Quantum optical measurements in ultracold gases: Macroscopic Bose-Einstein condensates

We consider an ultracold quantum degenerate gas in an optical lattice inside a cavity. This system represents a simple but key model for quantum optics with quantum gases, where a quantum description of both light and atomic motion is equally important. Due to the dynamical entanglement of atomic motion and light, the measurement of light affects the many-body atomic state as well. The conditional atomic dynamics can be described using the Quantum Monte Carlo Wave Function Simulation method. In this paper, we emphasize how this usually complicated numerical procedure can be reduced to an analytical solution after some assumptions and approximations valid for macroscopic Bose-Einstein condensates (BEC) with large atom numbers. The theory can be applied for lattices with both low filling factors (e.g. one atom per lattice site in average) and very high filling factors (e.g., a BEC in a double-well potential). The purity of the resulting multipartite entangled atomic state is analyzed.