Dynamical transition of the generalized Jaynes-Cummings model: Multiparticle and interparticle interaction effects

How environments affect the 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 of boson modes. Previous analytic solutions show that, starting from the initial condition that the particle is in its excited state and the boson modes are in their vacuum state, the time evolution of the probability that the particle occupies the excited state exhibits a dynamical 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 dynamical transition for the case that $N$ particles are initially excited with the boson modes in their vacuum state. The boson modes are assumed to follow a spectral function $J(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{s}$ up to a cutoff frequency ${\ensuremath{\omega}}_{c}$ with $s>0$. 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 nonperturbative numerical renormalization group method. We find that the value of the critical coupling for the transition exhibits a maximum at a finite $N$ and decreases with sufficiently large $N$, and is suppressed (enlarged) by a ferromagnetic (antiferromagnetic) Ising interaction. Our results enrich understanding of environmental effects on interacting quantum systems.