The phenomenon of Bose-Einstein condensation of dilute gases in traps is reviewed from a theoretical perspective. Mean-field theory provides a framework to understand the main features of the condensation and the role of interactions between particles. Various properties of these systems are discussed, including the density profiles and the energy of the ground state configurations, the collective oscillations and the dynamics of the expansion, the condensate fraction and the thermodynamic functions. The thermodynamic limit exhibits a scaling behavior in the relevant length and energy scales. Despite the dilute nature of the gases, interactions profoundly modify the static as well as the dynamic properties of the system; the predictions of mean-field theory are in excellent agreement with available experimental results. Effects of superfluidity including the existence of quantized vortices and the reduction of the moment of inertia are discussed, as well as the consequences of coherence such as the Josephson effect and interference phenomena. The review also assesses the accuracy and limitations of the mean-field approach.

The physics of quantum degenerate atomic Fermi gases in uniform as well as in harmonically trapped configurations is reviewed from a theoretical perspective. Emphasis is given to the effect of interactions which play a crucial role, bringing the gas into a superfluid phase at low temperature. In these dilute systems interactions are characterized by a single parameter, the s-wave scattering length, whose value can be tuned using an external magnetic field near a broad Feshbach resonance. The BCS limit of ordinary Fermi superfluidity, the Bose-Einstein condensation (BEC) of dimers and the unitary limit of large scattering length are important regimes exhibited by interacting Fermi gases. In particular the BEC and the unitary regimes are characterized by a high value of the superfluid critical temperature, of the order of the Fermi temperature. Different physical properties are discussed, including the density profiles and the energy of the ground-state configurations, the momentum distribution, the fraction of condensed pairs, collective oscillations and pair breaking effects, the expansion of the gas, the main thermodynamic properties, the behavior in the presence of optical lattices and the signatures of superfluidity, such as the existence of quantized vortices, the quenching of the moment of inertia and the consequences of spin polarization. Various theoretical approaches are considered, ranging from the mean-field description of the BCS-BEC crossover to non-perturbative methods based on quantum Monte Carlo techniques. A major goal of the review is to compare the theoretical predictions with the available experimental results.

We study the Fermi gas at unitarity and at T 0 by assuming that, at high polarizations, it is a normal Fermi liquid composed of weakly interacting quasiparticles associated with the minority spin atoms. With a quantum Monte Carlo approach we calculate their effective mass and binding energy, as well as the full equation of state of the normal phase as a function of the concentration x n # =n " of minority atoms. We predict a first order phase transition from normal to superfluid at x c 0:44 corresponding, in the presence of harmonic trapping, to a critical polarization P c N " ÿ N # = N " N # 77%. We calculate the radii and the density profiles in the trap and predict that the frequency of the spin dipole mode will be increased by a factor of 1.23 due to interactions. DOI: 10.1103/PhysRevLett.97.200403 PACS numbers: 05.30.Fk, 03.75.Hh, 03.75.Ss Recent experiments on degenerate gases of 6 Li with a mixture of two hyperfine species have explored the physics of Fermi gases [1,2] and have led to a number of theoretical analyses [3][4][5][6][7]. One of the major experimental observations has been the occurrence of phase separation if the mixture contains more atoms of one species than of the other, i.e., if the gas is polarized. Some of the experiments suggest that in the unitary limit of strong interactions there are three phases: an unpolarized superfluid phase, a mixed phase which exhibits a partial polarization, and a fully polarized gas. We now have a good understanding of the superfluid phase which has been the subject of numerous theoretical and experimental studies while the fully polarized phase is an ideal Fermi gas since atoms in the same spin state do not interact with each other. However, for intermediate polarizations, when both species are present, the nature of the mixed phase is not understood.Here we study the mixed phase in the unitary limit by adopting an approach inspired by the theory of dilute solutions of 3 He in 4 He [8]. We will assume that the majority species (") forms a background experienced by the minority species (#) and that the latter behaves as a gas of weakly interacting fermionic quasiparticles even though the " ÿ # atomic interaction is very strong, being characterized by an infinite scattering length. In other words, we will assume that the system is a normal Fermi liquid, which will allow us to characterize the energy of the gas in terms of a few parameters and, by calculating these with a quantum Monte Carlo approach, to make various predictions of experimental relevance.We begin by writing the expression for the energy E of a homogeneous system in the limit of very dilute mixtures and at zero temperature. The concentration of # atoms is given by the ratio of the densities x n # =n " , and we will take it to be small. If only " atoms are present, then the energy is that of an ideal Fermi gas E x 0 3=5E F" N " , where N " is the total number of " atoms and E F" @ 2 =2m 6 2 n " 2=3 is the ideal gas Fermi energy. When we add a # atom with a momentum p (jpj p F" ), we shall as...

