Spontaneous emission of a moving atom in a waveguide of rectangular cross section

We study the spontaneous emission (SE) of an excited nonrelativistic two-level system (TLS) interacting with the vacuum in a waveguide of rectangular cross section. All TLS's transitions and the center-of-mass motion of the TLS are taken into account. The SE rate and the carried frequency of the emitted photon for the TLS initially being at rest are obtained, it is found that in the first order of the mass M, the frequency of the emitted photon is smaller than the transition frequency of the TLS and the SE rate is smaller than the SE rate Gamma (f) of the TLS fixed in the same waveguide. The SE rate for the TLS initially being moving is obtained in the second order of the mass M. The SE rate is smaller than Gamma (f) but it is dependent not only on the atomic mass but also on the initial momentum. The carried frequency of the emitted photon is decreased when it travels along the direction of the initial momentum, whereas it is increased when it travels in the opposite direction of the initial momentum.