Condensation of pairs of fermionic atoms near a Feshbach resonance

Reentrant BCS-BEC crossover and a superfluid-insulator transition in optical lattices

We study the thermodynamics of a two-species Feshbach-resonant atomic Fermi gas in a periodic potential, focusing in a deep optical potential where a tight binding model is applicable. We show that for a more than half-filled band the gas exhibits a reentrant crossover with decreased detuning (increased attractive interaction), from a paired BCS superfluid to a Bose-Einstein condensate (BEC) of molecules of holes, back to the BCS superfluid, and finally to a conventional BEC of diatomic molecules. This behavior is associated with the nonmonotonic dependence of the chemical potential on detuning and the concomitant Cooper-pair or molecular size, larger in the BCS and smaller in the BEC regimes. For a single filled band we find a quantum phase transition from a band insulator to a BCS-BEC superfluid, and map out the corresponding phase diagram.

Triplet pair correlations in s-wave superfluids as a signature of the Fulde-Ferrell-Larkin-Ovchinnikov state

We show that antiparallel triplet pairing correlations are generated in superfluids with purely s-wave interactions whenever population imbalance enforces anisotropic Fulde-Ferrell (FF) or inhomogeneous Larkin-Ovchinikov (LO) states. These triplet correlations appear in the Cooper pair wave function, while the triplet part of the gap remains zero. The same set of quasiparticle states contributes to the triplet component and to the polarization, thus spatially correlating them. In the LO case, this set forms a narrow band of Andreev states centered on the nodes of the s-wave order parameter. This picture naturally provides a unifying explanation of previous findings that attractive p-wave interaction stabilizes FFLO states. We also study a similar triplet mixing which occurs when a balanced two-component system displays FFLO-type oscillations due to a spin-dependent optical lattice. We discuss how this triplet component can be measured in systems of ultracold atoms using a rapid ramp across a p-wave Feshbach resonance. This should provide a smoking gun signature of FFLO states.

Universality of modulation length and time exponents

We study systems with a crossover parameter λ, such as the temperature T, which has a threshold value λ(*) across which the correlation function changes from exhibiting fixed wavelength (or time period) modulations to continuously varying modulation lengths (or times). We introduce a hitherto unknown exponent ν(L) characterizing the universal nature of this crossover and compute its value in general instances. This exponent, similar to standard correlation length exponents, is obtained from motion of the poles of the momentum (or frequency) space correlation functions in the complex k-plane (or ω-plane) as the parameter λ is varied. Near the crossover (i.e., for λ→λ(*)), the characteristic modulation wave vector K(R) in the variable modulation length "phase" is related to that in the fixed modulation length "phase" q via |K(R)-q|[proportionality]|T-T(*)|(νL). We find, in general, that ν(L)=1/2. In some special instances, ν(L) may attain other rational values. We extend this result to general problems in which the eigenvalue of an operator or a pole characterizing general response functions may attain a constant real (or imaginary) part beyond a particular threshold value λ(*). We discuss extensions of this result to multiple other arenas. These include the axial next-nearest-neighbor Ising (ANNNI) model. By extending our considerations, we comment on relations pertaining not only to the modulation lengths (or times), but also to the standard correlation lengths (or times). We introduce the notion of a Josephson time scale. We comment on the presence of aperiodic "chaotic" modulations in "soft-spin" and other systems. These relate to glass-type features. We discuss applications to Fermi systems, with particular application to metal to band insulator transitions, change of Fermi surface topology, divergent effective masses, Dirac systems, and topological insulators. Both regular periodic and glassy (and spatially chaotic behavior) may be found in strongly correlated electronic systems.

Structural properties and energetics of diffuse 87Rb clusters in three-dimension

A correlated two-body basis function is used to describe the three-dimensional bosonic clusters interacting via two-body van der Waals potential. We calculate the ground state and the zero orbital angular momentum excited states for Rb(N) clusters with up to N = 40. We solve the many-particle Schrödinger equation by potential harmonics expansion method, which keeps all possible two-body correlations in the calculation and determines the lowest effective many-body potential. We study energetics and structural properties for such diffuse clusters both at dimer and tuned scattering length. The motivation of the present study is to investigate the possibility of formation of N-body clusters interacting through the van der Waals interaction. We also compare the system with the well studied He, Ne, and Ar clusters. We also calculate correlation properties and observe the generalised Tjon line for large cluster. We test the validity of the shape-independent potential in the calculation of the ground state energy of such diffuse cluster. These are the first such calculations reported for Rb clusters.

