Studies of Classically Chaotic Quantum Systems Within the PseudoProbability Formalism
Abstract
The evolution of classically chaotic quantum systems is analyzed within the formalism of Quantum PseudoProbability Distributions. Due to the deep connections that a quantum system shows with its classical correspondent in this representation, the PseudoProbability formalism appears to be a useful method of investigation in the field of "Quantum Chaos". In the first part of the thesis we generalize this formalism to quantum systems containing spin operators. It is shown that a classicallike equation of motion for the pseudoprobability distribution rho_ {rm W} can be constructed, partialrho_{rm W}/partial {rm t} = (L_{ rm CL} + L_{rm QGD})rho_{rm W}, which is rigorously equivalent to the quantum von Neumann Liouville equation. The operator L_{ rm CL} is undistinguishable from the classical operator that generates the semiclassical equations of motion. In the case of the spinboson system this operator produces semiclassical chaos and is responsible for quantum irreversibility and the fast growth of quantum uncertainty. Carrying out explicit calculations for a spinboson Hamiltonian the joint action of L_{rm CL } and L_{rm QGD} is illustrated. It is shown that the latter operator, L_{rm QGD} makes the spin system "remember" its quantum nature, and competes with the irreversibility induced by the former operator. In the second part we test the idea of the enhancement of the quantum uncertainty triggered by the classical chaos by investigating the analogous effect of diffusive excitation in periodically kicked quantum systems. The classical correspondents of these quantum systems exhibit, in the chaotic region, diffusive behavior of the unperturbed energy. For the Quantum Kicked Harmonic Oscillator, in the case of quantum resonances, we provide an exact solution of the quantum evolution. This proves the existence of a deterministic drift in the energy increase over time of the system considered. More generally, this "superdiffusive" excitation of the energy is due to coherent quantum mechanical tunnelling between degenerate tori of the classical phase space. In conclusion we find that some of the quantum effects resulting from this fast increase do not have any classical counterpart, they are mainly tunnelling processes. This seems to be the first observation of an effect of this kind.
 Publication:

Ph.D. Thesis
 Pub Date:
 January 1992
 Bibcode:
 1992PhDT........90R
 Keywords:

 QUANTUM CHAOS;
 Physics: General