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AIP conference proceedings

The Quantum to Classical Transition in Entangled Systems via Continuous Measurements

Shohini Ghose

Published Jan 1, 2004• 0 citations
Publisher

Abstract

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.

DOI
10.1063/1.1834383

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Canadian Journal of Physics•Jun 1, 2007•1 citation
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.
We study the role of continuous measurement in the quantum to classical transition for a system with coupled internal (spin) and external (motional) degrees of freedom. Even when the measured motional degree of freedom can be treated classically, entanglement between spin and motion causes strong measurement back action on the quantum spin subsystem so that classical trajectories are not recovered in this mixed quantum-classical regime. The measurement can extract localized quantum trajectories that behave classically only when the internal action also becomes large relative to \ensuremath{\Elzxh}.
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 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.
Summary form only given. The recovery of the classical limit in quantum systems that have underlying chaotic dynamics is a key question in current studies of the quantum to classical transition. Interaction with an environment results in decoherence that can cause the quantum Wigner function to approach the corresponding classical phase space distribution by suppressing quantum interference effects, but it fails to extract localized classical trajectories from the quantum dynamics. Such trajectories are crucial for quantifying chaos both theoretically and in experiments through the measure of the Lyapunov exponents. One can recover trajectories from the quantum dynamics through the process of continuous measurements when the record is retained. Ehrenfest s theorem guarantees that quantum systems that are well localized by the measurement effectively obey classical mechanics. We have studied the role of continuous measurement in recovering the classical chaotic dynamics of laser cooled atoms in a magneto-optical lattice of double wells.
The evolution of a quantum system subject to measurements can be described by stochastic quantum trajectories of pure states. Instead, the ensemble average over trajectories is a mixed state evolving via a master equation. Both descriptions lead to the same expectation values for linear observables. Recently, there is growing interest in the average entanglement appearing during quantum trajectories. The entanglement is a nonlinear observable that is sensitive to so-called measurement-induced phase transitions, namely, transitions from a system-size dependent phase to a quantum Zeno phase with area-law entanglement. Intriguingly, the mixed steady-state description of these systems is insensitive to this phase transition. Together with the difficulty of quantifying the mixed state entanglement, this favors quantum trajectories for the description of the quantum measurement process. Here, we study the entanglement of a single particle under continuous measurements (using the newly developed configuration coherence) in both the mixed state and the quantum trajectories descriptions. In both descriptions, we find that the entanglement at intermediate time scales shows the same qualitative behavior as a function of the measurement strength. The entanglement engenders a notion of coherence length, whose dependence on the measurement strength is explained by a cascade of underdamped-to-overdamped transitions. This demonstrates that measurement-induced entanglement dynamics can be captured by mixed states.
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