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arXiv:cond-mat/0411228 · Physical Review A

Phase-matched four wave mixing and quantum beam splitting of matter waves in a periodic potential

Karen Marie Hilligsøe, Klaus Mølmer

Published Apr 12, 2005• 48 citations
arXivHTMLPDF

Abstract

We show that the dispersion properties imposed by an external periodic potential ensure both energy and quasimomentum conservation such that correlated pairs of atoms can be generated by four wave mixing from a Bose-Einstein condensate moving in an optical lattice potential. In our numerical solution of the Gross-Pitaevskii equation, a condensate with initial quasimomentum ${k}_{0}$ is transferred almost completely $(>95%)$ into a pair of correlated atomic components with quasimomenta ${k}_{1}$ and ${k}_{2}$, if the system is seeded with a smaller number of atoms with the appropriate quasimomentum ${k}_{1}$.

DOI
10.1103/physreva.71.041602

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We show that the dispersion properties imposed by an external periodic potential ensure both energy and quasimomentum conservation such that correlated pairs of atoms can be generated by four wave mixing from a Bose-Einstein condensate moving in an optical lattice potential. In our numerical solution of the Gross-Pitaevskii equation, a condensate with initial quasimomentum k0{k}_{0}k0​ is transferred almost completely (>95(>95%)(>95 into a pair of correlated atomic components with quasimomenta k1{k}_{1}k1​ and k2{k}_{2}k2​, if the system is seeded with a smaller number of atoms with the appropriate quasimomentum k1{k}_{1}k1​.
A recent experiment [Deng et al., Nature (London) 398, 218 (1999)] demonstrated four-wave mixing of matter wave packets created from a Bose-Einstein condensate. The experiment utilized light pulses to create two high-momentum wave packets via Bragg diffraction from a stationary Bose-Einstein condensate. The high-momentum components and the initial low-momentum condensate interact to form a new momentum component due to the nonlinear self-interaction of the bosonic atoms. We develop a three-dimensional quantum- mechanical description, based on the slowly-varying-envelope approximation, for four-wave mixing in Bose-Einstein condensates using the time-dependent Gross-Pitaevskii equation. We apply this formulation to describe the experimental observations and to make predictions. We examine the role of phase-modulation, momentum, and energy conservation (i.e., phase matching), and particle number conservation in four-wave mixing of matter waves, and develop simple models for understanding our numerical results.
We demonstrate atomic four-wave mixing of two-component matter waves in a collinear geometry. Starting from a single-species Bose-Einstein condensate, seed and pump modes are prepared through microwave state transfer and state-selective Kapitza-Dirac diffraction. Four-wave mixing then populates the initially empty output modes. Simulations based on a coupled-mode expansion of the Gross-Pitaevskii equation are in very good agreement with the experimental data. We show that four-wave mixing can play an important role in studies of bosonic mixtures in optical lattices. Moreover, our system should be of interest in the context of quantum atom optics.
The advent of the laser as an intense, coherent light source gave birth to nonlinear optics, which now plays an important role in many areas of science and technology. One of the first applications of nonlinear optics was the production of coherent light of a new frequency by multi-wave mixing of several optical fields in a nonlinear medium. Until the experimental realization of Bose-Einstein Condensation (BEC) there had been no intense coherent source of matter-waves analogous to the optical laser. FEC has already been exploited to produce a matter-wave `laser' leading to the threshold of a new field of physics: nonlinear atom optics. Recently the first experiment in nonlinear atom optics was reported: the observation of coherent four wave mixing in which three sodium matter waves mix to produce a fourth. The experiment utilized light pulses to create two high-momentum wavepackets via Bragg diffraction from a stationary Bose- Einstein condensate. The high-momentum components and the remaining zero momentum condensate component interact to form a new momentum component due to the nonlinear self- interaction of the bosonic atoms. We develop a quantum mechanical description, based on the slowly-varying-envelope approximation to the time-dependent nonlinear Schrodinger equation (also called the Gross-Pitaevskii equation), to describe four-wave mixing in Bose-Einstein condensates and apply this description to understand the experimental observations and to make new predictions. We examine the role of phase-modulation, momentum and energy conservation (i.e., phase-matching), and particle number conservation in four-wave mixing of matter waves.
We study the existence of modulated amplitude waves with non-trivial phase of a quasi-1D multicomponent Bose–Einstein condensate (BEC) in the presence of an external periodic potential. Mathematically, such coherent structures are doubly periodic solutions, in space and time, of a coupled system of Gross–Pitaevskii equations. For a binary BEC, the weak interaction regime is tackled by means of the averaging method and regular perturbation theory. The case of strong particle interaction is covered by a simple rescaling argument. One of the components is stationary, while the second component has non-trivial phase, meaning that there is circulation of matter only of the second component. The case of three components is briefly discussed as well.
The advent of the laser as an intense, coherent light source gave birth to nonlinear optics, which now plays an important role in many areas of science and technology. One of the first applications of nonlinear optics was the production of coherent light of a new frequency by multi-wave mixing of several optical fields in a nonlinear medium. Until the experimental realization of Bose-Einstein Condensation (BEC) there had been no intense coherent source of matter-waves analogous to the optical laser. BEC has already been exploited to produce a matter-wave 'laser' atom optics was reported: the observation of coherent four wave mixing in which three sodium matter waves mix to produce a fourth. The experiment utilized light pulses to create two high-momentum wavepackets via Bragg diffraction from a stationary Bose- Einstein condensate. The high-momentum components and the remaining zero momentum condensate component interact to form a new momentum component due to the nonlinear self- interaction of the bosonic atoms. We develop a quantum mechanical description, based on the slowly-varying-envelope approximation to the time-dependent nonlinear Schroedinger equation, to describe four-wave mixing in Bose-Einstein condensates and apply this description to understand the experimental observations and to make new predictions. We examine the role of phase-modulation, momentum and energy conservation, and particle number conservation in four-wave mixing of matter waves.