  For a system strongly coupled to a heat bath, the quantum coherence of the
system and the heat bath plays an important role in the system dynamics. This
is particularly true in the case of non-Markovian noise. We rigorously
investigate the influence of system-bath coherence by deriving the reduced
hierarchal equations of motion (HEOM), not only in real time, but also in
imaginary time, which represents an inverse temperature. It is shown that the
HEOM in real time obtained when we include the system-bath coherence of the
initial thermal equilibrium state possess the same form as those obtained from
a factorized initial state. We find that the difference in behavior of systems
treated in these two manners results from the difference in initial conditions
of the HEOM elements, which are defined in path integral form. We also derive
HEOM along the imaginary time path to obtain the thermal equilibrium state of a
system strongly coupled to a non-Markovian bath. Then, we show that the steady
state hierarchy elements calculated from the real-time HEOM can be expressed in
terms of the hierarchy elements calculated from the imaginary-time HEOM.
Moreover, we find that the imaginary-time HEOM allow us to evaluate a number of
thermodynamic variables, including the free energy, entropy, internal energy,
heat capacity, and susceptibility. The expectation values of the system energy
and system-bath interaction energy in the thermal equilibrium state are also
evaluated.
