In the collisionless regime (where the collision rate is much smaller than the oscillation frequencies of the trapped atoms) we have suggested to detect experimentally the emergence of a phononic branch in the excitation spectrum. Such a branch, already predicted by Bogoliubov and Anderson, appears when the condensate of pairs forms, and is specific of a system of neutral particles. We have checked that such a branch persists in presence of harmonic confinement, when the quantum of oscillation is lower than k_B T_c where T_c is the critical temperature.

This early proposal is however not relevant for present experiments, that are performed close to a Feshbach resonance and are in the hydrodynamic regime (so that even the normal phase of the gas has a phononic branch).

A. Minguzzi, G. Ferrari, Y. Castin, "Dynamic structure factor of a superfluid Fermi gas", Eur. Phys. J. D 17, 49 (2001).

In the low temperature and weakly interacting regime, it is now well established that the Gross-Pitaevskii equation gives the correct description of a static condensate. In the time dependent case, this is not quite the case, as the excitations of a condensate are damped by coupling to the thermal cloud, an effect not included in the usual Gross-Pitaevskii equation.

Can one extend the validity of the time dependent Gross-Pitaevskii equation to the finite temperature case ? The intuitive idea that we have explored is to add initially some noise to the condensate field, mimicking the thermal component. One then integrates the usual time dependent non-linear Schrödinger equation, and its non-linearity will lead to an interaction between the condensate part and the noisy part of the field, which may provide damping.

This idea can be formalized with the help of the Wigner representation of the many-body density operator, which provides a quasi-probability distribution for the complex field psi. The above mentioned idea consists in neglecting third order derivatives with respect to psi in the equation of evolution for the Wigner distribution, which is intuitively satisfactory when psi is large, that is in the degenerate regime.

This procedure raises two questions, that we have addressed.

A practical one: how to sample the Wigner distribution for a given initial thermal equilibrium ? We have given a precise procedure to do so, based on Bogoliubov theory, and which can receive a Monte Carlo formulation, without the need for diagonalisation of the Bogoliubov operator.
<>A fundamental one: what is the validity condition of this truncated Wigner approximation ? Our answer is that the main limitation comes from the fact that the initial Wigner distribution, representing the quantum state of the system at finite temperature, is not exactly stationary under evolution by a classical field equation, that is by the Gross-Pitaevskii equation. This means that part of the noise in the initial field mimicking the quantum noise will be redistributed among the various modes by the classical field evolution, so as to thermalize the field in the classical statistical physics sense. This can lead to artifacts in the predictions. The way out is to put an energy cut-off on the order of a few k_B T, to reduce this effect, which excludes the domain of too low temperatures.

Alice Sinatra, Carlos Lobo, Yvan Castin, "Classical-Field Method for Time Dependent Bose-Einstein Condensed Gases", Phys. Rev. Lett. 87, 210404 (2001).

Alice Sinatra, Carlos Lobo, Yvan Castin, "The truncated Wigner method: limits of validity and applications", J. Phys. B 35, 3599-3631 (2002).

A rotating superfluid cannot have a solid body rotation velocity field, but can support phase singularities called quantum vortices. How do these vortices form ? The quantum gases are ideal systems to answer this fundamental question, since they are not coupled to an external thermal bath, contrarily to superfluid helium in a container.

Vortex lattices in a quantum gas were first observed in the group of Jean Dalibard. The rotation frequencies leading to vortex formation were in the experiment much larger than the expected minimal value predicted from thermodynamics, and also were concentrated on a narrow interval of values, whereas no upper bound for the rotation frequency was given by thermodynamics! This was a puzzle for the scientific community.

We have identified a scenario for the nucleation of vortices not relying on thermodynamics, but on time dependent hydrodynamics. By solving analytically the superfluid hydrodynamics equations derived from the Gross-Pitaevskii equation in the Thomas-Fermi limit, we have found that dynamical instabilities occur in the time evolution of the stirred gas, in the state where no vortex is present. The corresponding rotation frequencies are in excellent agreement with the observed ones. An important consequence is that the nucleation frequencies are not universal but depend on the stirring method used in the experiment.

Subhasis Sinha, Yvan Castin, "Dynamic instability of a rotating Bose-Einstein condensate", Phys. Rev. Lett. 87, 190402 (2001).

Can the dynamical instability that we have discovered explain the formation of the vortex lattice ? Some authors answered no to this question and argued that an extra mechanism should be put forward. This conclusion was based on the expectation that a conservative equation like the Gross-Pitaevskii equation cannot lead to a crystallization of the vortex lattice, and that a phenomenological damping term should be added to the equation.

