We study the time evolution of two wave packets prepared at the same initial state, but evolving under slightly different Hamiltonians. For chaotic systems, we determine the circumstances that lead to an exponential decay with time of the wave packet overlap function. We show that for sufficiently weak perturbations, the exponential decay follows a Fermi golden rule, while by making the difference between the two Hamiltonians larger, the characteristic exponential decay time becomes the Lyapunov exponent of the classical system. We illustrate our theoretical findings by investigating numerically the overlap decay function of a two-dimensional dynamical system.
The classical Gaussian ensembles of random matrices can be constructed by maximizing Boltzmann-Gibbs-Shannon's entropy, SBGS = − dH[P (H)] ln[P (H)], with suitable constraints. Here we construct and analyze random-matrix ensembles arising from the generalized entropyq /(q − 1) (thus S1 = SBGS). The resulting ensembles are characterized by a parameter q measuring the degree of nonextensivity of the entropic form. Making q → 1 recovers the Gaussian ensembles. If q = 1, the joint probability distributions P (H) cannot be factorized, i.e., the matrix elements of H are correlated. In the limit of large matrices two different regimes are observed. When q < 1, P (H) has compact support, and the fluctuations tend asymptotically to those of the Gaussian ensembles. Anomalies appear for q > 1: Both P (H) and the marginal distributions P (Hij) show power-law tails. Numerical analyses reveal that the nearest-neighbor spacing distribution is also long-tailed (not Wigner-Dyson) and, after proper scaling, very close to the result for the 2×2 case -a generalization of Wigner's surmise. We discuss connections of these "nonextensive" ensembles with other non-Gaussian ones, like the so-called Lévy ensembles and those arising from soft-confinement.
We propose a mechanism to explain the fluctuations of the ground state energy in quantum dots in the Coulomb blockade regime. Employing the random matrix theory we show that shape deformations may change the adjacent peak spacing distribution from Wigner-Dyson to nearly Gaussian even in the absence of strong charging energy fluctuations. We find that this distribution is solely determined by the average number of anti-crossings between consecutive conductance peaks and the presence or absence of a magnetic field. Our mechanism is tested in a dynamical model whose underlying classical dynamics is chaotic. Our results are in good agreement with recent experiments and apply to quantum dots with spin resolved or spin degenerate states.
Abstract. 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.
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