arXiv:1909.03558 · Physical Review Letters

Fluctuations in Extractable Work Bound the Charging Power of Quantum Batteries

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

Published Jul 22, 2020• 133 citations

Abstract

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 ${H}_{\mathcal{W}}$ 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.

DOI: 10.1103/physrevlett.125.040601

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arXiv:1909.03558 · Physical Review Letters

Fluctuations in Extractable Work Bound the Charging Power of Quantum Batteries

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

Published Jul 22, 2020• 133 citations

arXiv HTML PDF

Abstract

{\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 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.

DOI: 10.1103/physrevlett.125.040601

Related papers

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

Stefano Cusumano, Łukasz Rudnicki

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.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.

arXiv:2102.05627Physical Review LettersJul 9, 20219 citations

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

Shang-Yung Wang

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 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.

arXiv:2102.04921Feb 9, 20210 citations

Fluctuations in Extractable Work and Bounds on the Charging Power of Quantum Batteries

Shang-Yung Wang

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Bounds on charging power of open quantum batteries

Shadab Zakavati, F. T. Tabesh, S. Salimi

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.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.

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We consider the dissipative charging process of quantum batteries in terms of a collisional model, where the batteries are coupled to a heat bath using nonenergy preserving interactions. First, we show that for low temperatures, the collective process can attain a charging power that increases polynomially with the number of batteries. The scaling we find is ${N}^{3}$ which, while being greater than the bound obtained for unitary processes, has a lower efficiency. Then, we study the dissipative charging process of a single battery using a time-dependent Hamiltonian that generates coherences in the energy basis. In this case, we find that the presence of coherence could enhance the charging power and also its efficiency. Finally, we show how this process can be used in a quantum heat engine that contains the charging process as one of its open strokes.We consider the dissipative charging process of quantum batteries in terms of a collisional model, where the batteries are coupled to a heat bath using nonenergy preserving interactions. First, we show that for low temperatures, the collective process can attain a charging power that increases polynomially with the number of batteries. The scaling we find is ${N}^{3}$ which, while being greater than the bound obtained for unitary processes, has a lower efficiency. Then, we study the dissipative charging process of a single battery using a time-dependent Hamiltonian that generates coherences in the energy basis. In this case, we find that the presence of coherence could enhance the charging power and also its efficiency. Finally, we show how this process can be used in a quantum heat engine that contains the charging process as one of its open strokes.

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Francesco Campaioli, Felix A. Pollock, Felix C. Binder +4 more

Can collective quantum effects make a difference in a meaningful thermodynamic operation? Focusing on energy storage and batteries, we demonstrate that quantum mechanics can lead to an enhancement in the amount of work deposited per unit time, i.e., the charging power, when N batteries are charged collectively. We first derive analytic upper bounds for the collective quantum advantage in charging power for two choices of constraints on the charging Hamiltonian. We then demonstrate that even in the absence of quantum entanglement this advantage can be extensive. For our main result, we provide an upper bound to the achievable quantum advantage when the interaction order is restricted; i.e., at most k batteries are interacting. This constitutes a fundamental limit on the advantage offered by quantum technologies over their classical counterparts.Can collective quantum effects make a difference in a meaningful thermodynamic operation? Focusing on energy storage and batteries, we demonstrate that quantum mechanics can lead to an enhancement in the amount of work deposited per unit time, i.e., the charging power, when N batteries are charged collectively. We first derive analytic upper bounds for the collective quantum advantage in charging power for two choices of constraints on the charging Hamiltonian. We then demonstrate that even in the absence of quantum entanglement this advantage can be extensive. For our main result, we provide an upper bound to the achievable quantum advantage when the interaction order is restricted; i.e., at most k batteries are interacting. This constitutes a fundamental limit on the advantage offered by quantum technologies over their classical counterparts.

arXiv:1612.04991Physical Review LettersApr 12, 2017531 citations

{\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}

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.

{N}^{3}

which, while being greater than the bound obtained for unitary processes, has a lower efficiency. Then, we study the dissipative charging process of a single battery using a time-dependent Hamiltonian that generates coherences in the energy basis. In this case, we find that the presence of coherence could enhance the charging power and also its efficiency. Finally, we show how this process can be used in a quantum heat engine that contains the charging process as one of its open strokes.