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arXiv:1901.06173 · New Journal of Physics

Four-wave mixing in spin–orbit coupled Bose–Einstein condensates

Nguyen Viet Hung, Piotr Szańkowski, Vladimir V Konotop, Marek Trippenbach

Published Mar 13, 2020• 8 citations
arXivHTMLPDF

Abstract

Abstract We describe possibilities of spontaneous, degenerate four-wave mixing (FWM) processes in spin–orbit coupled Bose–Einstein condensates. Phase matching conditions (i.e., energy and momentum conservation laws) in such systems allow one to identify four different configurations characterized by involvement of distinct spinor states in which such a process can take place. We derived these conditions from first principles and then illustrated dynamics with direct numerical simulations. We found, among others, the unique configuration, where both probe waves have smaller group velocity than pump wave and proved numerically that it can be observed experimentally under proper choice of the parameters. We also reported the case when two different FWM processes can occur simultaneously. The described resonant interactions of matter waves is expected to play an important role in the experiments of BEC with artificial gauge fields. Beams created by FWM processes are an important source of correlated particles and can be used in the experiments testing quantum properties of atomic ensembles.

DOI
10.1088/1367-2630/ab7fcb

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Abstract We describe possibilities of spontaneous, degenerate four-wave mixing (FWM) processes in spin–orbit coupled Bose–Einstein condensates. Phase matching conditions (i.e., energy and momentum conservation laws) in such systems allow one to identify four different configurations characterized by involvement of distinct spinor states in which such a process can take place. We derived these conditions from first principles and then illustrated dynamics with direct numerical simulations. We found, among others, the unique configuration, where both probe waves have smaller group velocity than pump wave and proved numerically that it can be observed experimentally under proper choice of the parameters. We also reported the case when two different FWM processes can occur simultaneously. The described resonant interactions of matter waves is expected to play an important role in the experiments of BEC with artificial gauge fields. Beams created by FWM processes are an important source of correlated particles and can be used in the experiments testing quantum properties of atomic ensembles.
We experimentally investigate four-wave mixing (FWM) of matter waves in two geometric configurations in 39^{39}39K Bose-Einstein condensates with the atomic interaction tuned via Feshbach resonances. For one configuration with the single-spin component, the FWM yield increases with a larger scattering length. For the two-spin component configuration, we specifically investigate FWM in both the droplet and gas parameter regimes. We find that the FWM yield reaches its maximum near the critical parameter region between the gas and droplet phases. Our research can help to optimize the FWM yield for matter-wave amplification and entangled atom pair generation, making it conducive to applications in quantum information processing and precision measurement.
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 nonlinear coupling term in the Gross-Pitiaevski equation which describes a Bose-Einstein condensate (BEC) can cause four- wave mixing (4WM) if three BEC wavepackets with momenta k1, k2, and k3 interact. The interaction will produce a fourth wavepacket with momentum k4 = k1 + k2. We study this process using numerical models and suggest that experiments are feasible. Conservation of energy and momentum have different consequences for 4WM with massive particles than in the nonlinear optics case because of the different energy-momentum dispersion relations.
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).
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.
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.