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...

We study the truncated Wigner method applied to a weakly interacting spinless Bose condensed gas which is perturbed away from thermal equilibrium by a time-dependent external potential. The principle of the method is to generate an ensemble of classical fields ψ(r) which samples the Wigner quasi-distribution function of the initial thermal equilibrium density operator of the gas, and then to evolve each classical field with the Gross-Pitaevskii equation. In the first part of the paper we improve the sampling technique over our previous work [Jour. of Mod. Opt. 47, 2629-2644] and we test its accuracy against the exactly solvable model of the ideal Bose gas. In the second part of the paper we investigate the conditions of validity of the truncated Wigner method. For short evolution times it is known that the time-dependent Bogoliubov approximation is valid for almost pure condensates. The requirement that the truncated Wigner method reproduces the Bogoliubov prediction leads to the constraint that the number of field modes in the Wigner simulation must be smaller than the number of particles in the gas. For longer evolution times the nonlinear dynamics of the noncondensed modes of the field plays an important role. To demonstrate this we analyse the case of a three dimensional spatially homogeneous Bose condensed gas and we test the ability of the truncated Wigner method to correctly reproduce the Beliaev-Landau damping of an excitation of the condensate. We have identified the mechanism which limits the validity of the truncated Wigner method: the initial ensemble of classical fields, driven by the time-dependent Gross-Pitaevskii equation, thermalises to a classical field distribution at a temperature T class which is larger than the initial temperature T of the quantum gas. When T class significantly exceeds T a spurious damping is observed in the Wigner simulation. This leads to the second validity condition for the truncated Wigner method, T class − T ≪ T , which requires that the maximum energy ǫmax of the Bogoliubov modes in the simulation does not exceed a few kBT .

We consider the problem of a single # atom in the presence of a Fermi sea of " atoms, in the vicinity of a Feshbach resonance. We calculate the chemical potential and the effective mass of the # atom using two simple approaches: a many-body variational wave function and a T-matrix approximation. These two methods lead to the same results and are in good agreement with existing quantum Monte Carlo calculations performed at unitarity and, in one dimension, with the known exact solution. Surprisingly, our results suggest that, even at unitarity, the effect of interactions is fairly weak and can be accurately described using single particle-hole excitations. We also consider the case of unequal masses. DOI: 10.1103/PhysRevLett.98.180402 PACS numbers: 05.30.Fk, 03.75.Ss, 71.10.Ca, 74.72.ÿh The investigation of ultracold Fermi gases with two unbalanced hyperfine states (which we shall denote as " and # ) has been through an impressive expansion last year. This subfield is of great interest both on practical and on theoretical grounds. Indeed, on one hand, it is related to other fields of physics, namely, superconductivity, astrophysics, and high-energy physics, where similar situations arise [1]. On the other hand, the additional parameter provided by the population imbalance should provide a tool to deepen our understanding of the Bose-Einstein condensation (BEC)-BCS crossover in these systems and contribute to improving our control of many-body theory. This recent activity has been started by striking experimental results [2] which have given rise to a considerable number of theoretical works [3]. Experiments performed on trapped systems have observed equal density superfluid states as well as partially and fully polarized regions.The analysis of the T 0 phase separation requires the knowledge of the properties of both the superfluid and the partially polarized normal phase [4 -6]. This has been developed in the recent work of Lobo et al. [6], where, at unitarity, the properties of the partially polarized phase have been obtained by calculating through a quantum Monte Carlo (QMC) approach the parameters which characterize a single # atom immersed in a Fermi sea of " atoms, with density n " k 3 F =62 . In this way, they were able to obtain very good agreement with experimental results.Here we consider the general problem of a single # atom in a completely polarized " atom Fermi sea. The " -# interaction is characterized by an s-wave scattering length a, whose value can be tuned via a Feshbach resonance from the BEC (1=k F a 1) to the BCS (1=k F a ÿ1) limits. The Fermi gas is ideal due to the suppression of higher angular momentum scattering at low temperatures. This problem is a much simpler one than the case of two equal spin populations in the BEC-BCS crossover, although still quite nontrivial due to the absence of a small parameter in the strongly interacting regime. It is the simplest realization of the moving impurity problem, and it bears a strong similarity with other old, famous, and notoriously difficult...