Realization of a resonant Fermi gas with a large effective range

We have measured the interaction energy and three-body recombination rate for a two-component Fermi gas near a narrow Feshbach resonance and found both to be strongly energy dependent. Even for de Broglie wavelengths greatly exceeding the van der Waals length scale, the behavior of the interaction energy as a function of temperature cannot be described by atoms interacting via a contact potential. Rather, energy-dependent corrections beyond the scattering length approximation are required, indicating a resonance with an anomalously large effective range. For fields where the molecular state is above threshold, the rate of three-body recombination is enhanced by a sharp, two-body resonance arising from the closed-channel molecular state which can be magnetically tuned through the continuum. This narrow resonance can be used to study strongly correlated Fermi gases that simultaneously have a sizable effective range and a large scattering length.

BCS-BEC crossover and topological phase transition in 3D spin-orbit coupled degenerate Fermi gases

We investigate the BCS-BEC crossover in three-dimensional degenerate Fermi gases in the presence of spin-orbit coupling (SOC) and Zeeman field. We show that the superfluid order parameter destroyed by a large Zeeman field can be restored by the SOC. With increasing strengths of the Zeeman field, there is a series of topological quantum phase transitions from a nontopological superfluid state with fully gapped fermionic spectrum to a topological superfluid state with four topologically protected Fermi points (i.e., nodes in the quasiparticle excitation gap) and then to a second topological superfluid state with only two Fermi points. The quasiparticle excitations near the Fermi points realize the long-sought low-temperature analog of Weyl fermions of particle physics. We show that the topological phase transitions can be probed using the experimentally realized momentum-resolved photoemission spectroscopy.

Functional renormalization for the Bardeen-Cooper-Schrieffer to Bose-Einstein condensation crossover

We review the functional renormalization group (RG) approach to the Bardeen-Cooper-Schrieffer to Bose-Einstein condensation (BCS-BEC) crossover for an ultracold gas of fermionic atoms. Formulated in terms of a scale-dependent effective action, the functional RG interpolates continuously between the atomic or molecular microphysics and the macroscopic physics on large length scales. We concentrate on the discussion of the phase diagram as a function of the scattering length and the temperature, which is a paradigm example for the non-perturbative power of the functional RG. A systematic derivative expansion provides for both a description of the many-body physics and its expected universal features as well as an accurate account of the few-body physics and the associated BEC and BCS limits.

Establishing the presence of coherence in atomic fermi superfluids: spin-flip and spin-preserving Bragg scattering at finite temperatures

We show how in ultracold Fermi gases the difference between the finite temperature T structure factors, called S_(ω,q), associated with spin and density, reflects coherent order at all ω, q, k(F)a, and T. This observation can be exploited in two photon Bragg scattering experiments on gases which are subject to variable attractive interactions. Our calculations incorporate spin and particle number conservation laws which lead to compatibility at general T with two f-sum rules. Because of its generality a measurement of S_(ω,q) can be a qualitative, direct, in situ approach for establishing superfluid order.

π phases in balanced fermionic superfluids on spin-dependent optical lattices

We study a balanced two-component system of ultracold fermions in one dimension with attractive interactions and subject to a spin-dependent optical lattice potential of opposite sign for the two components. We find states with different types of modulated pairing order parameters which are conceptually similar to π phases discussed for superconductor-ferromagnet heterostructures. Increasing the lattice depth induces sharp transitions between states of different parity. While the origin of the order parameter oscillations is similar to the Fulde-Ferrel-Larkin-Ovchinnikov phase for paired states with spin imbalance, the current system is intrinsically stable to phase separation. We discuss experimental requirements for creating and probing these novel phases.

Non-Fermi-liquid fixed point for an imbalanced gas of fermions in 1+ϵ dimensions

We consider a gas of two species of fermions with population imbalance. Using the renormalization group in d=1+ϵ spatial dimensions, we show that for spinless fermions and 2>ϵ>0 a fixed point appears at finite attractive coupling where the quasiparticle residue vanishes, and identify this with the transition to Larkin-Ovchinnikov-Fulde-Ferrell order (inhomogeneous superconductivity). When the two species of fermions also carry spin degrees of freedom we find a fixed point indicating a transition to spin density wave order.

Broad Feshbach resonance in the 6Li-40K mixture

We study the widths of interspecies Feshbach resonances in a mixture of the fermionic quantum gases 6Li and 40K. We develop a model to calculate the width and position of all available Feshbach resonances for a system. Using the model, we select the optimal resonance to study the {6}Li/{40}K mixture. Experimentally, we obtain the asymmetric Fano line shape of the interspecies elastic cross section by measuring the distillation rate of 6Li atoms from a potassium-rich 6Li/{40}K mixture as a function of magnetic field. This provides us with the first experimental determination of the width of a resonance in this mixture, DeltaB=1.5(5) G. Our results offer good perspectives for the observation of universal crossover physics using this mass-imbalanced fermionic mixture.

