Josephson Effect in Superconductors

• The Josephson effect is the coherent tunneling of Cooper pairs between superconductors, characterized by a direct relationship between supercurrent and phase difference.
• Key methodologies include analyzing current-phase relations that yield DC, AC, and fractional (4π-periodic) effects, providing insights into voltage-to-frequency conversion.
• Emerging regimes such as topological and nonreciprocal junctions reveal novel phenomena, including anomalous phase shifts and Josephson diode behavior.

The Josephson effect denotes the coherent quantum tunneling of Cooper pairs or other order-parameter excitations across a weak link (typically a thin insulator, normal metal, or topological channel) connecting two superconductors, or in a broader setting, two phases with a continuous broken symmetry. Central to the effect is the direct relationship between the supercurrent and the phase difference of the respective order parameters, leading to a rich spectrum of quantum-coherent phenomena, including DC and AC Josephson currents, voltage-to-frequency transduction, nontrivial current-phase relations, and emergent effects in both conventional and topological systems.

1. Fundamental Principles and Conventional Josephson Relations

In an archetypal Josephson junction, two superconductors (labeled 1 and 2) are weakly coupled, and the pair amplitude in each is described by a macroscopic phase $\theta_j$. The gauge-invariant phase difference $\varphi = \theta_2 - \theta_1$ governs transport. The standard Josephson relations are: $I_s(\varphi) = I_c \sin\varphi$

$\frac{d\varphi}{dt} = \frac{2e}{\hbar} V$

where $I_c$ is the critical current and $V$ the voltage across the junction. In the absence of voltage ($V=0$), the DC Josephson effect enables a dissipationless current up to $|I| < I_c$. When $V \ne 0$, the AC Josephson effect produces an oscillating current at frequency $\omega_J = 2eV/\hbar$ (Citro et al., 2024).

The Josephson coupling energy is $\varphi = \theta_2 - \theta_1$0 with $\varphi = \theta_2 - \theta_1$1, directly related to the junction's superconducting phase stiffness.

2. Extensions: Generalized Josephson Effects and Nonelectronic Systems

The Josephson effect is a universal phenomenon tied to spontaneous symmetry breaking. In its generalized form, the relevant current is the Noether current associated with the spontaneously broken continuous symmetry—U(1) for superconductors, but extensible to SU(N), translation symmetry in crystals, and more (Beekman, 2019). Key relations persist:

• For any G→H breaking, the DC Josephson effect corresponds to Noether charge flow due to relative orientation in the broken symmetry manifold.
• The AC effect emerges when an energetic bias (e.g., chemical potential, external field, or displacement) induces coherent oscillations in the order-parameter difference.

As an example, the Josephson effect between two crystalline solids manifests as a periodic force as a function of their relative displacement, quantized by lattice constants—reflecting the flow of the crystalline Noether charge (Beekman, 2019).

3. Current-Phase Relations and Novel Josephson Regimes

Standard and Mixed CPRs

The conventional supercurrent is $\varphi = \theta_2 - \theta_1$2, reflecting $\varphi = \theta_2 - \theta_1$3 periodicity. In topological or symmetry-broken systems, the CPR can acquire additional harmonics, leading to "fractional" Josephson effects. In a quantum spin Hall Josephson junction hosting Majorana modes, a $\varphi = \theta_2 - \theta_1$4-periodic contribution $\varphi = \theta_2 - \theta_1$5 arises. The general CPR becomes: $\varphi = \theta_2 - \theta_1$6 The $\varphi = \theta_2 - \theta_1$7 periodicity is protected by fermion parity; its observability hinges on suppression of parity-changing processes (e.g., quasiparticle poisoning). Sufficiently strong two-particle inelastic scattering can restore the $\varphi = \theta_2 - \theta_1$8 signal even when single-particle poisoning is forbidden by symmetry, producing a distinct peak at $\varphi = \theta_2 - \theta_1$9 in the power spectrum (Sticlet et al., 2018).

Anomalous and Nonreciprocal CPRs

Breaking time-reversal and inversion symmetry, or engineering coherent coupling between junctions, can yield anomalous Josephson effects where the supercurrent exists at zero phase difference or has a shifted CPR of the form $I_s(\varphi) = I_c \sin\varphi$0. Multiterminal and coupled-junction platforms enable nonlocal control of this phase offset and allow realization of φ₀-junctions and Josephson diodes with nonreciprocal critical currents (Matsuo et al., 2023, Zhong et al., 13 Apr 2025, Pillet et al., 2023).

Nonequilibrium and Fractional Effects

Rapid biasing or voltage pulses can drive Josephson junctions into nonequilibrium regimes. Under these conditions, interplay of retarded (cos φ) and standard (sin φ) components