# Optimal Work Extraction from Finite-Time Closed Quantum Dynamics

## TL;DR

This paper establishes a fundamental trade-off between power and efficiency in finite-time quantum work extraction, providing exact solutions for optimal protocols in many-body systems using Lie algebra techniques.

## Contribution

It introduces a solvable class of finite-time optimal work extraction problems in quantum systems, revealing that optimal protocols can be simplified to time-independent controls in the interaction picture.

## Key findings

- Trade-off relation between power and work in quantum systems
- Exact reduction of the optimal control problem to a nonlinear equation
- Optimal protocols are time-independent in the interaction picture

## Abstract

Extracting useful work from quantum systems is a fundamental problem in quantum thermodynamics. In scenarios where rapid protocols are desired -- whether due to practical constraints or deliberate design choices -- a fundamental trade-off between power and efficiency is yet to be established. Here, we investigate the problem of finite-time optimal work extraction from closed quantum systems, subject to a constraint on the magnitude of the control Hamiltonian. We first reveal the trade-off relation between power and work under a general setup, showing that these fundamental performance metrics cannot be maximized simultaneously. We then identify a solvable class of finite-time optimal work-extraction problems. This class includes nontrivial many-body models such as the Heisenberg model and the SU(n)-Hubbard model. The key assumption is that the control Hamiltonian is optimized over a Lie algebra preserved by the uncontrolled dynamics. Within this class, the optimal work-extraction problem admits an exact reduction to a nonlinear self-consistent equation, circumventing extensive search over time-dependent control paths. The resulting optimal protocol turns out to be particularly simple: it suffices to use a time-independent control Hamiltonian in the interaction picture, determined by that equation. By exploiting the Lie-algebraic structure of the controllable terms, our approach is applicable to quantum many-body systems through efficient numerical computation. Our results highlight the necessity of rapid protocols to achieve the maximum power and provide an exact route to finite-time optimal work extraction in many-body quantum systems.

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Source: https://tomesphere.com/paper/2508.20512