Dissipative spectral form factor for elliptic Ginibre unitary ensemble and applications

We investigate the dissipative spectral form factor (DSFF)--a widely used probe of non-Hermitian quantum chaos--in the elliptic Ginibre unitary ensemble (eGinUE), which interpolates between the non-Hermitian Ginibre unitary ensemble (GinUE) and the Hermitian Gaussian unitary ensemble (GUE) via a symmetry breaking parameter. We derive exact finite-dimensional results and large-dimensional approximations for the DSFF, revealing a scaling relationship that connects the DSFF of eGinUE to that of GinUE and the spectral form factor of GUE. This relation explains the distinct time scales underlying the characteristic \emph{dip-ramp-plateau} structure across GinUE, GUE, and crossover regimes. Additionally, we refine estimates of dip-ramp and ramp-plateau transition times for different symmetry regimes. We validate our results with Monte Carlo simulations and demonstrate applications to paradigmatic quantum-chaotic systems: the crossover Sachdev-Ye-Kitaev model and the crossover Power-law Banded random matrices. We highlight an analogy between eGinUE eigenvalues and the positions of a rotating fermionic gas in a two-dimensional anisotropic trap.