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arxiv: 2602.15235 · v1 · submitted 2026-02-16 · 🪐 quant-ph

Dissipative Quantum Battery in the Ultrastrong Coupling Regime Between Two Oscillators

Yu-qiang Liu , Yi-jia Yang , Zheng Liu , Bao-qing Guo , Ting-ting Ma , Zunlue Zhu , Wuming Liu , Xingdong Zhao
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Chang-shui Yu
This is my paper

Pith reviewed 2026-05-15 21:20 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum batteryultrastrong couplingergotropybosonic systemtwo-mode squeezingopen quantum systemsdissipative dynamicsvector potential term
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The pith

An open quantum battery using two ultrastrongly coupled oscillators stores more energy and ergotropy across wider temperatures.

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

The paper proposes an open quantum battery built from a two-mode ultrastrongly coupled bosonic system in which one oscillator couples to an independent heat reservoir. It shows that charging energy and ergotropy both rise markedly once the coupling enters the ultra-strong regime, and this improvement holds over a broader temperature window during the transient phase. The gain stems from the simultaneous presence of beam-splitter and two-mode squeezing interactions together with the squared electromagnetic vector potential term, which also blocks phase transitions in the deep-strong regime. Choosing the initial two-mode squeezed ground state produces unidirectional energy flow into the battery mode. Steady-state stored energy and ergotropy likewise increase with larger temperature and stronger coupling.

Core claim

In a dissipative two-mode ultrastrongly coupled bosonic system that functions as a quantum battery, both the charging energy and the ergotropy are enhanced inside the ultra-strong coupling regime and over a wider temperature range during transient evolution. The enhancement requires the joint action of beam-splitter and parametric amplification (squeezing) couplings; either interaction alone produces zero ergotropy. The squared electromagnetic vector potential term stabilizes the Hamiltonian, prevents phase transitions, and permits high performance even in the deep-strong coupling regime. Unidirectional energy flow is obtained by preparing the system in its two-mode squeezed ground state, so

What carries the argument

Two-mode ultrastrongly coupled bosonic Hamiltonian containing beam-splitter coupling, two-mode squeezing, and the squared vector potential term, with one mode coupled to a thermal reservoir.

If this is right

  • Charging energy and ergotropy increase with coupling strength once the ultra-strong regime is reached.
  • Steady-state stored energy and ergotropy improve at both larger temperatures and stronger couplings.
  • Pure beam-splitter or pure two-mode squeezing interaction yields zero ergotropy.
  • Initial preparation in the two-mode squeezed ground state produces unidirectional energy flow into the battery.
  • The squared vector potential term enables significant charging energy and high ergotropy in the deep-strong regime.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could reduce the need for cryogenic cooling in practical quantum energy storage devices.
  • Similar enhancements may appear in other coupled-oscillator platforms such as circuit QED or optomechanical systems.
  • The requirement for mixed beam-splitter and squeezing terms suggests a design rule for engineering ergotropy in bosonic batteries.
  • Transient dynamics rather than steady state may offer the most practical operating window for such devices.

Load-bearing premise

The two-mode Hamiltonian that includes the squared electromagnetic vector potential term remains valid and prevents phase transitions without additional corrections in the deep-strong coupling regime.

What would settle it

Direct observation of a phase transition or a drop in charging energy when the squared vector potential term is retained at deep-strong coupling strengths would falsify the claimed enhancement.

Figures

Figures reproduced from arXiv: 2602.15235 by Bao-qing Guo, Chang-shui Yu, Ting-ting Ma, Wuming Liu, Xingdong Zhao, Yi-jia Yang, Yu-qiang Liu, Zheng Liu, Zunlue Zhu.

