Vacuum fluctuations and Casimir force

Second lecture :the Casimir force between two flat mirrors at rest in vacuumeffects of imperfect reflection (finite conductivity)comparison between theory and experimentssearch for a violation of Newton force law at short distances

First lecture :a short history of quantum fluctuations …a brief introduction to modern quantum optics …

Third lecture :geometry of the plates – effect of roughnessmotion of the plates – “dynamical Casimir effect”

43

CasmPa10L

P

attractionmeasured as a pressure

AL

cF 4

2

Cas 240

Casimir formula (1949)

AL

cE 3

2

Cas 720

AF

PCas

CasA lot of simplifications

plane parallel mirrorsperfect reflectionzero temperatureperfectly flat surfaces

B. Duplantier in Poincaré Seminar on Vacuum Energy (2002)

Tests of the Casimir forceTests of the Casimir force

The Casimir force in the plane-sphere geometry

AxF

xdF PPPS

)(2

LR

LFor the plane-sphere geometry ( if R>>L )

Recent experiments performed in the plane-sphere geometry

theory uses the proximity force approximation

contributions of the surface elements added up independently

This is not a theorem : Casimir forces are not additive

AE

RF PPPS 2

Riverside experiment (Mohideen et al)

Atomic force microscope (AFM)

Plane-sphere geometrySphere (100µm) and planecovered with goldDistance 60-900nmOptical readoutExperimental accuracy better than 2% at thesmallest separations

Courtesy U. Mohideen

B.W. Harris et al Phys. Rev. A62 052109 (2003)

50 100 150 200 250 300 350-0.5

-0.4

-0.3

-0.2

-0.1

0.0

Experiment

Theory

Cas

imir

forc

e (1

0-9N)

Plate-sphere surface separation (nm)

large correctionsPlane-sphere geometryImperfect reflection

Correct agreement with theory …

and small correctionsRoom temperature Surface roughness

… after having accounted for

Courtesy U. Mohideen

A. Lambrecht & S. Reynaud in Poincaré Seminar (2002), quant-ph/0302073

Riverside results (Mohideen et al)

Casimir force is an important prediction of quantum theory which has been recently measured with a good precision

a force induced by vacuum fluctuations which is directly observable in the macroscopic world !accurate comparison with experiments allows for a test of theoretical predictionstheory must take into account the differences between the ideal Casimir configuration and real experiments

One of the few experimental ways forapproaching the puzzle of vacuum energysearching for hypothetical new forces at short distances

Why the tests are going on

Courtesy : J. Coy, E. Fischbach, R. Hellings, C. Talmadge, and E. M. Standish (2003)

log 1

0

log10 (m)

Windows remain open for deviations at short ranges

Satellites

Laboratory

Geophysical

LLR Planetary

The Search for Non-Newtonian Gravity, E. Fischbach & C. Talmadge (1998)

or long ranges

Yukawa correctionto the Newton law

Tests of the Newton force lawTests of the Newton force law

log 1

0

log10 (m)

Casimir force measurements

Gravity measurements

Exclusion domainin the ( , ) plane

Short range tests of the Newton law

Gravity measurementsat short distances

> a few 10µm

E. Adelberger et al Annu. Rev. Nucl. Part. Sci. (2003) hep-ph/0307284

At even shorter distances tests of Casimir force

Adelberger et al (U. Washington)

Eöt-Wash : “Missing-mass” torsion balanceTwo disks with holes ; the attractor is rotated uniformlySeparations down to 200µm : the best limits for 200µm< <3mm

C.D. Hoyle et al Phys. Rev. Lett. 86, 1418 (2001)

detectormass (Al)

mirror foropticalreadout

fibersuspension

source mass(Cu) createsa torque

The Newtonian signal is largely,but not completely, cancelled

Short distance gravity tests :Summaryof theresults

E. Adelberger et al Annu. Rev. Nucl. Part. Sci. (2003) hep-ph/0307284

Stanford

Colorado

Washington

Casimir

Newton

Yukawa m

mL

Coulomb force must also be controlled

For two Cu plates(1cm x 1cm x 1mm),Casimir dominates Newton at L<10µm

Newton vs Casimir : the challenge

As an hypothetical new force would appear as a difference between experimental and theoretical values, these two values have to be obtained independently with the same care and accuracy

Lamoreaux 1997 : Torsion pendulum with electrostatic compensation; Plane & sphere R=12.5cm at L=0.6-6µm; Accuracy ~5% (??)

