Adaptive Resummation of Markovian Quantum Dynamics

We introduce a method for obtaining analytic approximations to the evolution of Markovian open quantum systems. It is based on resumming a generalized Dyson series in a way that ensures optimal convergence even in the absence of a small parameter. The power of this approach is demonstrated by two benchmark examples: the spatial detection of a free particle and the Landau-Zener problem in the presence of dephasing. The derived approximations are asymptotically exact and exhibit errors on the per mill level over the entire parameter range.

Figure 1

Reflection probability of a free particle with kinetic energy $E_{\mathrm{in}}$ at the boundary of a projective left-right measurement with rate $\gamma$. Open dots: numerically exact values; solid line: prediction (14) of the adaptive jump expansion; dashed line: leading order approximation (11) [18]. The result (14) is asymptotically exact, with a maximal deviation (at $E_{\mathrm{in}}/\hslash \gamma \approx 0.2$) of below 1%.

Figure 2

Landau-Zener tunneling probability $P$ as a function of dephasing rate $\gamma$ and adiabaticity parameter $\delta$. Numerically exact results (dots) are compared to the prediction $P=\sum _n{P}_{2n+1}$ from the jump expansion (black lines), with the approximation (19) and parameters ${\Lambda }_0=2\pi \delta$, ${\Lambda }_1=\gamma {\tau }^{*}$, and ${\tau }^{*}$ in (20). The deviation does not exceed 0.5% over the entire parameter range.