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On the correspondence principle : implications from a study of the nonlinear dynamics of a macroscopic quantum device

Mark Everitt (1250502)

Published Aug 9, 2019• 20 citations
OpenAlex

Abstract

The recovery of classical nonlinear and chaotic dynamics from\nquantum systems has long been a subject of interest. Furthermore, recent work\nindicates that quantum chaos may well be significant in quantum information\nprocessing. In this paper, we discuss the quantum to classical crossover of\na superconducting quantum interference device (SQUID) ring. Such devices\ncomprise a thick superconducting loop enclosing a Josephson weak link and are\ncurrently strong candidates for many applications in quantum technologies. The\nweak link brings with it a nonlinearity such that semiclassical models of this\nsystem can exhibit nonlinear and chaotic dynamics. For many similar systems\nan application of the correspondence principle together with the inclusion of\nenvironmental degrees of freedom through a quantum trajectories approach can\nbe used to effectively recover classical dynamics. Here we show (i) that the\nstandard expression of the correspondence principle is incompatible with the\nring Hamiltonian and we present a more pragmatic and general expression which\nfinds application here and (ii) that practical limitations to circuit parameters\nof the SQUID ring prevent arbitrarily accurate recovery of classical nonlinear\ndynamics.

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The recovery of classical nonlinear and chaotic dynamics from\nquantum systems has long been a subject of interest. Furthermore, recent work\nindicates that quantum chaos may well be significant in quantum information\nprocessing. In this paper, we discuss the quantum to classical crossover of\na superconducting quantum interference device (SQUID) ring. Such devices\ncomprise a thick superconducting loop enclosing a Josephson weak link and are\ncurrently strong candidates for many applications in quantum technologies. The\nweak link brings with it a nonlinearity such that semiclassical models of this\nsystem can exhibit nonlinear and chaotic dynamics. For many similar systems\nan application of the correspondence principle together with the inclusion of\nenvironmental degrees of freedom through a quantum trajectories approach can\nbe used to effectively recover classical dynamics. Here we show (i) that the\nstandard expression of the correspondence principle is incompatible with the\nring Hamiltonian and we present a more pragmatic and general expression which\nfinds application here and (ii) that practical limitations to circuit parameters\nof the SQUID ring prevent arbitrarily accurate recovery of classical nonlinear\ndynamics.
The recovery of classical nonlinear and chaotic dynamics from quantum systems has long been a subject of interest.Furthermore, recent work indicates that quantum chaos may well be significant in quantum information processing.In this paper, we discuss the quantum to classical crossover of a superconducting quantum interference device (SQUID) ring.Such devices comprise a thick superconducting loop enclosing a Josephson weak link and are currently strong candidates for many applications in quantum technologies.The weak link brings with it a nonlinearity such that semiclassical models of this system can exhibit nonlinear and chaotic dynamics.For many similar systems an application of the correspondence principle together with the inclusion of environmental degrees of freedom through a quantum trajectories approach can be used to effectively recover classical dynamics.Here we show (i) that the standard expression of the correspondence principle is incompatible with the ring Hamiltonian and we present a more pragmatic and general expression which finds application here and (ii) that practical limitations to circuit parameters of the SQUID ring prevent arbitrarily accurate recovery of classical nonlinear dynamics.
The recovery of classical chaotic dynamics from quantum systems has long been a subject of interest. Furthermore, recent work indicates that quantum chaos may well be significant in quantum information processing. In this paper we discuss the quantum to classical crossover of a superconducting quantum inference device (SQUID) ring. Such devices comprise of thick superconducting loop enclosing a Josephson weak link. These devices are currently strong candidates for many applications in quantum technologies. The weak link brings with it a non-linearity such that semi-classical models of this system can exhibit chaotic dynamics. For many similar systems an application of the correspondence principle together with the inclusion of environmental degrees of freedom through a quantum trajectories approach can be used to effectively recover classical dynamics. Here we show (i) that the standard expression of the correspondence principle is incompatible with the ring Hamiltonian and we present a more pragmatic and general expression which finds application here and (ii) that practical limitations to circuit parameters of the SQUID ring prevent arbitrarily accurate recovery of classical chaotic dynamics.
We explore the quantum-classical crossover of two coupled, identical, superconducting quantum interference device (SQUID) rings. We note that the motivation for this work is based on a study of a similar system comprising two coupled Duffing oscillators. In that work we showed that the entanglement characteristics of chaotic and periodic (entrained) solutions differed significantly and that in the classical limit entanglement was preserved only in the chaotic-like solutions. However, Duffing oscillators are a highly idealised toy model. Motivated by a wish to explore more experimentally realisable systems we now extend our work to an analysis of two coupled SQUID rings. We observe some differences in behaviour between the system that is based on SQUID rings rather than on Duffing oscillators. However, we show that the two systems share a common feature. That is, even when the SQUID ring's trajectories appear to follow (semi) classical orbits entanglement persists.
The effect of dissipation on the recovery of the classical chaotic dynamics is discussed in the semiclassical regime of the quantum nonintegrable systems. In the classically chaotic Hamiltonian systems, the semiclassical approach breaks down within rather short time scale due to the interference of the wave function. It is shown that the dissipation suppresses the interference and confines the quantum fluctuation around the semiclassical trajectory which is described by the semiclassical Langevin equation. We also show that the semiclassical condition obtained by Schmid is equivalent to the condition for the suppression of interference. The discussion is valid regardless whether the classical motion is chaotic or not and therefore dissipation helps the recovery of the classical dynamics in the chaotic systems.
The apparent difficulty in recovering classical nonlinear dynamics and chaos from standard quantum mechanics has been the subject of a great deal of interest over the last 20 years. For open quantum systems—those coupled to a dissipative environment and/or a measurement device—it has been demonstrated that chaotic-like behaviour can be recovered in the appropriate classical limit. In this paper, we investigate the entanglement generated between two nonlinear oscillators, coupled to each other and to their environment. Entanglement—the inability to factorize coupled quantum systems into their constituent parts—is one of the defining features of quantum mechanics. Indeed, it underpins many of the recent developments in quantum technologies. Here, we show that the entanglement characteristics of two 'classical' states (chaotic and periodic solutions) differ significantly in the classical limit. In particular, we show that significant levels of entanglement are preserved only in the chaotic-like solutions.