We study charge detection and controlled dephasing of a mesoscopic system via
a quantum dot detector (QDD), where the mesoscopic system and the QDD are
capacitively coupled. The QDD is considered to have coherent resonant
tunnelling via a single level. It is found that the dephasing rate is
proportional to the square of the conductance of the QDD for the Breit-Wigner
model, showing that the dephasing is completely different from the shot noise
of the detector. The measurement rate, on the other hand, shows a dip near the
resonance. Our findings are peculiar especially for a symmetric detector in the
following aspect: The dephasing rate is maximum at resonance of the QDD where
the detector conductance is insensitive to the charge state of the mesoscopic
system. As a result, the efficiency of the detector shows a dip and vanishes at
resonance, in contrast to the single-channel symmetric non-resonant detector
that has always a maximum efficiency. We find that this difference originates
from a very general property of the scattering matrix: The abrupt phase change
exists in the scattering amplitudes in the presence of the symmetry, which is
insensitive to the detector current but {\em stores} the information of the
quantum state of the mesoscopic system.Comment: 7 pages, 3 figure
