Josephson Junction as Nonlinear Element

The Josephson junction provides the essential nonlinearity that makes superconducting qubits possible. Without it, superconducting circuits are harmonic oscillators — all transitions are degenerate, and you cannot address a single $|0⟩\leftrightarrow |1⟩$ transition.

The Josephson Relations

A tunnel junction between two superconductors obeys:

• Current-phase: $I=I_c\sin \phi$
• Voltage-phase: $V=\frac{{\Phi }_0}{2\pi }\dot{\phi }$

where $\phi$ is the superconducting phase difference and $I_c$ is the critical current.

As a Circuit Element

The junction acts as a nonlinear inductance: $L_J(\phi )=\frac{{\Phi }_0}{2\pi I_c\cos \phi }$

The potential energy $U=-E_J\cos \phi$ (with $E_J={\Phi }_0{I}_c/2\pi$) is the cosine potential that:

• Provides anharmonicity (non-equal level spacing)
• Enables selective addressing of the qubit transition
• Creates a periodic potential landscape for phase-based qubits

In Different Qubit Types

Qubit	Role of JJ
cooper-pair-box-charge-qubit	Couples charge states via Cooper pair tunneling
transmon	Same physics, but $E_J/E_C\gg 1$ flattens charge dispersion
fluxonium	JJ + superinductance creates double-well potential
gatemonium	Semiconductor weak link replaces tunnel junction
blochnium	Operates in phase-slip regime (dual to JJ tunneling)

References

• nakamura-1999-cooper-pair-box
• blais-2004-circuit-qed