Time correlation functions for non-Hermitian quantum systems

TL;DR

This paper develops a formalism for time-dependent correlation functions in non-Hermitian quantum systems, introducing two definitions and a potential method to assess density operator properties without eigenvalue computation.

Contribution

It presents a novel framework for defining and comparing two types of time correlation functions in non-Hermitian quantum systems, including a conjectured diagnostic tool.

Findings

Two physically consistent correlation functions are defined.

The two correlation functions are equivalent in certain cases.

A conjecture links the difference between the functions to the positive semi-definiteness of the density operator.

Abstract

We introduce a formalism for time-dependent correlation functions for systems whose evolutions are governed by non-Hermitian Hamiltonians of general type. It turns out that one can define two different types of time correlation functions. Both these definitions seem to be physically consistent while becoming equivalent only in certain cases. Moreover, when autocorrelation functions are considered, one can introduce another function defined as the relative difference between the two definitions. We conjecture that such a function can be used to assess the positive semi-definiteness of the density operator without computing its eigenvalues. We illustrate these points by studying analytically a number of models with two energy levels.