DFT-27a: Winding Sector Selection in dc SQUIDs: A Dual Description

MWells · Post by **MWells** » Sat Dec 06, 2025 8:45 pm

This note supplements DFT-27 by examining, in detail, how a dc SQUID may be described in two logically equivalent ways:

Standard circuit-theoretic formulation.
T-frame (DFT) winding sector formulation.

The objective is not to advocate one view over the other, but to establish a direct correspondence that allows interpretation of measurable interference patterns in terms of energy differences between discrete fluxoid branches (indexed by integer winding number).

This provides a technical basis for discussion of potential corrections arising from small sector-dependent curvature terms, such as those suggested in the BPG analysis.

1. Standard Formulation (S-frame)

A symmetric dc SQUID with critical current 𝐼_𝑐 and negligible loop inductance 𝐿 obeys the usual relation:

$I_{c,\mathrm{eff}}(\Phi_{\mathrm{ext}}) = 2 I_c \Big|\cos(\pi \Phi_{\mathrm{ext}}/\Phi_0)\Big|.$

This result is established experimentally and theoretically.

For finite 𝐿, the circulating current 𝐼_circ contributes an inductive term.
The standard approach is to solve self-consistently:

$\Phi_{\mathrm{ext}} + L I_{\mathrm{circ}} = \Phi_{\mathrm{loop}},$
$I_{\mathrm{circ}} = I_c \sin(\delta),$

together with minimization of the Josephson energy.
These are conventional superconducting circuit calculations.

2. Winding Sector Formulation (T-frame, DFT)

In DFT, the same SQUID is represented by a global T-frame phase Θ(𝑠) defined on a closed loop.
The only global requirement is:

$\oint d\Theta = 2\pi n, \qquad n\in\mathbb{Z}.$

The spatial flux is the projection:

$\Phi = \oint A_i dx^i.$

Combining the two gives the usual fluxoid relation:

$\oint(d\Theta - A_i dx^i) = 2\pi n.$

In this picture:

Different integers 𝑛 correspond to discrete winding sectors. In conventional SQUID terminology, these correspond exactly to the discrete fluxoid branches.
The observable state corresponds to the sector that minimizes total energy. This is precisely the standard fluxoid-branch stability condition found in inductive SQUID analyses.

The T-frame energy for sector 𝑛 is:

$E_T(n;\Phi_{\mathrm{ext}},L) = E_{\mathrm{Josephson}}(n;\Phi_{\mathrm{ext}},L) + E_{\mathrm{inductive}}(n;\Phi_{\mathrm{ext}},L).$

No additional assumptions are made.

In particular, the integer 𝑛 referred to here is the same fluxoid index that appears in standard treatments. The T-frame formulation does not introduce additional topological classes beyond those already implicit in conventional fluxoid quantization.

3. Equivalence for Small 𝐿

Under 𝐿→0, the T-frame formulation yields:

$I^{(T)}_c(\Phi_{\mathrm{ext}}) = I^{(S)}_c(\Phi_{\mathrm{ext}}) + \mathcal{O}(10^{-14}),$

i.e. numerical equivalence with machine precision.

This confirms:

The winding-sector picture and standard circuit picture describe the same device.
The agreement is not approximate; it holds to computational precision.

4. Energy-Based Sector Selection

For finite 𝐿, one can compute, for each (Φ_ext,𝐿),

the energy 𝐸_𝑇(𝑛) for each admissible sector 𝑛,
the sector 𝑛_gs(Φ_ext,𝐿) minimizing the energy.

This produces a sector map

$n_{\mathrm{gs}}(\Phi,L).$

With no DFT corrections included, this already reproduces expected finite-𝐿 distortions of 𝐼_𝑐(Φ) known from SQUID analysis.

Thus the T-frame description is not “different physics”;
it is an equivalent description that makes the discrete sector structure explicit.
What is referred to here as ‘sector energetics’ is exactly the branch-energy comparison usually performed between fluxoid minima in conventional inductive SQUID theory; the T-frame formulation simply expresses this as a discrete integer label.

