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Page 129FEATURE ARTICLEULTRACOLD SUPERFLUIDSBY JOSEPH H. THYWISSENT

he field of ultracold atoms was born in 1995,
when Eric Cornell and collaborators observed
Bose-Einstein condensation (BEC) of a gas ofmagnetically trapped alkali atoms. Today, roughly
one hundred laboratories work with quantum degenerate
neutral atoms, in seventeen countries [1] including Canada.
Theorists who contribute to the field come from a wide
range of disciplines, including condensed matter, atomic
physics, high energy physics, and quantum information.

It is safe to say that both the growth and theaccomplishments of cold atoms have exceeded all
expectations of the 1990’s. After all, what could a “plain
vanilla” Bose condensate contribute to 21 st-century
condensed matter physics? The study of superfluids andsuperconductors is a well established field, withimpressively advanced techniques and theories, as
reviewed throughout this Special Issue.
The power of the cold-atom approach is in its progressive
complexity. Superconducting heterostructures, such as
cuprates and pnictides, are inherently complex. In
contrast, cold atom experiments can be confined in simple
or sophisticated geometries, and can be tuned from weak
to strong interactions. In the next few pages, we explain
this approach and give a few examples of its success. We
do not attempt to give a complete set of citations here, but
we refer the reader to reviews [2–6] and textbooks [7–9] forfull references.

SUMMARY
Superconductors are roughly a billion times
more dense than neutral gases, but thephysical principles of gases and solids havea surprising similarity in their quantum
degenerate regimes. In this brief review, we
trace a historical path from the early days of
weakly interacting Bose condensates tocurrent research in strongly interacting
Fermi degenerate gases. Cold atoms can beviewed as quantum simulators able toaddress open questions about many-bodysystems. Also tantalizing: if the physics ofresonant ultracold superfluids could be
reproduced in solids, the critical temperatureof superconductivity would be roughly
1000 K, well above room temperature.

STRONG INTERACTIONS
A student who has taken a course on statistical mechanics
can open their textbook, apply Einstein’s criterion for
Bose condensation, and get quantitative agreement withthe critical temperature Tc of the 1995 experiments (seealso Fig. 1). A thermal gas is nearly ideal (noninteracting), in contract to liquid helium at its lambda
point, where Einstein’s criterion gives a Tc that is 50% too
high. Even in a Bose-condensed gas, interactions can betreated with a relatively simple approach. The GrossPitaevskii (GP) equation [5], a mean-field treatment fromthe 1960’s which never worked very well for liquids orsolids, is an excellent description of a Bose-condensed gasof neutral atoms. At low temperature the condensate
fraction of a Bose-condensed gas is nearly 100%, in
contrast to liquid helium where interactions deplete thecondensate to roughly 10% of the total density.
Although this state of affairs was celebrated for its
textbook-like clarity, it was unclear at first what
contribution ultracold gases could make to the cuttingedge of many-body physics. Open questions in condensed
matter typically concern systems in which the interactionenergy per particle is equal or greater than its singleparticle (kinetic plus potential) energy. These strong
interactions induce strong correlations between particles,
causing mean-field approaches such as the GP equation to
fail. In contrast, the first ultracold Bose condensates [10]and Fermi degenerate gases [11] were weakly interacting.
Strongly interacting gases were subsequently created withtwo basic strategies: decreasing single-particle energy
with optical lattices, or increasing interaction strengthwith Feshbach resonances. In an optical lattice, a spatiallyperiodic potential is created using the Stark effect of astanding wave of laser light. Neutral atoms that move inthis potential acquire a band structure, just as freeelectrons do in a crystalline solid. As the depth of the
optical lattice is increased, kinetic energy of the atoms isdecreased due to a flatter dispersion relation [12]. In thisway, the ratio of interaction energy to kinetic energy canbe increased into the strongly interacting regime. Note thatthe lattice depth is proportional to the laser power, so it canbe turned on and off dynamically — unlike the band
structure of a solid.

