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advanced28 min studyLesson 538 in Physics

A gauge-invariant phase difference carries current through a weak link

Connect dc and ac Josephson relations to the interference of two junctions in a low-inductance SQUID.

Before you begin
On this page

The question for today

What you’ll understand

  • Define the gauge-invariant phase difference across a charged weak link.
  • Apply the Josephson current and voltage relations with the pair-charge factor.
  • Compute balanced and imbalanced two-junction SQUID critical currents.

Two superconductors separated by a weak link can carry a coherent current without a steady voltage. The current depends on their relative phase, including the electromagnetic phase accumulated across the junction. A constant voltage makes this phase evolve and converts the current into an oscillation. Combining two junctions around a loop turns phase coherence into a sensitive flux measurement. We will declare the sign conventions and identify the assumptions behind the simplest SQUID interference formula.

The junction phase includes the vector potential

Label the electrodes one and two, and take positive conventional current from one to two. Define γ as θ₁−θ₂ minus 2e/ħ times the line integral of A from one to two. Under an electromagnetic gauge transformation, the endpoint phases and vector-potential integral change by cancelling amounts because the pair charge is −2e. Thus γ is observable through its effects even though each electrode’s absolute phase is gauge dependent. It is not enough to subtract two arbitrary local phases and ignore the electromagnetic path.

For a conventional weak tunnel junction with a leading sinusoidal coupling, the supercurrent is Ic sinγ and the Josephson energy is −EJ cosγ, with EJ=ħIc/(2e). At zero voltage a static γ can support current below the critical magnitude. A constant voltage V=V₁−V₂ produces phase rate 2eV/ħ under the compatible electrochemical convention. Reversing the electrode labels reverses the signed relation, while the oscillation frequency remains 2e|V|/h. Higher harmonics, unconventional junctions, and circuit dynamics can modify the simple current law.

γ=θ1−θ2−2eℏ∫12A⋅dl,I=Icsin⁡γ,γ˙=2eVℏ,fJ=∣V∣Φ0\gamma=\theta_1-\theta_2-\frac{2e}{\hbar}\int_1^2\mathbf A\cdot d\mathbf l,\qquad I=I_c\sin\gamma,\qquad \dot\gamma=\frac{2eV}{\hbar},\qquad f_J=\frac{|V|}{\Phi_0}
γ is dimensionless, Ic is in amperes, V=V₁−V₂ is the compatible junction voltage in volts, fJ is cycles per second, and Φ₀=h/(2e). The voltage relation refers to the gauge-invariant electrochemical phase evolution; the displayed sinusoidal current assumes an ordinary leading-harmonic weak link.↔ Scroll wide equations horizontally.

A loop constrains the relative junction phases

A dc SQUID contains two junctions in parallel around a superconducting loop. For negligible loop self-inductance, its enclosed flux is approximately the applied flux, and fluxoid consistency fixes the difference of the two junction phases modulo 2π. Adding the two sinusoidal currents and maximizing over their common phase gives the critical current. For identical junctions, it is twice the individual critical current multiplied by the absolute cosine of πΦ/Φ₀. The absolute value is essential because critical current denotes the maximum magnitude available after choosing the common phase.

At integer flux quanta the junction contributions can reinforce; at half-integer quanta equal contributions cancel in this ideal model. Unequal junctions leave a nonzero minimum: the maximum at integer flux is their sum and the minimum at half flux is the absolute difference. Finite loop inductance adds self-generated flux and can change the pattern. Individual junction magnetic diffraction, noise, capacitance, and resistive channels also require extra treatment. The two-junction cosine interference is distinct from the single-junction Fraunhofer envelope produced by phase variation across one extended junction.

Ic,SQ(Φ)=Ic12+Ic22+2Ic1Ic2cos⁡(2πΦ/Φ0),Ic1=Ic2=I0⇒Ic,SQ=2I0∣cos⁡(πΦ/Φ0)∣I_{c,\mathrm{SQ}}(\Phi)=\sqrt{I_{c1}^2+I_{c2}^2+2I_{c1}I_{c2}\cos(2\pi\Phi/\Phi_0)},\qquad I_{c1}=I_{c2}=I_0\Rightarrow I_{c,\mathrm{SQ}}=2I_0|\cos(\pi\Phi/\Phi_0)|
The currents are critical-current magnitudes in amperes. Φ is enclosed flux, approximately external flux when self-inductance is negligible. Both junctions are sinusoidal and their individual critical currents are taken independent of the applied flux over this range.↔ Scroll wide equations horizontally.

