Fluxoid and Flux Quantization
Node role: Concept.
Reader task: distinguish fluxoid, magnetic flux, applied flux, and the gauge-invariant loop phase before using a flux-quantization constraint.
Why it matters
Flux bias is a continuous design control, yet superconducting loops have integer-labeled states. Both statements are true because the generally quantized object is the fluxoid, not the externally applied magnetic flux by itself.
This distinction decides how a SQUID constraint is written, when self-flux and screening current matter, and whether a fitted “flux period” has the expected physical meaning.
The route below separates the four quantities first, integrates the London model next, and only then takes the thick-wall or Josephson-loop limit.
Four quantities that must stay separate
Reorganized explanation. The London fluxoid combines magnetic flux and supercurrent circulation (Introduction to Superconductivity, n.d.). The table distinguishes the imposed control from the loop response; it does not restrict a bias sweep.
| Quantity | Symbol | Can it vary continuously? |
|---|---|---|
| Applied flux produced by a bias source | \(\Phi_{\mathrm{ext}}\) | Yes. |
| Total magnetic flux through the selected loop surface | \(\Phi\) | Yes in general; it responds to applied and self-fields. |
| Phase-winding index | \(N\in\mathbb Z\) | No within one topological state; it changes through a phase slip. |
| Fluxoid | \(\Phi+\mu_0\lambda_L^2\oint_C\mathbf J_s\cdot d\mathbf l\) | Quantized as \(N\Phi_0\) under the London-model conditions below. |
| Symbol | Physical meaning |
|---|---|
| \(C\) | Oriented contour inside the superconducting material. |
| \(\mathbf J_s\) | Supercurrent density. |
| \(\lambda_L\) | London penetration depth. |
Derivation in one loop
This-page derivation. Under the London description, integrate the gauge-invariant momentum around an oriented superconducting contour (Introduction to Superconductivity, n.d.). Single-valuedness supplies the integer winding.
Single-valuedness of the superconducting order parameter requires
\[ \oint_C\nabla\theta\cdot d\mathbf l=2\pi N. \]
The gauge-invariant condensate momentum relates the phase gradient to the vector potential and supercurrent. Integrating that relation around \(C\) and using
\[ \oint_C\mathbf A\cdot d\mathbf l=\Phi \]
gives the London fluxoid condition
\[ \boxed{ \Phi+\mu_0\lambda_L^2 \oint_C\mathbf J_s\cdot d\mathbf l=N\Phi_0 }. \]
Orientation fixes the signs of both integrals. Reversing the contour reverses the signed flux and current circulation together.
When magnetic flux alone is quantized
Reorganized explanation. Magnetic-flux quantization is the negligible current-circulation limit of fluxoid quantization (Introduction to Superconductivity, n.d.).
If a contour can be chosen deep inside a thick superconducting wall where the current density is negligible, the current-circulation term vanishes and
\[ \Phi\approx N\Phi_0. \]
That is an important limit, not the general statement. In a thin ring, near a surface, or where kinetic inductance is appreciable, the current term can be significant even though the fluxoid remains tied to \(N\Phi_0\).
A bias line may set \(\Phi_{\mathrm{ext}}/\Phi_0\) to any real value. The loop responds through phase, screening current, self-flux, and possibly a change of winding state. “Flux quantization” does not quantize the laboratory control knob.
Loops interrupted by Josephson junctions
Reorganized explanation. Junction phase drops must be included in loop closure (Vool and Devoret 2017, sec. 4.1.3). The schematic arm-flux quantity below groups magnetic and kinetic contributions according to the chosen loop model.
A Josephson junction contributes a localized gauge-invariant phase drop. For one oriented loop, a useful schematic constraint is
\[ \sum_j s_j\delta_j +\frac{2\pi}{\Phi_0}\Phi_{\mathrm{arms}} =2\pi N, \]
| Symbol | Physical meaning |
|---|---|
| \(s_j\in\{-1,+1\}\) | Sign of junction \(j\) relative to the loop orientation. |
| \(\Phi_{\mathrm{arms}}\) | Magnetic and kinetic contribution assigned to the superconducting arms by the chosen model. |
For a two-junction dc SQUID with negligible loop inductance, the self-flux and arm phase accumulation are neglected. One common orientation then reduces the constraint to
\[ \delta_1-\delta_2 =2\pi\frac{\Phi_{\mathrm{ext}}}{\Phi_0} \pmod{2\pi}. \]
This is a controlled approximation to the loop constraint, not an ad-hoc “correction” applied after writing two unrelated junction phases.
Before using a loop constraint, check:
- Is the quoted flux applied flux, total flux, or a branch flux coordinate?
- Is the superconducting contour and its orientation explicit?
- Is the current-circulation or kinetic-inductance contribution negligible, modeled, or merely omitted?
- Does a Josephson loop use total flux including self-flux when loop inductance is finite?
- Is a change of winding index distinguishable from continuous tuning within one branch?
- Does a reported period use \(\Phi_0=h/2e\) and a calibrated effective loop area?
Typical failures are:
- Replacing “fluxoid” with “magnetic flux” before taking the zero-current limit hides kinetic inductance and screening.
- Calling \(\Phi_{\mathrm{ext}}\) quantized makes a continuous bias sweep appear conceptually impossible.
- Using external flux where the dynamics require total flux ignores circulating-current self-consistency.
- Omitting branch orientation makes the sign of a SQUID phase constraint unauditable.
- Treating a phase slip as smooth motion within one \(N\) branch misses the topological state change.
Connections
- Superconducting Order Parameter and Gauge-Invariant Phase defines the local phase and gauge-invariant branch phase.
- DC SQUID Flux Tunability applies the negligible-loop-inductance constraint to two junctions.
- Josephson Current, Phase, Energy, and Inductance defines each junction phase drop.
- A quantum engineer’s guide supplies the device-level flux-tunability context.
- source-owner parameter tables distinguishes external flux, the flux quantum, and SQUID-facing tunability fields.
References
- N. Byers and C. N. Yang, “Theoretical Considerations Concerning Quantized Magnetic Flux in Superconducting Cylinders,” Physical Review Letters 7, 46 (1961), doi:10.1103/PhysRevLett.7.46.
- B. S. Deaver, Jr. and W. M. Fairbank, “Experimental Evidence for Quantized Flux in Superconducting Cylinders,” Physical Review Letters 7, 43 (1961), doi:10.1103/PhysRevLett.7.43.
- R. Doll and M. Näbauer, “Experimental Proof of Magnetic Flux Quantization in a Superconducting Ring,” Physical Review Letters 7, 51 (1961), doi:10.1103/PhysRevLett.7.51.
| Field | Value |
|---|---|
| Status | Source-backed |
| Used by | DC SQUID Flux Tunability and Superconducting Order Parameter and Gauge-Invariant Phase. |
| Implementation links | Workbench flux handoff and Workbench flux-analysis workflow. |
| Open review question | Can a reviewer tell, without rereading the derivation, which flux is continuous and which loop quantity is quantized? |