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 ${\mathcal{S}}_i$ of the system. The resultant equation properly treats the leakage effect from ${\mathcal{S}}_i$ 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.

Figure 1

(a) (upper solid curve) $L(t)$ in units of ${\lambda }^2$ for an Harmonic oscillator, in the initial state, $|\varphi ⟩=\frac{1}{\sqrt{2}}(|0⟩+|1⟩)$ coupled to a bath of 1000 harmonic oscillators at $T=300\mathrm{K}$. Here, ${\omega }_d=0.01(fs{)}^{-1}\approx 1.5\Omega$. The dashed curve shows the dominant part, as discussed in the text. The three lower solid curves show $L(t)$ with $\Delta =0$, ${\varphi }_0=\pi$ and $\tau =\frac{\pi }{20\Omega }$, $\frac{\pi }{10\Omega }$ and $\frac{\pi }{5\Omega }$, in the order from bottom to top. (b) $L(t)$ with $\tau =\frac{\pi }{10\Omega }$ and ${\varphi }_0=\pi$, but different values of $\Delta$. Solid curve $\Delta =\tau /5$; Dashed curve: $\Delta =\tau /2$; Dotted curve: $\Delta =\tau$. (c) $L(t)$ with $\tau =\frac{\pi }{10\Omega }$ and $\Delta =\frac{\tau }{2}$ but different pulse intensities, ${\varphi }_0=\frac{\pi }{10}$, $\frac{\pi }{5}$, $\frac{\pi }{2}$ from top to bottom. (d) two lower curves, $L(t)$ with ${\varphi }_0=\pi /100$, $\tau ={\tau }_0/100$ (${\tau }_0=\frac{{\varphi }_0}{10\Omega }$) and ${\varphi }_0=\pi /10$ and $\tau ={\tau }_0/10$ all with $\Delta =\tau /2$. They have the same value of ${\Omega }_c$. The upper curves have the same values of ${\varphi }_0$ and $\tau$ but different signs of ${\varphi }_0$.