FR32 · Casimir–Lifshitz force
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.
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
BackgroundTwo 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 mattersChan 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
The proximity-force approximation treats the sphere of radius as a stack of plate pairs, valid for . is the energy per area of two perfect mirrors and is the Lifshitz reduction for real gold, as tabulated by Lambrecht and Reynaud (2000).
With and , 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 reaches , the spring can no longer hold the plate and it snaps to the sphere.
This is Eq. 5 of Chan et al. Here and come from the second and third derivatives of the force, and is the damping rate. A cubic in has up to three positive roots, which is where the memory comes from.
Typical misconceptionReal metals do not feel a Casimir force; it is an idealisation.
Better mental modelReal 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.
Typical misconceptionThe swing depends on history because the sphere drags on the plate.
Better mental modelThe 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.
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.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.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.Fluctuational electrodynamics · Reading path
Articles on this path that use this experiment, followed by the other experiments they suggest.