Casimir momentum in crossed electromagnetic fields. QED correction to Abraham force?

Transcription

1 Casimir momentum in crossed electromagnetic fields. QED correction to Abraham force? PHOTONIMPULS Bart van Tiggelen and Geert Rikken Sébastien Kawka (Ph.D Grenoble ENS Pisa) James Babington (postdoc( ANR Grenoble) Casimir Workshop ; Leiden, March

2 Casimir energy. Isotropic radiation with power spectrum ω 3 is Lorentz-invariant (Einstein, 97); ˆ dω dk ω ω, = [ ρ,] 3 casi c h hk is an invariant four vector. Van der Waals force /r 6 (London, 93). Relation to Cosmological constant (Pauli, 934, Davies, 984) 3. Casimir Polder Force /r 7 (947) 4. Attraction between metallic plates (Casimir, 948) 5. Lifshitz theory for dielectric media (Lifshitz( Lifshitz,, 956, Dzyalonishiniskii 96). Observation of Casimir effect (Sparnaay,, 958, Lamoureux (5%), 997), Chan etal,, (%), ). Stability of the electron (Casimir, 956, Boyer, 968). Unruh effect & Hawking radiation (Hawking( 974, Unruh 976) 3. Bag model for Hadrons (Jaffe etal,, 974) 4. Sign of the Cosmological constant (Weinberg, 983) 5. Sonoluminescence (Schwinger, 993, Eberlein,, 996) 6. Quantum friction and sheering the quantum vacuum (Pendry( Pendry,, 998) 7. Casimir momentum in magneto-electric electric media (Feigel( Feigel,, 4)

3 The Casimir effect. ħω L ( ) E (L )= ħ A L 3 F(L) = E = L ha L ( L) Lˆ 3 4 Negative pressure No momentum exchange between matter and radiation

4 UV catastrophe in sonoluminescence (> 934) Schwinger (993) ΔE ( bubble)= d 3 r { d 3 k ħ ω k (bubble in water) d 3 k ħ ω k (water no bubble) } ω 3 4 ha c 3 c ε MeV cut-off in the UV? Dimensional regularisation? ΔE 3 bubble ) = 536π. ev ( ε ) c ( a

5 «Momentum from Nothing» E B ε,μ,g,g P=mv hω,k' hω, k

6 D H Fresnel dispersion law ω det ε k c Bi-anisotropic Media ( ω ) = ε( ω) E( ω ) + g( ω) B( ω) ( ) T ( ) ( ) ω = g ω E( ω) + μ ω B( ω) + kk ω g c ω c ( ε k) + ( ε k) g * = g ij (ω )=iω gδ ij Rotatory power g ij (ω )=( ε)ε ijl v l Fizeau effect c g ij ( ) ( ω = g E B B E ) Magneto-electric electric birefringence i j i j v k y E x B k x

7 phenomenological continuum theory t ( ρ v+ε E B) = T T ij = 8π ε E + μ B δ ij 4π ε E E i j + μ B B i j Observed in X-ray X E B 4πc c c d 3 k k ( ω) B [ ε ( ω) ] 3 d hωk hω k g E hω 4 c = π c v c v = ρ casi g E B Photonic momentum in dielectric media? classical «Abraham» contribution already controversial UV catastrophe of vacuum energy? Lorentz invariance of quantum vacuum? Inertia of quantum vacuum?

