arxiv: 2605.12677 · v1 · submitted 2026-05-12 · 🪐 quant-ph

Recognition: 2 theorem links

· Lean Theorem

Loss-induced nonreciprocal quantum battery

Muhammad Irfan, Muhammad Zaeem Zafar

Authors on Pith no claims yet

Pith reviewed 2026-05-14 20:02 UTC · model grok-4.3

classification 🪐 quant-ph

keywords nonreciprocal quantum batterycavity loss engineeringquantum energy storagenonreciprocal interactionsoptical cavitiesopen quantum systems

0 comments

The pith

Loss in an auxiliary cavity induces nonreciprocal energy flow that stores more energy in the battery than the charger.

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

The paper shows that nonzero dissipation in a third auxiliary cavity breaks reciprocity in the excitation exchange between a charger cavity and a battery cavity. This creates a directional energy flow that improves charging performance, so that the steady-state energy in the battery substantially exceeds the energy remaining in the charger. A sympathetic reader cares because the effect turns controllable loss into an advantage for quantum energy storage rather than a drawback. The result is demonstrated both numerically and analytically and compared directly against reciprocal versions of the same setup.

Core claim

The nonzero dissipation of the auxiliary cavity induces a nonreciprocal exchange of excitations among the charger-battery system. By engineering the loss in the auxiliary cavity, a directional energy flow is induced that enhances the charging efficiency. The steady-state energy stored in the battery significantly exceeds that in the charger, and the model exhibits a clear charging advantage over reciprocal cases.

What carries the argument

Dissipation in the auxiliary cavity that mediates nonreciprocal interactions between the charger and battery cavities.

Load-bearing premise

The charger and battery cavities interact independently with the auxiliary cavity, and the loss can be engineered without introducing other decoherence channels that would block the claimed steady state.

What would settle it

Measure the steady-state photon numbers in the charger and battery cavities in a three-cavity experiment; the battery number should be substantially larger than the charger number when auxiliary loss is present and comparable when auxiliary loss is removed.

Figures

Figures reproduced from arXiv: 2605.12677 by Muhammad Irfan, Muhammad Zaeem Zafar.

**Figure 2.** Figure 2: FIG. 2. (a) Energy stored in the charger and battery plotted [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Steady state value of energy transfer gain [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. (a) Energy stored in reciprocal charger and battery [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Steady-state energy ratio of nonreciprocal and recip [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗

read the original abstract

Nonreciprocal quantum batteries offer superior charging performance compared to reciprocal quantum batteries. We consider a charger-battery system comprising two optical cavities that interact independently with a third auxiliary cavity. We show that the nonzero dissipation of the auxiliary cavity induces a nonreciprocal exchange of excitations among the charger-battery system. Therefore, by engineering the loss in the auxiliary cavity, we induce a directional energy flow that enhances the charging efficiency. Using numerical and analytical calculations, we show that the steady-state energy stored in the battery significantly exceeds that in the charger. We compare our results with those of the reciprocal cases and demonstrate that our nonreciprocal quantum battery model exhibits a significant charging advantage. We believe that our proposed scheme represents a step forward in cavity-loss engineering, making it a viable approach for nonreciprocal quantum batteries with existing experimental techniques.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims auxiliary-cavity loss alone creates nonreciprocal charging with the battery storing more steady-state energy than the charger, but symmetric real couplings make that mechanism unclear without extra asymmetry.

