Chaos 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 systems.

Transition from wave packet collapse to explosion

The quantum dynamics of chaotic Hamiltonian systems is characterized by 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 This conclusion is no longer valid in an open system.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 [1] 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 [1]).

References

[1]  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).