We have therefore performed a full 3D numerical integration of the plain time dependent Gross-Pitaevskii equation, with no added damping term, in a harmonic trap with a rotation frequency adiabatically increased to some limiting value. When this limiting value is below the analytically predicted threshold for dynamical instability, no vortex enters the condensate in the simulation. For higher rotation frequencies, the classical field becomes turbulent, vortices enter rapidly and after some time, settle in a regular lattice.

A conservative equation and with time reversal symmetry, like the Gross-Pitaevskii equation, can therefore predict the formation of an ordered structure like a vortex lattice! This is due to the fact that the dynamical instability creates turbulent component of the field, which can act as a reservoir and take energy out of the condensate. An important point is then to realize that the field psi(r,t), solution of the Gross-Pitaevskii equation, no longer represents a pure condensate: the first order coherence function shall now be defined as the time average of psi*(r,t) psi(r',t), so that the noisy part of the field does not belong anymore to the condensate mode.

We have also studied the vortex lattice formation for an initially non zero temperature, by adding an initial noise to the initial field to represent the thermal component. Now the thermodynamic scenario is obtained: the nucleation frequency is the Landau frequency, and the vortices enter one by one and spiral slowly towards the trap center. This scenario was not observed experimentally yet, probably because a non-rotating anisotropy of the trap prevents the thermal cloud from thermalizing in the rotating frame.

Carlos Lobo, Alice Sinatra, Yvan Castin, "Vortex Lattice Formation in Bose-Einstein Condensates", Phys. Rev. Lett. 92, 020403 (2004).

We have developed for bosons an alternative to the Path Integral Monte Carlo method. Rather than working in position space, the new method obtains directly the full many body density operator of the gas at thermal equilibrium as an average of Hartree-Fock dyadics, each Hartree-Fock state experiencing a random evolution, similar to a Brownian motion.

The advantages with respect to the Path Integral Method are the following:

this formulation is directly made in second quantized formalism, so that it is straightforward to extend to fermions: as done by P. Chomaz and O. Juillet (LPC/ISMRA and GANIL), one simply replaces the bosonic Hartree-Fock states by fermionic Hartree-Fock states.
the method does not care of the fact that the Hamiltonian has real matrix elements or not in real space, so that it can be used as it is for a rotating system.

The Path Integral Monte Carlo remains however more efficient for the case of bosons at thermal equilibrium with a real Hamiltonian, as it has polynomial complexity in the number of particles, whereas our more general algorithm is exponential.

We have applied our method to simple models in 1D, e.g. to determine the exact probability distribution of the number of condensate particles as a function of temperature, including the vicinity of the critical temperature, and also to study the superfluidity of the 1D Bose gas.

Inspired by recent experiments on the superfluid to Mott insulator phase transition and on the Tonks-Girardeau gas of 1D impenetrable bosons, we have extended our method, replacing the stochastic Hartree-Fock states by the stochastic Gutzwiller states (the Gutzwiller state being the best simple starting point at the mean field level for correlated bosons in e.g. the Mott phase). We have successfully studied the Tonks-Girardeau gas with the new algorithm.

I. Carusotto, Y. Castin, J. Dalibard, "The N boson time dependent problem: a reformulation with stochastic wave functions", Phys. Rev. A 63, 023606 (2001).

I. Carusotto, Y. Castin, "An exact stochastic field method for the interacting Bose gas at thermal equilibrium", J. Phys. B 34, 4589-4608 (2001).

I. Carusotto, Y. Castin, "Condensate statistics in one-dimensional interacting Bose gases: exact results", Phys. Rev. Lett. 90, 030401 (2003).

I. Carusotto, Y. Castin, "Exact reformulation of the bosonic many-body problem in terms of stochastic wavefunctions: convergence issues", Laser Physics 13, 509-516 (2003).

I. Carusotto, Y. Castin, "An exact reformulation of the Bose-Hubbard model in terms of a stochastic Gutzwiller ansatz", New J. Phys. 5, 91 (2003).

The first matter wave bright soliton was observed by the group of Christophe Salomon, using a Bose-Einstein condensate with attractive interactions. Such an observation was expected to be difficult since a matter wave soliton has a size in the micrometer range, comparable to the optical wavelength of the light used to image it.

Our contribution to this observation was to suggest to put the soliton in an expulsive harmonic potential. This dramatically enhances the fact that a soliton does not spread spatially: whereas the ideal Bose gas was observed to explode exponentially fast in the expulsive potential, the soliton was found not to explode.

