Semiclassical theory of chaotic quantum resonances

States supported by chaotic open quantum systems fall into two categories: a majority showing instantaneous ballistic decay, and a set of quantum resonances of classically vanishing support in phase space. We present a theory describing these structures within a unified semiclassical framework. Emphasis is put on the quantum diffraction mechanism which introduces an element of probability and is crucial for the formation of resonances. Our main result is boundary conditions on the semiclassical propagation along system trajectories. Depending on whether the trajectory propagation time is shorter or longer than the Ehrenfest time, these conditions describe deterministic escape, or probabilistic quantum decay.

Figure 1

Schematic illustration of the models discussed in the text: (a) One-dimensional toy model consisting of a clean cavity (grey region) and leads connected to it (white regions). There is no backscattering at the cavity-lead interfaces and separation into cavity and leads is largely arbitrary. (b) Two-dimensional clean cavity with chaotic boundary scattering (grey region). Contact to the “outside world” is due to fully transmitting openings connecting to the reservoirs (white regions). Transitions from cavity to reservoirs are again smooth.

Figure 2

On the definition of the different time parameters relevant to the trajectory dynamics. A trajectory ${\gamma }_0$ (indicated by a straight line along the upper front corner of the box) enters or exits the system through phase-space interfaces $S_{i/o}$. Spanning the neighborhood of ${\gamma }_0$ by a stable (unstable) coordinate $u$ ($s$), nearby trajectories contract towards (depart from) ${\gamma }_0$ in the respective coordinate directions. The exit out of the cavity neighborhood of ${\gamma }_0$ then is through the system interface $S_0$ (solid trajectories), or through the “internal interface,” $C_o$ (dashed trajectories.) A similar distinction applies to the entry points. Depending on the entry (exit) variants, each trajectory neighboring ${\gamma }_0$ gets assigned entry (exit) points ${\mathbf{x}}_i$ (${\mathbf{x}}_o$), a timelike progression parameter $\tau$, and a phase-space parametrization $\mathbf{x}\left(\tau \right)$.

Figure 3

The information of Fig. 2 collapsed to the two-dimensional sections spanned by the stable and the unstable coordinate, and the trajectory parameter $q$, respectively. A phase-space point $(q,u,s)$ in the vicinity of ${\gamma }_0$ propagates along a unique classical trajectory $\gamma$. It will exit the cavity either through the interface $S_o$ or within the cavity through the surface $C_o$. Similarly, the union of the left vacuum interface $S_i$ together with the manifold $C_i$ defines the incoming interface.