Collective-dissipation-induced dark and metastable-like states for enhanced quantum battery performance

Abderrahim El Allati; Achraf Khoudiri; Asghar Ullah; \"Ozg\"ur E. M\"ustecapl{\i}o\u{g}lu

REVIEW 3 major objections 6 minor 55 references

Collective-dissipation-induced dark and metastable-like states for enhanced quantum battery performance

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Collective dissipation improves open quantum battery charging by creating symmetry-protected dark states and frozen metastable states, and the antiferromagnetic phase wins because its protected states sit at low energies.

desk verdict The spectral-location idea is real and worth a referee, but the frozen-state count is undefined until they fix the time window and initial state. read the letter →

arxiv 2608.09693 v1 pith:VK676ADP submitted 2026-08-10 quant-ph

Achraf Khoudiri , Asghar Ullah , Abderrahim El Allati , Özgür E. Müstecapl{i}oğlu This is my paper · ORCID

classification quant-ph

keywords quantumbatteriescollectivedissipationdarkstatesfrozentransverse-fieldIsingmodelergotropyCatalannumbersdecoherence-freesubspace

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that collective dissipation, rather than local dissipation, can improve the charging of a transverse-field Ising quantum battery by generating symmetry-protected dark states together with a much larger set of frozen metastable states. It derives that for an even number of qubits the dark-state count follows the Catalan sequence, while odd-sized systems have no dark states. The central claim is that extractable work is governed by where these protected sectors sit in the many-body energy spectrum: the antiferromagnetic and ferromagnetic phases have the same number of protected states, but the antiferromagnetic phase yields far larger ergotropy because its protected states occupy low-energy regions. If this is right, engineered dissipation and magnetic ordering become practical design tools for robust quantum energy storage.

What carries the argument

The central objects are the collective spin operators $J_{\pm} = \sum_i \sigma^{\pm}_i$; dark states are states annihilated by both $J_+$ and $J_-$, which forces $S=M=0$ and is possible only for even $N$. Their multiplicity follows from the total-spin decomposition $d_S = \binom{N}{N/2-S} - \binom{N}{N/2-S-1}$, giving $d_0 = C_{N/2}$. Frozen states are defined dynamically through $\max_t |dP_k/dt| < \varepsilon$, expressing a balance between population gain and loss, and together with dark states they form the protected sector. Hamiltonian leakage, $\eta = \|(I-\hat{\Pi}) H_B \hat{\Pi}\|_F / \|H_B \hat{\Pi}\|_F$, quantifies whether the dark subspace is invariant under the battery Hamiltonian, and the active Hilbert-space fraction $f_{\rm active} = 1 - N_F/2^N$ measures how much of the Hilbert space remains available for charging.

What would settle it

Run the same master equation for fixed $N$ and temperature and count frozen states with Eq. (43) over increasingly long integration windows; if $N_F$ keeps rising with the window length, the reported active Hilbert-space fraction is an artifact of the observation time. Alternatively, shift the protected states in energy while keeping their count fixed, for instance by tuning $h$ or adding a staggered field: if ergotropy does not track the spectral placement, the AFM-versus-FM explanation fails.

Watch

Extended reading notes

Core claim

For the autonomous open quantum battery with Hamiltonian $H_B = -J \sum_i \sigma^z_i \sigma^z_{i+1} - h \sum_i \sigma^x_i$, coupled to a thermal reservoir through either local or collective Lindblad operators, the paper claims that collective dissipation creates an extended protected Hilbert space. Strict dark states are exactly the total-spin singlet sector, $S=M=0$, which exists only when $N$ is even, and their multiplicity is the $n$-th Catalan number $C_n = \frac{1}{n+1}\binom{2n}{n}$ with $n=N/2$. The dark subspace is an exact decoherence-free subspace only for $N=2$; for larger even $N$, Hamiltonian-induced leakage is nonzero but the states remain dark with respect to the collective jump operators. A numerically identified set of frozen states, defined by approximate population conservation, is much larger than the dark sector and exists for both parities. Both ferromagnetic ($J>0$) and antiferromagnetic ($J<0$) configurations have identical protected-state counts, yet the antiferromagnetic phase shows considerably larger ergotropy because its protected states are predominantly located at low energies, where they suppress dissipative losses and preserve stored work. The paper further shows that collective dissipation generally enhances ergotropy and charging power relative to local dissipation, with the advantage depending on temperature, phase, and system size, and introduces the active Hilbert-space fraction as a diagnostic of the tradeoff between protection and active charging pathways.

