Universal dynamics of the superradiant phase transition in the anisotropic quantum Rabi model

Abstract

We investigate the universal nonequilibrium dynamics of the superradiant phase transition in the anisotropic quantum Rabi model. By introducing position and momentum operators, we find the generators and the displaced operators to diagonalize the anisotropic Hamiltonian, and then obtain the ground states and their excitation gaps. According to the Kibble-Zurek mechanism, we find that the excitation gap near the critical point vanishes as $\ensuremath{\omega}\ensuremath{\sim}{|\ensuremath{\epsilon}|}^{\ensuremath{\nu}z}$ with $\ensuremath{\nu}z=1/2$, and the length scale diverges according to $\ensuremath{\zeta}\ensuremath{\sim}{\ensuremath{\epsilon}}^{\ensuremath{-}\ensuremath{\nu}}$, which determines the universality class $(z=2,\ensuremath{\nu}=1/4)$ of the anisotropic quantum Rabi model. Moreover, by simulating the real-time dynamics across the critical point, we numerically extract the critical exponents from the phase transition delay ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}_{d}\ensuremath{\sim}{\ensuremath{\tau}}_{Q}^{\ensuremath{-}1/(1+\ensuremath{\nu}z)}$ and the diverging length scale $\mathrm{\ensuremath{\Delta}}x$, $\mathrm{\ensuremath{\Delta}}p\ensuremath{\sim}{\ensuremath{\tau}}_{Q}^{\ensuremath{\nu}/(1+\ensuremath{\nu}z)}$ (where ${\ensuremath{\tau}}_{Q}$ is the quench time). The numerical critical exponents are well consistent with the analytical ones. Our study provides a dynamic way to explore universal critical behaviors in the anisotropic quantum Rabi model.