A Universality Emerging in a Universality: Derivation of the Ericson Transition in Stochastic Quantum Scattering and Experimental Validation

Abstract

At lower energies, the resonances in scattering experiments are often isolated. In quantum chaotic many-body, disordered or generically stochastic systems, the resonances overlap at larger energies. Eventually, the Ericson regime is reached in which the cross section behaves like a random function. The scattering matrix elements then follow a universal Gaussian distribution. For more than sixty years, the emergence of this robust additional universal behavior on top of the universal system stochasticity awaits a concise analytical treatment. We derive the transition to the Ericson regime in the universal Heidelberg approach and prove the universal Gaussian distribution by a proper asymptotic expansion. We also obtain explicit formulae for the moments of the distributions. We compare with microwave experiments and numerical simulations.

Figures (5)

• Figure 1: Experimental and analytical results for the distributions of rescaled scattering matrix elements (top) and of cross sections (bottom) for $\Xi=1.424$ and $M=52$. Blue error bars indicate the empirical standard deviation around each point of measurement.
• Figure 2: Monte Carlo simulations and analytical results for the distributions of rescaled scattering matrix elements (top) and of cross sections (bottom) for $\Xi=1.424$ and $M=52$.
• Figure 3: Experimental data and the subleading terms for the distributions of rescaled scattering matrix elements (top) and of cross sections (bottom) for $\Xi=1.424$ and $M=52$.
• Figure 4: Full cumulative distribution $F(\xi_1)$ of the scattering matrix elements (top) and subleading term $F^{(sl)}(\xi_1)$ (bottom) for $\Xi=1.424$ and $M=52$.
• Figure 5: Analytical results and Monte Carlo simulations for the distributions $P(\xi_1)$, $P(\xi_2)$ rescaled real and imaginary parts of the scattering matrix elements (top) and for the distribution $p(\widetilde{\sigma}_{21})$ of the rescaled cross--sections (bottom) for $\Xi=9.55$ and $M=60$.