  We have investigated the quantum admittance of an ultrasmall tunnel junction
with arbitrary tunneling strength under an electromagnetic environment. Using
the functional integral approach a close analytical expression of the quantum
admittance is derived for a general electromagnetic environment. We then
consider a specific controllable environment where a resistance is connected in
series with the tunneling junction, for which we derived the dc quantum
conductance from the zero frequency limit of the imaginary part of the quantum
admittance. For such electromagnetic environment the dc conductance has been
investigated in recent experiments, and our numerical results agree
quantitatively very well with the measurements. Our complete numerical results
for the entire range of junction conductance and electromagnetic environmental
conductance confirmed the few existing theoretical conclusions.

Mesoscale and Nanoscale Physics