Casimir and Casimir-Polder forces in chiral and non-reciprocal media

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1 Casimir and Casimir-Polder forces in chiral and non-reciprocal media Stefan Yoshi Buhmann, Stefan Scheel, David Butcher Quantum Optics and Laser Science Blackett Laboratory, Imperial College London, UK Electronic address: Homepage: sbuhmann

2 Outline Introduction: Dispersion forces and unusual materials Dispersion forces Repulsive forces? Unusual material properties QED in nonlocal, nonreciprocal media Ohm s law Field quantisation QED in local bianisotropic media Constitutive relations Commutation relations Duality Casimir Polder potential of chiral molecules Perturbation theory Perfect mirror

3 Introduction: Dispersion forces and unusual materials

4 Dispersion forces Dispersion force: Effective electromagnetic force between neutral, polarizable objects F. London, Z. Phys. 63, 245 (1930),

5 Dispersion forces Dispersion force: Effective electromagnetic force between neutral, polarizable objects London (1930): Dispersion force due to charge fluctuations ˆd = 0 _ _ + _ +_ + F. London, Z. Phys. 63, 245 (1930),

6 Dispersion forces Dispersion force: Effective electromagnetic force between neutral, polarizable objects London (1930): Dispersion force due to charge fluctuations ˆd = 0, but ˆd 2 = 0 Dipole-dipole force _ + _ + + _ + _ F. London, Z. Phys. 63, 245 (1930), H.B.G. Casimir, Proc. K. Ned. Akad. Wet. 51, 793 (1948)

7 Dispersion forces Dispersion force: Effective electromagnetic force between neutral, polarizable objects London (1930): Dispersion force due to charge fluctuations ˆd = 0, but ˆd 2 = 0 Dipole-dipole force Casimir (1948): Dispersion force due to field fluctuations Ê = 0 F. London, Z. Phys. 63, 245 (1930), H.B.G. Casimir, Proc. K. Ned. Akad. Wet. 51, 793 (1948)

8 Dispersion forces Dispersion force: Effective electromagnetic force between neutral, polarizable objects London (1930): Dispersion force due to charge fluctuations ˆd = 0, but ˆd 2 = 0 Dipole-dipole force Casimir (1948): Dispersion force due to field fluctuations Ê = 0, but Ê 2 = 0 Radiation pressure force F. London, Z. Phys. 63, 245 (1930), H.B.G. Casimir, Proc. K. Ned. Akad. Wet. 51, 793 (1948)

9 Dispersion forces Dispersion force: Effective electromagnetic force between neutral, polarizable objects London (1930): Dispersion force due to charge fluctuations ˆd = 0, but ˆd 2 = 0 Dipole-dipole force Casimir (1948): Dispersion force due to field fluctuations Ê = 0, but Ê 2 = 0 Radiation pressure force Physical origin: Correlated charge and/or field fluctuations F. London, Z. Phys. 63, 245 (1930), H.B.G. Casimir, Proc. K. Ned. Akad. Wet. 51, 793 (1948)

10 Repulsive dispersion forces? Direction of ground-state dispersion forces between electric objects in vacuum typically attractive!

11 Repulsive dispersion forces? Direction of ground-state dispersion forces between electric objects in vacuum typically attractive! Repulsive dispersion forces: Objects in a medium ε < ε < ε J.N. Munday, F. Capasso, V.A. Parsegian, Nature 457, 170 (2009);

12 Repulsive dispersion forces? Direction of ground-state dispersion forces between electric objects in vacuum typically attractive! Repulsive dispersion forces: Objects in a medium Excited or amplifying objects * * * * * * * * * J.N. Munday, F. Capasso, V.A. Parsegian, Nature 457, 170 (2009); A. Sambale, S.Y.B., D.-G. Welsch, T.D. Ho, PRA 80, (R) (2009)

13 Repulsive dispersion forces? Direction of ground-state dispersion forces between electric objects in vacuum typically attractive! Repulsive dispersion forces: Objects in a medium Excited or amplifying objects Magnetoelectric objects ε µ J.N. Munday, F. Capasso, V.A. Parsegian, Nature 457, 170 (2009); A. Sambale, S.Y.B., D.-G. Welsch, T.D. Ho, PRA 80, (R) (2009)

