Vacuum fluctuations and Casimir force

Vacuum fluctuations and Casimir force
First lecture :
a short history of quantum fluctuations … a brief introduction to modern quantum optics …
Tests of the Casimir force
Casimir formula (1949)
FCas c 2 A 240 L4
ECas
Second lecture :
the Casimir force between two flat mirrors at rest in vacuum effects of imperfect reflection (finite conductivity) comparison between theory and experiments search for a violation of Newton force law at short distances
c 2 A 720 L3
attraction measured as a pressure
A lot of simplifications
plane parallel mirrors perfect reflection zero temperature perfectly flat surfaces
P Cas
3
F Cas A
m L
4
Third lecture :
geometry of the plates – effect of roughness motion of the plates – “dynamical Casimir effect”
P Cas
10 Pa
B. Duplantier in Poincaré Seminar on Vacuum Energy (2002)
The Casimir force in the plane-sphere geometry
Recent experiments performed in the plane-sphere geometry
theory uses the proximity force approximation contributions of the surface elements added up independently
Riverside experiment (Mohideen et al)
Atomic force microscope (AFM)
Plane-sphere geometry Sphere (100μm) and plane covered with gold Distance 60-900nm Optical readout Experimental accuracy better than 2% at the smallest separations
Courtesy U. Mohideen B.W. Harris et al Phys. Rev. A62 052109 (2003)
R L
L
For the plane-sphere geometry ( if R>>L )
FPS
d 2x
FPP ( x) A
FPS 2 R
EPP A
This is not a theorem : Casimir forces are not additive

Riverside results (Mohideen et al)
Correct agreement with theory …
Casimir force (10 -9N)
0.0
Why the tests are going on
Casimir force is an important prediction of quantum theory which has been recently measured with a good precision
Experiment Theory
… after having accounted for large corrections
Plane-sphere geometry Imperfect reflection
-0.1
-0.2
-0.3
-0.4
a force induced by vacuum fluctuations which is directly observable in the macroscopic world ! accurate comparison with experiments allows for a test of theoretical predictions theory must take into account the differences between the ideal Casimir configuration and real experiments
-0.5
and small corrections
Room temperature Surface roughness
50
100
150
200
250
300
350
One of the few experimental ways for
approaching the puzzle of vacuum energy searching for hypothetical new forces at short distances
Plate-sphere surface separation (nm)
Courtesy U. Mohideen
A. Lambrecht & S. Reynaud in Poincaré Seminar (2002), quant-ph/0302073
Tests of the Newton force law
Yukawa correction to the Newton law
Short range tests of the Newton law
Exclusion domain in the ( , ) plane Gravity measurements at short distances Casimir force measurements Gravity measurements
log10
Geophysical Laboratory Satellites
> a few 10μm
Windows remain open for deviations at short ranges
log10
log10 (m)
LLR
Planetary
At even shorter distances tests of Casimir force
log10 (m)
or long ranges
Courtesy : J. Coy, E. Fischbach, R. Hellings, C. Talmadge, and E. M. Standish (2003)
The Search for Non-Newtonian Gravity, E. Fischbach & C. Talmadge (1998)
E. Adelberger et al Annu. Rev. Nucl. Part. Sci. (2003) hep-ph/0307284

Adelberger et al (U. Washington)
E?t-Wash : “Missing-mass” torsion balance Two disks with holes ; the attractor is rotated uniformly Separations down to 200μm : the best limits for 200μm< <3mm
Short distance gravity tests : Summary of the results
Stanford
fiber suspension
mirror for optical readout detector mass (Al)
source mass (Cu) creates a torque
Colorado
The Newtonian signal is largely, but not completely, cancelled C.D. Hoyle et al Phys. Rev. Lett. 86, 1418 (2001)
Washington E. Adelberger et al Annu. Rev. Nucl. Part. Sci. (2003) hep-ph/0307284
Newton vs Casimir : the challenge
For two Cu plates (1cm x 1cm x 1mm), Casimir dominates Newton at L<10μm
Recent Casimir force measurements
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 cylinders R=10mm at L=20-100nm Capasso et al 2001 : Micro-Electro-Mechanical Systems Plane & sphere R=100μm at L=100-500nm Onofrio et al 2002 : Casimir geometry Two parallel planes at L=0.5-3μm; Accuracy ~15%
Casimir
Newton Yukawa m
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
L
m Coulomb force must also be controlled
Decca et al 2003-... : Measurement between dissimilar metals Plane & sphere R=600μm at L=200-1200nm; Accuracy 1-2%

