Matter-Wave Mixing by Optical Cavity Fields

Abstract

Recently, there has been a growing interest in matter-wave mixing involving Bose-Einstein condensates (BEC) 1, 2, 3, 4]. We consider a model [1] consisting of a cavity field (probe), an opposing non-depletion pump field, and a matter wave traveling along x normal to the propagation dimension z of the optical fields, which are tuned far away from the atomic transition. Our model is a matter wave version of the collective atomic recoil laser(CARL) 5, 6, 7], which shares the same gain mechanism with recoil-induced resonance (RIR) [7, 8]. Here, virtual two-photon transitions couple the |0> (injected atomic mode) to the |±K>z-momentum modes, where ℏK is the two-photon momentum. If the scattering of the atoms from |±ℏK> to higher order modes is ignored, the equations of motion are reduced into $$ \left[ {\frac{\partial }{{{\partial_t}}} + {\upsilon_{{0x}}}\frac{\partial }{{{\partial_x}}}} \right]{\Phi_0}\left( {x,t} \right) = i{A^{*}}{\Phi_{{ + 1}}}\left( {x,t} \right) + iA{\Phi_{{ - 1}}}\left( {x,t} \right) $$ (1a) $$ \left[ {\frac{\partial }{{{\partial_t}}} + {\upsilon_{{0x}}}\frac{\partial }{{{\partial_x}}}} \right]{\Phi_{{ - 1}}}\left( {x,t} \right) = - i{\omega_{{2r}}}{\Phi_{{ - 1}}}\left( {x,t} \right) + i{A^{*}}{\Phi_0}\left( {x,t} \right) $$ (1b) $$ \left[ {\frac{\partial }{{{\partial_t}}} + {\upsilon_{{0x}}}\frac{\partial }{{{\partial_x}}}} \right]{\Phi_{{ + 1}}}\left( {x,t} \right) = - i{\omega_{{2r}}}{\Phi_{{ + 1}}}\left( {x,t} \right) + iA{\Phi_0}\left( {x,t} \right) $$ (1c) $$ \frac{{dA(t)}}{{dt}} = i{\delta_c}A(t) + i\alpha \frac{1}{{{L_x}}}\int_0^{{{L_x}}} {dx\left[ {\Phi_0^{*}\left( {x,t} \right){\Phi_{{ + 1}}}\left( {x,t} \right){\Phi_0}\left( {x,t} \right)} \right]} - \kappa A(t) $$ (2) where (in0)(x,t) and ±(x,t) are, respectively, the |0> and |±ℏK components of the matter wave, ω2r the two-photon recoil frequency shift, v0x the speed of the injected wave, A(t)the scaled cavity field amplitude, δC the pump frequency shift relative to the cavity mode, α the gain coefficient, κ the decay rate of the cavity, and Lx the effective interaction distance along x dimension. In addition to the probe field, two momentum side modes will be made into “lasing” as a result of nonlinear coupling between the optical and atomic fields. For this reason, we treat this problem as a multimode “laser” operation, where “lasing” can mean both optical and matter waves. The requirement that the beat frequencies from both optical and atomic oscillations must be mode-locked at the steady state means the following steady-state ansatz