14 Electric vs magnetic objects Distance Retarded Nonretarded Objects e e e m e e e m r AB 1 r 8 AB + 1 r 8 AB 1 r 7 AB + 1 r 5 AB r A 1 r 8 A + 1 r 8 A 1 r 7 A + 1 r 5 A z A 1 z 6 A + 1 z 6 A 1 z 5 A + 1 z 3 A z A 1 z 5 A + 1 z 5 A 1 z 4 A + 1 z 2 A z 1 z z 4 1 z z

15 Electric vs magnetic objects Distance Retarded Nonretarded Objects e e e m e e e m Dual objects m m m e m m m e r AB r A z A z A 1 r 8 AB 1 r 8 A 1 z 6 A 1 z 5 A + 1 rab ra za za 5 1 r 7 AB 1 r 7 A 1 z 5 A 1 z 4 A + 1 rab ra za za 2 z 1 z z 4 1 z z Duality invariance: α β/c 2, ε µ (e m, m e) S.Y.B., S. Scheel, Phys. Rev. Lett. 102, (2009)

16 Electric vs magnetic objects Distance Retarded Nonretarded Objects e e e m e e e m Dual objects m m m e m m m e r AB r A z A z A 1 r 8 AB 1 r 8 A 1 z 6 A 1 z 5 A + 1 rab ra za za 5 1 r 7 AB 1 r 7 A 1 z 5 A 1 z 4 A + 1 rab ra za za 2 z 1 z z 4 1 z z Duality invariance: α β/c 2, ε µ (e m, m e) Opposites repel! S.Y.B., S. Scheel, Phys. Rev. Lett. 102, (2009)

17 Unusual material properties Simplest magnetoelectric: permittvity ε(r, ω), permeability µ(r, ω)

18 Unusual material properties Simplest magnetoelectric: permittvity ε(r, ω), permeability µ(r, ω) Generalisations: nonlocal media: ε(r, r, ω) ( metals)

19 Unusual material properties Simplest magnetoelectric: permittvity ε(r, ω), permeability µ(r, ω) Generalisations: nonlocal media: ε(r, r, ω) ( metals) anisotropic media: ε(r, ω) ( crystals, metamaterials, birefringence)

20 Unusual material properties Simplest magnetoelectric: permittvity ε(r, ω), permeability µ(r, ω) Generalisations: nonlocal media: ε(r, r, ω) ( metals) anisotropic media: ε(r, ω) ( crystals, metamaterials, birefringence) bi(an)isotropic media: cross-susceptibilitities ξ(r, ω), ζ(r, ω) ( chiral media)

21 Unusual material properties Simplest magnetoelectric: permittvity ε(r, ω), permeability µ(r, ω) Generalisations: nonlocal media: ε(r, r, ω) ( metals) anisotropic media: ε(r, ω) ( crystals, metamaterials, birefringence) bi(an)isotropic media: cross-susceptibilitities ξ(r, ω), ζ(r, ω) ( chiral media) nonreciprocal media: violation of Onsager theorem ( external magnetic fields, spontaneous symmetry breaking, moving media)

22 Onsager theorem Onsager reciprocity: current j along e 1 at r 1 causes field E along e 2 at r 2 j r 1 e 1 e E r 2 2

23 Onsager theorem Onsager reciprocity: current j along e 1 at r 1 causes field E along e 2 at r 2 current j along e 2 at r 2 causes field E along e 1 at r 1 E r 1 e 1 e j r 2 2

24 Onsager theorem Onsager reciprocity: current j along e 1 at r 1 causes field E along e 2 at r 2 current j along e 2 at r 2 causes field E along e 1 at r 1 Formally: E(r, ω) = iµ 0 ω d 3 r G(r, r, ω) j(r, ω), G T (r, r, ω) = G(r, r, ω)

25 QED in nonlocal, nonreciprocal media

26 Maxwell equations + Ohm s law Maxwell equations: ˆB = 0, Ê + ˆB = 0, Ê = ˆρ in ε 0, ˆB 1 c 2 Ê = µ 0 ĵ in

27 Maxwell equations + Ohm s law Maxwell equations: ˆB = 0, Ê + ˆB = 0, Ê = ˆρ in ε 0, Linear constitutive relations: ĵ in (r, t) = dτ ˆB 1 c 2 Ê = µ 0 ĵ in d 3 r Q(r, r, τ) Ê(r, t τ) + ĵ N (r, t), Q(r, r, τ)=0 for cτ < r r (causality)