By using a mean field approach, based on the Popov approximation, we calculate the temperature dependence of the condensate fraction of an interacting Bose gas confined in an anisotropic harmonic trap. For systems interacting with repulsive forces we find a significant decrease of the condensate fraction and of the critical temperature with respect to the predictions of the non-interacting model. These effects go in the opposite direction compared to the case of a homogeneous gas. An analytic result for the shift of the critical temperature holding to first order in the scattering length is also derived.02.70. Lq, 67.40.Db The recent experiments on Bose-Einstein condensation (BEC) in magnetically trapped atomic vapours [1] have stimulated a new interest in the theoretical study of inhomogeneous Bose gases. Although the atom clouds realized in these experiments are very dilute, the effects due to the interatomic forces are known to be important at low temperature. In particular, the shape and the energy of the condensate cloud [2,3] as well as the dispersion law of the elementary excitations [4] are strongly affected by the interaction. In the very recent experiments by the Boulder and MIT groups [5][6][7], the measured release energy and excitation frequencies of the collective modes have been found to be in good agreement with the theoretical predictions, thereby revealing important features of the trapped Bose condensed gases which are undoubtedly connected to the interparticle interaction. The question of how two-body forces affect the thermodynamic properties of these systems has been also the object of several theoretical investigations [8]. The critical temperature of the BEC transition in an homogeneous dilute gas has been recently calculated [9] using the renormalization group theory. The result is a shift towards higher temperatures, with respect to the prediction of the ideal Bose gas. Similar results have been also obtained with path integral Monte Carlo simulations [10]. However, no definitive conclusions have been so far drawn on the behavior of the condensate fraction, nor of the critical temperature in the presence of a confining potential [11]. First experimental data on these relevant quantities are now becoming available [5,12].Finite size effects on the temperature dependence of the condensate fraction and on the critical temperature in the presence of an external trap have been recently investigated by several authors within the non-interacting model [13,14]. In the presence of an anisotropic harmonic potential of the form V ext = m(ω 2 x x 2 + ω 2 y y 2 + ω 2 z z 2 )/2 this model predicts, in the large N limit, the well known results for the critical temperature 1

Quasi-one-dimensional (quasi-1d) two-component Fermi gases with effectively attractive and repulsive interactions are characterized for arbitrary interaction strength. The ground-state properties of the gas confined in highly elongated harmonic traps are determined within the local density approximation. For strong attractive effective interactions the existence of a molecular TonksGirardeau gas is predicted. The frequency of the lowest breathing mode is calculated as a function of the coupling strength for both attractive and repulsive interactions. PACS numbers:The study of cold quasi-1d atomic quantum gases presents a very active area of research. So far, most of the experimental [1] and theoretical [2,3,4,5,6] investigations have been devoted to quasi-1d Bose gases and, in particular, to the strongly-interacting Tonks-Girardeau gas, which can be mapped to a gas of non-interacting fermions [2,7,8]. Quasi-1d atomic Fermi gases have not been realized experimentally yet.The role of interactions in quasi-1d atomic Fermi gases has been studied mainly in connection with Luttinger liquid theory [9,10]. Recati et al. [10] investigate the properties of a two-component Fermi gas with repulsive interspecies interactions confined in highly-elongated harmonic traps. In the limit of weak and strong coupling these authors relate the parameters of the Luttinger Hamiltonian, which describe the low-energy properties of the gas, to the microscopic parameters of the system. The Luttinger model is used to analyze the manifestations of the uncoupled dynamics of spin and density waves (spin-charge separation).This Letter investigates the properties of inhomogeneous quasi-1d two-component Fermi gases under harmonic confinement with attractive and repulsive interspecies interactions. The present study is based on the exact equation of state of a homogeneous 1d system of fermions with zero-range attractive [11,12] and repulsive [13] interactions treated within the local density approximation (LDA). We calculate the energy per particle, the size of the cloud, and the frequency of the lowest compressional mode as a function of the effective 1d coupling constant, including infinitely strong attractive and repulsive interactions. Moreover, for attractive interactions we discuss the cross-over from the weak-to the strongcoupling regime and point out the possibility of forming a mechanically stable molecular Tonks-Girardeau gas.Quasi-1d two-component Fermi gases with effectively attractive and repulsive 1d interspecies interactions can be realized in highly-elongated traps. The behavior of atomic gases tightly-confined in two directions can, if the confinement is chosen properly, be characterized to a very good approximation by an effective 1d coupling constant, g 1d , which encapsulates the atom-atom interaction strength. This coupling constant can be tuned to essentially any value, including zero and ±∞, by varying the 3d s-wave scattering length a 3d through application of an external magnetic field in the proximity of a Feshbach resona...

