Can the Breaking of Time-Reversal Symmetry Introduce Bifurcations in Quantum Mechanics?

Bifurcation is a common phenomenon in nonlinear dynamics and a first step into the domain of chaos. In quantum mechanics, which is supposedly a linear theory, nonlinear equations also occur. So the dynamics of position and momentum uncertainties can be described by a nonlinear differential equation of Riccati-type. Despite the nonlinearity of this equation, no bifurcation occurs, as long as the conservative standard problems that can be solved analytically are considered. However, the situation changes drastically if the influence of a dissipative environment is taken into account in an effective way (via a nonlinear Schrödinger equation). The introduction of an additional parameter - the friction coefficient - leads to bifurcation in the dynamics of the quantum fluctuations. The consequences, e.g. for the wave-packet solutions of the damped free motion, will be discussed.