Thermalization dynamics of entanglement and non-locality of filtered two-mode squeezed states

We explore how entanglement and non-locality evolve between specific spectral components of two-mode squeezed states in thermal environments. These spectral components are extracted from output modes using filters that are frequently utilized in optomechanical systems. We consider two distinct thermalization scenarios: one occurring in the vacuum state prior to entering the nonlinear crystal for squeezing and another after the generation of the two-mode squeezed vacuum but before passing through filters and detectors. Entanglement and non-locality generally remain at their peak when identical filters are applied throughout. In the first scenario, higher initial squeezing causes the entanglement dissipation to slow down at the beginning of the time evolution, followed by a progressive acceleration of entanglement dissipation over time. However, the dissipation rate of non-locality, even though it changes over time it moreover remains the same irrespective of the initial degree of squeezing. In the second scenario, greater squeezing results in a more rapid loss of both entanglement and non-locality. We identify the evolution of specific boundaries for entanglement and non-locality and the conditions for their optimization. Finally, for all the cases, increasing the thermal population of the environment enhances the rate of dissipation, whereas stronger interaction slows dissipation in a normalized dimensionless time scale.