wide dynamic range of this method is a result of
the fact that, in contrast to the gapped spectrum of the bulk
one-component Mott insulator, the energy of the spin excitations can be tuned by adjusting the strength of themagnetic field gradient. The addition of a field gradientand a second spin component does not change the bulk
properties of the Mott insulator and can be regarded as‘‘attaching’’ a general thermometer to the first component.
The two-component Mott insulator in a field gradient is
a rich system which can provide experimental access tonovel spin physics as well as thermometry. In the work
presented here, we have always allowed the spin distribution to equilibrate in the gradient before ramping up the
optical lattice. However, changing the gradient after theatoms were already loaded into the lattice should open up
several interesting scientific opportunities, in which thegradient is used to manipulate or perturb the atoms rather
than as a diagnostic tool. If, for example, the gradient weresuddenly changed after lattice ramp-up, one could probe
nonequilibrium spin dynamics in a many-body quantum
system. If the gradient were instead lowered adiabaticallyafter ramp-up, adiabatic cooling of the Mott insulator could

week ending
11 DECEMBER 2009potentially be performed which, in contrast to [20], would
not involve spin-flip collisions.
In conclusion, we have proposed and demonstrated a
new method of thermometry for ultracold atoms in opticallattices. We have used the new method to measure temperatures in the Mott insulator as low as 1 nK. Thistemperature is to the best of our knowledge the lowest
ever measured in a lattice, and it indicates that the system
is deep in the quantum Mott regime.
The authors thank Eugene Demler, Takuya Kitagawa,David Pekker, and Lode Pollet for helpful discussions. This
work was supported by the NSF, through a MURI program,and under ARO Grant No. W911NF-07-1-0493 with funds
from the DARPA OLE program. H. M. acknowledges support from the NSF.

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885 (2008).
[2] M. Lewenstein et al., Adv. Phys. 56, 243 (2007).
[3] F. Gerbier, Phys. Rev. Lett. 99, 120405 (2007).
[4] T.-L. Ho and Q. Zhou, Phys. Rev. Lett. 99, 120404 (2007).
[5] L. Duan, E. Demler, and M. Lukin, Phys. Rev. Lett. 91,
090402 (2003).
[6] E. Altman, W. Hofstetter, E. Demler, and M. Lukin,
New J. Phys. 5, 113 (2003).
[7] M. Popp, J.-J. Garcia-Ripoll, K. G. Vollbrecht, and J. I.
Cirac, Phys. Rev. A 74, 013622 (2006).
[8] P. Rabl, A. J. Daley, P. O. Fedichev, J. I. Cirac, andP. Zoller, Phys. Rev. Lett. 91, 110403 (2003).
[9] J.-S. Bernier et al., Phys. Rev. A 79, 061601 (2009).
[10] T.-L. Ho and Q. Zhou, Proc. Natl. Acad. Sci. U.S.A. 106,6916 (2009).
[11] L. Pollet, C. Kollath, K. V. Houcke, and M. Troyer, New J.
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[12] D. McKay, M. White, and B. DeMarco, Phys. Rev. A 79,
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[13] A. Hoffmann and A. Pelster, Phys. Rev. A 79, 053623(2009).
[14] B. Capogrosso-Sansone, E. Kozik, N. Prokof’ev, andB. Svistunov, Phys. Rev. A 75, 013619 (2007).
[15] For other atomic species, this treatment can readily be
extended to include effects of finite U ¼ U   U"# . The
leading correction changes the magnetic field gradient at
the center of the sample B0 to an effective gradient B0 ð1 þ
ðn  1ÞU=kB TÞ.
[16] E. W. Streed et al., Rev. Sci. Instrum. 77, 023106 (2006).
[17] C.-L. Hung, X. Zhang, N. Gemelke, and C. Chin, Phys.
Rev. A 78, 011604 (2008).
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Phys. Rev. Lett. 97, 060403 (2006).
[20] M. Fattori et al., Nature Phys. 2, 765 (2006).

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Appendix B
Spin Gradient Demagnetization
Cooling of Atoms in an Optical
LatticeThis appendix contains a reprint of Ref. [5]: Patrick Medley, David M. Weld, Hirokazu
Miyake, David E. Pritchard, and Wolfgang Ketterle, Spin gradient demagnetization
cooling of atoms in an optical lattice. This paper is under revision and will be submitted for publication shortly.

103

Spin gradient demagnetization cooling of atoms in an optical lattice
BEC IV1

1

MIT-Harvard Center for Ultracold Atoms, Research Laboratory of Electronics, and Departmentof Physics, Massachusetts Institute of Technology, Cambridge MA 02139
Attainment of lower temperatures has often led to discoveries of new physical phenomena.
This observation has driven the search for new cooling methods which can be applied toneutral atoms in optical lattices. Such systems are expected to exhibit correlated magnetic
quantum phases1 , but only below a Curie or Néel temperature which is typically on the orderof 100 picokelvin. Realization of this low temperature in a lattice is a major goal of atomic
physics. Here we present a general method of refrigeration which works in an optical lattice. We show that a time-varying magnetic field gradient applied to a lattice-trapped spin
mixture can substantially cool the system. Cooling can be achieved either by nonadiabatic
preparation of thermally isolated spin distributions at arbitrary (including negative) temperatures or by adiabatic reduction of the temperature of an equilibrated sample in a manner
analogous to adiabatic demagnetization refrigeration2, 3 . We have prepared spin temperatures of +75 picokelvin and -75 picokelvin, and have used spin gradient demagnetization
cooling to adiabatically reduce the temperature of an apparently equilibrated sample to 350
picokelvin, although the possibility of long-lived metastable excitations which do not couple
to our thermometer cannot be ruled out. These are the lowest temperatures which have been1

