Spatially overlapped partners in quantum field theory

In quantum field theory particles are physically defined as what Unruh-DeWitt particle detectors observe. By detecting a particle mode $A$, a reduced density operator for a quantum state of $A$ is constructed. Even if the entire quantum state of the quantum field is pure, the state of $A$ is not pure but mixed due to entanglement between other subsystems. The partner mode $B$ of the field is defined as a purification partner of $A$ such that the $AB$ system in a pure state. We show that, without any fine-tuning of the particle detector design of $A$, the weighting function of partner $B$ has spatial overlap of that of $A$. We show a general formula of partner $B$ associated with arbitrarily fixed $A$ of a free field in a general Gaussian state. We demonstrate an example of memory effects in an expanding Freedman-Roberson-Walker universe.