The Casimir effectThe Casimir effect Physics 250 Spring 2006 Physics 250 Spring 2006

Dr BudkerDr BudkerEric CorsiniEric Corsini

CasimirCasimirPatron Saint of Poland Patron Saint of Poland and Lithuania and Lithuania (March 4(March 4thth))

Hendrik Casimir (1909-2000)Hendrik Casimir (1909-2000)Dutch theoretical physicistDutch theoretical physicistPredicted the “force from nowhere” in Predicted the “force from nowhere” in 19481948

AbstractAbstractThe Casimir ForceThe Casimir Force

The Casimir Force was first predicted by The Casimir Force was first predicted by Dutch theoretical physicist Hendrik Dutch theoretical physicist Hendrik Casimir and was first effectively measured by Steve Lamoreaux in 1995.Casimir and was first effectively measured by Steve Lamoreaux in 1995.

The boundary conditions imposed on the electromagnetic fields by metallic The boundary conditions imposed on the electromagnetic fields by metallic surfaces lead to a spatial redistribution of the zero-point energy mode density surfaces lead to a spatial redistribution of the zero-point energy mode density with respect to free space, creating a spatial gradient of the zero-point energy with respect to free space, creating a spatial gradient of the zero-point energy density and hence a net force between the metals. That force is the most density and hence a net force between the metals. That force is the most significant force between neutral objects for distances <100nmsignificant force between neutral objects for distances <100nm

Because of that dependence on boundary conditions, the Casimir Force Because of that dependence on boundary conditions, the Casimir Force spatial dependence and sign can be controlled by tailoring the shape of the spatial dependence and sign can be controlled by tailoring the shape of the interacting surfaces.interacting surfaces.

In this presentation I briefly review the formalism pertaining to the zero point In this presentation I briefly review the formalism pertaining to the zero point energy and summarize the recent experiment By Bell and Lucent labs, energy and summarize the recent experiment By Bell and Lucent labs, investigating the effect of the Casimir Force on a dynamic system.investigating the effect of the Casimir Force on a dynamic system.

Origin of the Casimir forceOrigin of the Casimir forceThe short answerThe short answer

The vacuum cannot have absolute zero energyThe vacuum cannot have absolute zero energy

that would violatethat would violate

Heisenberg uncertainty principle.Heisenberg uncertainty principle.

The long answer The long answer “green” book approach“green” book approach

We show a 1-1 relationship: SHO ↔ We show a 1-1 relationship: SHO ↔ E&ME&M Field Field

Maxwell + Coulomb gauge Maxwell + Coulomb gauge ((.A=0).A=0) (no local current/charge) (no local current/charge)

General sol to wave equation General sol to wave equation

ThenThen

02

22

t

)ˆ)(ˆ)((1

),(

)ˆˆ(1

),(

*).(*).(

*).(*).(

rkirki

trkitrki

etCetCV

tr

eCeCV

tr

}|)](Im[||)](Re[{|2

1)(

8

1 222

222 tCtC

cdVBE Substitute

V

Consider the SHOConsider the SHO

Note:Note:

Then there is a 1-1 relationThen there is a 1-1 relation

If we set If we set ααoo to be such thatto be such that

Then, per mode Then, per mode ωω we have: we have:

)(222

221

,:Re222

PQqm

m

pSHO

Qm

qPmpscale

SHO

)sin()(),cos()( ttPttQthendt

dQP

dt

dqmp oo

)()](Im[

)()](Re[

tPtC

tQtC

)()(2

iPQc

tC

)(2

22& PQME

We can then apply the SHO mechanics We can then apply the SHO mechanics to the E&M fieldto the E&M field

Eigenstates |n>Eigenstates |n> Eigenvalues EEigenvalues Enn = ħ = ħωω(n+(n+11//22))

In particular EIn particular Eoo= ħ= ħωω/2 ≠ 0 for mode /2 ≠ 0 for mode ωω

HoweverHowever

1

2/

But we are only concerned in the But we are only concerned in the difference in energy densitydifference in energy density

Between two conducting parallel Between two conducting parallel plates only virtual photons whose plates only virtual photons whose wavelengths fit a whole number wavelengths fit a whole number of times between the plates of times between the plates contribute to the vacuum energy contribute to the vacuum energy there is a force drawing the there is a force drawing the plates together.plates together.

