Josephson Current, Phase, Energy, and Inductance
Node role: Concept.
Reader task: keep the Josephson current law, cosine energy, and operating-point inductance physically distinct.
Why it matters
A Josephson junction is not a fixed inductor. Its sine current-phase relation creates a cosine energy landscape; a linear inductance appears only after an operating point and perturbation scale have been chosen.
Keeping those layers separate lets a reviewer tell whether an implementation is solving the full nonlinear branch, using a local linear surrogate, or adding capacitance and dissipation through a larger junction model.
Follow the element law into energy, add capacitance as a separate branch, then linearize at a stated operating point. The final model comparison shows which questions those three operations can answer.
Symbols and orientation
Engineering choice. These symbols and branch directions are this page’s notation; literature with other signs must be translated consistently.
Choose one branch orientation. All electrical quantities below use that same direction.
| Symbol | Meaning | Units |
|---|---|---|
| \(I\) | Oriented branch current | A |
| \(V\) | Oriented branch voltage | V |
| \(\Phi_J\) | Oriented generalized branch flux | Wb |
| \(\Phi_0=h/(2e)\) | Superconducting flux quantum | Wb |
| \(\varphi_0=\Phi_0/(2\pi)=\hbar/(2e)\) | Reduced flux quantum | Wb |
| \(I_c\) | Positive critical-current amplitude | A |
| \(E_J=\varphi_0 I_c\) | Josephson energy | J |
| \(\delta=(\Phi_J-\Phi_{J,\mathrm{offset}})/\varphi_0\) | Oriented gauge-invariant phase difference | dimensionless |
Reversing the branch reverses \(I\), \(V\), \(\Phi_J\), and \(\delta\). A different sign convention is valid only when all four change consistently.
The ideal Josephson element
Reorganized explanation. The ideal sinusoidal current law and voltage–phase relation are the conventional Josephson-element starting model (Vool and Devoret 2017, sec. 4.1). This-page derivation: integrating that law produces the cosine energy written below.
For a conventional tunnel junction with a sinusoidal current-phase relation,
\[ \boxed{I=I_c\sin\delta} \]
and the voltage-phase relation is
\[ \boxed{V=\varphi_0\dot\delta}. \]
Because \(V=\dot\Phi_J\) and current is the derivative of stored energy with respect to branch flux,
\[ I=\frac{\partial U_J}{\partial\Phi_J} =\frac{1}{\varphi_0}\frac{\partial U_J}{\partial\delta}. \]
Integrating the current-phase relation gives
\[ \boxed{U_J(\delta)=-E_J\cos\delta}, \qquad E_J=\varphi_0 I_c. \]
The additive constant in energy is arbitrary; the curvature and energy differences are physical.
Independent parallel junction capacitance
Reorganized explanation. A tunnel junction’s capacitive branch is distinct from its ideal Josephson element (Vool and Devoret 2017, sec. 1.1). Engineering choice: independently declaring or omitting that branch makes the model input explicit; it is not a fabrication relation inferred from inductance.
The ideal Josephson element above specifies only the nonlinear constitutive branch. A capacitively shunted Josephson junction adds an independent capacitance \(C_J\) across the same two terminals and with the same branch voltage:
\[ \boxed{ I=I_c\sin\delta+C_J\dot V =I_c\sin\delta+C_J\varphi_0\ddot\delta }. \]
The added electric energy is
\[ U_{C_J}=\frac12C_JV^2. \]
\(C_J\) is a separate model input. It is not inferred from \(I_c\), \(E_J\), a zero-bias \(L_{J0}\), or a bias-point \(L_J(\delta_0)\). A fabrication model may relate junction area to both \(I_c\) and \(C_J\), but that relation needs its own declared technology evidence; the Josephson constitutive law alone does not supply it. Setting \(C_J=0\) means the parallel capacitance is explicitly omitted from this circuit model. It must not trigger an implicit estimate from \(L_J\).
Inductance belongs to an operating point
This-page derivation. Differentiate the stated sinusoidal constitutive law at the selected bias. The resulting differential inductance is local, unlike the complete cosine model (Vool and Devoret 2017, sec. 4.1).
