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Optics Letters

Fully quantized theory of four-wave mixing with bosonic matter waves

Ying Wu, Xiaoxue Yang

Published Feb 1, 2005• 21 citations
Publisher

Abstract

We solve exactly a set of fully quantized coupled equations that describe the quantum dynamics, including the evolution of quantum fluctuations, entanglements, and correlations, of four-wave mixing (FWM) with matter waves. The analytical solution reveals Rabi oscillations, collapses and revivals, and agrees with FWM experiments. It also applies to optical FWM and the Rayleigh superradiance in Bose-Einstein condensates.

DOI
10.1364/ol.30.000311

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We solve exactly a set of fully quantized coupled equations that describe the quantum dynamics, including the evolution of quantum fluctuations, entanglements, and correlations, of four-wave mixing (FWM) with matter waves. The analytical solution reveals Rabi oscillations, collapses and revivals, and agrees with FWM experiments. It also applies to optical FWM and the Rayleigh superradiance in Bose-Einstein condensates.
Starting from a second-quantized Hamiltonian of many-particle systems, we derive the Gross-Pitaevskii (GP) equation in momentum space, which is suitable for studying the multi-wave mixing processes of coherent matter waves. The coupling equations are then applied to study ideal four-wave mixing (4WM), in which only four waves with definite wavevectors are involved. Some interesting problems of 4WM, such as the phase-matching condition, the collapse and revival behaviour, the effects of relative phase difference, and the conversion efficiency are discussed in detail. We also show that the main characters of recent 4WM experiment [Deng et al, Nature 398, 218 (1999)] can be undersood in the present simplified model.
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.
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 investigate the four-wave mixing of matter wave packets created from a Bose–Einstein condensate, realized experimentally by utilizing light pulses to create two high-momentum wave packets via Bragg diffraction from a stationary condensate. Based on the Gross–Pitaevskii equation, a set of nonlinearly coupled envelope equations including self- and cross-phase modulational effects are derived systematically using a method of multiple-scales. The exact and explicit analytical solutions are provided for the coupled envelope equations and the evolution of the wave packets after turning off trapping potential is discussed and compared with experiment.
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.