THE CASIMIR FORCE : THEORY

THE CASIMIR FORCE :
THEORY-EXPERIMENT COMPARISON
Cyriaque Genet?
Astrid Lambrecht
Serge Reynaud
Laboratoire Kastler Brossel
UPMC et ENS, Paris
“Quantum Fluctuations and Relativity”
http://www.spectro.jussieu.fr/Vacuum
?
Present address : Huygens Laboratory
Leiden University, The Netherlands
Casimir 1948
FCas =
ECas =
FCas ∼ 10−7N
h̄cπ 2A
240L4
h̄cπ 2A
720L3
for L = 1µm, A = 1cm2
Harris, Chen, Mohideen 2000
U. California, Riverside
Courtesy of U. Mohideen
Atomic Force Microscope (AFM)
Sphere (100µm) and plate, covered with gold
for distances L =100-900nm
Experimental precision ∼ 1 %
THEORY-EXPERIMENT
COMPARISON
Theory-experiment agreement at the 1% level
when corrections are taken into account
⇒ Plane-sphere geometry
⇒ Imperfect reflection
⇒ Non-zero temperature
⇒ Surface roughness
THE FABRY-PEROT CAVITY
q
Vacuum radiation pressure is different
outside
h̄ω
cos2 θ
2
and inside the cavity
h̄ω
2
cos2 θ × g [ω]
Airy function of the cavity
¯
¯
2ikz L ¯2
¯
1 − r1r2e
g [ω] =
2
|1 − r1r2e2ikz L|
kz =
ω
c
cos θ
THE CASIMIR FORCE
The Casimir force is the difference between
radiation pressures outside and inside the cavity
given as an integral over imaginary frequencies
(ω = iξ)
Z
Z
h̄A X d2k
∞
F =
π
p
4π 2
dξ κ
0
rkp 1 [iξ] rkp 2 [iξ] e−2κL
1 − rkp 1 [iξ] rkp 2 [iξ] e−2κL
s
κ=
k2 +
ξ2
c2
Regular expression for physical scattering
amplitudes
F → FCas
pour
rkp 1rkp
2
→1
THE DRUDE MODEL
ε [ω] = 1 −
ωP2
ω (ω + iΓ)
ηF =
λP =
2πc
ωP
Γ ¿ ωP
F
FCas
0
10
ηF
-1
10
plasma
L << l P
-2
10
10
-2
-1
10
0
L/l
L À λP
L ¿ λP
1
10
10
10
2
P
→ ηF ' 1
→ ηF ' 1.193 λL
P
→
→
F ∼
F ∼
1
L4
1
λP L 3
NON-ZERO TEMPERATURE
1
2
µ
h̄ω →
¶
1
+ n [ω] h̄ω
2
λT =
h̄c
kB T
,
1
n [ω] =
e
h̄ω
kB T
−1
∼ 7µm at T = 300K
3.0
2.0
ηF
1.0
0.9
0.8
0.7
0.6
0.5
0.1
1.0
10.0
L[µm]
L ¿ λT
L À λT
ηF ∼ 1
L
ηF ∼
λT
GLOBAL CORRECTION FACTOR
3.0
2.0
T=0K, plasma λP=136nm
T=300K, miroirs parfaits
T=300K, plasma λP=136nm
ηF
1.0
0.9
0.8
0.7
0.6
0.5
0.1
1.0
L[µm]
10.0
PLANE-SPHERE GEOMETRY
Proximity Force Approximation (PFA)
Z
F =
F
PS
(L) =
d2r
A
2πR
A
F PP (L (r))
E PP (L) for L ¿ R
ROUGHNESS CORRECTIONS
00
PFA
:
EPP (L) ¡­
EPP (L) = EPP (L) +
2
Z 2
­ 2®
d k
h
←−
ρ [k] σ [k]
4π 2
h21
5
4
3
ρ
2
1
0
0
1
2
3
4
5
6
7
8
9
|k|L
• |k| L ≤ 1
ρ [k] ' 1
• |k| L ≥ 1
ρ [k] ' β |k| L
PFA sector
Long distances
L À λP
β = 1/3
Short distances
L ¿ λP
β ' 0.45
10
®
­ 2®¢
+ h2
CONCLUSION
Plane-sphere geometry
- proximity force approximation
Imperfect reflection
- measured and compared to theoretical evaluations
Temperature
- not measured yet :
small contribution at short distances
Surface roughness
- small contribution < 1% within the PFA
- deviations from the PFA :
surface spectra measurements and
general evaluation of the sensitivity factor