The Spontaneous Emission Of A Motionless Atom In An Infinite Rectangular Waveguide

In this paper,we study a stationary two-level atom which is put in an infinitely long rectangular waveguide,and discuss the spontaneous emission problem of a two-level atom that is in an excited state in a single excitated space.The dispersion relation of a rectangular waveguide is nonlinear,and there are two types of modes in the waveguide,namely the transverse electric mode(TE wave)and the transverse magnetic mode(TM wave).For each mode,there is an infinite number of transverse modes,and each transverse mode corresponds to a cutoff frequency.Since the direction of the electric dipole moment of the atom moves along the z-axis,the atom can only interact with the transverse magnetic mode in the waveguide.When the atomic transition frequency is between the cutoff frequency of the lowest transverse mode and the cutoff frequency of the second transverse mode,and away from the cutoff frequency of the transverse mode,the atom is only coupled to a single transverse mode.The dispersion relationship can be expanded once,that is,linear approximation.We find that the probability that an atom is in an excited state decays exponentially with time.As the atomic transition frequency increases,the number of transverse modes coupled to the atom increases.Under the condition that the linear approximation is still true,the probability of the excited state of the atom decays faster with time,as the number of transverse modes coupled with the atom increases.When the atomic transition frequency is near the cutoff frequency of the lowest transverse mode,the atom interacts with a single transverse mode at this time,and the dispersion relationship needs to be quadratically expanded,that is,the rectangular waveguide is approximated as a quadratic waveguide.We find that the probability of an atom in an excited state is first attenuated by a small oscillation and then stabilized at a small probability value.This shows that when the dispersion relation is quadratic,there is a bound state corresponding to a real solution,and there is no real solution when making a linear approximation.