Induced interactions and the superfluid transition temperature in a three-component Fermi gas

We study many-body contributions to the effective interaction between fermions in a three-component Fermi mixture. We find that effective interactions induced by the third component can lead to a phase diagram different from that predicted if interactions with the third component are neglected. As a result, in a confining potential a superfluid shell structure can arise even for equal populations of the components. We also find a critical temperature for the BCS transition in a 6Li mixture which can deviate strongly from the one in a weakly interacting two-component system.

Collective coordinate description of anisotropically trapped degenerate Fermi gases

The solution of the many-body Schrödinger equation using an adiabatic treatment of the hyperradius is generalized to treat two components of a hyperspherical vector adiabatically. This treatment has advantages in certain physical situations, such as the description of a degenerate Fermi gas or Bose-Einstein condensate in an anisotropic trapping potential. A first application to the zero-temperature anisotropic Fermi gas is compared with predictions of the Hartree-Fock method.

p-wave resonant bose gas: a finite-momentum spinor superfluid

We show that a degenerate gas of two-species bosonic atoms interacting through a p-wave Feshbach resonance (as realized in, e.g., a (85)Rb -- (87)Rb mixture) exhibits a finite-momentum atomic-molecular superfluid (AMSF), sandwiched by a molecular p-wave (orbital spinor) superfluid and by an s-wave atomic superfluid at large negative and positive detunings, respectively. The magnetic field can be used to tune the modulation wave vector of the AMSF state, as well as to drive quantum phase transitions in this rich system.

Stabilization of the p-wave superfluid state in an optical lattice

It is hard to stabilize the p-wave superfluid state of cold atomic gas in free space due to inelastic collisional losses. We consider the p-wave Feshbach resonance in an optical lattice, and show that it is possible to have a stable p-wave superfluid state where the multiatom collisional loss is suppressed through the quantum Zeno effect. We derive the effective Hamiltonian for this system, and calculate its phase diagram in a one-dimensional optical lattice. The results show rich phase transitions between the p-wave superfluid state and different types of insulator states induced either by interaction or by dissipation.

Nonequilibrium dynamics of weakly and strongly paired superconductors

We study small oscillations of the order parameter in weakly and strongly paired superconductors driven slightly out of equilibrium, in the collisionless approximation. While it was known for quite some time that the amplitude of the oscillations in a weakly paired superconductor decays as t(-1/2), we show that in a superconductor sufficiently strongly paired so that its fermions form bound states usually referred to as molecules, these oscillations decay as t(-3/2). The transition between these two regimes happens when the chemical potential of the superconductor vanishes; thus, the behavior of the oscillations can be used to distinguish weakly and strongly paired superconductors. Finally, we interpret the result in the strongly paired superconductor as the probability of the molecular decay as a function of time.

Possible efimov trimer state in a three-hyperfine-component lithium-6 mixture

We consider the Efimov trimer theory as a possible framework to explain recently observed losses by inelastic three-body collisions in a three-hyperfine-component ultracold mixture of lithium 6. Within this framework, these losses would arise chiefly from the existence of an Efimov trimer bound state below the continuum of free triplets of atoms, and the loss maxima (at certain values of an applied magnetic field) would correspond to zero-energy resonances where the trimer dissociates into three free atoms. Our results show that such a trimer state is indeed possible given the two-body scattering lengths in the three-component lithium mixture and gives rise to two zero-energy resonances. The locations of these resonances appear to be consistent with observed losses.

Quantum liquid crystals in an imbalanced Fermi gas: fluctuations and fractional vortices in Larkin-Ovchinnikov states

We develop a low-energy model of an unidirectional Larkin-Ovchinnikov (LO) state. Because the underlying rotational and translational symmetries are broken spontaneously, this gapless superfluid is a smectic liquid crystal, that exhibits fluctuations that are qualitatively stronger than in a conventional superfluid, thus requiring a fully nonlinear description of its Goldstone modes. Consequently, at nonzero temperature the LO superfluid is an algebraic phase even in 3D. It exhibits half-integer vortex-dislocation defects, whose unbinding leads to transitions to a superfluid nematic and other phases. In 2D at nonzero temperature, the LO state is always unstable to a nematic superfluid. We expect this superfluid liquid-crystal phenomenology to be realizable in imbalanced resonant Fermi gases trapped isotropically.

