Quantum synchronization in disordered superconducting metamaterials

I report a theoretical study of collective coherent quantum-mechanical oscillations in disordered superconducting quantum metamaterials (SQMs), i.e artificially fabricated arrays of interacting qubits (two-levels system). An unavoidable disorder in qubits parameters results in a substantial spread of qubits frequencies, and in the absence of electromagnetic interaction between qubits these individual quantum-mechanical oscillations of single qubits manifest themselves by a large number of small resonant drops in the frequency dependent transmission of electromagnetic waves propagating through disordered SQM, $D(\omega)$. We show that even a weak electromagnetic interaction between adjacent qubits can overcome the disorder and establish completely or partially \emph{synchronized} quantum-mechanical dynamic state in the disordered SQM. In such a state a large amount of qubits displays the collective quantum mechanical oscillations, and this collective behavior manifests itself by a few giant resonant drops in the $D(\omega)$ dependence. The size of a system $r_0$ showing the collective (synchronized) quantum-mechanical behavior is determined in the one-dimensional SQMs as $r_0 \simeq a [K/\delta \Delta]^2$, where $K$, $\delta \Delta$, $a$ are the energy of nearest-neighbor interaction, the spread of qubits energy splitting, and the distance between qubits, accordingly. We show that this phenomenon has an origin in the Anderson localization of spinon-type excitations arising in the SQM. Our analysis is also in a good accord with recent experiments on the electrodynamics of the disordered 1D SQMs.