---
title: "Josephson Switching Statistics Interactive Lab | Sandbox Physics"
description: "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."
canonical_url: "https://sandboxphysics.com/labs/josephson-switching-statistics/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/josephson-switching-statistics/"
last_modified: "2026-10-24"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

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

Nobel · Condensed matter

## What did the original experiments establish?

Follow the apparatus, observations and interpretation in the historical evidence. Each article explains where this teaching model fits.

[Read the historical evidence](https://sandboxphysics.com/stories/condensed-matter/macroscopic-quantum-circuit/) [All six reading paths](https://sandboxphysics.com/stories/condensed-matter/)

Physics tutorial

## From a metastable phase to switching currents

**Background** A current bias tilts the Josephson phase potential. The zero-voltage state is a metastable minimum separated from running phase by a finite barrier.

**Why it matters** One switching event cannot distinguish escape mechanisms. Repeated ramps reveal how the distribution changes with temperature and capacitance.

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 misconception** The animated current carriers are the object tunnelling through the potential diagram.

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

### Run the experiment

1. 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.

2. 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.

3. 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.

Physical conclusion

Compare thermal and quantum alternatives with a calibrated barrier; the low-temperature WKB branch is not a crossover theory.

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)