Stochastic Mean-field Theory for Conditional Spin Squeezing by Homodyne Probing of Atom-Cavity Photon Dressed States
Projective measurements of collective observables can be employed to herald the preparation of entangled states of quantum systems, and the resulting conditional dynamics is usually handled by stochastic master equation (SME) for small systems, and by an approximate Gaussian-state formalism for large systems. In this work, we present an alternative technique by developing a stochastic variant of cumulant mean-field theory, benchmark it against an exact stochastic collective density matrix approach by the simulations of hundreds of identical two-level atoms. More importantly, we demonstrate its full power by studying the conditional spin squeezing of thousands of three-level atoms coupled strongly with an optical cavity subject to individual decay and dephasing, and by simulating the experimental protocol to reveal formation and detection of the spin squeezed state. The proposed technique might be further extended to study more exotic quantum-measurement effects of large quantum systems, such as deterministic spin squeezing with quantum feedback, spin squeezing of optical clock transitions, and retrodictive spin squeezing by posterior measurements, and so on.