Research Projects

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Research conducted by: Steven Tomsovic, Nicholas Cerruti

Introduction

The sensitivity of a wave field's evolution to small perturbations is of fundamental interest. A measure of the sensitivity is the fidelity which is the squared overlap of an initial state propagated forward in time with two slightly different Hamiltonians. The fidelity is important when comparing evolving wave packets that are propagated through a time-varying medium. Well known examples include acoustic waves propagating through the ocean, electrons diffusing through a metal and qubits interacting with the environment. The functional form of the decay with time of the fidelity depends upon the strength of the perturbation and whether the system is regular or chaotic. In regular systems the decay is Gaussian and in chaotic systems its form is either Gaussian for weak perturbations or exponential for strong perturbations.

Our work

For chaotic systems, two main decay regimes of either Gaussian or exponential behavior have been identified depending on the strength of the perturbation. For perturbation strengths intermediate between the two regimes, the fidelity displays both forms of decay. By applying a complementary combination of random matrix and semiclassical theory, a uniform approximation can be derived that covers the full range of perturbation strengths. The time dependence is entirely fixed by the density of states and the so-called transition parameter, which can be related to the phase space volume of the system and the classical action diffusion constant, respectively.

For integrable systems, the fidelity is dependent upon the initial wave packet. The rate of decay is determined by a weighted change in the derivatives of the eigenenergies due to the perturbation. Power law behavior of the fidelity has also been observed.

Figures

Example of Gaussian and exponential decays. The quantum standard map curves are solid and the theoretical curves are dashed. The upper plot is for epsilon = .002 and the lower plot is for epsilon =.0001. The other parameters are N = 1000 and lambda = 18.

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Crossover decay. The upper panel demonstrates the agreement between the quantum standard map (solid curve) and the theoretical results (dashed curve). The middle and lower panel shows each piece of the fidelity (diagonal and off-diagonal) separately and the corresponding theoretical results (Gaussian and exponential). The parameters are N = 1000, lambda = 18 and epsilon =.0005.