Non-Markovian homodyne-mediated feedback on a two-level atom: a quantum trajectory treatment

Quantum feedback can stabilize a two-level atom against decoherence (sponta neous emission), putting it into an arbitrary (specified) pure state. This requires perfect homodyne detection of the atomic emission, and instantaneo us feedback. Inefficient detection was considered previously by two of us. Here we allow for a non-zero delay time tau in the feedback circuit. Becaus e a two-level atom is a non-linear optical system, an analytical solution i s not possible. However, quantum trajectories allow a simple numerical simu lation of the resulting non-Markovian process. We find the effect of the ti me delay to be qualitatively similar to chat of inefficient detection. The solution of the non-Markovian quantum trajectory will not remain fixed, so that the time-averaged state will be mixed, not pure. In the case where one tries to stabilize the atom in the excited state, an approximate analytica l solution to the quantum trajectory is possible. The result, that the puri ty (P = 2Tr[rho (2)] - 1) of the average state is given by P = 1 - 4y tau ( where gamma is the spontaneous emission rate) is found to agree very well w ith the numerical results. (C) 2001 Elsevier Science B.V. All rights reserv ed.