Dynamic transition of the generalized Jaynes-Cummings model: multi-particles and inter-particle interaction effects

Abstract: How environments affect dynamics of quantum systems remains a central question in understanding transitions between quantum and classical phenomena and optimizing quantum technologies. A paradigm model to address the above question is the generalized Jaynes-Cummings model, in which a two-level particle is coupled to its environment modeled by a continuum boson modes. Previous analytic solution shows that, starting from the initial state that the particle is in its excited state and the boson modes in their vacuum state, the time evolution of the probability that the particle occupies the excited state exhibits a dynamic transition as the system-environment coupling varies; when the coupling is weak, the probability decays to zero monotonically, while a finite weight of the particle is localized in the excited state when the coupling is sufficiently strong. Here, we study the dynamic transition for the case that $N$ particles are initially excited with the boson modes in their vacuum state. In particular, we access the effects of an all to all Ising type interaction we introduce between the particles. Our calculation is carried out by the non-perturbative time-dependent numerical renormalization group method. We find that the critical coupling for the transition decreases with $N$, and is suppressed (enlarged) by the anti-ferromagnetic (ferromagnetic) Ising interaction. Our results enrich understanding on environmental effects on interacting quantum systems.