A M Ozorio de Almeida

Symplectic evolution of Wigner functions in Markovian open systems.

The Wigner function is known to evolve classically under the exclusive action of a quadratic Hamiltonian. If the system also interacts with the environment through Lindblad operators that are complex linear functions of position and momentum, then the general evolution is the convolution of a non-Hamiltonian classical propagation of the Wigner function with a phase space Gaussian that broadens in time. We analyze the consequences of this in the three generic cases of elliptic, hyperbolic, and parabolic Hamiltonians. The Wigner function always becomes positive in a definite time, which does not depend on the initial pure state. We observe the influence of classical dynamics and dissipation upon this threshold. We also derive an exact formula for the evolving linear entropy as the average of a narrowing Gaussian taken over a probability distribution that depends only on the initial state. This leads to a long time asymptotic formula for the growth of linear entropy. We finally discuss the possibility of recovering the initial state.

On the propagation of semiclassical Wigner functions

We establish the difference between the propagation of semiclassical Wigner functions and classical Liouville propagation. First we rediscuss the semiclassical limit for the propagator of Wigner functions, which on its own leads to their classical propagation. Then, via stationary phase evaluation of the full integral evolution equation, using the semiclassical expressions of Wigner functions, we provide the correct geometrical prescription for their semiclassical propagation. This is determined by the classical trajectories of the tips of the chords defined by the initial semiclassical Wigner function and centred on their arguments, in contrast to the Liouville propagation which is determined by the classical trajectories of the arguments themselves.

Integrable approximation to the overlap of resonances

Abstract We study the interaction of resonances with the same order in families of integrable Hamiltonian systems. This can occur when the unperturbed Hamiltonian is at least cubic in the actions. An integrable perturbation coupling the action-angle variables leads to the disappearance of an island through the coalescence of stable and unstable periodic orbits and originates a complex orbit plus an isolated cubic resonance. The chaotic layer that appears when a general term is added to the Hamiltonian survives even after the disappearance of the unstable periodic orbit.

Periodic orbit theory for the quantized baker's map

Abstract The semiclassical limit for an iteration of the bakers map is constructed by quantizing the corresponding iteration of the classical map. The resulting propagator can be expressed in terms of the classical generating function, leading to explicit expressions for the actions of all the periodic orbits. The periodic orbit sum for the smoothed density of quasi-energy levels is derived taking full account of the discreteness of the underlying phase space. Comparison with exact results shows excellent agreement for smoothings which are much larger than the average level spacing.

On the quantisation of homoclinic motion

The density of states of a classically chaotic system can be represented as a sum over its periodic orbits. An unstable periodic orbit may be the limit of homoclinic orbits, that in turn accumulate infinite families of satellite periodic orbits with arbitrarily long periods. The Birkhoff-Moser theorem, guaranteeing the convergence of normal forms in a neighbourhood of the homoclinic orbits, is the basis of explicit formulae for the actions of the satellite orbits with large periods. The conclusion of a study of the combined contribution of the satellite orbits to the periodic orbit sum is that homoclinic motion can support a quantum state if the central periodic orbit is quantised by Bohr-Sommerfeld rules and its Lyapunov exponent alpha is sufficiently small. The periodic orbits included in the sum range over the entire homoclinic region, but the wave intensity for the state that they support displays a heavy scar mainly along the central periodic orbit.

Analytical determination of unstable periodic orbits in area preserving maps

Abstract The Birkhoff normal form, for the neighbourhood of an unstable fixed point of an analytical area preserving map, was proved by Moser to converge. We here show that the region of convergence in fact stretches along a narrow strip surrounding the stable and the unstable manifolds. Consequently the normal form can be used to compute homoclinic points and unstable periodic orbit families that accumulate on them. This is verified for quadratic maps: we find unstable orbits which return to themselves within an accuracy of twenty-one significant figures. A pair of linear equations is derived, which supply approximately all the periodic orbits accumulating on a given homoclinic point. This explicit formula is asymptotically valid in the limit of large periods.

A variational principle for actions on symmetric symplectic spaces

Abstract We present a definition of generating functions of canonical relations, which are real functions on symmetric symplectic spaces, discussing some conditions for the presence of caustics. We show how the actions compose by a neat geometrical formula and are connected to the hamiltonians via a geometrically simple variational principle which determines the classical trajectories, discussing the temporal evolution of such “extended hamiltonians” in terms of Hamilton–Jacobi-type equations. Simplest spaces are treated explicitly.

Quantum time delay in chaotic scattering: a semiclassical approach

We study the universal fluctuations of the Wigner-Smith time delay for systems which exhibit chaotic dynamics in their classical limit. We present a new derivation of the semiclassical relation of the quantum time delay to properties of the set of trapped periodic orbits in the repeller. As an application, we calculate the energy correlator in the crossover regime between preserved and fully broken time reversal symmetry. We discuss the range of validity of our results and compare them with the predictions of random matrix theories.

The Wigner function for two dimensional tori: Uniform approximation and projections

Abstract The Wigner representation of a quantum state, corresponding to a classically integrable Hamiltonian, has been shown to be intimately tied to a classical phase space torus of the same energy. The fact that the semiclassical approximation of the Wigner function there derived turns out to be singular on the torus, as well as on the “Wigner caustic” which contains it, is due to well known limitations of the stationary phase method. The uniform approximation, here derived, does indeed ascribe to the Wigner function a high amplitude along the Wigner caustic, but this is modulated by rapid oscillations except at the torus itself. Asymptotic expansion away from the torus leads back to the semiclassical approximation. Close to the torus the Wigner function is described by a simple transitional approximation which can be resolved into a product of Wigner functions corresponding to one dimensional tori. These results permit one to explicitly project the Wigner function onto any (Lagrangian) coordinate plane so as to obtain the corresponding wave intensity.

Path integral for the quantum baker's map

We derive a formally exact sum of path integrals for the quantum propagator of the bakers transformation. The phases depend only on the classical actions as in usual phase space path integrals and the sums are over all the symbolic orbits. The deduction depends on multiple Poisson transformations, which lead to a further infinite sum of integrals, but our computations for the propagator and its trace for two iterations show that this is rapidly convergent. Explicit formulae for the quantum corrections to the semiclassical propagator are presented for this case.