We investigate the prospects of atomic interference using samples of Bose condensed atoms. First we show the ability of two independent Bose condensates to create an interference pattern. This holds even if both condensates are described by Fock states. Thus, the existence of an experimental signature for a broken gauge symmetry, seen in a single run of the experiment, is not necessarily reflected by a broken symmetry on the level of the quantum mechanical state vector. Based on these results, we simulate numerically a recent experiment with two independent Bose condensates [K. B. Davis et al., Phys. Rev. Lett. 75, 3969 (1995)]. The existence of interference fringes is predicted based on the nonlinear Schr"odinger equation. Finally we study theoretically the influence of finite temperatures on the visibility of the interference in a double pinhole configuration. \textcopyright{} 1996 The American Physical Society.
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.
We study theoretically the macroscopic interference of two independent Bose condensates released from a double potential trap. The observation of fringes could serve as a test for the paradigm of broken gauge symmetry. By numerical solution of the nonlinear Schr\"odinger equation in three dimensions, the consecutive stages of expansion, overlap, and interference are investigated in order to facilitate the design of future experiments. The phase-space dynamics of the condensates is analyzed by means of the Wigner function. It turns out that the distance of interference fringes grows linearly in time with a velocity inversely proportional to the initial distance of the two condensates. The collisional reduction of the fringe visibility is estimated.
The macroscopic interference of two Bose condensates released from a double minimum potential has been demonstrated recently [M. R. Andrews et al., Science 275, 637 (1997)]. In this Letter we show the excellent agreement between those experiments and theoretical predictions based on the nonlinear Schr\"odinger equation. In addition, the transition from interference of coupled condensates, comparable with the Josephson effect in superconductors, to the interference of independent Bose condensates is studied.
We investigate the quantum interference between two Bose-Einstein condensates formed in small atomic samples composed of a few thousand atoms both by imposing Bose broken gauge symmetry from the outset and also using an explicit model of atomic detection. In the former case we show that the macroscopic wave function collapses and revives in time, and we calculate the characteristic times for current experiments. Collapses and revivals are also predicted in the interference between two Bose-Einstein condensates which are initially in Fock states, a relative phase between the condensates being established via atomic detections corresponding to uncertainty in the number difference between them.
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
T, 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
T, 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=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, meaning that interference pattern is certainly observed on every snapshot below
Tc. The fact that the number of atoms is fixed in the canonical ensemble is crucial to this vanishing fluctuation.
The visibility of the density interference pattern of two Bose-Einstein condensates, which are produced in traps and overlap after the trapping potential is switched off, is investigated. Coherent wave packets are used to describe the order parameter in a second-quantization formalism. This results in a decrease of the visibility of the interference fringes with increasing time delay between the formation of the condensates and the observation of interference. In the two limiting cases of ideal and very dense gases the correlation time increases, τ →∞, and the result is identical to that obtained using an approach based on the Gross-Pitaevskii equation. Under the conditions of the experiment performed by M. R. Andrews, C. G. Townsend, H. J. Miesner et al. , Science 275 , 6367 (1997), the computed correlation time τ ≈0.2 s is much longer than the confinement time of the condensate, and it is possible to observe the predicted decrease of visibility of the interference fringes of the density of the atoms.