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arXiv:2102.05627 · Physical Review Letters

Comment on “Fluctuations in Extractable Work Bound the Charging Power of Quantum Batteries”

Stefano Cusumano, Łukasz Rudnicki

Published Jul 9, 2021• 9 citations
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Abstract

In the abstract of [Phys. Rev. Lett. 125, 040601 (2020)] one can read that ``[$\dots{}$] to have a nonzero rate of change of the extractable work, the state ${\ensuremath{\rho}}_{\mathcal{W}}$ of the battery cannot be an eigenstate of a `free energy operator,' defined by $\mathcal{F}\ensuremath{\equiv}{H}_{\mathcal{W}}+{\ensuremath{\beta}}^{\ensuremath{-}1}\mathrm{log}({\ensuremath{\rho}}_{\mathcal{W}})$, where ${H}_{\mathcal{W}}$ is the Hamiltonian of the battery and $\ensuremath{\beta}$ is the inverse temperature [$\dots{}$].'' Contrarily to what is presented below Eq. (17) of the Letter, we observe that the above conclusion does not hold when the battery is subject to nonunitary dynamics.

DOI
10.1103/physrevlett.127.028901

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García-Pintos, Hamma, and del Campo Reply:

Luis Pedro García-Pintos, Alioscia Hamma, Adolfo del Campo

We acknowledge that a derivation reported in Phys. Rev. Lett. 125, 040601 (2020)PRLTAO0031-900710.1103/PhysRevLett.125.040601 is incorrect as pointed out by Cusumano and Rudnicki. We respond by giving a correct proof of the claim "fluctuations in the free energy operator upper bound the charging power of a quantum battery" that we made in the Letter.

Physical Review Letters•Jul 9, 2021•4 citations

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In the abstract of [Phys. Rev. Lett. 125, 040601 (2020)] one can read that ``[…\dots{}…] to have a nonzero rate of change of the extractable work, the state \ensuremathρW{\ensuremath{\rho}}_{\mathcal{W}}\ensuremathρW​ of the battery cannot be an eigenstate of a `free energy operator,' defined by F\ensuremath≡HW+\ensuremathβ\ensuremath−1log(\ensuremathρW)\mathcal{F}\ensuremath{\equiv}{H}_{\mathcal{W}}+{\ensuremath{\beta}}^{\ensuremath{-}1}\mathrm{log}({\ensuremath{\rho}}_{\mathcal{W}})F\ensuremath≡HW​+\ensuremathβ\ensuremath−1log(\ensuremathρW​), where HW{H}_{\mathcal{W}}HW​ is the Hamiltonian of the battery and \ensuremathβ\ensuremath{\beta}\ensuremathβ is the inverse temperature […\dots{}…].'' Contrarily to what is presented below Eq. (17) of the Letter, we observe that the above conclusion does not hold when the battery is subject to nonunitary dynamics.
We study the connection between the charging power of quantum batteries and the fluctuations of the extractable work. We prove that in order to have a nonzero rate of change of the extractable work, the state ${\ensuremath{\rho}}_{\mathcal{W}}$ of the battery cannot be an eigenstate of a ``free energy operator,'' defined by $\mathcal{F}\ensuremath{\equiv}{H}_{\mathcal{W}}+{\ensuremath{\beta}}^{\ensuremath{-}1}\mathrm{log}({\ensuremath{\rho}}_{\mathcal{W}})$, where HW{H}_{\mathcal{W}}HW​ is the Hamiltonian of the battery and $\ensuremath{\beta}$ is the inverse temperature of a reference thermal bath with respect to which the extractable work is calculated. We do so by proving that fluctuations in the free energy operator upper bound the charging power of a quantum battery. Our findings also suggest that quantum coherence in the battery enhances the charging process, which we illustrate on a toy model of a heat engine.
In an article by García-Pintos et al. [Rev. Lett. 125, 040601 (2020)] the connection between the charging power of a quantum battery and the fluctuations of a "free energy operator" whose expectation value characterizes the maximum extractable work of the battery is studied. The result of the closed-system analysis shows that for a general charging process the battery will have a nonzero charging power if and only if the state of the battery is not an eigenstate of the free energy operator. In this Comment, we point out a few mistakes in the analysis and obtain the correct bound on the charging power. Consequently, the result for closed-system dynamics is in general not correct.
In general, quantum systems most likely undergo open-system dynamics due to their smallness and sensitivity. Energy storage devices, so-called quantum batteries, are not excluded from this phenomenon. Here, we study fundamental bounds on the power of open quantum batteries from the geometric point of view. By defining an activity operator, a tight upper bound on the charging power is derived for the open quantum batteries in terms of the fluctuations of the activity operator and the quantum Fisher information. The variance of the activity operator may be interpreted as a generalized thermodynamic force, while the quantum Fisher information describes the speed of evolution in the state space of the battery. The thermodynamic interpretation of the upper bound is discussed in detail. As an example, a model for the battery, taking into account the environmental effects, is proposed, and the effect of dissipation and decoherence during the charging process on both the stored work and the charging power is investigated. Our results show that the upper bound is saturated in some time intervals. Also, the maximum value of both the stored work and the corresponding power is achieved in the non-Markovian underdamped regime.
One of the most fundamental tasks in quantum thermodynamics is extracting energy from one system and subsequently storing this energy in an appropriate battery. Both of these steps, work extraction and charging, can be viewed as cyclic Hamiltonian processes acting on individual quantum systems. Interestingly, so-called passive states exist, whose energy cannot be lowered by unitary operations, but it is safe to assume that the energy of any not fully charged battery may be increased unitarily. However, unitaries raising the average energy by the same amount may differ in qualities such as their precision, fluctuations, and charging power. Moreover, some unitaries may be extremely difficult to realize in practice. It is hence of crucial importance to understand the qualities that can be expected from practically implementable transformations. Here, we consider the limitations on charging batteries when restricting to the feasibly realizable family of Gaussian unitaries. We derive optimal protocols for general unitary operations as well as for the restriction to easier implementable Gaussian unitaries. We find that practical Gaussian battery charging, while performing significantly less well than is possible in principle, still offers asymptotically vanishing relative charge variances and fluctuations.