# Quantum Synchronization and Entanglement of Dissipative Qubits Coupled to a Resonator
## 6 Beyond Semiclassical Analysis and Rate Equations, Entangled Synchronization
To confirm that this quantum synchronization and the entanglement of the two qubits is not an artifact of the RWA, we performed direct simulations of the time-dependent Lindblad dynamics of Equations (4) and (5). Figure 8 presents a comparison between RWA and dynamical Equations (4) and (5), showing almost perfect agreement for \(\omega/\Delta_{1}=30\). Interestingly, even if larger deviations from RWA appear when \(\omega/\Delta_{1}=30\) is lowered, the minimum \(\langle\hat{S}^{2}\rangle\) remains constant, indicating that the anti-locking of the qubits are robust to non-RWA effects, which break the (anti)symmetry between the qubits, shifting the frequency at which the minimum \(\langle\hat{S}^{2}\rangle\) is achieved from exact resonance. Even if the degree of cancellation between the qubit spins is not perfect, it is sufficient to generate steady states violating Bell inequalities [5]; this is shown in Figure 9. In the calculation of the Bell inequality, the spin projection has to be measured in two directions by two observers, giving the freedom to choose four possible spin projection measurement directions for the computation of the correlators. Since the entangled steady state is not a pure singlet state, we find that the violation of the Bell inequality is maximized with a \(\sim\)20\({}^{\circ}\) rotation of the projection directions compared to the rotation of 45\({}^{\circ}\), which is used for Bell inequality tests on pure Bell (EPR) states. To avoid any dependence on the projection angles, we also show the quantum negativity (see, e.g., [50]) of the reduced qubit pair density matrix (obtained from the total density matrix by tracing over the cavity). The negativity at resonance in Figure 9 is 0.36, which indicates a substantial degree of entanglement between the two-qubit pair. We emphasize that, as opposed to entangled states generated by pulse sequences, which have finite lifetimes, this stationary entangled state is preserved in time in the presence of dissipation and decoherence within our model. The preservation of the entangled state of two qubits is also seen in numerical studies with quantum trajectory descriptions, but in those studies, the dissipation was present only for the cavity and not for qubits [22]. Figure 8: When dissipative rates are reduced compared to Figure 7, the reduction of \(\langle S^{2}\rangle\) for antisymmetric qubit-cavity detunings \(\Delta_{1}=-\Delta_{2}=\lambda\) becomes stronger and the singlet state of the qubit pair becomes the most probable state (75% singlet probability at the minimum of \(\langle S^{2}\rangle\)). Here, the driving strength is set to \(F/\lambda=0.25\), the dissipative rates are \(\gamma=0.3\lambda\) with a weak qubit dissipation \(\gamma_{s}=10^{-3}\gamma\) (for these parameters, at resonance, \(\langle n\rangle\simeq 4F^{2}/\gamma^{2}\simeq 3\)). To confirm that the singlet formation is not an artifact of the RWA (black curve), we performed direct integration of the time-dependent Lindblad dynamics up to the total simulation time \(3\gamma_{s}^{-1}\) for increasing RWA parameter \(\omega_{0}/\lambda\) (color curves with symbols). The singlet formation is robust to non-RWA effects with the minimum \(\langle S^{2}\rangle\) remaining unchanged as \(\omega_{0}/\lambda\) is varied by an order of magnitude. Only weak non-RWA effects are visible as a small shift of the minimum from \(\omega=\omega_{0}\) and an asymmetric \(\langle S^{2}\rangle\) dependence since non-RWA effects break the symmetry between two anti-symmetrically detuned qubits. Figure 9: Bell inequality violation and negativity of the steady-state RWA qubit pair with the reduced density matrix (trace done over the cavity) for the parameters of Figure 8. Since the qubit pair is in a mixture of singlet and triplet states, the polarization choice for the Bell inequality has to be adjusted to observe a Bell inequality violation (see Appendix A and Figure A2).