Analytic level statistics of the quantum Rabi model

Abstract

We derive an analytic expression for the distribution of level spacings in the quantum Rabi model. At high energies, the spectrum in a fixed parity sector takes the asymptotic form of a uniformly spaced ladder modulated by a decaying oscillation. The alternating sign of this oscillation separates successive spacings into two branches. For a finite window containing many oscillations, counting how often the two branches exceed a given spacing yields an analytic expression for the spacing density at leading order. The oscillation amplitude at the lower end of the window sets the range of spacings, and its value at the upper end fixes the positions of two cusps that form twin peaks. Spacing histograms from exact diagonalization agree with the analytic density, including its spacing range and twin peak positions. At leading order, the same spectral analysis also yields an expression for the ratio of consecutive spacings, a statistic that requires no spectral unfolding. The resulting ratio statistics agree closely with those from exact diagonalization. For windows that continue to contain many oscillations, the spacing and ratio peaks move toward unity at higher energies as the spectrum approaches a uniformly spaced ladder. The same counting approach can be explored in other systems with controlled spectral asymptotics, slowly varying spacing branches, and windows containing many oscillations.