3 research outputs found
Exact and semiclassical Husimi distributions of Quantum Map Eigenstates
The projector onto single quantum map eigenstates is written only in terms of
powers of the evolution operator, up to half the Heisenberg time, and its
traces. These powers are semiclassically approximated, by a complex generating
function, giving the Husimi distribution of the eigenstates. The results are
tested on the Cat and Baker maps.Comment: 10 pages, 6 figure
Wavepacket Dynamics in Nonlinear Schr\"odinger Equations
Coherent states play an important role in quantum mechanics because of their
unique properties under time evolution. Here we explore this concept for
one-dimensional repulsive nonlinear Schr\"odinger equations, which describe
weakly interacting Bose-Einstein condensates or light propagation in a
nonlinear medium. It is shown that the dynamics of phase-space translations of
the ground state of a harmonic potential is quite simple: the centre follows a
classical trajectory whereas its shape does not vary in time. The parabolic
potential is the only one that satisfies this property. We study the time
evolution of these nonlinear coherent states under perturbations of their
shape, or of the confining potential. A rich variety of effects emerges. In
particular, in the presence of anharmonicities, we observe that the packet
splits into two distinct components. A fraction of the condensate is
transferred towards uncoherent high-energy modes, while the amplitude of
oscillation of the remaining coherent component is damped towards the bottom of
the well
Wavefunctions, Green's functions and expectation values in terms of spectral determinants
We derive semiclassical approximations for wavefunctions, Green's functions
and expectation values for classically chaotic quantum systems. Our method
consists of applying singular and regular perturbations to quantum
Hamiltonians. The wavefunctions, Green's functions and expectation values of
the unperturbed Hamiltonian are expressed in terms of the spectral determinant
of the perturbed Hamiltonian. Semiclassical resummation methods for spectral
determinants are applied and yield approximations in terms of a finite number
of classical trajectories. The final formulas have a simple form. In contrast
to Poincare surface of section methods, the resummation is done in terms of the
periods of the trajectories.Comment: 18 pages, no figure