Spectral Statistics of Non-Hermitian Matrices and Dissipative Quantum Chaos

Abstract

We propose a measure, which we call the dissipative spectral form factor (DSFF), to characterize the spectral statistics of non-Hermitian (and non-Unitary) 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 (and time) scale. Specifically, we provide the exact solution of DSFF for the 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 DSFF exhibits an exact rotational symmetry in its complex time argument $\tau$. Analogous to the spectral form factor (SFF) behaviour for GUE, DSFF for GinUE shows a ``dip-ramp-plateau'' behavior in $|\tau |$: DSFF initially decreases, increases at intermediate time scales, and saturates after a generalized Heisenberg time which scales as the inverse mean level spacing. Remarkably, for large matrix size, the ``ramp'' of DSFF for GinUE increases quadratically in $|\tau |$, in contrast to the linear ramp in SFF for Hermitian ensembles. For dissipative quantum integrable systems, we show that DSFF takes a constant value except for a region in complex time whose size and behavior depends on the eigenvalue density. Numerically, we verify the above claims and show that DSFF for real and quaternion real Ginibre ensembles coincides with the GinUE behaviour except for a region in 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, and show that these models fall under the Ginibre universality class.

8+12 pages, 3+50 figures. To appear in PRL. v2: Added appendices E,F and G

References in corpus (14)

• Making Sense of Non-Hermitian Hamiltonians
• Cold atoms in cavity-generated dynamical optical potentials
• Quantum trajectories and open many-body quantum systems
• Light fields in complex media: mesoscopic scattering meets wave control
• Dynamics of Loschmidt echoes and fidelity decay
• Critical properties of the measurement-induced transition in random quantum circuits
• Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model
• Chaos, Complexity, and Random Matrices
• Universal Signature from Integrability to Chaos in Dissipative Open Quantum Systems
• Anderson transition in three-dimensional systems with non-Hermitian disorder
• Quantum-gas microscopes - A new tool for cold-atom quantum simulators
• Random matrices beyond the Cartan classification
• Integrable nonunitary open quantum circuits
• General Eigenvalue Correlations for the Real Ginibre Ensemble

Cited by in corpus (2)

• Many-Body Open Quantum Systems
• Non-equilibrium boundary driven quantum systems: models, methods and properties