# Exact spectral decomposition of a time-dependent one-particle reduced density matrix

## TL;DR

This paper derives an exact spectral decomposition of a time-dependent one-particle reduced density matrix for a two-particle system, revealing conditions for purity and entanglement measures during abrupt interaction changes.

## Contribution

It provides an exact analytical solution for the spectral decomposition of a time-dependent reduced density matrix in a correlated two-particle system with abrupt interaction changes.

## Key findings

- Purity equals the overlap-square only if interactions change, confinement remains constant.
- Von Neumann entropy can exhibit periodic, logarithmic, or constant behavior depending on Hamiltonian modifications.
- Derived precise conditions for entanglement measures during system evolution.

## Abstract

We determine the exact time-dependent non-idempotent one-particle reduced density matrix and its spectral decomposition for a harmonically confined two-particle correlated one-dimensional system when the interaction terms in the Schr\"odinger Hamiltonian are changed abruptly. Based on this matrix in coordinate space we derivea precise condition for the equivalence of the purity and the overlap-square of the correlated and non-correlated wave functions as the system evolves in time. This equivalence holds only if the interparticle interactions are affected, while the confinement terms are unaffected within the stability range of the system. Under this condition we also analyze various time-dependent measures of entanglement and demonstrate that, depending on the magnitude of the changes made in the Schr\"odinger Hamiltonian, periodic, logarithmically incresing or constant value behavior of the von Neumann entropy can occur.