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

Sameer Al-Khawaja

Published Jul 6, 2026• 0 citations

Abstract

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 ϕi(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.

DOI: 10.20944/preprints202607.0388.v1

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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 ϕi(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.

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We investigate the quantum entanglement dynamics of two coupled harmonic oscillators with a time-dependent interaction. Using the Lewis-Riesenfeld invariant method, we derive the exact analytical wave functions without any perturbative or adiabatic approximations and combine this with a phase-space analysis using the Wigner function to provide a complete description of the system's quantum state evolution. We obtain general expression form for the purity and the linear entropy

S_L=1-\mathcal{P}

for arbitrary excitation numbers

(n,m)

, allows a systematic study of entanglement for a large class of quantum states. We show that the entanglement dynamics is very sensitive to the interplay between the detuning parameters

θ

and

\vartheta_2

, the frequency parameter

β_0

and the coupling strength

ε

: the increase of detuning takes the system from slow irregular oscillations to fast and regular periodic behavior, and the stronger coupling systematically enhances both the amplitude and the average value of the linear entropy. Most importantly, for the resonance case

ω_1=ω_2=1

and strong couplings

ε

, the system shows robust undamped synchronized periodic oscillations of the linear entropy for all quantum states considered, indicating preserved quantum coherence without saturation. Our findings demonstrate that linear entropy is a sensitive and practical entanglement witness, and we establish explicit analytical relations between the coupling parameters of the system and its entanglement properties, which are directly relevant to quantum information processing and the control of quantum correlations in continuous-variable systems.

Abstract The non-dissipative nonlinearity of Josephson junctions1 converts macroscopic superconducting circuits into artificial atoms2, enabling some of the best-controlled qubits today3, 4. Three fundamental types of superconducting qubit are known5, each reflecting a distinct behaviour of quantum fluctuations in a Cooper pair condensate: single-charge tunnelling (charge qubit6, 7), single-flux tunnelling (flux qubit8) and phase oscillations (phase qubit9 or transmon10). Yet, the dual nature of charge and flux suggests that circuit atoms must come in pairs. Here we introduce the missing superconducting qubit, ‘blochnium’, which exploits a coherent insulating response of a single Josephson junction that emerges from the extension of phase fluctuations beyond 2π (refs. 11–14). Evidence for such an effect has been found in out-of-equilibrium direct-current transport through junctions connected to high-impedance leads15–19, although a full consensus on the existence of extended phase fluctuations is so far absent20–22. We shunt a weak junction with an extremely high inductance—the key technological innovation in our experiment—and measure the radiofrequency excitation spectrum as a function of external magnetic flux through the resulting loop. The insulating character of the junction is manifested by the vanishing flux sensitivity of the qubit transition between the ground state and the first excited state, which recovers rapidly for transitions to higher-energy states. The spectrum agrees with a duality mapping of blochnium onto a transmon, which replaces the external flux by the offset charge and introduces a new collective quasicharge variable instead of the superconducting phase23, 24. Our findings may motivate the exploration of macroscopic quantum dynamics in ultrahigh-impedance circuits, with potential applications in quantum computing and metrology.

We numerically investigated the quantum-classical transition in rf-superconducting quantum interference device (SQUID) systems coupled to a dissipative environment. It is found that chaos emerges and the degree of chaos, the maximal Lyapunov exponent lambda(m), exhibits nonmonotonic behavior as a function of the coupling strength D. By measuring the proximity of quantum and classical evolution with the uncertainty of dynamics, we show that the uncertainty is a monotonic function of lambda(m)/D. In addition, the scaling holds in SQUID systems to a relatively smaller variant Planck's over [symbol: see text], suggesting the universality for this scaling.

For a variety of superconducting qubits, tunable interactions are achieved through mutual inductive coupling to a coupler circuit containing a nonlinear Josephson element. In this paper, we derive the general interaction mediated by such a circuit under the Born-Oppenheimer approximation. This interaction naturally decomposes into a classical part, with origin in the classical circuit equations, and a quantum part, associated with the coupler's zero-point energy. Our result is nonperturbative in the qubit-coupler coupling strengths and in the coupler nonlinearity. This can lead to significant departures from previous, linear theories for the interqubit coupling, including nonstoquastic and many-body interactions. Our analysis provides explicit and efficiently computable series for any term in the interaction Hamiltonian and can be applied to any superconducting qubit type. We conclude with a numerical investigation of our theory using a case study of two coupled flux qubits, and in particular study the regime of validity of the Born-Oppenheimer approximation.