Casimir Casimir MomentumMomentum in in ComplexComplexMedia?Media?
Bart van TiggelenBart van Tiggelen
CollaboratorsCollaborators::
•• Geert Geert RikkenRikken (LNCMI Grenoble/Toulouse)(LNCMI Grenoble/Toulouse)•• SSéébastien bastien KawkaKawka ((Ph.DPh.D Grenoble Grenoble ENS PisaENS Pisa))••James Babington (James Babington (postdocpostdoc ANR Grenoble)ANR Grenoble)
Costas Costas SoukoulisSoukoulis 60 60 yearsyears, , JuneJune 20112011
Grenoble
MomentumMomentum fromfrom NothingNothing
εε,,μμ,g,g
k,ωh k,ωh', kωhE0
B0
( )0
v)1(cl
ijlij εεωχ −=
( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )ωωμωω
ωωωωω
BEχHBχED
+⋅=
⋅+=*
ε
( ) ( )0000jijiij EBBEg −=ωχ
MagnetoMagneto--electricelectric birefringencebirefringence
( ) ( ) 0*det00
220
2
=⎟⎟⎠
⎞⎜⎜⎝
⎛⋅⋅+⋅⋅−+− χpεpεχpp
ccp
cωωω
ε( ) lnmlinm
piΦ ε=p
Fresnel dispersion Fresnel dispersion lawlaw
kx
BiBi--anisotropicanisotropic MediaMedia
Fizeau Fizeau effecteffect
ky
vv
( ) ijij gi δωωχ =
EE0 0 x Bx B00
RotatoryRotatory powerpower
1010--15151010--881010--22
( )
( )[ ]⎪⎪⎩
⎪⎪⎨
⎧
−×
××
∝×
∫
∫
0
3
0
003
0
1211
211
00
cd
c
gdc
k
k
vk
BEkBE
ωεω
ωω
h
h 0040
3
4
32 BE ×= g
cc
πωh
cutcut--off in Xoff in X--ray ?ray ?
phenomenologicalphenomenological continuum continuum theorytheory
PhotonicPhotonic momentummomentum in in dielectricdielectric media? media? classicalclassical «« AbrahamAbraham »» contribution contribution alreadyalready controversialcontroversial
UV catastrophe of vacuum UV catastrophe of vacuum energyenergy ??Lorentz invariance of quantum vacuum?Lorentz invariance of quantum vacuum?InertiaInertia of quantum vacuum? of quantum vacuum?
( ) ( )jijiijij
t
BBEET
c
+−+=
⋅−∇=⎟⎟⎠
⎞⎜⎜⎝
⎛×+∂
πδ
π
πρ
41
81
41
220
0
0
BE
TBEv
vcasiρ=
InertialInertial mass of quantum vacuum?mass of quantum vacuum?
⎭⎬⎫
⎩⎨⎧ −=Δ ∫∫∫ )bubble nowater (
21)in water bubble(
21)bubble( 333
kk kkr ωω hh dddE
MeV101130
43
≈⎟⎠⎞
⎜⎝⎛ −≈
εωca ch
Schwinger (1993)Schwinger (1993)
UV catastrophe in sonoluminescence(> 1934)
cutcut--off in the UV ?off in the UV ?( )
eV001.0
11536
23)bubble( 02
≈
−=Δ
acE ε
π
2
2
1)(ωωωε P−=− ∞== ∫
∞
2
23
030 ω
ωωωρ p
casi dch
Free Free electronelectron
++
Electric Electric quadruolequadruole
RizzoRizzo etaletal, 2003, 2003--2009, Babington & 2009, Babington & BAvTBAvT, 2011, 2011
∞=×= ∫∞
03
030
)( BEP 0ωωω gdccasihggMEME = 10= 10--1717---- 1010--1111
The UV catastrophe The UV catastrophe isis realreal
magneticmagnetic dipoledipole
gME(ω)/n
2222222
22 )(
2)(2)(
211),,( BEBEBEvvBE ⋅+−+−+−−=
ννρc
cL
( )( )ωω
BBBEEE
+=+=
0
0
00340
4*0 BEBE
×−=× Kνπ00
4*0 =×πHE
),()2(
1lim 04021
3 ΩΩ= ∫∫∞→ωρω
π π
ω
ωddK c
c
h
ZeroZero energyenergy flowflow infiniteinfinite momentummomentum densitydensity
Lorentz Lorentz scalarscalar
BiBi--anisotropicanisotropicLorentzLorentz--invariant vacuuminvariant vacuum
)'(2),',(Im20)','(),(0 2* ωωπδωωωω −×−= rrrr ijji GEE hFluctuationFluctuation--DissipationDissipation
Casimir Casimir momentummomentum, if , if infiniteinfinite, , isis Lorentz invariantLorentz invariant
BB00
εεEE00(t)(t) vv
( ) 003 )(
341)( BEv ×−= tatm πε0constant ==×− BPvρ
++ --EE00(t)(t)
BB00 vv
)()()()(
1222
1211
rqtqmrqtqmfBrErfBrEr−×−−=+×++=
&&&
&&&
02)(20constant2
20 ≈−×+=
==×+
xBRExBxR
ωmqtqmqm
&&&
&
0020
2
)(/2 BER ×= tmqmω
&
)(21
2,1 xaRr +±=
ClassicalClassical Abraham Abraham momentummomentum in in crossedcrossed EM EM fieldsfields