For a small phase perturbation \(\widetilde\delta\) around a bias point \(\delta_0\),
\[ \widetilde I \approx I_c\cos\delta_0\,\widetilde\delta, \qquad \widetilde\Phi_J=\varphi_0\widetilde\delta. \]
The bias-point differential inductance is therefore
\[ \boxed{ L_J(\delta_0) =\left.\frac{\partial I}{\partial\Phi_J}\right|_{\delta_0}^{-1} =\frac{\varphi_0}{I_c\cos\delta_0} }. \]
At a zero-current minimum, \(\delta_0=0\) modulo \(2\pi\) and
\[ L_{J0}=\frac{\varphi_0}{I_c}. \]
Near \(|\cos\delta_0|=0\), the linear inductance diverges and the next nonlinear terms dominate. A negative differential inductance signals negative local curvature of the isolated Josephson potential; stability then depends on the embedding circuit rather than on the junction alone.
Four different model contracts
| Model | Keeps | Omits | Suitable question |
|---|---|---|---|
| Ideal Josephson element | Full \(I_c\sin\delta\) and \(-E_J\cos\delta\). | Junction capacitance and dissipation. | What nonlinear current and energy follow from phase? |
| Capacitively shunted Josephson junction | Ideal Josephson element plus an independently supplied parallel \(C_J\). | Dissipation unless a separate loss element is added. | What conservative plasma or nonlinear capacitive dynamics follow from the declared \(I_c\) and \(C_J\)? |
| Bias-point inductance | \(L_J(\delta_0)\) for infinitesimal motion around one point. | Large-signal curvature, phase slips, harmonic generation. | What is the local resonance or small-signal response? |
| RCSJ-type junction | Josephson element plus shunt capacitance and resistance. | Microscopic quasiparticle detail unless added separately. | What switching, plasma, damping, or voltage-state dynamics occur? |
With a parallel resistance \(R_J\), the RCSJ branch current is
\[ I=I_c\sin\delta +C_J\varphi_0\ddot\delta +\frac{\varphi_0}{R_J}\dot\delta. \]
A physical tunnel junction has capacitance even when an ideal Josephson element is used as the conservative constitutive primitive. Naming the model and declaring \(C_J\) independently makes that abstraction visible.
This is an existing implementation description, not a freshly verified package-capability claim; current executable contracts remain package-owned.
- The Workbench Julia
JosephsonJunctionlowers a supplied Josephson inductance to a nonlinear JosephsonCircuitsLjrow. It is not an ordinary linear inductor. - The Workbench Python SQUID-frequency and \(Y_{11}\) formulas instead use \(L_{\mathrm{jun}}/2\) as the parallel combination of two identical supplied small-signal junction inductances. They contain no phase dynamics.
- A fitted \(L_J\) must therefore state whether it is \(L_{J0}\), a bias-point differential value, or an already-combined device inductance.
- Weak links can have a non-sinusoidal current-phase relation; then both \(U(\delta)\) and differential inductance must come from that actual relation.
- Replacing \(L_J(\delta_0)\) by \(L_{J0}\) at finite current hides the \(\cos\delta_0\) factor.
- Treating a linearized branch as a nonlinear Josephson element invents harmonics and mixing that the model cannot produce.
- Treating the full Josephson element as a fixed inductor removes the very nonlinearity used for anharmonicity and parametric processes.
- Inferring \(C_J\) from \(L_J\) without a separate technology relation invents a device model that the Josephson law does not contain.
- Reporting \(E_J\) in “GHz” without writing \(E_J/h\) confuses energy with frequency.
Connections
- Superconducting Order Parameter and Gauge-Invariant Phase defines \(\delta\) without equating it to an absolute condensate phase.
- Fluxoid and Flux Quantization constrains junction phases around a loop.
- Josephson Cosine and Quantum Anharmonicity shows when the cosine changes a quantized spectrum.
- DC SQUID Flux Tunability combines two junctions under a loop constraint.
- A quantum engineer’s guide connects these element parameters to qubit design.
- source-owner parameter tables maps \(I_c\), \(E_J/h\), and qubit-facing element semantics into target fields.
References
- B. D. Josephson, “Possible New Effects in Superconductive Tunnelling,” Physics Letters 1, 251 (1962), doi:10.1016/0031-9163(62)91369-0.
- U. Vool and M. H. Devoret, “Introduction to Quantum Electromagnetic Circuits,” International Journal of Circuit Theory and Applications 45, 897 (2017), arXiv:1610.03438, doi:10.1002/cta.2359.
| Field | Value |
|---|---|
| Status | Source-backed |
| Used by | Josephson Cosine and Quantum Anharmonicity, DC SQUID Flux Tunability, and Circuit Lagrangian, Hamiltonian, and Canonical Quantization. |
| Implementation links | Julia junction relation, Julia lowering, and Python small-signal surrogate. |
| Open review question | Does the page make it impossible to mistake \(L_{J0}\) for the full nonlinear element or for a finite-bias differential inductance? |