Historical research guide · 0703 · 2009
The Casimir force between real materials: Experiment and theory
G. L. Klimchitskaya · U. Mohideen · V. M. Mostepanenko
In one minute
Galina Klimchitskaya, Umar Mohideen and Vladimir Mostepanenko wrote the audit of the decade in which the Casimir force stopped being a demonstration and became metrology. The force exists because the electromagnetic field’s fluctuations at two separate points in empty space are correlated even though each one averages to zero; put real surfaces in that field and the correlation shows up as a pull you can weigh. Casimir’s textbook formula assumed perfect mirrors at absolute zero. This review is about what happens when the mirrors are gold, or doped silicon, or a rubidium atom near glass — rough, finite, conducting and warm. The authors work through every primary measurement of the previous ten years: a torsion pendulum good to five or ten percent, atomic force microscopes good to one or two, and a micromachined torsional oscillator that reached two parts in a thousand at a gap of 160 nanometres. And they set out the field’s live dispute, which is how the conduction electrons of a real metal should enter the temperature term.
Connection
Chapter 2 rests on the vacuum being a real medium that laboratories touch every day, and this is the review that shows how far that has been pushed: several techniques, several continents, agreement at the level of parts per thousand, plus a lateral force that pushes sideways and an optical pulse that changes the force in real time. Chapter 1 is the evidence ladder, and this paper is what a mature rung looks like — a published methodology for comparing force-distance data with theory, error budgets stated at named confidence levels, and a disagreement the authors name rather than smooth over. Kimball Milton’s companion review is at /library/stm-719d2e0a0b, the Mohideen and Roy atomic-force measurement at /library/stm-911b036fc2, the Lamoreaux torsion-pendulum measurement at /library/stm-208d347532, and the thermal result that comes out on the other side of the dispute at /library/stm-4fb469cffa.
From the source
The Casimir force and the van der Waals force are one phenomenon in two limits, and their origin is a correlation. The quantized electric field at a point in vacuum has zero average value, but the product of the field at two different points does not average to zero, so two neutral bodies acquire correlated fluctuating dipole moments and attract. Below a few nanometres this is the non-relativistic van der Waals force; at separations of order the absorption wavelength and beyond, retardation matters and it is the Casimir and Casimir-Polder force.Section I.A, Equation 1
For two ideal metal walls at zero temperature the Casimir energy per unit area is minus pi squared times h-bar c divided by 720 times the cube of the separation, and the pressure is minus pi squared times h-bar c divided by 240 times the fourth power of the separation. Lifshitz’s 1956 theory generalises this to real dissipative media in thermal equilibrium by way of the fluctuation-dissipation theorem, with the material entering only through its frequency-dependent dielectric permittivity, and it reproduces both the London and the Casimir limits.Section I.A, Equation 2; Section II.A, Equation 3
The measurement is now precision metrology. Lamoreaux’s 1997 torsion balance reached five to ten percent at separations near one micrometre; the atomic force microscope series beginning with Mohideen and Roy in 1998 reached one to two percent at the shortest separations, and needed the skin-depth and roughness corrections to agree; and Decca and colleagues, using a micromechanical torsional oscillator to extract the pressure between two parallel plates, reported a total experimental error of 0.2 percent at a separation of 160 nanometres — the first Casimir measurement in which the systematic error dominates the random one.Section I.C; Sections IV.A and IV.B
The vacuum force can be switched with a light beam. Shining 514-nanometre laser pulses on a four-micrometre silicon membrane raises its charge-carrier density, and an atomic force microscope measures the resulting change in the Casimir force on a gold-coated sphere directly, as the difference between the bright and dark phases of the pulse train, with the photon-pressure contribution accounted for at a few percent.Section V.B, Figure 17 and Equation 92
The vacuum also pushes sideways. Between a gold sphere and a gold plate carrying matched sinusoidal corrugations of 1.2 micrometre period, the lateral Casimir force varies as the sine of the phase shift between the two corrugation patterns, with a measured amplitude of 0.32 piconewtons at a separation of 221 nanometres and a total experimental error of 0.077 piconewtons at 95 percent confidence. Together with the normal force, this means a device element can be translated in any direction entirely by zero-point oscillations.Section VII.B, Equation 108
How a real metal’s conduction electrons enter the temperature term is the field’s open question, and it is an experimental one. Feeding the drift current of conduction electrons into Lifshitz theory by way of the Drude model, or feeding the small dc conductivity of a real dielectric into it, predicts an enormously large thermal correction below one micrometre — which the authors show violates the Nernst heat theorem, and which the micromechanical-oscillator pressure data exclude at 99.9 percent confidence, as do the silicon and the Casimir-Polder measurements. The authors state there is no consensus in the literature on this inference, and expect a future theory of dispersion forces to work from more general scattering characteristics that can carry spatial nonlocality rather than from a dielectric permittivity alone.Section II.D; Section VIII, Conclusions and Outlook
The way in
https://doi.org/10.1103/RevModPhys.81.1827How to cite it
G. L. Klimchitskaya, U. Mohideen, V. M. Mostepanenko (2009) The Casimir force between real materials: Experiment and theory doi:10.1103/RevModPhys.81.1827