Load-bearing premise

The quantitative claims about the protected sector rest on counting frozen states via the condition $\max_t |dP_k/dt| < \varepsilon$ over a finite simulation interval, but the paper does not specify the interval or the initial state when it reports $N_F$, so the count is not yet a well-defined property of the model.

Editorial extensions

If this is right

Collective dissipation becomes a resource: for the same battery Hamiltonian, replacing independent baths with one common bath can raise both ergotropy and peak charging power, with peak power growing roughly linearly with $N$ in the collective case.
Qubit-number parity matters: even-$N$ batteries have a guaranteed Catalan-counted dark sector, while odd-$N$ batteries have none, so parity is a design parameter even under symmetric dissipation.
Antiferromagnetic ordering is preferable for energy storage: with identical protected-state counts, the low-energy placement of protected states in the AFM phase gives substantially larger ergotropy than the FM phase, especially at low temperature.
Protection and charging pathways compete: a larger frozen sector reduces the active Hilbert-space fraction, so optimal charging requires balancing dissipative protection against available active pathways.
The $N=2$ dark state is an exact decoherence-free subspace, but for larger even $N$ the protection is only approximate, so practical gains at larger sizes rest on metastable behavior rather than exact DFS invariance.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper does not pursue this, but if spectral placement is the controlling factor, tuning the transverse field $h$ or adding a staggered field should move the protected states in energy and shift ergotropy even when $N_D$ and $N_F$ stay fixed, making the mechanism directly testable.
The Catalan counting is tied to symmetric collective coupling; analogous counting for non-uniform or anisotropic collective couplings would give different multiplicities, and the protected-sector-as-resource logic could be ported to those models.
The frozen-sector picture suggests a possible connection to prethermalization: if the frozen populations correspond to approximate conserved quantities, the charging dynamics could be coarse-grained into an effective model on the active Hilbert space alone.
The results indicate that protected sectors could be engineered in other autonomous quantum thermodynamic devices, such as refrigerators or heat engines, wherever decoherence must be suppressed while population mobility is preserved.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague, here's my take on 2608.09693. The genuinely new thing isn't the dark-state counting—that's textbook SU(2) combinatorics, and their Catalan derivation in Appendix A is correct but standard. It's the claim that what matters for battery performance is where the protected subspaces sit in the energy spectrum, not how many protected states exist. The N=2 analysis in Appendix B makes that concrete: AFM and FM have the same dark state, but in AFM it protects a low-lying level, and their numerics show much larger AFM ergotropy. That is a real, potentially useful design principle for open quantum batteries.

What the paper does well: the master-equation setup is standard, the local vs collective comparison is sensible, and the N=2 transition-pathway analysis in Fig. 8 is a nice explicit illustration. They also correctly flag that for N≥4 the dark subspace is not Hamiltonian-invariant, so it is not an exact DFS beyond N=2.

The soft spot is where the stress-test lands, and I think it's accurate. N_F, the count of frozen states, is defined in Eq. (43) as max_t |dP_k/dt| < ε with ε=1e-9, but no time interval and no initial state are specified. Since every population derivative eventually decays to zero in a Lindblad evolution, extending the simulation window makes N_F grow toward 2^N. Restricting to the early transient makes the threshold essentially unattainable. They need to state the window and initial state, and then show N_F is stable against reasonable variations. Without that, the exponential growth, the AFM/FM equality, and the active-fraction correlation in Fig. 7 are not well-defined. The circularity note is also fair: f_active comes from the same simulated populations that define ergotropy, so Fig. 7 is partly restating the simulation; that doesn't kill the point, but it should be framed more carefully.

Minor: Figs. 2 and 3 are computed only for FM (J=1); the text claims AFM and FM have identical N_F, but that's not numerically shown for N>2. Symmetry makes it plausible for dark states, but frozen states depend on the eigenbasis, so that claim needs its own figure or argument.

