共振谱中例外点的谐波反演分析
Harmonic inversion analysis of exceptional points in resonance spectra
摘要：
例如，开放量子系统的光谱通常被给出为具有洛伦兹线形的共振的叠加，其中每个共振与复能量域中的简单极点相关。然而，在例外点处，两个或更多个共振是简并的，并且所得到的非洛伦兹线形状与复能量域中的高阶极点有关。在傅里叶变换时域中，$n$阶例外点的特征在于非指数衰减时间信号，该信号表示为指数函数和$n-1$次多项式的乘积。非简并共振的复数位置和振幅可以通过应用非线性谐波反演方法来高精度地确定实值共振谱。我们扩展谐波反演方法，包括例外点的分析。该技术产生，在能量域中，有助于频谱的高阶极点的振幅，并且，在时域中，表征时间信号的非指数衰减的多项式的系数。本文用两个例子说明了扩展的谐波反演方法，即分析氢原子在磁场和电场交叉作用下共振谱中的例外点，以及分析含时谐波阱中单个粒子动力学中的例外点。
英文摘要：
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.
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