Nonadiabatic elimination of auxiliary modes in continuous quantum measurements

In quantum measurement or control processes, there are often auxiliary modes coupling to the quantum system that we are interested in—they together form a bath or an environment for the system. The bath can have finite memory (non-Markovian), and simply ignoring its dynamics (i.e., adiabatically eliminating it) will prevent us from predicting the true quantum behavior of the system. We generalize the technique introduced by Strunz et al. [Phys. Rev. Lett. 82, 1801 (1999)], and develop a formalism that allows us to eliminate the bath nonadiabatically in continuous quantum measurements, and obtain a non-Markovian stochastic master equation for the system that we focus on. This formalism also illuminates how to design the bath—acting as a quantum filter—to effectively probe interesting system observables (e.g., the quantum-nondemolition observable).

Figure 1

Schematics of our measurement process. The system is coupled to the bath, which in turn couples to an external probe field. The output probe field amplitude is projectively measured by a detector.

Figure 2

Schematics showing the atom-cavity system. A two-level atom (or a qubit) interacts with a cavity mode that is coupled to an external continuous optical field which is measured via homodyne detection.

Figure 3

The top panel shows simulation results for the time evolution of $\langle {\sigma }_x\rangle$, $\langle {\sigma }_y\rangle ,$ and $\langle {\sigma }_z\rangle$ for one realization of $dW$. The bottom panel is the convergency of the accumulated numerical difference between the SSE and SME simulation results given different number of grid points for the cavity mode.