achieved in any system4–6 . This new refrigeration technique appears to be capable of coolingbelow the Curie temperature of spin-ordered phases. These results open a realistic path tothe observation of magnetic quantum phase transitions in optical lattices.
Application of a magnetic field gradient to a spin mixture will result in spatial segregation of
the two spin components. In our experiments, these two spin domains always remain in thermal
contact, with a “domain wall” of intermixed spins between them. In the Mott insulating state,
the equilibrium spin distribution depends on the applied gradient ∇B and the temperature T as
hs(x)i = tanh(−β · ∆µ · B(x)/2), where hs(x)i is the expectation value of the spin at positionx, β is 1/kB T , and ∆µ is the difference between the magnetic moments of the two states. The
recently developed technique of spin gradient thermometry7 is based on the fact that the resulting
domain wall width is proportional to T . The technique works because the two-component Mott
insulator in a field gradient has a spectrum of soft and easily measurable spin excitations, theenergy of which can be tuned by adjusting the strength of the magnetic field gradient.

This coupling between the applied gradient and the energy spectrum allows exploration of
the system’s response to a time-varying gradient. The gradient can be varied either quickly or
slowly with respect to the spin relaxation timescale. If the gradient is changed in the deep Mott
insulating state, the spin relaxation rate due to coupling to other degrees of freedom, which scales
as J 2 /U where J is the tunneling energy and U is the interaction energy, is so slow that the gradientcan easily be varied much faster than the spin system can respond. This very slow relaxation
enables the production of negative temperature distributions, and distributions with a very low

2positive temperature typically set by the accuracy with which the field gradient can be measured.
Conversely, if the gradient is changed at shallow or zero lattice depth, the spin relaxation is fastenough that adiabatic adjustment to the lowering of the gradient occurs. Fig. 1 shows these two
possible paths in experimental phase space. Experiments performed in these two regimes give very
different results.

Figure 1: Details of two experimental cooling protocols. a: Experimental “phase diagram” oflattice depth vs. applied gradient. Dashed lines show two different paths along which one canmove between the high-gradient superfluid state and the low-gradient Mott insulating state. b:
Lattice depth (solid line) and gradient strength (dashed line) versus time for the lower path in panela. c: Lattice depth (solid line) and gradient strength (dashed line) versus time for the upper path inpanel a. The shape of the lattice rampup is designed to ensure maximum equilibration.

If instead of ensuring equilibration, we prevent it entirely, by lowering the gradient quickly
deep in the Mott insulator, then the response of the system is so slow that we can achieve essen-

3

tially arbitrary spin temperatures. This includes negative temperatures. Figure 2 shows the resultsof such an experiment. Spin distributions at 75 pK and -75 pK have been prepared. Since thetotal energy is monotonic in −1/T , these temperatures represent the most extreme (coldest andhottest) thermodynamic states ever measured in spin systems4, 5 . The inset of Fig. 2 shows temperature versus hold time for both positive and negative spin distributions. If a negative temperature
distribution is held for several seconds in the lattice, its temperature becomes more negative, as
expected.

If the gradient is instead changed adiabatically, another intriguing possibility exists, which
is based on an analogy with a common technique in experimental condensed matter physics: adiabatic demagnetization refrigeration2, 3 . Adiabatic demagnetization refrigeration in condensed matter systems typically makes use of a paramagnetic material, which is placed in a strong magnetic
field and allowed to come to thermal equilibrium. The field is then slowly decreased, which reduces the energy of spin flip excitations and results in an effective increase in the heat capacityof the material. Energy and entropy flow from other degrees of freedom into the spin degrees of
freedom; the non-magnetic degrees of freedom are thus cooled. In the two component Mott insulator, the total magnetization is fixed, so our proposed technique of “spin gradient demagnetization
cooling” proceeds slightly differently. Instead of a magnetic field, a magnetic field gradient is reduced, which results in a decrease of the slope of the spin distribution. This decrease in slope canbe thought of as an increase in the width of the domain wall between the spin-up and spin-down
regions of the sample. Since each lattice site in the domain wall carries entropy, this pumps entropy
into the domain wall from other (e.g. particle-hole) degrees of freedom, cooling them. If the initial
4

Figure 2: Preparation of arbitrary spin temperatures. Main Plot: measured spin temperature versus
final gradient, for the case of rapid gradient change in the Mott insulating state. Inset: Temperature
versus hold time in the lattice of spin distributions at negative (dashed line) and positive (solid line)
initial temperature. Note that, as expected, the temperature of the negative distribution becomes
more negative as it heats. Error bars are statistical.

entropy is low enough (less than about kB ln(2) per lattice site), then at some decreased value ofthe gradient all of the entropy of the system will be pumped into the domain wall. In principle, theentropy in the spin system could be removed at this point by optical pumping, resulting in a very
cold single-component Mott insulator. Further adiabatic reduction of the gradient below this point
cannot widen the domain wall further, and will thus linearly decrease the temperature of the spin5

degrees of freedom.

The initial state of the system is a two-component BEC at zero lattice depth in a strong
gradient. The gradient is adiabatically decreased, which cools the system as described above, andthe lattice is then raised past the superfluid–Mott insulator transition. If the atoms are in a Mott
insulator which is initially at a temperature low enough for the particle-hole approximation to hold,the maximum entropy per lattice site is about kB ln(2). The total entropy per site will be smaller
by a factor of the ratio of conducting “shell” volume to total volume 8 . The maximum entropy persite for the spin degree of freedom in a very small gradient is kB ln(n + 1), where n is the local
number of indistinguishable bosons per site. For temperatures below the melting point of the Mott
phase, the maximum heat capacity of the spin system is thus strictly larger than the heat capacityof the kinetic (i.e. particle-hole) degrees of freedom. This suggests that substantial cooling of theparticle-hole degrees of freedom can be achieved with this method, even in a one-shot (non-cyclic)
experiment. We have made a more quantitative analysis of the proposed technique by calculating
entropy-versus-temperature curves of the two-component Mott insulator in various field gradients,using a model which is exact in the limit of no tunneling (manuscript in preparation). The results ofthese calculations (see theory curves in Figs. 3 and 4) confirm that spin gradient demagnetization
cooling is in principle capable of reaching extremely low temperatures well below the expected
Curie temperatures of spin-ordered states.

