Photon angular momentum and geometric gauge Margaret Hawton, Lakehead University Thunder Bay, Ontario, Canada William Baylis, U. of Windsor, Canada Outline photon r operators and their localized eigenvectors leads to transverse bases and geometric gauge transformations, then to orbital angular momentum of the bases, connection with optical beams conclude Notation: momentum space ˆ z pz or z ˆ p ˆ ˆ ˆ ˆ ˆ ˆ φ ~ z p; θ φ p Use CP basis vectors: ˆ q θ py e(0) 1 2 ˆ θ i φ ˆ f ˆ φ e0 p ˆ px is the p-space gradient. (Will use and when in r-space.) Is the position of the photon an observable? In quantum mechanics, any observable requires a Hermitian operator 1948, Pryce obtained a photon position operator, a a pS ˆ rP i p p p • a =1/2 for F=E+icB ~ p1/2 as in QED to normalize • last term maintains transversality of rP(F) • but the components of rP don’t commute! • thus “the photon is not localizable”? A photon is asymptotically localizable 1) Adlard, Pike, Sarkar, PRL 79, 1585 (1997) ˆ A ~ q r a a is an arbitrarily large integer; power law 2) Bialynicki-Birula, PRL 80, 5247 (1998) ˆ Z ~ m exp (r / r0 ) 1 to satisfy Paley-Wiener theorem, r0 arbitrarily small; exponentially localized Is there a photon position operator with commuting components and exactly localized eigenvectors? It has been claimed that since the early day of quantum mechanics that there is not. Surprisingly, we found a family of r operators, Hawton, Phys. Rev. A 59, 954 (1999). Hawton and Baylis, Phys. Rev. A 64, 012101 (2001). and, not surprisingly, some are sceptical! Euler angles of basis iS p iS zf iS yq pz De e e ˆ φ O DOD 1 q ˆ θ F DF p py f i r( ) D i D1 px i is the position operator for m>0; ri , p j i ij etc. are preserved by above unitary transformation. New position operator becomes: ( ) ( ) r rP a S p where S p p.S ˆ a (0) cos q φ for θ / φ basis ˆ ˆ ˆ p sin q ( ) a a (0) • its components commute • eigenvectors are exactly localized states • it depends on “geometric gauge”, , that is on choice of transverse basis Like a gauge transformation in E&M A A a a a p A r 2 a p p + string so this looks ˆ exactly like the B-field of a magnetic monopole, complete with the Dirac string singularity to return the flux. Topology: You can’t comb the hair on a fuzz ball without creating a screw dislocation. Phase discontinuity at origin gives -function string when differentiated. Geometric gauge transformation cos q 0, a (0) ˆ φ p sin q eg =-f cos q 1 f , a ( f ) ˆ φ p sin q no +z singularity ˆ θ ˆ φ since p ˆ p p q p sin q f ( ) i (0) e e e is rotation by about p : ˆ e( f ) 1 θ iφ cos f i sin f 2 ˆ ˆ 1 2 cos fθ sin fφ ˆ ˆ i sin f θ cos f φ 1 2 ˆ ˆ qp ˆ φ ˆ Rotated about z by f f p.z f cos q ˆˆ q0 ˆ f at q =0, =f at q =p θ Is the physics -dependent? Localized basis states depend on choice of , e.g. e(0) or e(-f) localized eigenvectors look physically different in terms of their vortices. This has been given as a reason that our position operator may be invalid. The resolution lies in understanding the role of angular momentum (AM). Note: orbital AM rxp involves photon position. “Wave function”, e.g. F=E+icB Any field can be expanded in plane wave using the transverse basis determined by : 3 d p F r, t f p e e ( ) i p.r pct / 2p 3 f(p) will be called the (expansion) coefficient. For F describing a specific physical state, change of e() must be compensated by change in f. For an exactly localized state f p Np e a ipr ' Optical angular momentum (AM) Helicity : e( ) 1 2 ˆ ˆ θ iφ ei Spin sz : e( ) ~ 1 2 x isz y ˆ ˆ Usual orbital AM: L z i p z i f If coefficient f p ~ e ilzf Lz eilzf lz eilzf and lz is OAM Interpretation for helicity 1, single valued, dislocation on -ve z-axis e( f ) cosq 1 x iy ˆ ˆ sz=1, lz= 0 1 2 2 cosq 1 x iy exp ˆ ˆ sz= -1, lz= 2 2 2 2if 1 sin q exp if sz=0, lz= 1 2 Basis has uncertain spin and orbital AM, definite jz=1. Position space ;l pr e ipr / 4p i Yl , Yl q , f jl l n n* l 0; n l 2p Yl n* q , f eimf df ~ n ,m eim 0 eimf dependence in p-space eim in r-space There is a similar transfer of q dependence, and the factor jl (pr / ) is picked up. Beams Any Fourier expansion of the fields must make use of some transverse basis to write 3 d p F r, t f p e e ( ) i p.r pct / 2p 3 and the theory of geometric gauge transformations presented so far in the context of exactly localized states applies - in particular it applies to optical beams. Some examples involving beams follow: Bessel beam, fixed q0 , azimuthal and radial (jz =0): Volke-Sepulveda et al, J. Opt. B 4 S82 (2002). A has z and z terms. ˆ ˆ e1 e(0) (0) φ ˆ 1 i 2 1 x iy eif 1 x iy eif i 2 ˆ ˆ ˆ ˆ 2 i 2 2 ˆ 1 cos q x iy eif 1 cos q x iy eif sin q z θ ˆ ˆ ˆ ˆ ˆ 2 2 2 2 The basis vectors contribute orbital AM. e1f ) and e(f1) have same l z 1 ( Nonparaxial optical beams Barnett&Allen, Opt. Comm. 110, 670 (1994) get x iy ˆ ˆ 2 cos q 1 z sin q eif 2 ˆ cos q 1 e( f ) + cos q 1 e 2if e( f ) 2 1 2 1 Elimination of e2if term requires linear combination of RH and LH helicity basis states. Partition of J between basis and coefficient ( ) ( ) r e 0 since eigenvector at r ' 0. L( ) r ( ) p, L( )e( ) =0, L( ) acts only on coefficient. S ( ) ( ) JL a ( ) p p S p gives AM of basis. ˆ J S ( ) L( ) is invariant under geometric gauge transformations, e.g. e( ) e ) eimf and f fe imf ( for a fixed F describing a physical state. to rotate axis is also possible, but inconvenient. Commutation relations L(i ) , L(j ) i ijk L(k ) ; r, L( ) 0 Si( ) J i , rj( ) i ijk rk( ) i p j S z( ) J z r ( ) 1 0 since S ( ) = m j 2 p j z L() is a true angular momentum. Confirms that localized photon has a definite z-component of total angular momentum. Summary • Localized photon states have orbital AM and integral total AM, jz, in any chosen direction. • These photons are not just fuzzy balls, they contain a screw phase dislocation. • A geometric gauge transformation redistributes orbital AM between basis and coefficient, but leave jz invariant. • These considerations apply quite generally, e.g. to optical beam AM. Position and orbital AM related through L=rxp.
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