  Quantum optics with giant atoms provides a new approach for implementing
optical memory devices at the atomic scale. Here, we theoretically study the
relaxation dynamics of a single driven three-level atom interacting with a
one-dimensional waveguide, via two coupling points. Under certain conditions,
after the long-time dynamics, we found that the population of giant atom can
either maintain stable values or exhibit regular periodic oscillation behavior,
while photons can be trapped in the region of giant atoms. This phenomenon is
not achievable using a two-level atom with two legs. It is worth noting that
the atomic excitation probability of a stable bound state is a constant value,
which is determined by the size of the atom. Crucially, the size of the atom
(the distance between the two coupling points) is much larger than the
wavelength of the light field, which is a necessary condition for the existence
of oscillating bound states.