28 Maxwell equations + Ohm s law Maxwell equations: ˆB = 0, Ê + ˆB = 0, Ê = ˆρ in ε 0, Linear constitutive relations: ĵ in (r, t) = dτ ˆB 1 c 2 Ê = µ 0 ĵ in d 3 r Q(r, r, τ) Ê(r, t τ) + ĵ N (r, t), Q(r, r, τ)=0 for cτ < r r (causality) Frequency components: ĵ in (r, ω) = d 3 r Q(r, r, ω) Ê(r, ω) + ĵ N (r, ω)

29 Maxwell equations + Ohm s law Maxwell equations: ˆB = 0, Ê + ˆB = 0, Ê = ˆρ in ε 0, Linear constitutive relations: ĵ in (r, t) = dτ ˆB 1 c 2 Ê = µ 0 ĵ in d 3 r Q(r, r, τ) Ê(r, t τ) + ĵ N (r, t), Q(r, r, τ)=0 for cτ < r r (causality) Frequency components: ĵ in (r, ω) = d 3 r Q(r, r, ω) Ê(r, ω) + ĵ N (r, ω) Solution: Ê(r, ω) = iµ 0 ω d 3 r G(r, r, ω) ĵ N (r, ω)

30 Green tensor Helmholtz equation: [ ω2 c 2 ] G(r, r, ω) iµ 0 ω d 3 sq(r, s, ω) G(s, r, ω) = δ(r r )

31 Green tensor Helmholtz equation: [ ω2 c 2 ] G(r, r, ω) iµ 0 ω d 3 sq(r, s, ω) G(s, r, ω) = δ(r r ) General properties: Reciprocal media [Q T (r, r, ω)=q(r, r, ω)] Schwarz reflection principle: G (r, r, ω)=g(r, r, ω ) Onsager reciprocity: G T (r, r, ω) = G(r, r, ω) Integral relation: µ 0 ω d 3 s d 3 s G(r, s, ω) Re Q(s, s, ω) G (s, r, ω) = Im G(r, r, ω)

32 Green tensor General properties: Nonreciprocal media [Q T (r, r, ω) Q(r, r, ω)] Schwarz reflection principle: G (r, r, ω)=g(r, r, ω ) Onsager reciprocity: G T (r, r, ω) G(r, r, ω) Integral relation: µ 0 ω d 3 s d 3 s G(r, s, ω) ReQ(s, s, ω) G (s, r, ω) = ImG(r, r, ω), ReT(r, r ) = 1 2[ T(r, r ) + T (r, r) ], ImT(r, r ) = 1 2i [ T(r, r ) T (r, r) ]

33 Commutation relations: [ ĵ N (r, ω), ĵ ] N (r, ω ) Field quantisation = ω π ReQ(r, r, ω)δ(ω ω )

34 Commutation relations: [ ĵ N (r, ω), ĵ ] N (r, ω ) Field quantisation = ω π ReQ(r, r, ω)δ(ω ω ) Fundamental variables: [ ˆf(r, ω), ˆf (r, ω ) ] = δ(r r )δ(ω ω ) with ĵ N (r, ω) = ω π d 3 r R(r, r, ω) ˆf(r, ω), d 3 s R(r, s, ω) R (r, s, ω) = ReQ(r, r, ω)

35 Commutation relations: [ ĵ N (r, ω), ĵ ] N (r, ω ) Field quantisation = ω π ReQ(r, r, ω)δ(ω ω ) Fundamental variables: [ ˆf(r, ω), ˆf (r, ω ) ] = δ(r r )δ(ω ω ) with ĵ N (r, ω) = ω π d 3 r R(r, r, ω) ˆf(r, ω), d 3 s R(r, s, ω) R (r, s, ω) = ReQ(r, r, ω) Medium field Hamiltonian: Ĥ F = d 3 r 0 dω ω ˆf (r, ω) ˆf(r, ω)

36 Commutation relations: [ ĵ N (r, ω), ĵ ] N (r, ω ) Field quantisation = ω π ReQ(r, r, ω)δ(ω ω ) Fundamental variables: [ ˆf(r, ω), ˆf (r, ω ) ] = δ(r r )δ(ω ω ) with ĵ N (r, ω) = ω π d 3 r R(r, r, ω) ˆf(r, ω), d 3 s R(r, s, ω) R (r, s, ω) = ReQ(r, r, ω) Medium field Hamiltonian: Ĥ F = d 3 r 0 dω ω ˆf (r, ω) ˆf(r, ω) Consistency: Maxwell equations free-space QED fluctuation dissipation theorem

