Cavity nonlinear optics with few photons and ultracold quantum particles

The light force on particles trapped in the field of a high-Q cavity mode depends on the quantum state of field and particle. Different photon numbers generate different optical potentials anddifferent motional states induce different field evolution. Even for weak saturation and linear polarizability the induced particle motion leads to nonlinear field dynamics. We derive a corresponding effective field Hamiltonian containing all the powers of the photon number operator, which predicts nonlinear phase shifts and squeezing even at the few-photon level. Wave-function simulations of the full particle-field dynamics confirm this and show significant particle-field entanglement in addition.