Quantum jump dynamics in cavity QED

Abstract

We study the stochastic dynamics of the electromagnetic field in a lossless cavity interacting with a beam of two-level atoms, given that the atomic states are measured after they have crossed the cavity. The atoms first interact at the exit of the cavity with a classical laser field ℰ and then enter into a detector which measures their states. Each measurement disentangles the field and the atoms and changes in a random way the state |ψ(t)〉 of the cavity field. For weak atom-field coupling, the evolution of |ψ(t)〉 when many atoms cross the cavity and the detector is characterized by a succession of quantum jumps occurring at random times, separated by quasi-Hamiltonian evolutions, both of which depend on the laser field ℰ. For ℰ=0, the dynamics is the same as in the Monte Carlo wave function model of Dalibard et al. [Phys. Rev. Lett. 68, 580 (1992)] and Carmichael, An Open System Approach to Quantum Optics, Lecture Notes in Physics Vol. 18 (Springer, Berlin, 1991)]. The density matrix of the quantum field, obtained by averaging the projector |ψ(t)〉〈ψ(t)| over all results of the measurements, is independent of ℰ and follows the master equation of the damped harmonic oscillator at finite temperature. We provide numerical evidence showing that for large ℰ, an arbitrary initial field state |ψ(0)〉 evolves under the monitoring of the atoms and the measurements toward squeezed states |α,re2iφ〉, moving in the α-complex plane but with almost constant squeezing parameters r and φ. The values of r and φ are determined analytically. On the other hand, for ℰ=0, the dynamics transforms the initial state into Fock states |n〉 with fluctuating numbers of photons n, as shown in Kist et al. [J. Opt. B: Quantum Semiclassical Opt. 1, 251 (1999)]. In the last part, we derive the quantum jump dynamics from the linear quantum jump model proposed in Spehner and Bellissard [J. Stat. Phys. 104, 525 (2001)], for arbitrary open quantum systems having a Lindblad-type evolution. A careful derivation of the infinite jump rates limit, where the dynamics can be approximated by a diffusion process of the quantum state, is also presented.