Quantum fluctuations and entanglement in the collective atomic recoil laser using a Bose-Einstein condensate

We present a quantum description of the interaction between a Bose-Einstein condensate and a single-mode quantized radiation field in the presence of a strong far-off-resonant pump laser. In the linear regime, the atomic medium is described approximately by two momentum states coupled to the radiation mode. We calculate the evolution of the operators in the Heisenberg picture and their expectation values, such as average and variance of the occupation numbers, atom-atom and atom-field correlations, and two-mode squeezing parameters. Then, we disentangle the evolution operator and obtain the exact evolution of the state vector in the linear regime. This allows us to demostrate that the system can be atom-atom or atom-field thermally entangled. We define the quasiclassical and the quantum recoil limits, for which explicit expressions of the average population numbers are obtained.