Interference of Bose-Einstein Condensates in Momentum Space

Abstract

We suggest an experiment to investigate interference effects between two spatially separated Bose-Einstein condensates. In the presence of coherence, the dynamic structure factor, measurable through inelastic photon scattering, is predicted to exhibit, at high momentum transfer q, interference fringes with frequency period Dn q͞md, where d is the distance between the condensates. We show that this coherent configuration corresponds to an eigenstate of the physical observable measured in the experiment and that the relative phase of the condensates can be, hence, created through the measurement process. PACS numbers: 05.30.Jp, Dilute Bose-Einstein condensed gases behave as classical matter waves. This remarkable feature has been directly confirmed by several recent experiments [1][2][3][4]. In particular, very clean interference patterns generated by two overlapping condensates have been observed through absorption imaging techniques [1]. Interference phenomena produced by matter waves are key features underlying the quantum mechanical behavior of matter, so it is of considerable interest to understand the new role played by Bose-Einstein condensation (BEC).Bose-Einstein condensed gases can be regarded as classical objects because, according to Bogoliubov prescription, the corresponding field operator can be replaced by a classical field, resembling the classical limit of quantum electrodynamics. Differently from the case of the electromagnetic field which is governed by the Maxwell equations, the field associated with a Bose-Einstein condensate obeys equations of quantum nature which reduce, for dilute and cold gases, to the Gross-Pitaevskii equation [5]:where V ext is the external potential confining the gas, and g 4ph 2 a͞m is the interaction coupling constant, fixed by the s-wave scattering length a. The field C has the meaning of an order parameter and is often called the "wave function of the condensate." As a consequence of the quantum nature of Eq. (1), key features of the field C, as, for example, interference patterns, depend explicitly on the value of the Planck constant. The fact that two overlapping condensates behave as coherent matter waves giving rise to interference is not, however, obvious. In a similar context Anderson [6] raised the intriguing question: "Do two superfluids which have never seen one another possess a definitive phase?" This question has been the object of theoretical speculations [7] and has been more recently reconsidered [8][9][10][11][12] after the experimental realization of BEC in trapped atomic gases. The point of view shared by most authors is that the relative phase between two condensates is "created" during the measurement. In other words, even if the initial configuration is not coherent and the relative phase between the condensates is not fixed, one can still observe interference in a single realization of the experiment, as explicitly proven in [1]. However, measuring fringe patterns in the density profiles requires overlapping of the condensates in coordinate spac...

Expansion of a Coherent Array of Bose-Einstein Condensates

“…The momentum distribution of the whole system, given by n(p x ) =| Ψ(p x ) | 2 , is affected in a profound way by the lattice structure and exhibits distinctive interference phenomena. Actually the effects of coherence are even more dramatic than in the case of two separated condensates [6]. Indeed, in the presence of the lattice the momentum distribution is characterized by sharp peaks at the values p x = n2πh/d with n integer (positive or negative) whose weight is modulated by the function n 0 (p x ).…”

Phase coherence and fragmentation in weakly interacting bosonic gases

“…If atoms originating from the initially independent clouds interact, it is not even clear in what sense, at the measurement time, we can talk about condensates that have not seen each other. Since fragmented states are not robust to certain classes of perturbations, it could well happen that the interaction itself would lead to phase localization [10]. Moreover, for repulsive interactions a typically phase-coherent ground state is energetically favored [9,11].…”

Thermal vs Quantum Decoherence in Double Well Trapped Bose-Einstein Condensates

“…For example if one calculates the in situ momentum distribution of two separated condensates with a fixed relative phase Φ, one finds fringes in momentum space given by the expression [9] …”