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Communications in Theoretical Physics

Atomic Coherent State, Entangled State and Phase Operator for Studying Interference Between Two Bose–Einstein Condensates

Xubing Tang, Fan Hong‐Yi

Published Nov 1, 2008• 1 citation
Publisher

Abstract

For studying the interference between two Bose–Einstein condensates we introduce the atomic coherent state (ACS) in the Schwinger bosonic realization along with the phase operator to directly calculate the interference pattern with steady relative phase cos ϕ. Eigenstates of the density operator of condensates are classified as ACS is also demonstrated. The entangled state representation is used in some calculations.

DOI
10.1088/0253-6102/50/5/27

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For studying the interference between two Bose–Einstein condensates we introduce the atomic coherent state (ACS) in the Schwinger bosonic realization along with the phase operator to directly calculate the interference pattern with steady relative phase cos ϕ. Eigenstates of the density operator of condensates are classified as ACS is also demonstrated. The entangled state representation is used in some calculations.
Bose–Einstein condensates of particles behave in many ways like coherent radiation fields, and the realization of Bose–Einstein condensation in dilute gases has opened up the experimental study of interactions between coherent matter waves. In addition, the existence of these dilute trapped quantum gases has prompted a re-examination of a number of theoretical issues. In Sec. 13.1 we discuss Josephson tunnelling of a condensate between two wells and the role of fluctuations in particle number and phase. The number and phase variables play a key role in the description of the classic interference experiment, in which two clouds of atoms are allowed to expand and overlap (Sec. 13.2). Rather surprisingly, an interference pattern is produced even though initially the two clouds are completely isolated. Density fluctuations in a Bose gas are studied in Sec. 13.3, where we relate atomic clock shifts to the two-particle correlation function. The ability to manipulate gaseous Bose–Einstein condensates by lasers has made possible the study of coherent matter wave optics and in Sec. 13.4 we describe applications of these techniques to observe solitons, Bragg scattering, and non-linear mixing of matter waves. How to characterize Bose–Einstein condensation in terms of the density matrix is the subject of Sec. 13.5, where we also consider fragmented condensates.
Summary form only given.Due to interatomic collisions, the ground state of a Bose-Einstein condensate of dilute atoms is not, as is commonly assumed, a coherent state. First recognizing this point, Lewenstein and You concluded that the condensate is actually an amplitude quadrature eigenstate. We have extended this work by studying the quantum state for two condensates with a linear coupling provided by either spatial tunneling or laser coupling between magnetic substates. As the system is closed, the ground state is unambiguously a pure state, and as we deal only with relative phases, there is no need to invoke broken symmetry.
Bose–Einstein condensates of particles behave in many ways like coherent radiation fields, and the realization of Bose–Einstein condensation in dilute gases has opened up the experimental study of many aspects of interactions between coherent matter waves. In addition, the existence of these dilute trapped quantum gases has prompted a re-examination of a number of theoretical issues. This field is a vast one, and in this chapter we shall touch briefly on selected topics. In Sec. 13.1 we describe the classic interference experiment, in which two clouds of atoms are allowed to expand and overlap. Rather surprisingly, an interference pattern is produced even though initially the two clouds are completely isolated. We shall analyse the reasons for this effect. The marked decrease in density fluctuations in a Bose gas when it undergoes Bose–Einstein condensation is demonstrated in Sec. 13.2. Gaseous Bose–Einstein condensates can be manipulated by lasers, and this has made possible the study of coherent matter wave optics. We describe applications of these techniques to observe solitons, Bragg scattering, and non-linear mixing of matter waves in Sec. 13.3. The atom laser and amplification of matter waves is taken up in Sec. 13.4. How to characterize Bose–Einstein condensation microscopically is the subject of Sec. 13.5, where we also consider fragmented condensates. Interference of two condensates One of the striking manifestations of the wave nature of Bose–Einstein condensates is the observation of an interference pattern when two condensed and initially separated clouds are allowed to overlap. An example is shown in Fig. 13.1.
We study two aspects of measurement theory in spinor Bose-Einstein condensates of F=1F=1F=1 atoms: the probability of obtaining a certain outcome of the measurement and the evolution of the state of the condensate due to the measurement. We also study the interference patterns arising from the spatial overlap of two spinor condensates. We show that neither a measurement on a small number of escaping atoms nor an interference experiment can distinguish between an antiferromagnetic coherent state condensate, i.e., a condensate in which all the atoms have Sz=0{S}_{z}=0Sz​=0 along an a priori unknown direction, and a spin-singlet condensate, i.e., a condensate with Stotal=0.{S}_{\mathrm{total}}=0.Stotal​=0. We also show that a singlet-state condensate evolves into a coherent state as a result of the measurement.
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.