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The Casimir effect: a force from nothingAstrid Lambrecht

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Physics WorldSeptember 2002

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P H Y S I C S W O R L D S E P T E M B E R 2 0 0 2 p h y s i c s w e b . o r g 29

WHAT happens if you take two mirrorsand arrange them so that they are facingeach other in empty space? Your firstreaction might be “nothing at all”. Infact, both mirrors are mutually attractedto each other by the simple presence of the vacuum. This startling phenom-enon was first predicted in 1948 by the Dutch theoretical physicist HendrikCasimir while he was working at PhilipsResearch Laboratories in Eindhoven on– of all things – colloidal solutions (seebox on page 30). The phenomenon isnow dubbed the Casimir effect, whilethe force between the mirrors is knownas the Casimir force.

For many years the Casimir effect waslittle more than a theoretical curiosity.But interest in the phenomenon hasblossomed in recent years. Experimentalphysicists have realized that the Casimirforce affects the workings of microma-chined devices, while advances in instru-mentation have enabled the force to bemeasured with ever-greater accuracy.

The new enthusiasm has also beenfired by fundamental physics. Many the-orists have predicted the existence of“large” extra dimensions in 10- and 11-dimensional unified field theories of thefundamental forces. These dimensions,they say, could modify classical New-tonian gravitation at sub-millimetre dis-tances. Measuring the Casimir effectcould therefore help physicists to test thevalidity of such radical ideas.

Understanding the Casimir forceAlthough the Casimir force seems completely counterintu-itive, it is actually well understood. In the old days of classicalmechanics the idea of a vacuum was simple. The vacuumwas what remained if you emptied a container of all its parti-cles and lowered the temperature down to absolute zero. The

arrival of quantum mechanics, how-ever, completely changed our notion ofa vacuum. All fields – in particular elec-tromagnetic fields – have fluctuations.In other words at any given momenttheir actual value varies around a con-stant, mean value. Even a perfect va-cuum at absolute zero has fluctuatingfields known as “vacuum fluctuations”,the mean energy of which correspondsto half the energy of a photon.

However, vacuum fluctuations are notsome abstraction of a physicist’s mind.They have observable consequencesthat can be directly visualized in experi-ments on a microscopic scale. For ex-ample, an atom in an excited state willnot remain there infinitely long, but will return to its ground state by spon-taneously emitting a photon. This phe-nomenon is a consequence of vacuumfluctuations. Imagine trying to hold apencil upright on the end of your finger.It will stay there if your hand is perfectlystable and nothing perturbs the equilib-rium. But the slightest perturbation willmake the pencil fall into a more stableequilibrium position. Similarly, vacuumfluctuations cause an excited atom to fallinto its ground state.

The Casimir force is the most famousmechanical effect of vacuum fluctu-ations. Consider the gap between twoplane mirrors as a cavity (figure 1). Allelectromagnetic fields have a character-

istic “spectrum” containing many different frequencies. In afree vacuum all of the frequencies are of equal importance.But inside a cavity, where the field is reflected back and forthbetween the mirrors, the situation is different. The field is am-plified if integer multiples of half a wavelength can fit exactlyinside the cavity. This wavelength corresponds to a “cavity res-onance”. At other wavelengths, in contrast, the field is sup-pressed. Vacuum fluctuations are suppressed or enhanced

The attractive force between two surfaces in a vacuum – first predicted by Hendrik Casimirover 50 years ago – could affect everything from micromachines to unified theories of nature

The Casimir effect: a force from nothing

Astrid Lambrecht

1 The Casimir force

d

F

It was the Dutch theoretical physicist HendrikCasimir (1909–2000) who first realized thatwhen two mirrors face each other in a vacuum,fluctuations in the vacuum exert “radiationpressure” on them. On average the externalpressure (red arrows) is greater than the internalpressure (green arrows). Both mirrors aremutually attracted to each other by what istermed the Casimir force. The force F ~ A/d 4,where A is the area of the mirrors and d is thedistance between them.

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depending on whether their frequency corresponds to a cavityresonance or not.

