CasimirCasimir momentum in crossed momentum in crossed electromagnetic fields.electromagnetic fields.

QED correction to Abraham force?QED correction to Abraham force?

•• SSéébastien bastien KawkaKawka ((Ph.DPh.D Grenoble Grenoble ENS Pisa)ENS Pisa)••James Babington (James Babington (postdocpostdoc ANR Grenoble)ANR Grenoble)

Bart van Bart van TiggelenTiggelen and and Geert Geert RikkenRikken

Casimir Workshop ; Casimir Workshop ; LeidenLeiden , March 2012, March 2012

PHOTONIMPULSPHOTONIMPULS

Casimir Casimir energyenergy1.1. IsotropicIsotropic radiation radiation withwith power power spectrumspectrum ωω33 isis LorentzLorentz--invariantinvariant

(Einstein, 1917); (Einstein, 1917);

1.1. Van der Van der WaalsWaals force 1/rforce 1/r66 (London, 1930)(London, 1930)2.2. Relation to Relation to CosmologicalCosmological constant (Pauli, 1934, Davies, 1984) constant (Pauli, 1934, Davies, 1984) 3.3. Casimir Polder Force 1/rCasimir Polder Force 1/r77 (1947) (1947) 4.4. Attraction Attraction betweenbetween metallicmetallic plates (Casimir, 1948)plates (Casimir, 1948)5.5. LifshitzLifshitz theorytheory for for dielectricdielectric media (media (LifshitzLifshitz, 1956, , 1956, DzyalonishiniskiiDzyalonishiniskii

1961)1961)1.1. Observation of Casimir Observation of Casimir effecteffect ((SparnaaySparnaay, 1958, , 1958,

Lamoureux (5%), 1997), Chan Lamoureux (5%), 1997), Chan etaletal, (1%), 2001), (1%), 2001)1.1. StabilityStability of the of the electronelectron (Casimir, 1956, Boyer, 1968) (Casimir, 1956, Boyer, 1968) 2.2. Unruh Unruh effecteffect & & HawkingHawking radiation (radiation (HawkingHawking 1974, Unruh 1976)1974, Unruh 1976)3.3. Bag model for Hadrons (Jaffe Bag model for Hadrons (Jaffe etaletal, 1974), 1974)4.4. SignSign of the of the CosmologicalCosmological constant (Weinberg, constant (Weinberg, …… 1983) 1983) 5.5. Sonoluminescence (Schwinger, 1993, Sonoluminescence (Schwinger, 1993, EberleinEberlein, 1996), 1996)6.6. Quantum friction and Quantum friction and sheeringsheering the quantum vacuum (the quantum vacuum (PendryPendry, 1998), 1998)7.7. Casimir Casimir momentummomentum in in magnetomagneto--electricelectric media (media (FeigelFeigel, 2004), 2004)

isis an invariant four an invariant four vectorvector[ ],021

21ˆ1 2

30

casiρ=ω,ωkddωc ⎥⎦

⎤⎢⎣⎡∫ ∫ khh

The Casimir The Casimir effecteffect……..

LL

ħħωω

E (L )=∞−(⋯)

ℏ AL3

( ) LLA=

LE=F(L) ˆ3 4

hL−

∂∂

−

No No momentummomentum exchange exchange betweenbetween mattermatter and radiationand radiation

NegativeNegative pressurepressure

( )

eV001.0

11536

23)bubble( 02

≈

−=Δ

acE ε

π

ΔE ( bubble )=∫d 3 r {∫ d3 k 12ℏωk (bubble in water )−∫ d 3k 1

2ℏωk (water no bubble )}

Schwinger (1993)Schwinger (1993)

UV catastrophe in sonoluminescenceUV catastrophe in sonoluminescence(> 1934)(> 1934)

cutcut--off in the UV ? off in the UV ?

DimensionalDimensional regularisationregularisation??

