Master equation and control of an open quantum system with leakage.

Given a multilevel system coupled to a bath, we use a Feshbach P, Q partitioning technique to derive an exact trace-nonpreserving master equation for a subspace S_{i} of the system. The resultant equation properly treats the leakage effect from S_{i} into the remainder of the system space. Focusing on a second-order approximation, we show that a one-dimensional master equation is sufficient to study problems of quantum state storage and is a good approximation, or exact, for several analytical models. It allows a natural definition of a leakage function and its control and provides a general approach to study and control decoherence and leakage. Numerical calculations on an harmonic oscillator coupled to a room temperature harmonic bath show that the leakage can be suppressed by the pulse control technique without requiring ideal pulses.