The force that forgets Planck's constant

How many ways there are to vibrate found the Casimir pressure between two mirrors by counting the modes of the field that fit between them. At zero temperature each mode carries its zero-point energy, half of $\hbar\omega$; fewer modes fit in the gap than outside it, and the difference in energy pushes the mirrors together with a pressure of $\pi^2\hbar c/240\,a^4$. The attraction that weakens when light is too slow arrived at the same family of forces from two atoms, and found that past a distance set by how fast the source fluctuates, the field’s travel time cuts the slow fluctuations off from the fast ones and the force loses a power of the distance. It ended on a dispute: what happens when the mirrors are not at absolute zero.

The question looks like a matter of small corrections, since room temperature is a modest energy next to the photons that matter at a micrometre. It is not. At a few micrometres the thermal part of the force overtakes the vacuum part. Beyond that it dominates, and it has a remarkable property: Planck’s constant is not in it. The force between two warm mirrors far apart is a classical force, the same one that a nineteenth-century physicist with Maxwell’s equations and Boltzmann’s statistics could have computed. And half of it comes from one particular fluctuation, which a real metal may or may not reflect.

Counting modes at a temperature #

The zero-temperature argument puts half a quantum in every mode. At a temperature $T$ each mode also carries its thermal occupation, $\hbar\omega/(e^{\hbar\omega/kT} - 1)$, which is large for modes below $kT/\hbar$ and negligible above. The thermal photons push on the mirrors from both sides, and between the plates their spectrum is cut into the same discrete set of modes as the zero-point field. So the warm force is the cold force plus a radiation pressure difference, and the question is how large the difference is.

Lifshitz worked it out in 1956, and the form of his answer is what makes the physics visible. The field’s fluctuations at temperature $T$ are not summed over all real frequencies but over a comb of imaginary ones, the Matsubara frequencies,

$$\xi_n = \frac{2\pi n kT}{\hbar}, \qquad n = 0, 1, 2, \ldots$$

each contributing a term that dies across the gap as $e^{-2\xi_n a/c}$. That exponential is the retardation of the previous essay in a new costume: a fluctuation at $\xi_n$ cannot correlate the two plates if the light carrying it takes longer than $1/\xi_n$ to cross and return. When the plates are close, the comb’s teeth are so finely spaced compared with the cut-off that the sum is effectively an integral, and the integral is the zero-temperature result. When they are far apart, every term but the static one is cut off, and the force is whatever the static term says.

The spacing of the comb sets the scale. The first nonzero frequency corresponds to a wavelength $2\pi c/\xi_1 = \hbar c/kT$, the thermal length, which at room temperature is 7.63 micrometres. Plates much closer than that see a continuum of fluctuations; plates much farther apart see only $n = 0$.

The figure computes the sum for two perfectly reflecting plates at 300 kelvin and shows how the pressure is shared among the terms. At half a micrometre the shares fall off slowly — fifteen per cent from $n = 0$, twenty-seven from $n = 1$, and a long tail — which is the signature of a sum behaving like an integral. At two micrometres the static term carries fifty-nine per cent and the next one most of the rest. At eight micrometres the static term carries all but a few parts in a million. There the force is entirely the $n = 0$ term.

The term with no quantum in it #

The $n = 0$ term is the static limit: fluctuations of the field at zero frequency. In the Matsubara sum it carries a weight of a half, and its size is fixed without any reference to Planck’s constant. For two ideal mirrors it gives

$$P_{\text{cl}} = \frac{\zeta(3)\,kT}{4\pi a^3},$$

with $\zeta(3) = 1.202$, a pure number. There is $k$ and $T$ and the separation, and nothing else. The ħ that sets the Matsubara spacing cancels out of the static term exactly, because a fluctuation at zero frequency has no quantum: a static field mode in thermal equilibrium holds $kT$ of energy whatever its frequency, which is equipartition applied to the electromagnetic field. The same counting gives half a kT in a piece of wire, the Johnson noise of a resistor, which contains no ħ either until the frequency rises to $kT/\hbar$.

So the far-field Casimir force at a temperature is a classical statistical force. The mirrors constrain which static field configurations can exist between them, fewer configurations fit in the gap than outside, and the thermal free energy of the field is lower when the mirrors are close. The pressure is the derivative of that free energy. Its power law, the inverse cube, is one power shallower than the vacuum force’s inverse fourth, because the vacuum force carries an extra factor of $\hbar c/a$ — the energy of a photon whose wavelength fits the gap — where the classical force carries $kT$.

