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arXiv:0804.1944 · Europhysics Letters (EPL)

Counting statistics of interfering Bose-Einstein condensates

Andrei L. Shelankov, J. Rammer

Published Sep 1, 2008• 5 citations
arXivHTMLPDF

Abstract

A method is presented that is able to predict the probability of outcomes of snapshot measurements, such as the images of the instantaneous particle density distribution in a quantum many-body system. It is shown that a gauge-like transformation of the phase of the many-body wave function allows one to construct a probability generating functional, the Fourier transform of which with respect to the “gauge” field returns the joint probability distribution to detect any given number of particles at various locations. The method is applied to the problem of interference of two independent clouds of Bose-Einstein condensates, where the initially separated clouds with fixed boson numbers expand and the density profile image of the overlapping clouds is registered. In the limit of large particle numbers, the probability to observe a particular image of the density profile is shown to be given by a sum of partial probability distributions, each of which corresponds to a noisy image of interference of two matter waves with definite phase difference. In agreement with earlier theoretical arguments, interference fringes are, therefore, expected in any single shot measurement, the fringe pattern randomly varying from run to run. These results conform to the physical picture where the Bose-Einstein clouds are in spontaneously symmetry broken states, the hidden phases of which are revealed by the density profile measurement via the position of the interference fringes.

