---
title: "Casimir Force Oscillator: Hysteresis From the Vacuum Interactive Lab | Sandbox Physics"
description: "How can the Casimir force give a tiny oscillator a memory? By pulling harder the closer the plate swings."
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page_type: "lab"
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last_modified: "2026-10-30"
---

FR32 · Casimir–Lifshitz force

# Casimir Force Oscillator: Hysteresis From the Vacuum

**How can the Casimir force give a tiny oscillator a memory?** By pulling harder the closer the plate swings. A gold sphere near a rocking plate softens its spring and lowers its resonance; large swings feel a softer spring than small ones. The resonance curve leans until one drive frequency allows two steady swings. In a model of the 2001 Bell Labs device, the swing at 97 nm is about 7 or 35 nm, depending on whether the sphere approached or retracted.

Bring a gold sphere toward a rocking plate and watch the Casimir force pull its resonance down until two different swings are stable.

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

Model assumptions

The device follows Chan et al. (2001): a torsional plate with resonance 2753.47 Hz and quality factor 7150, and a gold-coated sphere of radius 100 µm at lever arm 131 µm. The sphere–plate force is the Lifshitz plate–plate energy times the proximity-force factor, with gold as a Drude metal, a lossless plasma metal or a perfect mirror. The oscillator obeys Eqs. 1 and 5 of Chan et al., which keep the force derivatives up to third order. Surface roughness, patch potentials and the residual electrostatic force are ignored, so the numbers are not a fit to the measured data.

Resonance curve at this gap

Swing while the gap is scanned

Resonance shift for three mirror models

More numbers and how they are computed

- **Force gradient**: $0$
- **Plate–plate force over perfect mirrors**: $0$
- **Plate–plate energy over perfect mirrors**: $0$
- **Nonlinear coefficient of Eq. 5**: $0$
- **Shift from Eq. 2 alone**: $0$
- **Steady swings at this drive**: $0$

The Lifshitz integrals run over imaginary frequency and wavevector with Gauss–Legendre quadrature. The second and third force derivatives are five-point differences of the plate pressure with a step of 1% of the gap. The hysteresis traces step the gap by 0.25 nm and keep the root nearest the previous swing. Model version fr32-casimir-oscillator-v1.

Physics tutorial

## A vacuum force that gives an oscillator memory

**Background** Two uncharged metal surfaces a fraction of a micron apart attract each other. The same field fluctuations that carry heat across a small gap also carry momentum, and the net result is the Casimir–Lifshitz force. Between a gold sphere and plate it is more than a hundred piconewtons at 100 nm, far too small to feel but large enough to change how a microscopic spring rings.

**Why it matters** Chan and colleagues at Bell Labs built a gold-coated polysilicon plate that rocks on two torsion rods and brought a gold-coated sphere close to one side. The Casimir force shifted its resonance, bent the resonance curve and finally made it bistable: the swing depended on whether the sphere had approached or retracted.

Start with the essentials

- **Focus question**: How can the Casimir force give a tiny oscillator a memory?
- **One-sentence intuition**: The force gradient softens the torsion spring and pulls the resonance down. Because the force grows steeply as the gap shrinks, a large swing feels a softer spring than a small one, so the resonance curve leans toward low frequency. Once it leans far enough, one drive frequency meets the curve three times and two of those swings are stable.

### Core mathematical model

#### Force between a sphere and a plate

$$
F(z)=2\pi R\,\eta_E(z)\,E_{\mathrm{pp}}(z),\qquad E_{\mathrm{pp}}(z)=-\frac{\pi^{2}\hbar c}{720\,z^{3}}
$$

The proximity-force approximation treats the sphere of radius $R$ as a stack of plate pairs, valid for $z\ll R$. $E_{\mathrm{pp}}$ is the energy per area of two perfect mirrors and $\eta_E\le1$ is the Lifshitz reduction for real gold, as tabulated by Lambrecht and Reynaud (2000).

#### Resonance shift from the force gradient

$$
\omega_1=\sqrt{\omega_0^{2}-\frac{b^{2}F'(z)}{I}}\approx\omega_0\left[1-\frac{b^{2}F'(z)}{2I\omega_0^{2}}\right]
$$

With $b=131\,\mu\mathrm{m}$ and $I=7.1\times10^{-17}\,\mathrm{kg\,m^{2}}$, the shift is about −1.2 Hz at 141 nm and −4.3 Hz at 98 nm for Drude gold. The approximate form is Eq. 2 of Chan et al. (2001). When $b^{2}F'$ reaches $I\omega_0^{2}$, the spring can no longer hold the plate and it snaps to the sphere.

#### Amplitude of the bent resonance

$$
A^{2}\left[(\omega-\omega_1-\kappa A^{2})^{2}+\lambda^{2}\right]=\frac{\tau^{2}}{4I^{2}\omega_1^{2}},\qquad \kappa=\frac{3\beta}{8\omega_1}-\frac{5\alpha^{2}}{12\omega_1^{3}}
$$

This is Eq. 5 of Chan et al. Here $\alpha=b^{3}F''/2I$ and $\beta=-b^{4}F'''/6I$ come from the second and third derivatives of the force, and $\lambda$ is the damping rate. A cubic in $A^{2}$ has up to three positive roots, which is where the memory comes from.

### Common difficulties

#### The Casimir force needs perfect mirrors

**Typical misconception** Real metals do not feel a Casimir force; it is an idealisation.

**Better mental model** Real gold reflects imperfectly, so the force is smaller, about half the perfect-mirror value near 100 nm in this model, but it is the same physics and is what the experiment measures.

#### Hysteresis means friction

**Typical misconception** The swing depends on history because the sphere drags on the plate.

**Better mental model** The model has no extra friction. The memory comes from the conservative force being nonlinear: two steady swings both balance the same drive, and the oscillator stays on whichever it reached first.

### Run the experiment

1. 01

   #### Start with the sphere far away

   Choose Sphere far away and look at the resonance chart.

   **What to observe:** At 3.3 µm the curve is a symmetric peak at 2753.47 Hz with a 39 nm swing, like any lightly damped oscillator.

2. 02

   #### Predict, then bring it close

   Guess which way the peak moves, then choose Resonance pulled down.

   **What to observe:** At 141 nm the peak sits about 1.2 Hz lower and leans toward low frequency.

3. 03

   #### Explain the memory

   Run Hysteresis scan at a fixed 2748 Hz and watch both traces.

   **What to observe:** Approaching, the swing stays small until about 95 nm, then jumps up. Retracting, it stays large until about 98 nm, then drops. In between, the two stable swings differ by up to about six times.

Physical conclusion

The Casimir force is strong enough at 100 nm to shift and bend the resonance of a microscopic oscillator, and its steep distance dependence makes two swings stable at once.

## Related experiments and articles

Fluctuational electrodynamics · Reading path

### From thermal emission to heat and force across a nanometre gap

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How the Casimir force gives a micromachined plate two swings**](https://sandboxphysics.com/stories/fluctuational-electrodynamics/casimir-force-and-a-bistable-oscillator/)
2. [Lab **Dynamical Casimir Photon Pairs** Let the boundary move: from attraction to photon pairs](https://sandboxphysics.com/labs/dynamical-casimir-photon-pairs/)

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)