(Walker Nature, 1976)(Walker Nature, 1976)
sec/nm32
)0(vabr ≈=pmEBα
sec/nm02.04
v 4Feigel ≈= gEBh
cρλπ
( )
sec/nm0.0
10158.0v 20
−≈
−−= gEBac
regula εh
sec/nm08.0log34vv abr ≈×=
e
atQED m
mαπ
ClassicalClassical abrahamabrahamforceforce
FeigelFeigel QED QED withwith cutcut--off off 0.1 nm0.1 nm
RegularizationRegularization of of vacuum vacuum energyenergy in a=10 cmin a=10 cm
(Milton, 2000)(Milton, 2000)
QED QED harmonicharmonic oscillatoroscillator((KawkaKawka, 2010), 2010)
E=450 V/mm; B=1 T
( )
m/VT10017.0)T(roomkg/m17.0
)6.16(/VCm1022.00
22
3
30
240
−
−
=
=
=
g
a
ρ
αEx: Ex: HeliumHelium
BEp ×= )0(α( ) LntBEPP ××××××+= ωωαω cos)0(0
AcousticAcousticpressurepressure
dpdp//dtdt=Abraham force=Abraham force
ExperimentExperiment: Geert : Geert RikkenRikken αα(0)(0)
p/(EB) p/(EB)
E=450 V/mm;E=450 V/mm;B=1 T; B=1 T;
f= 7.6 kHz f= 7.6 kHz
V= 8 V= 8 nm/secnm/sec++-- 0.80.8FeigelFeigel : 2 nm/sec: 2 nm/sec
rErBA ⋅−=×= 000 21 φ
( ) ( )
221
20210
22202
2
21101
1
21
)()(2
1)()(21
rrE
rArAprArAp
μω+⋅+
+++−−=
e
eem
eem
H
Casimir Casimir momentummomentum: : 1/41/4QED of QED of harmonicharmonic oscillatoroscillator in in crossedcrossed fieldsfields
EE00
BB00+e+e --ee
( )21* ++∑ iii
iaaωh
)()(
20212
10111
rAvprAvp
emem−=+=
Casimir Casimir momentummomentum: : 2/42/4QED of QED of harmonicharmonic oscillatoroscillator in in crossedcrossed fieldsfields
0],[ =HK
rBPrBppK ×+=×++= 021021 21ˆ ee kin
PseudoPseudo--momentummomentum isis conservedconserved
ConjugateConjugate momentamomenta≠≠ kinetickinetic momentummomentum
EE00
BB00+e+e --ee
pcmppdpm
mmmmmmM
ii +=⎟⎟
⎠
⎞⎜⎜⎝
⎛+
=+= ∫∞
2/34)( 20
21
2121 α
πδδδμδδ
Casimir Casimir momentummomentum: : 3/43/4QED of QED of harmonicharmonic oscillatoroscillator in in crossedcrossed fieldsfields
EE00
BB00
( )21*)()(ˆ
210212211 ++×+−++= ∑ iiii
aaeeemm krBrArAvvK h
210020
2
0020
2
001 KKBEBEvv ++×⎟⎟⎠
⎞⎜⎜⎝
⎛+×−+=ΨΨ
ωμδ
μωδ eeMMK
+e+e --ee
220
002 )0( αμωαα ∝×∝c
hBEK
Casimir Casimir momentummomentum: : 4/44/4QED of QED of harmonicharmonic oscillatoroscillator in in crossedcrossed fieldsfields
( ) ( )
2
1
12
1200
12
22
012
12001
log34)0(
/2//2/64)0(
mm
mmmm
cmppp
cmpppdp
mmmm
απ
α
απ
α
+−
×=
⎥⎦
⎤⎢⎣
⎡+
−++
−×= ∫
∞
BE
BEKhh
KK1 1 : 2 % QED correction to Abraham force: 2 % QED correction to Abraham forceKK22: 0.01 % QED correction : 0.01 % QED correction
KawkaKawka & Van & Van TiggelenTiggelen, EPL 2010, EPL 2010
EE00
BB00+e+e --ee
BB00
Faraday RotationFaraday Rotation
A quantum vacuum force F= g dB/A quantum vacuum force F= g dB/dtdt ??
( )0
20
220
204,
γωσωωγ
ωπσωα
iVBic
++−=
Chiral Chiral geometrygeometry withwith electricelectric polarizabilitiespolarizabilities
00000 =×=×⇒= ∫∫ HErBErHB dd
εε
εε εεεε
BB00
Faraday RotationFaraday Rotation
A quantum vacuum force F= g dB/A quantum vacuum force F= g dB/dtdt ??
( ) ( )γωσωω
ωχσωχiVBi ++−
= 20
2
200,
000 =×∫ HErd
Chiral Chiral geometrygeometry withwith magneticmagnetic polarizabilitiespolarizabilities
000 BBEr gd =×∫Na Na TetraederTetraeder L=10 nm L=10 nm g/m = 1 nm/sec/Tg/m = 1 nm/sec/T
µµµµµµ
µµ
momentummomentum of quantum vacuum toof quantum vacuum toshed new light on the shed new light on the controversialcontroversial
nature of nature of zerozero--point point energyenergy
CorsicaCorsica,,20062006
Congratulations Costas! Congratulations Costas!