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Physical Review A

Second-harmonic generation of Bogoliubov excitations in a two-component Bose-Einstein condensate

Guoxiang Huang, Xin-Qi Li, Jacob Szeftel

Published Jun 24, 2004• 26 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.69.065601

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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 consider a possible second harmonic generation (SHG) of propagating collective excitations in a two-component Bose–Einstein condensate (BEC) with repulsive atom–atom interactions. We show that the phase-matching condition for the SHG can be fulfilled if the wave vectors and frequencies of the excitations are chosen adequately from different dispersion branches. We solve the nonlinear amplitude equations for the SHG derived using a method of multiple-scales and provide SHG solutions similar to those obtained for a SHG in nonlinear optical media. A possible experimental realization of the SHG for the propagating collective modes in a cigar-shaped two-component BEC is also discussed.
Mode-mixing of coherent excitations of a trapped Bose-Einstein condensate is modeled using the Bogoliubov approximation. Calculations are presented for second-harmonic generation between the two lowest-lying even-parity m=0m=0m=0 modes in an oblate spheroidal trap. Hybridization of the modes of the breather (l=0)(l=0)(l=0) and surface (l=4)(l=4)(l=4) states leads to the formation of a Bogoliubov dark state near phase-matching resonance so that a single mode is coherently populated. Efficient harmonic generation requires a strong coupling rate, sharply-defined and well-separated frequency spectrum, and good phase matching. We find that in all three respects the quantal results are significantly different from hydrodynamic predictions. Typically the second-harmonic conversion rate is half that given by an equivalent hydrodynamic estimate.
We theoretically investigate quadrupolar collective excitations in two-component Bose-Einstein condensates and their nonlinear dynamics associated with harmonic generation and mode coupling. Under the Thomas-Fermi approximation and the quadratic polynomial ansatz for density fluctuations, the linear analysis of the superfluid hydrodynamic equations predicts excitation frequencies of three normal modes constituted from monopole and quadrupole oscillations, and those of three scissors modes. These six modes are bifurcated into in-phase and out-of-phase modes by the intercomponent interaction, yielding the nonlinear dynamics that are absent in a single-component condensate. We obtain analytically the resonance conditions for the second-harmonic generation in terms of the trap aspect ratio and the strength of intercomponent interaction. The numerical simulation of the coupled Gross-Pitaevskii equations vindicates the validity of the analytical results and reveals the dynamics of the second-harmonic generation and nonlinear mode coupling that lead to nonlinear oscillations of the condensate with damping and recurrence reminiscent of the Fermi-Pasta-Ulam problem.
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.
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.
Nonlinear excitations in two-component Bose-Einstein condensates (BECs) described by two coupled Gross-Pitaevskii equations are investigated analytically and numerically. The beating phenomenon, the higher-harmonic generation, and the mixing of the excited modes are revealed by both variational approximation and numerical method. The strong excitations induced by the parametric resonance are also studied by time-periodic modulation for the intercomponent interaction. The resonance conditions in terms of the modulation frequency and the strength of intercomponent interaction are obtained analytically, which are confirmed by numerical method. Direct numerical simulations show that, when the resonance takes place, periodic phase separation and multisoliton configurations (including soliton trains, soliton pairs, and multidomain walls) can be excited. In particular, we demonstrate a method for formation of multisoliton configurations through parametric resonance in two-component BECs.