Unitary evolution of a pair of Unruh-DeWitt detectors calculated efficiently to an arbitrary perturbative order. (arXiv:1608.08274v3 [hep-th] UPDATED)
Unruh-DeWitt Hamiltonian couples a scalar field with a two-level atom serving
as a particle detector model. Two such detectors held by different observers
following general trajectories can be used to study entanglement behavior in
quantum field theory. Lacking other methods, the unitary evolution must be
studied perturbatively which is considerably time-consuming even to a low
perturbative order. Here we completely solve the problem and present a simple
algorithm for a perturbative calculation based on a solution of a system of
linear Diophantine equations. The algorithm runs polynomially with the
perturbative order. This should be contrasted with the number of perturbative
contributions of the scalar phi^4 theory that is known to grow factorially.
Speaking of the phi^4 model, a welcome collateral result is obtained to
mechanically (almost mindlessly) calculate the interacting scalar phi^n theory
without resorting to Feynman diagrams. We demonstrate it on a typical textbook
example of two interacting fields for n=3,4.