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Physical Review A

Enhancement of four-wave mixing in Bose-Einstein condensate by introducing a mismatch

E. Infeld, Marek Trippenbach

Published Jul 22, 2003• 3 citations
Publisher

Abstract

A recent experiment demonstrated four-wave mixing of wave packets in a sodium Bose-Einstein condensate [Deng et al., Nature (London) 398, 218 (1999)]. This was followed by a theoretical and numerical treatment of the experiment [Trippenbach et al., Phys. Rev. A 62, 023608 (2000)]. In the experiment, a short time of free expansion of the condensate, after it was released from the magnetic trap, was followed by a set of two Bragg pulses, which created moving wave packets. These wave packets, due to nonlinear interaction and under phase-matching conditions created a new momentum component in a four-wave mixing process. Here we suggest that, instead of exactly matching the frequencies as in the above-mentioned experiments, we admit a small mismatch in energies, and therefore frequencies $\ensuremath{\Delta}\ensuremath{\omega}.$ We show that such a small mismatch can compensate for the initial phases built on the condensate during free expansion. A physical explanation is offered. This compensation can be beneficial for the efficiency of the four-wave mixing, in some cases even increasing it by a factor of 2.

DOI
10.1103/physreva.68.015603

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A recent experiment demonstrated four-wave mixing of wave packets in a sodium Bose-Einstein condensate [Deng et al., Nature (London) 398, 218 (1999)]. This was followed by a theoretical and numerical treatment of the experiment [Trippenbach et al., Phys. Rev. A 62, 023608 (2000)]. In the experiment, a short time of free expansion of the condensate, after it was released from the magnetic trap, was followed by a set of two Bragg pulses, which created moving wave packets. These wave packets, due to nonlinear interaction and under phase-matching conditions created a new momentum component in a four-wave mixing process. Here we suggest that, instead of exactly matching the frequencies as in the above-mentioned experiments, we admit a small mismatch in energies, and therefore frequencies \ensuremathΔ\ensuremathω.\ensuremath{\Delta}\ensuremath{\omega}.\ensuremathΔ\ensuremathω. We show that such a small mismatch can compensate for the initial phases built on the condensate during free expansion. A physical explanation is offered. This compensation can be beneficial for the efficiency of the four-wave mixing, in some cases even increasing it by a factor of 2.
A recent experiment demonstrated four-wave mixing of wavepackets in a sodium Bose–Einstein condensate (Deng et al 1999 Nature 398 218). This was followed by a theoretical and numerical treatment of the experiment (Trippenbach et al 2000 Phys. Rev. A 62 02368). In the experiment, a short period of free expansion of the condensate, after release from the magnetic trap, was followed by a set of two Bragg pulses which created moving wavepackets. These wavepackets, due to nonlinear interaction and under phase-matching conditions, created a new momentum component in a four-wave mixing process. We propose simple mathematical models for this process. Next we suggest that, instead of exactly matching the frequencies as in the abovementioned experiments, we introduce a small mismatch in the energies, and therefore the frequencies Δω. We show that such a small mismatch can compensate for the initial phases that are built on the condensate during free expansion. A physical explanation is offered. This compensation can improve the efficiency of four-wave mixing; in some cases even increasing it by a factor of 2. We also deal with the situation where two strong wavepackets are accompanied by a weak input beam applied as a seed both with and without a mismatch. Here the influence of the mismatch is less obviously beneficial. We also comment on recent work by Ketterle's group (Vogels et al 2002 Phys. Rev. Lett. 89 020401).
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. 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 calculate the four-wave mixing (FWM) in a Bose-Einstein condensate system having multiple spin wave packets that are initially overlapping in physical space, but have nonvanishing relative momentum that causes them to recede from one another. Three receding condensate atom wave packets can result in production of a fourth wave packet by the process of FWM due to atom-atom interactions. We consider cases where the four final wave packets are composed of one, two, three, and four different internal spin components. FWM with one or two-spin state wave packets is much stronger than three- or four-spin state FWM, wherein two of the coherent moving Bose-Einstein condensate wave packets form a spin-polarization grating that rotates the spin projection of the third wave into that of the fourth diffracted wave (as opposed to the one- or two-spin state case where a regular density grating is responsible for the diffraction). Calculations of FWM for 87Rb^{87}\mathrm{Rb}87Rb and 23Na^{23}\mathrm{Na}23Na condensate systems are presented.
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.