# Quantum Synchronization and Entanglement of Dissipative Qubits Coupled to a Resonator
## 7 Synchronization of Several Qubits  
In [21], using the method of quantum trajectories, it was shown that the phase of rotating qubits can be synchronized with the phase of microwave driving. We confirmed this by integrating the quantum Lindblad evolution Equations (4) and (5) (see Figure A3 of Appendix A). In fact, the quantum trajectories always produce the results with noise and it is difficult to directly see the properties of the system's steady state. In contrast with the Lindblad description, we directly obtain the system's steady state. The results of Figure A3 in the Appendix A directly show that in the steady state, the phase of the qubit is synchronized with the phase of monochromatic driving that corresponds to the synchronization phenomenon [2]. We study this synchronization effect for up to four qubits. Our results are shown in Figures 11 and 12, where the RWA steady state is compared with semiclassical (Figure 11) and rate equation theories (Figure 12). All the qubits in this simulation are detuned from each other, and without external driving, their in-plane magnetization will process at different frequencies, leading to an average cancellation of the total in-plane magnetization. When the cavity is driven, two synchronization peaks appear, where the total spin projection in the rotating frame \(\langle S_{x}\rangle\) grows linearly with the number of detuned qubits. The first peak is when the cavity is excited near resonance, the resonant cavity vibrations excite the qubits, which then all press at the same phase and frequency. Surprisingly, a second synchronization peak appears at a higher frequency detuned from both cavity and qubit resonances. The semiclassical approximation correctly reproduces the two synchronization peaks; however, the agreement with RWA results is only qualitative. This is because this is a weak dissipation regime with strong cavity-qubit coupling \(\Delta_{1}=\lambda\) and \(\gamma/\lambda=0.2\). In Figure A4 of Appendix A, we show that the agreement becomes almost perfect for the simpler weak coupling strong dissipation limit. We note that, in this simple regime,  
Figure 10: We test here if our two semi-analytic approaches can reproduce singlet formation for antisymmetric detuning presented in Figures 8 and 9. The left panel shows the RWA spin projection \(\langle S_{1x}\rangle\) for the first qubit compared with the rate equation series and the variational approximation, and the right panel shows the mean total spin \(\langle S^{2}\rangle\). While both approaches reproduce some qualitative features, they both fail to describe the singlet formation at \(\omega-\omega_{0}=0\). the linear growth of \(\langle S_{x}\rangle\) with the number of qubits is observed only at cavity resonance. Figure 11 suggests that the agreement between RWA and semiclassical approximation tends to improve with the number of qubits; this may be due to the fact that our semiclassical approximation can also be viewed as a mean-field theory whose accuracy improves with more interacting qubits. The improvement in the accuracy of the semiclassical theory with a larger number of qubits (at resonance) can also be seen from the decreasing values of the semiclassical functional at its minimum (see Figure A5 of Appendix A). With this method, it is, thus, possible to obtain controlled semiclassical results for a larger number of qubits for which the exact quantum computation is no longer possible. The summation of the rate equation series reproduces the RWA results exactly away from the cavity resonance in Figure 12; however, it seems that the diverging region around resonance grows slowly with the number of qubits. To summarize, we find that this regime of quantum synchronization, where the total rotating in-plane spin grows with the number of qubits, is well described by both semiclassical and rate equation approaches.