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arXiv:1903.04245 · Physical Review A

Expansion of the strongly interacting superfluid Fermi gas: Symmetries and self-similar regimes

E. A. Kuznetsov, M. Yu. Kagan, Andrei V. Turlapov

Published Apr 16, 2020• 12 citations
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Abstract

We consider an expansion of the strongly interacting superfluid Fermi gas in a vacuum in the so-called unitary regime when the chemical potential $\ensuremath{\mu}\ensuremath{\propto}{\ensuremath{\hbar}}^{2}{n}^{2/3}/m$, where $n$ is the density of the Bose-Einstein condensate of Cooper pairs of fermionic atoms. At low temperatures $T\ensuremath{\rightarrow}0$, such an expansion can be described in the framework of the Gross-Pitaevskii equation (GPE). For such a dependence of the chemical potential on the density, the GPE has additional symmetries, resulting in the existence of the virial theorem, connecting the mean size of the gas cloud and its Hamiltonian. It leads asymptotically at $t\ensuremath{\rightarrow}\ensuremath{\infty}$ to the gas cloud expansion, linearly growing in time. We study such asymptotics and reveal the perfect match between the quasiclassical self-similar solution and the asymptotic expansion of the noninteracting gas. This match is governed by the virial theorem, derived through utilizing the Talanov transformation, which was first obtained for the stationary self-focusing of light in media with a cubic nonlinearity due to the Kerr effect. In the quasiclassical limit, the equations of motion coincide with three-dimensional hydrodynamics for the perfect monatomic gas with $\ensuremath{\gamma}=5/3$. Their self-similar solution describes, within the background of the gas expansion, the angular deformities of the gas shape in the framework of the Ermakov-Ray-Reid--type system.

