Selective interactions in the quantum Rabi model

We demonstrate the emergence of selective $k$-photon interactions in the strong and ultrastrong coupling regimes of the quantum Rabi model with a Stark coupling term. In particular, we show that the interplay between the rotating and counterrotating terms produces multiphoton interactions whose resonance frequencies depend, due to the Stark term, on the state of the bosonic mode. We develop an analytical framework to explain these $k$-photon interactions by using time-dependent perturbation theory. Finally, we propose a method to achieve the quantum simulation of the quantum Rabi model with a Stark term by using the internal and vibrational degrees of freedom of a trapped ion, and demonstrate its performance with numerical simulations considering realistic physical parameters.

Figure 1

One-photon selective interactions of the Rabi-Stark model. Hamiltonian (1) acts during a time $t=\pi /2{\mathrm{\Omega }}_n$ and we calculate $〈a^{†}a〉$ for different ratios of ${\omega }_0/\omega$ and initial states $|e,0\rangle$ (blue), $|e,1\rangle$ (orange), $|e,2\rangle$ (purple), and $|e,3\rangle$ (green) with fixed couplings $\gamma /\omega =-0.25$ and $g/\omega =0.02$ (solid lines). If $\gamma =0$, all JC peaks would be at $\omega -{\omega }_0=0$ (dashed lines).

Figure 2

Three-photon selective interactions of the Rabi-Stark model. (a) Resonance spectrum of anti-JC-like three-photon process for state $|g,5\rangle$. After a time $t=\pi /2{\mathrm{\Omega }}_{5+}^{\left(3\right)}$, $〈a^{†}a〉$ is shown for $\gamma /\omega =-0.4$. The peaks appear shifted from ${\delta }_{5+}^{\left(3\right)}=0$ (dashed line) at ${\tilde{\delta }}_{5+}^{\left(3\right)}=0$ which corresponds to the dark curve in the $XY$ plane representing the lower values of ${log}_{10}|{\delta }_{5+}^{\left(3\right)}|$. (b) Time evolution of populations $P_{\mathrm{g},4}$ (solid) and $P_{\mathrm{e},7}$ (dashed) for initial state $|g,4\rangle$ (green) and populations $P_{g,5}$ and $P_{e,8}$ for initial state $|g,5\rangle$ (red) for $g/\omega =0.05$ (up) and $g/\omega =0.1$ (down).

Figure 3

Five-photon selective interactions of the Rabi-Stark model. (a) On the left, $\langle {\sigma }_{+}{\sigma }_{-}\rangle$ is shown after a time $t=\pi /2{\mathrm{\Omega }}_{2-}^{\left(5\right)}$, for different values of ${\omega }_0/\omega$ around ${\omega }_0^c=5\omega -\gamma (2\times 2+5)$ and initial state $|g,7\rangle$. Here, $g/\omega =0.1$ and $\gamma /\omega =0.9$. On the right, time evolution of populations $P_{e,2}$ and $P_{g,7}$ for initial state $|g,7\rangle$ and populations $P_{e,1}$ and $P_{g,6}$ for initial state $|g,8\rangle$, for ${\omega }_0/\omega =-3.227$. (b), (c) The same procedure with initial state $|g,8\rangle$ and $|g,9\rangle$, where the peaks appear for ${\omega }_0/\omega =-5.072$ and ${\omega }_0/\omega =-6.918$.

Figure 4

Selective one-photon and three-photon interactions with a trapped ion. (a) Time evolution of the mean number of phonons and populations $P_{+,2}$ (solid) and $P_{-,3}$ (dashed) starting from state $|+,2\rangle$ for $g/{\omega }^{\mathrm{R}}=0.05$, $\gamma /{\omega }^{\mathrm{R}}=-0.4$, and ${\omega }_0^{\mathrm{R}}/{\omega }^{\mathrm{R}}=3$. (b) Time evolution of the mean number of phonons and populations $P_{+,3}$ (solid) and $P_{-,0}$ (dashed) starting from state $|+,3\rangle$ for $g/{\omega }^{\mathrm{R}}=0.3$, $\gamma /{\omega }^{\mathrm{R}}=-0.1$ and ${\omega }_0^{\mathrm{R}}/{\omega }^{\mathrm{R}}=-2.4385$. Solid green lines evolve according to Eq. (10) while black squares evolve according to Eq. (8).

Figure 5

Three-photon interactions with dissipative effects for $g/\omega =0.1$, $\gamma /\omega =-0.4$, ${\omega }_0/\omega =2.317$, and initial state $|g,5\rangle$. In (a), (d), and (g) the time evolution of populations $P_{g,5}$ (solid) and $P_{e,8}$ (dashed) is shown for $\kappa /\omega =0$, $\kappa /\omega ={10}^{-4}$ and $\kappa /\omega ={10}^{-3}$, respectively. In (b) and (c), (e) and (f), and (h) and (i) the time evolution of populations $P_{e,n}$ and $P_{g,n}$ is shown, for $\kappa /\omega =0$, $\kappa /\omega ={10}^{-4}$ and $\kappa /\omega ={10}^{-3}$, respectively.