  We investigate speeding up of relaxation of Markovian open quantum systems
with the Liouvillian exceptional point (LEP), where the slowest decay mode
degenerate with a faster decay mode. The degeneracy significantly increases the
gap of the Liouvillian operator, which determines the timescale of such systems
in converging to stationarity, and hence accelerates the relaxation process. We
explore an experimentally relevant three level atomic system, whose
eigenmatrices and eigenspectra are obtained completely analytically. This
allows us to gain insights in the LEP and examine respective dynamics with
details. We illustrate that the gap can be further widened through Floquet
engineering, which further accelerates the relaxation process. Finally, we
extend this approach to analyze laser cooling of trapped ions, where vibrations
(phonons) couple to the electronic states. An optimal cooling condition is
obtained analytically, which agrees with both existing experiments and
numerical simulations. Our study provides analytical insights in understanding
LEP, as well as in controlling and optimizing dissipative dynamics of atoms and
trapped ions.