Overall: the central mechanistic claim—spectral location of protected sectors, not their multiplicity, governs charging—holds up as a hypothesis and is supported by N=2 and by the qualitative ergotropy curves. The frozen-state diagnostic is currently too loose to support the quantitative part. This deserves a serious referee, but it needs a revision that pins down the frozen-state definition and either adds AFM frozen counts or drops the equality claim.

Referee Report

3 major / 6 minor

Summary. The manuscript studies autonomous charging of a transverse-field Ising quantum battery of N qubits coupled to thermal reservoirs, comparing local and collective dissipation. It derives analytically that the number of collective dark states for even N is the Catalan number C_{N/2}, that such states are absent for odd N, and that for N=2 the dark subspace is an exact decoherence-free subspace. It then identifies numerically a much larger set of 'frozen' energy levels whose populations remain approximately constant under collective dissipation, and it reports that the FM and AFM configurations have identical numbers of dark and frozen states but substantially different ergotropy, which it attributes to the spectral location of the protected subspaces. The paper also introduces the active Hilbert-space fraction f_active = 1 - N_F/2^N and argues that metastable protection suppresses dissipative losses while preserving charging pathways.

Significance. If the central claims are correct, the manuscript makes a useful conceptual contribution: it separates the multiplicity of protected subspaces from their spectral placement and proposes that spectral placement governs charging performance in open quantum batteries. The analytic Catalan derivation in Appendix A is clean and standard, the N=2 analysis in Appendix B is explicit and self-contained, and the numerical master-equation results cover a useful range of system sizes, temperatures, and magnetic phases. The paper also states its modeling assumptions clearly, including the engineered-bath interpretation of T and the fixed-rate approximation. However, the main mechanistic conclusion is not yet fully well-defined because the central quantity N_F is introduced without specifying the initial state and time window used in its definition; this ambiguity propagates into f_active and the interpretation of Fig. 7. With that definition fixed, the paper would be a solid contribution to the open-quantum-battery literature.

major comments (3)

[Sec. IV A, Eq. (43)] The number of frozen states is defined by max_t |dP_k/dt| < epsilon with epsilon = 10^-9, but the time interval and the initial state entering the max are not specified at the point where N_F is introduced. In any finite-dimensional Lindblad evolution every population derivative decays to zero as the stationary state is approached, so extending the integration window makes all levels frozen and N_F tends to 2^N; including the early transient instead makes the condition essentially unattainable at the O(gamma) population currents. Since N_F and f_active underpin Figs. 2, 3, and 7 and the protected-sector claims in the abstract and Sec. V, the manuscript must state the initial state and the time window used in Eq. (43), and ideally show that N_F is robust to reasonable variations of both. The initial ground state is specified only later, in Sec. V A, for the ergotropy dynamics, not for the N_F computation.
[Sec. IV B and Fig. 7] The active fraction f_active = 1 - N_F/2^N is computed from the same simulated populations that produce the ergotropy curves, so plotting the enhancement ratio W_colmax/W_locmax against f_active in Fig. 7 partly restates the simulation output rather than providing an independent test of the proposed mechanism. The authors should either define f_active from an independent construction (for example, from the spectrum of the Liouvillian or from a state-independent connectivity condition) or explicitly frame Fig. 7 as a presentation of simulation data and add error bars or convergence checks, including the integration window and the tolerance used in Eq. (43). Without such checks, the statement that metastable protection suppresses losses while preserving charging pathways is a summary of the numerics rather than a demonstrated mechanism.
[Sec. IV A and Sec. V A] The text asserts that the FM and AFM configurations have identical numbers of dark and frozen states for a given system size, but Fig. 2 is computed only for J=1 (its caption states 'We set J=1 and h=0.1'), and no AFM data for N_F are shown. The equality of dark-state counts follows analytically from the spin-symmetry argument and does not depend on J, but the equality of N_F across the two magnetic phases is a numerical claim and is load-bearing for the spectral-location argument in Sec. V A. The authors should show the AFM computation explicitly or restrict the claim to the parameter values actually computed.