10nm)(d objects neutral obetween tw forceStrongest

1atmPacal10 Pressure 10nmd

Pascal10Pressureor N10 m1d,1cm A

d

A

480

c

5

3-7-2

4

F

F

NotesNotes

Bosons Bosons attractive Casimir force attractive Casimir force Fermions Fermions repulsive Casimir force repulsive Casimir force With supersymmetry there is a fermion for With supersymmetry there is a fermion for

each Boson each Boson no Casimir effect. no Casimir effect. Hence if supersymmetry exists it must be a Hence if supersymmetry exists it must be a

broken symmetrybroken symmetry

Casimir ForceCasimir ForceFrom theory to experimentFrom theory to experiment

Predicted by Dutch physicist Hendrick Casimir in 1948.Predicted by Dutch physicist Hendrick Casimir in 1948. First attempt to measure the Casimir Force: 1958 by M.J.SparnaayFirst attempt to measure the Casimir Force: 1958 by M.J.Sparnaay

- Used the attraction between a pair of parallel plates.- Used the attraction between a pair of parallel plates.- But irreducible systematic errors - But irreducible systematic errors measurements had a 100% measurements had a 100% uncertainty, (but it fit the expectations)uncertainty, (but it fit the expectations)

Sparnaay gave three guidelines;Sparnaay gave three guidelines;- The plates should be free of any dust or debris, with as little surface roughness as possible- The plates should be free of any dust or debris, with as little surface roughness as possible- Static electrical charges should be removed (electrostatic force can easily swamp the weak Casimir - Static electrical charges should be removed (electrostatic force can easily swamp the weak Casimir attraction). attraction). - The plates should not have different surface potentials- The plates should not have different surface potentials- Ref: "Measurements of Attractive Forces Between Flat Plates“- Ref: "Measurements of Attractive Forces Between Flat Plates“(Sparnaay, 1958) Physica, (Sparnaay, 1958) Physica, 2424 751-764 751-764

2nd attempt and first successful results: 1996 by Steven Lamoreaux: - In 2nd attempt and first successful results: 1996 by Steven Lamoreaux: - In agreement with theory to within uncertainty of 5%.agreement with theory to within uncertainty of 5%.

Several other successful experiments since.Several other successful experiments since.

Steve Lamoreaux Steve Lamoreaux (University of (University of Washington – Washington – Seattle)Seattle)

Measured the Measured the Casimir force Casimir force between a 4 cm between a 4 cm diameter spherical diameter spherical lens and an optical lens and an optical quartz plate about quartz plate about 2.5 cm across, both 2.5 cm across, both coated with copper coated with copper and gold. The lens and gold. The lens and plate were and plate were connected to a connected to a torsion pendulum. torsion pendulum.

Steven Lamoreaux’ experimental set upSteven Lamoreaux’ experimental set up

There are only a few dozen published experimental measurements of the Casimir force

But there are more than 1000 theoretical papers And citations of Casimir’s 1948 paper are growing

exponentially.

Effects of edgesEffects of edgesshape of decay function is strongly dependent on size and separation of surfacesshape of decay function is strongly dependent on size and separation of surfaces

ref:http://images.google.com/imgres?imgurl=http://www.sr.bham.ac.uk/yr4pasr/project/casimir/currentthumb.jpg&imgrefurl=http://www.sr.bham.ac.uk/yr4pasr/project/casimir/ref:http://images.google.com/imgres?imgurl=http://www.sr.bham.ac.uk/yr4pasr/project/casimir/currentthumb.jpg&imgrefurl=http://www.sr.bham.ac.uk/yr4pasr/project/casimir/&h=275&w=275&sz=41&tbnid=Buy2QDUNZEvi6M:&tbnh=109&tbnw=109&hl=en&start=20&prev=/images%3Fq%3Dcasimir%2Beffect%26svnum%3D10%26hl%3Den%26lr%3D%26sa%3DG&h=275&w=275&sz=41&tbnid=Buy2QDUNZEvi6M:&tbnh=109&tbnw=109&hl=en&start=20&prev=/images%3Fq%3Dcasimir%2Beffect%26svnum%3D10%26hl%3Den%26lr%3D%26sa%3DG

Dist > 25µm: dome shapeDist > 25µm: dome shapeThe Casimir force occurs when The Casimir force occurs when

virtual photons are virtual photons are restricted.restricted.

The force is reduced where The force is reduced where virtual photons are virtual photons are diffracted into the gap diffracted into the gap between the platesbetween the plates

Unshaded areas correspond to Unshaded areas correspond to higher Casimir forceshigher Casimir forces

Casimir force is decreased at Casimir force is decreased at the edges of the plates the edges of the plates

Prior experiments have focused on static FPrior experiments have focused on static FCC and adhesion Fand adhesion FC C

This experiment investigates the dynamic This experiment investigates the dynamic effect of Feffect of FC:C:

A Hooke’s law spring provides the restoring A Hooke’s law spring provides the restoring forceforce

FFCC between a movable plate and a fixed between a movable plate and a fixed sphere provides the anharmonic forcesphere provides the anharmonic force

