We study first and second order coherence of trapped dilute Bose gases using appropriate correlation functions. Special attention is given to the discussion of second order or density correlations. Except for a small region around the surface of a Bose-Einstein condensate the correlations can be accurately described as those of a locally homogeneous gas with a spatially varying chemical potential. The degrees of first and second order coherence are therefore functions of temperature, chemical potential, and position. The second order correlation function is governed both by the tendency of bosonic atoms to cluster and by a strong repulsion at small distances due to atomic interactions. In present experiments both effects are of comparable magnitude. Below the critical temperature the range of the bosonic correlation is affected by the presence of collective quasi-particle excitations. The results of some recent experiments on second and third order coherence are discussed. It is shown that the relation between the measured quantities and the correlation functions is much weaker than previously assumed.Comment: RevTeX, 25 pages with 7 Postscript figure
We use continuous measurement theory to describe the evolution of two Bose condensates in an interference experiment. It is shown how in a single run the system evolves into a state with a fixed relative phase, without violating particle number conservation. ͓S1050-2947͑96͒07810-9͔ PACS number͑s͒: 03.75.Fi, 03.65.Bz, 05.30.Jp, 42.50.Ar Recent observation of Bose-Einstein condensation ͑BEC͒ ͓1-3͔ has initiated theoretical discussions regarding the properties of Bose condensates ͓4͔. Of particular interest have been questions related to the phase of the condensate ͓5͔.The assumption of such a phase as a result of a broken gauge symmetry allows for a natural explanation of many physical phenomena, but also implies that the state of the condensate is a linear superposition of states with different particle numbers. However, in second quantized formalism of nonrelativistic quantum mechanics, as is usually employed to describe BEC, all observables commute with the atomic number operator N , which thus plays the role of a superselection rule. As a consequence, starting from a state with fixed atomic number or a mixed state, which is diagonal in the atomic number basis, no atomic superpositions and coherences will develop ͓6͔.In order to resolve this seeming contradiction, we use in this paper the language of continuous measurement theory ͓7͔ to describe a single realization of an interference experiment between two independent condensates. We will discuss how the state of the two condensates evolves as atoms are detected. In particular, from our analysis it follows how a state of well defined relative phase builds up dynamically in a single experimental run as a consequence of the von Neumann projection postulate of quantum theory. We emphasize that our description does not contradict atomic number superselection rules, and allows one to understand the coexistence of both particle number conservation and the phase of a condensate in a general situation. The problem addressed in the present paper is related to very recent work by Javanainen and Yoo ͓8͔, and Naraschewski et al. ͓9͔, where it is shown that two independent Bose condensates prepared in Fock states may form a measurable interference pattern. However, here we will show how the interference pattern is formed dynamically, and, what is more important, how the state of the two condensates collapses during the process of detection of the many individual atoms.We consider the situation depicted in Fig.
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., PRL 75, 3969 (1995)]. The existence of interference fringes is predicted based on the nonlinear Schrödinger equation. Finally we study theoretically the influence of finite temperatures on the visibility of the interference in a double pinhole configuration. 03.75.Fi,05.30.Jp
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