C-X4 · Repeating a current ramp

Josephson Switching Statistics

Why does a biased Josephson junction switch before its critical current? Below the critical current, the phase is trapped behind a finite barrier. Heat can lift it over the barrier, and at low temperature quantum tunnelling can carry it through. Repeated current ramps therefore switch at a distribution of currents below the threshold.

Repeat a rising current ramp and record when a junction develops voltage. Compare thermal activation with a ground-state tunnelling approximation using the same potential barrier.

Mean switching current $\text{Loading}$

Standard deviation $\text{Loading}$

Recorded ramps $\text{Loading}$

Escape model $\text{Loading}$

A small change, a different state

Change the physical setting

Escape approximation

Bath temperature

0.06

T/\mathrm{K}

Junction capacitance

0.3

C/\mathrm{pF}

Current ramp rate

2

\dot I/(\mathrm{\mu A}\,\mathrm{s}^{-1})

The default experiment is recorded automatically.

Model and scale

A synthetic junction has a fixed critical current of one microampere and quality factor twenty. Thermal escape uses the attempt-frequency Arrhenius approximation; quantum escape uses a weak-damping cubic-barrier WKB rate. The two alternatives are compared separately; no crossover interpolation is claimed. Rates are integrated along a linear current ramp, then sampled into switching currents. The circuit animation replays those samples with exaggerated time. Heating, retrapping, environmental noise, microwave excitation and excited-state populations are omitted. The quantum branch is temperature independent and applies only well below crossover with a sufficiently deep metastable well.

Bath temperature controls the thermal comparison. The ground-state quantum approximation has no temperature parameter in its rate.

From the experiment

Observe, then compare

Cyan: sampled switching histogram. Amber: thermal alternative at the same temperature. Both use synthetic junction parameters.

Recorded switching currents

The phase potential at mean switching

Start with the essentials

Focus question: Why does a biased Josephson junction switch before its critical current?
One-sentence intuition: A finite escape rate accumulated during the ramp produces a distribution below the critical current.

Core mathematical model

Tilted Josephson potential

U(\varphi)=-E_J(\cos\varphi+i\varphi),\quad i=I/I_c

Phase is the collective circuit coordinate; the potential diagram belongs in the data row, not on the physical chip.

Exact sinusoidal barrier

\Delta U=2E_J\left[\sqrt{1-i^2}-i\arccos i\right]

The barrier shrinks to zero as the current reaches the critical value.

Survival during a ramp

S(i)=\exp\!\left[-\int_0^i\frac{\Gamma(u)}{\dot i}\,du\right]

The histogram is sampled from the probability lost from this survival function. The displayed standard deviation is calculated from the stored samples.

Common difficulties

Reading the model correctly

Typical misconceptionThe animated current carriers are the object tunnelling through the potential diagram.

Better mental modelThe escaping coordinate is the collective phase difference. The apparatus animation replays recorded voltage switching; it does not resolve microscopic pair paths.

Run the experiment

01
Compare escape mechanisms
Select Cold tunnelling, then Cold thermal comparison.
What to observe: Thermal activation becomes exponentially rare in the cold; the ground-state tunnelling approximation retains a finite rate.
02
Warm the bath
Select Warm thermal escape and change Bath temperature.
What to observe: Thermal switching moves to a lower current as escape becomes easier.
03
Change the phase inertia
Compare Cold tunnelling and Larger capacitance.
What to observe: Larger capacitance lowers the plasma frequency and suppresses the WKB rate at the same bias.