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Mixing-demixing and spontaneous symmetry breaking in trapped Bose-Fermi mixtures: Weak-coupling to unitarity crossover

Sandeep Gautam, Sadhan K. Adhikari

Published Jun 4, 2019• 0 citations
OpenAlex

Abstract

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 ($|a|\to \infty$). We illustrate our results using the Bose-Fermi $^7$Li-$^6$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.

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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.

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Chinese Physics B•Jan 1, 2012•7 citations
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∣|a|∣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∣|a|∣a∣), whereas the present beyond-mean-field Lagrangian is finite in the unitarity limit (∣a∣→∞|a|\to \infty∣a∣→∞). We illustrate our results using the Bose-Fermi 7^77Li-6^66Li 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.
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∣|a|∣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∣|a|∣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.
We investigate the zero-temperature properties of a superfluid Bose-Fermi mixture by introducing a set of coupled Galilei-invariant nonlinear Schr\"odinger equations valid from weak coupling to unitarity. The Bose dynamics is described by a Gross-Pitaevskii-type equation including beyond-mean-field corrections possessing the correct weak-coupling and unitarity limits. The dynamics of the two-component Fermi superfluid is described by a density-functional equation including beyond-mean-field terms with correct weak-coupling and unitarity limits. The present set of equations is equivalent to the equations of generalized superfluid hydrodynamics, which take into account also surface effects. The equations describe the mixture properly as the Bose-Bose repulsive (positive) and Fermi-Fermi attractive (negative) scattering lengths are varied from zero to infinity in the presence of a Bose-Fermi interaction. The present model is tested numerically as the Bose-Bose and Fermi-Fermi scattering lengths are varied over wide ranges covering the weak-coupling to unitarity transition.
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^{40}\mathrm{K}40K and 87Rb^{87}\mathrm{Rb}87Rb. Numerical results are represented by dependencies of asymmetry parameters for both components on particle numbers of the mixture, NF{N}_{\mathrm{F}}NF​ and NB{N}_{\mathrm{B}}NB​, and by phase diagrams in the (NF,NB)({N}_{\mathrm{F}},{N}_{\mathrm{B}})(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{N}_{\mathrm{F}}NF​ and NB{N}_{\mathrm{B}}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.
We study the ground states of one-dimensional Bose-Bose mixtures under harmonic confinement. As we vary the interspecies coupling strength up to the limit of infinite repulsion, we observe a generalized, composite-fermionization crossover. The initially coexisting phases demix as a whole for weak intraspecies interactions, whereas the atoms localize individually for strong intraspecies repulsion. By symmetry, the two components end up with strongly overlapping profiles, albeit sensitive to symmetry-breaking perturbations. Different pathways emerge if the two components have different atom numbers, different intraspecies interactions, or different masses and/or trap frequencies.
We study an ultracold and dilute superfluid Bose-Fermi mixture confined in a strictly one-dimensional (1D) atomic waveguide by using a set of coupled nonlinear mean-field equations obtained from the Lieb-Liniger energy density for bosons and the Gaudin-Yang energy density for fermions. We consider a finite Bose-Fermi interatomic strength gbf{g}_{bf}gbf​ and both periodic and open boundary conditions. We find that with periodic boundary conditions---i.e., in a quasi-1D ring---a uniform Bose-Fermi mixture is stable only with a large fermionic density. We predict that at small fermionic densities the ground state of the system displays demixing if gbf>0{g}_{bf}>0gbf​>0 and may become a localized Bose-Fermi bright soliton for gbf<0{g}_{bf}<0gbf​<0. Finally, we show, using variational and numerical solutions of the mean-field equations, that with open boundary conditions---i.e., in a quasi-1D cylinder---the Bose-Fermi bright soliton is the unique ground state of the system with a finite number of particles, which could exhibit a partial mixing-demixing transition. In this case the bright solitons are demonstrated to be dynamically stable. The experimental realization of these Bose-Fermi bright solitons seems possible with present setups.
We study the spontaneous symmetry breaking of a superfluid Bose—Fermi mixture in a two-dimensional double-well potential. The mixture is described by a set of coupled Gross—Pitaevskii equations. The symmetry breaking phenomenon is demonstrated in the two-dimensional double-well potential in the mixture. The results are summarized in the phase diagrams of the mixture particle numbers, which are divided into symmetric and asymmetric regions by the asymmetry ratios. The dynamical pictures of the spontaneous symmetry breaking induced by a gradual transformation of the single-well potential into a double-well one are also illustrated. The properties of the quantum degenerate mixture are explored using the realistic parameters for a 40 K− 87 Rb system.