Behavior of quantum coherence in the ultrastrong and deep strong coupling regimes of light-matter system

The ultrastrong and deep strong coupling regimes exhibit a variety of intriguing physical phenomena. In this work, we utilize the Hopfield model of a two-mode bosonic system, with each mode interacts with a heat reservoir, to research the behavior of quantum coherence. Our results indicate that a coupled oscillator system can exhibit significant quantum coherence in the ultrastrong and deep strong coupling regimes. In the ground state, the photon-mode and the matter-mode coherences are equal. The larger coherences that encompass the photon mode, the matter mode, and the overall system are achieved at lower optical frequencies and with increased coupling strengths. Notably, the the beam-splitter and phase rotation terms alone does not generate coherences for either total coherence or subsystem coherences; instead, the generation of quantum coherences originates from the one-mode and two-mode squeezing terms. When heat environments are present, the total coherence can be enhanced by the the beam-splitter and phase rotation terms, while it has no effect on subsystem coherences. Moreover, when the one-mode and two-mode squeezing terms and the the beam-splitter and phase rotation terms are considered together, the total coherence increases with stronger coupling. We also observe that lower frequencies maximize total coherence in the deep strong coupling regime. These results demonstrate that the ultrastrong and deep strong coupling regimes give rise to novel characteristics of quantum coherence. This work provides valuable insights into the quantum coherence properties, particularly in the ultrastrong and deep strong coupling regimes between light and matter and may have potential applications in quantum information processing.

💡 Research Summary

In this work the authors investigate quantum coherence in light‑matter systems operating in the ultrastrong (g/ω > 0.1) and deep‑strong (g/ω > 1) coupling regimes. They adopt the full Hopfield model for two bosonic modes—a photon mode a and a matter mode b—each coupled to its own thermal reservoir at temperatures T_a and T_b. The Hamiltonian is split into three parts: the free part H₀, the resonant exchange part H_res (beam‑splitter and phase‑rotation terms), and the anti‑resonant part H_anti (one‑mode and two‑mode squeezing terms). The diamagnetic term D = g²/ω_b is retained, which is essential for gauge‑invariant descriptions in the ultrastrong regime.

The system‑environment interaction is treated with a global master equation derived under the Born‑Markov‑secular approximation, yielding a Lindblad form with Ohmic spectral densities (γ ω for the photon bath and κ ω for the matter bath). The master equation provides differential equations for the second‑order moments of the polariton operators, which are solved to obtain the steady‑state covariance matrix σ.

Quantum coherence is quantified using the relative‑entropy‑based measure. For each mode the authors compute C_a and C_b, while the total coherence of the bipartite Gaussian state is C_tot = S(ρ_th) − S(ρ), where S denotes the von Neumann entropy expressed through the symplectic eigenvalues of σ.

Key findings are as follows:

1. Role of Interaction Terms – When only H_res (beam‑splitter and phase‑rotation) is present, both total and subsystem coherences remain essentially zero; these terms alone do not generate coherence. In contrast, the squeezing terms in H_anti alone produce substantial coherence, indicating that one‑mode and two‑mode squeezing are the primary sources of quantum coherence in this model.

2. Dependence on Coupling Strength – As the normalized coupling g/ω increases from the weak to the ultrastrong regime, the subsystem coherences C_a and C_b rise together and become nearly equal. In the deep‑strong regime (g/ω ≫ 1) the total coherence C_tot grows dramatically, reflecting the enhanced non‑classical correlations induced by strong squeezing.

3. Frequency Effects – Lower optical frequencies (ω_a < ω_b) favor larger total coherence. The authors show that for a fixed coupling strength, decreasing the photon frequency enhances the squeezing parameters, thereby maximizing C_tot.

4. Thermal Environment – Identical bath temperatures (T_a = T_b) lead to symmetric occupation numbers N(ω_j). Raising the temperature increases the diagonal elements of σ, which raises the entropy of the state and reduces C_tot, while the off‑diagonal squeezing‑induced elements remain temperature‑independent. When H_res is added to the full Hamiltonian, the presence of thermal baths can slightly increase C_tot, because the beam‑splitter and phase‑rotation terms facilitate energy exchange that amplifies the off‑diagonal correlations, although subsystem coherences remain unaffected.

5. Combined Interaction – With both H_res and H_anti present, C_tot continues to increase with coupling strength, especially in the deep‑strong regime where the two squeezing mechanisms reinforce each other. This synergistic effect is absent when either term is omitted.

6. Practical Implications – The study suggests that optimal quantum‑coherence resources are achieved at low temperatures, strong coupling, and low photon frequencies. Since the squeezing terms are the essential drivers, experimental platforms that allow controllable parametric interactions—such as superconducting circuit QED, plasmon‑polariton hybrids, and semiconductor quantum‑well structures—are promising candidates for harnessing these coherences in quantum information processing, quantum sensing, and non‑classical light generation.

Overall, the paper provides a thorough theoretical analysis of how ultrastrong and deep‑strong light‑matter coupling shapes quantum coherence, highlighting the central role of squeezing interactions and offering guidance for future experimental exploitation of coherence as a quantum resource.