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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 !