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Journal of Modern Optics

Quantum dynamics of the phase of a Bose–Einstein condensate

Pierre Villain, Maciej Lewenstein, Ralph Dum, Yvan Castin, Li You, Ataç İmamoğlu, T. A. B. Kennedy

Published Oct 1, 1997• 29 citations
Publisher

Abstract

In this review we discuss the dynamics of the phase of trapped Bose–Einstein condensates. In particular we consider the phenomena of phase decoherence (termed also as phase collapse, or diffusion), and phase revival in systems of interacting atoms. We analyse the dependence of the collapse and revival times on the trap potential, dimensionality of the gas, atom number fluctuations, and on the coherent dynamics of the condensate. We show that in a class of experimentally relevant systems, the collapse time is relatively short, and in some cases vanishes in the limit of a large number of atoms, implying that the trapped Bose gas cannot sustain a well-defined quantum phase, and that the phase memory is lost on a relatively short time scale. Furthermore, we calculate the relative atom number fluctuations or a model of two interacting condensates, and show that the fluctuations are generically sub-Poissonian.

DOI
10.1080/09500349708231846

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In this review we discuss the dynamics of the phase of trapped Bose–Einstein condensates. In particular we consider the phenomena of phase decoherence (termed also as phase collapse, or diffusion), and phase revival in systems of interacting atoms. We analyse the dependence of the collapse and revival times on the trap potential, dimensionality of the gas, atom number fluctuations, and on the coherent dynamics of the condensate. We show that in a class of experimentally relevant systems, the collapse time is relatively short, and in some cases vanishes in the limit of a large number of atoms, implying that the trapped Bose gas cannot sustain a well-defined quantum phase, and that the phase memory is lost on a relatively short time scale. Furthermore, we calculate the relative atom number fluctuations or a model of two interacting condensates, and show that the fluctuations are generically sub-Poissonian.
We discuss some recent work on the dephasing of Bose-Einstein condensates of interacting atoms due to fluctuations in the chemical potential μ. At vanishing temperature such fluctuations are due to the uncertainty of the number of particles in the condensate caused by the depletion effect. The dephasing is in this case a non-diffusive (and in principle reversible) effect called ‘collapse’ (and ‘revival’). Above a certain cross-over temperature the dephasing proceeds mainly via a diffusive process which follows in time a short and ineffective phase collapse. The phase-diffusion constant is dominated by the temperature dependent condensate-number fluctuations caused by the scattering of low-energy excitations off the condensate. We also discuss the (in general subdominant) contribution of the occupation-number fluctuations in quasi-particle states in the thermal cloud. For simplicity we discuss here results for a box-like trapping potential.
Abstract We report on the dynamics of a Bose–Einstein condensate in one and two dimensions driven by the time-dependent harmonic trapping potential. Without the inter-particle interaction, the condensate exhibits the coherent behavior with the time-oscillating density distribution. When the inter-particle interaction is taken into consideration, the phase fluctuations, the multi-peak structure of the density distribution, and the coherence revival phenomenon, apart from the well-studied density oscillation behavior, can be observed in the condensate. Furthermore, it is demonstrated that due to the dimensional restriction these effects are more stable in the two and three-dimensional system, if compared to the one-dimensional case.
Bose-Einstein condensates of quasiparticles such as exciton-polaritons, magnons and massive photons allow for new experimental possibilities as compared to the atomic Bose-Einstein condensates. In this Thesis we focus on a condensate of photons, first created at Bonn University in 2010. One of the recent achievements is the measurement of number fluctuations in such a condensate of photons. We present a general theory to calculate these number fluctuations in a harmonically trapped interacting Bose gas and apply this to the available experimental results on the condensate of photons, finding good quantitative agreement. Additionally, we investigate the fundamental phenomenon of phase diffusion, which is based on the fact that a Bose-Einstein condensate can be described as a symmetry-broken phase. However, the symmetry-broken phase is only well defined in the thermodynamic limit, such that in a finite system the phase of the Bose-Einstein condensate can have nontrivial dynamics. We propose a new type of interference experiment involving a condensate of photons to measure this dynamical behavior of the phase. By calculating the effects of quantum and thermal fluctuations we find different time scales on which the interference pattern vanishes. Based on these time scales, we conclude that phase diffusion is experimentally observable within the precision of current devices.
In the field of ultracold atoms in optical lattices a plethora of phenomena governed by the hopping energy JJJ and the interaction energy UUU have been studied in recent years. However, the trapping potential typically present in these systems sets another energy scale and the effects of the corresponding time scale on the quantum dynamics have rarely been considered. Here we study the quantum collapse and revival of a lattice Bose-Einstein condensate (BEC) in an arbitrary spatial potential, focusing on the special case of harmonic confinement. Analyzing the time evolution of the single-particle density matrix, we show that the physics arising at the (temporally) recurrent quantum phase revivals is essentially captured by an effective single-particle theory. This opens the possibility of preparing exotic nonequilibrium condensate states with a large degree of freedom by engineering the underlying spatial lattice potential.