Mohideen et al 1998-... : Atomic force microscope (AFM)Plane & sphere R=100µm at L=100-500nm; Accuracy 1-2%

Ederth 2000 : Crossed cylindersR=10mm at L=20-100nm

Capasso et al 2001 : Micro-Electro-Mechanical SystemsPlane & sphere R=100µm at L=100-500nm

Onofrio et al 2002 : Casimir geometryTwo parallel planes at L=0.5-3µm; Accuracy ~15%

Decca et al 2003-... : Measurement between dissimilar metalsPlane & sphere R=600µm at L=200-1200nm; Accuracy 1-2%

Recent Casimir force measurementsRecent Casimir force measurements

Pivot Point

AdjustmentRods

ScrewsPiezoStacks

VacuumCan

Compensator Plates

SolenoidPlunger

Lamoreaux (Seattle)

S. Lamoreaux Phys. Rev. Lett. 78 (1997)

The first “modern” experimentTorsion pendulum with largeelectrostatic force used for compensation

Correct agreement at shortest distances

Accuracy announced at ~5% but not mastered

Temperature effect not seen at largest distances (up to 6µm)

Courtesy S. Lamoreaux

Capasso et al (Bell Labs)

Micro-electro-mechanical systems (MEMS)Poly-silicon plate hold by a torsional rod

Sphere (100µm) and plane covered with goldDistance 100-500nmCapacitive readout

Courtesy F. Capasso

H.B. Chan et al Science 291, 1941 (2001), PRL 87, 211801 (2001)

Courtesy F. CapassoR. Decca et al Phys. Rev. D68 (2003)

Fischbach et al (Purdue)

Au-coated sphere (R=100-600µm)

Cu-coated platemounted on atorsional MEMS

Capacitive readout

L = 260-1200nm

Static or dynamicmeasurements

Bressi et al (Padova)

Bressi, Carugno, Onofrio, Ruoso PRL 88 (2002)

The only recent experiment in the Casimir geometry

Two parallel planes at 0.5-3µm

Dynamical measurement (shift of the vibration frequency of the flexible plate)

Readout through an optical fiber interferometer

Accuracy ~15%

Courtesy R. Onofrio

Casimir tests of the Newton law :Summaryof theresults

E. Adelberger et al Annu. Rev. Nucl. Part. Sci. (2003) hep-ph/0307284

Stockholm

Riverside

SeattlePurdue

Such mirrors are described by scattering amplitudes r,twhich depend on

frequency incidence angle polarization p=TE,TM

The Casimir force can be expressed in terms of the reflection amplitudes

Perfectly plane, parallel and flat (but not perfectly reflecting) mirrors obey lateral translation invarianceand show specular scattering

Jaekel, Reynaud, J. Physique (1991), arXiv:quant-ph/0101067

incident

reflectedtransmitted

Imperfectly reflecting mirrorsImperfectly reflecting mirrors

2

21

2

21

21

21

Likerr

Likerrg

z

z

cosc

kz

Cavity with imperfectly reflecting mirrors

free field

The spectral density is multiplied by the Airy function for the intracavity field

r1 and r2 are the reflection amplitudes seen by the intracavity fieldkz is the longitudinal wave-vector intracavity field

This is a theorem which has been provenfor lossy as well as lossless mirrors

C. Genet, A. Lambrecht & S. Reynaud, Phys. Rev. A67 (2003)

For each field mode, vacuum radiation pressures differ

on the outer sideandon the inner side

Radiation pressure on the mirrors

Likerr

Likerrf

z

z

21

2

21

21

gn 2cos2

kzc L

g

1

The net pressure is proportional to

*1 ffg

attractive contributions

repulsivecontributions

2cos2

n

Fluctuations-dissipation theorem has been used

gnF 1cos2modes

2PP

Sum over the modes

This formula is valid for arbitrary mirrorsthe reflection amplitudes are still to be specified through measurements or modeling of mirrors

we have only assumed specular reflection

Long-range force: electronic influence between the mirrors negligible

The force is an integral over all field modes

The sum over the modes includes evanescent waves as well as ordinary propagating waves

It goes to the Casimir formula for and

The final scattering formula is regularno need for further regularization

for any causal mirrors

Final expression of the Casimir force …

Using causality propertiesthe formula can also be written as an integral over imaginary frequencies