There are, however, both practical and theoretical limits on the temperatures which can beattained with spin gradient demagnetization cooling. In traditional magnetic refrigeration exper-

6

iments, the minimum temperature is often set by the minimum achievable magnetic field or thepresence of unavoidable internal fields due to spin ordering in the material9 . Analogues of boththese limits are relevant to spin gradient demagnetization refrigeration. Practically, the ratio between the highest and lowest magnetic field gradients which can be applied to the system is anupper bound on the ratio between the initial and final temperatures. In our experiment, the maxi∇Bi
is about 1000, which would give a minimum temperature below 10 picokelvinmum value of ∇Bfassuming a typical initial temperature of 7 nanokelvin. Another limit on the lowest achievabletemperature stems from the effects of spin coherence, which will start to become important near
the Curie temperature of the spin-ordered phases. Spin correlations reduce the heat capacity ofthe spin degrees of freedom, because a magnetic domain containing multiple sites can only holdas much entropy as a single site could in the absence of correlations. These practical and theoretical limits on spin gradient demagnetization cooling do not appear to preclude cooling below theCurie temperature of the spin-ordered phases. This technique thus provides a specific and realistic
method of realizing magnetic phase transitions in lattice-trapped ultracold atoms.

Figures 3 and 4 show the results of initial spin gradient demagnetization cooling experiments.
As the field gradient is reduced, the width of the domain wall is observed to increase (see Fig. 3,
indicating the transfer of entropy from the kinetic degrees of freedom to the spins. The width
increases much less steeply than would be expected for an isothermal sample; this demonstrates
cooling. The observed domain wall width can be converted to a temperature using spin gradientthermometry. The measured temperature falls rapidly as the gradient is lowered (see Fig. 4). The
lowest measured temperature is 350±50 picokelvin, which is within a factor of 2 of the expected7Curie temperature of the XY ferromagnet 10 . Theoretical curves in Figs. 3 and 4 show good
agreement with the data. These curves were fitted to the measured data using only one variable
parameter: the initial temperature at the maximum gradient. The initial temperature inferred fromthis fit is 7 nanokelvin. In light of these data, it seems possible that the 1 nanokelvin temperatures
measured in our earlier work7 , which were observed at low field gradients, were in fact the resultof adiabatic demagnetization cooling.

The idea behind spin gradient demagnetization cooling is straightforward, and if adiabaticity
can be maintained then it is clear that cooling can be achieved. Our experimental protocol was
designed to allow the system to equilibrate as much as possible at low lattice depths where the response times are short. However, equilibration timescales get very long as the system approaches
the Mott insulator transition, and this can make it difficult to demonstrate perfect adiabaticity. We
have tested the reversibility of the gradient ramps used in spin gradient demagnetization cooling bydecreasing, then increasing, then decreasing the gradient, and we observe no detectable difference
between the resulting data and data produced by a single gradient ramp. This indicates that thegradient ramps used in spin gradient demagnetization cooling in the superfluid state are adiabatic,
as is expected based on the trap frequencies (36, 141, and 156 Hz) and total demagnetization times(100 ms). Equilibration in the Mott insulator is more difficult to demonstrate, although the previously demonstrated agreement between spin gradient thermometry and trap width thermometry athigh temperatures7 indicates that the kinetic and spin degrees of freedom are equilibrated in that
regime. The fact that the spin distribution fits well to the hyperbolic tangent form expected of an
equilibrated spin system is also indirect evidence for equilibration, as is the good one-parameter8

Figure 3: Demonstration of spin gradient demagnetization cooling. Circles represent measured
domain wall width vs. final magnetic field gradient, for the case of adiabatic gradient loweringin the superfluid state followed by lattice rampup. Solid lines connect the points, and error bars
are statistical. Insets show examples of spin images at the indicated point. The dashed line is an
isotherm, meaning that it represents the expected widening behavior assuming no cooling. The
dash-dotted line shows the minimum measurable width, given our optical resolution. The dottedcurve is the theoretical prediction, assuming adiabatic demagnetization from an initial temperatureof 7 nK and including the effects of finite optical resolution.

fit to our theoretical predictions (which assume adiabaticity). However, if the lattice is raised, thenlowered to zero, then raised again, heating is generally observed. Thus, we cannot rule out the9Figure 4: Demonstration of spin gradient demagnetization cooling. Circles represent measured
temperature vs. final magnetic field gradient, for the case of adiabatic gradient lowering in the
superfluid state followed by lattice rampup. Error bars are statistical. These measurements are thesame as those plotted in Fig. 3. The dashed line follows the isothermal trajectory and the dashdotted line shows the optical resolution limit. The solid line is the theoretical prediction, assuming
an initial temperature of 7 nK and including the effects of finite optical resolution. The dotted lineis the same theoretical prediction without the effects of optical resolution.

existence of long-lived metastable excitations in the Mott insulating state which do not couple tothe spin degrees of freedom and thus do not influence our temperature measurement. Indeed, otherexperiments have seen evidence of long equilibration times in the Mott insulator11, 12 . However,10the long life of such excitations may mean they are so decoupled from the spin degrees of freedom
as to be irrelevant to the bulk magnetic properties of the spin system. In this case, they would notprevent spin ordering. The slowest relevant timescale for the formation of spin-ordered domains is
the second-order tunneling rate J 2 /U (although dynamics in the single-component Mott insulator
at the faster timescale of U have recently been observed)13 . For 87 Rb at the highest lattice depths
used in our system, this time is of order 1 Hz, although faster ordering could be achieved if, as is
expected, some equilibration takes place at lower lattice depths. Additionally, equilibration time ofa lighter atomic species (e.g. 7 Li) would be faster by up to an order of magnitude due to the higher
recoil energy, making such species ideal candidates for spin gradient demagnetization cooling.

