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

Abstract

Quantum feedback can stabilize a two-level atom against decoherence (spontaneous emission), putting it into an arbitrary (specified) pure state. This requires perfect homodyne detection of the atomic emission, and instantaneous feedback. Inefficient detection was considered previously by two of us. Here we allow for a non-zero delay time $\tau$ in the feedback circuit. Because a two-level atom is a nonlinear optical system, an analytical solution is not possible. However, quantum trajectories allow a simple numerical simulation of the resulting non-Markovian process. We find the effect of the time delay to be qualitatively similar to that 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 analytical solution to the quantum trajectory is possible. The result, that the purity ($P=2\mathrm{Tr}[{\rho }^2]-1$) of the average state is given by $P=1-4\gamma \tau$ (where $\gamma$ is the spontaneous emission rate) is found to agree very well with the numerical results.

Changed content, Added references and Corrected typos

References in corpus (1)

• Quantum Trajectories and Quantum Measurement Theory

Cited by in corpus (8)

• Quantum control theory and applications: A survey
• Laser driven atoms in half-cavities
• Quantum feedback for rapid state preparation in the presence of control imperfections
• Quantum measurements in continuous time, non Markovian evolutions and feedback
• Quantum trajectories: memory and continuous observation
• Robust control of decoherence in realistic one-qubit quantum gates
• Behavior of two-level quantum system driven by non-classical inputs
• Targeting qubit states using open-loop control