Lamoreaux (Seattle)
The first “modern” experiment
Torsion pendulum with large electrostatic 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)
Pivot Point Compensator Plates Solenoid Plunger Piezo Stacks Screws
Capasso et al (Bell Labs)
Micro-electro-mechanical systems (MEMS) Poly-silicon plate hold by a torsional rod
Adjustment Rods
Vacuum Can
Sphere (100μm) and plane covered with gold Distance 100-500nm Capacitive readout
Courtesy S. Lamoreaux S. Lamoreaux Phys. Rev. Lett. 78 (1997)
Courtesy F. Capasso H.B. Chan et al Science 291, 1941 (2001), PRL 87, 211801 (2001)
Fischbach et al (Purdue)
Au-coated sphere (R=100-600μm) Cu-coated plate mounted on a torsional MEMS Capacitive readout
Static or dynamic measurements
L = 260-1200nm Courtesy F. Capasso
R. Decca et al Phys. Rev. D68 (2003)

Bressi et al (Padova)
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%
Casimir tests of the Newton law : Summary of the results
Stockholm
Courtesy R. Onofrio
Riverside
Purdue Bressi, Carugno, Onofrio, Ruoso PRL 88 (2002)
Seattle
E. Adelberger et al Annu. Rev. Nucl. Part. Sci. (2003) hep-ph/0307284
Imperfectly reflecting mirrors
Perfectly plane, parallel and flat (but not perfectly reflecting) mirrors obey lateral translation invariance and show specular scattering Such mirrors are described by scattering amplitudes r,t which depend on
frequency incidence angle polarization p=TE,TM
transmitted reflected
Cavity with imperfectly reflecting mirrors
The spectral density is multiplied by the Airy function for the intracavity field
incident
1 g 1
r1 r2 e r1 r2 e
2 ik z L 2 ik z L
2
2
free field
r1 and r2 are the reflection amplitudes seen by the intracavity field kz is the longitudinal wave-vector kz
c
cos
intracavity field
The Casimir force can be expressed in terms of the reflection amplitudes
Jaekel, Reynaud, J. Physique (1991), arXiv:quant-ph/0101067
This is a theorem which has been proven for lossy as well as lossless mirrors
C. Genet, A. Lambrecht & S. Reynaud, Phys. Rev. A67 (2003)

Radiation pressure on the mirrors
For each field mode, vacuum radiation pressures differ
on the outer side and on the inner side
Sum over the modes
The force is an integral over all field modes
2 2
n n
cos2 cos2 g
attractive contributions repulsive contributions
FPP
modes
2
n
cos2
1 g
The net pressure is proportional to
g
The sum over the modes includes evanescent waves as well as ordinary propagating waves
g
f 1
1
f
f
*
This formula is valid for arbitrary mirrors
1
the reflection amplitudes are still to be specified through measurements or modeling of mirrors
r1 r2 e
2 ik z L 2 ik z L
r1 r2 e
cL
kz
we have only assumed specular reflection
Long-range force: electronic influence between the mirrors negligible
Fluctuations-dissipation theorem has been used
Final expression of the Casimir force …
Using causality properties
the formula can also be written as an integral over imaginary frequencies
… in the plane-sphere geometry
At zero temperature in the plane-sphere geometry
The final scattering formula is regular
no need for further regularization for any causal mirrors
FPS
E L 2 R PP A
EPP L
L
R
L
dL' FPP L'
It goes to the Casimir formula 1 and T 0 for r1 r2
FCas
c 2 A 240 L4
These expressions represent the QED prediction to be compared with measurements
reflection amplitudes still to be specified
C. Genet, A. Lambrecht & S. Reynaud, Phys. Rev. A67 (2003)
Proximity force approximation used for the plane-sphere geometry

