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Canadian Journal of Physics

Nonseparability of continuously measured quantum systems in the classical limit

Shohini Ghose, Barry C. Sanders, Paul M. Alsing, Ivan Deutsch

Published Jun 1, 2007• 1 citation
Publisher

Abstract

We analyze the question of separability in a continuously measured quantum system as it approaches the classical limit. We show that the record of position measurements can approach the classical limit even when the system is described by highly nonseparable states. In particular, in systems with a chaotic classical limit, chaos can work to enhance the entanglement in the system in the classical regime. This coexistence of nonclassical states and classical dynamics can be understood by analyzing the conditioned evolution of the measured system and the conditions for the quantum-to-classical transition. PACS Nos.: 03.65.Ta, 03.65.Ud, 03.67.Mn, 05.45.Mt, 03.67.–a

DOI
10.1139/p07-071

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We analyze the question of separability in a continuously measured quantum system as it approaches the classical limit. We show that the record of position measurements can approach the classical limit even when the system is described by highly nonseparable states. In particular, in systems with a chaotic classical limit, chaos can work to enhance the entanglement in the system in the classical regime. This coexistence of nonclassical states and classical dynamics can be understood by analyzing the conditioned evolution of the measured system and the conditions for the quantum-to-classical transition. PACS Nos.: 03.65.Ta, 03.65.Ud, 03.67.Mn, 05.45.Mt, 03.67.–a
The dynamical status of isolated quantum systems is unclear as conventional measures fail to detect chaos in such systems. However, when quantum systems are subjected to observation--as all experimental systems must be--their dynamics is no longer linear and, in the appropriate limit(s), the evolution of expectation values, conditioned on the observations, closely approaches the behavior of classical trajectories. Here we show, by analyzing a specific example, that microscopic continuously observed quantum systems, even far from any classical limit, can have a positive Lyapunov exponent, and thus be truly chaotic.
We study the quantum to classical transition in bipartite entangled systems in which one system is continuously coupled to a measurement apparatus as in the von Neumann model of quantum measurement. As an example, we study the open system dynamics of a particle in a harmonic well whose motion in the well is coupled to the internal spin. This system provides a rich illustration of the quantum to classical transition in weakly measured coupled systems. We analyze and derive conditions for which the dual constraints of strong localization/small noise required for the quantum-classical transition are satisfied for both regular and chaotic dynamics. We also study the dynamics of bipartite entanglement in the regime where classical trajectories emerge in the measurement record. Our analysis shows the surprising result that bipartite entanglement can persist in the classical limit.
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.
Continuous observation of a quantum system yields a measurement record that faithfully reproduces the classically predicted trajectory provided that the measurement is sufficiently strong to localize the state in phase space but weak enough that quantum backaction noise is negligible. We investigate the conditions under which classical dynamics emerges, via a continuous position measurement, for a particle moving in a harmonic well with its position coupled to internal spin. As a consequence of this coupling, we find that classical dynamics emerges only when the position and spin actions are both large compared to $\ensuremath{\hbar}$. These conditions are quantified by placing bounds on the size of the covariance matrix which describes the delocalized quantum coherence over extended regions of phase space. From this result, it follows that a mixed quantum-classical regime (where one subsystem can be treated classically and the other not) does not exist for a continuously observed spin-12\frac{1}{2}21​ particle. When the conditions for classicality are satisfied (in the large-spin limit), the quantum trajectories reproduce both the classical periodic orbits as well as the classically chaotic phase space regions. As a quantitative test of this convergence, we compute the largest Lyapunov exponent directly from the measured quantum trajectories and show that it agrees with the classical value.
We study the role of continuous measurement in the quantum to classical transition for a particle in a harmonic well whose motion in the well is coupled to the internal spin. This system provides a rich illustration of the quantum to classical transition in weakly measured coupled systems. We demonstrate the emergence of classically regular as well as chaotic dynamics from the measured quantum trajectories when the external and internal actions are large relative to ℏ. We also show the breakdown of the conditions for the quantum‐classical transition in a mixed quantum‐classical regime where the spin is treated quantum mechanically but the motion in the well can be treated classically. The conditions for the transition can be quantified by examining the covariance matrix. We explore the behavior of entanglement between spin and motion in the (classical) limit of large actions and find that whereas the measured quantum trajectories recover classical dynamics, the conditioned evolution can lead to highly non‐classical entangled states. This is due to the finite resolution of the measurement.