MeV101130

43

≈⎟⎠⎞

⎜⎝⎛ −≈

εω

ca ch

«« MomentumMomentum fromfrom NothingNothing »»

εε,,μμ,g,gkω,h'ω,kh

E0

B0

P=P=mvmv

gij (ω )=(1−ε )εijl

vl

c0

( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )ωωμ+ω=ω

ωω+ωωε=ωT BEgH

BgED1)( −⋅

⋅

ω

( ) ( )0000jijiij EBBEg=ωg −

MagnetoMagneto--electricelectric birefringencebirefringence

Fresnel dispersion Fresnel dispersion lawlaw

kx

BiBi--anisotropicanisotropic MediaMedia

Fizeau Fizeau effecteffect

ky

vv

gij (ω )=iω gδij

EE0 0 x Bx B00

RotatoryRotatory powerpower

1010--15151010--881010--22

( ) ( ) 0*det00

220

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅+⋅⋅−+− gkεkεgkk

cck

cωωω

ε

0040

3

4

32 BE ×g

cπω= ch

ObservedObserved in Xin 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

BBμ

+EEεδBμ

+Eε=T

=ε+ρ

00

2

0

20

0

00

14π11

8π1

TBEv

( )

( )[ ]⎪⎪⎩

⎪⎪⎨

⎧

−×

××

∝×

∫

∫

0

3

0

003

0

0 1211

211

04

0

cd

c

gdc

ck

k

vk

BEkBE

ωεω

ωω

πh

h

vcasiρ=

The Abraham ForceThe Abraham Force

fΤG −⋅∇∂ =+Mt

μHεE=

=M

∇−∇−

×22f

BDG

MacroscopicMaxwell

MinkowskiMinkowski

The Abraham ForceThe Abraham Force

NelsonNelson

AbrahamAbraham

Abraham Abraham momentummomentum = = kinetickinetic momentummomentum, , Minkowski Minkowski momentummomentum = = conjugateconjugate momentummomentum

PeierlsPeierls

[ ] [ ](t)(t)(t)(t)= tt 000052... BEBEf ∂×−×∂−−

( ) (t)(t)Vε= t 001 BEF ×∂− Walker & Walker, Walker & Walker, Nature 1976Nature 1976

0

0

GBEG

=ε=N ×

S

HEG

0

00

1c

=

με=A ×

exp

Barnett Barnett (PRL 2010): (PRL 2010):

)()μ(εε=+ trrAt BEfTG ×∂−−−⋅∇∂ /10

)()(εε=+ trAt BEfTG ×∂−−−⋅∇∂ 10

Nelson Nelson momentummomentum = pseudo = pseudo momentummomentum NelsonNelson(PRA 1991)(PRA 1991)

theo

The Abraham Force (The Abraham Force (ourour version)version)

fTG −⋅∇∂ =+t

)(+=+ρ tt BPfUv ×∂⋅∇∂

( ) 000 =+ε+ρt TBEv ⋅∇×∂

+

μHεE==

∇−∇−

×22f

BDG

MacroscopicMaxwell

Maxwell-Lorentz forceon induced polarization

and current

MicroscopicMaxwell

The Abraham Force (The Abraham Force (ourour version)version)

)( BPfTG ×−∂−⋅∇∂ tt =+

)(+=+ρ tt BPfUv ×∂⋅∇∂

( ) 000 =+ε+ρt TBEv ⋅∇×∂

+

μHεE==

∇−∇−

×22f

BDG

MacroscopicMaxwell

Maxwell-Lorentz forceon induced polarization

and current

MicroscopicMaxwell

++ --EE00(t)(t)

BB00 vv

)(rq(t)q=m)(r+q+(t)qm

1222

1211 +=fBrErfBrEr−×−−

×&&&

&&&

02q2q0constant2m

20 ≈−×

×

xBRExBxR

mω+(t)=m==q+

&&&

&

0020

2 /2m BER ×(t)ω

mq=&

)(±= xRr21

1,2

ClassicalClassical Abraham Abraham momentummomentum in in crossedcrossed EM EM fieldsfields

