Optimal Control to Minimize Dissipation and Fluctuations in Open Quantum Systems Beyond Slow and Rapid Regimes

TL;DR

The paper develops a time-local optimal-control framework for open quantum systems described by Lindblad dynamics to minimize both dissipated work and TPM-based work fluctuations beyond traditional slow- and rapid-driving limits. By introducing an auxiliary operator Y(t), the inherently nonlocal TPM variance is converted into a local running cost, enabling gradient-based optimization (GRAPE) of a composite objective that trades off dissipation, fluctuations, and terminal control constraints. Numerical demonstrations on a coherent spin-boson system and a quantum-dot model reveal regime-dependent behaviors, including switches between locally optimal protocol families and multi-step control structures that depart from rapid-drive intuition. The approach provides a practical route to design driving protocols that balance dissipation and fluctuations in open quantum systems, with potential extensions to larger Hilbert spaces and alternative cost functionals.

Abstract

Optimal control is a central problem in quantum thermodynamics. While control theories in the rapid-driving and slow-driving limits have been developed, to the best of our knowledge there is no general optimization method applicable to intermediate timescales. We introduce an optimal-control framework to minimize dissipated work and work variance, defined via the two-point measurement scheme, in open quantum systems governed by time-dependent Lindblad master equations. By introducing an auxiliary operator, we convert the history-dependent work variance into a time-local integral, enabling efficient gradient-based optimization beyond slow or rapid driving regimes. Applying our method, we find that in the coherent spin-boson model the optimized protocol can switch discontinuously between distinct locally optimal solutions as the relative weight between dissipation and fluctuations is varied. Moreover, for a single-level quantum dot coupled to a fermionic reservoir, the optimized fluctuation-minimizing protocol develops a qualitatively different multi-step structure that is not captured by approaches based on slow- or rapid-driving limits.

Figures (7)

• Figure 1: Optimized control field $u(t)$ obtained by the GRAPE algorithm when $\Delta=0$ for several values of the weight parameter $\alpha$ in the cost functional, as indicated in the legend, for protocol duration $T=1$
• Figure 2: Optimized control field $u(t)$ obtained by the GRAPE algorithm when $\Delta=0$ for several values of the weight parameter $\alpha$ in the cost functional, as indicated in the legend, for protocol duration $T=10$
• Figure 3: Optimized control field $u(t)$ obtained by the GRAPE algorithm when $\Delta=1$ for several values of the weight parameter $\alpha$ in the cost functional, as indicated in the legend, for protocol duration $T=1$
• Figure 4: Pareto front for the spin-boson model at $T=1, \Delta=1$ in the $(W_{\rm diss},\,\beta\sigma_w^2)$ plane. Each marker corresponds to an optimized protocol (obtained by GRAPE initialized from a linear ramp) for a given $\alpha$, where $\alpha$ ranges from $0$ to $1$ in steps of $0.05$.
• Figure 5: FIG. 5. Optimized control field u(t) obtained by the GRAPE algorithm for the quantum-dot model at $\beta=1$ for several values of the weight parameter $\alpha$ in the cost functional, as indicated in the legend, for protocol duration T=1.