Photon angular momentum and geometric gauge

Margaret Hawton, Lakehead University

Thunder Bay, Ontario, Canada

William Baylis, U. of Windsor, Canada

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

Outline

Notation: momentum space

1(0)

0

2

ˆˆ ˆ ˆ ˆˆ~

ˆ

;

Use CP basis vectors:

ˆ

is the -space gradien

ˆ

.

t

i

e θ

φ z p θ

φ

φ p

e p

p

(Will use and when in -space.) r

φ

θ

px

pz or z

py

pz

Is the position of the photon an observable?

• =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”?

Pryce photon position operator1948, obtained a ,

ˆP i p p p

p Sr

In quantum mechanics, any observable requires a Hermitian operator

A photon is asymptotically localizable

ˆ

1) Adlard, Pike, Sarkar, PRL , 1585 (1997)

is an arbitrarily large integer; power aw

~

l

r

A

79

0

0

2) Bialynicki-Birula, PRL , 5247 (1998)

1 to satisfy Paley-Wiener theorem,

arbitrarily s

ˆ~ exp ( / )

exponentially localmall; izedr

r r

8

Z m

0

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!

px

pz

py

Euler angles of basis

1

p yziS iSiSD e e e

O DOD

D

F F

i j is the position operator for m>0; ,

etc. are preserved by above unitary transformation.

iji r p i

pφ

θ

( ) 1D i Di r

New position operator becomes:

• its components commute

• eigenvectors are exactly localized states

• it depends on “geometric gauge”, that is on choice of transverse basis

( ) (

(

( ) (

0

)

)

0

) ˆwhere .cos ˆˆ ˆ for / basis

s

in

P p pSS

p

r r a

φ φ

a

θ

a

p S

a

Like a gauge transformation in E&M

2 + so this looks

like the B-field of a magnetic

, complete with the Dirac

to return t

string

string singularity he flux.

ˆ

monopole

exactly

p

p

A A

a a

a p A r

a

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

(0)

( )

cos

sin

eg =-

cos 1

sin

ˆ0,

ˆ,

p

p

a φ

a φ

no +z singularity

ˆ ˆˆsince

sinp p p

θ φ

p

φ

θ

( ) 1

2

12

12

ˆ ˆ cos sin

ˆ ˆcos sin

ˆ ˆsin cos

i i

i

e θ φ

θ φ

θ φ

ˆ is rotation by about : p

( ) (0)ie

e e

at = at

ˆRotated about by

ˆ ˆ. cos

, =0 =

z

p z

Is the physics -dependent?

Localized basis states depend on choice of , e.g. e(0) or e(-) 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

3

. /( )32

, i pctd pt f e

p rF r p e

'if Np e p rpFor an exactly localized state

Any field can be expanded in plane wave using the transverse basis determined by

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.

Optical angular momentum (AM)

( ) 1

2

ˆ ˆHelicity : ii e e θ φ

( ) 1z z2

ˆ ˆSpin : ~s is e x y

z: Usual or Lbital AMz

i i

p

z

~

is OA

If coefficient

L n Ma d

z

z zil ilz

il

z

f e

e e ll

p

( )1 2 2

2 2

1

2

ˆ ˆcos 1

ˆ ˆcos 1exp 2

sin exp

i

ii

i

x ye

x y

Interpretation for helicity , single valued, dislocation on -ve z-axis

sz=, lz=

sz= -1, lz=

sz=0, lz=

Basis has uncertain spin and orbital AM, definite jz=.

Position space

;/ *

0;

2 *,0

4 , ,

, ~

dependence in p-space in r-space

There is a similar transfer of dependence,

and the factor ( / ) is picked up.

li l n n

l l ll n l

n i

i

m iml n m

l

m im

pre i Y Y j

Y

e e

e d e

j pr

p r

Beams

Any Fourier expansion of the fields must make use of some transverse basis to write

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:

3

. /( )32

, i pctd pt f e

p rF r p e

0

(0) (0)1 1

2

2 2 2 2

2 2 2

, azimuthal and radial ( =0):

Volke-Sepulveda et al, J. Opt. B S82 (2002).

ˆ ˆ has and terms.

ˆ

ˆ

Bessel beam, fixed

ˆ ˆ ˆ1 1

ˆ ˆ ˆ1 1ˆ cos cos

z

i i

i

j

i

i ie ei i

ie

4

A z z

e eφ

x y x y

x y xθ

2

ˆˆsinii

e

yz

The basis vectors contribute orbital AM.

( ) ( )1 1

2 2

( ) 2 ( )1 1

Nonparaxial optical beams

Barnett&Allen, Opt. Comm. 110, 670 (1994) get

ˆ ˆ 1 ˆco

and have same 1

s sin

cos 1 cos 1+2 2

i

z

i

ie

e

l

x y

e

z

e

e e

Elimination of e2i term requires linear combination of RH and LH helicity basis states.

Partition of J between basis and coefficient

( )

( )

( )

( ) ( )

( ) ( ) ( ) (

( )

( )

)

) (

0 since eigenvector at ' 0.

, =0, .

ˆ .

under geometric gauge

transformati

acts only on coefficient

gives AM of basis

is invariant

o

pS

r e r

L r p L e

J L a p p

L

S

J S L( ) ( )

for a fixe

ns, e.

d

g. e

de

and

scribing a physical s te.ta

eim imf f

F

e e

to rotate axis is also possible, but inconvenient.

Commutation relations( ) ( ) ( ) ( )

( )( ) ( )

( )( ) ( )

, ; , 0

,

1 0 since = 2

i j ijk k

ii j ijk k

j

zz j z

j

L L i L

SJ r i r i

p

SJ r S m

p

r L

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.