No No controversycontroversy existsexists in in microscopicmicroscopic descriptiondescriptionConsistent Consistent withwith Abrahams and Nelson versionAbrahams and Nelson version

gME(ω)

ε (ω )−1=−ωP

2

ω2ρcasi=

ℏc0

3∫0

∞dωω3 ω p

2

ω2 =∞Free Free electronelectron ((electricelectric dipoledipole))

Electric Electric quadrupolequadrupole

RizzoRizzo etaletal, 2003, 2003--2009, Babington & 2009, Babington & BAvTBAvT, EPJD 2011, EPJD 2011

Pcasi=ℏc0

3∫dr∫0

∞dωω3 g (ω )E0×B0=∞

magneticmagnetic dipoledipole

Pcasi=ℏ c0 g (ε (0 )−1)

he UV catastrophe he UV catastrophe isis real in real in macroscopicmacroscopic ‘‘descriptiondescription

aE0×B0 ?

DimensionalDimensional regularizationregularization for for objectobject of size a?of size a?BAvTBAvT EPJD 2009EPJD 2009

222222 )(2

)(2)(21),( BEBEBEBE ⋅+−+−=

ννL

( )( )ωω

BBBEEE

+=+=

0

0

000 3

404*0 BEBE

×−=× Kc

νπ00*

40 0 =×HE

πc

),()2(

10402

13 ΩΩ= ∫∫

∞ωρω

π πddK 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

nm/sec3.02m

00abr ≈

p

)EBα(ε=v

vFeigel=π4

hρλc

4 gEB≈0. 02 nm /sec

ClassicalClassical abrahamabrahamforceforce

vQED∝vabr×( α)2≈0 . 0002 nm /sec

SemiSemi--classicalclassical QED QED withwith cutcut--off off 0.1 nm (0.1 nm (FeigelFeigel) )

RigorousRigorous QED (QED (KawkaKawka, 2010,2012), 2010,2012)

E0=450 V/mm; B0=1 T

α (0)=0 . 22 10−40Cm2/V (16 . 6a03 )

ρ=0 .17 kg/m3 (room T )g=0. 017 10−22m/VT

Ex: Ex: HeliumHelium(SI units)

NFN

NF

at13

abr

32abr

10

107−

−

∝

⋅≈

nm/sec3.02m

00abr ≈

p

)EBα(ε=v

vFeigel=π4

hρλc

4 gEB≈0. 02 nm /sec

ClassicalClassical Abraham Abraham Force Force

vQED∝vabr×( α)2≈0 . 001 nm /sec

SemiSemi--classicalclassical QED QED withwith cutcut--off off 0.1 nm (0.1 nm (FeigelFeigel) )

RigorousRigorous QED (QED (KawkaKawka, 2010,2012), 2010,2012)

E0=450 V/mm; B0=1 T

α (0)=0 . 22 10−40Cm2/V (16 . 6a03 )

ρ=0 .17 kg/m3 (room T )g=0. 017 10−22m/VT

Ex: Ex: HeliumHelium(SI units)

Z?

NFN

NF

at13

abr

32abr

10

107−

−

∝

⋅≈

dpdt=α(0) dE

dt×B

P (ω )=P0+ α (0 )×E×B×ω×cosωt×n×LAcousticAcousticpressurepressure

Abraham forceAbraham force

RikkenRikken / Van / Van TiggelenTiggelen, PRL 2011, PRL 2011

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 nm/sec+V= 8 nm/sec+-- 0.80.8FeigelFeigel correction: 2 nm/secorrection: 2 nm/se

dpdt=ε0(εr−1) d

dt (E×B )

