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arXiv:1905.07328 · Physical Review Letters

Work Fluctuations in Slow Processes: Quantum Signatures and Optimal Control

Harry J. D. Miller, Matteo Scandi, Janet Anders, Martí Perarnau-Llobet

Published Dec 4, 2019• 104 citations
arXivHTMLPDF

Abstract

An important result in classical stochastic thermodynamics is the work fluctuation-dissipation relation (FDR), which states that the dissipated work done along a slow process is proportional to the resulting work fluctuations. We show that slowly driven quantum systems violate this FDR whenever quantum coherence is generated along the protocol, and we derive a quantum generalization of the work FDR. The additional quantum terms in the FDR are found to lead to a non-Gaussian work distribution. Fundamentally, our result shows that quantum fluctuations prohibit finding slow protocols that minimize both dissipation and fluctuations simultaneously, in contrast to classical slow processes. Instead, we develop a quantum geometric framework to find processes with an optimal trade-off between the two quantities.

DOI
10.1103/physrevlett.123.230603

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An important result in classical stochastic thermodynamics is the work fluctuation-dissipation relation (FDR), which states that the dissipated work done along a slow process is proportional to the resulting work fluctuations. We show that slowly driven quantum systems violate this FDR whenever quantum coherence is generated along the protocol, and we derive a quantum generalization of the work FDR. The additional quantum terms in the FDR are found to lead to a non-Gaussian work distribution. Fundamentally, our result shows that quantum fluctuations prohibit finding slow protocols that minimize both dissipation and fluctuations simultaneously, in contrast to classical slow processes. Instead, we develop a quantum geometric framework to find processes with an optimal trade-off between the two quantities.
In slowly driven classical systems, work is a stochastic quantity and its probability distribution is known to satisfy the work fluctuation-dissipation relation, which states that the mean and variance of the dissipated work are linearly related. Recently, it was shown that generation of quantum coherence in the instantaneous energy eigenbasis leads to a correction to this linear relation in the slow-driving regime. Here, we go even further by investigating nonclassical features of work fluctuations in setups with more than one system. To do this, we first generalize slow control protocols to encompass multipartite systems, allowing for the generation of quantum correlations during the driving process. Then, focusing on two-qubit systems, we show that entanglement generation leads to a positive contribution to the dissipated work, which is distinct from the quantum correction due to local coherence generation known from previous work. Our results show that entanglement generated during slow control protocols, e.g., as an unavoidable consequence of qubit crosstalk, comes at the cost of increased dissipation. Published by the American Physical Society 2024
Abstract To achieve efficient and reliable control of microscopic systems one should look for driving protocols that mitigate both the average dissipation and stochastic fluctuations in work. This is especially important in fast driving regimes in which the system is driven far out of equilibrium, potentially creating large amounts of unwanted entropy production. Here we characterise these optimal protocols in rapidly driven classical and quantum systems and prove that they consist of two discontinuous jumps in the full set of control variables. These jumps can be tuned to interpolate between processes with either minimal dissipation or minimal fluctuations, and in some situations allow for simultaneous minimisation. We illustrate our general results with rapidly driven closed quantum systems, classical bit erasure and a dissipative Ising chain driven close to a quantum phase transition.
Optimal control is a central problem in quantum thermodynamics. When minimizing dissipated work and work fluctuations defined via the two-point measurement scheme in open quantum systems, existing approaches largely focus on the rapid- and slow-driving limits, leaving the behavior at intermediate timescales elusive. In this work, by numerically optimizing the driving protocols, we demonstrate that the open quantum systems exhibit distinct optimal structures not captured by the conventional limits. Specifically, in the coherent spin-boson model, we find that the optimal protocol switches discontinuously between distinct locally optimal solutions as the relative weight between dissipation and fluctuations is varied. Furthermore, for a single-level quantum dot coupled to a fermionic reservoir, the optimized protocol develops a characteristic multi-step structure.
In thermodynamics, entropy production and work quantify irreversibility and the consumption of useful energy, respectively, when a system is driven out of equilibrium. For quantum systems, these quantities can be identified at the stochastic level by unravelling the system's evolution in terms of quantum jump trajectories. We here derive a general formula for computing the joint statistics of work and entropy production in Markovian driven quantum systems, whose instantaneous steady states are of Gibbs form. If the driven system remains close to the instantaneous Gibbs state at all times, then we show that the corresponding two-variable cumulant generating function implies a joint detailed fluctuation theorem so long as detailed balance is satisfied. As a corollary, we derive a modified fluctuation-dissipation relation (FDR) for the entropy production alone, applicable to transitions between arbitrary steady states, and for systems that violate detailed balance. This FDR contains a term arising from genuinely quantum fluctuations, and extends an analogous relation from classical thermodynamics to the quantum regime.
Recent developments of the analysis of the second law of thermodynamics by using the information theoretic quantities has led to the discovery of the so-called nonequilibrium free energy and has allowed us to formulate thermodynamic relations for arbitrary nonequilibrium initial and final states.By using the nonequilibrium free energies, we can quantify thermodynamic costs of information encoding and erasure processes as well as to quantify the extractable work from information heat engines in a unified way.In those general situations, reducing dissipation allows us to increase the extractable work from the system (or to reduce the work cost), and reducing work fluctuation allows us to narrow the work probability distribution and to extract a fluctuation-free work from the system.Suppressing both work fluctuation and dissipation as much as possible is vital to control mesoscopic and nano systems that work at the level of thermal fluctuations.Previous studies have searched for a vanishing work fluctuation protocol by using techniques of quantum information theory, known as the single-shot statistical mechanics (and sometimes called the deterministic work extraction protocol) and the vanishing dissipation protocol, known as the thermodynamically reversible protocol by using the second law of thermodynamics.In this thesis, we address issues concerning a method to control nonequilibrium systems by using fluctuation theorems and trade-off relation between work fluctuation and fluctuation in dissipation.We derive quantum fluctuation theorems under measurement and feedback control, which allow us to study how to suppress dissipation during a general quantum measurement and a feedback control.We then consider a method to suppress work fluctuation and fluctuation in dissipation simultaneously as much as possible by showing the trade-off relation between work and dissipation in terms of their standard deviations and derive an explicit protocol that achieves the lower bound of the trade-off relation.We show that the explicit protocol reproduces the deterministic work extraction protocol in the limit of vanishing work fluctuation and the thermodynamically reversible protocol in the limit of vanishing fluctuation in dissipation.
The fluctuation-dissipation theorem is a fundamental result in statistical physics that establishes a connection between the response of a system subject to a perturbation and the fluctuations associated with observables in equilibrium. Here we derive its version within a resource-theoretic framework, where one investigates optimal quantum state transitions under thermodynamic constraints. More precisely, we first characterize optimal thermodynamic distillation processes, and then we prove a relation between the amount of free energy dissipated in such processes and the free-energy fluctuations of the initial state of the system. Our results apply to initial states given by either asymptotically many identical pure systems or an arbitrary number of independent energy-incoherent systems, and they allow not only for a state transformation but also for the change of Hamiltonian. The fluctuation-dissipation relations we derive enable us to find the optimal performance of thermodynamic protocols such as work extraction, information erasure, and thermodynamically free communication, up to second-order asymptotics in the number N of processed systems. We thus provide a first rigorous analysis of these thermodynamic protocols for quantum states with coherence between different energy eigenstates in the intermediate regime of large but finite N.