Figure 1
Figure 1. Figure 1: Schematic representation of the bipartite quantum battery model. It consists of two oscillators with cavity mode a (the charger) and matter b mode (the battery) that are directly coupled to one another, and the mode a is coupled to an independent reservoir. Here, we isolate the battery from the heat reservoir, but the charger is driven by a heat reservoir at temperature T. In this context, we propose an op… view at source ↗
Figure 2
Figure 2. Figure 2: The behavior of the stored energy, ∆Eb, and its corresponding ergotropy, E are presented as a function of time t for various coupling strengths in panels (a) and (b), and different temperatures in panels (c) and (d) under resonant condition when considering the initial state ρ(0) = |00⟩ab⟨00|. In panels (c) and (d), the red solid, blue dashed, green dashed￾dotted, and magenta dotted lines correspond to tem… view at source ↗
Figure 3
Figure 3. Figure 3: The behavior of the stored energy ∆Eb, and its corresponding ergotropy E are presented as a function of time t for various coupling strengths in panels (a) and (b), and different temperatures in panels (c) and (d) under resonant condition considering the initial state |G⟩ = |0⟩+|0⟩−. The following parameters are used: (a) and (b) Ta = ωb; (c) and (d) g = 0.3ωb. Other parameters are ωa = ωb = ω, and γ a = 1… view at source ↗
Figure 4
Figure 4. Figure 4: The squeezing parameters r± are plotted as a function of the coupling strength g. The parameter can take ωa = ωb = ω. As illustrated in [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Behavior of the stored energy ∆Eb and corre￾sponding ergotropy E as functions of the coupling strengths gbs and gsq (a, b), and temperature Ta and coupling strength g (c, d) under resonant condition. The parameters are set as Ta = ωb for panels (a) and (b), gbs = gsq = g = 0.3ωb for pan￾els (c) and (d), and other parameters can take ωa = ωb = ω, with all parameters expressed in units of the frequency ωb. T… view at source ↗
Figure 6
Figure 6. Figure 6: The x (a) and ergotropy (b) as functions of tem￾perature for different coupling strengths. The parameter can take ωa = ωb = ω, with all parameters expressed in units of the frequency ωb. To optimize the quantum battery’s performance, one can rewrite the interaction Hamiltonian of the charger and battery as HI = gbs(a † b + ab† ) + gsq(a † b † + ab), (25) where the terms proportional to gbs and gsq represen… view at source ↗
Figure 7
Figure 7. Figure 7: Behavior of the stored energy ∆Eb (a, c) and ergotropy E (b, d) as a function of the time t for different coupling strengths gbs and gsq at resonance based on initial state |G⟩ = |0⟩+|0⟩−. In panels (a) and (b), the squeezing coupling parameter is set to gsq = 0, while in panels (c) and (d), the beam-splitter coupling parameter is set to gbs = 0. Other parameters are ωa = ωb = ω, γ a = 10−3ωb and Ta = ωb, … view at source ↗
Figure 8
Figure 8. Figure 8: Behavior of the stored energy ∆Eb for the Hopfield model as shown in Eq. (28), depicted in panels (a) and (c), and its corresponding ergotropy E shown in panels (b) and (d), are plotted as functions of time t for various coupling strengths g and temperatures Ta. In panels (a) and (b), the temperature parameter is set to Ta = ωb, and in panels (c) and (d), the coupling strength is fixed at g/ωb = 1. The rem… view at source ↗
Figure 9
Figure 9. Figure 9: The ratio of the ergotropy to the stored energy E/∆Eb as a function of the coupling strength g for two cou￾pled oscillators without A2 term (a) and the Hopfield model (refer to Eq. (28)) (b). The parameter can take ωa = ωb = ω and Ta/ωb = 1. Besides, coupled oscillators systems can be imple￾mented using current solid-state technologies, such as su￾perconducting quantum circuits [27, 45, 46, 50], interact￾i… view at source ↗
read the original abstract

In this work, we propose an open quantum battery that stores and releases energy by employing a two-mode ultrastrongly coupled bosonic system, with one mode (the charger) coupled to an independent heat reservoir. Our results demonstrate that both the charging energy and ergotropy of the quantum batteries can be significantly enhanced within the ultra-strong coupling regime and across a broader temperature range in transient time. A unidirectional energy flow is achieved by controlling the system's initial state through its two-mode squeezed ground state. Furthermore, we show that the steady-state stored energy, along with its corresponding ergotropy, can be enhanced at larger temperatures and stronger coupling strengths. Notably, a purely beam-splitter or two-mode squeezing interaction yields zero ergotropy. These findings indicate that the enhanced stored energy and ergotropy of the quantum battery arises principally from the combined effects of beam-splitter and parametric amplification (squeezing) couplings. In addition, the presence of the squared electromagnetic vector potential term can prevent a phase transition and achieve a significant charging energy and high ergotropy in the deep-strong coupling regime. The results presented herein enhance our understanding of the operating principles of open bosonic quantum batteries.

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.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes an open quantum battery realized by two ultrastrongly coupled bosonic modes, with one mode (the charger) coupled to an independent thermal reservoir. It reports that both charging energy and ergotropy are substantially enhanced in the ultrastrong-coupling regime and over a wider temperature window during transient dynamics, that unidirectional energy flow is obtained by preparing the two-mode squeezed ground state, and that the diamagnetic A² term prevents a phase transition while enabling high performance in the deep-strong regime. The enhancement is attributed to the simultaneous presence of beam-splitter and parametric-amplification couplings; purely beam-splitter or purely two-mode-squeezing interactions are stated to yield zero ergotropy. Steady-state stored energy and ergotropy are also claimed to increase with temperature and coupling strength.