minor comments (6)

[Sec. III B] The sentence 'Numerically, we identify the dark states in our scenario using using the condition in Eq. (19)' contains a duplicated 'using'.
[Fig. 2 and Fig. 7 captions] The captions set B=0.1, while the model in Eq. (1) and the text elsewhere use h=0.1 and never define B; please harmonize the notation.
[Sec. IV A, Eq. (43)] The notation 'epsilon -> 0' with a fixed epsilon = 10^-9 is not a limit but a tolerance choice; please write the condition with a small positive tolerance and state its value.
[Sec. IV A] The phrase 'We compute the number of frozen states number in Fig. 2' has a duplicated noun and should be reworded.
[Eq. (45) and Fig. 3] The notation f^{loc(col)}_{D(F)} is overloaded; spelling out f_D^col, f_F^col, f_D^loc, f_F^loc or defining the subscript convention explicitly would improve readability.
[Sec. II B] The model uses fixed rates set by a single omega0 rather than the actual Bohr frequencies of H_B; this is a legitimate engineered-bath choice, but the text should state explicitly that the bath is not resonant with the individual transitions of the battery.

Circularity Check

1 steps flagged · score 4.0 of 10

The dark-sector Catalan derivation and N=2 spectral analysis are independent, but the frozen-state/active-fraction explanation is partly defined from the same simulated populations it is invoked to explain.

self definitional [Sec. IV A, Eq. (43); Sec. IV B, Eq. (48); Sec. V B, Fig. 7]
"We define a frozen energy level |E_k⟩ as a level whose population remains approximately constant during the system evolution. ... The number of frozen states, denoted by N_F, is determined according to max_t |dP_k(t)/dt| < ε, ε→0. ... The fraction of active states ... is given by f^{loc(col)}_{active}=1−N_F/2^N. ... We analyze this enhancement factor W^{col}_{max}/W^{loc}_{max} as a function of f^{col}_{active} in Fig. 7, to highlight the role of the active levels in the Hilbert space in maximal ergotropy production."

N_F is defined by thresholding the time derivatives of energy-level populations P_k(t) in the very same simulated trajectories that determine W(t) and W_max. f_active is then defined as 1−N_F/2^N, so Fig. 7 is a scatter plot of two derived quantities from one trajectory, not a test of an independent mechanism. The explanatory claim that metastable protection preserves stored energy or charging pathways is therefore largely a restatement of the criterion used to label levels as frozen. The definition is also incomplete: Eq. (43) does not specify the time interval or initial state over which the max is taken, so N_F and f_active are not well-defined; in the long-time limit every population derivative tends to zero and N_F approaches 2^N.

full rationale

The main analytical result—the Catalan multiplicity of the collective dark sector—is derived from the total-spin decomposition (Appendix A, Eqs. A12–A21) and does not presuppose the ergotropy or charging results. The N=2 comparison of AFM and FM spectral positions (Appendix B) is an exact diagonalization and is likewise independent of the frozen-state counting. The collective-vs-local simulation comparison is direct numerical evidence. The partial circularity is confined to the frozen-state/active-fraction analysis: because N_F is counted from the same population derivatives that shape the ergotropy curves, plotting the enhancement factor against f_active (Fig. 7) re-expresses the simulation output rather than confirming a mechanism. Eq. (43) also omits the time window and initial state, leaving N_F and f_active indeterminate; this limitation should be fixed before the active-fraction claims are used. The assertion that FM and AFM have identical N_F is not supported by an AFM computation in Fig. 2, which is stated for J=1 only, but this is a missing-evidence issue rather than circularity. No load-bearing self-citation chain or imported uniqueness theorem was found.

