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.