Current oscillations and circuit voltages are different observables

A voltage-biased ideal junction has an oscillating supercurrent, but a real measurement includes its electromagnetic environment. A current-biased junction above its critical current generally develops voltage through dynamics involving capacitance and dissipative channels. The static interference formula does not predict that complete current–voltage curve. Microwave phase locking can convert the phase evolution into voltage steps, while SQUID magnetometry often monitors a voltage response under bias. The common ingredient is coherent phase evolution with charge magnitude 2e, not a classical conductance that becomes infinite at every voltage.

The waves lab isolates amplitude interference. Reset and set both controls to phase difference zero and amplitude ratio one, predicting summed amplitude two. Reset and set phase difference 180 degrees and ratio one, predicting zero. These reproduce the reinforcing and cancelling phasor limits of equal sinusoidal junction contributions. The controls directly set a classical wave phase and amplitude ratio; they do not set magnetic flux or junction voltage. Playback translates both equal-frequency waves and is not an ac Josephson frequency calculation.

Work it through

Evaluate a balanced SQUID and a voltage-biased junction

Each junction of an ideal low-inductance SQUID has I₀=0.50 mA. Find its critical current at Φ=Φ₀/3. Separately, find the Josephson oscillation frequency of a junction held at V=2.068 μV, using Φ₀=2.068×10⁻¹⁵ Wb.

  1. The SQUID argument is πΦ/Φ₀=π/3, whose cosine is one half. Use the critical-current magnitude formula, including its absolute value.

  2. The available critical current is 2×0.50×(1/2)=0.50 mA. This is the maximum over the common junction phase at the specified flux.

  3. For the separate voltage-biased junction, fJ=|V|/Φ₀=(2.068×10⁻⁶)/(2.068×10⁻¹⁵)=1.0×10⁹ Hz. Weber equals volt-second, so the units give inverse seconds.

The result

The SQUID critical current is 0.50 mA at one-third flux quantum; the separately voltage-biased junction oscillates at 1.0 GHz.

The living notebook

Explore the model

Interactive lab

Interactive lab · waves

Reproduce the two equal-junction interference limits

Reset and set BOTH controls to phase difference 0° and amplitude ratio 1, predicting summed amplitude 2. Reset and set phase difference 180° and ratio 1, predicting zero. This compares phasor limits; the lab has no flux, voltage, or Josephson-frequency control.

Reproduce the two equal-junction interference limitsTwo waves share the same frequency and wave number. Playback advances t in sin(2x − t) + a sin(2x − t + phase). Current readings: Sum at x = 0: 0.866; Resultant amplitude: 1.732; Phase difference: 60°. All parameters are adjustable with the labeled controls below.−3.14−1.5701.573.14−3.4−1.701.73.4x · radiansAmplitude
SumFirst componentSecond component
Sum at x = 0
0.866
Resultant amplitude
1.732
Phase difference
60°
t = 0 s

Two waves share the same frequency and wave number. Playback advances t in sin(2x − t) + a sin(2x − t + phase). Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forThe SQUID interference comes from a constrained relative phase; its conversion from flux and voltage requires the superconducting relations.

Pause & reason

Check your understanding

An ideal low-inductance SQUID has unequal sinusoidal junction critical currents I₁ and I₂. What is its minimum critical current over flux?

Your turn

Try it yourself

A low-inductance SQUID has junction critical currents 3 μA and 1 μA. Compute its maximum and minimum critical currents, identify their flux locations, and decide whether a nonzero minimum alone proves lost phase coherence.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Evaluate the general formula at cosine +1 and −1. Junction imbalance remains present even with fully coherent phases.

Compare with the solution

At Φ=NΦ₀ the maximum is 3+1=4 μA. At Φ=(N+1/2)Φ₀ the minimum is |3−1|=2 μA. A nonzero minimum alone does not prove decoherence: this fully coherent imbalanced-junction model already predicts it. Other effects such as finite loop inductance require separate checks before interpreting an experimental interference pattern.

Keep the idea

Take it with you

  • A junction current depends on a gauge-invariant phase difference.
  • The ac Josephson frequency contains charge magnitude 2e.
  • SQUID minima depend on junction balance and loop assumptions as well as coherence.

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