Assumptions & free parameters 2 free parameters · 4 assumptions · 2 invented entities

The only parameters genuinely fitted by hand are the tolerance epsilon and the unstated simulation time window that enters the frozen-state count. The model parameters h, J, gamma, and omega0 are physical inputs, not fit parameters. The main axioms are the Markovian master equation with an engineered reservoir and the operational definition of frozen states. The paper introduces no new particles or forces; its invented entities are a numerical classification of states and a derived metric.

free parameters (2)

epsilon (tolerance) = 1e-9
Hand-chosen threshold used in Eqs. (27), (43), and (44) to define dark, frozen, and strict frozen states; the count N_F is sensitive to this value.
Simulation time window for frozen states = unspecified
The frozen-state condition in Eq. (43) evaluates the maximum time derivative over an unstated interval; because all populations become constant at steady state, the count N_F depends on this choice.

assumptions (4)

domain assumption GKSL Markovian master equation with time-independent rates
The entire dynamics is described by Eq. (5); the paper states the reservoir is an engineered pump-and-decay bath, not a thermal bath in equilibrium with H_B.
ad hoc to paper Bath rates depend on a fixed omega0 rather than the battery energy gaps
Stated in Sec. II B; this choice means the stationary state is not the thermal state of H_B, and the results may depend on this modeling decision.
standard math Total spin decomposition and singlet-sector counting
Appendix A uses standard SU(2) representation theory; the Catalan result is a known identity and is derived correctly.
ad hoc to paper Frozen states are defined by approximate population conservation over finite time
Eq. (43) is the operational definition; it is not a dynamical invariant and depends on initial state and time window.

invented entities (2)

Frozen (metastable-like) states
purpose: To identify energy levels whose populations are approximately constant and to define the protected sector responsible for suppressing dissipative loss.
The concept is defined only within this paper's numerical framework; no independent observable is predicted that would confirm their existence outside the model.
Active Hilbert-space fraction f_active
purpose: To quantify the fraction of levels not protected and to correlate it with charging performance.
A derived metric that is a deterministic function of N_F; it carries no independent predictive content.

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Cite this review

Pith. "Pith review of Collective-dissipation-induced dark and metastable-like states for enhanced quantum battery performance." pith.science (2026). https://pith.science/paper/VK676ADP

@misc{pith2026260809693,
  author       = {Pith},
  title        = {Pith review of: Collective-dissipation-induced dark and metastable-like states for enhanced quantum battery performance},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VK676ADP}},
  note         = {Machine review of arXiv:2608.09693}
}

read the original abstract

We investigate the role of symmetry-protected dark states and metastable-like frozen states in the autonomous charging dynamics of open quantum batteries described by a transverse-field Ising model. By comparing local and collective dissipation over a range of system sizes, temperatures, and magnetic phases, we demonstrate that collective dissipation generates symmetry-protected dark states together with a much larger set of frozen (metastable) states, forming an extended protected Hilbert space. We derive the multiplicity of the collective dark sector analytically, showing that it follows the Catalan sequence for even system sizes, while such states are absent for odd sizes. Our results show that collective dissipation can enhance ergotropy and charging power, with its advantage depending on temperature, magnetic phase, and system size. While the number of dark and frozen states is identical in the ferromagnetic and antiferromagnetic phases, the achievable ergotropy differs substantially because of the different spectral locations of these protected states. In particular, the antiferromagnetic configuration exhibits considerably larger extractable work owing to the favorable positioning of the protected subspaces within the many-body energy spectrum. Finally, we analyze the active Hilbert-space fraction and show that metastable protection provides an effective mechanism for suppressing dissipative losses while preserving efficient charging pathways. These results establish the dark-state and frozen-state sectors as key resources for optimizing the performance of open quantum batteries through engineered dissipation.

Figures

Figures reproduced from arXiv: 2608.09693 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3. Fractions [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: FIG. 4. Dynamics of the ergotropy [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: FIG. 5. Maximal charging power [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: FIG. 6. Stationary-state ergotropy per qubit, [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: FIG. 7. Enhancement factor [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 8.** Figure 8: FIG. 8. Energy level diagrams and transition pathways for the (a) AFM and (b) FM configurations. Blue dotted lines represent the transition [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]

**Figure 9.** Figure 9: FIG. 9. Dynamics of [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]

**Figure 10.** Figure 10: shows the time evolution of the ergotropy for the initial state given in Eq. (B13), in both the AFM [Fig. 10a] and FM [Fig. 10b] regimes. In both cases, the ergotropy W(ρ(t)) is consistently higher under collective dissipation than under local dissipation. It increase…

**Figure 11.** Figure 11: FIG. 11. Comparison between the energy dynamics of the state [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]

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Definition and Hilbert Space Decomposition Dark states are defined as the quantum states|ψ 0⟩that are completely annihilated by the collective ladder operatorsJ ±: J±|ψ 0⟩=0,J †|ψ0⟩=0,(A1) where the collective spin operators for a system ofNqubits are defined as: J± = N ∑ i=1 σ± i ,J β = N ∑ i=1 σβ i forβ∈ {x,y,z}.(A2) The total squared spin operator is g...

