Collective quantum coherent oscillations in a globally coupled array of superconducting qubits
Pavel A. Volkov, M. V. Fistul · Physical Review B · 2014
We report a theoretical study of coherent collective quantum dynamic effects in an array of $N$ qubits (two-level systems) incorporated into a low-dissipation resonant cavity. Individual qubits are characterized by energy level differences ${\ensuremath{\Delta}}_{i}$ and a spread of ${\ensuremath{\Delta}}_{i}$ is taken into account. Noninteracting qubits display coherent quantum beatings with $N$ different frequencies, i.e., ${\ensuremath{\omega}}_{i}={\ensuremath{\Delta}}_{i}/\ensuremath{\hbar}$. Virtual emission and absorption of cavity photons provides a long-range interaction between qubits. In the presence of such interaction we analyze quantum correlation functions of individual qubits ${C}_{i}(t)$ to obtain two collective quantum-mechanical coherent oscillations, characterized by frequencies ${\ensuremath{\omega}}_{1}=\overline{\ensuremath{\Delta}}/\ensuremath{\hbar}$ and ${\ensuremath{\omega}}_{2}={\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{R}$, where ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{R}$ is the resonant frequency of the cavity renormalized by interaction. The amplitude of these oscillations can be strongly enhanced in the resonant case when ${\ensuremath{\omega}}_{1}\ensuremath{\simeq}{\ensuremath{\omega}}_{2}$. These collective quantum oscillations can be directly observed, e.g., by measurements of frequency dependent transmission coefficient $D(\ensuremath{\omega})$ of electromagnetic field propagating in a transmission line coupled to the system.