# Quantum Synchronization and Entanglement of Dissipative Qubits Coupled to a Resonator
## 6 Beyond Semiclassical Analysis and Rate Equations, Entangled Synchronization  
We found that the rate equation and semiclassical approaches capture complementary regimes of the dissipative quantum dynamics with, respectively, low and high damping regimes. It is, thus, interesting if we can find a regime that is not accessible to these approaches, where an exact solution of dissipative quantum dynamics is required. This regime would have to be at a weak damping near resonance drive since this is the only regime that is outside the range of validity of both approaches. However, a strong cavity excitation at resonance can lead to an effectively semiclassical overheated regime, without special quantum coherence properties. We, thus, consider the following setting, where two qubits are coupled to the cavity with opposite detunings from the cavity frequency \(\Delta_{1}=-\Delta_{2}\) (where \(\Delta_{1}=\Omega_{1}-\omega_{0}\) and \(\Delta_{2}=\Omega_{2}-\omega_{0}\) are the qubit-cavity detunings). The cavity excitation at frequency \(\omega=\omega_{0}\) does not break the symmetry between the qubits in RWA, and this seems to be a good regime to induce the entanglements between qubits even in the presence of damping. In the following, we analyze this model in detail with the RWA approach (6) and the exact time-dependent Lindblad dynamics (4) and (5). We also consider the validity of our two approximate approaches in this regime. To monitor the entangled state of the two detuned qubits, we consider their mean total spin operator \(\hat{S}^{2}\)\(\hat{S}^{2}\), whose expectation value is \(S(S+1)=2\) for the triplet \(S=1\) state of the two qubits and zero for the singlet configuration. The dependence of \(\langle\hat{S}^{2}\rangle\) on \(\omega-\omega_{0}\) is shown in Figure 7 for two cases of qubit-detuning from the cavity, comparing the antisymmetric detuning introduced above and more generic values with \((\Omega_{1}-\omega_{0})/\lambda=2,(\Omega_{2}-\omega_{0})/\lambda=3\). For antisymmetric detuning, a strong reduction of \(\langle\hat{S}^{2}\rangle\) is observed when the cavity is excited at resonance; this corresponds to the dynamics of the two qubits where they rotate together, synchronizing in opposite directions and canceling (at least partially) their total spin. This reduction is not observed for the generic detuning values for which \(\langle\hat{S}^{2}\rangle\) stays close to 2, which is its equilibrium value. Figure 6: This figure compares the spin polarization \(\langle S_{x}\rangle\) for one qubit coupled to a cavity for increasing dissipation rates. The polarization is shown as a function of \((\omega-\omega_{0})/\lambda\) for \(F=\lambda\), \(\Omega-\omega_{0}=2\lambda\). Smooth thin curves show the RWA steady state while dashed curves show the semiclassical theory. For the lowest dissipation, \(\gamma_{s}=0,\gamma=0.3\lambda\), the semiclassical theory predicts a reversal of \(\langle S_{x}\rangle\) in the range of \((\omega-\omega_{0})/\lambda\in(0,2)\), which is not present in RWA, which gives the exact quantum result. But the agreement is good outside of this range. Adding some dissipation to the spin \(\gamma_{s}=\gamma=0.3\lambda\) is enough to observe the polarization reversal in RWA. The agreement becomes very good for \(\gamma_{s}=\gamma=2\lambda\). As expected, at a higher friction, the system behaves in a more semiclassical way. Figure 7, corresponds to equal damping rates for cavities and qubits, which are below (but comparable to) the coupling strength \(\gamma/\lambda=0.2\). The entanglement between the two qubits in the asymmetric detuning case can be enhanced when the damping rate of the qubits is reduced. Such a situation is shown in Figure 8 for \(\gamma/\lambda=0.3\) and \(\gamma_{s}/\lambda=3\times 10^{-3}\). The total spin \(\langle\hat{S}^{2}\rangle\) then reduces to \(\simeq\)0.5 at resonance, indicating a higher degree of anti-locking and compensation between the two qubits.