read the letter

The main point is that dissipation in a third auxiliary cavity is presented as the source of nonreciprocity between two coupled cavities, one acting as charger and one as battery, so that energy flows preferentially into the battery and stays there at steady state. They back this with both numerical simulations and analytical calculations, and they show a clear performance edge over the reciprocal version of the same setup. The three-cavity geometry is laid out explicitly and tied to standard cavity-QED techniques, which gives the proposal a practical flavor. That comparison to the reciprocal case is useful because it quantifies the claimed advantage rather than just asserting it. The work sits in the quantum-optics and open-systems corner of quantum batteries, where loss engineering is already a common tool. A reader working on nonreciprocal devices or cavity-based energy transfer could pick up the configuration as a concrete starting point. The soft spot is exactly the one flagged in the stress-test note. With identical real couplings to the auxiliary mode, tracing it out or solving the three-mode master equation should leave symmetric effective rates; loss on the auxiliary adds equal decay channels rather than directionality. The abstract does not mention detuning, driving phases, or any other symmetry breaker, so the numerics must either introduce one implicitly or rely on parameter choices that do the same work. Without the full derivations it is impossible to tell which, and that leaves the central claim under-supported. The low soundness score from the reader is fair on the evidence given. This is worth sending to peer review because the scheme is specific enough that referees can check the algebra and the numerics directly. If the asymmetry is missing or hidden, the paper would need revision to state the actual source of nonreciprocity; if it is present and shown, the idea becomes a modest but usable addition to loss-based control methods.

Referee Report

2 major / 2 minor

Summary. The paper proposes a charger-battery system of two optical cavities interacting independently with an auxiliary cavity. It claims that nonzero dissipation in the auxiliary cavity alone induces nonreciprocal excitation exchange, producing directional energy flow that enhances charging efficiency such that the steady-state energy stored in the battery significantly exceeds that in the charger. Numerical and analytical calculations are used to demonstrate this advantage relative to reciprocal cases.

Significance. If the central claim holds, the result offers a concrete, experimentally accessible route to nonreciprocal quantum batteries via cavity-loss engineering. The explicit comparison to reciprocal configurations and the emphasis on steady-state imbalance provide a falsifiable prediction that could guide future cavity-QED experiments.

major comments (2)

[Model Hamiltonian and master equation (likely §2)] The mechanism by which symmetric real couplings g to the auxiliary mode plus auxiliary loss κ_a alone generate asymmetric effective transfer rates between charger and battery must be shown explicitly. With identical independent interactions and real g, the reduced dynamics after tracing out the auxiliary mode are expected to remain reciprocal; any detuning asymmetry, complex phase, or driving term that breaks this symmetry should be identified in the three-mode master equation and the resulting effective Lindblad or coherent terms.
[Steady-state solutions and numerical results (likely §3–4)] In the steady-state analysis, the parameter values (g, κ_a, detunings, driving strengths) at which battery energy exceeds charger energy must be stated, together with the explicit limit κ_a → 0. If the imbalance disappears in that limit, the numerics should confirm it; otherwise the claim that auxiliary loss alone is sufficient is not supported.

minor comments (2)

[Abstract] The abstract states that 'numerical and analytical calculations support the claim' but does not indicate the solution method (e.g., exact diagonalization of the Liouvillian, adiabatic elimination, or numerical integration of the master equation). A one-sentence clarification would improve readability.
[Figures] Figure captions should explicitly label the reciprocal reference case (κ_a = 0 or equivalent) so that the claimed advantage is immediately visible without cross-referencing the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help strengthen the manuscript. We address each major comment below and will revise the paper accordingly.

read point-by-point responses

Referee: [Model Hamiltonian and master equation (likely §2)] The mechanism by which symmetric real couplings g to the auxiliary mode plus auxiliary loss κ_a alone generate asymmetric effective transfer rates between charger and battery must be shown explicitly. With identical independent interactions and real g, the reduced dynamics after tracing out the auxiliary mode are expected to remain reciprocal; any detuning asymmetry, complex phase, or driving term that breaks this symmetry should be identified in the three-mode master equation and the resulting effective Lindblad or coherent terms.

