Work-minimizing protocols in driven-dissipative quantum systems: An impulse-ansatz approach

TL;DR

The paper tackles the problem of minimizing work in finite-time, driven quantum systems by examining isothermal transitions between thermal states within a fully quantum, non-Markovian framework. It models the environment with a bath-oscillator (Drude) spectral density and computes the work functional $W_ au[\lambda]=\int_0^\tau dt\,{\rm tr}_S[\dot H_S(\lambda(t))\rho_S(t)]$ against the free-energy difference $\Delta F$, using numerically exact hierarchical equations of motion (HEOM). A three-parameter impulse ansatz (IMP3), allowing boundary-like impulses with fixed width $\delta$, is introduced to approximate the optimal protocol, and its performance is benchmarked against linear, polynomial (POLY3), and brute-force optimal (B-F) protocols across a broad range of bath parameters. The results show that impulse-like features remain near-optimal in the quantum, non-Markovian short-time regime, with IMP3 closely matching the numerical optimum at short durations, while Markovian GKSL-based approaches can fail even at weak coupling. The study also finds cases where POLY3 outperforms IMP3 at longer times and emphasizes the need for fully quantum optimization methods, providing a practical, fast route to near-optimal finite-time thermodynamic protocols in open quantum systems.

Abstract

The second law of thermodynamics sets a lower bound on the work required to drive a system between thermal equilibrium states, with equality attained in the quasistatic limit. For finite-time processes, part of the extractable work is inevitably dissipated, motivating the search for driving protocols that minimize the work. While classical stochastic systems have been extensively explored, quantum analyses remain limited and often rely on Markovian master equations valid only in the weak-coupling regime. Here, we study minimal work protocols for representative two-level systems coupled to a harmonic-oscillator bath using a numerically exact method. Inspired by known optimal solutions for Brownian oscillators, we introduce an impulse ansatz that incorporates possible boundary impulses and test it across a wide range of bath parameters. We find that impulse-like features remain nearly optimal in the quantum, non-Markovian regime, at short times. We also identify cases in which the widely used Markovian master equation fails even at weak coupling, underscoring the need for fully quantum approaches to finite-time thermodynamic optimization.

Figures (5)

• Figure 1: Schematic illustration of the problem setting. The system is driven between thermal equilibrium states at $\lambda_i$ and $\lambda_f$ by a control field $\lambda(t)$ varied only during a finite interval $\tau$. The work depends on the protocol $\{\lambda(t)\}_{0\le t\le \tau}$, and our goal is to identify the protocol that minimizes it.
• Figure 2: Impulse ansatz characterized by three parameters $(\alpha_1,\alpha_2,h)$, with a fixed impulse width $\delta$ treated as a hyperparameter.
• Figure 3: Excess work and optimal protocols for the driven two-level system. All dimensional quantities are in units of $\epsilon = \hbar = 1$. (a)–(d) Excess work $W_{\tau}^{\rm ex}[u]=W_\tau[u]-\Delta F$ as a function of $\tau$ for the linear (green dashed), IMP3 (red), POLY3 (blue), and B-F (black stars) protocols. The coupling is fixed at $\xi=1$, while $\beta$ and $\gamma$ vary as indicated along the top and right edges of the panels. (e) Normalized optimal protocols at $\tau=0.5$ for IMP3 (red) and B-F (black dashed) for $(\beta,\gamma,\xi)=(0.2,5,1)$. The vertical axis shows $\lambda^*(t)/\Lambda$, where $\Lambda=\max_{0\le t\le 0.5}|\lambda^*(t)|$, with $\Lambda=9.3$ (IMP3) and $\Lambda=59$ (B-F).
• Figure 4: Excess work and optimal protocols for the tunable two-level system with $(\beta,\gamma,\xi)=(5,5,0.2)$. All quantities are in units of $\epsilon=\hbar=1$. (a) Excess work as a function of $\tau$ for the linear (green dashed), IMP3 (red), POLY3 (blue), and B-F (black stars) protocols. (b) Optimal protocols at $\tau=15$ for IMP3 (red) and POLY3 (blue).
• Figure 5: Comparison of methods for the driven two-level system. All quantities are in units of $\epsilon=\hbar=1$. (a) Work as a function of $\tau$ for IMP3 obtained with HEOM (red), TCL2 (blue circles), and A-GKSL (green) for $(\beta,\gamma,\xi)=(0.2,5,0.002)$. (b) IMP3 optimal protocol at $\tau=0.5$ obtained with A-GKSL for $(\beta,\gamma,\xi)=(0.2,0.2,0.2)$. The inset shows the corresponding TCL2 result.