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Unruh-DeWitt Hamiltonian couples a scalar field with a two-level atom servingas a particle detector model. Two such detectors held by different observersfollowing general trajectories can be used to study entanglement behavior inquantum field theory. Lacking other methods, the unitary evolution must bestudied perturbatively which is considerably time-consuming even to a lowperturbative order. Here we completely solve the problem and present a simplealgorithm for a perturbative calculation based on a solution of a system oflinear Diophantine equations. The algorithm runs polynomially with theperturbative order. This should be contrasted with the number of perturbativecontributions of the scalar phi^4 theory that is known to grow factorially.  Speaking of the phi^4 model, a welcome collateral result is obtained tomechanically (almost mindlessly) calculate the interacting scalar phi^n theorywithout resorting to Feynman diagrams. We demonstrate it on a typical textbookexample of two interacting fields for n=3,4.

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