Three-body recombination in a three-state Fermi gas with widely tunable interactions

We investigate the stability of a three spin state mixture of ultracold fermionic 6Li atoms over a range of magnetic fields encompassing three Feshbach resonances. For most field values, we attribute decay of the atomic population to three-body processes involving one atom from each spin state and find that the three-body loss coefficient varies by over 4 orders of magnitude. We observe high stability when at least two of the three scattering lengths are small, rapid loss near the Feshbach resonances, and two unexpected resonant loss features. At our highest fields, where all pairwise scattering lengths are approaching a_{t}=-2140a_{0}, we measure a three-body loss coefficient L_{3} approximately 5x10;{-22} cm;{6}/s and a trend toward lower decay rates for higher fields indicating that future studies of color superfluidity and trion formation in a SU(3) symmetric Fermi gas may be feasible.

Superfluidity in a gas of strongly interacting fermions

After an introduction into 100 years of research on superfluidity and the concept of the BCS-BEC crossover, we describe recent experimental studies of a spin-polarized Fermi gas with strong interactions. Tomographically resolving the spatial structure of an inhomogeneous trapped sample, we have mapped out the superfluid phases in the parameter space of temperature, spin polarization, and interaction strength. Phase separation between the superfluid and the normal component occurs at low temperatures, showing spatial discontinuities in the spin polarization. The critical polarization of the normal gas increases with stronger coupling. Beyond a critical interaction strength all minority atoms pair with majority atoms, and the system can be effectively described as a boson-fermion mixture. Pairing correlations have been studied by rf spectroscopy, determining the fermion pair size and the pairing gap energy in a resonantly interacting superfluid.

Bragg spectroscopy of a strongly interacting Fermi gas

We present a comprehensive study of the Bose-Einstein condensate to Bardeen-Cooper-Schrieffer (BEC-BCS) crossover in fermionic 6Li using Bragg spectroscopy. A smooth transition from molecular to atomic spectra is observed with a clear signature of pairing at and above unitarity. These spectra probe the dynamic and static structure factors of the gas and provide a direct link to two-body correlations. We have characterized these correlations and measured their density dependence across the broad Feshbach resonance at 834 G.

Optical feshbach resonance using the intercombination transition

We report control of the scattering wave function by an optical Feshbach resonance effect using ytterbium atoms. The narrow intercombination line (1S0-3P1) is used for efficient control as proposed by Ciuryło et al. [Phys. Rev. A 71, 030701(R) (2005)10.1103/PhysRevA.71.030701]. The manipulation of the scattering wave function is monitored with the change of a photoassociation rate caused by another laser. The optical Feshbach resonance is especially efficient for isotopes with large negative scattering lengths such as 172Yb, and we have confirmed that the scattering phase shift divided by the wave number, which gives the scattering length in the zero energy limit, is changed by about 30 nm.

Critical temperature and condensate fraction of a fermion pair condensate

We report on measurements of the critical temperature and the temperature dependence of the condensate fraction for a fermion pair condensate of 6Li atoms. Bragg spectroscopy is employed to determine the critical temperature and the condensate fraction after a fast magnetic field ramp to the molecular side of the Feshbach resonance. Our measurements reveal evidence of level off of the critical temperature and limiting behavior of condensate fraction near the unitarity limit.

Probing fermionic condensates by fast-sweep projection onto Feshbach molecules

Fast-sweep projection onto Feshbach molecules has been widely used as a probe of fermionic condensates. By determining the exact dynamics of a pair of atoms in time-varying magnetic fields, we calculate the number of condensed and noncondensed molecules created after fast magnetic field sweeps from the BCS to the Bose-Einstein condensate side of the resonances in 40K and 6Li, for different sweep rates and a range of initial and final fields. We discuss the relation between the initial fermionic condensate fraction and the molecular condensate fraction measured after the sweep.

Bragg spectroscopy of a strongly interacting 85Rb Bose-Einstein condensate

We report on measurements of the excitation spectrum of a strongly interacting Bose-Einstein condensate. A magnetic-field Feshbach resonance is used to tune atom-atom interactions in the condensate and to reach a regime where quantum depletion and beyond mean-field corrections to the condensate chemical potential are significant. We use two-photon Bragg spectroscopy to probe the condensate excitation spectrum; our results demonstrate the onset of beyond mean-field effects in a gaseous Bose-Einstein condensate.

Collisional properties of p-wave Feshbach molecules

We have observed p-wave Feshbach molecules for all three combinations of the two lowest hyperfine spin states of 6Li. By creating a pure molecular sample in an optical trap, we measured the inelastic collision rates of p-wave molecules. We have also measured the elastic collision rate from the thermalization rate of a breathing mode which was excited spontaneously upon molecular formation.

