Entanglement-Mediated Quantum Synchronisation in Nonlinear Coupled SQUID Circuits via a Full TDSE Wavefunction Approach

We present a fully quantum-mechanical investigation of synchronisation in two inductively coupled superconducting oscillators modelled as nonlinear rf-SQUID circuits. Unlike semiclassical or hybrid treatments, both SQUID degrees of freedom are evolved as a joint two-body wavefunction by solving the time-dependent Schrödinger equation (TDSE) using a split-operator spectral method. This enables direct access to coherent flux dynamics, entanglement generation, and quantum phase-space structure without introducing dissipation, measurement back-action, or classical approximations. By varying the mutual inductive coupling strength, we identify a transition from effectively desynchronised tunnelling to quasi-periodic phase locking and, at stronger coupling, chaos-assisted synchronised tunnelling. Synchronisation is quantified using three complementary diagnostics: expectation-value locking of the flux variables ϕ_i(t), von Neumann entanglement entropy S(t), and Husimi phase-space distributions Q(ϕ,p). The results show that coherent quantum synchronisation can emerge solely from Hamiltonian evolution, with entanglement acting as a dynamical correlation channel rather than simply reaching a maximal value. At moderate coupling, bounded entropy oscillations accompany stable phase locking, while stronger coupling produces broadened Husimi structures and irregular but correlated flux dynamics characteristic of chaos-assisted coherence. These findings establish coupled SQUID oscillators as a physically relevant platform for studying Hamiltonian quantum synchronisation, entanglement-mediated phase control, and nonlinear quantum transport. The work also suggests possible routes towards synchronisation-based control in superconducting qubit networks, quantum metrology, and coherent neuromorphic superconducting architectures where dissipation is deliberately minimised.