21 research outputs found
Stability of Quantum Motion: Beyond Fermi-golden-rule and Lyapunov decay
We study, analytically and numerically, the stability of quantum motion for a
classically chaotic system. We show the existence of different regimes of
fidelity decay which deviate from Fermi Golden rule and Lyapunov decay.Comment: 5 pages, 5 figure
Recurrence of fidelity in near integrable systems
Within the framework of simple perturbation theory, recurrence time of
quantum fidelity is related to the period of the classical motion. This
indicates the possibility of recurrence in near integrable systems. We have
studied such possibility in detail with the kicked rotor as an example. In
accordance with the correspondence principle, recurrence is observed when the
underlying classical dynamics is well approximated by the harmonic oscillator.
Quantum revivals of fidelity is noted in the interior of resonances, while
classical-quantum correspondence of fidelity is seen to be very short for
states initially in the rotational KAM region.Comment: 13 pages, 6 figure
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
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
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
Sensitivity to perturbations in a quantum chaotic billiard
The Loschmidt echo (LE) measures the ability of a system to return to the
initial state after a forward quantum evolution followed by a backward
perturbed one. It has been conjectured that the echo of a classically chaotic
system decays exponentially, with a decay rate given by the minimum between the
width of the local density of states and the Lyapunov exponent. As the
perturbation strength is increased one obtains a cross-over between both
regimes. These predictions are based on situations where the Fermi Golden Rule
(FGR) is valid. By considering a paradigmatic fully chaotic system, the
Bunimovich stadium billiard, with a perturbation in a regime for which the FGR
manifestly does not work, we find a cross over from to Lyapunov decay.
We find that, challenging the analytic interpretation, these conjetures are
valid even beyond the expected range.Comment: Significantly revised version. To appear in Physical Review E Rapid
Communication
Short time decay of the Loschmidt echo
The Loschmidt echo measures the sensitivity to perturbations of quantum
evolutions. We study its short time decay in classically chaotic systems. Using
perturbation theory and throwing out all correlation imposed by the initial
state and the perturbation, we show that the characteristic time of this regime
is well described by the inverse of the width of the local density of states.
This result is illustrated and discussed in a numerical study in a
2-dimensional chaotic billiard system perturbed by various contour deformations
and using different types of initial conditions. Moreover, the influence to the
short time decay of sub-Planck structures developed by time evolution is also
investigated.Comment: 7 pages, 7 figures, published versio
Decay of the classical Loschmidt echo in integrable systems
We study both analytically and numerically the decay of fidelity of classical
motion for integrable systems. We find that the decay can exhibit two
qualitatively different behaviors, namely an algebraic decay, that is due to
the perturbation of the shape of the tori, or a ballistic decay, that is
associated with perturbing the frequencies of the tori. The type of decay
depends on initial conditions and on the shape of the perturbation but, for
small enough perturbations, not on its size. We demonstrate numerically this
general behavior for the cases of the twist map, the rectangular billiard, and
the kicked rotor in the almost integrable regime.Comment: 8 pages, 3 figures, revte
Dynamical fidelity of a solid-state quantum computation
In this paper we analyze the dynamics in a spin-model of quantum computer.
Main attention is paid to the dynamical fidelity (associated with dynamical
errors) of an algorithm that allows to create an entangled state for remote
qubits. We show that in the regime of selective resonant excitations of qubits
there is no any danger of quantum chaos. Moreover, in this regime a modified
perturbation theory gives an adequate description of the dynamics of the
system. Our approach allows to explicitly describe all peculiarities of the
evolution of the system under time-dependent pulses corresponding to a quantum
protocol. Specifically, we analyze, both analytically and numerically, how the
fidelity decreases in dependence on the model parameters.Comment: 9 pages, 6 figures, submitted to PR
Loschmidt Echo and Lyapunov Exponent in a Quantum Disordered System
We investigate the sensitivity of a disordered system with diffractive
scatterers to a weak external perturbation. Specifically, we calculate the
fidelity M(t) (also called the Loschmidt echo) characterizing a return
probability after a propagation for a time followed by a backward
propagation governed by a slightly perturbed Hamiltonian. For short-range
scatterers we perform a diagrammatic calculation showing that the fidelity
decays first exponentially according to the golden rule, and then follows a
power law governed by the diffusive dynamics. For long-range disorder (when the
diffractive scattering is of small-angle character) an intermediate regime
emerges where the diagrammatics is not applicable. Using the path integral
technique, we derive a kinetic equation and show that M(t) decays exponentially
with a rate governed by the classical Lyapunov exponent.Comment: 9 pages, 7 figure