Manipulating counter-rotating interactions in the quantum Rabi model via modulation of the transition frequency of the two-level system

We propose a theoretical approach to manipulate the counter-rotating (CR) interactions in the quantum Rabi model by introducing a sinusoidal modulation to the transition frequency of the quantum two-level system. By choosing appropriate modulation frequency and amplitude, enhancement and suppression of the CR interactions can be achieved in the Jaynes-Cummings regime (including both weak- and strong-coupling cases) as well as the ultrastrong-coupling regime. In particular, we calculate the output excitation emission of the bosonic vacuum state under enhanced CR terms. Our results show that continuous and steady bosonic excitation emission from the bosonic vacuum can be observed in the Jaynes-Cummings regime as a consequence of the enhancement. Our approach is general and system independent, and hence it works for various physical systems described by the quantum Rabi model. As an example, we discuss the implementation of this scheme with superconducting quantum circuits.

Figure 1

Schematic of the modulated quantum Rabi model. A quantum two-level system with an energy separation ${\omega }_0$ between its excited state $|e\rangle$ and ground state $|g\rangle$ is coupled to a bosonic field mode with resonance frequency ${\omega }_c$. The coupling between the two-level system and the bosonic mode has the form $g{\sigma }_x(a+a^{†})$. The energy separation of the two-level system is modulated with a term $(1/2)\xi \nu cos\left(\nu t\right){\sigma }_z$.

Figure 2

Plots of the absolute value of the Bessel function of the first kind $|J_m\left(\xi \right)|$, and the ratios $|g_c/{\mathrm{\Delta }}_{m_0}|$ and $|{\mathrm{\Delta }}_{m_0}/{\omega }_0|$ vs the normalized modulation amplitude $\xi$ and the scaled modulation frequency $\nu /{\omega }_0$. (a) Plot of $|J_m\left(\xi \right)|$ as a function of $\xi$ at $m=0,-1$, and $-2$. (b) Dependence of $|g_c/{\mathrm{\Delta }}_{m_0}|$ on $\xi$ at $\nu =1.01{\omega }_0$ and $1.99{\omega }_0$. (c) The ratio $|{\mathrm{\Delta }}_{m_0}/{\omega }_0|$ as a function of $\nu /{\omega }_0$. (d) The ratio $|g_c/{\mathrm{\Delta }}_{m_0}|$ vs $\nu /{\omega }_0$ at $\xi =2$, 4, and 5. The insets in panels (c) and (d) show the detail of the curves for smaller values of $\nu /{\omega }_0$. Other parameters are given by $g=0.5{\omega }_0$ and ${\omega }_c={\omega }_0$.

Figure 3

(a) Fidelity $F\left(t\right)$ defined in Eq. (16) vs the time for various values of the modulation frequency: $\nu =0.2{\omega }_0=4g$ (black solid curve), $\nu =0.6{\omega }_0=12g$ (red solid curve), and $\nu ={\omega }_0=20g$ (blue solid curve). (b) Fidelity $F\left(t_s\right)$ at time $t_s=2\pi /g$ as a function of $\nu$. Other parameters are $\xi =2.40483,{\omega }_c={\omega }_0$, and $g=0.05{\omega }_0$. The initial state of the system is $(1/\sqrt{2})\left(\right|g\rangle +|e\rangle )|\alpha \rangle$, where $|\alpha \rangle$ is a coherent state with $\alpha =0.1$.

Figure 4

Time dependence of the probabilities $P_{|g,0\rangle }\left(t\right)$ (blue solid curves) and $P_{|e,1\rangle }\left(t\right)$ (red solid curves) obtained by the exact Hamiltonian $\tilde{H}\left(t\right)$ when the modulation frequency $\nu$ satisfies the sideband resonance condition at two different values: (a) $\nu ={\omega }_0$ $\left(m_0=-2\right)$ and (b) $\nu =2{\omega }_0$ $\left(m_0=-1\right)$. The black short-dashed curves are the analytical results in Eq. (18) obtained by the effective Hamiltonian ${\tilde{H}}_{\mathrm{aJC}}\left(t\right)$. Other parameters are $J_0\left(\xi \right)=0$ $\left(\xi =2.40483\right)$, $g=0.05{\omega }_0$, and ${\omega }_c={\omega }_0$. The initial state of the system is $|g,0\rangle$.

