Quantum Phase Transition and Universal Dynamics in the Rabi Model

We consider the Rabi Hamiltonian, which exhibits a quantum phase transition (QPT) despite consisting only of a single-mode cavity field and a two-level atom. We prove QPT by deriving an exact solution in the limit where the atomic transition frequency in the unit of the cavity frequency tends to infinity. The effect of a finite transition frequency is studied by analytically calculating finite-frequency scaling exponents as well as performing a numerically exact diagonalization. Going beyond this equilibrium QPT setting, we prove that the dynamics under slow quenches in the vicinity of the critical point is universal; that is, the dynamics is completely characterized by critical exponents. Our analysis demonstrates that the Kibble-Zurek mechanism can precisely predict the universal scaling of residual energy for a model without spatial degrees of freedom. Moreover, we find that the onset of the universal dynamics can be observed even with a finite transition frequency.

Figure 1

Top panel: Exact solutions of the Rabi model in the $\mathrm{\Omega }/{\omega }_0\to \infty$ limit as a function of the dimensionless coupling strength $g/g_c$ for (a) the rescaled ground state energy $e_G$ (solid line) and $d^2{e}_G/d^{2}g$ (red dashed line), (b) the excitation energy $\epsilon$ (solid line) and the energy difference between the ground and the first excited state (red dashed line) showing the ground state degeneracy for $g/g_c\ge 1$, and (c) the variance of position $\mathrm{\Delta }x$ (solid line) and momentum $\mathrm{\Delta }p$ (red dashed line) quadrature of the cavity field, and $\mathrm{\Delta }x\mathrm{\Delta }p$ (dotted line). In (b) and (c), the scaling relation near the critical point is indicated. Bottom panel: A leading-order correction for finite $\mathrm{\Omega }/{\omega }_0$ at $g=g_c$ for $\mathrm{\Delta }p$, $\epsilon$, the order parameter $n_c$, and $e_G$ from top to bottom, respectively. The analytical results (lines) predict precisely the exact diagonalization results (points) for all observables. The finite-frequency scaling exponents for each observable are indicated.

Figure 2

Residual energy $E_r$ as a function of the quench time ${\tau }_q$ obtained by solving a nearly adiabatic dynamics for (a) different values of the final coupling strength $g_f$ ranging from $g_f=0.9g_c$ to $g_f=g_c$ (from bottom to top) in the $\mathrm{\Omega }/{\omega }_0\to \infty$ limit and (b) different ratios $\mathrm{\Omega }/{\omega }_0$ ranging from $\mathrm{\Omega }/{\omega }_0={10}^2$ to $\mathrm{\Omega }/{\omega }_0\to \infty$ (from bottom to top) with a fixed final coupling strength $g_f=g_c$. For $g_f=g_c$ in the $\mathrm{\Omega }/{\omega }_0\to \infty$ limit, it precisely follows the universal scaling relation (solid line), predicted by the Kibble-Zurek mechanism. Both moving $g_f$ away from $g_c$ and reducing the ratio $\mathrm{\Omega }/{\omega }_0$ result in a crossover from the universal scaling to ${\tau }_q^{-2}$ scaling (dashed line).

Figure 3

Exponents of power-law scaling ${\tau }_q^{\mu }$ for the residual energy $E_r$ presented in Fig. 2. (a) Fits obtained for different ranges of the quench time $[{\tau }_{q1},{\tau }_{q2}]$. The values for ${\tau }_{q1(q2)}$ are indicated in the figure. The exponent $\mu$ converges to the universal scaling exponent $-1/3$ for finite $\mathrm{\Omega }/{\omega }_0$. (b) Fits obtained for a range of the quench time $[{\tau }_q\mathrm{\Delta }{\tau }_q,{\tau }_q/\mathrm{\Delta }{\tau }_q]$ as a function of ${\tau }_q$ with a fixed log-scale interval ${\log }_{10}\mathrm{\Delta }{\tau }_q=6.25\times 10^{-2}$. The crossover from $\mu =-2$ to $\mu =-1/3$ as one increases $\mathrm{\Omega }/{\omega }_0$ is clearly demonstrated.