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Physical Review A

Sensitivity to measurement errors in the quantum kicked top

J. K. Breslin, Gerard J Milburn

Published Mar 1, 1999• 4 citations
Publisher

Abstract

We address the problem of chaos in quantum systems in terms of the sensitivity of the evolution of individual quantum states to errors in measurements. Specifically, we generate quantum trajectories of the quantum kicked top by considering discrete measurements on individual states as they evolve. If an error occurs when recording the results of the measurements we can calculate the difference between the true quantum state and the state inferred from the measurement results. Not surprisingly, the results depend on the strength of the measurement back action and also on whether the initial state is centered in a regular or chaotic region of the corresponding classical phase space. For measurements with a strong back action we find that the initial chaotic state shows sensitivity to measurement errors, but an initial regular state does not.

DOI
10.1103/physreva.59.1781

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We address the problem of chaos in quantum systems in terms of the sensitivity of the evolution of individual quantum states to errors in measurements. Specifically, we generate quantum trajectories of the quantum kicked top by considering discrete measurements on individual states as they evolve. If an error occurs when recording the results of the measurements we can calculate the difference between the true quantum state and the state inferred from the measurement results. Not surprisingly, the results depend on the strength of the measurement back action and also on whether the initial state is centered in a regular or chaotic region of the corresponding classical phase space. For measurements with a strong back action we find that the initial chaotic state shows sensitivity to measurement errors, but an initial regular state does not.
We analyze the interplay of chaos, entanglement, and decoherence in a system of qubits whose collective behavior is that of a quantum kicked top. The dynamical entanglement between a single qubit and the rest can be calculated from the mean of the collective spin operators. This allows the possibility of efficiently measuring entanglement dynamics in an experimental setting. We consider a deeply quantum regime and show that signatures of chaos are present in the dynamical entanglement for parameters accessible in an experiment that we propose using cold atoms. The evolution of the entanglement depends on the support of the initial state on regular versus chaotic Floquet eigenstates, whose phase-space distributions are concentrated on the corresponding regular or chaotic eigenstructures. We include the effect of decoherence via a realistic model and show that the signatures of chaos in the entanglement dynamics persist in the presence of decoherence. In addition, the classical chaos affects the decoherence rate itself.
Quantum chaos, a phenomenon that began to be studied in the last century, still does not have a rigorous understanding. By virtue of the correspondence principle, the properties of the system that lead to chaotic dynamics at the classical level must also be present in the underlying quantum system. In the classical case, the exponential divergence of nearby trajectories in time is described in terms of the Lyapunov exponent. However, in the quantum case, a similar description of chaos is, strictly speaking, impossible due to the absence of trajectories. There are different approaches to remedy this situation, but the universal criterion of quantum chaos is absent. We propose a quantum chaos definition in a manner similar to the classical one using decoherent histories as a quantum analog of trajectories. For this purpose, we consider the model of an open quantum kicked top interacting with the environment, which is a bosonic bath, and illustrate this idea. Here, the environment plays the role of a trajectory-recording device. For the kicked-top model at the classical level, depending on the kick strength, crossover occurs between the integrable and chaotic regimes. We show that for such a model, the production of entropy of decoherent histories is radically different in integrable and chaotic regimes. Thus, the entropy of an ensemble of quantum trajectories can be used as a signature of quantum chaos.
Laser-cooled atoms offer an excellent platform for testing new ideas of quantum control and measurement. I will discuss experiments where we use light and magnetic fields to drive and monitor non-trivial quantum dynamics of a large spin-angular momentum associated with an atomic hyperfine ground state. We can design Hamiltonians to generate arbitrary spin states and perform a full quantum state reconstruction of the results. We have implemented and verified time optimal controls to generate a broad variety of spin states, including spin-squeezed states useful for metrology. Yields achieved are of the range 0.8-0.9.We present a first experimental demonstration of the quantum kicked top, a popular paradigm for quantum and classical chaos. We make `movies' of the evolving quantum state which provides a direct observation of phase space dynamics of this system. The spin dynamics seen in the experiment includes dynamical tunneling between regular islands, rapid spreading of states throughout the chaotic sea, and surprisingly robust signatures of classical phase space structures. Our data show differences between regular and chaotic dynamics in the sensitivity to perturbations of the quantum kicked top Hamiltonian and in the average electron-nuclear spin entanglement during the first 40 kicks. The difference, while clear, is modest due to the small size of the spin.
For the quantum kicked top we study numerically the distribution of Hilbert-space vectors evolving in the presence of a small random perturbation. For an initial coherent state centered in a chaotic region of the classical dynamics, the evolved perturbed vectors are distributed essentially like random vectors in Hilbert space. In contrast, for an initial coherent state centered near an elliptic (regular) fixed point of the classical dynamics, the evolved perturbed vectors remain close together, explore only a few dimensions of Hilbert space, and do not explore them randomly. These results support and extend the results of earlier studies, thereby providing additional support for a characterization of quantum chaos that uses concepts from information theory.
The emergence of classical chaos from an underlying quantum mechanics remains a challenging question due to the differences between dynamics driven by Schrödinger's equation versus Newton's equations. We present an infinite family of purely quantum recurrences that are not present in the classical limit of a chaotic system. They take the form of stroboscopic unitary evolutions in the quantum kicked top that act as the identity after a finite number of kicks. These state-independent recurrences are present in all finite dimensions and depend on the strength of the chaoticity parameter of the top. We further discuss the relationship of these periodicities to the quantum kicked rotor dynamics and the phenomenon of quantum antiresonance. Published by the American Physical Society 2024