Harmonic inversion analysis of exceptional points in resonance spectra

The spectra of, e.g. open quantum systems are typically given as the superposition of resonances with a Lorentzian line shape, where each resonance is related to a simple pole in the complex energy domain. However, at exceptional points two or more resonances are degenerate and the resulting non-Lorentzian line shapes are related to higher order poles in the complex energy domain. In the Fourier-transform time domain an $n$-th order exceptional point is characterised by a non-exponentially decaying time signal given as the product of an exponential function and a polynomial of degree $n-1$. The complex positions and amplitudes of the non-degenerate resonances can be determined with high accuracy by application of the nonlinear harmonic inversion method to the real-valued resonance spectra. We extend the harmonic inversion method to include the analysis of exceptional points. The technique yields, in the energy domain, the amplitudes of the higher order poles contributing to the spectra, and, in the time domain, the coefficients of the polynomial characterising the non-exponential decay of the time signal. The extended harmonic inversion method is demonstrated on two examples, viz. the analysis of exceptional points in resonance spectra of the hydrogen atom in crossed magnetic and electric fields, and an exceptional point occurring in the dynamics of a single particle in a time-dependent harmonic trap.