Quantum chaos in open systems

It is show that the quantum relaxation process in a classically chaotic open dynamical system is characterized by a quantum relaxation time scale $t_q$ [95]. This scale is much shorter than the Heisenberg time and much larger than the Ehrenfest time: $t_q \propto g^\alpha$ where $g$ is the conductance of the system and the exponent $\alpha$ is close to $1/2$. As a result, quantum and classical decay probabilities remain close up to values $P \sim \exp{ \left( -\sqrt{g} \right) }$ similarly to the case of open disordered systems. The analysis of this behaviour was done for the kicked rotator model with absorption which was introduced in [50]. Later this result was confirmed by methods of random matrix theory (D.Savin and V.Sokolov) and by supersymmetry approach (K.Frahm). This result can be also understand on the basis of weak localization correction to diffusion in disordered systems (A.Mirlin, B.Muzikantskii and D.Khmelnitskii).

In the chaotic regime the quantum eigenstates of nonunitary evolution operator reveal a fractal structure in the phase space [97] corresponding to a underlying classical strange repeller. An example of such fractal quantum eigenstate in Husimi representation is shown at the frontal page of this report (the color changes from red (maximal probability) to blue (zero probability)). It is conjectured that quantum strange attractors, once identified, should have a similar structure.

The quantum effects for Poincaré recurrences in divided phase space (see sections 3.2.1 and 3.6.1) are investigated in [100]. It is shown that quantum effects modify the decay rate of Poincaré recurrences $P(\tau)$ in classical chaotic systems with hierarchical structure of phase space. The exponent $p$ of the algebraic decay $P(\tau) \propto 1/\tau^p$ is shown to have the universal value $p=1$ due to tunneling and localization effects. Experimental evidence of such decay should be observable in mesoscopic systems, Rydberg and cold atoms.

This direction of research forms the basis of the thesis "Quantum chaos in open systems" of G.Maspero (Univ. of Milano at Como) finished in January 1999.