By using the diffusion Monte Carlo method we calculate the one-and two-body density matrix of an interacting Fermi gas at T = 0 in the BCS-BEC crossover. Results for the momentum distribution of the atoms, as obtained from the Fourier transform of the one-body density matrix, are reported as a function of the interaction strength. Off-diagonal long-range order in the system is investigated through the asymptotic behavior of the two-body density matrix. The condensate fraction of fermionic pairs is calculated in the unitary limit and on both sides of the BCS-BEC crossover. PACS numbers:The physics of the crossover from Bardeen-CooperSchrieffer (BCS) superfluids to molecular Bose-Einstein condensates (BEC) in ultracold Fermi gases near a Feshbach resonance is a very exciting field that has recently attracted a lot of interest, both from the experimental [1,2] and the theoretical side [3]. An important experimental achievement is the observation of a condensate of pairs of fermionic atoms on the side of the Feshbach resonance where no stable molecules would exist in vacuum [4,5]. Although the interpretation of the experiment is not straightforward, as it involves an out-of-equilibrium projection technique of fermionic pairs onto bound molecules [6], it is believed that these results strongly support the existence of a superfluid order parameter in the strongly correlated regime on the BCS side of the resonance [5].The occurrence of off-diagonal long-range order (ODLRO) in interacting systems of bosons and fermions was investigated by C.N. Yang in terms of the asymptotic behavior of the one-and two-body density matrix [7]. In the case of a two-component Fermi gas with N ↑ spin-up and N ↓ spin-down particles, the one-body density matrix (OBDM) for spin-up particles, defined asdoes not possess any eigenvalue of order N ↑ . This behavior implies for homogeneous systems the asymptotic condition ρ 1 (rIn the above expression ψ † ↑ (r) (ψ ↑ (r)) denote the creation (annihilation) operator of spin-up particles. The same result holds for spin-down particles. ODLRO may occur instead in the two-body density matrix (TBDM), that is defined asFor an unpolarized gas with N ↑ = N ↓ = N/2, if ρ 2 has an eigenvalue of the order of the total number of particles N , the TBDM can be written as a spectral decomposition separating the largest eigenvalue,2 containing only eigenvalues of order one. The parameter α ≤ 1 in Eq. (3) is interpreted as the condensate fraction of pairs, in a similar way as the condensate fraction of single atoms is derived from the OBDM.The spectral decomposition (3) yields for homogeneous systems the following asymptotic behavior of the TBDMThe complex function ϕ is proportional to the order parameter ψ ↑ (r 1 )ψ ↓ (r 2 ) = αN/2ϕ(|r 1 − r 2 |), whose appearance distinguishes the superfluid state of the Fermi gas. Equation (4) should be contrasted with the behavior of Bose systems with ODLRO, where ρ 1 has an eigenvalue of order N [8], and consequently the largest eigenvalue of ρ 2 is of the order of N 2 . In thi...

We investigate the phase diagram of asymmetric two-component Fermi gases at zero temperature as a function of polarization and interaction strength. The equations of state of the uniform superfluid and normal phase are determined using quantum Monte Carlo simulations. We find three different mixed states, where the superfluid and the normal phase coexist in equilibrium, corresponding to phase separation between (a) the polarized superfluid and the fully polarized normal gas, (b) the polarized superfluid and the partially polarized normal gas, and (c) the unpolarized superfluid and the partially polarized normal gas.

We investigate the crossover from Bardeen-Cooper-Schrieffer (BCS) superfluidity to Bose-Einstein condensation (BEC) in a two-dimensional Fermi gas at T = 0 using the fixed-node diffusion Monte Carlo method. We calculate the equation of state and the gap parameter as a function of the interaction strength, observing large deviations compared to mean-field predictions. In the BEC regime our results show the important role of dimer-dimer and atom-dimer interaction effects that are completely neglected in the mean-field picture. Results on Tan's contact parameter associated with short-range physics are also reported along the BCS-BEC crossover. The study of ultracold atomic Fermi gases has become an active and rich field of research [1]. Important areas of investigation include the BCS-BEC crossover in a superfluid gas with resonantly enhanced interactions, the Chandrasekhar-Clogston instability of the superfluid state when spin polarization is increased, the possible onset of itinerant ferromagnetism in a gas with repulsive interactions [2] and the realization of the Hubbard model for fermions loaded in optical lattices [3].Low dimensional configurations of degenerate Fermi gases have also been the object of experimental and theoretical studies [1,3]. In particular, a two-dimensional (2D) ultracold Fermi gas has been recently realized using a highly anisotropic pancake-shaped potential, and the density profile of the cloud has been measured using in situ imaging [4]. On the theoretical side, the evolution from a superfluid state with large Cooper pairs to one with tight molecules in a 2D system of attractive fermions was first investigated by Miyake [5] and later by Randeria and coworkers [6] aiming to describe high-T c superconductors. More recent studies address the problem of the superfluid transition [7,8], of harmonic trapping [9] and of population and mass imbalance [10]. These studies are in general based on perturbative or mean-field (MF) approaches that are suitable in the regime of weak coupling, but are bound to break down for stronger interactions.In this Letter we provide the first determination using quantum Monte Carlo methods of the equation of state at T = 0 of a homogeneous 2D Fermi gas in the BCS-BEC crossover. We also obtain results for the pairing gap and the contact parameter as a function of the interaction strength. In the strong-coupling regime the emergence of interaction effects involving dimers produce large deviations compared to MF predictions. A similar study carried out in 3D [11] has provided an important benchmark against which experimental determination of the equation of state, using measurements of the dispersion of collective modes [12] or of in situ density profiles [13], have been successfully compared. Hopefully, our results will stimulate more experimental efforts towards the realization of a 2D Fermi gas in the strong-coupling regime by means, for example, of a Feshbach resonance to increase the interaction parameter [4].We consider a homogeneous two-component Fermi gas d...