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Communications in Theoretical Physics

Three-Wave Resonant Interaction of Bogoliubov Excitations in a Bose–Einstein Condensate with Diffraction

Sun Chun-Liu, Guoxiang Huang

Published May 15, 2004• 2 citations
Publisher

Abstract

We investigate the three-wave resonant interaction (TWRI) of Bogoliubov excitations in a disk-shaped Bose–Einstein condensate with the diffraction of the excitations taken into account. We show that the phase-matching condition for the TWRI can be satisfied by a suitable selection of the wavevectors and the frequencies of the three exciting modes involved in the TWRI. Using a method of multiple-scales we derive a set of nonlinearly coupled envelope equations describing the TWRI process and give some explicit solitary-wave solutions.

DOI
10.1088/0253-6102/41/5/699

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We investigate the three-wave resonant interaction (TWRI) of Bogoliubov excitations in a disk-shaped Bose–Einstein condensate with the diffraction of the excitations taken into account. We show that the phase-matching condition for the TWRI can be satisfied by a suitable selection of the wavevectors and the frequencies of the three exciting modes involved in the TWRI. Using a method of multiple-scales we derive a set of nonlinearly coupled envelope equations describing the TWRI process and give some explicit solitary-wave solutions.
A three-wave resonant interaction of collective modes and related soliton excitations in a disk-shaped Bose–Einstein condensate are investigated. The phase-matching conditions for the resonant interaction are satisfied by suitably choosing the wavevectors and the frequencies of the collective modes. A set of nonlinearly coupled envelope equations describing the spatio-temporal evolution of the three-wave resonant interaction are derived by using a method of multiple-scales, and some explicit (2+1)-dimensional three-wave soliton solutions are also presented and discussed.
We make a systematic study of the resonant mode coupling of the collective excitations at zero temperature in a Bose-Einstein condensate (BEC). (i) Based on the Gross-Pitaevskii equation we derive a set of nonlinearly coupled envelope equations for a three-mode resonant interaction (TMRI) by means of a method of multiple scales. (ii) We calculate the coupling matrix elements for the TMRI and show that the divergence appearing in previous studies can be eliminated completely by using a Fetter-like variational approximation for the ground-state wave function of the condensate. (iii) We provide the selection rules in mode-mode interaction processes [including TMRI and second-harmonic generation (SHG)] according to the symmetry of the excitations. (iv) By solving the nonlinearly coupled envelope equations we obtain divergence-free nonlinear amplitudes for the TMRI and SHG processes and show that our theoretical results on the shape oscillations of the condensate agree well with the experimental ones. We suggest also an experiment to check the theoretical prediction of the present study on the TMRI of collective excitations in a BEC.
A second-harmonic generation (SHG) is predicted for the Bogoliubov excitations in a two-component Bose-Einstein condensate. It is shown that, because the linear dispersion curve of the excitations displays two branches, the phase-matching condition for the SHG can be fulfilled if the wave vectors and frequencies of fundamental and second-harmonic waves are selected suitably from different branches. The nonlinearly coupled envelope equations for the SHG are derived by using a method of multiple scales. The explicit solutions of these envelope equations are provided and the conversion efficiency of the SHG is also discussed.
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 resonant interaction between collective modes in a quasi-one-dimensional (Q1D) attractive Bose–Einstein condensate (BEC) is considered. Nonlinearly coupled amplitude equations are derived using a method of multiple scales. Coupling matrix elements describing mode–mode resonances in both Landau and Beliaev mechanisms are calculated analytically. In particular, second-harmonic generation and three-mode resonant interaction of the collective modes in Q1D BEC are investigated in detail. The result shows that the physics of the resonant mode couplings in attractive Q1D BEC is richer than that of repulsive Q1D BEC.