Figure 5

Maximum probability $P_{|e,1\rangle }^{\max }$ of state $|e,1\rangle$ vs the modulation frequency $\nu$. Other parameters are the same as those in Fig. 4.

Figure 6

Time dependence of the output bosonic excitation flux rate ${\mathrm{\Phi }}_{\mathrm{out}}\left(t\right)$ in various cases with resonant CR interactions $\left({\mathrm{\Delta }}_{m_0}=0\right)$ but different coupling strengths $g_r$ and $g_c$. Corresponding to $\nu ={\omega }_0$ $\left(2{\omega }_0\right)$, the sideband parameter of the resonant term is $m_0=-2$ $\left(-1\right)$. (a) The parameter $\xi$ is chosen such that the coupling strength $|g_c|=g|J_{m_0}\left(\xi \right)|$ corresponding to the resonant $m_0$ term is maximized. (b) The parameter $\xi =2.40483$ is chosen such that the rotating terms in Eq. (13) disappear $[J_0\left(\xi \right)=0]$ and the Hamiltonian is given by Eq. (17). In these two panels, the unmodulated case $\nu =0$ is presented for comparison. Other parameters are $\delta =0,g=0.05{\omega }_0$, and ${\gamma }_a={\gamma }_c=0.02{\omega }_0$. The initial state of the system is $|g,0\rangle$.

Figure 7

Steady-state output excitation flux rate ${\mathrm{\Phi }}_{\mathrm{out}}^{\mathrm{ss}}$ vs the modulation frequency $\nu$ for $\xi =1.84118$ and 2.40483. Other parameters are $\delta =0,g=0.05{\omega }_0$, and ${\gamma }_a={\gamma }_c=0.02{\omega }_0$. The initial state of the system is $|g,0\rangle$.

Figure 8

(a) Time dependence of the fidelity $F\left(t\right)$ at various values of modulation frequency: $\nu =0$ (black solid curve), $\nu =5{\omega }_0$ (red solid curve), and $\nu =30{\omega }_0$ (blue solid curve). (b) Fidelity $F\left(t_s\right)$ at time $t_s=2\pi /g$ as a function of $\nu$. Other parameters are $\xi =2.21868$ $[J_0\left(\xi \right)=0.1],{\omega }_c={\omega }_0$, and $g=0.5{\omega }_0$. The initial state of the system is $(1/\sqrt{2})\left(\right|g\rangle +|e\rangle )|\alpha \rangle$, where $|\alpha \rangle$ is a coherent state with $\alpha =0.1$.

Figure 9

Time dependence of the probability $P_{|g,0\rangle }\left(t\right)$ of the system in state $|g,0\rangle$ at various values of $\nu$: $\nu =0$ (black dashed curve), $\nu =5{\omega }_0$ (blue short-dashed curve), and $\nu =30{\omega }_0$ (red solid curve). Other parameters are $\xi =2.21868$ $[J_0\left(\xi \right)=0.1],g=0.5{\omega }_0$, and ${\omega }_c={\omega }_0$. The initial state of the system is $|g,0\rangle$.

Figure 10

Time dependance of the probabilities $P_{|e,0\rangle }\left(t\right)$ (blue solid curves) and $P_{|g,1\rangle }\left(t\right)$ (red solid curves) obtained by the exact Hamiltonian (8) at several values of $\nu$: (a) $\nu =0$, (b) $\nu =10{\omega }_0$, and (c) $\nu =30{\omega }_0$. The black short-dashed curves in panels (b) and (c) are the Rabi oscillations between the states $|e,0\rangle$ and $|g,1\rangle$ governed by the effective JC Hamiltonian $\left(27\right)$. Other parameters are $g=0.5{\omega }_0,\xi =2.21868$ $[J_0\left(\xi \right)=0.1]$, and ${\omega }_c={\omega }_0$. The initial state of the system is $|e,0\rangle$.