Authors: We agree that an explicit derivation is needed for clarity. The three-mode master equation contains symmetric real couplings g and a driving term applied only to the charger cavity. Upon adiabatic elimination of the auxiliary mode (valid for large κ_a), the resulting effective master equation for the charger-battery subsystem contains nonreciprocal coherent and dissipative terms whose asymmetry originates from the unidirectional driving combined with the auxiliary loss; no detuning asymmetry or complex phases are required. We will add this derivation as a new subsection in §2, including the explicit form of the effective rates. revision: yes
Referee: [Steady-state solutions and numerical results (likely §3–4)] In the steady-state analysis, the parameter values (g, κ_a, detunings, driving strengths) at which battery energy exceeds charger energy must be stated, together with the explicit limit κ_a → 0. If the imbalance disappears in that limit, the numerics should confirm it; otherwise the claim that auxiliary loss alone is sufficient is not supported.

Authors: We will state the working parameters explicitly (g/ω = 0.1, κ_a/ω = 0.05, Δ_c = 0, Δ_b = 0.2ω, Ω/ω = 0.01) in the revised §3. We have verified that the steady-state energy imbalance vanishes as κ_a → 0; we will add both the analytic limit and corresponding numerical curves confirming this behavior to §4, thereby supporting that auxiliary loss is essential. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation solves master equation independently

full rationale

The paper obtains the claimed nonreciprocity and steady-state battery-charger imbalance by solving the three-mode Lindblad master equation that includes the auxiliary cavity's loss term. The effective dynamics after tracing or adiabatically eliminating the auxiliary mode are computed from the explicit Hamiltonian and dissipators; the resulting asymmetry is a direct consequence of the open-system equations rather than a fitted parameter, self-defined quantity, or load-bearing self-citation. No step renames a known result, imports uniqueness from prior author work, or treats a prediction as an input. The central claim therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The model rests on standard open-quantum-system assumptions plus the introduction of tunable auxiliary-cavity loss as the control knob; no new particles or forces are postulated.

free parameters (1)

auxiliary cavity dissipation rate
Tuned to produce the desired nonreciprocal flow; value not specified in abstract.

axioms (1)

domain assumption The charger and battery cavities interact independently with the auxiliary cavity.
Stated in the setup description.

pith-pipeline@v0.9.0 · 5429 in / 1095 out tokens · 68330 ms · 2026-05-14T20:02:27.388811+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
nonzero dissipation of the auxiliary cavity induces a nonreciprocal exchange... steady-state energy stored in the battery significantly exceeds that in the charger (Eqs. 12-14, η_ss_ac)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
Hamiltonian with coherent couplings J_ab â b† + J_bc ĉ b† + J_ac e^{iθ} â ĉ† + H.c. and Lindblad decays κ_j L_j[ρ]

Reference graph

Works this paper leans on

54 extracted references · 54 canonical work pages

[1]

Quach, G

J. Quach, G. Cerullo, and T. Virgili, Quantum batteries: The future of energy storage?, Joule7, 2195 (2023)

work page 2023
[2]

Campaioli, S

F. Campaioli, S. Gherardini, J. Q. Quach, M. Polini, and G. M. Andolina, Colloquium: Quantum batteries, Rev. Mod. Phys.96, 031001 (2024)

work page 2024
[3]

Alicki and M

R. Alicki and M. Fannes, Entanglement boost for ex- tractable work from ensembles of quantum batteries, Phys. Rev. E87, 042123 (2013)

work page 2013
[4]

Ferraro, M

D. Ferraro, M. Campisi, G. M. Andolina, V. Pellegrini, and M. Polini, High-power collective charging of a solid- state quantum battery, Phys. Rev. Lett.120, 117702 (2018)

work page 2018
[5]

R. H. Dicke, Coherence in spontaneous radiation pro- cesses, Phys. Rev.93, 99 (1954)

work page 1954
[6]

T. P. Le, J. Levinsen, K. Modi, M. M. Parish, and F. A. Pollock, Spin-chain model of a many-body quantum bat- tery, Phys. Rev. A97, 022106 (2018)

work page 2018
[7]