For z>dFor z>dCRITICALCRITICAL system is bistable system is bistable PE has a local + global minimaPE has a local + global minima FFCC makes the shape of local min anharmonic makes the shape of local min anharmonic Note: Note: chosing a sphere as one of the chosing a sphere as one of the

surfaces avoids alignment problemssurfaces avoids alignment problems

The Casimir force: FThe Casimir force: FCC

on Microelectromechanical systems on Microelectromechanical systems (MEMS)(MEMS)

(PRL: H. B. Chan et al – Bell Lab & Lucent Tech –Published Oct 2001)(PRL: H. B. Chan et al – Bell Lab & Lucent Tech –Published Oct 2001)

Mock set upMock set upK= 0.019 Nm-1 K= 0.019 Nm-1 Sphere radius = 100Sphere radius = 100μμmmddEQUILIBRIUM EQUILIBRIUM = 40nm= 40nm

3

3

4 d

R

360

c

d

A

480

c

FF to

The actual set upThe actual set up Oscillator: 3.5-mm-thick, 500-mmOscillator: 3.5-mm-thick, 500-mm2, 2, gold plated (on top), polysilicon plategold plated (on top), polysilicon plate Room temp – 1 milli TorrRoom temp – 1 milli Torr A driving voltage A driving voltage VVACAC excites the torsional mode of oscillation excites the torsional mode of oscillation

((VVDC1DC1: bias): bias)

VVdc: dc: bias to one of the two electrodes under the plate to linearize the voltage bias to one of the two electrodes under the plate to linearize the voltage

dependence of the driving torque dependence of the driving torque VVDC2: DC2: detection electrode detection electrode

Note: amplitude increases with VNote: amplitude increases with VACAC = 35.4 = 35.4μμV to 72.5 V to 72.5 μμVV

Torsional Spring constant: k=2.1 10-8 Nrad-1Torsional Spring constant: k=2.1 10-8 Nrad-1Fund res. Freq. = 2753.47 HzFund res. Freq. = 2753.47 HzI = 7.1 10-17 kgmI = 7.1 10-17 kgm22

System behaves linearly w/o sphereSystem behaves linearly w/o sphere

Add a gold plated polystyrene sphere radius = 200Add a gold plated polystyrene sphere radius = 200μμmm

Equation of motion Equation of motion

I

Fb

I

Fb

torquedrivingofamplitude

coefdampingI

k

approachclosestofancedistzzz

zF

tzFI

b

toupzaboutpandexTaylorbzF

CC

o

o

6,

2

,

)(120

R c)(

cos)]([2

)(

43

141

3

322

2

3

]2

1[20

201

32

I

Fb

andintermstheIgnoring

C

Freq shift Freq shift ~ F~ FCC gradient (F gradient (FCC’)’)

z (equil dist sph-plate w/o FC)

Due to FC

Due to Electrostatic force

FFCC anharmonic behavior anharmonic behavior

I: Sphere far away I: Sphere far away normal resonnance normal resonnance Sphere is moved closer to plate I Sphere is moved closer to plate I IV IV Res. freq shifts as per model Res. freq shifts as per model At close distance At close distance hysteresis occurs hysteresis occurs

ie: amplitude A has up to 3 roots:ie: amplitude A has up to 3 roots:

linearitynonzescharacteri

IAA

2

311

21

2

222

12

12

5

8

3

4][(

]2

1[20

201

I

Fb C

Or we can keep a constant excitation freq Or we can keep a constant excitation freq (2748Hz), vary sphere-plate distance, and (2748Hz), vary sphere-plate distance, and measure amplitude.measure amplitude.

Freq < resonant freq Freq > resonant freq

Depends on history

Is repulsive Casimir force physical ?Is repulsive Casimir force physical ?

Plate-plate: attractivePlate-plate: attractive Sphere-plate: attractiveSphere-plate: attractive Concave surface – concave surface: can be Concave surface – concave surface: can be

repulsive or attractive depending on separation repulsive or attractive depending on separation pendulum pendulum

Plate-plate with specific dielectric properties Plate-plate with specific dielectric properties can be repulsive can be repulsive nanotech applications nanotech applications

ReferencesReferences

Nonlinear Micromechanical Casimir Oscillator [PRL: published 31 October 2001H. B. Chan,* V. A. Aksyuk, R. N. Kleiman, D. J. Bishop, and Federico Capasso† Bell Laboratories, Lucent Technologies, Murray Hill, New Jersey 07974

Physics World article (Sept 2002) – Author:Astrid Lambrecht

REPORTS ON PROGRESS IN PHYSICSRep. Prog. Phys. 68 (2005) 201–236Steven Lamoreaux