Diffraction and spectral statistics in systems with a multilevel scatterer

A semiclassical framework to interpret the spectral rigidity of a system containing a scatterer with internal states is developed. Our prototype system is a scaled Rydberg molecule in an external magnetic field, where the core is a multilevel scatterer: the potential sheet in which the outer electron moves depends on the quantum state of the core. Thus the electron-core collision, interpreted in terms of the diffraction of the semiclassical waves associated with the outer electron on the core, can result in a change of the electron’s dynamical regime. We examine the contribution of the diffraction to the spectral rigidity by obtaining the diffractive Green’s function in the semiclassical limit. We concurrently determine this contribution from accurate quantum spectra and compare numerically the semiclassical and quantum results. Our findings indicate that, in a system with a multilevel scatterer, the diffractive contribution to the spectral rigidity cannot be accounted for by a simple universal expression, but rather depends on system specific nonuniversal terms: the quantum properties of the scatterer (reflected by the relative values of the phase shifts in the different channels) and the classical properties of the shortest periodic orbits in the different dynamical regimes.

Figure 1

(a) shows the potential sheet associated with the core scatterer in state $\mid j⟩$ and (b) shows another potential sheet lying lower in energy associated with the scatterer in state $\mid j^{\prime }⟩$. A unit source placed at ${\mathbf{r}}_1$ reaches ${\mathbf{r}}_2$ either by direct propagation [dashed line in (a)], by hitting the core while staying on the same potential sheet [solid line in (a), elastic scattering], or by changing to another potential sheet after the collision with the core [solid line from ${\mathbf{r}}_1$ to the core in (a) and from the core to ${\mathbf{r}}_2$ in (b), inelastic scattering]. (c) shows schematically the matching between the semiclassical waves outside the boundary circle and the quantum mechanical wave function in the inner zone (see text).

Figure 2

Spectral rigidity determined from calculated quantum spectra. (a) The black solid curve corresponds to the scatterer-free system whereas the curves underneath correspond to two examples of systems with equal quantum defects allowing only for elastic scattering: ${\mu }_{\Sigma }={\mu }_{\Pi }=-0.25$ (upper line) and below ${\mu }_{\Sigma }={\mu }_{\Pi }=0.5$ (lower line). The straight dashed line is the $L∕15$ limiting curve. (b) The black solid curve corresponds again to the scatterer-free system and the curves underneath to systems with scatterers allowing for inelastic scattering: from top to bottom ${\mu }_{\Sigma }=0.22{\mu }_{\Pi }=-0.06$ (dashed curve), ${\mu }_{\Sigma }=0.5{\mu }_{\Pi }=0$ (dotted line), and ${\mu }_{\Sigma }=0.5{\mu }_{\Pi }=-0.15$ (dot-dashed curve). In each case, the inset gives an enlarged view of the rigidity for small $L$.

Figure 3

Diffractive contribution to the spectral rigidity for systems with different scatterers whose properties depend on the values of the quantum defects: going from the most rigid to the less rigid spectrum we have ${\mu }_{\Sigma }=0.22{\mu }_{\Pi }=-0.06$ (dashed curve), ${\mu }_{\Sigma }=0.5{\mu }_{\Pi }=0$ (dotted curve), ${\mu }_{\Sigma }={\mu }_{\Pi }=-0.25$ (upper solid line), ${\mu }_{\Sigma }=0.5{\mu }_{\Pi }=-0.15$ (dot-dashed curve), and ${\mu }_{\Sigma }={\mu }_{\Pi }=0.5$ (lower solid line). (a) shows the results obtained from quantum calculations by subtracting the curves for systems with scatterers in Fig. 2 from the scatterer-free system line. (b) displays the semiclassical results obtained within the approximations discussed in the text.

Figure 4

Example of a higher-order process with multiple scattering: a wave travels from ${\mathbf{r}}_1$ to ${\mathbf{r}}_2$ in the same potential sheet after scattering twice with the core and having followed a closed orbit in another potential sheet.