  Collisions with background gas particles can shift the resonance frequencies
of atoms in atomic clocks. The internal quantum states of atoms can also become
entangled with their motional states due to the recoil imparted by a collision,
which leads to a further shift of the clock frequency through the relativistic
Doppler shift. It can be complicated to evaluate the Doppler and collisional
frequency shifts for clock atoms in such entangled states, but estimates of
these shifts are essential in order to improve the accuracy of optical atomic
clocks. We present a formalism that describes collisions and relativistic
Doppler shifts in a unified manner, and can therefore be used to accurately
estimate collisional frequency shifts in trapped-atom clocks.
