COMPLEX QUANTUM SYSTEMS

The dynamics of quantum systems is of central importance for our understanding of the properties and the control of matter at the microscopic level. The study of quantum dynamics finds applications in such fields as mesoscopic electronics and wave scattering, reaction-rate theory and femtochemistry, and quantum condensation phenomena, among others. In this context, new results have been obtained on different aspects of complexity in quantum systems, as well as on the methods to control their dynamics.

Statistical properties of energy spectra have been found such as universal power laws in the distribution of the curvatures of the energy levels. Semiclassical methods have been developed, especially, for the quantization of chaotic scattering systems. The spectra of quantum scattering resonances have been studied in great detail for the three- and four-disk scatterers, as well as for the ultrafast photodissociation of HgI₂ and CO₂, leading to a fundamental understanding of these processes and their reaction rates. Optimal control theory has been applied to the driving-field control of quantum systems and, in particular, of chemical reactions.

Currently, the coupling of a quantum system to a thermal bath is investigated by developing methods based on a non-Markovian stochastic Schrödinger equation. The dynamics of Bose-Einstein condensates is being theoretically studied in gases of atoms with repulsive and attractive interaction. The coherent control of quantum systems by high intensity lasers is also being investigated.

Selected publications:

P. Gaspard, S. A. Rice, H. J. Mikeska, and K. Nakamura, "Parametric Motion of Energy Levels: Curvature Distribution", Phys. Rev. A 42 , 4015-4027 (1990).

P. Gaspard, D. Alonso, and I. Burghardt, "New Ways of Understanding Semiclassical Quantization", Adv. Chem. Phys. 90, 105-364 (1995).

I. Burghardt and P. Gaspard, "Resonances in the Photodissociation of CO₂: Periodic Orbit and Wavepacket Analyses", Chem. Phys. 225, 259-298 (1997).

P. Gaspard and M. Nagaoka, "NonMarkovian Stochastic Schrödinger Equation", J. Chem. Phys. 111, 5676-5690 (1999).

C. Huepe, S. Métens, G. Dewel, P. Borckmans, and M. E. Brachet, "Decay Rates in Attractive Bose-Einstein Condensates", Phys. Rev. Lett. 82, 1616-1619 (1999).

Approximate resonance eigenfunctions of the main UV absorption band of carbon dioxide, depicted in the configuration space of both CO bond lengths: (a) even-parity resonance at energy E = 2347 cm^-1 with the lifetime of 12 fs (femtosecond); (b) even-parity resonance at E = 10,695 cm^-1 with 13 fs. As energy increases, the nodal lines of the eigenfunctions bifurcate from the mode of symmetric stretching to the one of asymmetric stretching, which corresponds to a nonlinear bifurcation in the classical dynamics.

(a) A periodic and a nonperiodic trajectory of a point particle elastically bouncing between three fixed disks. This mechanical system is a model of classically chaotic scattering. (b) Spectrum of the quantum scattering resonances of the three-disk system calculated by periodic-orbit theory. The resonances are depicted as points in the complex plane of the wavenumber: the horizontal axis gives the inverse wavelength of each resonance while the vertical axis gives the width of each resonance. The resonances fall into the different irreducible representations (A₁, A₂, E) of the C_3v symmetry group of the system. The semiclassical A₁ resonances are in red, the A₂ resonances in blue, and the double E resonances in green. The crosses are the exact resonances calculated by wavefunction method.