We have studied theoretically what happens to a soliton in an expulsive trap. First, excitations of the soliton are found to damp exponentially, which also allows to obtain rapidly an almost pure soliton. Second, the soliton can experience quantum evaporation; it then looses particles until its mean field cannot compensate the expulsive potential, in which case it explodes.

L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L.D. Carr, Y. Castin, C. Salomon, "Formation of a matter-wave bright soliton", Science 296, 1290 (2002).

L. Carr, Y. Castin, "Dynamics of a matter-wave bright soliton in an expulsive potential", Phys. Rev. A 66, 063602 (2002).

Perez-Garcia et Garcia-Ripoll have discovered numerically that the ground state solution of the Gross-Pitaveskii equation in a rotating frame can be not a straight, but a bent vortex line, when the trapping potential is cigar shaped. We have provided an analytical explanation of this fact, by deriving an approximate energy functional for the vortex line, valid in the Thomas-Fermi limit and for an elongated trap. One the sees that the condensate can be seen as the superposition of 2D slices orthogonal to the rotation axis z, each slice having a density depending on z.

In 2D, for a given rotation frequency, one finds that the vortex core is located in the trap center when the chemical potential, that is the density, is large enough; otherwise, the core moves to infinity. So in a cigar in 3D: for values of z close to 0, the density is high so the vortex line
remains essentially on the rotation axis; for larger values of |z|, the density drops so the local 2D vortex core would like to move to infinity: in 3D, the vortex line bends sharply and moves radially towards infinity. This explains the bending. The agreement with the numerics is even quantitative.

Furthermore, with our simplified energy functional, we have a discovered a saddle point, corresponding to a stationary vortex line shape that is not a minimum of energy in the rotating frame, and so could not be discovered numerically. We have found that this saddle point is actually a minimum of energy for a fixed angular momentum, rather than for a fixed rotation frequency, which illustrates the non-equivalence of two corresponding thermodynamic ensembles, canonical and grand-canonical like with respect to the angular momentum. This saddle point solution was observed experimentally in the group of Jean Dalibard.

Michele Modugno, Ludovic Pricoupenko, Yvan Castin, "Bose-Einstein condensates with a bent vortex in rotating traps", Eur. Phys. J. D 22, 235-257 (2003).

In a 2D or 1D gas, or a 3D gas in a very elongated trap, no Bose-Einstein condensate necessarily occurs even at temperatures and densities where the gas is degenerate and weakly interacting. One then cannot use the Bogoliubov theory to predict the state of the gas. However, when density fluctuations are weak (the regime of the quasi-condensates), one can still use the relative density fluctuations as a small parameter: one writes the field operator in terms of its modulus and its quantum phase, one then expands the field and the Hamiltonian in powers of the density fluctuations.Such an idea, present in the literature, however leads to ultraviolet divergences for some observables, because of the non-rigorous use of the phase operator. People then put a momentum cut-off, but the predictions then become cut-off dependent.

We have developed a simple, though rigorous way, of implementing the procedure, so that no cut-off dependence will spoiled the results at the end. The atomic positions are discretized on a lattice. The lattice spacing is larger than the healing and the thermal de Broglie wavelength, in order to recover the physics of the continuous space. The mean number of atoms per lattice site is large enough, so that the phase operator of the field can be defined safely. The on-site interaction is given by a well chosen coupling constant, depending on the lattice constant. We then expand the Hamiltonian up to third order in the density fluctuations, in a way very similar to the Bogoliubov method, but without supposing weak phase fluctuations.

We have tested our predictions in 1D, by successful comparison to results obtained from the Bethe ansatz and from the much more difficult Popov theory. We have proposed the first expression for the first order coherence function of the field valid in 1D, 2D and 3D, with no cut-off dependence, and agreeing with the usual Bogoliubov theory in 3D.

This approach was extended to the case of a 1D Bose gas on a torus. In this case, the existence of several quasi-condensate states differing by their winding number has to be included. We have used this to address the question of the superfluidity of the 1D Bose gas. The analytics is in excellent agreement with our Quantum Monte Carlo. The literature contains both predictions, that the 1D Bose gas is superfluid, and that it is not superfluid. We have figured out that the answer depends dramatically on the fact that the various winding numbers are taken into account or not.

