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arXiv:2003.06437 · Physical Review Research

Work as an external quantum observable and an operational quantum work fluctuation theorem

Konstantin Beyer, Kimmo Luoma, Walter T. Strunz

Published Sep 28, 2020• 41 citations
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Abstract

We propose a definition of externally measurable quantum work in driven systems.Work is given as a quantum observable on a control device which is forcing the system and can be determined without knowledge of the system Hamiltonian ĤS .We argue that quantum work fluctuation theorems which rely on the knowledge of ĤS are of little practical relevance, contrary to their classical counterparts.Using our framework, we derive a fluctuation theorem which is operationally accessible and could in principle be implemented in experiments to determine bounds on free energy differences of unknown systems.

DOI
10.1103/physrevresearch.2.033508

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We propose a definition of externally measurable quantum work in driven systems.Work is given as a quantum observable on a control device which is forcing the system and can be determined without knowledge of the system Hamiltonian ĤS .We argue that quantum work fluctuation theorems which rely on the knowledge of ĤS are of little practical relevance, contrary to their classical counterparts.Using our framework, we derive a fluctuation theorem which is operationally accessible and could in principle be implemented in experiments to determine bounds on free energy differences of unknown systems.
In this Letter we propose a framework to define work as a quantum observable on a control device which is driving a system. Motivated by the conceptual difference between well known quantum versions of the Jarzynski equality (JE), where the initial and final system Hamiltonian have to be known in advance, and their classical counterparts where such knowledge is not needed, we construct a scheme where a work increment is given by an expectation value of a quantum observable on an external system. We further show that such a definition of work satisfies a fluctuation theorem which allows the free energy difference to be estimated operationally without knowledge about the Hamiltonian of the system at the end of the driving protocol. However, knowledge about the basis of the initial Hamiltonian is in general still needed. Providing two examples we describe how bounds on free energy differences could approximately be determined in experiments.
Estimation of work at the quantum scale remains a central challenge in quantum thermodynamics. Canonical approaches have a fundamental drawback: they assume classical control of the quantum system, as implicit in a time-dependent Hamiltonian. Yet the energetic cost of implementing this control is omitted and can exceed the system's energy scale by orders of magnitude, calling into question the operational significance of work values. We address this by autonomizing the controlled energy transfer, embedding the driven dynamics into an energy-conserving evolution on a larger quantum system. Work is then unambiguously identified with the energy transferred between the two systems, singling out a unique observable on the driven system: the work operator. We show that the quantum work operator evades a no-go theorem by deriving a quantum fluctuation theorem that recovers the Jarzynski equality in classical scenarios. We incorporate imperfect control and quantify corrections to work statistics. Extending the framework to open quantum systems, we obtain an operatorial first law in which work, heat, and internal-energy changes are represented by distinct operators on the reduced system. Together, these results settle the long-standing debate over whether work is a quantum observable: it is, once the controlling system is included in the description rather than treated as external.
The classical Jarzynski equality establishes an exact relation between the stochastic work performed on a system driven out of thermal equilibrium and the free energy difference in a corresponding quasistatic process. This fluctuation theorem bears experimental relevance, as it enables the determination of the free energy difference through the measurement of externally applied work in a nonequilibrium process. In the quantum case, the Jarzynski equality only holds if the measurement procedure of the stochastic work is drastically changed: it is replaced by a so-called two-point measurement scheme that requires the knowledge of the initial and final Hamiltonian and therefore lacks the predictive power for the free energy difference that the classical Jarzynski equation is known for. Here, we propose a quantum fluctuation theorem that is valid for externally measurable quantum work determined during the driving protocol. In contrast to the two-point measurement case, the theorem also applies to open quantum systems and the scenario can be realized without knowing the system Hamiltonian. Our fluctuation theorem comes in the form of an inequality and therefore only yields bounds to the true free energy difference. The inequality is saturated in the quasiclassical case of vanishing energy coherences at the beginning and at the end of the protocol. Thus, there is a clear quantum disadvantage.
Fluctuation theorems for work provide universal constraints on nonequilibrium fluctuations, yet their quantum generalizations often rely on externally prescribed classical driving protocols. While for classical systems, fluctuation theorems have been extended to autonomous work, where the dynamics of the work source is subject to the backaction of the system, their generalization to the quantum regime is constrained by the uncertainty principle. Here, we extend fluctuation theorems for autonomous work from the classical regime to the quantum regime. By performing successive projective measurements over the work source and the system, we derive Jarzynski-type and Crooks-type fluctuation theorems for autonomous inclusive work from initial mixed thermal states. These relations are analogous to fluctuation theorems for autonomous work in the classical regime and explicitly incorporate the fluctuations of the work source. However, quantum noncommutativity prevents a consistent reduction to the nonautonomous counterparts, even in the limit of a large work source and correspondingly negligible backaction. By contrast, under the exclusive work definition, the nonautonomous limit is recovered when the measured observable of the work source commutes with its bare Hamiltonian and the backaction of the system on the work source is negligible. Our results are illustrated with the Dicke model, where a single-mode radiation field and an ensemble of two-level atoms act as the system of interest and the work source, respectively.
We consider work fluctuation theorems for isolated driven systems and the possibility to probe them in mesoscopic systems. In this context nonequilibrium fluctuation theorems (NFTs) relate work performed on a system as its Hamiltonian varies with time, to equilibrium data of the initial and final states. In a classical context the system energy can be directly measured, while a quantum implementation requires the incorporation of a work agent. Here, as a work agent we consider a dynamical single-coordinate object, which exchanges energy with the system. The work done on the system is defined as the energy reduction of the work agent, which requires an energy measurement of the agent (only) at the end of the process. To justify the applicability of the NFT we require the agent's trajectory to be weakly affected by the energy exchange with the system. We furthermore argue that the uncertainty in the energy measurements imposes an inherent quantum limitation on the validity of the NFT. We demonstrate our findings for a two-level system, and discuss applications to more complex mesoscopic systems.