Decoherence

1998

Gibbs entropy is invariant for the Baker map. A Jordan basis spectral decomposition of the Baker Frobenius-Perron operator suggests that any initial density evolves to the stationary density that has maximal entropy. This entropy conundrum is resolved by considering the difference between weak and strong convergence. A binary representation is used to make these points transparent.

Annales Henri Poincaré, 2007

We study the baker's map and its Walsh quantization, as a toy model of a quantized chaotic system. We focus on localization properties of eigenstates, in the semiclassical régime. Simple counterexamples show that quantum unique ergodicity fails for this model. We obtain, however, lower bounds on the entropies associated with semiclassical measures, as well as on the Wehrl entropies of eigenstates. The central tool of the proofs is an "entropic uncertainty principle".

Physical Review Letters, 2000

We show that for an open quantum system which is classically chaotic (a quartic double well with harmonic driving coupled to a sea of harmonic oscillators) the rate of entropy production has, as a function of time, two relevant regimes: For short times it is proportional to the diffusion coefficient (fixed by the system-environment coupling strength). For longer times (but before equilibration) there is a regime where the entropy production rate is fixed by the Lyapunov exponent. The nature of the transition time between both regimes is investigated.

2001

We study the decoherence process for an open quantum system which is classically chaotic (a quartic double well with harmonic driving coupled to a sea of harmonic oscillators). We carefully analyze the time dependence of the rate of entropy production showing that it has two relevant regimes: For short times it is proportional to the diffusion coefficient (fixed by the system-environment coupling strength). For longer times (but before equilibration) it is fixed by dynamical properties of the system (and is related to the Lyapunov exponent). The nature of the transition time between both regimes is investigated and the issue of quantum to classical correspondence is addressed. Finally, the impact of the interaction with the environment on coherent tunneling is analyzed.

Physica D: Nonlinear Phenomena, 1994

We develop semiclassical (h---~0) treatments of the quantum baker's map paying special attention to (a) the discrete and finite nature of its Hilbert space, and (b) the asymptotic analysis of the true quantum objects themselves. New techniques are described at the theoretical level (exact symbolic decomposition of the propagator) and at the numerical level (color graphics display of operators in the Kirkwood representation), both of which are of general interest for other quantized chaotic maps. Periodic orbit theory is tested numerically, and we find that log(h) corrections need to be included before an accurate calculation of the spectrum from periodic orbits can be achieved. However, the asymptotic analysis of the traces of the propagator reveals unexpected departures from standard semiclassical theory; those anomalies, which we compute explicitly in a few cases, are traced ultimately to the discontinuity of the map and to the compactness of the phase space. The analysis is far from complete but the results point towards the necessity of careful assessment of semiclassical "folklore" when applied to chaotic maps.

Annals of Physics, 1993

We introduce and study the classical and quantum mechanics of certain non hyperbolic maps on the unit square. These maps are modifications of the usual baker's map and their behaviour ranges from chaotic motion on the whole measure to chaos on a set of measure zero. Thus we have called these maps "lazy baker maps." The aim of introducing these maps is to provide the simplest models of systems with a mixed phase space, in which there are both regular and chaotic motions. We find that despite the obviously contrived nature of these maps they provide a good model for the study of the quantum mechanics of such systems. We notice the effect of a classically chaotic fractal set of measure zero on the corresponding quantum maps, which leads to a transition in the spectral statistics. Some periodic orbits belonging to this fractal set are seen to scar several eigenfunctions.

Physics Letters A, 2001

Physics Letters B, 2012

We formulate a novel approach to decoherence based on neglecting observationally inaccessible correlators. We apply our formalism to a renormalised interacting quantum field theoretical model. Using out-of-equilibrium field theory techniques we show that the Gaussian von Neumann entropy for a pure quantum state increases to the interacting thermal entropy. This quantifies decoherence and thus measures how classical our pure state has become. The decoherence rate is equal to the single particle decay rate in our model. We also compare our approach to existing approaches to decoherence in a simple quantum mechanical model. We show that the entropy following from the perturbative master equation suffers from physically unacceptable secular growth.

Communications in Mathematical Physics, 2006

We prove a Egorov theorem, or quantum-classical correspondence, for the quantised baker's map, valid up to the Ehrenfest time. This yields a logarithmic upper bound for the decay of the quantum variance, and, as a corollary, a quantum ergodic theorem for this map.

The classical Bernoulli and baker maps are two simple models of deterministic chaos. On the level of ensembles, it has been shown the time evolution operator for these maps admits generalized spectral representations in terms of decaying eigenfunctions. vVe introduce the quantum version of the Bernoulli map. We define it as a projection of the quantum baker map. We construct a quantum analogue of the generalized spectral• representation, yielding quantum decaying states represented by density matrices. The quantum decaying states develop a quasi-fractal shape limited by the quantum uncertainty.

Open Systems & Information Dynamics, 2018

We examine the dynamics of entanglement entropy of all parts in an open system consisting of a two-level dimer interacting with an environment of oscillators. The dimer-environment interaction is almost energy conserving. We find the precise link between decoherence and production of entanglement entropy. We show that not all environment oscillators carry significant entanglement entropy and we identify the oscillator frequency regions which contribute to the production of entanglement entropy. For energy conserving dimer-environment interactions the models are explicitly solvable and our results hold for all dimer-environment coupling strengths. We carry out a mathematically rigorous perturbation theory around the energy conserving situation in the presence of small non-energy conserving interactions.

2005

Discrete maps play an important role in the investigation of dynamical features of complex classical systems, especially within the theory of chaos. Similarly, quantum maps have proven to be a very useful mathematical tool within the study of complex quantum dynamical systems. Using the decoherent histories formulation of quantum mechanics we consider a particular framework for studying quantum maps which is motivated by the method of classical symbolic dynamics. Symbolic dynamics is known to be a very powerful method specifically invented for the purpose of representing classical dynamical systems by a discrete model that is suitable for information theoretic studies. Our framework uses the decoherent histories formalism which, similarly to classical symbolic dynamics, allows one to introduce information theoretic quantities with respect to system dynamics. Our research within this framework can be viewed as a contribution towards the development of a general theory of “quantum sym...

Physical Review A, 2005

For a solvable pure-decoherence model, we confirm by an explicit model calculation that the decay of entanglement of two two-state systems ͑two qubits͒ is approximately governed by the product of the suppression factors describing decoherence of the subsystems, provided that they are subject to uncorrelated sources of quantum noise. This demonstrates an important physical property that separated open quantum systems can evolve quantum mechanically on time scales larger than the times for which they remain entangled.

The classical Bernoulli and baker maps are two simple models of deterministic chaos. On the level of ensembles, it has been shown that the time evolution operator for these maps admits generalized spectral representations in terms of decaying eigenfunctions. We introduce the quantum version of the Bernoulli map. We define it as a projection of the quantum baker map. We construct a quantum analogue of the generalized spectral representation, yielding quantum decaying states represented by density matrices. The quantum decaying states develop a quasi-fractal shape limited by the quantum uncertainty.

2002

The following statements belonging to the folklore of the theory of environmental decoherence are shown to be incorrect: 1) linear coupling to harmonic oscillator bath is a universal model of decoherence, 2) chaotic environments are more efficient decoherers.