Continuous Measurement, Nonlinear Dynamics, and the Quantum-Classical Transition

Shohini Ghose

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.

Shohini Ghose, Barry C. Sanders

A. J. Scott, Gerard J Milburn

Classical dynamics is formulated as a Hamiltonian flow in phase space, while quantum mechanics is formulated as unitary dynamics in Hilbert space. These different formulations have made it difficult to directly compare quantum and classical nonlinear dynamics. Previous solutions have focused on computing quantities associated with a statistical ensemble such as variance or entropy. However a more direct comparison would compare classical predictions to the quantum predictions for continuous simultaneous measurement of position and momentum of a single system. In this paper we give a theory of such measurement and show that chaotic behavior in classical systems can be reproduced by continuously measured quantum systems.

ARMEN E. ALLAHVERDYAN, Roger Balian, Theo M. Nieuwenhuizen

Shohini Ghose, Paul M. Alsing, Ivan Deutsch +3 more

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.

S. Ia. Kilin