Dissipative Quantum Battery in the Ultrastrong Coupling Regime Between Two Oscillators

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.

💡 Research Summary

In this work the authors propose and theoretically analyze an open quantum battery consisting of two directly coupled harmonic oscillators, labeled a (the charger) and b (the battery). Oscillator a is weakly coupled to an independent thermal reservoir at temperature T_a, while oscillator b is isolated from the environment. The system Hamiltonian is

H_S = ω_a a†a + ω_b b†b + g (a† + a)(b† + b) ,

with g comparable to the bare frequencies, placing the device in the ultrastrong (0.1 ≤ g/ω < 1) or deep‑strong (g/ω > 1) coupling regimes. Because the counter‑rotating terms a†b† and ab cannot be neglected, the authors diagonalize H_S via a Hopfield transformation, obtaining normal‑mode operators A_± and frequencies ω_± that depend on g, ω_a, and ω_b. The model exhibits a critical coupling g_c = √(ω_a ω_b)/2 beyond which the bare Hamiltonian becomes unstable; however, inclusion of the diamagnetic A² term removes this instability, allowing stable operation even in the deep‑strong regime.

The open‑system dynamics are described by a global master equation derived under the Born‑Markov‑secular approximation, which remains valid for ultrastrong coupling. The dissipator acts on the normal‑mode operators A_j with rates Γ_a(±ω_j) = γ_a ω_j N(ω_j) where N(ω) is the Bose‑Einstein occupation at temperature T_a. This framework captures both energy exchange with the reservoir and the internal coherent dynamics of the coupled oscillators.

Energy stored in the battery is defined as E_b(t) = ω_b ⟨b†b⟩, while the extractable work (ergotropy) is 𝔈(t) = E_b(t) – E_β(t), where E_β(t) is the passive‑state energy obtained by minimizing over all local unitaries. For Gaussian states the passive energy can be expressed analytically in terms of the second moments ⟨b†b⟩, ⟨b⟩, and ⟨b²⟩. The authors emphasize that non‑zero ergotropy requires the simultaneous presence of beam‑splitter (a†b + h.c.) and two‑mode squeezing (a†b† + h.c.) interactions; either term alone yields zero ergotropy.

Two initial conditions are examined. (i) The global vacuum |00⟩ab leads to Rabi‑like oscillations of both stored energy and ergotropy, which gradually damp to a steady state. The steady‑state energy is independent of the dissipation rate and grows with increasing g and with higher reservoir temperature. (ii) The normal‑mode vacuum |G⟩ = |0⟩+|0⟩- eliminates all coherence terms ⟨A+†A_-⟩ while preserving population terms ⟨A_j†A_j⟩. Consequently, energy flows unidirectionally from the charger to the battery without back‑flow, and the system reaches a monotonic charging curve that saturates at a g‑dependent value. This demonstrates that appropriate state preparation can enforce stable, one‑way charging even in regimes where weak‑coupling models would predict reciprocal energy exchange.

In the deep‑strong coupling regime, the A² term prevents the super‑radiant phase transition that would otherwise occur at g > g_c, thereby maintaining dynamical stability. Numerical results show that both stored energy and ergotropy continue to increase with g and temperature, confirming that the combined beam‑splitter and squeezing mechanisms remain effective deep into the strong‑coupling domain.

The paper concludes that an open bosonic quantum battery operating in the ultrastrong or deep‑strong coupling regime can achieve significantly enhanced energy storage and non‑zero ergotropy by exploiting the interplay of beam‑splitter and parametric‑amplification interactions. Proper initialization (e.g., the two‑mode squeezed ground state) guarantees unidirectional, stable charging, while the inclusion of the diamagnetic term ensures stability across all coupling strengths. These findings broaden the design space for quantum energy devices and suggest practical routes toward high‑performance quantum batteries based on superconducting circuits, trapped‑ion platforms, or other implementations capable of reaching ultra‑strong light‑matter coupling.