37 Local bianisotropic media

38 Local bianisotropic media Maxwell equations: ˆB = 0, Ê + iω ˆB = 0, ˆD = 0, Ĥ + iω ˆD = 0

39 Local bianisotropic media Maxwell equations: ˆB = 0, Ê + iω ˆB = 0, ˆD = 0, Ĥ + iω ˆD = 0 Constitutive relations: ˆD =ε 0 Ê+ ˆP, Ĥ = ˆB/µ 0 ˆM ˆP = ε 0 (ε ξ µ 1 ζ I) Ê + 1 Z 0 ξ µ 1 ˆB + ˆP N, ˆM = 1 Z 0 µ 1 ζ Ê + 1 µ 0 (I µ 1 ) ˆB + ˆM N

40 Local bianisotropic media Maxwell equations: ˆB = 0, Ê + iω ˆB = 0, ˆD = 0, Ĥ + iω ˆD = 0 Constitutive relations: ˆD =ε 0 Ê+ ˆP, Ĥ = ˆB/µ 0 ˆM ˆP = ε 0 (ε ξ µ 1 ζ I) Ê + 1 Z 0 ξ µ 1 ˆB + ˆP N, ˆM = 1 Z 0 µ 1 ζ Ê + 1 µ 0 (I µ 1 ) ˆB + ˆM N Alternative form: ˆD = ε 0 ε Ê + 1 c ξ Ĥ + ˆP N + 1 c ξ ˆM N, ˆB = 1 c ζ Ê + µ 0 µ Ĥ + µ 0 µ ˆM N

41 Relation to conductivity Conductivity: Q(r, r, ω) = 1 iµ 0 ω (µ 1 I) δ(r r ) + 1 Z 0 µ 1 ζ δ(r r ) + 1 Z 0 ξ µ 1 δ(r r ) iε 0 ω(ε ξ µ 1 ζ I) δ(r r ) Noise terms: ĵ N = iω ˆP N + ˆM N Green tensor: [ µ 1 iω c µ 1 ζ + iω c ξ µ 1 ] ω2 c 2 (ε ξ µ 1 ζ) G(r, r, ω) = δ(r r )

42 Commutation relations [ ˆP N (r, ω), ˆP N (r, ω ) ] = ε 0 π Im(ε ξ µ 1 ζ) δ(r r )δ(ω ω ), [ ˆM N (r, ω), ˆM N (r, ω ) ] = πµ 0 Imµ 1 δ(r r )δ(ω ω ), [ ˆP N (r, ω), ˆM N (r, ω ) ] = 2πiZ 0 (ζ µ 1 ξ µ 1 ) δ(r r )δ(ω ω )

43 Commutation relations [ ˆP N (r, ω), ˆP N (r, ω ) ] = ε 0 π Im(ε ξ µ 1 ζ) δ(r r )δ(ω ω ), [ ˆM N (r, ω), ˆM N (r, ω ) ] = πµ 0 Imµ 1 δ(r r )δ(ω ω ), [ ˆP N (r, ω), ˆM N (r, ω ) ] = 2πiZ 0 (ζ µ 1 ξ µ 1 ) δ(r r )δ(ω ω ) Basic variables: [ ˆf λ (r, ω), ˆf λ (r, ω ) ] = δ λλ δ(r r )δ(ω ω ) with R R = ( ) ˆP N (r, ω) ˆM N (r, ω) = ε 0 Im(ε ξ µ 1 ζ) µ 1 ζ µ 1 ξ 2πiZ 0 ( ) ˆf π R e (r, ω) ˆf m (r, ω) ζ µ 1 ξ µ 1 2iZ 0 Imµ 1 µ 0

44 Duality invariance Maxwell equations in dual-pair notation: ( ) ( ) Z0 ˆD 0 =, ˆB 0 ( Z0 ˆD ˆB ( Ê ) ) iω Z 0 Ĥ = 1 ( ε ξ c ζ µ ( )( Ê Z 0 Ĥ )( Z0 ˆD ˆB ) + ) ( µ = ( 0 0 ), )( Z0 ˆP N µ 0 ˆM N )

