We introduce a nonlinear Schr\"odinger equation to describe the dynamics of a superfluid Bose gas in the crossover from the weak-coupling regime, where $a{n}^{1/3}\ensuremath{\ll}1$ with
a the interatomic
s-wave scattering length and
n the bosonic density, to the unitarity limit, where $a\ensuremath{\rightarrow}+\ensuremath{\infty}$. We call this equation the unitarity Schr\"odinger equation (USE). The zero-temperature bulk equation of state of this USE is parametrized by the Lee-Yang-Huang low-density expansion and Jastrow calculations at unitarity. With the help of the USE we study the profiles of quantized vortices and vortex-core radius in a uniform Bose gas. We also consider quantized vortices in a Bose gas under cylindrically symmetric harmonic confinement and study their profile and chemical potential using the USE and compare the results with those obtained from the Gross-Pitaevskii-type equations valid in the weak-coupling limit. Finally, the USE is applied to calculate the breathing modes of the confined Bose gas as a function of the scattering length.
We introduce a quasianalytic nonlinear Schr\"odinger equation with beyond mean-field corrections to describe the dynamics of a zero-temperature dilute superfluid Fermi gas in the crossover from the weak-coupling Bardeen-Cooper-Schrieffer (BCS) regime, where
kF∣a∣⪡1 with
a the
s-wave scattering length and
kF the Fermi momentum, through the unitarity limit ${k}_{F}a\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}$ to the Bose-Einstein condensation (BEC) regime where
kFa>0. The energy of our model is parametrized using the known asymptotic behavior in the BCS, BEC, and the unitarity limits and is in excellent agreement with accurate Green's-function Monte Carlo calculations. The model generates good results for frequencies of collective breathing oscillations of a trapped Fermi superfluid.
We study the spontaneous symmetry breaking (SSB) of a superfluid Bose-Fermi (BF) mixture in a double-well potential (DWP). The mixture is described by the Gross-Pitaevskii equation (GPE) for the bosons, coupled to an equation for the order parameter of the Fermi superfluid, which is derived from the respective density functional in the unitarity limit (a similar model applies to the BCS regime, too). Straightforward SSB in the degenerate Fermi gas loaded into a DWP is impossible, as it requires an attractive self-interaction, and the intrinsic nonlinearity in the Fermi gas is repulsive. Nonetheless, we demonstrate that the symmetry breaking is possible in the mixture with attraction between fermions and bosons, like
40K and
87Rb. Numerical results are represented by dependencies of asymmetry parameters for both components on particle numbers of the mixture,
NF and
NB, and by phase diagrams in the
(NF,NB) plane, which displays regions of symmetric and asymmetric ground states. The dynamical picture of the SSB, induced by a gradual transformation of the single-well potential into the DWP, is reported too. An analytical approximation is proposed for the case when the GPE for the boson wave function may be treated by means of the Thomas-Fermi (TF) approximation. Under a special linear relationship between
NF and
NB, the TF approximation allows us to reduce the model to a single equation for the fermionic function, which includes competing repulsive and attractive nonlinear terms. The latter one directly displays the mechanism of the generation of the effective attraction in the Fermi superfluid, mediated by the bosonic component of the mixture.
The usual treatment of a Bose-Fermi mixture relies on weak-coupling Gross-Pitaevskii (GP) and density-functional (DF) Lagrangians, often including the more realistic perturbative Lee-Huang-Yang (LHY) corrections. We suggest analytic non-perturbative beyond-mean-field Bose and Fermi Lagrangians valid along the crossover from weak- to strong-coupling limits of intraspecies interactions consistent with the LHY corrections and the strong-coupling (unitarity) limit for small and large scattering lengths
∣a∣, respectively, and use these to study the Bose-Fermi mixture. We study numerically mixing-demixing and spontaneous symmetry breaking in Bose-Fermi mixtures in spherically symmetric and quasi-one-dimensional traps, while the intraspecies Bose and Fermi interactions are varied from weak-coupling to strong-coupling limits. The LHY correction is appropriate for medium to weak atomic interactions and diverges for stronger interactions (large scattering length
∣a∣), whereas the present beyond-mean-field Lagrangian is finite in the unitarity limit ($|a|\ensuremath{\rightarrow}\ensuremath{\infty}$). We illustrate our results using the Bose-Fermi ${}^{7}\mathrm{Li}{\text{\ensuremath{-}}}^{6}\mathrm{Li}$ mixture under a spherically-symmetric and a quasi-one-dimensional trap. The results obtained with the present model for density distribution of the Bose-Fermi mixture along the crossover could be qualitatively different from the usual GP-DF Lagrangian with or without LHY corrections. Specifically, we identified spontaneous symmetry breaking and demixing in the present model not found in the usual model with the same values of the parameters.
