in dissipative quantum systems
Technological progress leads to the investigation of physical phenomena
at smaller and smaller scales, where both quantum and dissipative
effects play a very important role. Moreover, it is interesting to
consider the quantum dynamics of classically chaotic dissipative
Transition from wave packet collapse to explosion
The quantum dynamics of chaotic Hamiltonian systems is
an exponentially fast spreading of the quantum wave packet. The
spreading rate is given by the Lyapunov exponent, which measures the
rate of exponential instability of classical chaotic motion. This
implies that the classical concept of trajectory becomes meaningless
after the so-called Ehrenfest time scale. This time scale is
logarithmically short in the effective Planck constant of the system.
This conclusion is no longer valid in an open system. We have shown 
that the coupling to a dissipative environment can restore the true
chaos typical of classical mechanics, characterized by positive
Kolmogorov-Sinai entropy and exponential divergence of nearby
trajectories. We show that, for strong dissipation, the quantum wave
function in the phase space collapses onto a compact packet which
follows classical chaotic dynamics and whose area is proportional to
the Planck constant. In contrast, at weak dissipation the exponential
instability of Hamiltonian quantum dynamics up to the Ehrenfest time
scale dominates and leads to wave packet explosion. The transition from
collapse to explosion takes place when the dissipation time scale
exceeds the Ehrenfest time.
Transition from wave packet collapse
collapse (top left) to explosion (top right); Classical and quantum
Poincare' sections are shown in the two bottom plots (see ).
 G.G. Carlo, G. Benenti and D.L. Shepelyansky, Dissipative quantum chaos: transition from
wave packet collapse to explosion,
Phys. Rev. Lett. 95
, 164101 (2005).