45 Duality invariance Maxwell equations in dual-pair notation: ( ) ( ) Z0 ˆD 0 =, ˆB 0 ( Z0 ˆD ˆB ( Ê ) ) iω Z 0 Ĥ = 1 ( ε ξ c ζ µ ( )( Ê Z 0 Ĥ )( Z0 ˆD ˆB ) + ) ( µ = ( 0 0 ), )( Z0 ˆP N µ 0 ˆM N ) Duality transformation: ( ) ( x x = D(θ) y y ), D(θ) = ( cos θ sin θ sin θ cos θ )

46 Transformation of response functions ε ξ ζ µ = cos 2 θ sin θ cos θ sin θ cos θ sin 2 θ sin θ cos θ cos 2 θ sin 2 θ sin θ cos θ sin θ cos θ sin 2 θ cos 2 θ sin θ cos θ sin 2 θ sin θ cos θ sin θ cos θ cos 2 θ ε ξ ζ µ

47 Transformation of response functions ε ξ ζ µ = cos 2 θ sin θ cos θ sin θ cos θ sin 2 θ sin θ cos θ cos 2 θ sin 2 θ sin θ cos θ sin θ cos θ sin 2 θ cos 2 θ sin θ cos θ sin 2 θ sin θ cos θ sin θ cos θ cos 2 θ ε ξ ζ µ Discrete vs continuous symmetry: Isotropic media (ε=εi, µ=µi, ξ =ζ =0): discrete Biisotropic media (ε=εi, µ=µi, ξ =ξi, ζ =ζi): continuous Anisotropic media (ξ = ζ = 0): discrete Reciprocal media (ε T =ε, ξ T = ζ, µ T =µ): discrete

48 Casimir Polder potential of chiral molecules

49 Perturbation theory Idea: System in ground state g = 0 {0} Interaction Ĥ AF = ˆd Ê(r A ) ˆm ˆB(r A ) energy shift E Casimir-Polder potential U(r A )= E(r A ) 2 nd -order perturbation theory: E = ψ E g E ψ g Ĥ AF ψ 2 H.B.G. Casimir, D. Polder, PRA (1948)

50 Perturbation theory Idea: System in ground state g = 0 {0} Interaction Ĥ AF = ˆd Ê(r A ) ˆm ˆB(r A ) energy shift E Casimir-Polder potential U(r A )= E(r A ) U(r A ) = µ 0 π 0 dξ ξχ(iξ) Tr[ G (1) (r A, r A, iξ) ] Chiral polarisability: χ(ω) = 2 3 lim ɛ 0 k iωr 0k ω 2 k ω2 iωɛ, R 0k = Im(d 0k m k0 ) Scattering Green tensor: G (1) (r, r, ω) H.B.G. Casimir, D. Polder, PRA (1948)

51 Example: Atom near perfect mirror Curie principle: need chiral body to detect chiral molecule Perfect chiral mirror: r s p = 1, r p s = ±1

52 Example: Atom near perfect mirror Curie principle: need chiral body to detect chiral molecule Perfect chiral mirror: r s p = 1, r p s = ±1 Chiral Casimir Polder potential: U(z A ) = ± 8π 2 ε 0 za 3 0 dξ χ(iξ) c ( e 2ξz A/c ξz ) A c

53 Example: Atom near perfect mirror Curie principle: need chiral body to detect chiral molecule Perfect chiral mirror: r s p = 1, r p s = ±1 Chiral Casimir Polder potential: U(z A ) = ± 8π 2 ε 0 za 3 0 dξ χ(iξ) c ( e 2ξz A/c ξz ) A c Long-/short-distance limits: U(z A ) = 1 16π 2 ε 0 za 5 1 ± 12π 2 ε 0 cza 3 R 0k k ωk 2 k, R 0k ln(ω k z A /c)

54 Summary Macroscopic quantum electrodynamics Nonlocal, nonreciprocal media: conductivity Local bianisotropic media: cross-susceptibilities Duality: continuous (if Onsager violation allowed) Chiral Casimir-Polder potential Perfectly chiral plate: left- vs right-handed molecules Outlook Local duality: force maximal for opposite dual pairs? Nonreciprocal media: CP-violating systems? Moving systems: symmetrise Green tensor? Further Reading: S.Y.B., Dispersion Forces I + II, Springer Tracts in Modern Physics, to appear spring 2012

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