# Quantum Synchronization and Entanglement of Dissipative Qubits Coupled to a Resonator
## 3 Numerical and Semi-Analytical Results
### RWA Validity Analysis  
To probe the validity of the RWA steady state, we consider an example of Lindblad dynamics for a single non-dissipative qubit coupled to a dissipative cavity. This model was analyzed previously in [21] using the quantum trajectory method. This method showed the bistability of the quantum trajectory wave function corresponding to two-qubit orientations even for relatively large detuning between the qubit and cavity eigenfrequencies compared to their coupling strength \(\lambda\). This nontrivial regime seems to be a good test case to investigate RWA validity. The bistability discussed in [21] manifests as two branches of the mean cavity occupation \(\langle a^{+}a\rangle\) for the two possible qubit eigenstates. This can be seen as two distinct peaks in the probability distribution of the cavity occupation number for spin-down, \(P_{-}(n)=\langle n,-|\hat{\rho}|n,-\rangle\), and spin up, respectively \(P_{+}(n)=\langle n,+|\hat{\rho}|n,+\rangle\), where \(n\) is the cavity eigenstate index. This bistability is reproduced by integrating the time-dependent Lindblad evolution in Figure 1, in the top two panels, showing \(P_{-}(n)\) and \(P_{+}(n)\) as functions of the detuning \(\omega_{r}=\omega_{0}-\omega\) between the cavity and the driving field. Other parameters are fixed to \(\Delta=2\lambda\), where we introduce \(\Delta=\Omega-\omega_{0}\) as the detuning between the qubit and the cavity frequencies, \(F=\lambda\) and \(\gamma=0.3\lambda\). The cavity frequency is fixed to \(\omega_{0}/\lambda=10\). The bottom panels in Figure 1 show the same quantities for the RWA steady state with good agreement between both methods. Figure 1: Distribution of the oscillator occupation number, \(n\), as a function of the detuning from the cavity resonance \((\omega-\omega_{0})/\lambda\) and \(z\)-axis spin projection. The color scale shows the amplitude \(|\rho(n,s;n,s)|\) where \(s=\uparrow,\downarrow\) is the spin up/down state. We remind readers that \(\lambda\) is the strength of the spin cavity interaction; we use it as the energy scale to facilitate the comparison between direct time integration and RWA, which depends only on the difference \(\Delta=\Omega-\omega_{0}\) between the Zeeman splitting \(\Omega\) and the cavity frequency \(\omega_{0}\), which is set to \(\Delta=2\lambda\) in this figure. Here, the driving strength is set to \(F=\lambda\), cavity dissipation is \(\gamma=0.3\lambda\), and qubit is dissipationless \(\gamma_{s}=0\). The top rows in the figures are obtained by direct time integration for \(\omega_{0}/\lambda=10\), while the bottom rows present the results from RWA. For the Lindblad time integration, numerical data show the density matrices after \(\tau_{p}=2\times 10^{4}\) excitation periods, starting from the system’s ground state for \(F=0\) at \(t=0\) (we use the same \(\tau_{p}\) for the results presentations from other cases of the time-dependent Lindblad equation). The oscillator phase spaces in the simulations were truncated to the first 100 oscillator levels (usually used for other cases). While RWA qualitatively reproduces the bistability behavior, some deviations are also visible. To be more quantitative about the agreement between RWA and the exact dynamics, we consider how the mean qubit/cavity quantity dependencies on \(\omega_{r}=\omega_{0}-\omega\) are changed with the increasing RWA parameter, \(\omega_{0}/\lambda\), which is expected to determine the RWA validity. The results are shown in Figure 2 for similar parameters as in Figure 1. While some deviations between the integration of the time-dependent Equation (1) and RWA are visible for \(\omega/\lambda=10\), the deviation rapidly drops as \(\omega/\lambda\) increases. For \(\langle\hat{S}_{x}\rangle\), \(\langle\hat{S}_{z}\rangle\), and \(\langle\hat{a}^{+}\hat{a}\rangle\), the convergence is rapid with identical results for \(\omega/\Delta=10\), while the qubit projection \(\hat{S}_{y}\) converges more slowly.