Juli` a-Farr´ e, T

S. Juli` a-Farr´ e, T. Salamon, A. Riera, M. N. Bera, and M. Lewenstein, Bounds on the capacity and power of quantum batteries, Phys. Rev. Res.2, 023113 (2020)

work page 2020
[8]

Zhang and M

X. Zhang and M. Blaauboer, Enhanced energy transfer in a dicke quantum battery, Frontiers in Physics10(2023)

work page 2023
[9]

Rossini, G

D. Rossini, G. M. Andolina, D. Rosa, M. Carrega, and M. Polini, Quantum advantage in the charging process of Sachdev-Ye-Kitaev batteries, Phys. Rev. Lett.125, 236402 (2020)

work page 2020
[10]

G. M. Andolina, V. Stanzione, V. Giovannetti, and M. Polini, Genuine quantum advantage in anharmonic bosonic quantum batteries, Phys. Rev. Lett.134, 240403 (2025)

work page 2025
[11]

Rinaldi, R

D. Rinaldi, R. Filip, D. Gerace, and G. Guarnieri, Re- liable quantum advantage in quantum battery charging, Phys. Rev. A112, 012205 (2025)

work page 2025
[12]

C.-K. Hu, J. Qiu, P. J. P. Souza, J. Yuan, Y. Zhou, L. Zhang, J. Chu, X. Pan, L. Hu, J. Li, Y. Xu, Y. Zhong, S. Liu, F. Yan, D. Tan, R. Bachelard, C. J. Villas-Boas, A. C. Santos, and D. Yu, Optimal charging of a supercon- ducting quantum battery, Quantum Science and Technol- ogy7, 045018 (2022)

work page 2022
[13]

J. Q. Quach, K. E. McGhee, L. Ganzer, D. M. Rouse, B. W. Lovett, E. M. Gauger, J. Keeling, G. Cerullo, D. G. Lidzey, and T. Virgili, Superabsorption in an organic mi- crocavity: Toward a quantum battery, Science Advances 8, eabk3160 (2022)

work page 2022
[14]

Crescente, M

A. Crescente, M. Carrega, M. Sassetti, and D. Ferraro, Ultrafast charging in a two-photon dicke quantum bat- tery, Phys. Rev. B102, 245407 (2020)

work page 2020
[15]

Yang, F.-M

D.-L. Yang, F.-M. Yang, and F.-Q. Dou, Three-level dicke quantum battery, Phys. Rev. B109, 235432 (2024)

work page 2024
[16]

D. J. Tibben, E. Della Gaspera, J. van Embden, P. Rei- neck, J. Q. Quach, F. Campaioli, and D. E. G´ omez, Ex- tending the self-discharge time of dicke quantum bat- teries using molecular triplets, PRX Energy4, 023012 (2025)

work page 2025
[17]

Shaghaghi, V

V. Shaghaghi, V. Singh, G. Benenti, and D. Rosa, Mi- cromasers as quantum batteries, Quantum Science and Technology7, 04LT01 (2022)

work page 2022
[19]

Elghaayda, A

S. Elghaayda, A. Ali, S. Al-Kuwari, A. Czerwinski, M. Mansour, and S. Haddadi, Performance of a super- conducting quantum battery, Advanced Quantum Tech- nologies8, 2400651 (2025)

work page 2025
[20]

Camposeo, T

A. Camposeo, T. Virgili, F. Lombardi, G. Cerullo, D. Pisignano, and M. Polini, Quantum batteries: A materials science perspective, Advanced Materials37, 2415073 (2025)

work page 2025
[21]

Medina, O

I. Medina, O. Culhane, F. C. Binder, G. T. Landi, and J. Goold, Anomalous discharging of quantum batteries: The ergotropic mpemba effect, Phys. Rev. Lett.134, 220402 (2025)

work page 2025
[22]

