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Chinese Physics B

Quantum backreaction of quantum fluid in Bose–Einstein condensates

Yan Xu, Xiong Zu-Zhou, Bing Chen, Zhaoxin Li, Lei Tan

Published Nov 1, 2009• 1 citation
Publisher

Abstract

In this paper, with the full field operator expressed in terms of a particle-number-conserving mean-field ansatz, we investigate the dynamical behaviour of Bose–Einstein condensates from microscopic physics. Including the first-order term correction from single-particle excitation and the remaining higher-order term correction from collective excitations simultaneously, we obtain the formulation for a closed local expression of quantum backreaction Q , and discuss the influence on static Bose–Einstein condensates. Even though the quantum backreaction is small, it still has some influence on its dynamics.

DOI
10.1088/1674-1056/18/11/021

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In this paper, with the full field operator expressed in terms of a particle-number-conserving mean-field ansatz, we investigate the dynamical behaviour of Bose–Einstein condensates from microscopic physics. Including the first-order term correction from single-particle excitation and the remaining higher-order term correction from collective excitations simultaneously, we obtain the formulation for a closed local expression of quantum backreaction Q , and discuss the influence on static Bose–Einstein condensates. Even though the quantum backreaction is small, it still has some influence on its dynamics.
For many physical systems which can be approximated by a classical background field plus small (linearized) quantum fluctuations, a fundamental question concerns the correct description of the backreaction of the quantum fluctuations onto the dynamics of the classical background. We investigate this problem for the example of dilute atomic/molecular Bose-Einstein condensates, for which the microscopic dynamical behavior is under control. It turns out that the effective-action technique does not yield the correct result in general and that the knowledge of the pseudo-energy-momentum tensor $⟨{\stackrel{^}{T}}_{\ensuremath{\mu}\ensuremath{\nu}}⟩$ is not sufficient to describe quantum backreaction.
We provide a number-conserving approach to the backreaction problem of small quantum fluctuations onto a classical background for the exactly soluble dynamical evolution of a quasi-one-dimensional Bose-Einstein condensate, experimentally realizable in the ultracold gas laboratory. A force density exerted on the gas particles, which is of quantum origin, is uniquely identified as the deviation from the classical Eulerian force density. The backreaction equations are then explored for the specific example of a finite-size uniform density condens- ate initially at rest. By assuming that the condensate starts from a noninteracting regime and in its ground state, we fix a well-defined initial vacuum condition, which is driven out of equilibrium by instantaneously turning on the interactions. The assumption of this initial vacuum accounts for the ambiguity in choosing a vacuum state for interacting condensates, which is due to phase diffusion and the ensuing condensate collapse. As a major finding, we reveal that the time evolution of the condensate cloud leads to condensate density corrections that cannot in general be disentangled from the quantum depletion in measurements probing the power spectrum of the total density. Furthermore, while the condensate is initially at rest, quantum fluctuations give rise to a nontrivial condensate flux, from which we demonstrate that the quantum force density attenuates the classical Eulerian force. Finally, the knowledge of the particle density as a function of time for a condensate at rest determines, to order N0{N}^{0}N0 (where NNN is the total number of particles), the quantum force density, thus offering a viable route for obtaining experimentally accessible quantum backreaction effects.
The dynamics of the Bose–Einstein condensate (BEC) in a double-well potential is often investigated under the mean-field theory (MFT). This works successfully for large particle numbers with dynamical stability. But for dynamical instabilities, quantum corrections to the MFT becomes important [J. R. Anglin and A. Vardi, Phys. Rev. A64, 013605 (2001)]. Recently the adiabatic dynamics of the double-well BEC is investigated under the MFT in terms of a dark variable [C. Ottaviani et al., Phys. Rev. A81, 043621 (2010)], which generalizes the adiabatic passage techniques in quantum optics to the nonlinear matter-wave case. We give a fully quantized version of it using second-quantization and introduce new correction terms from higher order interactions beyond the on-site interaction, which are interactions between the tunneling particle and the particle in the well and interactions between the tunneling particles. If only the on-site interaction is considered, this reduces to the usual two-mode BEC.
Bose-Einstein condensates are suitable systems for studying fundamental aspects of quantum backreaction. Here, the backreaction problem in one-dimensional condensates is considered from the perspective of energy and momentum conservation. By assuming the validity of Bogoliubov theory, the backreaction equations are used to identify the contributions to the system energy and momentum coming from quantum fluctuations and condensate corrections. The backreaction is solved for a condensate trapped in a ring configuration such that particle interactions are gradually switched on. It is shown that the energy in the condensate cannot be addressed without taking into account how the system entered the interacting regime, and even for homogeneous condensates the power transferred to the condensate by quantum fluctuations showcases an intricate nonmonotonic pattern.
We formulate particle-creation phenomena in Bose-Einstein condensates in terms of conserving gapless mean-field theory for weakly interacting Bose gases. The particle-creation spectrum is calculated by rediagonalizing the Bogoliubov--de Gennes (BdG) Hamiltonian in mean-field theory. The conservation implies that quasiparticle creation is accompanied by quantum back reaction to the condensates. Particle creation in this mean-field theory is found to be equivalent to that in quantum field theory (QFT) in curved space-time. An expression is obtained for an effective metric affected by quantum back reaction. The formula for the particle-creation spectrum obtained in terms of QFT in curved space-time is shown to be the same as that given by rediagonalizing the BdG Hamiltonian.
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.