RikkenRikken / Van / Van TiggelenTiggelen, , submittedsubmitted 20122012

fE+fB=650 Hz

ε r=1. 7 105Y5V ceramic

rErBA ⋅−=×= 000 21 φ

( ) ( )

)(

)()(2

1)()(21

12210

22202

2

21101

1

rVe

eem

eem

H

+⋅+

+++−−=

rE

rArAprArAp

Casimir Casimir momentummomentum: : 1/61/6QED of QED of atomatom in in crossedcrossed fieldsfields

EE00

BB00+e+e --ee

( )21* ++∑ iii

iaaωh

Coulomb Gauge

)()(

20212

10111

rAvprAvp

emem−=+=

Casimir Casimir momentummomentum: : 2/62/6QED of QED of harmonicharmonic oscillatoroscillator in in crossedcrossed fieldsfields

0],[ =HK

rBPrBppK ×+=×++= 021021 21ˆ ee kin

Pseudo Pseudo momentummomentum isisconservedconserved

ConjugateConjugate momentamomenta≠≠ kinetickinetic momentummomentum

EE00

BB00+e+e --ee

Coulomb Gauge

GroundGround state changes due to state changes due to couplingcoupling withwith quantum vacuumquantum vacuum

0 .84α3

000 =|| ⟩⟨ A

α (ω=0, μ+ δμ )⟩⟨ 00 ψ|e|ψ

+δMAv

Casimir Casimir momentummomentum: : 3/63/6QED of QED of hydrogenhydrogen atomatom in in crossedcrossed fieldsfields

EE00

BB00

( )12ˆ210212211 +aa+e+)(e)(e+m+m=K iii

i

∗∑×− krBrArAvv h

21000000

20

000

0038

KKEBEB

vvvK

++)α(δμε+)α(ε+cE+Mδ+M=Ψ||Ψ

μ ×∂×

⟩⟨

+e+e --ee

No No multipolemultipole approximation inapproximation in cc+a)(i)(gk

k .exp gkkrgrA ∑∝

δM =δ (m1+ m2 ) δμ=δ( m1 m2

m1+ m2) δmi=4

3παℏ∫0

∞dk ℏ k

ℏ2 k 2/2mi+ ℏ kc

Casimir Casimir momentummomentum: : 4/64/6QED of QED of hydrogenhydrogen atomatom in in crossedcrossed fieldsfields

EE00

BB00

⎟⎠⎞

⎜⎝⎛×− ∗∑ 2

1ˆ210212211 +aa+e+)(e)(e+m+m=K iii

ikrBrArAvv h

( )R

μ

+++

)α(εμδ+)α(ε+cE

cE+Mδ+M=Ψ||Ψ

KKK

EBEB

vvvvK

21

000000

20

020

000

0035

38

×∂×

−⟩⟨

+e+e --ee

2idem2c

212

0

211 +

mv

v−

δM =δ (m1+ m2 ) δμ=δ( m1 m2

m1+ m2) δmi=4

3παℏ∫0

∞dk ℏ k

ℏ2 k 2/2mi+ ℏ kc

Casimir Casimir momentummomentum: : 5/65/6QED of QED of hydrogenhydrogen in in crossedcrossed fieldsfields

EE00

BB00+e+e --ee

Continuous spectrumassuming

plane waves for electrons

Quantum vacuum contribution:

Discrete Rydberg states

( )( ) A

n0n

α=+α)α(ε=

||nEE

n|rπε

e|µca

e=

KEB

rrEBK

22000

20

2

2200

22

001

0.210.00450.2080

01ˆ4

031

−×−

⟩⟨⋅−

⋅⟩⟨×− ∑h

( ) A

nn0

α+α)α(ε

||nEE

n||aπε

eα)α(ε

KEB

rrEBK

22000

020

22

0002

+0.1=0.0180.0790+=

01ˆ0427

10+=

×

⟩⟨⋅−

⋅⟩⟨× ∑

Casimir Casimir momentummomentum: : 6/66/6QED of QED of hydrogenhydrogen in in crossedcrossed fieldsfields

