Coherent Exciton Dynamics in Semiconductor Superlattices:A Quasi by gnAMGr1

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									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     pS
                     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                   De              e             e
                       ˆ
                       φ       O  DOD 1
          q            
                   ˆ
                   θ           F  DF
               p
                       py
          f                 i   r(  )  D i  D1
     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  iφ   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                   qp
                                                          ˆ
                                                          φ
              ˆ
Rotated about z by                              f
 f  p.z  f cos q
       ˆˆ                                               q0
                                                    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 ipr '
 Optical angular momentum (AM)
 Helicity : e(  ) 
              
                            1
                            2
                                   ˆ     ˆ   
                                    θ  iφ ei

   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   ipr /
              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 eif
      i 2
           ˆ ˆ            ˆ ˆ
             2        i 2   2


  ˆ  1 cos q x  iy eif  1 cos q x  iy eif  sin q z
  θ
              ˆ ˆ                  ˆ ˆ
                                                       ˆ
       2         2          2         2


The basis vectors contribute orbital AM.
e1f ) 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   ( )       ( )
            JL       
                       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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