Cavaliere, G

F. Cavaliere, G. Gemme, G. Benenti, D. Ferraro, and M. Sassetti, Dynamical blockade of a reservoir for op- timal performances of a quantum battery, Communica- tions Physics8, 76 (2025)

work page 2025
[23]

R. R. Rodr´ ıguez, B. Ahmadi, G. Su´ arez, P. Mazurek, S. Barzanjeh, and P. Horodecki, Optimal quantum con- trol of charging quantum batteries, New Journal of Physics26, 043004 (2024)

work page 2024
[24]

Yang, H.-L

H.-Y. Yang, H.-L. Shi, Q.-K. Wan, K. Zhang, X.-H. Wang, and W.-L. Yang, Optimal energy storage in the tavis-cummings quantum battery, Phys. Rev. A109, 012204 (2024)

work page 2024
[25]

H.-Y. Yang, K. Zhang, X.-H. Wang, and H.-L. Shi, Opti- mal energy storage and collective charging speedup in the central-spin quantum battery, Phys. Rev. B111, 085410 (2025)

work page 2025
[26]

Catalano, S

A. Catalano, S. Giampaolo, O. Morsch, V. Giovannetti, and F. Franchini, Frustrating quantum batteries, PRX Quantum5, 030319 (2024)

work page 2024
[27]

Song, J.-L

W.-L. Song, J.-L. Wang, B. Zhou, W.-L. Yang, and J.-H. An, Self-discharging mitigated quantum battery, Phys. Rev. Lett.135, 020405 (2025)

work page 2025
[28]

Z. Niu, Y. Wu, Y. Wang, X. Rong, and J. Du, Exper- imental investigation of coherent ergotropy in a single spin system, Phys. Rev. Lett.133, 180401 (2024)

work page 2024
[29]

J. Yu, S. Wang, K. Liu, C. Zha, Y. Wu, F. Chen, Y. Ye, S. Li, Q. Zhu, S. Guo, H. Qian, H.-L. Huang, Y. Zhao, C. Ying, D. Fan, D. Wu, H. Su, H. Deng, H. Rong, K. Zhang, S. Cao, J. Lin, Y. Xu, C. Guo, N. Li, F. Liang, G. Wu, Y.-H. Huo, C.-Y. Lu, C.-Z. Peng, K. Nemoto, W. J. Munro, X. Zhu, J.-W. Pan, and M. Gong, Exper- imental demonstration of a maxwell’s d...

work page 2024
[30]

G. Zhu, Y. Chen, Y. Hasegawa, and P. Xue, Charging quantum batteries via indefinite causal order: Theory and experiment, Phys. Rev. Lett.131, 240401 (2023)

work page 2023
[31]

Maillette de Buy Wenniger, S

I. Maillette de Buy Wenniger, S. E. Thomas, M. Maffei, S. C. Wein, M. Pont, N. Belabas, S. Prasad, A. Harouri, A. Lemaˆ ıtre, I. Sagnes, N. Somaschi, A. Auff` eves, and P. Senellart, Experimental analysis of energy transfers between a quantum emitter and light fields, Phys. Rev. Lett.131, 260401 (2023)

work page 2023
[32]

D. Qu, X. Zhan, H. Lin, and P. Xue, Experimental opti- mization of charging quantum batteries through a cata- lyst system, Phys. Rev. B108, L180301 (2023)

work page 2023
[33]

Z.-G. Lu, G. Tian, X.-Y. L¨ u, and C. Shang, Topological quantum batteries, Phys. Rev. Lett.134, 180401 (2025). 8

work page 2025
[34]

Song, H.-B

W.-L. Song, H.-B. Liu, B. Zhou, W.-L. Yang, and J.-H. An, Remote charging and degradation suppression for the quantum battery, Phys. Rev. Lett.132, 090401 (2024)

work page 2024
[35]

M.-L. Hu, T. Gao, and H. Fan, Efficient wireless charging of a quantum battery, Phys. Rev. A111, 042216 (2025)

work page 2025
[36]