Quantum fluctuations in the time-dependent BCS-BEC crossover

We describe the time-dependent formation of a molecular Bose-Einstein condensate from a BCS state of fermionic atoms as a result of slow sweeping through a Feshbach resonance. We apply a path integral approach for the molecules, and use two-body adiabatic approximations to solve the atomic evolution in the presence of the classical molecular fields, obtaining an effective action for the molecules. In the narrow resonance limit, the problem becomes semiclassical, and we discuss the growth of the molecular condensate in the saddle point approximation. Considering this time-dependent process as an analogue of the cosmological Zurek scenario, we compare the way condensate growth is driven in this rigorous theory with its phenomenological description via time-dependent Ginzburg-Landau theory.

Gravity duals for nonrelativistic conformal field theories

We attempt to generalize the anti-de Sitter/conformal field theory correspondence to nonrelativistic conformal field theories which are invariant under Galilean transformations. Such systems govern ultracold atoms at unitarity, nucleon scattering in some channels, and, more generally, a family of universality classes of quantum critical behavior. We construct a family of metrics which realize these symmetries as isometries. They are solutions of gravity with a negative cosmological constant coupled to pressureless dust. We discuss realizations of the dust, which include a bulk superconductor. We develop the holographic dictionary and find two-point correlators of the correct form. A strange aspect of the correspondence is that the bulk geometry has two extra noncompact dimensions.

Vibrational cooling of spin-stretched dimer states by He buffer gas: quantum calculations for Li2(a 3Sigma(u)+) at ultralow energies

The interaction between the triplet state of the lithium dimer, (7)Li(2), with (4)He is obtained from accurate ab initio calculations where the vibrational dependence of the potential is newly computed. Vibrational quenching dynamics within a coupled-channel quantum treatment is carried out at ultralow energies, and large differences in efficiency as a function of the initial vibrational state of the targets are found as one compares the triplet results with those of the singlet state of the same target.

Suppression of molecular decay in ultracold gases without Fermi statistics

We study inelastic processes for ultracold three-body systems in which only one interaction is resonant. We show that at ultracold temperatures three-body recombination in such systems leads mainly to the formation of weakly bound molecules. In addition, and perhaps more importantly, we have found that the decay rates for weakly bound molecules due to collisions with other atoms can be suppressed not only without fermionic statistics but also when bosonic statistics applies. These results indicate that recombination in three-component atomic gases can be used as an efficient mechanism for molecular formation, allowing the achievement of high molecular densities.

Kinetics of the phase separation transition in cold-atom boson-fermion mixtures

We study the kinetics of the first order phase separation transition in boson-fermion cold-atom mixtures. At sufficiently low temperatures such a transition is driven by quantum fluctuations responsible for the formation of critical nuclei of a stable phase. Based on a microscopic description of interacting boson-fermion mixtures we derive an effective action for the critical droplet and obtain an asymptotic expression for the nucleation rate in the vicinity of the phase transition and near the spinodal instability of the mixed phase. We also discuss effects of dissipation which play a dominant role close to the transition point, and identify the regimes where quantum nucleation can be experimentally observed in cold-atom systems.

Do mixtures of bosonic and fermionic atoms adiabatically heat up in optical lattices?

Mixtures of bosonic and fermionic atoms in optical lattices provide a promising arena to study strongly correlated systems. In experiments realizing such mixtures in the quantum-degenerate regime the temperature is a key parameter. We investigate the intrinsic heating and cooling effects due to an entropy-preserving raising of the optical lattice, identify the generic behavior valid for a wide range of parameters, and discuss it quantitatively for the recent experiments with 87Rb and 40K atoms. In the absence of a lattice, we treat the bosons in the Hartree-Fock-Bogoliubov-Popov approximation, including the fermions in a self-consistent mean-field interaction. In the presence of the full three-dimensional lattice, we use a strong coupling expansion. We find the temperature of the mixture in the lattice to be always higher than for the pure bosonic case, shedding light onto a key point in the analysis of recent experiments.

Dynamical creation of bosonic cooper-like pairs

We propose a scheme to create a metastable state of paired bosonic atoms in an optical lattice. The most salient features of this state are that the wave function of each pair is a Bell state and that the pair size spans half the lattice, similar to fermionic Cooper pairs. This mesoscopic state can be created with a dynamical process that involves crossing a quantum phase transition and which is supported by the symmetries of the physical system. We characterize the final state by means of a measurable two-particle correlator that detects both the presence of the pairs and their size.

Quantum gases

Ultracold quantum gases are proving to be a powerful model system for strongly interacting electronic many-body systems. This Perspective explores how such atomic ensembles can help to unravel some of the outstanding open questions in the field.