0T121 rr

C. Genet, A. Lambrecht & S. Reynaud, Phys. Rev. A67 (2003)

AL

cF 4

2

Cas 240

'F'd PPPP LLLEL

… in the plane-sphere geometry

At zero temperaturein the plane-sphere geometry

These expressions represent the QED prediction to be compared with measurements

reflection amplitudes still to be specified

Proximity force approximation used for the plane-sphere geometry

ALE

RF PPPS 2

LR

10-2 10-1 100 101 102

L/λP

10-2

10-1

100

ηF

plasmaL << λP

CasFF

F

P/L

reductionfactor forthe plasmamodel

Plasma frequencyand wavelength

Models of “real” mirrorsModels of “real” mirrorsSimplest model for metallic mirrors

Fresnel reflection lawsplasma model for the gasof conduction electrons

2

2P1)(

PP2 c

Reduction of the force calculated with respect to the ideal Casimir formula

mirrorsperfect1F

limitplasmon0F

More complete description of metals

and causality relations

1012 1013 1014 1015 1016 1017

ω[rad/s]

10-2

10-1

100

101

102

103

104

105

106

ε’’(

ω)

AlAuCu

)(i

]rad/s[

Al

Au

Cu

dissipative partof the dielectric

function for some metals

as a function of frequency

)()()( ir i

220

)(d21)(xx

x ir

Plane, parallel and flat mirrors showing specular reflection

Scattering amplitudes deduced from Fresnel lawsOptical response of electrons describedby tabulated dielectricfunction

0.1 1 10L[µm]

0.4

0.6

0.8

1.0

ηF

plasma model optical data

A. Lambrecht & S. Reynaud, Eur. Phys. J. D8 309 (2000)µmL

Au-coated mirrors(optical data)

Results for two Au-covered mirrors

plasma modelP = 136nm

integration oftabulated optical dataneeded for obtainingaccurate predictions

CasFF

F

global behaviourwell reproduced

by the plasma model

50 100 150 200 250 300 350-0.5

-0.4

-0.3

-0.2

-0.1

0.0

Experiment

Theory

Cas

imir

forc

e (1

0-9N)

Plate-sphere surface separation (nm)

Radiation pressure of thermal fluctuations has to be added to that of vacuum fluctuations

122 B/ Tke

][ mL

KT 0model,plasma

KT 300mirror,perfect

KT 300model,plasma

0.1 1.0 10.00.5

0.6

0.70.80.91.0

2.0

3.0

temperature & plasma corrections are correlated at intermediate distancesL 1-3 m

C. Genet, A. Lambrecht & S. Reynaud, Phys. Rev. A62 (2000)

Finite temperature correction

at T=300K,important correctionsfor L>3 m

Conclusions (at the moment …)

The Casimir force is now measured with a good experimental accuracy ~ 1-2%

Theory and experiment agree at the same level in the distance range 100nm < L < 500nm

The effect of imperfect reflection is precisely measured and accurately calculated

Theoretical discussions are still extremely active about the effect of non zero temperature (for dissipative mirrors)

And the effect of thermal fluctuations has still to be detected unambiguously in experiments

10−2 10−1 100 101 1020.0

0.1

1.0

P/L

Casimir and Casimir-PolderCasimir and Casimir-Polder

at short distances, different scaling law

1E

P

LE

2P

2 1480 L

AcE

Variation of the reduction factor for the energy as a function of distance

This is reminiscent of the Casimir-Polder law for atomic forces

CasEE

E

3

2

Cas1

720 LAc

E

193.1

at large distances,limit of perfect mirrors

RetardedRetarded interaction interaction forfor

InstantaneousInstantaneous interaction interaction forfor

RetardedRetarded interaction interaction forfor

Casimir-Polder crossover

From the Van der Waals interaction between neutral atoms in their ground states

CasimirCasimir--PolderPolder 19481948

InstantaneousInstantaneous interaction interaction for for

London 1930London 1930

Casimir 1949Casimir 1949

6A

2

rc

E 7

20

P-C 423

rc

E

3

2

Cas 720LAc

E2P

2

480 LAc

E

to the Casimir interaction between two mirrors

LifshitzLifshitz 19561956

Surface plasmons collective electron excitations coupled to evanescent fields and propagating on the interface between each metallic bulk and vacuum

For each metallic bulk,dispersion relation versus transverse wavevector k

Plasmons on the two bulksare coupled by Coulomb law

Casimir force and surface plasmons

At short distances, the Casimir force can be seen as the effect of Coulomb interaction between surface plasmons

2k4k2 444

P222

P2sp

cc

2)1( k2

P2Le

see also F. Intravaia & A. Lambrecht, Phys. Rev. Lett. (2005)