In this paper, we study the interaction-modulated tunneling dynamics of a Bose-Fermi superfluid mixture, where a Bose-Einstein condensate (BEC) with weak repulsive interaction is confined in a symmetric deep double-well potential and an equally populated two-component Fermi gas in a harmonic potential symmetrically is positioned in the center of the double-well potential. The tunneling between the two wells is modulated by fermions trapped in a harmonic potential. When the temperature is adequately low and the bosonic particle number is adequately large, we can employ the mean-field theory to describe the evolution of the BEC in the double-well potential through the time-dependent Gross-Pitaevskii equation. For the Fermi gas in the harmonic potential trap, we consider the case where the inter-fermion interaction is tuned on the deep Bose-Einstein condensate of the inter-fermion Feshbach resonance, where two fermions of spin-up and spin-down form a two-body bound state. Within the regime, the Fermi gas is well described by a condensate of these fermionic dimers, and hence can be simulated as well by a Gross-Pitaevskii equation of dimers. The inter-species interactions couple the dynamics of the two species, which results in interesting features in the tunneling oscillations. The dynamic equations of the BEC in the double-well potential is described by a two-mode approximation. Coupling it with time-dependent Gross-Pitaevskii equation of the harmonically potential trapped molecular BEC, we numerically investigate the dynamical evolution of the Boson-Fermi hybrid system under different initial conditions. It is found that the interaction among fermions in a harmonic potential leads to strong non-linearity in the oscillations of the bosons in the double-well potential and enriches the tunneling dynamics of the bosons. Especially, it strengthens macroscopic quantum self-trapping. And the macroscopic quantum self-trapping can be expressed in three forms: the phase tends to be negative and monotonically decreases with time, the phase evolves with time, and the phase tends to be positive and increases monotonically with time. This means that it is possible the tunneling dynamics of the BEC in double-well potential is adjustable. Our results can be verified experimentally in a Bose-Fermi superfluid mixture by varying different interaction parameters via Feshbach resonance and confinement-induced resonance.
We study numerically mixing-demixing and spontaneous symmetry breaking in Bose-Fermi mixtures in spherically-symmetric and quasi-one-dimensional traps while the intra-species Bose and Fermi interactions are varied from weak-coupling to strong-coupling limits. The usual treatment of a Bose-Fermi mixture relies on weak-coupling Gross-Pitaevskii (GP) and density-functional (DF) Lagrangians, often including the more realistic perturbative Lee-Huang-Yang (LHY) corrections. We suggest analytic non-perturbative beyond-mean-field Bose and Fermi Lagrangians valid along the crossover from weak- to strong-coupling limits of intra-species interactions consistent with the LHY corrections and the strong-coupling (unitarity) limit for small and large scattering lengths
∣a∣, respectively, and use these to study the Bose-Fermi mixture. The LHY correction is appropriate for medium to weak atomic interactions and diverges for stronger interactions (large scattering length
∣a∣), whereas the present beyond-mean-field Lagrangian is finite in the unitarity limit (
∣a∣→∞). We illustrate our results using the Bose-Fermi
7Li-
6Li mixture under a spherically-symmetric and a quasi-one-dimensional trap. The results obtained with the present model for density distribution of the Bose-Fermi mixture along the crossover could be qualitatively different from the usual GP-DF Lagrangian with or without LHY corrections. Specifically, we identified spontaneous symmetry breaking and demixing in the present model not found in the usual model with the same values of the parameters.