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Physical Review A

Evolution of a Bose gas in anisotropic time-dependent traps

Yu. Kagan, E.L. Surkov, Georgy V. Shlyapnikov

Published Jan 1, 1997• 206 citations
Publisher

Abstract

We find scaling solutions for the evolution of a Bose gas in time-dependent anisotropic traps and reveal the phenomenon of stochastization in the evolution of a condensate. In contrast to collisionless gases, the evolution of the thermal gas in the hydrodynamic regime proves to be in many aspects similar to that of the condensate. We emphasize that clear signatures of Bose-Einstein condensation and the effects of the loss of coherence can be found in local correlation properties of the evolving gas through the measurement of the rates of intrinsic or light-induced inelastic collisional processes.

DOI
10.1103/physreva.55.r18

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We find scaling solutions for the evolution of a Bose gas in time-dependent anisotropic traps and reveal the phenomenon of stochastization in the evolution of a condensate. In contrast to collisionless gases, the evolution of the thermal gas in the hydrodynamic regime proves to be in many aspects similar to that of the condensate. We emphasize that clear signatures of Bose-Einstein condensation and the effects of the loss of coherence can be found in local correlation properties of the evolving gas through the measurement of the rates of intrinsic or light-induced inelastic collisional processes.
We derive a symmetry property for the Fourier-transform of the density fluctuation modes of Bose condensates in anisotropic traps in the hydrodynamic limit connected with a somewhat hidden conservation law.. We discuss its possible observation by dispersive light scattering, its range of validity, and also experiments using inelastic scattering of light. PACS: 03.75.Fi,05.30.Jp,42.50.Gy Since the achievement of Bose-Einstein condensates in trapped alkali gases [1–3] at ultralow temperatures numerous aspects of these systems have been investigated both from the experimental and the theoretical side. Adding a time dependent perturbation to the trap potential the lowest frequencies of the density fluctuation spectrum were measured by observing condensate shape oscillations either by time of flight measurements [4–6] or by dispersive light scattering [7]. In this paper we want to report on a property of the mode functions of the low lying density fluctuations of condensates in anisotropic traps. Mode functions can be observed in dispersive light scattering experiments [7,8], but also in inelastic light scattering. We discuss how the derived property could be observed in both experiments and discuss the limits of both approaches. For a Bose gas below the Bose-Einstein transition the usual separation of the particle field 1 operator into a classical part φ0(r), the condensate wave function, and a residual operator ˆφ(r), describing fluctuations, yields for the number-density operator ˆρ(r) = |φ0(r) | 2 + ( φ ∗ 0(r) ˆ φ(r) + φ0(r) ˆ φ † (r)) + ˆ φ † (r) ˆ φ(r). (0.1) Thermal averaging of the last term gives the condensate depletion at T = 0 plus the density of thermally excited atoms. At low temperatures for a weakly interacting system the condensate density |φ0(r) | 2 contains most of the particles, and the second term in (0.1) dominates by far the last term. Now we restrict ourselves to the low-lying density fluctuations with ¯hω ≪ µ, which can be described by the hydrodynamic approach to trapped Bose systems [9,10]. In this limit the dominant density fluctuations in (0.1) take the form [10,11] φ ∗ 0 (r) ˆ φ(r) + φ0(r) ˆ φ † (r) = ∑ j
The anisotropic density distribution of Bose particles after expansion from a magnetic trap has been taken as an important signature of the formation of Bose-Einstein condensate. In this letter we show that even for a non-condensed trapped gas one can get an anisotropic distribution due to anisotropic collision rates along radial and axial directions. We demonstrate that the mean energies along those directions, different from the energy in the trap, have opposite evolution tendencies with the expansion. This difference is particularly important for the case where a large number of bosons is trapped. We also show that the velocity distribution of the non-condensed gas in the trap, assumed initially isotropic and Gaussian, becomes anisotropic and with linewidth dependent on the expansion direction.
We develop the idea of selectively manipulating the condensate in a trapped Bose-condensed gas, without perturbing the thermal cloud. The idea is based on the possibility to modify the mean field interaction between atoms (scattering length) by nearly resonant incident light or by spatially uniform change of the trapping magnetic field. For the gas in the Thomas-Fermi regime we find analytical scaling solutions for the condensate wave function evolving under arbitrary variations of the scattering length aaa. The change of aaa from positive to negative induces a global collapse of the condensate, and the final stages of the collapse will be governed by intrinsic decay processes.
We discuss the collective modes of a trapped Bose gas in the hydrodynamic regime where atomic collisions ensure local thermal equilibrium for the distribution function. Starting from the conservation laws, in the linearized limit we derive a closed equation for the velocity fluctuations in a trapped Bose gas above the Bose-Einstein transition temperature. Explicit solutions for a parabolic trap are given. We find that the surface modes above the transition have the same dispersion relation as the one recently obtained by Stringari for the oscillations of the condensate at T=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0 within the Thomas-Fermi approximation. Results are also given for the monopole ``breathing'' mode as well as for the m=0m\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0m=0 excitations which result from the coupling of the monopole and quadrupole modes in an anisotropic parabolic well.
We analyze the expansion process of a thermal trapped gas taking place during the detection of Bose-Einstein condensation through an expansion of the condensate. By extending our Monte Carlo method to study this phenomenon, we demonstrate that the velocity distributions of the thermal trapped atoms are anisotropic due to anisotropic explosion of atoms. The radial and axial components of translational temperature are different from the trap temperature and have opposite evolution tendencies with the gas expansion.
We study experimentally the dissipative dynamics of ultracold bosonic gases in a dynamic disorder potential with tunable correlation time. First, we measure the heating rate of thermal clouds exposed to the dynamic potential and present a model of the heating process, revealing the microscopic origin of dissipation from a thermal, trapped cloud of bosons. Second, for Bose-Einstein condensates, we measure the particle loss rate induced by the dynamic environment. Depending on the correlation time, the losses are either dominated by heating of residual thermal particles or the creation of excitations in the superfluid, a notion we substantiate with a rate model. Our results illuminate the interplay between superfluidity and time-dependent disorder and on more general grounds establish ultracold atoms as a platform for studying spatiotemporal noise and time-dependent disorder.