# Quantum Synchronization and Entanglement of Dissipative Qubits Coupled to a Resonator
## 2 Model Description  
We consider a model of several qubits (spin half systems) interacting with one harmonic cavity driven by an external monochromatic field. The Hamiltonian of this system is as follows:  
\[\hat{H}(t)=\hbar\omega_{0}\hat{a}^{+}\hat{a}+\hbar\sum_{l}\lambda_{l}\hat{ \sigma}_{x,l}(\hat{a}+\hat{a}^{+})+\sum_{l}\frac{\hbar\Omega_{l}}{2}\hat{ \sigma}_{z,l}+2F(\hat{a}+\hat{a}^{+})\cos\omega t \tag{1}\]  
where \(\omega_{0}\) is the frequency of the cavity, \(\omega\) is the frequency of the driving field, and \(F\) is the driving strength. The operators \(\hat{a}^{+}\) and \(\hat{a}\) are the rising/lowering operators for the cavity photons. The qubits in this model are indexed by \(l\) (we consider various numbers of qubits, from 1 to 4); their contributions to the Hamiltonian are given by the Pauli matrix operators, where \(\Omega_{l}\) is the energy splitting (Zeeman splitting for spin systems) and \(\lambda_{l}\) denotes their coupling strengths with the cavity. For \(F=0\) and one qubit, the Hamiltonian (1) is reduced to the Jaynes-Cummings model [35], appearing in many systems of quantum optics and other physical systems [36; 37]. It was experimentally realized with Rydberg atoms in a resonance cavity [38]. The unitary behavior of the Jaynes-Cummings model with a monochromatic-driven cavity was studied in [39]. However, here, we consider the model with one or several qubits in the presence of driving and dissipation, which corresponds to a real situation of superconducting qubits [6]. In the rotating wave approximation (RWA), the Hamiltonian in Equation (1) is as follows:  
\[\hat{H}(t)=\hbar\omega_{0}\hat{a}^{+}\hat{a}+\hbar\sum_{l}\lambda_{l}(\hat{a} \hat{\sigma}_{l}^{+}+\hat{a}^{+}\hat{\sigma}_{l}^{-})+\sum_{l}\frac{\hbar \Omega_{l}}{2}\hat{\sigma}_{z,l}+F(\hat{a}e^{i\omega t}+\hat{a}^{+}e^{-i\omega t }) \tag{2}\]  
where we kept only the resonant terms in both the driving field and spin-cavity interactions. The Pauli matrices \(\hat{\sigma}_{l}^{+}\) and \(\hat{\sigma}_{l}^{-}\) are the rising/lowering spin operators for spin/qubit \(l\). For \(F=0\) and one qubit, the Hamiltonian (2) is reduced to the RWA form of the Jaynes-Cummings model [35]. An advantage of the RWA is that the time-dependent Hamiltonian in Equation (2) can be made stationary by moving to the rotating frame using the unitary transformation, as follows: \(\hat{U}(t)=\exp(i\omega\hat{a}^{+}\hat{a}t+i\omega\sum_{l}\hat{\sigma}_{l}^{+ }\hat{\sigma}_{l}^{-}t)\) which acts on the density matrix \(\hat{\rho}\), moving it into the rotating frame through \(\hat{\rho}_{\mathcal{R}}=\hat{U}(t)\hat{\rho}\hat{U}^{+}(t)\). This leads to a stationary rotating frame Hamiltonian [36; 37]:  
\[\hat{H}_{\mathcal{R}}=\hbar(\omega_{0}-\omega)\hat{a}^{+}\hat{a}+\hbar\lambda \sum_{l}(\hat{a}\hat{\sigma}_{l}^{+}+\hat{a}^{+}\hat{\sigma}_{l}^{-})+\sum_{l} \frac{\hbar(\Omega_{l}-\omega)}{2}\hat{\sigma}_{l,z}+F(\hat{a}+\hat{a}^{+}) \tag{3}\]  
for simplicity, we assume that the coupling strength between each qubit and cavity is the same for all qubits; thus, \(\lambda_{l}=\lambda\) for all \(l\). We describe the dissipative effects in the framework of the time-dependent Lindblad equation [14] with zero temperature dissipative terms for both qubits and cavity dynamics, as follows:  
\[\partial_{t}\hat{\rho}=-\frac{i}{\hbar}[\hat{H},\hat{\rho}]+\mathcal{L}_{d}(\rho) \tag{4}\]  
where \(\mathcal{L}_{d}\) gives the dissipative part of the Lindblad dynamics:  
\[\mathcal{L}_{d}(\hat{\rho})=\gamma\bigg{(}\hat{a}\hat{\rho}\hat{a}^{+}-\frac{ 1}{2}\hat{a}^{+}\hat{a}\hat{\rho}-\frac{1}{2}\hat{\rho}\hat{a}^{+}\hat{a}\bigg{)} +\gamma_{s}\sum_{l}\bigg{(}\hat{\sigma}_{l}^{-}\hat{\rho}\hat{\sigma}_{l}^{+}- \frac{1}{2}\hat{\sigma}_{l}^{+}\hat{\sigma}_{l}^{-}\hat{\rho}-\frac{1}{2}\hat {\rho}\hat{\sigma}_{l}^{+}\hat{\sigma}_{l}^{-}\bigg{)} \tag{5}\]  
with \(\gamma\) being the dissipation rate of the cavity and all qubits have the same dissipation rate \(\gamma_{s}\).