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Chinese Physics B

Measure synchronization in hybrid quantum–classical systems

Haibo Qiu, Yuanjie Dong, Huangli Zhang, Jing Tian

Published Apr 20, 2022• 6 citations
Publisher

Abstract

Measure synchronization in hybrid quantum–classical systems is investigated in this paper. The dynamics of the classical subsystem is described by the Hamiltonian equations, while the dynamics of the quantum subsystem is governed by the Schrödinger equation. By increasing the coupling strength in between the quantum and classical subsystems, we reveal the existence of measure synchronization in coupled quantum–classical dynamics under energy conservation for the hybrid systems.

DOI
10.1088/1674-1056/ac685c

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Measure synchronization in hybrid quantum–classical systems is investigated in this paper. The dynamics of the classical subsystem is described by the Hamiltonian equations, while the dynamics of the quantum subsystem is governed by the Schrödinger equation. By increasing the coupling strength in between the quantum and classical subsystems, we reveal the existence of measure synchronization in coupled quantum–classical dynamics under energy conservation for the hybrid systems.
Hamiltonian formulation of quantum dynamics and nonlinear constraints are used to derive dynamical equations of a hybrid classical-quantum system. Starting with a compound quantum system in the Hamiltonian formulation, conditions for classical behavior are imposed on one of its subsystems and the corresponding hybrid dynamical equations are derived. The dynamical equations for hybrid systems in pure and in mixed states indicate that the hybrid systems have properties that are not exhausted by those of quantum and classical systems.
Constrained Hamiltonian description of the classical limit is utilized in order to derive consistent dynamical equations for hybrid quantum-classical systems. Starting with a compound quantum system in the Hamiltonian formulation, conditions for classical behavior are imposed on one of its subsystems and the corresponding hybrid dynamical equations are derived. The presented formalism suggests that the hybrid systems have properties that are not exhausted by those of quantum and classical systems.
General statistical ensembles in the Hamiltonian formulation of hybrid quantum-classical systems are analyzed. It is argued that arbitrary probability densities on the hybrid phase space must be considered as the class of possible physically distinguishable statistical ensembles of hybrid systems. Nevertheless, statistical operators associated with the hybrid system and with the quantum subsystem can be consistently defined. Dynamical equations for the statistical operators representing the mixed states of the hybrid system and its quantum subsystem are derived and analyzed. In particular, these equations irreducibly depend on the total probability density on the hybrid phase space.
We present a generalized information-theoretic measure of synchronization in quantum systems.This measure is applicable to dynamics of anharmonic oscillators, few-level atoms, and coupled oscillator networks.Furthermore, the new measure allows us to discuss synchronization of disparate physical systems such as coupled hybrid quantum systems and coupled systems undergoing mutual synchronization that are also driven locally.In many cases of interest, we find a closed-form expression for the proposed measure.
Consistent dynamics which couples classical and quantum degrees of freedom exists, provided it is stochastic. This dynamics is linear in the hybrid state, completely positive and trace preserving. One application of this is to study the back-reaction of quantum fields on space-time which does not suffer from the pathologies of the semi-classical equations. Here we introduce several toy models in which to study hybrid classical-quantum evolution, including a qubit coupled to a particle in a potential, and a quantum harmonic oscillator coupled to a classical one. We present an unravelling approach to calculate the dynamics, and provide code to numerically simulate it. Unlike the purely quantum case, the trajectories (or histories) of this unravelling can be unique, conditioned on the classical degrees of freedom for discrete realisations of the dynamics, when different jumps in the classical degrees of freedom are accompanied by the action of unique operators on the quantum system. As a result, the “measurement postulate'' of quantum theory is not needed; quantum systems become classical because they interact with a fundamentally classical field.
The concept of measure synchronization between two coupled quantum many-body systems is presented. In general terms we consider two quantum many-body systems whose dynamics gets coupled through the contact particle-particle interaction. This coupling is shown to produce measure synchronization, a generalization of synchrony to a large class of systems which takes place in absence of dissipation. We find that in quantum measure synchronization, the many-body quantum properties for the two subsystems, e.g., condensed fractions and particle fluctuations, behave in a coordinated way. To illustrate the concept we consider a simple case of two species of bosons occupying two distinct quantum states. Measure synchronization can be readily explored with state-of-the-art techniques in ultracold atomic gases and, if properly controlled, be employed to build targeted quantum correlations in a sympathetic way.