DC SQUID Flux Tunability
Node role: Concept.
Reader task: understand how two junction phases and applied loop flux produce an effective tunable Josephson element, and where that reduction ends.
Why it matters
A dc SQUID can tune a qubit, coupler, or parametric circuit with external flux. The familiar cosine formula is powerful only inside a declared device model. Without its factor of two, asymmetry, stable phase, and loop-inductance limits, a plausible frequency curve can correspond to the wrong circuit.
Read the zero-loop model through its phase reduction, asymmetry and stable minimum. The finite-loop section then changes the topology rather than merely adjusting one coefficient. Keep that distinction visible when choosing a model.
Two noninterchangeable SQUID models
Reorganized explanation. The negligible-loop parallel-junction reduction starts from the Josephson loop construction (Vool and Devoret 2017, sec. 4.1.3.1). The finite-loop circuit below is a separately declared engineering realization.
| Model | Explicit circuit content | Intended use |
|---|---|---|
| Negligible-loop effective-cosine surrogate | Two Josephson energies under an algebraic external-flux constraint; no loop self-inductance or pump-inductor branch. | Static tunability and a local effective Josephson coefficient inside the assumptions below. |
| Finite-loop mutually pumped topology | Two oriented Josephson branches, loop self-inductance, a pump inductor, and signed mutual inductance; optional \(C_J\) remains explicit on each junction. | Executable dc, pump, and small-signal circuit analysis without replacing the loop by one cosine. |
The next sections derive the first model. The finite-loop topology later on is a different circuit contract, not a more accurate parameter substitution inside the same formula.
Negligible-loop effective-cosine assumptions
This page first treats:
- two parallel junctions with sinusoidal current-phase relations;
- static externally applied flux;
- negligible geometric and kinetic loop inductance;
- a superconducting zero-voltage state; and
- junction capacitances omitted from the static potential reduction.
Finite loop inductance, self-flux, capacitance, dissipation, and driven voltage states are explicit extensions, not hidden corrections.
Reduce two phases to one
This-page derivation. Combine the two sinusoidal-junction potentials under the static, negligible-inductance phase constraint (Vool and Devoret 2017, sec. 4.1.3.1). The asymmetry sign and phase-offset definition are local to the equations below.
Define
\[ \theta=\pi\frac{\Phi_{\mathrm{ext}}}{\Phi_0}, \qquad \delta_1=\delta+\theta, \qquad \delta_2=\delta-\theta, \]
using a loop and branch orientation consistent with the negligible-inductance loop constraint. The total Josephson potential is
\[ U(\delta) =-E_{J1}\cos(\delta+\theta) -E_{J2}\cos(\delta-\theta). \]
The total junction energy and asymmetry are
\[ E_{J\Sigma}=E_{J1}+E_{J2}, \qquad d=\frac{E_{J2}-E_{J1}}{E_{J\Sigma}} =\frac{I_{c2}-I_{c1}}{I_{c1}+I_{c2}}. \]
Then the two cosines combine into one shifted cosine,
\[ U(\delta)=-E_{J,\mathrm{eff}} \cos(\delta-\delta_0), \]
with the positive amplitude
\[ \boxed{ E_{J,\mathrm{eff}} =E_{J\Sigma} \sqrt{\cos^2\theta+d^2\sin^2\theta} } \]
and phase offset
\[ \delta_0=\operatorname{atan2} \left(d\sin\theta,\cos\theta\right). \]
Because the Josephson relation gives \(E_J=\varphi_0 I_c\), the effective critical-current amplitude is
\[ I_{c,\mathrm{eff}} =(I_{c1}+I_{c2}) \sqrt{\cos^2\theta+d^2\sin^2\theta}. \]
Asymmetry prevents complete cancellation at half flux and shifts the stable collective phase.
Symmetric SQUID and the factor of two
This-page derivation. Set the two junction energies equal in the preceding potential, then differentiate at its stable minimum. The absolute amplitude and phase shift must be handled together.
