A lossy transmission line as a quantum open system in the standard quantum limit

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.