8 The Abraham Force Macroscopic Maxwell G Τ= f t M + G M = D B f= E ε H μ Minkowski

9 G G A = c N = G =ε μ E S =ε E H B The Abraham Force G T= f ε (ε / μ ) ( E B) t A+ r r t G T= f ε (ε ) ( E B) t A+ r t... B 5 [ E (t)] B (t) E (t) [ (t)] = f t t Abraham Nelson exp ( ε ) V E (t) B (t) F = t Peierls Walker & Walker, Nature 976 theo Abraham momentum = kinetic momentum, Minkowski momentum = conjugate momentum Barnett (PRL ): Nelson momentum = pseudo momentum Nelson (PRA 99)

10 The Abraham Force (our( version) Macroscopic Maxwell Maxwell-Lorentz force on induced polarization and current G+ T= f t t ρv + U= f + ( P B) t G= D B f= E ε H μ + Microscopic Maxwell t ( ρv+ε E B) + T =

11 The Abraham Force (our( version) Macroscopic Maxwell Maxwell-Lorentz force on induced polarization and current G + T f ( P B) t t = ρv + U= f + ( P B) t t G= D B f= E ε H μ + Microscopic Maxwell t ( ρv+ε E B) + T =

12 Classical Abraham momentum in crossed EM fields m&& r m&& r = + qe(t)+q r& = qe (t) qr& E (t) B B+ f(r B f(r ) ) + - v r, = R± ( x) mr& +qx B= constant= m&& x= qe(t)+ qr& B mω x / mr & q m = E(t) B ω No controversy exists in microscopic description Consistent with Abrahams and Nelson version

13 e UV catastrophe is real in macroscopic description Free electron (electric dipole) g ME (ω) ε(ω) = ω P ω ρ casi = ħ c 3 3 ω p dωω ω = magnetic dipole P casi = ħ c 3 dr dωω 3 g (ω)e B = Electric quadrupole P casi = ħ c g (ε( ) ) a Rizzo etal,, 3-9, 9, Babington & BAvT,, EPJD E B? Dimensional Dimensional regularization for object of size a? BAvT EPJD 9

14 Casimir momentum,, if infinite, is Lorentz invariant L ν ( ( E, B) = E B + E B + E B ( ) ν ( ) ) Fluctuation- Dissipation E = B = E( r, ω) E i * j E B + E + B ( r', ω') ( ω) ( ω) Bi-anisotropic Lorentz-invariant vacuum = hω ImG ( r, r', ω) πδ( ω ω') ij c E* H = 4π Zero energy flow E* B 4 = νk E B 4πc K 3 infinite momentum density = ω Ω ρ( ω, Ω) 3 (π ) h d d 4π Lorentz scalar

15 Ex: Helium E=45 V/mm; B= T α ()=. 4 Cm /V (6.6a 3 ) ρ=.7 kg/m 3 (room T ) g=.7 m/vt (SI units) v abr = ε v Feigel = π 4 α( )EB m p h ρλ c 4.3 nm/sec geb. nm /sec Classical abraham force F N abr at 7 F abr 3 3 N Semi-classical QED with cut-off. nm (Feigel( Feigel) N v QED v abr ( α). nm /sec Rigorous QED (Kawka( Kawka,,,)

16 Ex: Helium E=45 V/mm; B= T α ()=. 4 Cm /V (6.6a 3 ) ρ=.7 kg/m 3 (room T ) g=.7 m/vt (SI units) v abr = ε v Feigel = π 4 α( )EB m p h ρλ c 4.3 nm/sec geb. nm /sec Classical Abraham Force F N abr at 7 F abr 3 3 N Semi-classical QED with cut-off. nm (Feigel( Feigel) N v QED v abr ( Z? α). nm/sec Rigorous QED (Kawka( Kawka,,,)

17 Acoustic pressure dp de =α() dt dt B Abraham force P(ω )=P + α( ) E B ω cosωt n L V= 8 nm/sec+-.8 Feigel correction: nm/se P/(EB) E=45 V/mm; B= T; f= 7.6 khz α() () ikken / Van Tiggelen,, PRL

18 dp dt =ε (ε r ) d dt ( E B) ε r =.7 5 Y5V ceramic fe+fb=65 Hz ikken / Van Tiggelen, submitted

19 Casimir momentum: /6 QED of atom in crossed fields +e -e E B A = B r φ = E r Coulomb Gauge H = m ( p ea ( r ) ea( r )) + ( p + ea ( r ) + ea( r )) + ee r + V ( r m ) + hωi i i i ( ) * a a +

