Quantum dynamical phase transition in a system with many-body interactions

We introduce a microscopic Hamiltonian model of a two level system with many-body interactions with an environment whose excitation dynamics is fully solved within the Keldysh formalism. If a particle starts in one of the states of the isolated system, the return probability oscillates with the Rabi frequency $\omega_{0}$. For weak interactions with the environment $1/\tau_{\mathrm{SE}}<2\omega_{0},$ we find a slower oscillation whose amplitude decays with a decoherence rate $1/\tau_{\phi}=1/(2\tau_{\mathrm{SE}})$. However, beyond a finite critical interaction with the environment, $1/\tau_{\mathrm{SE}}>2\omega_{0}$, the decoherence rate becomes $1/\tau_{\phi}\propto(\omega_{0}^{2})\tau_{\mathrm{SE}}$. The oscillation period diverges showing a \emph{quantum dynamical phase transition}to a Quantum Zeno phase.Comment: 5 pages, 3 figures, minor changes, fig.2 modified, added reference

Estimating purity in terms of correlation functions

We prove a rigorous inequality estimating the purity of a reduced density matrix of a composite quantum system in terms of cross-correlation of the same state and an arbitrary product state. Various immediate applications of our result are proposed, in particular concerning Gaussian wave-packet propagation under classically regular dynamics.Comment: 3 page

On general relation between quantum ergodicity and fidelity of quantum dynamics

General relation is derived which expresses the fidelity of quantum dynamics, measuring the stability of time evolution to small static variation in the hamiltonian, in terms of ergodicity of an observable generating the perturbation as defined by its time correlation function. Fidelity for ergodic dynamics is predicted to decay exponentially on time-scale proportional to delta^(-2) where delta is the strength of perturbation, whereas faster, typically gaussian decay on shorter time scale proportional to delta^(-1) is predicted for integrable, or generally non-ergodic dynamics. This surprising result is demonstrated in quantum Ising spin-1/2 chain periodically kicked with a tilted magnetic field where we find finite parameter-space regions of non-ergodic and non-integrable motion in thermodynamic limit.Comment: Slightly revised version, 4.5 RevTeX pages, 2 figure

Evolution of entanglement under echo dynamics

Echo dynamics and fidelity are often used to discuss stability in quantum information processing and quantum chaos. Yet fidelity yields no information about entanglement, the characteristic property of quantum mechanics. We study the evolution of entanglement in echo dynamics. We find qualitatively different behavior between integrable and chaotic systems on one hand and between random and coherent initial states for integrable systems on the other. For the latter the evolution of entanglement is given by a classical time scale. Analytic results are illustrated numerically in a Jaynes Cummings model.Comment: 5 RevTeX pages, 3 EPS figures (one color) ; v2: considerable revision ;inequality proof omitte

Decoherence from a Chaotic Environment: An Upside Down "Oscillator" as a Model

Chaotic evolutions exhibit exponential sensitivity to initial conditions. This suggests that even very small perturbations resulting from weak coupling of a quantum chaotic environment to the position of a system whose state is a non-local superposition will lead to rapid decoherence. However, it is also known that quantum counterparts of classically chaotic systems lose exponential sensitivity to initial conditions, so this expectation of enhanced decoherence is by no means obvious. We analyze decoherence due to a "toy" quantum environment that is analytically solvable, yet displays the crucial phenomenon of exponential sensitivity to perturbations. We show that such an environment, with a single degree of freedom, can be far more effective at destroying quantum coherence than a heat bath with infinitely many degrees of freedom. This also means that the standard "quantum Brownian motion" model for a decohering environment may not be as universally applicable as it once was conjectured to be.Comment: RevTeX, 29 pages, 5 EPS figures. Substantially rewritten analysis, improved figures, additional references, and errors fixed. Final version (to appear in PRA

Decoherence as Decay of the Loschmidt Echo in a Lorentz Gas

Classical chaotic dynamics is characterized by the exponential sensitivity to initial conditions. Quantum mechanics, however, does not show this feature. We consider instead the sensitivity of quantum evolution to perturbations in the Hamiltonian. This is observed as an atenuation of the Loschmidt Echo, $M(t)$, i.e. the amount of the original state (wave packet of width $\sigma$) which is recovered after a time reversed evolution, in presence of a classically weak perturbation. By considering a Lorentz gas of size $L$, which for large $L$ is a model for an {\it unbounded} classically chaotic system, we find numerical evidence that, if the perturbation is within a certain range, $M(t)$ decays exponentially with a rate $1/\tau_{\phi}$ determined by the Lyapunov exponent $\lambda$ of the corresponding classical dynamics. This exponential decay extends much beyond the Eherenfest time $t_{E}$ and saturates at a time $t_{s}\simeq \lambda^{-1}\ln (\widetilde{N})$, where $\widetilde{N}\simeq (L/\sigma)^2$ is the effective dimensionality of the Hilbert space. Since $\tau _{\phi}$ quantifies the increasing uncontrollability of the quantum phase (decoherence) its characterization and control has fundamental interest.Comment: 3 ps figures, uses Revtex and epsfig. Major revision to the text, now including discussion and references on averaging and Ehrenfest time. Figures 2 and 3 content and order change

The arrow of time: from universe time-asymmetry to local irreversible processes

In several previous papers we have argued for a global and non-entropic approach to the problem of the arrow of time, according to which the ''arrow'' is only a metaphorical way of expressing the geometrical time-asymmetry of the universe. We have also shown that, under definite conditions, this global time-asymmetry can be transferred to local contexts as an energy flow that points to the same temporal direction all over the spacetime. The aim of this paper is to complete the global and non-entropic program by showing that our approach is able to account for irreversible local phenomena, which have been traditionally considered as the physical origin of the arrow of time.Comment: 48 pages, 8 figures, revtex4. Accepted for publication in Foundations of Physic

Universality of the Lyapunov regime for the Loschmidt echo

The Loschmidt echo (LE) is a magnitude that measures the sensitivity of quantum dynamics to perturbations in the Hamiltonian. For a certain regime of the parameters, the LE decays exponentially with a rate given by the Lyapunov exponent of the underlying classically chaotic system. We develop a semiclassical theory, supported by numerical results in a Lorentz gas model, which allows us to establish and characterize the universality of this Lyapunov regime. In particular, the universality is evidenced by the semiclassical limit of the Fermi wavelength going to zero, the behavior for times longer than Ehrenfest time, the insensitivity with respect to the form of the perturbation and the behavior of individual (non-averaged) initial conditions. Finally, by elaborating a semiclassical approximation to the Wigner function, we are able to distinguish between classical and quantum origin for the different terms of the LE. This approach renders an understanding for the persistence of the Lyapunov regime after the Ehrenfest time, as well as a reinterpretation of our results in terms of the quantum--classical transition.Comment: 33 pages, 17 figures, uses Revtex