J.H. Thywissen<joseph.thywissen@utoronto.ca>,
Associate Professor
and Canada
Research Chair,
Department of
Physics, University ofToronto, 60 St.
George, Toronto, ON
M5S1A7;and Member,
Quantum Materials
Program, Canadian
Institute for Advanced
Research, Toronto,
ON, M5G 1Z8

The second strategy to create strongly interacting neutral
gases is to tune their interaction energy. Interaction
potentials between neutral atoms have a length scale of a

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RESONANT SUPERFLUIDS

Fig. 1

Bose-Einstein Condensation. Absorption images of a cloud of atomic
rubidium crossing the critical temperature in Toronto. The color scale (fromblue to red) represents integrated density. a) At 960 nK, a cloud of 1.2 × 105
atoms is thermal; b) At 360 nK, a cloud of 7 × 104 atoms is below the criticaltemperature, and a Bose condensate appears, mixed with a thermal cloud;c) At even lower temperature, no thermal fraction is evident, and thus all
4.5 × 104 atoms are Bose condensed. Images are 0.5 mm across, and taken
after 10 ms of free-flight expansion. For further details, see Ref. [13].

nanometer — hundreds of times smaller than the typical interparticle distance in a gas, and thus interactions can beconsidered pair-wise. At the sub-microkelvin temperatures ofquantum degeneracy, two colliding atoms have too little energy
to overcome a centrifugal barrier in their center-of-mass frame,
so interactions are restricted to those with no orbital angularmomentum, i.e. to “s-wave” collisions. Together these two
conditions simplify interactions tremendously: the only
memory an atom has of a collision is the phase it accumulated.
This phase can be parameterized with a “scattering length”.
Once measured, this single parameter tells a theorist or an
experimentalist all they need to know about the interaction.
Furthermore, if one can modify that phase, one is able to tune
the interaction strength.
In the presence of a magnetic field, free atoms and molecules
experience different Zeeman shifts since their magnetic dipole
moments are not equal. At certain serendipitous values of the
field, this differential Zeeman shift can bring a bound dimer
state into resonance with the energy of two asymptotically free
atoms. This condition is called a Feshbach resonance [4]. Even
though atoms do not actually form a dimer, the phase shift oftheir s-wave collision is modified by the proximity to theresonance [14]. As a result, by using a magnetic field, theultracold gas can be tuned from weakly to strongly interacting.
Using these two approaches, several lines of work have been
pursued. In the context of this Special Issue, we will focus onconceptual issues concerning superfluidity. However inpassing we will mention that other areas of study include
magnetism [15], artificial gauge fields, controlled disorder [16],insulating states of bosons and fermions [2,3], few-body boundstates, quantum gates, quantum memory, cold molecules [17],low-dimensional systems, and non-equilibrium physics.

130 C PHYSICS IN CANADA / VOL. 67, NO. 2 ( APR.-June 2011 )The BCS theory of superconductivity tells us thatthe critical temperature is proportional to thepairing gap. Although BCS is valid only for weakinteractions (compared to the Fermi energy, EF), it
motivates the exploration of high pair energy insearch of high-temperature superfluids. With
Feshbach control, ultracold fermionic atoms canbe tuned to unitarity-limited interactions — thestrongest allowed by conservation of probability
in a scattering event. In this limit, Deborah Jin [18]and colleagues observed superfluidity that occurs
at around 0.2TF, where TF = EF /kB, and kB is theBoltzmann constant. If this same behavior could
be replicated in solids, where the typical TF is tensof thousands of kelvin, the critical temperature ofsuperconductivity would be 1000 K. In otherwords, if the goal of room-temperature
superconductivity were scaled to the Fermi energyof the system, it has already been achieved(ironically) in nanokelvin gases.

The physics of this resonant superfluid has an interesting kindof universality because its properties do not depend on theparticular type of interaction in question. The same
thermodynamics, critical temperature, etc. would occur inneutrons stars or metals, if their interactions (whose
microscopic origins differ from neutral atoms) could be tuned
to resonance. Perhaps the most provocative connection is to thedense quark-gluon plasma, currently being studied at RHIC
(relativistic heavy ion collider). Both systems may be creating
perfect fluids that probe the quantum limits of viscosity, as
conjectured by string theory [19].
The resonant regime also provides an important conceptual
unification between fermionic and bosonic superfluids.
Sweeping across the Feshbach resonance, experimentalists candynamically transform a paired BCS superfluid to a BoseEinstein condensate of molecular dimers. This shows us thatthe only distinction between Cooper pairs and molecules is theordering of the interatomic and inter-pair length scales, but thephysics is continuous across the BEC-BCS crossover [9].
What could break such a strong superfluid? Since composite
bosons require fermions of distinct spins, one can certainly
prevent superfluidity by polarizing the sample: a Fermi gas of
one spin state is simply noninteracting. However a question
debated in the literature was whether a critical polarization(called the Chandrasekhar-Clogston limit) existed, beyond
which superfluidity could not occur [20]. This question was
answered by Wolfgang Ketterle and collaborators, observing a
Chandrasekhar-Clogston limit of 36% [21]. Above this limit, theFermi surfaces are too far apart to be bridged by the pairing
interaction.
However not all superfluids need to find the same number of
up- and down-spin partners. Forty years ago, Fulde and Ferrell,Apr11-final.qxp

