Entanglement between uncoupled modes with time-dependent complex frequencies

Abstract

In this work we present the general unified description for the unitary time evolution generated by time-dependent non-Hermitian Hamiltonians embedding the bosonic representations of $\mathfrak{su}(1,1)$ and $\mathfrak{su}(2)$ Lie algebras. We take into account a time-dependent Hermitian Dyson maps written in terms of the elements of those algebras with the relation between non-Hermitian and its Hermitian counterpart being independent of the algebra realization. As a direct consequence, we verify that a time-evolved state of uncoupled modes modulated by a time-dependent complex frequency may exhibits a nonzero entanglement even when the cross operators, typical of the interaction between modes, are absent. This is due the nonlocal nature of the nontrivial dynamical Hilbert space metric encoded in the time-dependent parameters of the general Hermitian Dyson map, which depend on the imaginary part of the complex frequency. We illustrate our approach by setting the $\mathcal{PT}$-symmetric case where the imaginary part of frequency is linear on time for the two-mode bosonic realization of Lie algebras.