We propose a method to study the time evolution of Bose-Einstein condensed gases perturbed from an initial thermal equilibrium, based on the Wigner representation of the N-body density operator. We show how to generate a collection of random classical fields sampling the initial Wigner distribution in the number conserving Bogoliubov approximation. The fields are then evolved with the time dependent Gross-Pitaevskii equation. We illustrate the method with the damping of a collective excitation of a one-dimensional Bose gas. DOI: 10.1103/PhysRevLett.87.210404 PACS numbers: 03.75.Fi, 05.10.Gg, 42.50. -p Since the first experimental demonstrations of BoseEinstein condensation in atomic gases [1], the role played by finite temperature effects in the physics of BoseEinstein condensed alkali gases has drawn increasing attention. For example, in thermal equilibrium, both the spatial density of the condensate atoms [2] and the distribution of number of particles in the condensate [3] are modified due to the presence of the thermal atoms. Similarly, we must take into account the noncondensed atoms in order to explain time dependent phenomena such as the damping and frequency shifts of collective modes [4,5] or the evolution of the recently created vortices [6], where the dissipation of the condensate motion is provided by the noncondensed atoms [7].The widely used Gross-Pitaevskii equation, the nonlinear Schrödinger equation for the condensate wave function, is not able to describe these effects since it neglects the interaction between the condensate and the noncondensed atoms [8]. One possibility to go beyond the GrossPitaevskii equation is to use Bogoliubov theory, which is a perturbative method valid for a small noncondensed fraction. The Bogoliubov method can be applied to thermal equilibrium but also to time dependent situations: in a U(1) symmetry breaking point of view, to zeroth order one solves the time dependent Gross-Pitaevskii equation for the condensate field c 0 ͑ r, t͒, to first order one linearizes the quantum field equations around the classical field c 0 ͑ r, t͒ to get the dynamics of noncondensed particles, and to second order one includes the backaction of noncondensed particles on the condensate. However, in this way one can predict only small corrections to the Gross-Pitaevskii equation. Additionally, if the number of noncondensed particles increases during the evolution of the system, the Bogoliubov approach is valid only for short times [9]. Another existing approach is the mean field HartreeFock-Bogoliubov approximation. This approach is known however to present consistency problems and is still the object of research [10].In this Letter we propose an alternative method to study the time evolution of Bose condensed gases perturbed from an initial thermal equilibrium, based on the classical field approximation in the Wigner representation, the so-called truncated Wigner approximation. The classical field approximation amounts to evolving a set of initially randomly distributed atomic fields w...

We show that the formation of a vortex lattice in a weakly interacting Bose condensed gas can be modeled with the nonlinear Schrö dinger equation for both T 0 and finite temperatures without the need for an explicit damping term. Applying a weak rotating anisotropic harmonic potential, we find numerically that the turbulent dynamics of the field produces an effective dissipation of the vortex motion and leads to the formation of a lattice. For T 0, this turbulent dynamics is triggered by a rotational dynamic instability of the condensate. For finite temperatures, noise is present at the start of the simulation and allows the formation of a vortex lattice at a lower rotation frequency, the Landau frequency. These two regimes have different vortex dynamics. We show that the multimode interpretation of the classical field is essential. DOI: 10.1103/PhysRevLett.92.020403 PACS numbers: 03.75.Lm Vortex lattices exist in many domains of physics, from neutron stars to superconductors or liquid helium. In none of these systems has the formation of the lattice been understood at the level of a microscopic theory. Several groups have recently observed the formation of a vortex lattice in weakly interacting Bose gases [1][2][3][4] and are able to monitor this formation in real time. This gives us the chance to understand the problem of lattice formation in a relatively simple system. Indeed there have been theoretical attempts to understand the formation process [5][6][7][8] with simulations of the Gross-Pitaevskii equation for the condensate wave function. All of them stress the need for explicitly including a damping term representing the noncondensed modes to which the vortices have to give away energy to relax to a lattice configuration. In this Letter, we consider this problem in the framework of the classical theory of a complex field [9] whose exact equation of motion is the nonlinear Schrö dinger equation (NLSE). First, we show that lattice formation is predicted within this framework without the addition of damping terms. Second, we provide two distinct scenarios of vortex lattice formation (dynamics, temperature dependence of the formation time, and critical rotation frequency) that can be directly compared with the experiments. We study the formation of the lattice in 3D from an initially nonrotating Bose condensed gas both at T 0 and at finite temperature. Contrary to the common belief, we find that the dynamic instability, which was predicted in [10] to occur above a certain threshold value of the trap rotation frequency, leads to the formation of a vortex lattice. The formation time is in this case only weakly dependent of the temperature and the observed scenario and time scales are comparable to those seen in present experiments. For a lower trap rotation frequency corresponding to the Landau frequency, but only at finite temperature, we identify a new scenario not yet observed experimentally in which the vortices enter a few at a time and gradually spiral towards the center.We start our simulations with...

We study the zero temperature properties of a trapped polarized Fermi gas at unitarity by assuming phase separation between an unpolarized superfluid and a polarized normal phase. The effects of the interaction are accounted for using the formalism of quasi-particles to build up the equation of state of the normal phase with the Monte Carlo results for the relevant parameters. Our predictions for the Chandrasekhar-Clogston limit of critical polarization and for the density profiles, including the density jump at the interface, are confirmed with excellent accuracy by the recent experimental results at MIT. The role of interaction on the radial width of the minority component, on the gap of spectral functions and on the spin oscillations in the normal phase is also discussed. Our analysis points out the Fermi liquid nature of these strongly interacting spin polarized configurations.PACS numbers:

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We propose an efficient stochastic method to implement numerically the Bogolubov approach to study finite-temperature Bose-Einstein condensates. Our method is based on the Wigner representation of the density matrix describing the non condensed modes and a Brownian motion simulation to sample the Wigner distribution at thermal equilibrium. Allowing to sample any density operator Gaussian in the field variables, our method is very general and it applies both to the Bogolubov and to the Hartree-Fock Bogolubov approach, in the equilibrium case as well as in the time-dependent case. We think that our method can be useful to study trapped Bose-Einstein condensates in two or three spatial dimensions without rotational symmetry properties, as in the case of condensates with vortices, where the traditional Bogolubov approach is difficult to implement numerically due to the need to diagonalize very big matrices.