Excitation of an inertial Unruh detector in the Minkowski vacuum: A numerical calculation using spherical modes

We consider the excitation of a finite-length inertial Unruh detector in the Minkowski vacuum with an adiabatic switch on of the interaction in the infinite past and a sudden switch-off at finite times and obtain the excitation probability via a numerical calculation using the expansion of the quantum field in spherical modes. We evaluate first the excitation probabilities for the final states of the field with one particle per mode and then we sum over the modes. An interesting feature is that, despite of the inertial trajectory and of the vacuum state of the field, the multipole components of the excitation probability are time-dependent quantities. We make clear how the multipole sum yields the time-independent probability characteristic to an inertial trajectory. The procedure can be extended to obtain the response of the detector along radial trajectories in a generic spherically symmetric spacetimes.