# Quantum Synchronization and Entanglement of Dissipative Qubits Coupled to a Resonator
## 2 Model Description
### Semiclassical Approximation
In the case of one-qubit with the RWA Hamiltonian Equation (3), we generalize the trial form of the density matrix to a product state:  
\[\hat{\rho}=\frac{1}{2}|\alpha\rangle\langle\alpha|\big{(}1+b_{x}\hat{\sigma}_ {x}+b_{y}\hat{\sigma}_{y}+b_{z}\hat{\sigma}_{z}\big{)} \tag{37}\]  
which is now parameterized by three real numbers, \(b_{x,y,z}\), given the Bloch sphere coordinates of the spin density matrix, in addition to the complex number \(\alpha\) parameterizing the cavity coherent state. Evaluating Equation (31), we find the following:  
\[S=\frac{1+b_{x}^{2}+b_{y}^{2}+b_{z}^{2}}{2}S_{0}(\alpha_{x},\alpha_{y})+\frac {\Omega_{r}^{2}(b_{x}^{2}+b_{y}^{2})}{2}+S_{\lambda}+S_{\gamma_{s}} \tag{38}\]  
where the first term corresponds to the single cavity described previously. In the second term, the frequency \(\Omega_{r}=\Omega-\omega\) gives the detuning between the qubit level spacing \(\Omega\) and the driving frequency. Here, we omit the qubit index, where \(\Omega=\Omega_{l}\), and \(\lambda\) denotes the qubit-cavity interaction strength. This term just describes the qubit eigenstates without qubit-cavity coupling corresponding to \(b_{x,y}=0\). The third term, \(S_{\lambda}\), describes the Hamiltonian part of the qubit-cavity coupling and all terms are proportional to \(\lambda\):  
\[S_{\lambda} =2\lambda\big{[}Fb_{x}+(\alpha_{x}b_{x}-\alpha_{y}b_{y})(\omega_{r}- b_{z}\Omega_{r})\big{]}+\gamma\lambda(\alpha_{y}b_{x}+\alpha_{x}b_{y}) \tag{39}\] \[+\frac{\lambda^{2}}{2}\Big{[}4b_{z}^{2}\Big{(}\alpha_{x}^{2}+ \alpha_{y}^{2}\Big{)}+\Big{(}4\alpha_{x}^{2}+1\Big{)}b_{y}^{2}+8\alpha_{x} \alpha_{y}b_{x}b_{y}+\Big{(}4\alpha_{y}^{2}+1\Big{)}b_{x}^{2}+(b_{z}+1)^{2} \Big{]}\]  
The final term \(S_{\gamma_{s}}\) contains all terms, which are induced by the qubit damping, as follows:  
\[\hat{H} =\omega_{r}\hat{a}^{+}\hat{a}+F(\hat{a}+\hat{a}^{+}) \tag{32}\]0  
where the first term is minimized by the qubit ground states \(b_{x},b_{y}=0\) and \(b_{z}=-1\), while the second term mixes qubit-cavity coupling and qubit relaxation. The cavity-qubit steady state is then obtained by minimizing \(S\) as a function of five parameters \(\alpha_{x,y}\), \(b_{x,y,z}\). For the case of two qubits, we generalize the trial form of the density matrix to the following:  
\[\hat{H} =\omega_{r}\hat{a}^{+}\hat{a}+F(\hat{a}+\hat{a}^{+}) \tag{32}\]1  
the expression for \(S\) for this trial function is given in the Appendix A. Our trial form for the density matrix does not include any entanglement between qubits and cavities since the trial density matrix is taken as a tensor product. The cavity's steady state is given by a coherent state and the qubits by their Bloch sphere components. We, thus, call this the semiclassical approximation to the cavity-qubit steady state. While the minimization of the semiclassical function can no longer be performed analytically in general cases, this minimization is computationally less demanding compared to full quantum calculations. It is, thus, important to identify the regimes of the Lindblad equation, where the above semi-analytical approaches can correctly reproduce its full quantum dissipative solution. Above, we describe the main elements of semiclassical approximation. Its applications to our model are described in the following sections. The important feature of this approximation is that it allows us to understand if the steady state of the system is close to classical behavior or not. In this way, we can see how quantum is the synchronization of qubits with the driven cavity. This semi-analytical method provides a better vision of system properties in the steady state.