Probing the Dynamics of a Superradiant Quantum Phase Transition with a Single Trapped Ion

We demonstrate that the quantum phase transition (QPT) of the Rabi model and critical dynamics near the QPT can be probed in the setup of a single trapped ion. We first demonstrate that there exists equilibrium and nonequilibrium scaling functions of the Rabi model by finding a proper rescaling of the system parameters and observables, and show that those scaling functions are representative of the universality class to which the Rabi model belongs. We then propose a scheme that can faithfully realize the Rabi model in the limit of a large ratio of the effective atomic transition frequency to the oscillator frequency using a single trapped ion and, therefore, the QPT. It is demonstrated that the predicted universal functions can indeed be observed based on our scheme. Finally, the effects of realistic noise sources on probing the universal functions in experiments are examined.

Figure 1

Universal functions of the Rabi model. (a) Rescaled ground state population $S_s(G)$ as a function of $G=R|g-1{|}^{\gamma /\mu }$, with $\gamma =1$ and $\mu =2/3$, for different frequency ratio $R=50$, 100, 200, and 400 and coupling strength $0.9\le g\le 1$. For $G\ll 1$, it follows a power law $S_s(G)\propto G^{-\mu }$. (b) Rescaled residual atomic population $S_r(T,G)$ as a function of $T=R^{-1}{\tau }_q$ for a fixed value of the rescaled final coupling strength $G_f=R|g_f-1{|}^{\gamma /\mu }=0$, $1/2$, and 1. The quench time ${\tau }_q$ varies from $0.1/{\omega }_0$ to $100/{\omega }_0$ and $0.9\le g_f\le 1$. The same values of $R$ as in (a) have been used.

Figure 2

Universal functions $S_s(G)$ and $S_r(T,G)$ obtained from the trapped-ion Hamiltonians with the scheme of (a),(b) two lasers and (c),(d) four lasers, together with the result presented in Fig. 1 (solid line). The different points represent different $R$ with a fixed effective oscillator frequency ${\tilde{\omega }}_0/2\pi =200\mathrm{Hz}$. In (a) and (c), the quench time is ${\tau }_q=50/{\tilde{\omega }}_0=250\mathrm{ms}$. In (b) and (d), a range of quench time $0.5\le {\tau }_q\le 10\mathrm{ms}$ for three different values $G=0$, $1/2$, and 1 is used.

Figure 3

Effect of noises on the universal functions. The rates ${\mathrm{\Gamma }}_{dp}=0.1{\omega }_0$ and ${\mathrm{\Gamma }}_{c,a,h}=0.05{\omega }_0$ are used with the same system parameters used in the Fig. 1 to produce the solid line. (a) The graphs corresponding to the different $R$ do not collapse onto the universal curve; see Ref. [46] for a discussion. (b) A rather short range of quench time $0.1/{\omega }_0\le {\tau }_q\le 0.275/{\omega }_0$ is used, as the longer time evolution deviates from the universal behavior due to the effect of noise [46]; while this leads to smaller data points than Figs. 1 and 2, it nevertheless correctly reveals the substantial part of the universal function.