DOI
10.1209/0295-5075/83/60002

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A method is presented that is able to predict the probability of outcomes of snapshot measurements, such as the images of the instantaneous particle density distribution in a quantum many-body system. It is shown that a gauge-like transformation of the phase of the many-body wave function allows one to construct a probability generating functional, the Fourier transform of which with respect to the “gauge” field returns the joint probability distribution to detect any given number of particles at various locations. The method is applied to the problem of interference of two independent clouds of Bose-Einstein condensates, where the initially separated clouds with fixed boson numbers expand and the density profile image of the overlapping clouds is registered. In the limit of large particle numbers, the probability to observe a particular image of the density profile is shown to be given by a sum of partial probability distributions, each of which corresponds to a noisy image of interference of two matter waves with definite phase difference. In agreement with earlier theoretical arguments, interference fringes are, therefore, expected in any single shot measurement, the fringe pattern randomly varying from run to run. These results conform to the physical picture where the Bose-Einstein clouds are in spontaneously symmetry broken states, the hidden phases of which are revealed by the density profile measurement via the position of the interference fringes.
We have discussed the interference of two independently prepared Bose-Einstein condensates. When the number of atoms in each gas is fixed and the phases of the gases are uncertain, no interference is expected in the single-particle distribution. However, the interference is observed in each single snapshot. We have presented tools for studying such snapshot interference, the measurement-induced interference. The idea is to discard the randomness in the snapshot profiles (the spatial offsets of the interference patterns) and to focus on the typical feature in the interference patterns (the fringe spacing). If the sinusoidal patterns with a definite fringe spacing is actually typical, the fringe spectrum with the random offset discarded does not fluctuate, and we are allowed to speak of the snapshot interference through its average. We have described the two gases as canonical ensembles with the numbers of atoms fixed at N individually. We have shown that the covariance of the fringe spectrum over all possible snapshot profiles is vanishingly small below the critical temperature of the Bose-Einstein condensation: the interference pattern whose fringe contrast is characterized by the average fringe spectrum is certainly observed in every snapshot in the presence of condensates. The fringe contrast becomes stronger as the temperature is lowered and the condensation fraction is increased. This clarifies the importance of the Bose-Einstein condensation to the interference of independent Bose-Einstein condensates. The knowledge I acquired from Prof. Tasaki's lecture on how to characterize the canonical ensemble of bosonic atoms has enabled us to carry out this analysis. Like this example, the coherence phenomena of quantum many-body systems in general would be explained on the basis of the idea of the measurement-indued coherence, without resort to the idea of the spontaneous symmetry breaking. This is actually true also for fermionic systems: the relative phase between two independent superconductors is built up by measurement, even though their U(1) symmetries are not broken and their phases are uncertain beforehand. It would be interesting to explore the possibility of explaining various phenomena, in which the symmetry breaking plays an essential role, on the basis of the measurement-induced coherence.
We consider the interference of two overlapping ideal Bose-Einstein condensates. The usual description of this phenomenon involves the introduction of a condensate wave function with a definite phase. We investigate the origin of this phase and the theoretical basis of treating interference. It is possible to construct a phase state for which the particle number is uncertain, but the phase is known. How such a state would be prepared before an experiment is not obvious. We show that a phase can also arise from experiments using condensates with known particle numbers. The analysis of measurements in such states also gives us a prescription for preparing phase states. The connection of this procedure to questions of spontaneously broken gauge symmetry and to hidden variables is discussed.
A formalism for describing the coherence and interference properties of two atomic clouds of Bose-Einstein condensates (BEC) is presented, which is applicable even in the opposite limits when the BEC clouds are initially coherent and when they are initially independent. First, we develop a mean-field theory wherein one mean-field mode is used, and then, for fragmented (i.e., independent) condensates, we use a mean-field theory with two modes. We then develop a full two-mode field theory, with a field operator composed of a sum of two terms containing matter wave mode functions phi1 and phi2, that multiply the destruction operators of the modes, a1 and a2. When atom-atom interactions are present and when the mode functions overlap, the matter wave mode functions phi1 and phi2 develop components moving to the right and left, and this results in interference fringes in the density. At the many-body level, another source of interference arises from expectation values of the form (a(i)+a(j)) with i double dagger j, which become nonzero due to tunneling and interactions. We detail how these two sources of interference affect the density profile and the density-density correlation functions of Bose-Einstein condensates in the coherent and in the fragmented regimes.
Interference of two independently prepared ideal Bose gases is discussed, on the basis of the idea of measurement-induced interference. It is known that, even if the number of atoms in each gas is individually fixed finite and the symmetry of the system is not broken, an interference pattern is observed on each single snapshot. The key role is played by the Hanbury Brown and Twiss effect, which leads to an oscillating pattern of the cloud of identical atoms. Then, how essential is the Bose-Einstein condensation to the interference? In this work, we describe two ideal Bose gases trapped in two separate three-dimensional harmonic traps at a finite temperature TTT, using the canonical ensembles (with fixed numbers of atoms). We compute the full statistics of the snapshot profiles of the expanding and overlapping gases released from the traps. We obtain a simple formula valid for finite TTT, which shows that the average fringe spectrum (average fringe contrast) is given by the purity of each gas. The purity is known to be a good measure of condensation, and the formula clarifies the relevance of the condensation to the interference. The results for T=0T=0T=0, previously known in the literature, can be recovered from our analysis. The fluctuation of the interference spectrum is also studied, and it is shown that the fluctuation is vanishingly small only below the critical temperature Tc{T}_{c}Tc​, meaning that interference pattern is certainly observed on every snapshot below Tc{T}_{c}Tc​. The fact that the number of atoms is fixed in the canonical ensemble is crucial to this vanishing fluctuation.
This chapter describes the Bose--Einstein condensate (BEC) interferometry. We first introduce the optical interferometer, briefly discussing the role of fringe contrast in observing interference patterns. Next, we examine a BEC interferometer in a Michelson geometry that consists of a BEC in a trapping potential well. The BEC cloud in the potential well is put into a linear superposition of two clouds that travel along different paths in the trap and are recombined using the same light beams. By studying the population of atoms in the different motional states of the BEC clouds that emerge after recombination, we obtain information about the relative phase shift accumulated by the two traveling superposed condensates during the interferometry. We then characterize the atom distribution that is found in the output of the interferometer by finding the probability density distribution and calculating the features of the probability density such as the mean, the variance, and the interference fringe contrast. Finally, we parameterize the effect of two-body collisional interactions on the observed interference fringes in a way that can be directly controlled in an experiment.
Interference of an array of independent Bose-Einstein condensates, whose experiment has been performed recently, is theoretically studied in detail. Even if the number of the atoms in each gas is kept finite and the phases of the gases are not well-defined, interference fringes are observed on each snapshot. The statistics of the snapshot interference patterns, i.e. the average fringe amplitudes and their fluctuations (covariance), are computed analytically, and concise formulas for their asymptotic values for long time of flight are derived. Processes contributing to these quantities are clarified and the relationship with the description on the basis of the symmetry-breaking scenario is revealed.