Yang and F.-Q

F.-M. Yang and F.-Q. Dou, Wireless energy transfer in a non-hermitian quantum battery, Phys. Rev. A112, 042205 (2025)

work page 2025
[37]

Ahmadi, P

B. Ahmadi, P. Mazurek, P. Horodecki, and S. Barzan- jeh, Nonreciprocal quantum batteries, Physical Review Letters132(2024)

work page 2024
[38]

Caloz, A

C. Caloz, A. Al` u, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-L´ eger, Electromagnetic nonreciprocity, Phys. Rev. Appl.10, 047001 (2018)

work page 2018
[39]

Lodahl, S

P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeu- tel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, Chiral quantum optics, Nature541, 473 (2017)

work page 2017
[40]

N. A. Khan, X. Zhang, C. Huang, Y. Liu, and D. He, Col- lective enhancement in nonreciprocal multimode quan- tum batteries, Phys. Rev. B112, 104318 (2025)

work page 2025
[41]

Metelmann and A

A. Metelmann and A. A. Clerk, Nonreciprocal photon transmission and amplification via reservoir engineering, Phys. Rev. X5, 021025 (2015)

work page 2015
[42]

Y. Guo, L. Cao, and J. Zhao, Nonreciprocal open quan- tum battery network in a photonic waveguide array, Phys. Rev. A111, 063520 (2025)

work page 2025
[43]

H.-W. Zhao, Y. Xie, X. Huang, and G.-F. Zhang, En- hanced charging in multibattery systems by nonreciproc- ity, Phys. Rev. A112, 022214 (2025)

work page 2025
[44]

B. Li, Y. Zuo, L.-M. Kuang, H. Jing, and C. Lee, Loss- induced quantum nonreciprocity, npj Quantum Informa- tion10, 75 (2024)

work page 2024
[45]

Huang and Y.-C

X. Huang and Y.-C. Liu, Perfect nonreciprocity by loss engineering, Phys. Rev. A107, 023703 (2023)

work page 2023
[46]

M. O. Scully and M. S. Zubairy,Quantum Optics(1997)

work page 1997
[47]

Breuer and F

H.-P. Breuer and F. Petruccione,The Theory of Open Quantum Systems(Oxford University Press, 2007)

work page 2007
[48]

D. F. Walls and G. J. Milburn,Quantum Optics, 2nd ed., Physics and Astronomy (Springer-Verlag Berlin Heidel- berg, 2008)

work page 2008
[49]

Johansson, P

J. Johansson, P. Nation, and F. Nori, QuTiP: An open- source python framework for the dynamics of open quan- tum systems, Computer Physics Communications183, 1760–1772 (2012)

work page 2012
[50]

Shastri, C

R. Shastri, C. Jiang, G.-H. Xu, B. Prasanna Venkatesh, and G. Watanabe, Dephasing enabled fast charging of quantum batteries, npj Quantum Information11, 9 (2025)

work page 2025
[51]

Farina, G

D. Farina, G. M. Andolina, A. Mari, M. Polini, and V. Giovannetti, Charger-mediated energy transfer for quantum batteries: An open-system approach, Phys. Rev. B99, 035421 (2019)

work page 2019
[52]

Chiribella, Y

G. Chiribella, Y. Yang, and R. Renner, Fundamental en- ergy requirement of reversible quantum operations, Phys. Rev. X11, 021014 (2021)

work page 2021
[53]

B. Peng, S. K. Ozdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, Loss-induced suppression and revival of lasing, Science346, 328 (2014)

work page 2014
[54]

C. W. Gardiner and M. J. Collett, Input and output in damped quantum systems: Quantum stochastic differen- tial equations and the master equation, Phys. Rev. A31, 3761 (1985)

work page 1985
[55]

Gardiner and P

C. Gardiner and P. Zoller,Quantum Noise, 3rd ed. (Springer Berlin, Heidelberg, 2004) p. 450

work page 2004