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!