The figure draws the whole crossover. Below a micrometre the full pressure lies on the zero-temperature line, within two parts in a thousand at 1 μm, where it is 1.3 millipascals. Above ten micrometres it lies on the classical line, and at 10 μm it is 3.05 times what the vacuum alone would give. Between, the curve bends from the inverse fourth power to the inverse third. The two laws are equal at

$$a^* = \frac{\pi^3}{60\,\zeta(3)}\,\frac{\hbar c}{kT} = 0.43\,\frac{\hbar c}{kT},$$

which is 3.28 micrometres at room temperature. Beyond $a^*$ the plates attract more strongly because they are warm than because they are in a quantum vacuum.

The same crossing at every temperature #

Nothing in the argument is special to room temperature. The crossover distance is a fixed multiple of the thermal length, and the thermal length scales as one over the temperature.

In liquid helium, at 4.2 kelvin, the crossover is at 234 micrometres, a quarter of a millimetre: anything measured below that is the quantum vacuum force, almost untouched by temperature. In liquid nitrogen it is 13 micrometres. Near a hot filament at a thousand kelvin it is under a micrometre, so the force between two glowing wires a few micrometres apart is classical. The line in the figure divides the plane of temperature and separation into a quantum half, where the force is $\hbar c/a^4$ with a small thermal correction, and a classical half, where it is $kT/a^3$ with a small quantum one.

This is the same division that runs through the curve that would not come down, the blackbody problem. There the Rayleigh–Jeans law, classical and correct at long wavelengths, failed at short ones, and Planck’s constant appeared exactly where $\hbar\omega$ overtook $kT$. Here it is the other way round: the zero-point force, quantum and correct at short distances, fails at long ones, and Planck’s constant disappears exactly where $kT$ overtakes $\hbar c/a$. The crossover is a statement that the gap between the plates is acting as a filter on wavelengths, and the filter’s cut-off is the separation.

How much warmth adds #

For ideal mirrors the thermal force only ever adds. That is not what experiment cares about, which is how much it adds at the separations where the force can be measured, and the answer is very little where the measurements are precise.

The blue curves divide the warm pressure by the cold one. For ideal plates at room temperature the ratio is within a per cent of one up to about a micrometre and then rises steeply, reaching three at ten micrometres. Cooling to 77 kelvin pushes the rise out by a factor of four. The precision measurements of the Casimir force, which reach the per-cent level, are made between a few hundred nanometres and about a micrometre, because that is where the force is large enough to measure well. The ideal-mirror model says the thermal correction there is a fraction of a per cent, too small to argue about.

The green curves say something else, and they are the subject of the argument.

One fluctuation, and the metal’s choice #

A real metal is not an ideal mirror. At high frequencies its electrons cannot follow the field, which reduces the force below a few hundred nanometres; that correction is well understood and is not in dispute. The dispute is at the other end of the comb, at $\xi = 0$, and it concerns one of the two polarisations of the static field.

The $n = 0$ term has two parts. One is a static electric fluctuation, which any conductor reflects, since a conductor’s charges move to cancel any electric field inside it. The other — called transverse electric, because its electric field lies along the plates — becomes, at zero frequency, a static magnetic field. Whether a metal reflects a static magnetic field depends on what kind of metal it is.

An ordinary metal, with finite conductivity, reflects a magnetic field only by the currents the field induces when it changes, as how far a field gets into metal found for the skin depth. A field that does not change induces no lasting current, and it passes through the metal as though it were not there. In that case the static magnetic fluctuation is the same between the plates as outside, and its contribution to the force is zero. Removing it takes away exactly half of the classical term, $\zeta(3)kT/8\pi a^3$. This is the Drude description, named after the model of a metal’s electrons as a gas with a collision time, which is the description that fits every optical measurement on metals ever made.

A superconductor behaves the opposite way. It pushes a static magnetic field out, within its penetration depth, and so it reflects the static magnetic fluctuation just as an ideal mirror does. The “plasma” description of a metal — electrons with no collisions at all — does the same, since collisionless electrons screen a static magnetic field the way a superconductor’s pairs do. In that description the static term survives in full.

The two descriptions differ in nothing but how the electrons behave at frequencies below their collision rate, about $10^{13}$ per second in gold, and in this one term. At large separations the difference is a factor of two in the force. At a micrometre, in the ideal-mirror model, it is fifteen per cent, and it pulls the green curve in the earlier figure below one: a metal that lets the static magnetic fluctuation through attracts less when warm, by a third at three micrometres, before the remaining classical term takes over further out.

What the measurements have said #

The force has been measured with a torsion pendulum, with atomic-force-microscope cantilevers, and with micromechanical oscillators carrying a sphere close to a plate, and the measurements disagree about which description is right in a way that has not resolved.

