This chapter derives from first principles the time-dependent Gross–Pitaevskii equation, which describes the time-dependent behavior of the condensate wave function associated with the composite bosons that form on the BEC side of the BCS–BEC crossover at sufficiently low temperature. The derivation relies on the Green’s functions method for nonequilibrium problems developed before and explores the assumption that the fermionic chemical potential, associated with the initial preparation of the system at thermodynamic equilibrium, is the largest energy scale in the problem. The relation between the scattering length for composite bosons and the scattering length for the constituent fermions is also discussed.
We derive the time-independent Gross-Pitaevskii equation at zero temperature for condensed bosons, which form as bound-fermion pairs when the mutual fermionic attractive interaction is sufficiently strong, from the strong-coupling limit of the Bogoliubov-de Gennes equations that describe superfluid fermions in the presence of an external potential. Three-body corrections to the Gross-Pitaevskii equation are also obtained by our approach. Our results are relevant to the recent advances with ultracold fermionic atoms in a trap.
A well-known problem with the many-body approximations for interacting condensed bosons is the dichotomy between the "conserving" and "gapless" approximations, which either obey the conservations laws or satisfy the Hugenholtz-Pines condition for a gapless excitation spectrum, in the order. It is here shown that such a dichotomy does not exist for a system of composite bosons , which form as bound-fermion pairs in the strong-coupling limit of the fermionic attraction. By starting from the constituent fermions, for which conserving approximations can be constructed for any value of the mutual attraction according to the Baym-Kadanoff prescriptions, it is shown that these approximations also result into a gapless excitation spectrum for the boson-like propagators in the broken-symmetry phase.
We consider the current correlation function for a three-dimensional system of fermions embedded in a homogeneous background and mutually interacting via an attractive short-range potential, below the (superconducting) critical temperature. Diagrammatic contributions in the broken-symmetry phase are identified, which yield for the (wave-vector and frequency-dependent) current correlation function the fermionic BCS approximation in the weak-coupling limit and the bosonic Bogoliubov approximation in the strong-coupling limit (whereby composite bosons form as bound-fermion pairs). The temperature dependence of the superfluid density (from the BCS exponential behavior at weak coupling to a power-law behavior at strong coupling) and the form of the Pippard-like kernel at zero temperature are explicitly obtained from weak to strong coupling. Quite generally, it is shown that the Pippard-like kernel is the sum of a local (London-like) term and of a nonlocal component, the local term being dominant in the strong-coupling limit and the nonlocal component in the BCS (weak-coupling) limit. It is also shown that the range of the nonlocal component is determined by the coherence length measuring the spatial correlations of the amplitude of the order parameter, namely, the correlations among different Cooper pairs (or composite bosons), rather than between the fermionic partners of a given pair. In addition, a prescription is provided for mapping the fermionic onto the bosonic diagrammatic theories in the broken-symmetry phase, thus complementing what has been already done in the normal phase.
We consider the problem of the crossover from BCS superconductivity to Bose-Einstein condensation in three dimensions for a system of fermions with a mutual attractive interaction, for which we adopt the simplifying assumption of a suitably regularized point-contact interaction. We examine in a critical way the fermionic (self-consistent) T-matrix approximation, which has been widely utilized in the literature to describe this crossover above the superconducting critical temperature, and show that it fails to yield the correct behavior of the system in the strong-coupling limit, where composite bosons form as tightly bound fermion pairs. We then set up the correct approximation for a ``dilute'' system of composite bosons and show that a class of diagrams has to be considered in the place of the fermionic T-matrix approximation for the self-energy. This class of diagrams correctly describes both the weak- and strong-coupling limits, and consequently results in an improved interpolation scheme for the intermediate (crossover) region. In this context, we provide also a systematic mapping between the corresponding diagrammatic theories for the composite bosons and the constituent fermions. As a preliminary result to demonstrate the numerical effect of our class of diagrams on physical quantities, we calculate the value of the scattering length for composite bosons in the strong-coupling limit and show that it is considerably modified with respect to the result obtained within the self-consistent fermionic T-matrix approximation.
Abstract We study the dynamics of a system of N interacting bosons in a disc-shaped trap, which is realised by an external potential that confines the bosons in one spatial dimension to an interval of length of order
ε ε . The interaction is non-negative and scaled in such a way that its scattering length is of order
ε/N ε / N , while its range is proportional to
(ε/N)β ( ε / N ) β with scaling parameter
β∈(0,1] β ∈ ( 0 , 1 ] . We consider the simultaneous limit
(N,ε)→(∞,0) ( N , ε ) → ( ∞ , 0 ) and assume that the system initially exhibits Bose–Einstein condensation. We prove that condensation is preserved by the N-body dynamics, where the time-evolved condensate wave function is the solution of a two-dimensional non-linear equation. The strength of the non-linearity depends on the scaling parameter
β β . For
β∈(0,1) β ∈ ( 0 , 1 ) , we obtain a cubic defocusing non-linear Schrödinger equation, while the choice
β=1 β = 1 yields a Gross–Pitaevskii equation featuring the scattering length of the interaction. In both cases, the coupling parameter depends on the confining potential.