  We investigate parity-time reversal (PT) phase transitions in open quantum
systems and discuss a criterion of Liouvillian PT symmetry proposed recently by
J. Huber, P. Kirton, S. Rotter and P. Rabl. Using the third quantization, which
is a general method to solve the Lindblad equation for open quadratic systems,
we show, with a proposed criterion of PT symmetry, that the eigenvalue
structure of the Liouvillian clearly changes at the PT symmetry breaking point
for an open 2-spin model with exactly balanced gain and loss if the total spin
is large. Specially, in a PT unbroken phase, some eigenvalues are pure
imaginary numbers while in a PT broken phase, all the eigenvalues are real.
From this result, it is analytically shown for open quantum system including
quantum jumps that the dynamics in the long time limit changes from an
oscillatory to an overdamped behavior at the proposed PT symmetry breaking
point. Furthermore, we show a direct relation between the criterion of Huber et
al. of Liouvillian PT symmetry and dynamics of the physical quantities for
quadratic bosonic systems. Our results support validity of the proposed
criterion of Liouvillian PT symmetry.
