  We examine the Unruh-DeWitt particle detector coupled to a scalar field in an
arbitrary Hadamard state in four-dimensional curved spacetime. Using smooth
switching functions to turn on and off the interaction, we obtain a
regulator-free integral formula for the total excitation probability, and we
show that an instantaneous transition rate can be recovered in a suitable
limit. Previous results in Minkowski space are recovered as a special case. As
applications, we consider an inertial detector in the Rindler vacuum and a
detector at rest in a static Newtonian gravitational field. Gravitational
corrections to decay rates in atomic physics laboratory experiments on the
surface of the Earth are estimated to be suppressed by 42 orders of magnitude.
