$\cos(2\varphi)$ Element in Superconducting Circuits

A $\cos (2\phi )$ potential energy term can emerge in superconducting circuits through higher-order Cooper pair tunneling processes (coherent transport of pairs of Cooper pairs). This produces a potential with doubled periodicity compared to the standard Josephson $\cos (\phi )$.

Physical Origin

In circuits with two parallel Josephson junctions sharing a common island (e.g., the 0-π qubit or gatemonium), destructive interference can suppress the first-harmonic $\cos (\phi )$ term while preserving the second-harmonic $\cos (2\phi )$. This occurs when the two junctions have matched energies and the external flux is tuned to half a flux quantum.

Significance for Protected Qubits

A pure $\cos (2\phi )$ potential has degenerate minima at $\phi =0$ and $\phi =\pi$, which can encode qubit states with intrinsic protection against both charge and flux noise — a route toward hardware-protected quantum memory. This is the operating principle of the 0-pi-qubit family.

In Gatemonium

In gatemonium, voltage-tunable semiconductor weak links (replacing tunnel junctions) allow in-situ control of the $E_J$ ratio, enabling continuous tuning between the standard Josephson regime and the $\cos (2\phi )$-dominated regime.

References

• strickland-2024-gatemonium-a-voltage-tunable-fluxonium
• 0-pi-qubit