# Quantum Synchronization and Entanglement of Dissipative Qubits Coupled to a Resonator
## 1 Introduction  
The synchronization of two maritime pendulum clocks, discovered by [PERSON] in 1665 [1], forms the foundation for synchronization phenomena that appear in various systems ranging from clocks to fireflies, cardiac pacemakers, lasers, and Josephson junction (JJ) arrays (see historical survey and overview in [2; 3; 4]). With the modern development of quantum computation and quantum information (see, e.g., [5]), the exploration of quantum synchronization phenomena in the quantum realm has become significantly important. The phenomenon of quantum synchronization gains even more relevance for JJ arrays and superconducting qubits where dissipative and quantum effects are important (see, e.g., [4; 6]). A strong coupling regime involving several superconducting qubits with a microwave resonator has been experimentally realized [7; 8; 9; 10; 11], which requires investigating the effects of strong interactions in quantum dissipative-driven systems. The essential element of synchronization in classical mechanics is dissipation, which leads to the phase synchrony of coupled autonomous systems with energy supply [2]. Thus, the analysis of quantum synchronization requires the use of formalism of dissipative quantum mechanics. This formalism is known and based on the Lindblad master equation for the density matrix \(\rho\) of the whole system developed in [12; 13] and reviewed in [14]. It has fundamental grounds and allows us to study complex phenomena, such as quantum strange attractors [15; 16]. However, its numerical simulation requires integrating time propagation of the whole density matrix with a large number of \(N\times N\) components that are numerically rather heavy. Another approach to dissipative quantum evolution is based on the method of quantum trajectories (see, e.g., [17; 18; 19]), which stores only a stochastically evolving state vectorof size \(N\). The averaging over several realizations allows us to characterize the probabilities from the density matrix with statistical fluctuations. The application of quantum trajectories to the problem of quantum synchronization has been reported in [20], where it was shown that--at small dimensionless values of the Planck constant \(\hbar\)--the synchronization remains robust with respect to quantum fluctuations. These model studies of quantum synchronization [20] were shown to have close similarities with [PERSON]'s steps in the Josephson junction [4]. The quantum trajectory method has also been applied to investigations of quantum synchronization of one and two qubits coupled to a driven dissipative resonator [21; 22]. It has been shown that the driving phase can impose quantum synchronization of qubit phases and mutual synchronization and entanglement of two qubits. Regarding the synchronization of classical systems, this corresponds to the regime of the phase synchronization of oscillators by external driving [23]. Thus, the phases of oscillators rigidly follow the phase of driving a monochromatic field. In a similar way, we define a quantum synchronization of qubits as the regime in which a phase (or orientation angle) of each qubit follows the phase of an external monochromatic driving field applied to a resonator (cavity). The description of quantum trajectories involves significant quantum noise and fluctuations, making it difficult to obtain complete information about the evolution of the density matrix and its steady state using this approach. Therefore, it is crucial to study the phenomenon of quantum synchronization of qubits using the Lindblad master equation for the density matrix. In this work, we describe the results obtained with the Lindblad description. In recent years, there has been growing interest in various aspects of the quantum synchronization of various systems (see, e.g., [24; 25; 26; 27; 28; 29; 30]). There are even discussions about quantum synchronization for satellite networks [31; 32; 33].