An important physical quantity when discussing the Casi-mir force is the “field radiation pressure”. Every field – eventhe vacuum field – carries energy. As all electromagnetic fieldscan propagate in space they also exert pressure on surfaces,just as a flowing river pushes on a floodgate. This radiationpressure increases with the energy – and hence the frequency– of the electromagnetic field. At a cavity-resonance fre-quency the radiation pressure inside the cavity is strongerthan outside and the mirrors are therefore pushed apart. Outof resonance, in contrast, the radiation pressure inside thecavity is smaller than outside and the mirrors are drawntowards each other.

It turns out that, on balance, the attractive componentshave a slightly stronger impact than the repulsive ones. Fortwo perfect, plane, parallel mirrors the Casimir force is there-fore attractive and the mirrors are pulled together. The force,F, is proportional to the cross-sectional area, A, of the mirrorsand increases 16-fold every time the distance, d, between the

mirrors is halved: F ~ A/d 4. Apart from thesegeometrical quantities the force depends onlyon fundamental values – Planck’s constant andthe speed of light.

While the Casimir force is too small to be ob-served for mirrors that are several metres apart,it can be measured if the mirrors are withinmicrons of each other. For example, two mirrorswith an area of 1 cm2 separated by a distance of1 µm have an attractive Casimir force of about10–7 N – roughly the weight of a water dropletthat is half a millimetre in diameter. Althoughthis force might appear small, at distances belowa micrometre the Casimir force becomes thestrongest force between two neutral objects. In-deed at separations of 10 nm – about a hundredtimes the typical size of an atom – the Casimireffect produces the equivalent of 1 atmosphereof pressure.

Although we do not deal directly with suchsmall distances in our everyday lives, they areimportant in nanoscale structures and micro-electromechanical systems (MEMS). These are“intelligent” micron-sized devices in which me-chanical elements and moving parts, such astiny sensors and actuators, are carved into a sili-con substrate. Electronic components are thenwired on to the device to process informationthat it senses or to drive the movement of itsmechanical parts. MEMS have many possibleapplications in science and engineering, and arealready used as car air-bag pressure sensors.

As MEMS devices are fabricated on the mi-cron and submicron scale, the Casimir force cancause the tiny elements in a device to stick to-gether – as reported recently by Michael Roukesand co-workers at the California Institute ofTechnology (2001 Phys. Rev. B 63 033402). Butthe Casimir force can also be put to good use.Last year Federico Capasso and his group atLucent Technologies showed how the force canbe used to control the mechanical motion of a

MEMS device (2001 Science 291 1941). The researchers sus-pended a polysilicon plate from a torsional rod – a twistinghorizontal bar just a few microns in diameter (figure 2). Whenthey brought a metallized sphere close up to the plate, theattractive Casimir force between the two objects made theplate rotate. They also studied the dynamical behaviour of theMEMS device by making the plate oscillate. The Casimirforce reduced the rate of oscillation and led to nonlinear phe-nomena, such as hysteresis and bistability in the frequency re-sponse of the oscillator. According to the team, the system’sbehaviour agreed well with theoretical calculations.

Measuring the Casimir effectWhen the Casimir effect was first predicted in 1948 it wasvery difficult to measure using the equipment of the time.One of the first experiments was carried out in 1958 byMarcus Spaarnay at Philips in Eindhoven, who investigatedthe Casimir force between two flat, metallic mirrors madefrom either aluminium, chromium or steel. Spaarnay meas-ured the force using a spring balance, the extension of which

The fact that an attractive force exists between two conducting metal plates was firstpredicted in 1948 by Hendrik Casimir of Philips Research Laboratories in theNetherlands. At the time, however, Casimir was studying the properties of “colloidalsolutions”. These are viscous materials, such as paint and mayonnaise, that containmicron-sized particles in a liquid matrix. The properties of such solutions aredetermined by van der Waals forces – long-range, attractive forces that existbetween neutral atoms and molecules. One of Casimir’s colleagues, TheoOverbeek, realized that the theory that was used at the time to explain van der Waalsforces, which had been developed by Fritz London in 1932, did not properly explainthe experimental measurements on colloids. Overbeek therefore asked Casimir toinvestigate the problem. Working with Dirk Polder, Casimir discovered that theinteraction between two neutral molecules could be correctly described only if thefact that light travels at a finite speed was taken into account. Soon afterwards,Casimir noticed that this result could be interpreted in terms of vacuum fluctuations.He then asked himself what would happen if there were two mirrors – rather thantwo molecules – facing each other in a vacuum. It was this work that led to hisfamous prediction of an attractive force between reflecting plates.