Significance. If the central claims are substantiated, the work would clarify how ultrastrong coupling and the diamagnetic term can be exploited to improve the performance of dissipative bosonic quantum batteries, extending their operational temperature range and demonstrating a concrete route to unidirectional charging via initial-state engineering. The explicit contrast between combined versus single-interaction Hamiltonians supplies a useful design principle for future open-system quantum batteries.

major comments (3)
  1. [model Hamiltonian / Eq. (1)] The central claim that the two-mode Hamiltonian including the A² term remains valid and prevents phase transitions in the deep-strong-coupling regime (abstract and model section) is load-bearing for all reported enhancements. No explicit diagonalization, critical-coupling calculation, or comparison against higher-order corrections is supplied to confirm the absence of instabilities or the validity of the standard Lindblad form.
  2. [open-system dynamics] The master-equation derivation for the ultrastrong-coupling regime (open-system section) applies a standard Lindblad dissipator without USC-specific counter-rotating or bath-coupling corrections. This truncation directly affects the reported transient unidirectional flow and steady-state values; an explicit check of the secular approximation or a comparison with a USC-consistent master equation is required.
  3. [ergotropy calculation] The statements that pure beam-splitter or pure two-mode-squeezing interactions produce zero ergotropy while their combination yields finite ergotropy (results section) are central to the interpretation. The explicit expressions for the ergotropy (or the passive-state energy) used to obtain these conclusions are not provided, preventing verification that the enhancement is not an artifact of the chosen initial state or truncation.
minor comments (2)
  1. [figures] Figure captions and axis labels should explicitly state the units of energy and the precise definition of the coupling strength g/ω used in each panel.
  2. [results] The phrase “broader temperature range” is used without a quantitative benchmark; a reference curve for the weak-coupling limit should be added to the temperature-sweep plots.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. We have carefully addressed each of the major comments and revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [model Hamiltonian / Eq. (1)] The central claim that the two-mode Hamiltonian including the A² term remains valid and prevents phase transitions in the deep-strong-coupling regime (abstract and model section) is load-bearing for all reported enhancements. No explicit diagonalization, critical-coupling calculation, or comparison against higher-order corrections is supplied to confirm the absence of instabilities or the validity of the standard Lindblad form.

    Authors: We appreciate this observation. To substantiate the claim, we have added an appendix with the explicit diagonalization of the Hamiltonian in the presence of the A² term. This shows that the critical coupling for the phase transition is pushed to infinity, preventing instabilities in the deep-strong regime. We also include a brief discussion on the validity of the model and the Lindblad form within the parameter range considered. revision: yes

  2. Referee: [open-system dynamics] The master-equation derivation for the ultrastrong-coupling regime (open-system section) applies a standard Lindblad dissipator without USC-specific counter-rotating or bath-coupling corrections. This truncation directly affects the reported transient unidirectional flow and steady-state values; an explicit check of the secular approximation or a comparison with a USC-consistent master equation is required.

    Authors: We agree that this is an important point. In the revised version, we have performed a comparison with a master equation that accounts for USC effects (following the approach in relevant literature). The results confirm that the key features—enhanced charging energy, ergotropy, and unidirectional flow—persist, although quantitative values may shift slightly. We have added this analysis to the supplementary information and clarified the approximations used. revision: yes

  3. Referee: [ergotropy calculation] The statements that pure beam-splitter or pure two-mode-squeezing interactions produce zero ergotropy while their combination yields finite ergotropy (results section) are central to the interpretation. The explicit expressions for the ergotropy (or the passive-state energy) used to obtain these conclusions are not provided, preventing verification that the enhancement is not an artifact of the chosen initial state or truncation.

    Authors: We thank the referee for highlighting this omission. The ergotropy is calculated as the difference between the mean energy and the energy of the passive state, where the passive state is obtained by sorting the eigenvalues of the density matrix in decreasing order and assigning them to the Hamiltonian eigenvalues in increasing order. We have now included the explicit expressions and the calculations for the pure interaction cases in the main text, showing that the resulting states are passive, hence zero ergotropy, while the combined interaction leads to non-passive states with finite ergotropy. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper derives its claims about enhanced charging energy, ergotropy, and unidirectional flow directly from the time evolution of the proposed two-mode ultrastrongly coupled Hamiltonian (including the A² term) under an open-system master equation, starting from specified initial states such as the two-mode squeezed ground state. No parameters are fitted to data and then relabeled as predictions; no self-definitional loops appear where a quantity is defined in terms of the result it is claimed to produce; and no load-bearing uniqueness theorems or ansatzes are imported solely via self-citation. The central results follow from solving the dynamical equations under the stated model assumptions, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The model rests on standard bosonic-mode and open-system assumptions typical of quantum optics; no new entities are introduced and no parameters are explicitly fitted in the abstract.

axioms (2)
  • domain assumption Two bosonic modes interact via ultrastrong coupling that includes both beam-splitter and parametric-amplification terms plus the squared vector potential.
    Invoked to define the Hamiltonian whose dynamics produce the claimed enhancements.
  • domain assumption One mode couples to an independent thermal reservoir while the overall system remains Markovian.
    Required for the dissipative charging and steady-state analysis.

pith-pipeline@v0.9.0 · 5540 in / 1306 out tokens · 41047 ms · 2026-05-15T21:20:37.250953+00:00 · methodology

discussion (0)

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Reference graph

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