Ae

EE20e

R

Mmα

|)(p|Mc

e=

K

xBK

2

02

2

002m

∝

⟩×⟨−

EE00

BB00+e+e --ee

Relativistic contribution:

Casimir Casimir momentummomentum: : 6/66/6QED of QED of hydrogenhydrogen in in crossedcrossed fieldsfields

( ) )O(α+α+cE+= A

3220

0kin 0.11 KvKK −

EE00

BB00+e+e --ee

Casimir Casimir momentummomentum of H of H atomatom existsexistsand and slightlyslightly reducesreduces the the classicalclassical Abraham Abraham momentummomentum

BaVTBaVT, , KawkaKawka, , RikkenRikken, , submittedsubmitted to EPJDto EPJD

000 0 EBK ×)α(ε=A

•• ClassicalClassical Abraham force , Abraham force , linearlinear in Ein E0 0 and Band B0 0 , , isis observedobserved for for neutralneutral atomsatomsand for and for strongstrong dielectricsdielectrics

•• QED contribution by QED contribution by FeigelFeigel isis not not observedobserved•• UV UV divergenciesdivergencies disappeardisappear in mass in mass

renormalizationrenormalization or cancel. or cancel. NeedNeed to go to go beyondbeyond multipolemultipole approximationapproximation

•• Quantum vacuum Quantum vacuum contributescontributes to to Abraham Abraham momentummomentum in in orderorder --(1/137)^2(1/137)^2

Will Will thisthis bebe --(Z/137)^2 for Z > 1??(Z/137)^2 for Z > 1??

SUMMARYSUMMARYCasimir Casimir momentummomentum in in crossedcrossed E,BE,B

A A CasimirCasimir momentum with only magnetic field?momentum with only magnetic field?

<<EE x x BB> = gB> = gB0 0 ??

•• ClassicallyClassically no no equivalentequivalent Abraham version in charge Abraham version in charge neutralneutral systemssystems

•• g must g must bebe a pseudo a pseudo scalarscalarmedium must medium must bebe chiral (on chiral (on nanoscalenanoscale))

•• DescribeDescribe chiralitychirality microscopicallymicroscopically, , not not phenomenologicallyphenomenologically

via via ««magnetomagneto--chiral chiral «« index of index of refractionrefraction ((ΔΔn=g Bn=g B00.k.k))•• WouldWould separateseparate enantiomersenantiomers usingusing magneticmagnetic fieldsfields

= Pasteurs = Pasteurs dreamdream !!•• Medium must Medium must bebe magneticmagnetic sincesince <E x H> =0<E x H> =0

BB00

α (ω ,σ )=4πc0

2

ω02

γω2−ω0

2+ iσ VB+ i γω0

hiral hiral geometrygeometry withwith electricelectric polarizabilitiespolarizabilities withwith Zeeman Zeeman splittingsplitting

εε

εε εεεε

PasteurPasteur’’s dream with a s dream with a CasimirCasimir momentum P= gmomentum P= gBB0 0 ? ?

000000 =×∝×⇒= ∫∫ HErBErHB ddμ

BB00

χ (ω ,σ)= χ (0)ω0

2

ω2−ω02+ iσ VB+ i γω

Na Na TetraederTetraeder L=10 nm L=10 nm g/m = 1 nm/sec/Tg/m = 1 nm/sec/TBabington , Babington , BaVTBaVT, EPL 2011, EPL 2011

µµµµµµ

µµ

iral iral geometrygeometry withwith magneticmagnetic polarizabilitiespolarizabilities withwith Zeeman Zeeman splittinsplittin

A A CasimirCasimir momentum P= g Bmomentum P= g B0 0 ? Pasteur? Pasteur’’s s dream!dream!

000 =×∫ HErd

000 BBEr gd =×∫)0(χ

Thank you !