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arXiv:quant-ph/0501153 · Physical Review A

Model of a deterministic detector and dynamical decoherence

Jae Weon Lee, D. V. Averin, Giuliano Benenti, Dima L. Shepelyansky

Published Jul 12, 2005• 15 citations
arXivHTMLPDF

Abstract

We discuss a deterministic model of detector coupled to a two-level system (a qubit). The detector is a quasiclassical object whose dynamics is described by the kicked rotator Hamiltonian. We show that in the regime of quantum chaos the detector acts as a chaotic bath and induces decoherence of the qubit. We discuss the dephasing and relaxation rates and demonstrate that the main features of single-qubit decoherence due to a heat bath can be reproduced by our fully deterministic dynamical model. Moreover, we show that, for strong enough qubit-detector coupling, the dephasing rate is given by the rate of exponential instability of the detector's dynamics, that is, by the Lyapunov exponent of classical motion. Finally, we discuss the measurement in the regimes of strong and weak qubit-detector coupling. For the case of strong coupling the detector performs a measurement of the up∕down state of the qubit. In the case of weak coupling, due to chaos, the dynamical evolution of the detector is strongly sensitive to the state of the qubit. However, in this case it is unclear how to extract a signal from any measurement with a coarse-graining in the phase space on a size much larger than the Planck cell.

DOI
10.1103/physreva.72.012310

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We discuss a deterministic model of detector coupled to a two-level system (a qubit). The detector is a quasiclassical object whose dynamics is described by the kicked rotator Hamiltonian. We show that in the regime of quantum chaos the detector acts as a chaotic bath and induces decoherence of the qubit. We discuss the dephasing and relaxation rates and demonstrate that the main features of single-qubit decoherence due to a heat bath can be reproduced by our fully deterministic dynamical model. Moreover, we show that, for strong enough qubit-detector coupling, the dephasing rate is given by the rate of exponential instability of the detector's dynamics, that is, by the Lyapunov exponent of classical motion. Finally, we discuss the measurement in the regimes of strong and weak qubit-detector coupling. For the case of strong coupling the detector performs a measurement of the up∕down state of the qubit. In the case of weak coupling, due to chaos, the dynamical evolution of the detector is strongly sensitive to the state of the qubit. However, in this case it is unclear how to extract a signal from any measurement with a coarse-graining in the phase space on a size much larger than the Planck cell.
We study the dynamics of the entanglement between two qubits coupled to a common chaotic environment, described by the quantum kicked rotator model. We show that the kicked rotator, which is a single-particle deterministic dynamical system, can reproduce the effects of a pure dephasing many-body bath. Indeed, in the semiclassical limit the interaction with the kicked rotator can be described as a random phase kick, so that decoherence is induced in the two-qubit system. We also show that our model can efficiently simulate non-Markovian environments.
Fidelity is a convenient tool to study the sensitivity of quantum motion under Hamiltonian perturbations. In this paper we first show that classical chaos can produce the dephasing necessary to suppress quantum interference, even in the absence of any environment. To this end we consider the fidelity of mixed states, which takes into account interference amplitudes, and directly relate its decay to the decay of an appropriate classical correlation function, which is totally unrelated to quantum phases. We then discuss the dephasing in a two-qubit system, induced by the coupling to a single-particle, deterministic chaotic environment. The latter is shown to behave as a pure dephasing many-body object which induces decoherence in the system; memory effects are also taken into account.
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.
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.
We study the open system dynamics of a circuit QED model operating in the ultrastrong coupling regime. If the resonator is pumped periodically in time the underlying classical system is chaotic. Indeed, the periodically driven Jaynes-Cummings model in the Born-Oppenheimer approximation resembles a Duffing oscillator which in the classical limit is a well-known example of a chaotic system. Detection of the field quadrature of the output field acts as an effective position measurement of the oscillator. We address how such detection affects the quantum chaotic evolution in this bipartite system. We differentiate between single measurement realizations and ensembles of repeated measurements. In the former case a measurement/decoherence induced localization effect is encountered, while in the latter this localization is almost completely absent. This is in marked contrast to numerous earlier works discussing the quantum-classical correspondence in measured chaotic systems. This lack of a classical correspondence under relatively strong measurement induced decoherence is attributed to the inherent quantum nature of the qubit subsystem and in particular to the quantum correlations between the qubit and the field which persist despite the decoherence.
We study the open system dynamics of a circuit quantum electrodynamics (QED) model operating in the ultrastrong coupling regime. If the resonator is pumped periodically in time the underlying classical system is chaotic. Indeed, the periodically driven Jaynes-Cummings model in the Born-Oppenheimer approximation resembles a Duffing oscillator which in the classical limit is a well-known example of a chaotic system. Detection of the field quadrature of the output field acts as an effective position measurement of the oscillator. We address how such detection affects the quantum chaotic evolution in this bipartite system. We differentiate between single measurement realizations and ensembles of repeated measurements. In the former case a measurement/decoherence induced localization effect is encountered, while in the latter this localization is almost completely absent. This is in marked contrast to numerous earlier works discussing the quantum-classical correspondence in measured chaotic systems. This lack of a classical correspondence under relatively strong measurement induced decoherence is attributed to the inherent quantum nature of the qubit subsystem and in particular to the quantum correlations between the qubit and the field which persist despite the decoherence.