5. Incorporating a Small Sector-Dependent Correction

Motivated by the BPG discussion, a small correction term may be postulated:

$\Delta E_{\mathrm{DFT}}(n) = \kappa_{\mathrm{DFT}}\,f(n),$

where:

𝜅DFT is a small constant,
𝑓(𝑛) is a fixed function of sector index, constant across Φ and 𝐿,
no additional flux dependence is introduced.

This does not introduce new dynamical terms into the Josephson or inductive relations; it modifies the energy ordering between existing fluxoid branches.

6. Numerical Findings (Summary)

Using this formulation, the following statements can be made:

6.1 Sector selection results

Across the (Φ,𝐿) domain sampled, approximately half the grid exhibits

$n_{\mathrm{DFT}}(\Phi,L) \ne n_{\mathrm{noDFT}}(\Phi,L).$

For all 𝐿, the first discrepancy occurs at

$\Phi_{\mathrm{ext}} \approx \Phi_0/2,$

the point where sector energies normally cross. This crossing is the standard SQUID result where adjacent fluxoid branches interchange stability; the T-frame calculation reproduces this behavior because it is an equivalent representation of the same discrete branch landscape.

This confirms that changes in sector selection are concentrated in the expected region of competitive energetics.

6.2 Observable consequences

When the corresponding 𝐼_𝑐(Φ,𝐿) values are computed:

For small 𝐿, the influence is negligible, as expected.
For moderate 𝐿, measurable changes appear in the interference pattern.
For larger 𝐿, structured distortions occur (including sign changes).

These distortions arise not from alterations to Josephson or inductive relations, but from changes in which fluxoid branch (sector index 𝑛) minimizes the total energy. The T-frame formulation does not introduce new dynamics; it exposes the same branch selection that is present but implicit in standard SQUID analysis.

7. Interpretation and Scope

The results indicate that:

The global winding-sector picture and the standard circuit picture are computationally equivalent under conditions where equivalence is expected.
Finite-L distortions in interference patterns arise from the conventional competition between discrete fluxoid branches. The T-frame formulation provides an explicit representation of this competition, rather than introducing new physical content.
Small sector-dependent corrections do not produce arbitrary distortions, but only affect regions where sector competition is significant.

In particular, the T-frame formulation provides a useful separation between the conventional dynamical contributions (Josephson and inductive) and the discrete branch-selection structure (fluxoid sectors), allowing subtle sector-dependent energetic effects to be analyzed without altering or complicating standard SQUID physics.

No claim is made here regarding experimental observability of such corrections;
the present result is purely structural:

If the energy landscape over winding sectors is altered,
the interference pattern follows in a predictable way.

This establishes a framework for further analysis, either theoretical or experimental, without implying conclusions beyond what has been demonstrated.

8. Outlook

The T-frame formulation provides a compact way to reason about SQUID phase structure, especially in contexts where discrete sector behavior is relevant.
The analysis suggests potential avenues for comparing sector-based interpretations with precise interference data. Such comparisons can be carried out using standard SQUID calibration techniques because the quantities compared (branch-selective energetics and critical current modulation) are already conventional observables.
No assumptions have been introduced that depend on reciprocal-system interpretations; the mathematics is compatible with standard SQUID physics. i.e., nothing here requires nonstandard assumptions about superconductivity.

From a broader perspective, the sector-based description developed here has an important consequence for Dual-Frame Theory. The same structural ingredients—discrete winding sectors, small curvature-dependent energy offsets, and S-frame projection into measurable quantities—underlie the treatment of atomic spectra and the Lamb shift. Thus, dc SQUIDs provide a macroscopic, tunable laboratory for probing the curvature–energetics mechanism that manifests microscopically as spectral level shifts. This does not alter standard SQUID physics; it simply reveals a shared geometric structure behind fluxoid branch competition and atomic energy corrections, making that structure experimentally accessible in a device setting.