DOI
10.1103/physreva.101.043612

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We consider an expansion of the strongly interacting superfluid Fermi gas in a vacuum in the so-called unitary regime when the chemical potential \ensuremathμ\ensuremath∝\ensuremathℏ2n2/3/m\ensuremath{\mu}\ensuremath{\propto}{\ensuremath{\hbar}}^{2}{n}^{2/3}/m\ensuremathμ\ensuremath∝\ensuremathℏ2n2/3/m, where nnn is the density of the Bose-Einstein condensate of Cooper pairs of fermionic atoms. At low temperatures T\ensuremath→0T\ensuremath{\rightarrow}0T\ensuremath→0, such an expansion can be described in the framework of the Gross-Pitaevskii equation (GPE). For such a dependence of the chemical potential on the density, the GPE has additional symmetries, resulting in the existence of the virial theorem, connecting the mean size of the gas cloud and its Hamiltonian. It leads asymptotically at t\ensuremath→\ensuremath∞t\ensuremath{\rightarrow}\ensuremath{\infty}t\ensuremath→\ensuremath∞ to the gas cloud expansion, linearly growing in time. We study such asymptotics and reveal the perfect match between the quasiclassical self-similar solution and the asymptotic expansion of the noninteracting gas. This match is governed by the virial theorem, derived through utilizing the Talanov transformation, which was first obtained for the stationary self-focusing of light in media with a cubic nonlinearity due to the Kerr effect. In the quasiclassical limit, the equations of motion coincide with three-dimensional hydrodynamics for the perfect monatomic gas with \ensuremathγ=5/3\ensuremath{\gamma}=5/3\ensuremathγ=5/3. Their self-similar solution describes, within the background of the gas expansion, the angular deformities of the gas shape in the framework of the Ermakov-Ray-Reid--type system.
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Abstract A brief review of the results on the expansion of quantum and classical gases into vacuum based on the use of symmetries is presented. For quantum gases in the Gross–Pitaevskii (GP) approximation, additional symmetries arise for gases with a chemical potential μ that depends on the density n powerfully with exponent ν = 2/D, where D is the space dimension. For gas condensates of Bose atoms at temperatures T → 0, this symmetry arises for two-dimensional systems. For D = 3 and, accordingly, ν = 2/3, this situation is realized for an interacting Fermi gas at low temperatures in the so-called unitary limit (see, for example, L. P. Pitaevskii, Phys. Usp. 51, 603 (2008)). The same symmetry for classical gases in three-dimensional geometry arises for gases with the adiabatic exponent γ = 5/3. Both of these facts were discovered in 1970 independently by Talanov [V. I. Talanov, JETP Lett. 11, 199 (1970).] for a two-dimensional nonlinear Schrödinger (NLS equation, which coincides with the Gross–Pitaevskii equation), describing stationary self-focusing of light in media with Kerr nonlinearity, and for classical gases, by Anisimov and Lysikov [S. I. Anisimov and Yu. I. Lysikov, J. Appl. Math. Mech. 34, 882 (1970)]. In the quasiclassical limit, these GP equations coincide with the equations of the hydrodynamics of an ideal gas with the adiabatic exponent γ = 1 + 2/D. Self-similar solutions in this approximation describe the angular deformations of the gas cloud against the background of an expanding gas by means of Ermakov-type equations. Such changes in the shape of an expanding cloud are observed in numerous experiments both during the expansion of gas after exposure to powerful laser radiation, for example, on metal, and during the expansion of quantum gases into vacuum.
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 analyze strongly interacting Fermi gases in the unitary regime by considering the generalization to an arbitrary number NNN of spin-1∕21∕21∕2 fermion flavors with Sp(2N)\mathrm{Sp}(2N)Sp(2N) symmetry. For $N\ensuremath{\rightarrow}\ensuremath{\infty}$ this problem is exactly solved by the Bardeen-Cooper-Schrieffer--Bose-Einstein condensate mean-field theory, with corrections small in the parameter 1∕N1∕N1∕N. The large-NNN expansion provides a systematic way to determine corrections to mean-field predictions, allowing the calculation of a variety of thermodynamic quantities at (and in proximity to) unitarity, including the energy, the pairing gap, and the upper-critical polarization (in the case of a polarized gas) for the normal to superfluid instability. For the physical case of N=1N=1N=1, among other quantities, we predict in the unitarity regime, the energy of the gas to be $\ensuremath{\xi}=0.28$ times that for the noninteracting gas and the pairing gap to be 0.52 times the Fermi energy.
We derive a unified set of scaling equations for Bose-Einstein condensates in time-dependent harmonic traps, connecting the weakly interacting Gaussian regime to the strongly interacting Thomas-Fermi regime. The exact Gross-Pitaevskii ground-state density is taken as a fixed profile in rescaled coordinates, with its dynamics described by three scaling factors corresponding to compression or expansion along the three spatial directions. The resulting equations contain no adjustable parameter, since their coefficients are determined once from the initial state. They recover analytically the Gaussian variational and Thomas-Fermi scaling equations in their respective limits and satisfy an axis-resolved virial theorem. The approach also yields the global phase and the low-lying collective-mode frequencies of the condensate. We benchmark the model against three-dimensional Gross-Pitaevskii simulations and against measured expansion energies. The model remains accurate across all interaction regimes, in strongly anisotropic traps, and along time-dependent sequences including a relatively fast quench, up to the point where the underlying frozen-profile hypothesis, shared by all three scaling approaches, breaks down.
Abstract A monolayer of superfluid Fermi gas can be prepared within an optical dipole trap using the tight confinement along the chosen direction. In this case, Cooper pairs occupy the lowest state corresponding to the motion in the trapping potential. After switching off the trapping potential, the initially two-dimensional gas expands to the three-dimensional space. In the case of unitary s-wave interactions, the dynamics of Fermi gas expansion is treated in the framework of appropriately modified Gross–Pitaevskii equation. It is found that the superfluid gas expands significantly faster than the normal gas, in contrast to the situation characteristic of the initially three-dimensional gas. The available experimental data [P. Dyke et al., Phys. Rev. A 93, 011603 (2016)] are close to the predictions of the model under study.