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Physical Review E

Parameter scaling in the decoherent quantum-classical transition for chaotic rf superconducting quantum interference devices

Ting Mao, Yang Yu

Published Jan 21, 2010• 2 citations
Publisher

Abstract

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.

DOI
10.1103/physreve.81.016212

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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.
We numerically investigated the quantum-classical transition in rf-SQUID systems coupled to a dissipative environment. It is found that chaos emerges and the degree of chaos, the maximal Lyapunov exponent λmλ_{m}λm​, exhibits non-monotonic behavior as a function of the coupling strength DDD. By measuring the proximity of quantum and classical evolution with the uncertainty of dynamics, we show that the uncertainty is a monotonic function of λm/Dλ_{m}/Dλm​/D. In addition, the scaling holds in SQUID systems to a relatively smaller ℏeff\hbar_{eff}ℏeff​, suggesting the universality for this scaling.
The quantum to classical transition for a system depends on many parameters, including a scale length for its action, variant Planck's over 2 pi, a measure of its coupling to the environment, D, and, for chaotic systems, the classical Lyapunov exponent, lambda. We propose measuring the proximity of quantum and classical evolutions as a multivariate function of (Planck's over 2 pi,lambda,D) and searching for transformations that collapse this hypersurface into a function of a composite parameter zeta= Planck's over 2 pi alpha)lambda beta D gamma. We report results for the quantum Cat Map and Duffing oscillator, showing accurate scaling behavior over a wide parameter range, indicating that this may be used to construct universality classes for this transition.
The classical-quantum transition for chaotic systems is understood to be accompanied by the suppression of chaotic effects as the relative variant Planck's over 2pi is increased. We show evidence to the contrary in the behavior of the quantum trajectory dynamics of a dissipative quantum chaotic system, the double-well Duffing oscillator. The classical limit in the case considered has regular behavior, but as the effective variant Planck's over 2pi is increased we see chaotic behavior. This chaos then disappears deeper into the quantum regime, which means that the quantum-classical transition in this case is nonmonotonic in variant Planck's over 2pi.
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.
This thesis makes a unique contribution to the field of quantum chaos by theoretically demonstrating the effect that measurement has on the emergence of chaos from the quantum world and demonstrating a means to control the onset of chaos in the quantum system using adaptive measurements. Here we investigate how the choice of the continuous measurement strategy for an open quantum system affects the emergence of chaos in the transition from the quantum limit to the classical limit when the system is dissipative. We consider two models in our research. The Duffing oscillator is classically chaotic and also dissipative( ie. an open quantum system), whereas the driven top is classically a closed system; adding dissipation via continuous measurement therefore changes the behaviour in from the classical limit. The first half of this thesis presents the investigation of a dissipative system whose classical limit is chaotic. We explore the emergence of chaos from the open quantum system that is continuously monitored and investigate the dependence on the choice of monitoring by changing a single parameter in a homodyne measurement scheme, effectively changing the information gained by the measurement. We show that the emergence of chaos in the regime where quantum effects are still present can be determined solely by changing the measurement parameter. This is a result of the interplay between the quantum interference effects induced by the nonlinear dynamics and the localisation and decoherence that occurs due to the measurement. We also investigate the case where the classical limit is regular for the Duffing oscillator, and demonstrate the semiclassical effect of chaos induced by the measurement back-action. A certain choice of measurement leads to a noise which drives the system to large spread in the dimensionless position enabling a non-classical transition mechanism that is classically forbidden, inducing chaos in the system. These results are verified by the numerical calculation of the maximal Lyapunov exponent in the quantum regime. The second part of this thesis investigates the possibility of controlling the degree of chaos with quantum control. We design an effective control scheme to control the degree of chaos using the measurement dependency of the state. We propose an adaptive measurement scheme which changes the homodyne measurement angle in real time depending on the direction of the state's interference fringes in phase space. This is done using the knowledge gained by the measurement signal. We show that this control scheme can enhance or suppress chaos. By enhancing the degree of chaos we are also able to push the onset of chaos further into the quantum regime than was possible before. By suppressing chaos we generate highly non-classical states and regular motion. The feasibility of experimentally realising this control technique is discussed in detail. The final section of this thesis considers a chaotic system that is not dissipative in the classical limit: the driven top. We investigate the effect that opening the quantum system to decoherence has on the degree of chaos when we continuously measure the system. We demonstrate that the presence of decoherence suppresses the chaos and alters the dynamics of the quantum system. This is seen to worsen as the strength of the measurement is increased unless a particular measurement is chosen that perfectly cancels out the decoherence resulting in the Hamiltonian evolution in addition to noise from the measurement. These results are verified by the separation time between classical and quantum dynamics.