Quantum-Mechanical Lossy Transmission Lines-Analysis based on Damped Harmonic Oscillator GKSL Theory

Abstract

The linear partial differential equations(PDEs) of a lossy transmission line are first set up. We then fix the temporal frequency of operation and derive using spatial Fourier series representation of the voltage and current, a sequence of decoupled damped oscillator equations, which are then quantized using Lindblad/GKSL operators modeling the losses added to the lossless Harmonic Hamiltonian. The GKSL equations in Heisenberg form are shown to yield the correct damped quantum oscillator equations by making use of the canonical commutation relations for the creation and annihilation operators of the oscillators. We explain a scheme based on the Glauber-Sudarshan coherent state representation of how to transform the GKSL equation into partial differential equations for functions of time and complex variables. Quantum entropy computations are also made and finally MATLAB simulation of the GKSL is made which yield exponentially decaying graphs of a damped harmonic oscillator for the norms of observables evolving with time.