Unitary evolution of a pair of Unruh-DeWitt detectors calculated efficiently to an arbitrary perturbative order

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.