Although spin gradient demagnetization cooling was inspired by (and is locally similar to)magnetic refrigeration in condensed matter systems, there are important differences between thetechniques. For example, it was generally believed that demagnetization cooling required spinflips. However, spin gradient demagnetization cooling, because it uses a magnetic field gradientinstead of a spatially homogeneous field, proceeds via spin transport in a system with fixed magnetization. In contrast to a previously reported technique in atomic systems14 , spin gradient demagnetization cooling involves no spin-flip collisions, and thus avoids heating effects associatedwith such collisions. On a practical level, it is easier to achieve very small magnetic field gradients
than very small magnetic fields. In our system, an energy resolution of kB ×25 picokelvin can be
relatively easily achieved with a gradient of 1 milligauss/cm and an optical resolution of 5 µm,
while similar energy resolution in a homogeneous system would require control of the magnetic
field at the microgauss level. Thus our results extend the applicability of magnetic refrigeration11

techniques beyond their previously accepted limits.

The Hubbard Hamiltonian describing a mixture of pseudo-spin-1/2 bosons in a lattice canbe reduced to an XXZ Heisenberg spin hamiltonian1 . This observation has generated substantial
interest in the possibility of studying magnetic phase transitions in such a system. Recent quantum Monte Carlo calculations studying the XY -ferromagnetic ground state in a system with thesame parameters as our experiment indicate a critical entropy per particle of 0.35kB and a criticaltemperature of 200 picokelvin at U↑↓ /Uσ = 0.510 . The critical entropy for the antiferromagnetic
state is higher, at 0.5kB . The nonadiabatic cooling technique presented here has achieved spintemperatures and entropies well below these critical values, and the total entropies and temperatures we observe after adiabatic spin gradient demagnetization cooling are within reach of thecritical values. This indicates the feasibility of achieving spin-ordering using one of these coolingtechniques. It is not yet clear which of the two cooling methods is better suited to the productionof spin-ordered states; the answer will depend on the achievable initial entropy and the ratio of
heating timescales to ordering timescales.

Ultracold atoms trapped in optical lattices have the potential to be used as flexible quantum simulators of strongly interacting many-body systems. Progress in this field has depended
upon the development of a cooling method which works in a lattice. Many such methods havebeen proposed15–18 but await experimental realization. The method presented here can be immediately implemented in existing experiments. We have produced and measured spin distributionsat arbitrary spin temperatures as low as 75 picokelvin by rapidly changing the magnetic field gra-12

dient applied to a two-component Mott insulator. Using the same system, we have proposed anddemonstrated an adiabatic cooling technique for the Mott insulator, based on demagnetization of adomain wall. Using the adiabatic cooling we have measured final equilibrated temperatures of 350
picokelvin, although we cannot rule out the presence of long-lived metastable excitations which do
not couple to our thermometer. Ultimately, this technique is capable of cooling a two-component
Mott insulator below the superexchange temperature. This work opens a realistic path towards
experimental observation of superexchange-driven phase transitions in optical lattices.

Methods
The apparatus used to produce ultracold 87 Rb atoms is described elsewhere19 . After being trappedand pre-cooled with RF evaporation, the atoms are delivered to the experimental vacuum chamberby translation of a far-detuned optical tweezer beam. The atoms are then loaded into a crossed
optical dipole trap. Evaporation is performed by decreasing the power in the trapping beams, whichcools the atoms below Tc , the critical temperature for Bose-Einstein condensation. The atoms arethen placed into a mixture of the |F = 1, mF = −1i and |F = 2, mF = −2i hyperfine states by anonadiabatic magnetic field sweep through the microwave transition between the states. Further
evaporative cooling in a 2 Gauss/cm magnetic field gradient removes the entropy created by state
preparation, and shifts the two spin states to opposite sides of the trap. A three-dimensional cubic
optical lattice, formed by three retroreflected beams, overlaps the trapping region. This lattice israised using the profile shown in Fig. 1, which includes a pause at intermediate lattice depths. This
profile was observed to improve spin equilibration. Magnetic field gradients up to a few Gauss/cm

13

can be applied with external coils, and measured using Stern-Gerlach separation. The gradientis applied along the weak axis of the crossed dipole trap, and can point in either direction. The
zero-crossing point is measured by observing the point at which the |1, −1i and |2, −2i atoms
swap sides as the gradient is reversed in the superfluid state. Absorption imaging of the atoms istypically performed with a camera pointing down along the vertical axis, which is perpendicular tothe direction of applied field gradient. In order to resolve the density profile of the optically dense
thick atomic cloud, the saturation parameter of the imaging beam is typically greater than 1. The
saturated images are corrected and calibrated according to a standard procedure20 .

