Model for scattering with proliferating resonances: Many coupled square wells

We present a multichannel model for elastic interactions, composed of an arbitrary number of coupled finite square-well potentials, and derive semianalytic solutions for its scattering behavior. Despite the model's simplicity, it is flexible enough to include many coupled short-ranged resonances in the vicinity of the collision threshold, as is necessary to describe ongoing experiments in ultracold molecules and lanthanide atoms. We also introduce a simple but physically realistic statistical ensemble for parameters in this model. We compute the resulting probability distributions of nearest-neighbor resonance spacings and analyze them by fitting to the Brody distribution. We quantify the ability of alternative distribution functions, for resonance spacing and resonance number variance, to describe the crossover regime. The analysis demonstrates that the multichannel square-well model with the chosen ensemble of parameters naturally captures the crossover from integrable to chaotic scattering as a function of closed-channel coupling strength.

Figure 1

Illustration of the multichannel square-well model, defined in Eq. (3). Small offsets in the square-well radius are included only for visual clarity.

Figure 2

Results shown here are for an $s$-wave, $N=3$ channel scattering systems. Panel (a) has black vertical lines indicating the bound states of only the closed channel system, solutions to Eq. (30). The red vertical lines in (a), which lie nearly on top of the black lines, are the locations of resonances in this system identified as maxima of the time delay. Panel (b) shows the natural logarithm of both the time delay (black curve) and the closed channel population (red dashed curve). Sharp peaks occur at the resonance positions. Panel (c) shows ${sin}^2\left(\delta \right)$ (black). The dashed red curve in panel (c) corresponds to the background scattering for which the open channel is decoupled from the closed channel sector. The blue dashed curves are Fano line shapes plotted using Eq. (18).

Figure 3

The wave function is shown for the simple $s$-wave, $N=3$ channel scattering systems when the system is both (a) resonant and (b) nonresonant. The closed channels are shown as dashed red and dotted blue curves, and the open channel is shown as a solid black line. For the scattering wave functions, we use the normalization of $s^2+c^2=1$. Panel (a) shows the scattering wave function with an energy of $33.2{\epsilon }_0$. Panel (b) shows the scattering wave function with an energy of $60.0{\epsilon }_0$. Inset: Bound state for the closed sector is shown with energy of $32.9{\epsilon }_0$.

Figure 4

Results shown here are for two $s$-wave $N=41$ channel scattering systems. Panels (a)–(c) show scattering data for a system belonging to an ensemble with a Brody parameter $w=0.05\pm 0.01$, indicating an uncorrelated distribution of resonances. Panels (d)–(f) show data for the same system after the closed-channel coupling has been increased to produce an ensemble Brody parameter of $w=0.96\pm 0.03$, indicating strong level repulsion and chaos. Panels (a) and (d) have vertical lines indicating positions of closed-channel bound states found as solutions to Eq. (30). Panels (b) and (e) show the natural logarithm of the time delay (black curves) where sharp peaks occur at the resonance positions. Also shown is the closed-channel fraction (red dashed curve). Panels (c) and (f) show ${sin}^2\left(\delta \right)$ (black). The dashed red curve in panels (c) and (f) corresponds to the background scattering for which the open channel is decoupled from the closed-channel sector.

Figure 5

Three examples of probability distributions of energy level spacings calculated from an ensemble of 100 systems, each of which exhibits spectra like those shown in Fig. 4. The coupling values shown for each panel are summarized in Table 1. For each panel, the solid red curve shows the fitted Brody distribution. Panel (a) shows the distribution in the regime of Poisson statistics (weak $g_{cc}$). The dashed blue curve indicates the Poisson distribution. Panel (b) shows an intermediate case. The dashed blue curve is a plot of the semi-Poisson distribution. Panel (c) shows the chaotic limit. The dashed blue curve shows the Wigner-Dyson distribution.

Figure 6

The Brody parameter $w$ (black) is shown as a function of the closed channel coupling strength, which is increased from $g_{cc}={10}^{-5}{\epsilon }_0$ to $g_{cc}={10}^{-2}{\epsilon }_0$. It is rescaled here by the ensemble average energy level spacing $g_{cc}/\langle S\rangle$. The open-closed coupling is held fixed in model units at $g_{oc}={10}^{-3}{\epsilon }_0$. However, due to the repulsion of levels, it decreases when scaled by $\langle S\rangle$, as shown in the inset. The shaded region indicates the error calculated using Eq. (37).

Figure 7

The reduced chi-squared, ${\chi }_r^2$, values for the Brody and semi-Poisson distributions are plotted here. We use histograms with 25 bins of size $0.2\langle S\rangle$, like those shown in Fig. 5. Black circles show ${\chi }_r^2$ for level-spacing fits to the Brody distribution. The red squares show the ${\chi }_r^2$ value for level-spacing fits to the semi-Poisson distribution Eq. (39).

Figure 8

The variance in the number of resonances within an energy window $\mathrm{\Delta }E/\langle S\rangle$ is plotted for various closed-channel couplings that correspond to $w=0.05,0.51,0.96$ for the red, green, and blue curves, respectively. The dashed curves show the analytical results in the uncorrelated limit of Poisson statistics, and the GOE. The dotted curve indicates the level number variance for semi-Poisson statistics given by Eq. (43).