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arXiv:cond-mat/9811175 · Physical Review A

Quantum theory of atomic four-wave mixing in Bose-Einstein condensates

Elena V. Goldstein, Pierre Meystre

Published May 1, 1999• 75 citations
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Abstract

We present an exact quantum-mechanical analysis of collinear four-wave mixing in a multicomponent Bose-Einstein condensate consisting of sodium atoms in the $F=1$ ground state. Technically, this is achieved by taking advantage of the conservation laws of the system to cast its Hamiltonian in terms of angular momentum operators. We discuss explicitly the build-up of matter-wave side modes from quantum fluctuations, as well as the correlations between these modes. We show the appearance of a strong quantum entanglement between hyperfine states. We also demonstrate that for finite atomic numbers, the system exhibits periodic collapses and revivals in the exchange of atoms between different spin states.

DOI
10.1103/physreva.59.3896

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We present an exact quantum-mechanical analysis of collinear four-wave mixing in a multicomponent Bose-Einstein condensate consisting of sodium atoms in the F=1F=1F=1 ground state. Technically, this is achieved by taking advantage of the conservation laws of the system to cast its Hamiltonian in terms of angular momentum operators. We discuss explicitly the build-up of matter-wave side modes from quantum fluctuations, as well as the correlations between these modes. We show the appearance of a strong quantum entanglement between hyperfine states. We also demonstrate that for finite atomic numbers, the system exhibits periodic collapses and revivals in the exchange of atoms between different spin states.
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.
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.
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 consider a trapped multicomponent atomic Bose-Einstein condensate, concentrating specifically on condensates in the hyperfine ground state F=1,F=1,F=1, where spin exchange collisions result in a transfer of population between m=0m=0m=0 and $m=\ifmmode\pm\else\textpm\fi{}1$ internal states. Drawing an analogy with the optical situation, we show that this system can be regarded as a matter-wave analog of optical multiwave mixing. This opens up the way to realize matter-wave phase conjugation, whereby an incident atomic beam can be ``time reversed.'' In addition, matter-wave phase conjugation also offers novel diagnostic tools to study the coherence properties of condensates, as well as to measure the relative scattering lengths of hyperfine sublevels.
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.