Models of “real” mirrors
Simplest model for metallic mirrors
Fresnel reflection laws plasma model for the gas 0 of conduction electrons 10
F
More complete description of metals
F FCas
F
1
Plane, parallel and flat mirrors showing specular reflection
Scattering amplitudes deduced 6 from Fresnel laws 10 Optical response of 5 10 electrons described 4 10 by tabulated dielectric 3 function 10
i
perfect mirrors
( )
( ) 1
Plasma frequency and wavelength
P
2 P 2
dissipative part of the dielectric function for some metals
Cu Al
2 c
P
Reduction of the force calculated with respect to the ideal Casimir formula
reduction factor for plasma the << λ plasma L P model
10
-2
( )
r
( ) i i( )
x i( ) x2 2
ε’’(ω)
ηF
10
-1
10 10
2
1
and causality relations
r( ) 1
10 10 10
0
2
0
10
-2
dx
-1
Al Au Cu
12
Au
10
14
10
-1
10
0
10
1
10
2
-2
F
0
L/λP
L/
10
10
13
P
plasmon limit
as a function of frequency
10 ω[rad/s]
15
10
16
10
17
[rad/s]
Results for two Au-covered mirrors
global behaviour well reproduced by the plasma model integration of tabulated optical data needed for obtaining accurate predictions
0.0
Finite temperature correction
Radiation pressure of thermal fluctuations has to be added to that of vacuum fluctuations
3.0
1.0
F
F FCas
Au-coated mirrors (optical data)
0.8
2
2
e
/ k BT
1
2.0
perfect mirror, T
300 K
ηF
0.6
at T=300K, important corrections for L>3 m plasma model P = 136nm
plasma model optical data
plasma model, T
1.0 0.9 0.8 0.7 0.6 0.5 0.1
300 K
-0.1
Experiment Theory
Casimir force (10 -9N)
-0.2
-0.3
-0.4
1 L[μm] A. Lambrecht & S. Reynaud, Eur. Phys. J. D8 309 (2000)
50 100 150 200 250 300 350
-0.5
0.4
Plate-sphere surface separation (nm)
0.1
L μm
10
temperature & plasma corrections are correlated at intermediate distances L 1-3 m
plasma model, T
1.0
0K
10.0
L[ m]
C. Genet, A. Lambrecht & S. Reynaud, Phys. Rev. A62 (2000)

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
Casimir and Casimir-Polder
Variation of the reduction factor for the energy as a function of distance
at large distances, limit of perfect mirrors
1.0
E
E ECas
E
1
ECas
c 2A 1 720 L3
0.1
at short distances, different scaling law
E c 2A 1 480 P L2
1.193
0.0 ?2 10 10
?1
L
E P
0 1 2
10
10
10
L/
P
This is reminiscent of the Casimir-Polder law for atomic forces
Casimir-Polder crossover
From the Van der Waals interaction between neutral atoms in their ground states
Instantaneous interaction for Retarded interaction for
Casimir force and surface plasmons
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 bulks are coupled by Coulomb law
2 sp 2 P
2c 2k 2 2
2 P
E
c
A
2
r
6
EC - P
23 c 0 4 r7
2
4 P
4c 4k 4
London 1930
Casimir-Polder 1948 Casimir-
2
(1 e 2
kL
)
to the Casimir interaction between two mirrors
Instantaneous interaction for Retarded interaction for
At short distances, the Casimir force can be seen as the effect of Coulomb interaction between surface plasmons
E
c 2A 480 P L2
Lifshitz 1956
ECas
c 2A 720 L3
see also F. Intravaia & A. Lambrecht, Phys. Rev. Lett. (2005)
Casimir 1949

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