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Theoretical and Mathematical Physics

Semiclassical expansion of quantum gases into a vacuum

Evgeny A. Kuznetsov, M. Yu. Kagan

Published Mar 1, 2020• 9 citations
Publisher

Abstract

In the framework of the Gross-Pitaevskii equation, we consider the problem of the expansion of quantum gases into a vacuum. For them, the chemical potential μ has a power-law dependence on the density n with the exponent ν = 2/D, whereD is the space dimension. For gas condensates of Bose atoms as the temperature T → 0, s scattering gives the main contribution to the interaction of atoms in the leading order in the gas parameter. Therefore, the exponent ν = 1 for an arbitrary D. In the three-dimensional case, ν = 2/3 is realized for condensates of Fermi atoms in the so-called unitary limit. For ν = 2/D, the Gross-Pitaevskii equation has an additional symmetry under Talanov transformations of the conformal type, which were first found for the stationary self-focusing of light. A consequence of this symmetry is the virial theorem relating the average size R of an expanding gas cloud to its Hamiltonian. The quantity R asymptotically increases linearly with time as t →∞. In the semiclassical limit, the equations of motion coincide with those of the hydrodynamics of an ideal gas with the adiabatic exponent γ = 1+2/D. In this approximations, self-similar solutions describe angular deformations of the gas cloud against the background of the expanding gas in the framework of equations of the Ermakov-Ray-Reid type.

DOI
10.1134/s0040577920030125

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In the framework of the Gross-Pitaevskii equation, we consider the problem of the expansion of quantum gases into a vacuum. For them, the chemical potential μ has a power-law dependence on the density n with the exponent ν = 2/D, whereD is the space dimension. For gas condensates of Bose atoms as the temperature T → 0, s scattering gives the main contribution to the interaction of atoms in the leading order in the gas parameter. Therefore, the exponent ν = 1 for an arbitrary D. In the three-dimensional case, ν = 2/3 is realized for condensates of Fermi atoms in the so-called unitary limit. For ν = 2/D, the Gross-Pitaevskii equation has an additional symmetry under Talanov transformations of the conformal type, which were first found for the stationary self-focusing of light. A consequence of this symmetry is the virial theorem relating the average size R of an expanding gas cloud to its Hamiltonian. The quantity R asymptotically increases linearly with time as t →∞. In the semiclassical limit, the equations of motion coincide with those of the hydrodynamics of an ideal gas with the adiabatic exponent γ = 1+2/D. In this approximations, self-similar solutions describe angular deformations of the gas cloud against the background of the expanding gas in the framework of equations of the Ermakov-Ray-Reid type.
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 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 nnn 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.
A dilute homogeneous two-dimensional (2D) Bose gas at zero temperature is studied with the method developed earlier by the authors. This method allows for considering renormalization of an arbitrary pairwise potential in a self-consistent manner, without the pseudopotential delta-function representation. Low-density expansions are derived for the chemical potential, ground-state energy, pair distribution function, kinetic and interaction energies. The expansion parameter is found to be a dimensionless in-medium scattering amplitude u obeying the equation 1/u+ln u=-ln(na(2)pi)-2gamma, where na(2) and gamma are the gas parameter and the Euler constant, respectively. It is shown that the ground-state energy is mostly kinetic in the low-density limit. This result does not depend on a specific form of the pairwise interaction potential, contrary to the 3D case. A new form of the 2D Gross-Pitaevskii equation is proposed within our scheme.
The expansion of Bose-Einstein condensates with quantized vortices is studied by solving numerically the time-dependent Gross-Pitaevskii equation at zero temperature. For a condensate initially trapped in a spherical harmonic potential, we confirm previous results obtained by means of variational methods showing that, after releasing the trap, the vortex core expands faster than the radius of the atomic cloud. This could make the detection of vortices feasible, by observing the depletion of the density along the axis of rotation. We find that this effect is significantly enhanced in the case of anisotropic disk-shaped traps. The results obtained as a function of the anisotropy of the initial configuration are compared with the analytic solution for a noninteracting gas in three dimensions as well as with the scaling law predicted for an interacting gas in two dimensions.
This course on two-dimensional quantum gases, prepared for the S{ã}o Paulo ICTP-SAIFR IVth School on Light and Cold Atoms, is organized into three lectures. The first introductory lecture addresses Bose-Einstein condensation in an ideal gas, the effect of interactions in the weakly interacting limit, Gross-Pitaevskii equation and its hydrodynamic formulation, as well as superfluidity. The second lecture is devoted to the main features of two-dimensional quantum gases: importance of the interactions, enhanced phase fluctuations and the emergence of a quasi long-range order, superfluidity through the Berezinskii-Kosterlitz-Thouless mechanism, scaling symmetry. The third lecture has taken the form of a seminar on fast rotating quasi two-dimensional Bose gases and thermal melting of the vortex lattice. The slides accompanying these notes can be found on the school website: https://www.ictp-saifr.org/slca2025/.
In the mean-field approximation, a trapped Bose-Einstein condensate at zero temperature is described by the Gross-Pitaevskii equation for the condensate or, equivalently, by the hydrodynamic equations for the number density and current density. These equations receive corrections from quantum field fluctuations around the mean field. We calculate the semiclassical corrections to these equations for a general time-independent state of the condensate, extending previous work to include vortex states as well as the ground state. In the Thomas-Fermi limit, the semiclassical corrections can be taken into account by adding a local correction term to the Gross-Pitaevskii equation. At second order in the Thomas-Fermi expansion, the semiclassical corrections can be taken into account by adding local correction terms to the hydrodynamic equations.