1. Duan, L., Demler, E. & Lukin, M. Controlling spin exchange interactions of ultracold atoms
in optical lattices. Phys. Rev. Lett. 91, 090402 (2003).
2. Giauque, W. F. A thermodynamic treatment of certain magnetic effects. a proposed method of
producing temperatures considerably below 1◦ absolute. J. Amer. Chem. Soc. 49, 1864–1870(1927).
3. Debye, P. Einige bemerkungen zur magnetisierung bei tiefer temperatur. Ann. Phys. 386,
1154–1160 (1926).
4. Hakonen, P. J., Vuorinen, R. T. & Martikainen, J. E. Nuclear antiferromagnetism in rhodium
metal at positive and negative nanokelvin temperatures. Phys. Rev. Lett. 70, 2818–2821 (1993).
5. Tuoriniemi, J. T. et al. Double-spin-flip resonance of rhodium nuclei at positive and negative
spin temperatures. Phys. Rev. Lett. 84, 370–373 (2000).

14

6. Leanhardt, A. et al. Cooling bose-einstein condensates below 500 picokelvin. Science 301,
1513 (2003).
7. Weld, D. M. et al. Spin gradient thermometry for ultracold atoms in optical lattices. Phys.
Rev. Lett. 103, 245301 (2009).
8. Ho, T.-L. & Zhou, Q. Intrinsic heating and cooling in adiabatic processes for bosons in opticallattices. Phys. Rev. Lett. 99, 120404 (2007).
9. Oja, A. S. & Lounasmaa, O. V. Nuclear magnetic ordering in simple metals at positive andnegative nanokelvin temperatures. Rev. Mod. Phys. 69, 1–136 (1997).
10. Capogrosso-Sansone, B., Söyler, i. m. c. G., Prokof’ev, N. V. & Svistunov, B. V. Criticalentropies for magnetic ordering in bosonic mixtures on a lattice. Phys. Rev. A 81, 053622(2010).
11. Hung, C.-L., Zhang, X., Gemelke, N. & Chin, C. Slow mass transport and statistical evolutionof an atomic gas across the superfluid–mott-insulator transition. Phys. Rev. Lett. 104, 160403(2010).
12. Sensarma, R. et al.

Lifetime of double occupancies in the Fermi-Hubbard model.

arXiv:1001.3881 (2010).
13. Greiner, M. Private communication.
14. Fattori, M. et al. Demagnetization cooling of a gas. Nat Phys 2, 765–768 (2006).

15

15. Rabl, P., Daley, A. J., Fedichev, P. O., Cirac, J. I. & Zoller, P. Defect-suppressed atomic
crystals in an optical lattice. Phys. Rev. Lett. 91, 110403 (2003).
16. Bernier, J.-S. et al. Cooling fermionic atoms in optical lattices by shaping the confinement.
Phys. Rev. A 79, 061601 (2009).
17. Ho, T.-L. & Zhou, Q. Squeezing out the entropy of fermions in optical lattices. Proceedingsof the National Academy of Sciences 106, 6916–6920 (2009).
18. Ho, T. & Zhou, Q. Universal Cooling Scheme for Quantum Simulation. arXiv:0911.5506(2009).
19. Streed, E. W. et al. Large atom number bose-einstein condensate machines. Rev. Sci. Instrum.
77, 023106 (2006).
20. Reinaudi, G., Lahaye, T., Wang, Z. & Guéry-Odelin, D. Strong saturation absorption imaging
of dense clouds of ultracold atoms. Opt. Lett. 32, 3143–3145 (2007).

Acknowledgements It is a pleasure to acknowledge discussions with Eugene Demler, Takuya Kitagawa,David Pekker, and Aditi Mitra.

Competing Interests The authors declare that they have no competing financial interests.

Correspondence Correspondence and requests for materials should be addressed to A.B.C. (email: abc@abc.edu).

16

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A Timeline of
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Appendix C
Phase Diagram for a Bose-Einstein
Condensate Moving in an Optical
LatticeThis appendix contains a reprint of Ref. [53]: Jongchul Mun, Patrick Medley, Gretchen
K. Campbell, Luis G. Marcassa, David E. Pritchard, and Wolfgang Ketterle, Phase
Diagram for a Bose-Einstein Condensate Moving in an Optical Lattice. While notdiscussed in detail in this thesis, this paper is a good example of the sort of quantum
simulation work already performed in this lab, and the desire to extend this work
to two component systems led to the development of the cooling and thermometry
techniques this thesis presents.

121

PRL 99, 150604 (2007)

PHYSICAL REVIEW LETTERS

week ending
12 OCTOBER 2007

Phase Diagram for a Bose-Einstein Condensate Moving in an Optical Lattice
Jongchul Mun, Patrick Medley, Gretchen K. Campbell,* Luis G. Marcassa,† David E. Pritchard, and Wolfgang Ketterle
MIT-Harvard Center for Ultracold Atoms, Research Laboratory of Electronics, and Department of Physics, MIT,
Cambridge, Massachusetts 02139, USA(Received 26 June 2007; published 12 October 2007)
The stability of superfluid currents in a system of ultracold bosons was studied using a moving opticallattice. Superfluid currents in a very weak lattice become unstable when their momentum exceeds 0.5
recoil momentum. Superfluidity vanishes already for zero momentum as the lattice deep reaches the Mott
insulator (MI) phase transition. We study the phase diagram for the disappearance of superfluidity as afunction of momentum and lattice depth between these two limits. Our phase boundary extrapolates to thecritical lattice depth for the superfluid-to-MI transition with 2% precision. When a one-dimensional gas
was loaded into a moving optical lattice a sudden broadening of the transition between stable and unstablephases was observed.
DOI: 10.1103/PhysRevLett.99.150604