For \(I_{c1}=I_{c2}=I_{c0}\),
\[ U(\delta)=-2E_{J0}\cos\theta\cos\delta. \]
The signed cosine coefficient is \(2E_{J0}\cos\theta\). The positive effective amplitude and critical current are
\[ E_{J,\mathrm{eff}}=2E_{J0}|\cos\theta|, \qquad I_{c,\mathrm{eff}}=2I_{c0}|\cos\theta|. \]
When \(\cos\theta\) changes sign, the stable phase moves by \(\pi\); replacing the coefficient with an absolute value is valid only when that phase shift is also understood.
At the stable zero-transport-current point,
\[ \boxed{ L_{\mathrm{SQUID},0} =\frac{\varphi_0} {2I_{c0}|\cos(\pi\Phi_{\mathrm{ext}}/\Phi_0)|} }. \]
The factor of two is required when \(I_{c0}\) names one junction. It disappears only when the supplied current is the zero-flux aggregate
\[ I_{c,\Sigma}:=2I_{c0}. \]
For a transport-phase displacement \(\epsilon\) from the stable minimum, the local differential inductance gains another \(1/\cos\epsilon\) factor.
The Workbench Python frequency surrogate evaluates
\[ f=\frac{1}{2\pi \sqrt{\left(L_{\mathrm{jun}}/2+L_s\right)C}}. \]
This formula assumes \(L_{\mathrm{jun}}\) is the supplied small-signal inductance of each of two identical parallel branches. It does not calculate that value from flux, \(I_c\), phase, asymmetry, or loop inductance. Artifact producers must still confirm that their L_jun column has this per-junction meaning.
The Julia Core implements a nonlinear single JosephsonJunction solver row, but it currently has no dc-SQUID type or external-flux lowering. Two junction illustrations in component-library docs are therefore topology intent, not an executable SQUID capability.
Finite-loop mutually pumped dc SQUID
Engineering choice. The following branch list selects one explicit mutually pumped topology. Its loop equation is a this-page orientation-specific construction, not a universal dc-SQUID netlist. The pinned upstream example linked below supplies topology evidence, with its stated testing limitation.
Choose circuit nodes \(a\), \(b\), pump node \(p\), and reference node \(0\). Use the following oriented branches:
| Branch | Positive orientation | Role |
|---|---|---|
| \(J_1\) and optional \(C_{J1}\) | \(a\rightarrow0\) | First nonlinear arm. |
| \(L_\ell\) | \(a\rightarrow b\) | Explicit SQUID-loop self-inductance. |
| \(J_2\) and optional \(C_{J2}\) | \(b\rightarrow0\) | Second nonlinear arm. |
| \(L_p\) | \(p\rightarrow0\) | External dc/pump inductor. |
| \(M_{\ell p}\) | Oriented by \(I_\ell\) and \(I_p\) above | Mutual coupling between \(L_\ell\) and \(L_p\). |
Each \(C_{Jj}\) is the independent parallel junction capacitance; an exact zero means that branch is explicitly omitted.
For the declared current orientations, the reciprocal coupled-inductor contract is
\[ \begin{bmatrix} \lambda_\ell\\ \lambda_p \end{bmatrix} = \begin{bmatrix} L_\ell&M_{\ell p}\\ M_{\ell p}&L_p \end{bmatrix} \begin{bmatrix} I_\ell\\ I_p \end{bmatrix}, \qquad M_{\ell p}=k_{\ell p}\sqrt{L_\ell L_p}. \]
The sign of \(M_{\ell p}\) belongs to these orientations. Reversing one current coordinate flips the mutual term and its flux contribution together; it does not change the hardware. The complete sign, reciprocity, and passive-matrix contract is owned by Inductive Coupling Coefficient, Mutual Inductance, and Self Inductance.
With junction phases oriented along \(a\rightarrow0\) and \(b\rightarrow0\), one consistent loop constraint is
\[ \boxed{ \varphi_0(\delta_2-\delta_1) +\lambda_\ell=n\Phi_0, \qquad \lambda_\ell=L_\ell I_\ell+M_{\ell p}I_p. } \]
Thus \(L_\ell I_\ell\) is self-flux and \(M_{\ell p}I_p\) is the externally driven loop-flux contribution in this orientation. The full circuit must solve this constraint together with both Josephson relations and circuit KCL; it must not replace \(\lambda_\ell\) by external flux in the negligible-loop cosine.
