Phase Winding, Flux Quantization, and Josephson Effects

Flux quantization and the Josephson relations follow from the gauge-covariant phase of a charge-$2e$ pair field. Around a superconducting loop, single-valuedness quantizes the fluxoid; across a weak link, the only local phase variable is the phase difference corrected by the line integral of the vector potential. The familiar flux quantum $h/(2e)$ and Josephson frequency $2eV/h$ therefore test the transported charge and coherent phase dynamics, but fractional periodicities are not unique evidence for topology.

Required background. Gauge-invariant Meissner response fixes the charged stiffness. Vortices supplies winding and core physics.

Helpful background. Parallel transport and holonomy clarifies the gauge-invariant loop variable.

Fluxoid quantization

Write the long-wavelength free energy as

$$F=\frac{\rho_s}{2}\int\mathrm d^3x\, (\boldsymbol\nabla\theta-q_p\mathbf A)^2,$$

with pair charge $q_p=-2e$ for electrons. The supercurrent is proportional to $\nabla\theta-q_p\mathbf A$. Integrating around a contour inside a thick superconducting ring gives

$$2\pi n=\oint\boldsymbol\nabla\theta\mathbin{\cdot}\mathrm d\boldsymbol\ell =q_p\Phi+\frac{1}{q_p\rho_s}\oint\mathbf j_s\mathbin{\cdot}\mathrm d\boldsymbol\ell.$$

Restoring $\hbar$, this quantizes the fluxoid, not always the bare magnetic flux. Deep in a thick ring the current term can be negligible, yielding

$$\Phi=n\Phi_0, \qquad \Phi_0=\frac{h}{2e}.$$

Thin films, finite penetration depth, kinetic inductance, and multiply connected geometry retain the current contribution. Using $h/e$ corresponds to a charge-$e$ coherent field and is not a mere unit convention.

Gauge-invariant weak-link phase

Across a junction from point 1 to point 2 define

$$\varphi=\theta_2-\theta_1-q_p\int_1^2\mathbf A\mathbin{\cdot}\mathrm d\boldsymbol\ell.$$

The leading local, time-reversal-symmetric tunnel energy is

$$E_J(\varphi)=-E_J\cos\varphi, \qquad I=\frac{2e}{\hbar}\frac{\partial E_J}{\partial\varphi} =I_c\sin\varphi.$$

The temporal gauge-covariant relation gives

$$\hbar\dot\varphi=2eV.$$

At $V=0$ this permits a dc supercurrent; at constant voltage it produces oscillations at $f_J=2eV/h$. The phase and voltage signs depend on endpoint and charge conventions, but the magnitude of the frequency–voltage ratio is invariant. Josephson 1962, pp. 251–253 gives the original prediction.

In a superconducting loop with two junctions, the two gauge-invariant phases obey a flux constraint. For negligible loop inductance and equal critical currents,

$$I_c(\Phi)=2I_0\left\lvert\cos\frac{\pi\Phi}{\Phi_0}\right\rvert.$$

Finite inductance makes the flux self-consistent; junction asymmetry prevents exact nodes. These are calibration effects, not evidence against phase coherence.

Junction dynamics and nonstandard periodicity

A real junction has capacitance $C$ and dissipation $R$. The resistively and capacitively shunted model is

$$I=I_c\sin\varphi+\frac{V}{R}+C\dot V, \qquad V=\frac{\hbar}{2e}\dot\varphi.$$

Its plasma frequency, hysteresis, thermal activation, and quantum phase slips determine whether the ideal relations are visible. Microwave irradiation produces Shapiro steps at $V_n=nhf/(2e)$ under the conventional $2\pi$ relation. Tinkham 2004, chs. 6–7 gives the fluxoid, SQUID, and shunted-junction derivations.

A protected fermion-parity sector in an ideal topological junction can support a $4\pi$-periodic contribution. But Landau–Zener transitions, poisoning, nonequilibrium occupation, heating, and conventional higher harmonics can also suppress odd Shapiro steps or mimic an apparent fractional response. A fractional signal is therefore a hypothesis generator; topology requires the independent bulk, nonlocal, and parity tests on the Majorana evidence page.

The validity diagram shows where charge normalization, circuit dynamics, and alternative periodicity enter.

Flux and Josephson relations follow from charge-$2e$ phase covariance. Nonstandard periodicity becomes topological evidence only after circuit, poisoning, Landau–Zener, and conventional harmonic alternatives fail. Original schematic, not to scale.

The paired-matter claim test matrix preserves the full comparison.

Exercise

Derive the symmetric SQUID pattern. Maximize $I=I_0\sin\varphi_1+I_0\sin\varphi_2$ subject to $\varphi_2-\varphi_1=2\pi\Phi/\Phi_0$.

Solution

Set $\bar\varphi=(\varphi_1+\varphi_2)/2$ and $\delta=(\varphi_2-\varphi_1)/2=\pi\Phi/\Phi_0$. Then $I=2I_0\sin\bar\varphi\cos\delta$. Maximizing over $\bar\varphi$ gives $I_c=2I_0\lvert\cos(\pi\Phi/\Phi_0)\rvert$. The result assumes negligible self-inductance, sinusoidal junctions, and equal critical currents.

References

• Josephson, B. D. (1962). “Possible new effects in superconductive tunnelling.” Physics Letters 1, 251–253. doi:10.1016/0031-9163(62)91369-0.
• Tinkham, M. (2004). Introduction to Superconductivity, 2nd ed. Dover. Publisher record.