We propose a measure, which we call the dissipative spectral form factor (DSFF), to characterize the spectral statistics of non-Hermitian (and nonunitary) matrices. We show that DSFF successfully diagnoses dissipative quantum chaos and reveals correlations between real and imaginary parts of the complex eigenvalues up to arbitrary energy scale (and timescale). Specifically, we provide the exact solution of DSFF for the complex Ginibre ensemble (GinUE) and for a Poissonian random spectrum (Poisson) as minimal models of dissipative quantum chaotic and integrable systems, respectively. For dissipative quantum chaotic systems, we show that the DSFF exhibits an exact rotational symmetry in its complex time argument $\tau$. Analogous to the spectral form factor (SFF) behavior for Gaussian unitary ensemble, the DSFF for GinUE shows a “dip-ramp-plateau” behavior in $|\tau |$: the DSFF initially decreases, increases at intermediate timescales, and saturates after a generalized Heisenberg time, which scales as the inverse mean level spacing. Remarkably, for large matrix size, the “ramp” of the DSFF for GinUE increases quadratically in $|\tau |$, in contrast to the linear ramp in the SFF for Hermitian ensembles. For dissipative quantum integrable systems, we show that the DSFF takes a constant value, except for a region in complex time whose size and behavior depend on the eigenvalue density. Numerically, we verify the above claims and additionally show that the DSFF for real and quaternion real Ginibre ensembles coincides with the GinUE behavior, except for a region in the complex time plane of measure zero in the limit of large matrix size. As a physical example, we consider the quantum kicked top model with dissipation and show that it falls under the Ginibre universality class and Poisson as the “kick” is switched on or off. Lastly, we study spectral statistics of ensembles of random classical stochastic matrices or Markov chains and show that these models again fall under the Ginibre universality class.

中文翻译：

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非厄米矩阵和耗散量子混沌的谱统计

我们提出了一种度量，我们称之为耗散谱形状因子 (DSFF)，以表征非厄米（和非幺正）矩阵的谱统计。我们表明 DSFF 成功地诊断了耗散量子混沌，并揭示了高达任意能量尺度（和时间尺度）的复特征值的实部和虚部之间的相关性。具体来说，我们分别为作为耗散量子混沌和可积系统的最小模型的复杂 Ginibre 系综 (GinUE) 和泊松随机谱 (Poisson) 提供了 DSFF 的精确解。对于耗散量子混沌系统，我们表明 DSFF 在其复杂的时间参数中表现出精确的旋转对称性$\tau$. 类似于高斯酉系综的频谱形状因子 (SFF) 行为，Gi​​nUE 的 DSFF 在$|\tau |$：DSFF 最初减少，在中间时间尺度增加，并在广义海森堡时间后饱和，其尺度为逆平均水平间距。值得注意的是，对于大矩阵大小时，“斜坡”的用于GinUE的DSFF增加二次方在$|\tau |$，相对于线性Hermitian 系综的 SFF 中的斜坡。对于耗散量子可积系统，我们表明 DSFF 取常数值，但复杂时间中的区域除外，其大小和行为取决于特征值密度。在数值上，我们验证了上述主张，并另外表明实数和四元数实数 Ginibre 系综的 DSFF 与 GinUE 行为一致，除了在大矩阵大小限制下测量为零的复时间平面中的区域。作为一个物理示例，我们考虑具有耗散的量子踢顶模型，并表明它在“踢”打开或关闭时属于 Ginibre 普遍性类和泊松。最后，我们研究了随机经典随机矩阵或马尔可夫链的集合的谱统计，并表明这些模型再次属于 Ginibre 普遍性类别。