Variational generalized rotating-wave approximation in the two-qubit quantum Rabi model

We present an analytical method for the two-qubit quantum Rabi model. While still operating in the frame of the generalized rotating-wave approximation (GRWA), our method further embraces the idea of introducing variational parameters. The optimal value of the variational parameter is determined by minimizing the energy function of the ground state. Comparing with numerically exact results, we show that our method evidently improves the accuracy of the conventional GRWA in calculating fundamental physical quantities, such as energy spectra, mean photon number, and dynamics. Interestingly, the accuracy of our method allows us to reproduce the asymptotic behavior of a mean photon number in a large frequency ratio for the ground state and investigate the quasiperiodical structure of the time evolution, which the GRWA is incapable of predicting. The applicable parameter ranges cover the ultrastrong-coupling regime, which will be helpful to recent experiments.

Figure 1

The energy spectra of the system with (a) $\mathrm{\Omega }=1.0$ and (b) $\mathrm{\Omega }=2.0$. The numerical result is obtained by an exact diagonalization of Hamiltonian Eq. (1). The results for our method and the GRWA are obtained by solving Hamiltonian Eq. (11) with an optimal $\lambda =g/(\omega +\mathrm{\Omega })$ and $\lambda =g/\omega$, respectively.

Figure 2

Mean photon number as a function of $g$ for (a) $\mathrm{\Omega }=1$ and (b) $\mathrm{\Omega }=2$, and as a function of $\mathrm{\Omega }$ for (c) $g=0.1$ and (d) $g=0.3$. We compare our results [solid line, obtained by Eqs. (C2), (C3), and (C4)] with those obtained by the numerically exact diagonalization method (open circles) and GRWA (dashed line).

Figure 3

The comparison of the mean photon number for the ground state obtained by different methods. Here we chose $g=0.1$.

Figure 4

Time evolution of $\langle {\widehat{J}}_z\rangle$ with $g=0.2$ and $\mathrm{\Omega }=2$. For the initial state, we chose $\alpha =2$. (a) The results obtained by our method [see Eq. (D9)]. (b) The exact diagonalization results. (c) The GRWA results. (d) The deviations of the analytical results (our method and the GRWA) from the numerically exact ones.

Figure 5

Population of the qubits remaining in the initial state $\left|-1_z\right.〉$ with $g=0.2$ and $\mathrm{\Omega }=2$. For the initial state, we chose $\alpha =2$. (a) The results obtained by our method [see Eq. (D10)]. (b) The exact diagonalization results. (c) The GRWA results. (d) The deviations of the analytical results (our method and the GRWA) from the numerically exact ones.