Quantum phase transition and quench dynamics in the anisotropic Rabi model

We investigate the quantum phase transition (QPT) and quench dynamics in the anisotropic Rabi model when the ratio of the qubit transition frequency to the oscillator frequency approaches infinity. Based on the Schrieffer-Wolff transformation, we find an anti-Hermitian operator that maps the original Hamiltonian into a one-dimensional oscillator Hamiltonian within the spin-down subspace. We analytically derive the eigenenergy and eigenstate of the normal and superradiant phases and demonstrate that the system undergoes a second-order quantum phase transition at a critical border. The critical border is a straight line in a two-dimensional parameter space which essentially extends the dimensionality of QPT in the Rabi model. By combining the Kibble-Zurek mechanism and the adiabatic dynamics method, we find that the residual energy vanishes as the quench time tends to zero, which is a sharp contrast to the universal scaling where the residual energy diverges in the same limit.

Figure 1

The excitation energies of the anisotropic Rabi Hamiltonian as functions of the coupling strengths ${\lambda }_1$ and ${\lambda }_2$ in the limit $\mathrm{\Omega }/w_0\gg 1$. We set $\mathrm{\Omega }=100w_0$. The vanishing of $\epsilon$ at the critical border ${\lambda }_c=1$ indicates the appearance of the QPT.

Figure 2

(a) The rescaled ground-state energy $e_G$ and (b) its second derivative $d^2{e}_G/d{\lambda }_c^2$ as functions of ${\lambda }_1$ and ${\lambda }_2$. We set $\mathrm{\Omega }=100w_0$.

Figure 3

The rescaled cavity photon number of the anisotropic Rabi Hamiltonian as functions of the coupling strengths ${\lambda }_1$ and ${\lambda }_2$ in the limit $\mathrm{\Omega }/w_0\gg 1$. We set $\mathrm{\Omega }=100w_0$. The nonzero $n_c$ value at the critical border ${\lambda }_c=1$ indicates the appearance of the QPT.

Figure 4

Residual energy $E_r$ versus the quench time ${\tau }_q$ obtained by the Kibble-Zurek mechanism.