A part of a lumped transmission line model.We consider a transmission line in a lumped model. This will allow you to find the so-called telegrapher equation which describes the signal propagation along transmission lines in general. You will also see how such devices can even also be used as quantum bits.

Problem Statement

A part of a lumped transmission line model.We may regard a transmission line as made up of certain serially connected elements. Each of these elements has length \(\Delta x\) and obeys some lossy inductance \(L_{n}\) (with resistance \(R_{n,L}\)) and capacitance \(C_{n}\) (with conductance \(G_{n,C}=1/R_{n,C}\)), see figure:

1. Show that in a continuum limit \(\Delta x\rightarrow0\) both voltage and current follow a so-called telegrapher equation \[\text{(1)}\ \ \partial_{xx}U\left(x,t\right)-a_1 U\left(x,t\right)-a_2\dot{U}\left(x,t\right)-\frac{1}{v^{2}}\ddot{U}\left(x,t\right) = 0\ .\]

(a) Start with Kirchhoffs voltage and current laws to derive a finite-difference equation in space.
(b) Perform the continuum-limit to include spatial derivatives. You should arrive at coupled first-order partial differential equations for voltage and current. You may want to use the characteristic values per length like \(r_C=R_C/\Delta x\) etc.
(c) Use a representation in “frequency space” (Fourier transform for \(t\rightarrow\omega\)) to decouple the equations. Assume that the characteristic values are constants. Go back to a time representation to find equation (1) and identify the \(a_i\) and \(v\).

2. The lossless (\(r_{L}=g_{C}=0\)) telegrapher equation is equal to the wave equation\[\partial_{xx}U\left(x,t\right)-\frac{1}{v^{2}}\ddot{U}\left(x,t\right)    =    0\ .\]For which \(\alpha\) is \(U\left(x,t\right)=f\left(x\pm\alpha t\right)\) a solution to this equation? Discuss your result - what is the implication for signals broadcasted along the transmission line?



Background: Superconducting Transmission Lines for Quantum Computation

A superconducting transmission line as qubit.Let us think about some applications of transmission lines. There are the obvious ones in the radio-frequency domain but we may go a little bit deeper here to fundamental physics. We will see that superconducting transmission lines might be used as quantum bits in quantum computers.

First, a very handwaving summary of the Bardeen, Cooper and Schrieffer (BCS) theory of superconductivity. The interactions of phonons to electrons results in an effective attraction between electrons. Such coupled electrons are called a Cooper pair and are Bosons, not Fermions like a single electron. Bosons have quantum-mechanically different properties (no Pauli-principle!) and can propagate through a solid state body without resistance. (More on a description of superconductivity in terms of massive photons can be found in the background of Superconductors and Their Magnetostatic Fields.)
If one puts two superconductors close together, they form a cavity in-between. Through this cavity, Cooper-pairs can tunnel to the other side which is called Josephson effect. This effect enables a certain coupling between the two superconducting sides. In a classical picture, if two electrons were tunneling, there will be an electric field between them. Assuming that the tunneling is not very frequent such that only none or one pair is crossed at any time, the superconductors with Cooper pairs are basically a two-level system.
On the contrary, if an electric field is present between the two superconductors, this field may couple to the Cooper pairs. The interaction to the transmission line then allows for example to switch between the states of this two-level-system. This makes the superconductors an accessible quantum bit, the analog to a classical bit! This setup is indeed a promising candidate as a building block for quantum computers. Much more details can be found in “Cavity quantum electrodynamics for superconducting electrical circuits: an architecture for quantum computation” by Blais et al., Physical Review A 69 (2004).

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