Casimir and colloids

2 The Casimir force in microdevices

torsional rod

polysilicon plate

The Casimir force, which is most noticeable at submicron distances, can affectmicroelectromechanical systems, or MEMS. (a) This MEMS device consists of a polysilicon platesuspended by a torsional rod only a few micrometresin diameter. When a metallized sphere (purple)approaches the plate, the attractive Casimir forcebetween the two objects causes the plate to rotatearound the rod. (b) An electron micrograph of thedevice that shows the polysilicon plate. (c) A close-up of the rod.

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was determined by the capacitance ofthe two plates. To prevent the Casimirforce from being swamped by the elec-trostatic force, the mirrors had to bekept neutral by first touching themtogether before each measurement wasmade. Spaarnay also had to ensure thatthe plane mirrors were exactly parallelto each other, as the Casimir force isvery sensitive to changes in distance.Spaarnay managed to overcome thesedifficulties and concluded that hisresults “did not contradict Casimir’stheoretical prediction”.

Since those early days, however, so-phisticated equipment has made itmuch easier to study the Casimir effect.A new generation of measurementsbegan in 1997. Steve Lamoreaux, whowas then at the University of Washing-ton in Seattle, measured the Casimirforce between a 4 cm diameter sphericallens and an optical quartz plate about2.5 cm across, both of which werecoated with copper and gold. The lensand plate were connected to a torsionpendulum – a twisting horizontal barsuspended by a tungsten wire – placedin a cylindrical vessel under vacuum.When Lamoreaux brought the lens andplate together to within several micronsof each other, the Casimir force pulledthe two objects together and caused thependulum to twist. He found that hisexperimental measurements agreed with theory to an accu-racy of 5%.

Inspired by Lamoreaux’s breakthrough, many other re-searchers tried new Casimir measurements. Umar Mohi-deen and co-workers at the University of California atRiverside, for example, attached a polystyrene sphere 200µmin diameter to the tip of atomic force microscope (figure 3). Ina series of experiments they brought the sphere, which wascoated with either aluminium or gold, to within 0.1 µm of aflat disk, which was also coated with these metals. The result-ing attraction between the sphere and the disk was monitoredby the deviation of a laser beam. The researchers were ableto measure the Casimir force to within 1% of the expectedtheoretical value.

Thomas Ederth at the Royal Institute of Technology inStockholm, Sweden, has also used an atomic force micro-scope to study the Casimir effect. He measured the force be-tween two gold-coated cylinders that were arranged at 90° toeach other and that were as little as 20 nm apart. His resultsagreed to within 1% of theory (figure 4).

However, very few recent experiments have measured theCasimir force using the original configuration of two plane,parallel mirrors. The reason is that the mirrors have to bekept perfectly parallel during the experiment, which is dif-ficult. It is much easier to bring a sphere close up to a mirrorbecause the separation between the two objects is simply thedistance of closest approach. The only drawback of using asphere and a plane mirror is that the calculations of the Casi-

mir force are not as accurate as thosebetween two plane mirrors. In partic-ular it has to be assumed that the contri-butions to the force between the sphereand plate are completely independent ateach point. This is true only if the radiusof the sphere is much larger than thedistance between it and the plate.

The only recent experiment to rep-licate Casimir’s original set-up of twoplane, parallel mirrors was carried outby Gianni Carugno, Roberto Onofrioand co-workers at the University ofPadova in Italy. They measured theforce between a rigid chromium-coatedplate and the flat surface of a cantilevermade from the same material that wereseparated by distances ranging from0.5–3 µm (G Bressi et al. 2002 Phys. Rev.Lett. 88 041804). The researchers foundthat the measured Casimir force agreedto within 15% of the expected theor-etical value. This relatively poor fit re-flected the technical difficulties involvedin the experiment.

