WKB Propagation of Gaussian Wavepackets

We analyze the semiclassical evolution of Gaussian wavepackets in chaotic systems. We prove that after some short time a Gaussian wavepacket becomes a primitive WKB state. From then on, the state can be propagated using the standard TDWKB scheme. Complex trajectories are not necessary to account for the long-time propagation. The Wigner function of the evolving state develops the structure of a classical filament plus quantum oscillations, with phase and amplitude being determined by geometric properties of a classical manifold.Comment: 4 pages, 4 figures; significant improvement

Lyapunov exponent of many-particle systems: testing the stochastic approach

The stochastic approach to the determination of the largest Lyapunov exponent of a many-particle system is tested in the so-called mean-field XY-Hamiltonians. In weakly chaotic regimes, the stochastic approach relates the Lyapunov exponent to a few statistical properties of the Hessian matrix of the interaction, which can be calculated as suitable thermal averages. We have verified that there is a satisfactory quantitative agreement between theory and simulations in the disordered phases of the XY models, either with attractive or repulsive interactions. Part of the success of the theory is due to the possibility of predicting the shape of the required correlation functions, because this permits the calculation of correlation times as thermal averages.Comment: 11 pages including 6 figure

How do wave packets spread? Time evolution on Ehrenfest time scales

We derive an extension of the standard time dependent WKB theory which can be applied to propagate coherent states and other strongly localised states for long times. It allows in particular to give a uniform description of the transformation from a localised coherent state to a delocalised Lagrangian state which takes place at the Ehrenfest time. The main new ingredient is a metaplectic operator which is used to modify the initial state in a way that standard time dependent WKB can then be applied for the propagation. We give a detailed analysis of the phase space geometry underlying this construction and use this to determine the range of validity of the new method. Several examples are used to illustrate and test the scheme and two applications are discussed: (i) For scattering of a wave packet on a barrier near the critical energy we can derive uniform approximations for the transition from reflection to transmission. (ii) A wave packet propagated along a hyperbolic trajectory becomes a Lagrangian state associated with the unstable manifold at the Ehrenfest time, this is illustrated with the kicked harmonic oscillator.Comment: 30 pages, 3 figure

Semiclassical Description of Wavepacket Revival

We test the ability of semiclassical theory to describe quantitatively the revival of quantum wavepackets --a long time phenomena-- in the one dimensional quartic oscillator (a Kerr type Hamiltonian). Two semiclassical theories are considered: time-dependent WKB and Van Vleck propagation. We show that both approaches describe with impressive accuracy the autocorrelation function and wavefunction up to times longer than the revival time. Moreover, in the Van Vleck approach, we can show analytically that the range of agreement extends to arbitrary long times.Comment: 10 pages, 6 figure

Retrospectiva y retos de una década de gestión universitaria 2020-2012

De manera concreta, en estos últimos diez años, la Universidad ha realizado dos procesos de evaluación institucional con pares académicos centroamericanos, cuyos resultados han sido muy constructivos en la elaboración de un diagnóstico de cómo nos encontramos en cada uno de los ejes educativos, derivando en la implementación de planes de mejoramiento de las carreras y programas

Semiclassical approach to fidelity amplitude

The fidelity amplitude is a quantity of paramount importance in echo type experiments. We use semiclassical theory to study the average fidelity amplitude for quantum chaotic systems under external perturbation. We explain analytically two extreme cases: the random dynamics limit --attained approximately by strongly chaotic systems-- and the random perturbation limit, which shows a Lyapunov decay. Numerical simulations help us bridge the gap between both extreme cases.Comment: 10 pages, 9 figures. Version closest to published versio

Scaling laws for the largest Lyapunov exponent in long-range systems: A random matrix approach

We investigate the laws that rule the behavior of the largest Lyapunov exponent (LLE) in many particle systems with long range interactions. We consider as a representative system the so-called Hamiltonian alpha-XY model where the adjustable parameter alpha controls the range of the interactions of N ferromagnetic spins in a lattice of dimension d. In previous work the dependence of the LLE with the system size N, for sufficiently high energies, was established through numerical simulations. In the thermodynamic limit, the LLE becomes constant for alpha greater than d whereas it decays as an inverse power law of N for alpha smaller than d. A recent theoretical calculation based on Pettini's geometrization of the dynamics is consistent with these numerical results (M.-C. Firpo and S. Ruffo, cond-mat/0108158). Here we show that the scaling behavior can also be explained by a random matrix approach, in which the tangent mappings that define the Lyapunov exponents are modeled by random simplectic matrices drawn from a suitable ensemble.Comment: 5 pages, no figure

Semiclassical spatial correlations in chaotic wave functions

We study the spatial autocorrelation of energy eigenfunctions $\psi_n({\bf q})$ corresponding to classically chaotic systems in the semiclassical regime. Our analysis is based on the Weyl-Wigner formalism for the spectral average $C_{\epsilon}({\bf q^{+}},{\bf q^{-}},E)$ of $\psi_n({\bf q}^{+})\psi_n^*({\bf q}^{-})$, defined as the average over eigenstates within an energy window $\epsilon$ centered at $E$. In this framework $C_{\epsilon}$ is the Fourier transform in momentum space of the spectral Wigner function $W({\bf x},E;\epsilon)$. Our study reveals the chord structure that $C_{\epsilon}$ inherits from the spectral Wigner function showing the interplay between the size of the spectral average window, and the spatial separation scale. We discuss under which conditions is it possible to define a local system independent regime for $C_{\epsilon}$. In doing so, we derive an expression that bridges the existing formulae in the literature and find expressions for $C_{\epsilon}({\bf q^{+}}, {\bf q^{-}},E)$ valid for any separation size $|{\bf q^{+}}-{\bf q^{-}}|$.Comment: 24 pages, 3 figures, submitted to PR

Semiclassical Wigner distribution for two-mode entangled state

We derive the steady state solution of the Fokker-Planck equation that describes the dynamics of the nondegenerate optical parametric oscillator in the truncated Wigner representation of the density operator. We assume that the pump mode is strongly damped, which permits its adiabatic elimination. When the elimination is correctly executed, the resulting stochastic equations contain multiplicative noise terms, and do not admit a potential solution. However, we develop an heuristic scheme leading to a satisfactory steady-state solution. This provides a clear view of the intracavity two-mode entangled state valid in all operating regimes of the OPO. A nongaussian distribution is obtained for the above threshold solution.Comment: 9 pages, 5 figures. arXiv admin note: this contains the content of arXiv:0906.531

Coulomb blockade conductance peak fluctuations in quantum dots and the independent particle model

We study the combined effect of finite temperature, underlying classical dynamics, and deformations on the statistical properties of Coulomb blockade conductance peaks in quantum dots. These effects are considered in the context of the single-particle plus constant-interaction theory of the Coulomb blockade. We present numerical studies of two chaotic models, representative of different mean-field potentials: a parametric random Hamiltonian and the smooth stadium. In addition, we study conductance fluctuations for different integrable confining potentials. For temperatures smaller than the mean level spacing, our results indicate that the peak height distribution is nearly always in good agreement with the available experimental data, irrespective of the confining potential (integrable or chaotic). We find that the peak bunching effect seen in the experiments is reproduced in the theoretical models under certain special conditions. Although the independent particle model fails, in general, to explain quantitatively the short-range part of the peak height correlations observed experimentally, we argue that it allows for an understanding of the long-range part.Comment: RevTex 3.1, 34 pages (including 13 EPS and PS figures), submitted to Phys. Rev.