The official stable JosephsonCircuits v0.5.4 flux-pumped JPA example is a public executable instance of this topology: Lj1 || Cj1 connects node 3 to reference, L2 connects nodes 3 and 4, Lj2 || Cj2 connects node 4 to reference, and K1 couples L2 to pump inductor L3. This is topology and backend evidence, not validation of every pumped result. The upstream example itself describes flux biasing and three-wave mixing (3WM) as relatively untested, so those capabilities retain that limitation here.
When the one-cosine model stops being sufficient
Reorganized explanation. Finite loop self-flux requires retaining the loop constraint and current response (Introduction to Superconductivity, n.d.); it is outside the assumptions of the zero-loop reduction.
If loop inductance is finite, circulating current produces self-flux:
\[ \Phi_{\mathrm{tot}} =\Phi_{\mathrm{ext}}+L_{\mathrm{loop}}I_{\mathrm{circ}} \]
under a simple lumped sign convention. The loop constraint must then use \(\Phi_{\mathrm{tot}}\) and be solved self-consistently with both junction relations. Screening, hysteresis, multiple stable branches, and asymmetric arm inductances can no longer be represented by substituting external flux into the negligible-inductance cosine.
- Does \(I_c\) mean one junction or the zero-flux aggregate?
- Is the reported \(E_{J,\mathrm{eff}}\) a signed coefficient or positive amplitude?
- Is the stable phase shift retained when the signed coefficient changes sign?
- Is \(L_J\) evaluated at zero transport current or another declared bias point?
- Is loop inductance negligible by an explicit criterion, modeled, or merely absent?
- Is the circuit using the negligible-loop surrogate or the finite-loop topology, rather than mixing parameters from both?
- For a mutually pumped loop, are both inductor orientations and the signed \(M_{\ell p}\) contribution recorded?
- Does the implementation contain real external-flux semantics, or only a sweep of precomputed inductance values?
Connections
- Fluxoid and Flux Quantization supplies the loop constraint.
- Josephson Current, Phase, Energy, and Inductance supplies each junction relation and the bias-point inductance.
- Josephson Cosine and Quantum Anharmonicity explains what the flux-dependent cosine can do after quantization.
- Inductive Coupling Coefficient, Mutual Inductance, and Self Inductance owns the signed mutual-inductance and current-orientation contract used by the finite-loop pump topology.
- A quantum engineer’s guide connects split-junction tunability to qubit design.
- source-owner parameter tables records external flux, effective Josephson energy, asymmetry, and sweet-spot fields.
References
- R. C. Jaklevic, J. Lambe, A. H. Silver, and J. E. Mercereau, “Quantum Interference Effects in Josephson Tunneling,” Physical Review Letters 12, 159 (1964), doi:10.1103/PhysRevLett.12.159.
- J. Koch et al., “Charge-Insensitive Qubit Design Derived from the Cooper Pair Box,” Physical Review A 76, 042319 (2007), arXiv:cond-mat/0703002, doi:10.1103/PhysRevA.76.042319.
- C. D. Tesche and J. Clarke, “dc SQUID: Noise and Optimization,” Journal of Low Temperature Physics 29, 301 (1977), doi:10.1007/BF00655097.
- JosephsonCircuits
v0.5.4flux-pumped JPA example, stable tag commita88bafb8f57ad5b43ef302ae893288f0b75c11e3.
| Field | Value |
|---|---|
| Status | Source-backed |
| Used by | Fluxoid and Flux Quantization, Josephson Current, Phase, Energy, and Inductance, and the Workbench workflows below. |
| Implementation links | Python LC surrogate, Python admittance surrogate, SQUID fitting workflow, and the official stable JosephsonCircuits finite-loop flux-pumped example. |
| Open review question | Confirm the Workbench L_jun artifact contract before changing numerical behavior: per-junction \(L_{J0}\), finite-bias per-junction inductance, or already-combined SQUID inductance. |