Exact results for models of multichannel quantum nonadiabatic transitions

TL;DR

This paper derives exact constraints and explicit expressions for scattering probabilities in multichannel quantum nonadiabatic transition models with time-dependent Hamiltonians, leveraging the absence of the Stokes phenomenon and symmetries.

Contribution

It introduces a method to obtain exact results for scattering matrices in specific time-dependent quantum models, expanding understanding of nonadiabatic transitions.

Findings

Derived exact constraints on scattering matrix elements.

Provided explicit expressions for scattering probabilities.

Demonstrated utility in models with additional discrete symmetries.

Abstract

We consider nonadiabatic transitions in explicitly time-dependent systems with Hamiltonians of the form $\hat{H}(t) = \hat{A} +\hat{B} t + \hat{C}/t$, where $t$ is time and $\hat{A}$, $\hat{B}$, $\hat{C}$ are Hermitian $N\times N$ matrices. We show that in any model of this type, scattering matrix elements satisfy nontrivial exact constraints that follow from the absence of the Stokes phenomenon for solutions with specific conditions at $t \rightarrow -\infty$. This allows one to continue such solutions analytically to $t \rightarrow +\infty$, and connect their asymptotic behavior at $t \rightarrow -\infty$ and $t \rightarrow +\infty$. This property becomes particularly useful when a model shows additional discrete symmetries. In particular, we derive a number of simple exact constraints and explicit expressions for scattering probabilities in such systems.