# Quantum Synchronization and Entanglement of Dissipative Qubits Coupled to a Resonator
## 8 Discussion  
In this work, we used the Lindblad equation density matrix formalism for a description of a dissipative monochromatically driven resonator (oscillator cavity) interacting with one or several qubits with a very long or moderate dissipative lifetime. This system extends the seminal Jaynes-Cummings model [PERSON] (1953) to a dissipative regime, which is typical for superconducting qubits coupled to a driven cavity [PERSON] _et al._ (2008). The performed numerical simulations show that the results for the full-time-dependent Lindblad equations, Equations (4) and (5), are well reproduced by the solution of the stationary RWA Lindblad equation, Equation (6). Efficient and exact methods have been developed to numerically determine the steady state under the RWA with several qubits and a driven frequency close to cavity resonance, where many cavity eigenstates are excited. For the single qubit case, a regime of qubit bistability has been established, confirming previous results obtained using the method of quantum trajectories [PERSON] _et al._ (2016). We also developed and tested two semi-analytical approaches, which enabled us to obtain approximately accurate solutions for the exact RWA steady-state density matrix in regimes of relatively weak and strong dissipation. These correspond, respectively, to the rate equation and semiclassical approximations for the steady-state density matrix. The analytical steps of these two approaches significantly simplify the equations for the density matrix and facilitate consecutive numerical solutions, allowing us to investigate the system behavior in the vicinity of cavity resonance with many excited states. The numerical verification of these semi-analytical approaches is achieved by comparing them with the exact RWA solution, confirming their validity across a broad range of system parameters. We argue that the state of each spin is a combination of a semiclassical density matrix and quantum fluctuations. When considering the interaction of a cavity with many spins, the forces due to spin quantum fluctuations average out, and the semiclassical contribution becomes dominant. This resembles a mean-field description. At the same time, we demonstrate the existence of system behavior that cannot be described by these semi-analytical approaches. Thus, we find that for two dissipative qubits and a dissipative driven cavity, there exists a regime when qubits remain entangled, forming a singlet, in the steady-state density matrix. The existence of such a stationary regime of entangled synchronized qubits, created by a monochromatically driven cavity, is really surprising since both qubits and cavity are dissipative. Indeed, naively one could expect that the dissipation of cavities and qubits would kill the entanglement but our results  
Figure 12: The RWA calculation of the total spin projection \(\langle S_{x}\rangle\) from Figure 11 is compared to the result of the summation of the rate equation series (R)\({}^{s}\) (dotted curves). The rate equation series yield results that are nearly exact away from resonance, but they suffer from instabilities near \(\omega=\omega_{0}\). It seems that the convergence range decreases with the number of qubits with various multi-qubit resonances making the series unstable even at \(\omega-\omega_{0}=1.5\lambda\) for four qubits. show that the entanglement may be robust and survive even in the presence of dissipation. Due to the preservation of entanglement of qubits, such a regime can be considered as a real entangled quantum synchronization. We also identify regimes (see Figure 11) where up to four qubits are synchronized with the phase of monochromatic cavity driving, such that the total spin of the system increases proportionally to the number of qubits (spin halves) in the system. At the same time, we demonstrate that this synchronized regime is well described by the semiclassical approach.