PACS numbers: 05.60.Gg, 03.75.Kk, 03.75.Lm, 05.30.Jp

The realization of condensed matter systems using ultracold atoms brings the precision and control of atomic
physics to the study of many-body physics. Many studies
have focused on Mott insulator physics, an importantparadigm for the suppression of transport by particle correlations. Previous studies of the superfluid (SF)-to-Mott
insulator (MI) transition in optical lattices with ultracoldbosons [1–8] addressed the quenching of superfluidity
below a critical lattice depth. Here we extend these studies
into a second dimension by studying stability of superfluid
current as a function of momentum and lattice depth assuggested in Ref. [9]. These transport measurements show
the stability of superfluid at finite current, which is innonequilibrium.
Transport measurements extend previous work on stationary systems in two regards. First, superfluidity near theMI transition has only been indirectly inferred from coherence measurements, whereas in this work, we characterize
the superfluid regime by observing a critical current forsuperfluid flow through the onset of dissipation. Second,previous studies [1–8] were not able to precisely locate thephase transition, since the observed excitation spectrum
and atomic interference pattern did not abruptly change[3,5,6], partially due to the inhomogeneous density. In
contrast, the sudden onset of dissipation provides a clear
distinction between the two quantum phases. In the SF
phase, current flows without dissipation if the momentum
does not exceed a critical momentum, while in the MIphase the critical momentum vanishes and transport isdissipative.
Bosonic atoms in an optical lattice are often describedby the Bose-Hubbard Model where the tunneling between
nearest neighbor lattice sites is characterized by the hopping matrix element J and the repulsive interactions by theon-site matrix element U [1,10 –12]. The dimensionlessinteraction energy u  U=J determines the quantum phaseof the system. For u > uc , the system is in the MI phase, foru < uc , the SF phase. uc increases with the number ofatoms N per site.
0031-9007=07=99(15)=150604(4)For weak interactions (u ! 0), the system approaches
single-particle physics in a periodic potential well described by Bloch states and band structure. The critical
momentum for a stable current-carrying state is 0.5 pr(pr  h= is the recoil momentum of an atom, where is the wavelength of the optical lattice light) [13]. At thecritical momentum, it becomes possible for two atoms inthe same initial Bloch state to scatter into two other states
and conserve energy and quasimomentum [14,15]. Instabilities in a 1D optical lattice were studied theoreticallyusing a linear stability analysis of the Gross-Pitaevskii
equation [13,16], and experimentally [14,17,18]. The theoretical studies predicted that for increasing lattice depth orincreasing atomic interactions the stability of superfluid
flow should increase [13,16]: the dynamic instability
would stay near 0.5 pr , whereas the Landau critical velocity and therefore the energetic instability would shift tolarger momenta (For more discussions on dynamic and energetic instability, see Refs. [19,20]). However, theseanalyses neglect the growing importance of quantum correlations for larger lattice depth which leads to the SF-MI
phase transition, where the critical momentum for a superfluid current vanishes. In this Letter, we study the decreaseof the critical momentum from its value for the weaklyinteracting regime towards zero at the MI transition(Fig. 1).
Most studies of the SF-MI phase transition monitor thecoherence in the superfluid phase through an interference
pattern observed in the ballistic expansion resulting from asudden turn-off of the confining potential and lattice.
Previous observations of the phase transition found theexperimental transition point to lie in the range between
10 and 13 ER (with the recoil energy defined as ER 
p2r =2m, where m is the atomic mass) [3]. This uncertainty
is related to the inhomogeneous density profile of trapped
atoms and to the fact that the visibility of the interference
extends beyond the transition point due to short-range
coherence in the MI phase [6]. It has been suggested thatobserved kinks in the visibility are linked to the formation

150604-1© 2007 The American Physical Society

PHYSICAL REVIEW LETTERS

Critical Momentum p c [pr ]PRL 99, 150604 (2007)
0.5

dissipative flow

0.4

superfluid flow

0.3
0.2
0.1
0
0

0.2

0.4

0.6

0.8

1

Interaction u/uc
Strength

FIG. 1. Phase diagram showing the stability of superfluid flowin an optical lattice and the experimental procedure. The graycurve shows the predicted boundary between superfluid flow anddissipative flow phases for a three-dimensional gas with acommensurate filling of N  1 atom per site [9]. The solid(dashed) arrows illustrate the experimental trajectory used forsmall (large) lattice depths (see text for details).

of the MI shells with occupation numbers N  2 and 3 [6].
Several authors have suggested other features in the momentum distribution beyond coherent interference peaks asa more distinct signature of the phase transition [21,22].
Here we show that the disappearance of the critical momentum for superfluid flow provides such a signature andallows the determination of the transition point with highprecision.
Our measurement was not limited by the inhomogeneous density profile. For our range of lattice depths, low
critical momenta and the onset of dissipation occur only
near the formation of MI shells with integer occupation
numbers N [9]. The onset of dissipation related to the N 
1 domains occurs at smaller momentum than for other N
domains. For instance, with increasing momentum p theN  1 domain becomes unstable first, and this triggers
dissipation over the whole atomic cloud [9]. Therefore,
the breakdown of superfluid flow in the system was determined by the formation of the N  1 domain and was not
smeared out by the inhomogeneous density. Our criterion,
the sudden onset of dissipation, depended on the formationof an insulating shell surrounded by a superfluid region,
which occurs only in the inhomogeneous case.
In our experimental setup, a Bose-Einstein condensate(BEC) of 87 Rb atoms in the 5S1=2 j1; 1i state was prepared and trapped in a combination of an Ioffe-Pritchardmagnetic trap and an optical dipole trap. The number ofatoms in the BEC was typically 2  105 . The magnetic trap
frequencies were !x;y  40 Hz radially and !z  4:6 Hz
axially. The laser beam for the optical dipole trap was oriented along the x axis. This laser beam was retroreflected
and the polarization of the retroreflected beam was rotatedin order to minimize interference between the two beams.
Along the vertical direction (y axis) a lattice was formed bya retroreflected laser beam. For the z axis, a moving lattice

