Continuously Measured Chaotic Quantum Systems

Sarben Sarkar, J.S. Satchell

Barry C. Sanders, Gerard J Milburn

J. K. Breslin, Gerard J Milburn, Howard M. Wiseman

We define an ideal optimal quantum measurement as that measurement on the apparatus for which the average algorithmic information in the measurement record is minimized. We apply the definition to a chaotic system subject to continuous (Markov) quantum nondemolition measurements. For optimized measurements the average information in the record is much closer to the von Neumann entropy than in the nonoptimized case, but increases more quickly in the chaotic region than in the regular region.

M. S. Santhanam

Martin Schlautmann, R. Graham

We consider the behavior of a classically chaotic quantum system, the periodically driven pendulum, under the influence of a continuous measurement of its angular momentum. Without measurement the system shows dynamical localization, a quantum interference effect which suppresses the classical chaotic diffusion in momentum space. The coupling of the system to a measuring device destroys its coherence and thus leads to delocalization. This is studied on the level of individual systems including the recording of the measurement results. For that purpose we analyze the appropriate stochastic Schr\"odinger equation, from which a stochastic quantum map is derived as an effective tool for numerical simulation of measurement trajectories. We show that a continuous momentum measurement restores the diffusive behavior of the system in momentum space, but that for sufficiently low accuracy of measurement the corresponding diffusion constant is smaller than the classical one. This is reflected by an equal diffusive growth of the recorded measurement results. Thus we find signatures of the classical chaos and of the dynamical localization both in the behavior of the measured quantum system and in the corresponding signal of the measuring device.

Tanmoy Bhattacharya A, Salman Habib, Kurt A. Jacobs

We formulate the conditions under which the dynamics of a continuously measured quantum system becomes indistinguishable from that of the corresponding classical system. In particular, we demonstrate that even in a classically chaotic system the quantum state vector conditioned by the measurement remains localized and, under these conditions, follows a trajectory characterized by the classical Lyapunov exponent.