Isochronous bifurcations in a two-parameter twist map.

Isochronous islands in phase space emerge in twist Hamiltonian systems as a response to multiple resonant perturbations. According to the Poincaré-Birkhoff theorem, the number of islands depends on the system characteristics and the perturbation. We analyze, for the two-parameter standard map, also called two-harmonic standard map, how the island chains are modified as the perturbation amplitude increases. We identified three routes for the transition from one chain, associated with one harmonic, to the chain associated with the other harmonic, based on a combination of pitchfork and saddle-node bifurcations. These routes can present intermediate island chains configurations. Otherwise, the destruction of the islands always occurs through the pitchfork bifurcation.

Exact Markovian evolution of quantum systems with several degrees of freedom: Phase space representations.

The exact solution of the Lindblad equation with a quadratic Hamiltonian and linear coupling operators was derived within the chord representation, that is, for the Fourier transform of the Wigner function, also known as the characteristic function. It is here generalized for several degrees of freedom, so as to provide an explicit expression for the reduced density operator of any subsystem, as well as moments expressed as derivatives of this evolving chord function. The Wigner function is then the convolution of its straightforward classical evolution with a widening multidimensional Gaussian window, eventually ensuring its positivity. Futher on, positivity also holds for the Glauber-Sundarshan P function, which guarantees separability of the components. In the context of several degrees of freedom, a full dissipation matrix is defined, whose trace is equal to twice the previously derived dissipation coefficient. This governs the rate at which the phase space volume of the argument of the Wigner function contracts, while that of the chord function expands. Examples of Markovian evolution of a triatomic molecule and of an array of harmonic oscillators are discussed.

Quantum-Chaotic Evolution Reproduced from Effective Integrable Trajectories.

Classically integrable approximants are here constructed for a family of predominantly chaotic periodic systems by means of the Baker-Hausdorff-Campbell formula. We compare the evolving wave density and autocorrelation function for the corresponding exact quantum systems using semiclassical approximations based alternatively on the chaotic and on the integrable trajectories. It is found that the latter reproduce the quantum oscillations and provide superior approximations even when the initial coherent state is placed in a broad chaotic region. Time regimes are then accessed in which the propagation based on the system's exact chaotic trajectories breaks down.