week ending
12 OCTOBER 2007

was created by introducing a small frequency detuning fbetween the two counterpropagating laser beams using
acousto-optical modulators driven by phase-locked frequency generators. The 3D optical lattice was ramped up
exponentially in 160 ms. All lattice beams were derived
from the same laser operating at   1064 nm and had an
1=e2 waist of 100–200 m. The lattice depth was calibrated with 1% accuracy by applying a 12:5 s latticelaser pulse to a BEC and comparing the observed
Kapitza-Dirac diffraction pattern of a BEC to theory.
For transport measurements, we moved an optical lattice[17,23] which provides more flexibility to change themomentum than exciting a dipole oscillation by displacement of the BEC [24,25]. A moving optical lattice withvelocity v  f=2 was created along the long axis of the
BEC by introducing a small frequency detuning f between two counterpropagating lattice beams. If the velocityvt changes slowly enough not to induce interband excitations, the initial Bloch state jp  0i of the condensate
in the optical lattice adiabatically evolves into the currentcarrying state jpt  mvti where p is the quasimomentum. For increasing lattice depth, the effective mass of
atoms m  @2 Ep=@p2 1 increases, and the group velocity vg  m=m vt decreases. As a consequence,
atoms prepared in a moving lattice with quasimomentump  mv travel in the frame of the moving lattice with vg
and in the lab frame with velocity v  v vg  1 
m=m v, which approaches v in a deep lattice.
Consequently, we observed that in a deep moving lattice
atoms were dragged along to the edge of the trapping
region limiting the experimental time scale to probe fordissipation. This became an issue for larger values of p andwas addressed by first ramping up the lattice with p  0
and then alternating the velocity of the moving lattice, thus
performing a low-frequency ac transport measurement instead of dc.
We have used two sets of experimental procedures(Fig. 1), and our results were consistent for both. Closeto the SF-MI phase transition, the lattice was increased to
Vlatt with a fixed (and small) value of momentum p (dashedarrows in Fig. 1). After a variable hold time thold at Vlatt thelattice was ramped down to zero, and the magnetic trap
switched off. After 33 ms of ballistic expansion, the atoms
were imaged and the condensate fraction was determined
as a function of momentum by using a bimodal fitting
function. For smaller lattice depths, the lattice was ramped
up with p  0 (Fig. 1). Then a sinusoidal momentum
modulation of the moving lattice with amplitude pM wasapplied by modulating the frequency detuning f between
the counterpropagating lattice beams. The 10 ms period of
this momentum modulation was slow enough to meet the
adiabaticity condition, but fast enough to limit the displacement of the atomic cloud to less than a few m.
Both the trapping potential and the optical lattice werethen turned off suddenly. After 33 ms of ballistic expansion, the condensate fraction of the center peak of the150604-2

superfluid interference pattern was recorded as a functionof the momentum modulation amplitude pM . Several
cycles (typically, three to five) of the momentum modulation were applied to obtain a high contrast between thestable and dissipative regimes [Fig. 2(a)].
Figure 2(a) shows how the transition between superfluidand dissipative currents became sharper with increasing
number of cycles of the momentum modulation. The critical momentum was determined from a log-log plot of thecondensate fraction as a function of momentum p[Fig. 2(c)]. The intersection between two linear fit functions was taken as the critical momentum. Our result was
found to be independent of the time period and number ofcycles of the momentum modulation at a few percent level.
In the MI phase, stable superfluid flow is not possible
and the critical momentum should vanish. However, using
the procedure described above, we measured a small critical momentum of 0.02 pr for lattice depths Vlatt  14, 15,
16 ER . Up to this momentum, the SF-MI phase transition
remained reversible. We interpret the nonzero critical momentum as a finite-size effect. For our cloud size of

0.8

0.6

0.4

0.2
0.1

(c)

0.2

0.3

0 pr

0.12 p r

0.31 p r

0.7

0.5

0.6

0.3
0.1

0.32 p r

0.43 p r

0.7

C. F.

C. F.

0.4

Momentum [pr ]0.5

u /u c =0.22
0.1

0.2 0.3 0.5

p [ pr ]u /u c =0.53
0.3

0.1

0.2 0.3

p [ pr ]0.5

FIG. 2 (color online). Determination of the critical momentumof superfluid flow. Shown is the condensate fraction as a functionof a momentum p. (a) Condensate fraction with u=uc  0:61 fora variable number of cycles of the momentum modulation (onecycle:  and blue line, two cycles: 䊏 and purple line, threecycles: 䉱 and red line). A dashed vertical line indicates thecritical momentum where instability begins to occur. The twoand three-cycle data are offset vertically for clarity. These data
were fitted with an error function to guide the eye. (b) Images ofinterference patterns released from an optical lattice at u=uc 
0:61 moving with variable momentum. Instability occurred
between p  0:31pr and 0:32pr . Some of the triangular data
points in (a) were obtained from these images. (c) Condensate
fraction on a log-log scale for two different interaction strengths.

