# Quantum Synchronization and Entanglement of Dissipative Qubits Coupled to a Resonator
## 1 Introduction
These developments initiated our research on the quantum synchronization of several superconducting qubits coupled to a driven dissipative resonator performed in the frame of the Lindblad master equation. There are also works on the quantum synchronization of oscillators (see, e.g., [34]), but here, we only discuss the synchronization of qubits. They conducted extensive numerical simulations of tens of thousands of Lindblad equation components, enabling the establishment of various nontrivial regimes of quantum synchronization and entanglements involving one, two, three, or four qubits strongly coupled with a driven resonator in the presence of dissipation. It is shown that in regimes of either weak or strong dissipation, the system behavior can be well described by semi-analytical methods of weak perturbation theory or semiclassical theory, respectively. However, certain nontrivial dissipative regimes with preserved entanglement of dissipative qubits remain inaccessible for such semi-analytical descriptions. At the time when [PERSON] discovered clock synchronization in 1665, improving clock accuracy was a strategic military task for ship navigation, which saw significant enhancements based on this discovery (see [1; 2; 3])). We believe that quantum synchronization of qubits will also play a fundamental role in quantum computing. Indeed, with a decrease in size scales, the quantum and dissipative effects start to play more important roles (see, e.g., [4; 6]). In such a regime, a quantum synchronization of several qubits becomes important for qubit control and the operation of quantum gates. At the same time, it is imperative to understand to what extent the dynamics of dissipative qubits become classical and, thus, it can be described in the semiclassical approximation, meaning that dissipation suppresses quantum effects, or that quantum features and entanglement resist and remain present even for dissipative qubits. Our results are obtained in the frame of the Lindblad equation for a driven dissipative resonator coupled with dissipative qubits, establishing the existence of two regimes: one when the synchronization of qubits is well described by the semiclassical approach so that quantum features are suppressed by dissipation and another when entanglement is preserved for dissipative synchronized qubits, thus being essentially quantum. Our results are based on an extensive numerical and semi-analytical analysis of the Lindblad equation of the system of driven dissipative resonators coupled to up to four dissipative qubits. A certain advantage of the Lindblad description is that it allows obtaining the steady state of the system corresponding to infinite times while the approach of quantum trajectories always gives characteristics fluctuating with time and very long times are needed to obtain steady-state properties. This was the case of results presented in [21; 22] with one or two non-dissipative qubits; this is not the case in experiments with superconducting qubits (see, e.g., [6; 7; 8; 9; 10; 11]). Our results clearly show the presence of a synchronized steady state with up to four qubits. They also show that the entanglement of qubits can survive in the system's steady state. We begin this article with the model description, followed by the introduction of semi-analytical methods, which allow for a better understanding of the physical effects appearing in the direct numerical simulations of the Lindblad equation of the system. This article is organized as follows: Section 2 describes the model, numerical methods, and semi-analytical approximations. Section 3 presents the obtained numerical results and RWA validity tests of RWA; the results and validity of the weak damping rate equation approach are presented in Section 4 and those of the strong damping semiclassical regime are presented in Section 5. The regimes beyond the validity of semi-analytical approaches are analyzed in Section 6, the synchronization of several qubits is described in Section 7, and a discussion is presented in Section 8.