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However, these difficult experiments provide an
exciting perspective on superfluids and otherquantum many-body systems. In the traditionalcondensed matter approach, the voyage of
discovery begins with the observation of a newphenomenon — such as the observation ofsuperconductivity 100 years ago, or ofsuperconductivity in pnictides a few years ago.
Theoretical physicists search for a Hamiltonian,or a ground state, that might explain thephenomenon. In contrast, the cold atoms
approach starts with well known ingredients.
The system is then tuned to explore a specificregime, to test whether the Hamiltonian might
explain a certain class of behaviour, or to find a
postulated but as-yet unobserved phase.
Fig. 2Another way to see this approach is as aquantum simulation. For instance, stronglyinteracting fermions pose a ‘sign problem’ fornumerical Quantum Monte-Carlo calculations.
Whereas tens of thousands of bosons can be
simulated on supercomputers, these samemachines can handle only ten or so fermions –insufficient to characterize a macroscopic phaseat low temperature. From this perspective, a
cold atom experiment with a well known
Hamiltonian is a quantum simulator, exceeding
the abilities of a classical computer.
Experimental realizations of universal quantum
computers are currently limited to several (less than ten) qubits.
Although non-universal, a cold atom system can alreadyperform a quantum simulation requiring thousands of qubits.
These quantum simulators are now being applied to solve
problems involving strongly correlated fermions, such aswhether the Hubbard model can explain d-wave
superconductivity [2,3].

Observation of a partially polarized Fermi superfluid. Cold atoms in onedimensional tubes show density profiles that are consistent with a FFLO-type
pairing. Integrated axial density profiles of the tube bundles (black circles
represent the majority, the blue diamonds represent the minority, and the redsquares show the difference) are shown as functions of central polarization P.
a) At low P ( = 0.015), the edge of the cloud is fully paired and the density
difference is zero. The centre of the cloud is partially polarized. The density
difference has been multiplied by two for better visibility of the phase boundary(dashed black line); b) Near Pc (P = 0.10), where almost the entire cloud ispartially polarized; c) Well above Pc (P = 0.33), where the edge of the cloud isfully polarized and the minority density vanishes. For further details, see
Ref. [23].

and Larkin and Ovchinnikkov (FFLO) proposed an exotic
pairing mechanism in which the superfluid itself is spinimbalanced and overcomes the difference in Fermi energies bycreating pairs with finite momentum [22]. The FFLO state has
been elusive in condensed matter systems and occupies little ofthe phase diagram in three-dimensional cold atom systems.
However in one-dimensional systems, the FFLO state isprevalent. Randy Hulet and collaborators have studied stronglyinteracting one-dimensional Fermi gases and found partiallypolarized superfluids in which the FFLO paring mechanism
may be at work (see Fig. 2) [23]. This exciting regime paves theway for direct observation of FFLO pairing.

PERSPECTIVE
In order to trap and cool atoms to sub-microkelvintemperatures, an ultracold gas experiment combines a wide
range of technologies and techniques. Typical experimental
systems include a half-dozen lasers frequency stabilized to
1 part in 109, far-off-resonant lasers with tens of watts ofpower, CCD cameras operating at photon-shot-noise
sensitivity, nested magnetic traps requiring mT stability andsub-ms time response, ultra-high vacuum systems operating at
10-12 Torr, microwave and RF manipulation systems, and realtime sequencing control involving several hundred time steps
and up to one hundred channels. The majority of these systems
are not available commercially, so must be designed, built, andtested in-house.

In conclusion, a program of progressive complexity isunderway in the ultracold atom community. Starting fromtextbook-like illustrations of the basic principles of condensed
matter physics, we progress towards open questions by tuning
Hamiltonians, controlling internal states, and engineering
environments. Although quantum degenerate neutral gases
were realized 84 years after Kamerlingh Onnes observed
superconductivity, they are providing powerful new insights
into superfluidity and other topics of quantum many-body
physics.

ACKNOWLEDGEMENTS
Our research is supported by NSERC, the Canadian Institutefor Advanced Research, AFOSR, ARO, and the DARPA
Optical Lattice Emulation program. The author thanks
R. Hulet and Y. Liao for providing Figure 2 and thanks
B. Braverman, N. Cheng, G. Edge, D. Jervis, and D. McKay
for manuscript comments.

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