Improved calculationsThe problem with studying the Casimireffect is that real mirrors are not like the perfectly smooth plane mirrors thatHendrik Casimir originally considered.In particular, real mirrors do not reflectall frequencies perfectly. They reflectsome frequencies well – or even nearly

perfectly – while others are reflected badly. In addition, allmirrors become transparent at very high frequencies. Whencalculating the Casimir force the frequency-dependent reflec-tion coefficients of the mirrors have to be taken into account –a problem first tackled by Evgeny Lifshitz in the mid-1950s,and then by Julian Schwinger and many others.

It turns out that the measured Casimir force between realmetallic mirrors that are 0.1 µm apart is only half the the-oretical value predicted for perfect mirrors. If this discrep-ancy is not taken into account when comparing experimentaldata with theory, then an experimental measurement coulderroneously be interpreted as a new force. My colleagueSerge Reynaud and I have taken into account the real be-haviour of mirrors in our calculations using the physicalproperties of the metals themselves. We found that the simplesolid-state models of the mirror match the real behaviouronly above 0.5 µm.

Another problem with calculating the expected Casimirforce for a real system is the fact that experiments are nevercarried out at absolute zero – as originally envisaged in Casi-mir’s calculations – but at room temperature. This causesthermal – as well as vacuum – fluctuations to come into play.These thermal fluctuations can produce their own radiationpressure and create a bigger Casimir force than expected. Forexample, the Casimir force between two plane mirrors 7 µmapart is twice as large at room temperature than at absolutezero. Fortunately, thermal fluctuations at room temperatureare only important at distances above 1 µm, below which the

3 Casimir force tips the balance

laser

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This experiment measures the Casimir forcebetween a metallized plate and a metallizedsphere fixed to the tip of the cantilever of anatomic force microscope. When the sphere isbrought near to the plate, an attractive Casimirforce causes the cantilever to bend. This bendingis monitored by bouncing a laser off the top of thecantilever and using photodiodes to record thereflected light. The electron micrograph shows ametallized sphere attached to the triangularcantilever tip of an atomic force microscope.

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wavelength of the fluctuations is too bigto fit inside the cavity.

Although the temperature depend-ence of the Casimir force has not yetbeen studied in detail experimentally, itmust be included in calculations of theforce at separations above 1 µm. Manyresearchers have tackled this problemfor perfectly reflecting mirrors, inclu-ding Lifshitz and Schwinger back in the1950s. It has also been examined morerecently by Michael Bordag at LeipzigUniversity, Bo Sernelius at LinköpingUniversity in Sweden, Galina Klim-chitskaya and Vladimir Mostepanenkoat the University of Paraiba in Brazil,and by our group in Paris. Indeed thetemperature dependence of the Casi-mir force was for some time a matter ofhot debate in the community. The vari-ous contradictions, however, now seemto have been resolved, and this has givenan additional motivation to an experimental observation ofthe influence of temperature on the Casimir force.

A third and final problem in calculating the Casimir force isthat real mirrors are not perfectly smooth. Most mirrors aremade by coating a substrate with a thin metal film using thetechnique of “sputtering”. However, this produces films witha roughness of about 50nm. While such roughness is invisibleto the naked eye, it does affect measurements of the Casimirforce, which is very sensitive to small changes in distance.

Mohideen and his group in California have recently usedsurface deformations to show that two surfaces can also havea lateral Casimir force that acts in a parallel – rather than aperpendicular – direction to the surface of the mirrors. In theexperiments they prepared specially corrugated mirrors thesurfaces of which were sinusoidally curved. They then movedthe mirrors parallel to one another so that a peak of one mir-ror passed successively over the peaks and troughs of theother mirror. The researchers found that the lateral Casimirforce varied sinusoidally with the phase difference betweenthe two corrugations. The size of the force was about tentimes smaller than the ordinary Casimir force between twomirrors the same distance apart. The lateral force is also dueto vacuum fluctuations.

