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Physical Review Letters

Phase and Phase Diffusion of a Split Bose-Einstein Condensate

Juha Javanainen, Martin Wilkens

Published Jun 23, 1997• 205 citations
Publisher

Abstract

We analyze theoretically an experiment in which a trapped Bose-Einstein condensate is cut in half, and the parts are subsequently allowed to interfere. If the delay between cutting and atom detection is small, the interference pattern of the two halves of the condensate is the same in every experiment. However, for longer delays the spatial phase of the interference shows random fluctuations from one experiment to the other. This phase diffusion is characterized quantitatively.

DOI
10.1103/physrevlett.78.4675

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We analyze theoretically an experiment in which a trapped Bose-Einstein condensate is cut in half, and the parts are subsequently allowed to interfere. If the delay between cutting and atom detection is small, the interference pattern of the two halves of the condensate is the same in every experiment. However, for longer delays the spatial phase of the interference shows random fluctuations from one experiment to the other. This phase diffusion is characterized quantitatively.
Recently Javanainen and Wilkens [Phys. Rev. Lett. 78, 4675 (1997)] have analysed an experiment in which an interacting Bose condensate, after being allowed to form in a single potential well, is "cut" by splitting the well adiabatically with a very high potential barrier, and estimate the rate at which, following the cut, the two halves of the condensate lose the "memory" of their relative phase. We argue that, by neglecting the effect of interactions in the initial state before the separation, they have overestimated the rate of phase randomization by a numerical factor which grows with the interaction strength and with the slowness of the separation process.
We investigate the phase diffusion of a Bose–Einstein condensate (BEC) confined in the combined potential of a magnetic trap and a one-dimensional optical lattice. We show that the phase diffusion of the condensate in the whole optical lattice is evident and can be measured from the interference pattern of expanding subcondensates after the optical lattice is switched off. The maximum density of the interference pattern decreases significantly due to the mixing of the phase diffusion that appears in different subcondensates. This suggests a way to detect experimentally the notable phase diffusion of BECs.
We study phase diffusion in a Bose--Einstein condensate of light in a dye-filled optical microcavity, i.e., the spreading of the probability distribution for the condensate phase. To observe this phenomenon, we propose an interference experiment between the condensed photons and an external laser. We determine the average interference patterns, considering quantum and thermal fluctuations as well as dissipative effects due to the dye. Moreover, we show that a representative outcome of individual measurements can be obtained from a stochastic equation for the global phase of the condensate.
We show that, when two Bose-Einstein condensates are combined, a spatial interference pattern ensues even if the condensates are in number states with no spontaneously broken gauge symmetry. In our measurement-theoretical approach the interference pattern arises from correlations between the detected positions of the atoms. We also simulate numerically experiments intended to distinguish number-state condensates and spontaneous symmetry breaking, the latter being modelled by coherent-state condensates. To detect the difference in the pair correlation functions, the experiment has to be repeated about a thousand times even for condensates with as few as five atoms each.
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.