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arXiv:cond-mat/0112315 · Physical Review A

Continuous quantum measurement of a Bose-Einstein condensate: A stochastic Gross-Pitaevskii equation

Diego A. R. Dalvit, Jacek Dziarmaga, Roberto Onofrio

Published Apr 19, 2002• 34 citations
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Abstract

We analyze the dynamics of a Bose-Einstein condensate undergoing a continuous dispersive imaging by using a Lindblad operator formalism. Continuous strong measurements drive the condensate out of the coherent-state description assumed within the Gross-Pitaevskii mean-field approach. Continuous weak measurements allow us instead to replace, for time scales short enough, the exact problem with its mean-field approximation through a stochastic analog of the Gross-Pitaevskii equation. The latter is used to show the unwinding of a dark soliton undergoing a continuous imaging.

DOI
10.1103/physreva.65.053604

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We analyze the dynamics of a Bose-Einstein condensate undergoing a continuous dispersive imaging by using a Lindblad operator formalism. Continuous strong measurements drive the condensate out of the coherent-state description assumed within the Gross-Pitaevskii mean-field approach. Continuous weak measurements allow us instead to replace, for time scales short enough, the exact problem with its mean-field approximation through a stochastic analog of the Gross-Pitaevskii equation. The latter is used to show the unwinding of a dark soliton undergoing a continuous imaging.
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The phase of the mean-field wave function of a Bose-Einstein condensate can be recovered from a time series of images. We adapt an iterative retrieval method that has been successful in linear electron and optical imaging systems to solve the nonlinear Gross-Pitaevskii equation. We address a number of issues related to the successful application of this method to the nonlinear system, including the retrieval of wave functions with nonzero net topological charge (i.e., containing vortices) and the effects of repulsive and attractive interactions in the condensate on the convergence properties of the method. Recovering the phase from the continuity-of-density equation is also investigated. An understanding of these issues is of importance for the practical implementation of phase retrieval to Bose-Einstein condensates.
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In this thesis, we apply the stochastic Gross-Pitaevskii equation, using the formulation of Stoof, to the study of weakly interacting, finite temperature atomic Bose gases, both in and out of equilibrium. Explicitly maintaining the effects of phase and density fluctuations, this state-of-the-art method is well suited to the study of low dimensional Bose systems. As such, we consider the non-equilibrium dynamics of dark matter wave solitons, oscillating within harmonically trapped, one-dimensional phase-fluctuating condensates, through numerical simulations of the stochastic Gross-Pitaevskii equation. In treating the damping due to higher lying thermal modes of the system, we include the HartreeFock contribution of the equilibrium atomic density, which is a novel feature of our work. A stochastic approach is found to be essential in capturing the shot-to-shot variation in soliton trajectories observed experimentally. Neglecting stochastic effects, we find the well-known dissipative Gross-Pitaevskii equation to recover the average dynamics of the stochastic model, on retaining the microscopically justified form for the damping of this approach. Such a coupling is found to be necessary in modelling the dissipative nature of finite temperature soliton dynamics. Furthermore, we undertake a comparison between the finite temperature equilibrium state of the stochastic Gross-Pitaevskii equation, and that predicted within a number conserving Bogoliubov approach. Through a series of numerical experiments, we compare the equilibrium properties of a harmonically trapped Bose gas, considering our results in light of those of independent theoretical treatments, and additionally assessing the sensitivity of the numerical implementation of the former method to the numerical grid spacing. In the regime in which thermal effects dominate, the stochastic Gross-Pitaevskii approach proves to be a more consistent means of generating such equilibrium states.
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We study the time-evolution of initially trapped Bose-Einstein condensates in the Gross-Pitaevskii regime. Under a physically motivated assumption on the energy of the initial data, we show that condensation is preserved by the many-body evolution and that the dynamics of the condensate wave function can be described by the time-dependent Gross-Pitaevskii equation. With respect to previous works, we provide optimal bounds on the rate of condensation (i.e. on the number of excitations of the Bose-Einstein condensate). To reach this goal, we combine the method of \cite{LNS}, where fluctuations around the Hartree dynamics for NNN-particle initial data in the mean-field regime have been analyzed, with ideas from \cite{BDS}, where the evolution of Fock-space initial data in the Gross-Pitaevskii regime has been considered.