To clarify the debate, we have enlarged the usual criterion of superfluidity. This criterion considers the mean momentum of the gas at thermal equilibrium in a rotating frame. In 1D, a given mean value of the total momentum can however be obtained in dramatically different ways: as an average of similar contributions for different experimental realizations, or as an average of contributions that are widely differing for various experimental realizations. We have therefore calculated, with our extended Bogoliubov approach, the probability distribution of the total momentum. Which has revealed the existence of the so-called metastable currents that are individually superfluid, in the quasi-condensate regime, and which has allowed to conclude.

C. Mora, Y. Castin, "Extension of Bogoliubov theory to quasicondensates", Phys. Rev. A 67, 053615 (2003).

I. Carusotto, Y. Castin, "Superfluidity of the 1D Bose gas", C. R. Physique 5, 107-127 (2004).

The most popular cooling technique of a Fermi gas is the sympathetic cooling of fermions by e.g. bosons: the fermions are in thermal contact with the bosons, and the bosons are evaporatively cooled.

Is there a limit to the accessible low temperatures, due to inelastic losses of fermions by collisions with the background gas, as suggested by E. Timmermans ? To answer the question quantitatively, we have considered a model of sympathetic cooling of fermions by a Bose condensate, including losses. We have solved the corresponding Boltzmann-like equations (on the mean occupation numbers) but also the more complex master equation (on the probability distribution of these occupation numbers).

We found that the limit temperatures are simple functions of the ratio of the loss rate to the collision rate between one boson and one fermion.The correspond limits are lower than the lowest observed temperatures (which are routinely 0.1 or 0.05 times the Fermi temperature) which suggests that what limits the temperature in the experiment is more technical than fundamental. We note however that our models were in a box, whereas the experiments use harmonic traps.

A crucial experimental result of the recent years is that one can produce a stable and ultracold Fermi gas close to a Feshbach resonance. One can then study the crossover from BEC to BCS, and study the strongly interacting regime where k_F |a| >1, where k_F is the Fermi momentum and a is the scattering length. One has a=infinity in principle, right at the resonance. The quantum gases can then interact more strongly than a neutron star (for which k_F |a|<1) !

The first experimental results of Christophe Salomon were surprising. Strong atom losses were detected and were maximum not right on resonance, but on the a>0 side. Beyond this region of maximal losses, the gas became stable, but with an attractive effective interaction, whereas the scattering length was still positive. The mean field theory could not explain that.

We have produced a simple model, based on two interacting particles on a box with perfectly reflecting walls and of a radius comparable to the mean interparticle separation. This box mimics the effects of the other N-2 particles, whereas the two body correlations are taken into account non perturbatively. We can then explain the experimental results, by showing that two branches of the macroscopic state of the gas, varying continuously when 1/(k_F a) is varied from -infinity to + infinity, are involved. The ground branch connects a BEC of dimers to the BCS state. The first excited branch corresponds to the weakly repulsive Fermi gas on the side -1/(k_F a) tending to -infinity, which is the starting point in the experiment. When one increases the scattering length a towards +infinity, the three body loss rate increases and this leads to the formation of dimers: the gas moves from the first excited branch to the ground branch, for a finite value of a. The `losses' actually correspond to the formation of molecules, which were not detected in the first experiments. Our model gives then a pressure on the ground branch which is less that the one of the ideal Fermi gas, which explains the occurrence of effectively attractive interactions for a positive value of the scattering length.

We have started working on the unitary quantum gas, that is for an infinite scattering length. This problem is fascinating because it is expected to be universal: in the spatially homogeneous case, at zero temperature, the only length scale left is the interatomic distance so that the chemical potential of the interacting gas is proportional to the one of the ideal Fermi gas having the same density, with a numerical coefficient that is bounded from above by fixed node quantum Monte Carlo calculations done by Pandharipande and co-workers. In the case of an isotropic harmonic trap, we have found that the evolution of the N-body wavefunction (solving the N body Schrödinger equation for a contact binary interaction) is exactly given by a gauge plus a scaling transform, whatever the time evolution of the trap spring constant. An important consequence for experiments is that the density of the gas experiences a pure scaling transform during a time of flight, so that the time of flight acts as a perfect magnifying lens.

Finally, in order to identify the observables able to reveal the formation of a condensate of pairs on the a<0 side of the Feshbach resonance, we have used the fermionic version of our quantum Monte Carlo method to study a 1D model of rather strongly interacting fermions. We show that the opposite spin density-density correlation function, proposed in the literature to detect the transition, is not a good indicator, as correlations exist also above T_c. The best observable that we have found to reveal the condensation of pairs is the first order coherence function of the pair of fermions.