Precise measurements by Decca and colleagues between about 160 and 750 nanometres, at the per-cent level and below, have repeatedly agreed with the plasma description and disagreed with the Drude description — the one that fits the optical data. A later measurement by the same group, designed to cancel the electrostatic forces from patches of differing surface potential, which are the largest systematic error in every such experiment, gave the same verdict. In 2011 Sushkov, Lamoreaux and colleagues measured at 0.7 to 7 micrometres, where the thermal term is large, and reported a thermal force of the size the Drude description predicts. But to do it they subtracted an electrostatic background many times larger than the thermal force, with a shape fitted to the data, and whether that subtraction leaves the answer they reported or some other one has been argued ever since.

The disagreement has a deeper side. It has been argued that the Drude description, for a metal whose resistance falls to zero at absolute zero as a perfect crystal’s would, gives the Casimir free energy an entropy that does not vanish at zero temperature, in contradiction with the third law. Others have answered that any real crystal has impurities, that their residual resistance restores the third law, and that the contradiction is a property of an idealisation rather than of the Drude description. Neither side has persuaded the other.

Why the static field is the hard part #

The difficulty is that the term in question is the one no optical experiment can see directly. A metal’s response is measured with light, at real frequencies, and the response at zero frequency is reached by extrapolation. The Drude model extrapolates through the collision rate to a finite conductivity at zero frequency; the plasma model drops the collisions and extrapolates to an infinite one. At every frequency anyone can shine light at, the two are nearly indistinguishable. At $\xi = 0$ they differ completely, and the static magnetic term sits exactly there.

So the Casimir force at a few micrometres is a measurement of something optics cannot reach: whether a static magnetic fluctuation enters a metal at room temperature. In ordinary physics the answer is not in doubt — a bar magnet’s field passes straight through a copper sheet — but the thermal fluctuations of the field are not a bar magnet, and the question is whether the Lifshitz formula, with the measured conductivity put into it, is the right description of how a metal’s electrons respond to fluctuations at all. It may be that the formula needs a correction for how electrons near the surface behave that neither model contains, and several have been proposed. None has yet been confirmed.

A proposed way out uses superconductors. Cooling a metal film through its superconducting transition switches its response to the static magnetic field from transmitting to excluding, while changing its optical response hardly at all, so the change in the Casimir energy across the transition would isolate exactly the term in dispute. The change is tiny and the measurement is extremely hard, and it has been proposed rather than completed.

A classical force hidden inside a quantum one #

The surprising thing is where the classical force has been hiding. The Casimir effect is introduced, rightly, as the cleanest mechanical evidence of the zero-point energy of the field, and at the separations where it was first measured convincingly — a fraction of a micrometre — it is exactly that. Yet the same calculation, carried a few micrometres further at room temperature, delivers a force that Maxwell and Boltzmann could have computed, larger than the quantum one, and independent of Planck’s constant.

The two forces are the two ends of one sum. The zero-point force is what the sum gives when the teeth of the Matsubara comb are dense; the classical force is its static term alone. Which one a measurement sees depends on a single ratio, the separation against $\hbar c/kT$. The same ratio decides when a blackbody’s spectrum is classical, when a resistor’s noise is classical and when the heat crossing a gap too narrow for light is set by evanescent waves rather than by photons that travel, and the near-field heat flow and the thermal Casimir force are, in Lifshitz’s formulation, two outputs of one calculation.

What the pictures cannot show #

The figures use plates that reflect perfectly at every nonzero Matsubara frequency, to isolate the one term in dispute. A real gold plate reflects less well above its plasma frequency, which lowers the force below a few hundred nanometres by ten per cent and more, and has its own small temperature dependence; the figures leave both out. They are drawn for parallel plates, while almost every experiment uses a sphere near a plate, related to the plates by an approximation that is good to a fraction of a per cent at the distances used. And they show no electrostatic patch forces, roughness or residual gas, which in practice are larger than the thermal effect wherever it is large, and which are why its measurement is still argued over.

Still open: what a metal does with a static fluctuation #

The question has narrowed to one number: the reflection coefficient of a metal at zero frequency for the transverse electric polarisation, which the Drude description sets to zero and the plasma description to one. Optical data cannot decide it, and the force measurements that can have disagreed. What would settle it is a measurement of the thermal force at several micrometres with the electrostatic background controlled independently rather than fitted, or a measurement across a superconducting transition, or a theory of the electrons at a metal’s surface that predicts which description applies before it is measured. None exists yet.

The habit worth carrying away is to ask which term of a sum survives in a limit, and what it contains. A sum over the frequencies of a field’s fluctuations becomes, at large separation, its zero-frequency term alone, and that term carries kT per mode and no Planck constant — so the force between warm mirrors far apart is classical, with a crossover at 0.43 ħc/kT, 3.3 μm at room temperature. Half of that classical force is a static magnetic fluctuation, and whether a metal mirror reflects it is the one thing about the Casimir force that is still not known.