20 Casimir momentum: /6 QED of harmonic oscillator in crossed fields E +e -e B Conjugate momenta kinetic momentum Pseudo momentum is conserved p p = mv = m v + ea ea ( r ( r ) K ˆ = p + p+ eb r = Pkin + eb r ) [ K, H ] = Coulomb Gauge

21 Ground state changes due to coupling with quantum vacuum δmv+ ψ ea ψ.84α 3 α(ω=, μ+ δμ) A =

22 Casimir momentum: 3/6 QED of hydrogen atom in crossed fields +e -e E B Kˆ = m ( a a ) v + mv + ea( r ) ea( r )+ eb r + hk i i i + i No multipole approximation in Ψ + ε K Ψ α( ) B = Mv +δ M E +ε A( r) v+ 8 3 E c v δμ μ α( ) B E gk + K gk exp(ikr)a +c. c + K gk δm =δ (m + m ) δμ=δ( m m ) δm m + m i = 4 3π αħ dk ħ k ħ k /m i +ħkc

23 Casimir momentum: 4/6 QED of hydrogen atom in crossed fields +e -e E Kˆ = m v + mv + ea( r ) ea( r )+ eb r + hk i ai ai + i m v v idem c + Ψ +ε + K K Ψ α( ) B + K = Mv +δ E + K R +δ M ( μ) 8 v+ 3 ε μ E c B v α( ) B δm =δ (m + m ) δμ=δ( m m ) δm m + m i = 4 3π αħ dk 5 3 E c E v ħ k ħ k /m i +ħkc

24 Casimir momentum: 5/6 QED of hydrogen in crossed fields +e -e E Quantum vacuum contribution: B K = ε = B E α( ) B 3 E e h a c µ α n e 4πε rˆ n r ( E E ) (.8+.45) =.α K A n n r K = + ε α( ) B = + ε α( ) B E e E α rˆ n 7 4πε a α (.79+.8) = +.α K A n E n E n r Discrete Rydberg states Continuous spectrum assuming plane waves for electrons

25 Casimir momentum: 6/6 QED of hydrogen in crossed fields E +e -e B Relativistic contribution: K R α e = m Mc me M K e A E p ( B x ) E

26 Casimir momentum: 6/6 QED of hydrogen in crossed fields E +e -e B K E = K + v+. kin A c ( ) α K +O(α 3 ) K ) A =ε α( B E Casimir momentum of H atom exists and slightly reduces the classical Abraham momentum BaVT, Kawka, Rikken, submitted to EPJD

27 SUMMARY Casimir momentum in crossed E,B Classical Abraham force, linear in E and B, is observed for neutral atoms and for strong dielectrics QED contribution by Feigel is not observed UV divergencies disappear in mass renormalization or cancel. Need to go beyond multipole approximation Quantum vacuum contributes to Abraham momentum in order -(/37)^ Will this be -(Z/37)^ for Z >??

28 A Casimir momentum with only magnetic field? <E x B> > = gb? Classically no equivalent Abraham version in charge neutral systems g must be a pseudo scalar medium must be chiral (on nanoscale) Describe chirality microscopically, not phenomenologically via «magneto-chiral «index of refraction (Δn=g B.k) Would separate enantiomers using magnetic fields = Pasteurs dream! Medium must be magnetic since <E x H> =

29 Pasteur s s dream with a Casimir momentum P= g B? ε ε ε ε B iral geometry with electric polarizabilities with Zeeman splitting α (ω, σ )= 4πc ω γ ω ω + iσ VB+ i γω B = μ H dre B dre H =

30 A Casimir momentum P= g B? Pasteur s dream! µ µ µ µ χ (ω, σ)= χ () B ral geometry with magnetic polarizabilities with Zeeman splittin dre H = d re B = gb ω ω ω + iσ VB+ i γω χ() Na Tetraeder L= nm g/m = nm/sec/t Babington, BaVT,, EPL

31 Thank you!