Chaotic walking of a cold atom in an ideal cavity

Abstract

The interaction of cold atoms with coherent fields involves not only the internal atomic transitions and photon states, but also the center-of-mass motion of atoms.We consider the simple situation: a two-level atom moves inside a high-finesse Fabry-Perot cavity and interacts with a single-mode standing-wave electromagnetic field whose spatial variation creates an one-dimensional optical potential. The respective Hamiltonian $$ \hat{H} = \frac{{{{\hat{p}}^2}}}{{2m}} + \frac{1}{2}\hbar {\varpi_a}{\hat{\sigma }_z} + \hbar {\varpi_f}\left( {{{\hat{a}}^{\dag }}\hat{a} + \frac{1}{2}} \right)\hbar {\Omega_0}\sin (k\hat{x} + \varphi )({\hat{a}^{\dag }}\hat{\sigma } - \hat{a}\hat{\sigma }{}_{ + } $$ (1) generates in the semiclassical approximation the equations of motion $$ \mathop{\zeta }\limits^{.} = \alpha \rho, \mathop{{ \rho }}\limits^{.} = - u\sin \zeta, \mathop{u}\limits^{.} = \delta v,\mathop{{ v}}\limits^{.} = - \delta u + 2Nz\cos \zeta, \mathop{{ z}}\limits^{.} = - 2vocs\zeta $$ (2) for normalized expectation values of the atomic population inversion z= , the position ξ = k f , the momentum ρ =< P < /k, the collective atom-field variables and, where k f is the wave number of the standing wave. The operators are averaged over an initial quantum state of the full system. The set possesses the integral of motion, and three control parameters, the normalized recoil frequency the normalized detuning δ=(ωf−ωa)/Ω0, and the total number of excitations