  We systematically investigate how to quantize a transmission line resonator
(TLR) in a mesoscopic electrical circuits in the presence of the resistance and
the conductance of the dielectric media. Developed from the quantum bath based
effective Hamiltonian method for single mode harmonic oscillator, the approach
we presented in this article is a microscopic theory integrating quantum
fluctuation-dissipation relation. To qualitatively check the condition under
which the TLR can behave as a quantum object we study the classical-quantum
boundary characterized by the standard quantum limit.
