  We study the dynamics of a nanomechanical resonator (NMR) subject to a
measurement by a low transparency quantum point contact (QPC) or tunnel
junction in the non-Markovian domain. We derive the non-Markovian
number-resolved (conditional) and unconditional master equations valid to
second order in the tunneling Hamiltonian without making the rotating-wave
approximation and the Markovian approximation, generally made for systems in
quantum optics. Our non-Markovian master equation reduces, in appropriate
limits, to various Markovian versions of master equations in the literature. We
find considerable difference in dynamics between the non-Markovian cases and
its Markovian counterparts. We also calculate the time-dependent transport
current through the QPC which contains information about the measured NMR
system. We find an extra transient current term proportional to the expectation
value of the symmetrized product of the position and momentum operators of the
NMR. This extra current term, with a coefficient coming from the combination of
the imaginary parts of the QPC reservoir correlation functions, has a
substantial contribution to the total transient current in the non-Markovian
case, but was generally ignored in the studies of the same problem in the
literature. Considering the contribution of this extra term, we show that a
significantly qualitative and quantitative difference in the total transient
current between the non-Markovian and the Markovian wide-band-limit cases can
be observed. Thus, it may serve as a witness or signature of the non-Markovian
features in the coupled NMR-QPC system.