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Definition and Hilbert Space Decomposition Dark states are defined as the quantum states|ψ 0⟩that are completely annihilated by the collective ladder operatorsJ ±: J±|ψ 0⟩=0,J †|ψ0⟩=0,(A1) where the collective spin operators for a system ofNqubits are defined as: J± = N ∑ i=1 σ± i ,J β = N ∑ i=1 σβ i forβ∈ {x,y,z}.(A2) The total squared spin operator is g...

work page

[2] [2]

(A8) To satisfy the dark state criterionJ ±|ψ⟩=0, we substitute our state expansion into the ladder operator eigenvalue rela- tion: J±|ψ⟩= ∑ S,M,α aS,M,α q S(S+1)−M(M±1)|S,M±1,α⟩=0

Proof of the Dark State Condition The actions of the total spin operators on our basis states |S,M,α⟩(setting¯h=1) yield: J2|S,M,α⟩=S(S+1)|S,M,α⟩,(A6) Jz|S,M,α⟩=M|S,M,α⟩,(A7) J±|S,M,α⟩= q S(S+1)−M(M±1)|S,M±1,α⟩. (A8) To satisfy the dark state criterionJ ±|ψ⟩=0, we substitute our state expansion into the ladder operator eigenvalue rela- tion: J±|ψ⟩= ∑ S,M,...

work page

[3] [3]

LetN ↑ be the num- ber of spins pointing up (m i = 1

Derivation of Dark State Multiplicity via Catalan Numbers The total dimension of a given magnetization sectorMin the computational basis is denoted bygM. LetN ↑ be the num- ber of spins pointing up (m i = 1

work page

[4] [4]

|↑↑⟩ −J− p 4h2 +J 2 2h (|↑↓⟩+|↓↑⟩ ) +|↓↓⟩ # , (B8) |E4⟩= 1 N+

andN ↓ be the number of spins pointing down (mi =− 1 2), whereN=N ↑ +N ↓. The total magnetization is: M= 1 2 (N↑ −N ↓) = N 2 −N ↓ =⇒N ↓ = N 2 −M. (A14) The number of computational basis configurations possess- ing a specific total magnetization valueMis: gM = N N 2 −M .(A15) From the decomposition of the total Hilbert space, the di- mension can also be wr...

work page

[5] [5]

Energy dynamics under local and collective dissipation: Role of dark states The energy stored in the quantum battery can be expressed as a sum over all energy levelsk: E(ρ(t)) = 2N −1 ∑ k=0 Eα k (t),(B10) Eα k (t) =P α k (t)E α k .(B11) Because the Hamiltonian is time independent, the dynamics ofE(ρ(t))is fully determined by the evolution of the state ρ(t...

work page

[6] [6]

Un- der collective dissipation, the emergence of dark states mod- ifies the system dynamics by suppressing specific transition channels between energy levels

Impact of dark states on work extraction: Comparison between local and collective dissipation The extractable work from a quantum battery is quantified by the difference between its mean energy,E(ρ(t)), and the energy of the corresponding passive state,Epassive(ρ(t)). Un- der collective dissipation, the emergence of dark states mod- ifies the system dynam...

work page

[7] [7]

Thermodynamics in the quantum regime,

Felix Binder, Luis A Correa, Christian Gogolin, Janet An- ders, and Gerardo Adesso, “Thermodynamics in the quantum regime,” Fundamental Theories of Physics195(2018)

work page 2018

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A quantum mechanical open system as a model of a heat engine,

Ronnie Kosloff, “A quantum mechanical open system as a model of a heat engine,” J. Chem. Phys.80, 1625–1631 (1984)

work page 1984