1. Why the LC oscillator fails as a qubit

The previous chapter quantized the LC circuit as a macroscopic harmonic oscillator, with flux $\hat{\Phi}$ and charge $\hat{Q}$ obeying $[\hat{\Phi},\hat{Q}]=i\hbar$ and energy levels $E_n=\hbar\omega_r(n+\tfrac{1}{2})$. The equal spacing is the problem: a drive resonant with the $0\to1$ transition is equally resonant with $1\to2$, so the system cannot be confined to a two-level subspace.

A usable qubit therefore requires two properties:

1. Nonlinearity, to make the level spacings unequal and isolate a $\{|0\rangle,|1\rangle\}$ transition.
2. Negligible dissipation, so the linewidth $\kappa$ stays small.

One circuit element provides both at once. Following Devoret, the Josephson junction is the only known nonlinear, dissipationless circuit element.

2. The Josephson junction

[Figure from Ref. 7]

2.1 Structure

A Josephson junction (JJ) is two superconducting electrodes separated by a thin (~1 nm) insulating barrier. The canonical realization in cQED is the Al / AlO$_x$ / Al trilayer, two aluminum films with a native amorphous oxide barrier. Electrically, the junction is the tunneling element in parallel with an unavoidable parallel-plate capacitance $C_J$ set by the electrode geometry

Atomic interpretation of Josephson Junction at the interface from [7]

Each electrode is described by a single macroscopic order parameter $\Psi_{1,2}=\sqrt{n_{1,2}}\,e^{i\theta_{1,2}}$ (Ginzburg-Landau, neglecting number fluctuations), where $n_{1,2}$ is the Cooper-pair density and $\theta_{1,2}$ the condensate phase on each side. The junction dynamics are governed by the gauge-invariant phase difference $\varphi=\theta_2-\theta_1$.

Josephson junction schematic from [8]: The JJ consists of two superconductors separated by a thin layer of insulator. The Cooper pairs on each side can tunnel through the insulator and create a super-current \(I\). Remarkably, the current can be non-zero even when \(V = 0\). The highly nonlinear \(I\)-\(V\) characteristics of the JJ can be exploited for quantum circuits.

2.2 Two tunneling processes

Two distinct processes cross the barrier; only one is useful.