0.5

dissipative
flow

0.4

0.3

pc

0

60 m, the corresponding Heisenberg momentum uncertainty of 0.018 pr agrees with our measured critical momentum. In cold atom experiments, some sloshing motionof the cloud in the trapping potential is unavoidable. The
momentum uncertainty determined above indicates how
much sloshing motion can be tolerated without affecting
the observed phase transition.
The critical lattice depth for the SF-MI phase transition
can be determined as the point where the critical momentum vanishes. Using the predicted functional form [9] ofp
the approach towards zero, pc / 1  u=uc , as a fit function for the data points close to the SF-MI phase transition(the data points shown in the inset of Fig. 3) we determinedthe critical value uc  34:2 ( 2:0) corresponding to alattice depth of 13:5 0:2 ER . Our result agrees with themean-field theory prediction uc  5:8  6  34:8 for N 
1 SF-MI phase transition [1] and deviates by 2  from thepredictions of uc  29:342 of quantum Monte Carlo(QMC) simulation [26,27], which includes corrections
beyond the mean-field theory. This demonstrates that our
method has the precision to identify non-mean-field corrections. However, to turn precision into accuracy, experiments or QMC simulations [21,26,27] have to address
corrections due to finite size, finite temperature, and finite
time to probe the onset of the instability [27]. In our experiment, these corrections seemed to be small, but have notbeen characterized at the level of 1% in lattice depth.
The mean-field prediction for stable superfluid flow in
1D is similar to that for the 3D system [9]. However, it iswell known that fluctuations play a much more importantrole in 1D. For studying a 1D system, we prepared an arrayof one-dimensional gas tubes by ramping two pairs of
optical lattice beams up to lattice depths of Vx  Vy 
30 ER suppressing hopping between the tubes. After a holdtime of 10 ms, a moving optical lattice was ramped up
along the z axis. As in our 3D experiment, a momentum
modulation was applied, after which the moving opticallattice was ramped down to zero, followed by the other two
optical lattices. The condensate fraction was determined
after 33 ms of ballistic expansion as a function of the
Critical Momentum pc [p r]Condensate Fraction

(a)

(b)

week ending
12 OCTOBER 2007

PHYSICAL REVIEW LETTERSPRL 99, 150604 (2007)

0.2
0.1

0.3
0.2

superfluid
flow

0.0
0.7

0.8

0.9

1.0

u / uc

0.1
0
0

0.25

0.50

0.75

1.0

1.25

1.50

1.75

2.0

Interaction Strength u / uc

FIG. 3 (color online). Critical momentum for a condensate in a
3D lattice. The solid line shows the theoretical prediction for thesuperfluid region. The horizontal solid line is a fit to the data
points in the MI phase. (Inset) Fit of critical momenta near the
SF-MI phase transition.

150604-3

PHYSICAL REVIEW LETTERSPRL 99, 150604 (2007)(b)0.25 ER

0.75

C. F.

0.5
0.4

0

0.2

0.2

0.4

p [pr ]

0.75

superfluid
flow

0.1

0.50

0.25

dissipative
flow

0.3

C. F.

Critical Momentum p c [pr ]

(a)

0.6

0.75 ER

0.50
0.25

0

0.2

0.4

0.6

0.8

Interaction Strength u/u c

1

0

0.1

0.2

0.3

p [pr ]

FIG. 4 (color online). Critical momentum for a 1D gas in an
optical lattice. (a) The gray line indicates the mean-field theory
prediction. The interaction strengths are normalized by themean-field prediction for uc  5:8  2 [1,4]. Squares (crosses)
represent the measured critical momentum (the center of thetransition). Measurements were taken at lattice depths of 0.25,
0.50, 0.75, 1.0, 2.0 ER . The lines between crosses and squares
indicate the width of the transition region. (b) Condensate fraction measured at 0.25 ER and 0.75 ER . The data were fitted withan error function. Squares (the critical momentum) and crosses(the center of the transition) are indicated on the plots.

momentum modulation amplitude. The critical momentum, where the onset of dissipation begins, was identified
from a log-log plot as in the 3D case. Since the transitions
became very broad, we characterized them by an error
function fit, with the center of the fitted error function
taken as the center of the transition (Fig. 4).
In the 1D system, at a very shallow lattice depth of 0.25
ER (corresponding to u=uc  0:08) a sharp transition wasobserved, and the measured critical momentum agreed
very well with the prediction [9,28] of a critical momentumof 0.39 pr . However, a slight increase of the interaction
strength (to u=uc  0:09 at a lattice depth of 0.5 ER ) led toa significant decrease of the critical momentum as well as a
dramatic broadening of the transition as shown in Fig. 4.
For lattice depths larger than 2 ER , the transition becamevery broad and showed complex behavior, and we could
not obtain quantitative fits. Our results show a significant
deviation from the mean-field theory predictions and are inagreement with previous works [25,29,30].
The observed broadening of the transition confirms
theoretical studies which emphasize the importance ofquantum fluctuations in the 1D system. Quantum tunnelingout of metastable states which are ignored in the mean-field
description can lead to a decay of the superfluid current atvery low momenta [28]. In addition to quantum fluctuations, thermal fluctuations provide a mechanism for current
decay [28]. In our experiment, we used a ‘‘pure’’ BEC
without a discernible thermal component. The close agreement with T  0 predictions indicates that thermal fluctuations were not dominant.
In conclusion, we have used transport studies to connect
a well-known dynamical instability for weakly interacting
bosons with the equilibrium superfluid to Mott insulatortransition. A comparison of 3D and 1D systems confirms

week ending
12 OCTOBER 2007

the applicability of a mean-field description in three dimensions and the crucial importance of fluctuations in one
dimension. The disappearance of superfluid currents at the
SF-MI phase transition precisely located the phase transition. Our results illustrate the control and precision ofcondensed matter physics experiments done with ultracold
atoms and their suitability to test many-body theories.
This work was funded by NSF through the grant for
CUA. L. G. M. acknowledges support from Coordenacaode Aperfeicoamento de Pessoal de Nivel Superior. We
thank E. Demler and A. Polkovnikov for insightful discussions, and David Weld for a critical reading of themanuscript.

*Present address: JILA, Boulder, Colorado 80309, USA.
†
Permanent address: Instituto de Fisica de São Carlos,
University of São Paulo, São Carlos, 13560-970, SP,
Brazil.
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