# Quantum Synchronization and Entanglement of Dissipative Qubits Coupled to a Resonator
## 2 Model Description
### Semiclassical Approximation  
While the perturbative rate equation approach works at weak dissipation, its convergence fails in the opposite limit of strong damping. In this regime, an approximate description of the system's steady state can be obtained from a semiclassical trial density matrix. In this work, we adopt a variation of the usual semiclassical approach, where some quantum mechanical operators are replaced by their average values (for a recent example of this approach, see [48]). Instead, we apply the Lindblad superoperator to a semiclassical tensor product ansatz and minimize the residual over variational parameters, characterizing the state of the cavity and qubit polarizations. This approach provides variational approximations to the kernel of the Lindblad superoperator. The functional to be minimized can be computed analytically, and its value offers an estimation of the accuracy of the semiclassical ansatz. The possibility of checking the accuracy of the semiclassical approximation without comparing it to quantum results was one of our motivations for the variational strategy; another advantage is that since we compute the exact action of the Lindblad superoperator on the trial density matrix, some information on quantum fluctuations is preserved, which is not the case when operators are replaced by their average value. We start by presenting this approach in the simple model case of a driven cavity, for which the RWA steady state is a pure coherent state and, thus, its minimization gives the steady state exactly and then generalizes this approach to a cavity coupled to qubits. Our approach is to minimize the (square) matrix norm \(S\) of the Lindblad superoperator applied to some trial form of the density matrix, as follows:  
\[S=\mathrm{Tr}\left[\mathcal{L}(\hat{\rho})\right]^{+}\mathcal{L}(\rho) \tag{31}\]  
We first consider the simple model of a driven dissipative cavity in the RWA, as follows:  
\[\hat{H} =\omega_{r}\hat{a}^{+}\hat{a}+F(\hat{a}+\hat{a}^{+}) \tag{32}\] \[\mathcal{L}(\hat{\rho}) =\left[\mathcal{L}(\hat{\rho})\right]^{+}=-i[\hat{H},\hat{\rho}] +\gamma\left(\hat{a}\hat{\rho}\hat{a}^{+}-\frac{1}{2}\hat{a}^{+}\hat{a}\hat{ \rho}-\frac{1}{2}\hat{\rho}\hat{a}^{+}\hat{a}\right) \tag{33}\]  
where \(\omega_{r}=\omega_{0}-\omega\) is the detuning between the cavity frequency and the driving field, \(F\), at frequency \(\omega\). We consider a trial density matrix given by a pure coherent state, parameterized by a complex \(\alpha=\alpha_{x}+i\alpha_{y}\), where (\(\alpha_{x}\) and \(\alpha_{y}\) are the real and imaginary parts of \(\alpha\)):  
\[\hat{\rho}_{\alpha}=|\alpha\rangle\langle\alpha| \tag{34}\]  
where \(\hat{a}|\alpha\rangle=\alpha|\alpha\rangle\). The functional Equation (31) can then be evaluated as follows:  
\[S_{0}(\alpha)=S(|\alpha\rangle\langle\alpha|)=2\left[F^{2}+2F\alpha_{x}\omega _{r}+(\alpha_{x}^{2}+\alpha_{y}^{2})\omega_{r}^{2}\right]+2F\alpha_{y}\gamma+ \gamma^{2}\frac{\alpha_{x}^{2}+\alpha_{y}^{2}}{2} \tag{35}\]  
The first bracketed term comes from the Hamiltonian dynamics while the other terms include damping effects. We can check that the minimum \(S_{0}=0\) is achieved at the following:  
\[\alpha_{x}=-\frac{4F\omega_{r}}{\gamma^{2}+4\omega_{r}^{2}}\,\ \alpha_{y}=- \frac{2F\gamma}{\gamma^{2}+4\omega_{r}^{2}} \tag{36}\]  
which is the classical response function of an oscillator at frequency \(\omega_{0}\) excited at frequency \(\omega\) with a detuning of \(\omega_{r}=\omega_{0}-\omega\) and a relaxation time of \(\tau=2\gamma^{-1}\). Since \(S_{0}=0\), this solution is exact. We now generalize this approach to the case of a cavity coupled to one and then more qubits.