Mehran Kadar and co-workers at the Massachusetts In-stitute of Technology have calculated a theoretical value forthe force between two perfectly reflecting corrugated mirrors,while Mohideen and colleagues evaluated the lateral force formetallic mirrors and found good agreement with experiment.The lateral Casimir force may have yet another consequencefor micromachines.

New physics?The Casimir effect could also play a role in accurate forcemeasurements between the nanometre and micrometrescales. Newton’s inverse-square law of gravitation has beentested many times at macroscopic distances by observing themotion of planets. But no-one has so far managed to verifythe law at micron length scales with any great precision. Suchtests are important because many theoretical models thatattempt to unify the four fundamental forces of nature pre-

dict the existence of previously undis-covered forces that would act at suchscales. Any deviation between experi-ment and theory could hint at the ex-istence of new forces. But all is not losteven if both values agree: the measure-ments would then put new limits onexisting theories.

Jens Gundlach and colleagues atWashington, for example, have used atorsion pendulum to determine thegravitational force between two testmasses separated by distances from10mm down to 220µm. Their measure-ments confirmed that Newtonian gravi-tation operates in this regime but that theCasimir force dominates at shorter dis-tances. Meanwhile Joshua Long, JohnPrice and colleagues at the University of Colorado – together with EphraimFischbach and co-workers from PurdueUniversity – are trying to eliminate the

Casimir effect altogether from sub-millimetre tests of gravita-tion by carefully selecting the materials used in the experiment.

This article only gives a flavour of the many experimentaland theoretical studies of the Casimir effect. There are manyother exciting developments as well. Many groups are, forexample, looking at what would happen if the interactionbetween two mirrors is mediated not by an electromagneticfield – which is made up of massless bosons – but by fieldsmade of massive fermions, such as quarks or neutrinos.Other research teams, meanwhile, are studying the Casimireffect with different topologies, such as Möbius strips anddoughnut-shaped objects.

But despite the intensive efforts of researchers in the field,many unsolved problems about the Casimir effect remain. Inparticular the seemingly innocent question of the Casimirforce within a single hollow sphere is still a matter of livelydebate. People are not even sure if the force is attractive orrepulsive. Hendrik Casimir himself thought about this prob-lem as early as 1953 while looking for a stable model for theelectron. Half a century on, the mysteries of the Casimirforce are likely to keep us entertained for many years to come.

Further readingM Bordag, U Mohideen and V M Mostepanenko 2001 New developments in the

Casimir effect Phys. Rep. 353 1

H B Chan et al. 2001 Nonlinear micromechanical Casimir oscillator Phys. Rev.

Lett. 87 211801

F Chen and U Mohideen 2002 Demonstration of the lateral Casimir force

Phys. Rev. Lett. 88 101801

C Genet, A Lambrecht and S Reynaud 2000 Temperature dependence of the

Casimir force between metallic mirrors Phys. Rev. A 62 012110

S K Lamoreaux 1997 Demonstration of the Casimir force in the 0.6 to

6 micrometer range Phys. Rev. Lett. 78 5

K A Milton 2001 The Casimir Effect: Physical Manifestations of Zero-point

Energy (World Scientific, Singapore)

Astrid Lambrecht is in the Laboratoire Kastler Brossel, Université Pierre et Marie

Curie, Ecole Normale Supérieure, Centre National de Recherche Scientifique,

Campus Jussieu, Case 74, F-75252 Paris cedex 05, France, email lambrecht@

spectro.jussieu.fr

4 Cylindrical attraction

crossedcylinders

piezoelectrictube

LVDT

piezoelectricdeflectionsensor

This experiment measures the Casimir forcebetween two gold-coated cylinders positioned atright angles to one another. The upper cylindercan be lowered using the piezoelectric tube, whichchanges shape when a voltage is applied. Thelower cylinder is mounted on a piezoelectricdeflection sensor (known as a bimorph spring)that generates a charge when it is bent. When thetwo cylinders are close together, the Casimir forcecauses the lower cylinder to be attracted to theupper one, thereby deflecting the spring in theprocess. The linearly variable displacementtransducer (LVDT) monitors the nonlinearexpansion of the piezotube.