Transcript Photon localizability - Current research interest: photon position

Photon Position
Margaret Hawton, Lakehead University
Thunder Bay, Canada
It has long been claimed that there is no photon position
operator with commuting components and, as a
consequence, no basis of localized states and no position
space wave function, just fields and energy density.
In this talk I will argue that all of these limitations can be
overcome! This conclusion is supported by our position
operator publications starting in 1999.
Localizability
The literature starts before 1930 and is sometimes
confusing, in part because there are really 3 problems:
1) For any quantum particle ψ~e-iwt with +ve w= c k  k
and localizability is limited by FT theorems.
2) If all k's are equally weighted to localize the
number probability density, then energy density
(and fields in the case of photons) are not localized.
3) For 3D localization of the photon, transverse fields
don’t allow separation of spin and orbital AM and
this is reflected in the complexity of the r-operator.
Classical versus quantum
For a classical field one can take the real part
which is equivalent to including +ve and –ve
w's. Thus (1) does not limit localizability of a
classical pulse, but the math of (2) and (3) are
relevant to localizability of a classical field.
1) Localizability of quantum particles
For positive energy particles the wave function
ψ~e-iwt where w must be positive. Fourier
transform theory then implies that a particle can
be exactly localized at only one instant. This has
been interpreted as a violation of causality. Also,
the Paley-Wiener theorem limits localizability if
only +ve (or –ve) k's are included.
Paley-Wiener theorem
The Fourier transform g(r) of a square-integrable
function h(k) that vanishes for all negative values of k
(i.e. +ve k or +ve w only) must obey:

| log | g (r ) ||
hk    dr

2

1 r
This does not allow exact localization of a pulse travelling
in a well defined direction but does allow exponential and
algebraic localization, for example (Iwo Bialynicki-Birula,
PRL 80, 5247 (1998))
g (r ) ~ exp(  Ar  ) with   1
Hegerfeldt theorem
For a particle localized at a k, l a, l  0l k eik.a where
 depends on scalar product. In field theory 0l k  ~ k.
Consider a photon, helicity l, localized at r=0 at time t=0.
The probability amplitude to find it at a at time t is:
2 i
3 1
l
a, l  0,l t    d k  k 0 0 e

2
k  m 2 ct
eik.a

FT theory implies that an initially localized particle
immediately develops tails that are nonzero everywhere.
Hegerfeldt causality paradox
wave fronts
red particle localized at r=0 (or
in any finite region) at t=0 can
be found anywhere in space at
all other times.
propagation direction
These problems are not unique to 3D. I’ll
first consider the 1D analog of the
Hegerfeldt causality problem.
As an example consider an ultrafast photon
pulse whose description requires only one
spatial variable, z, if length<<area.
In 1D there is no problem to define a photon position
operator, it the same as for an electron.
zˆ a  a a
 a, t   a  t 
The probability density that particle is at a is |(a,t)|2.
Representations of the (1D) position operator are:
zˆ  z in coordinate space
 i k in k - space
The exactly localized states are Dirac d-functions in
position space and equally weighted in k-space:
z a  d z  a 
k a  1 exp  ika ; k a  constant
2
Exactly localized states cannot be realized numerically
or experimentally so I’ll include a factor e-ek:
d e z   21

e


e

ikz e k
dk
 d z  but for  ve k's only
z  e e 0
 ikz e k
iz  e
we get  e
dk  2
which is not localizabl e.
2
0
z e
2
2
Consider a traveling pulse with peak at Dz=0,
center wave vector k0 and width ~1/e:

f n (Dz )   k n exp[ ikDz  e (k  k0 )]dk
0
If k0=0 we get the simple forms (PV is the principal value):
localized
f 0 (Dz )  1 / e  iDz  e
iPV 1 / Dz   d Dz 
0
f1/ 2 (Dz )   / e  iDz 
1/ 2
is not localizabl e
A pair of pulses, one initially at –a travelling to the right (k's>0),
and the other at a travelling to the left (k's<0) is :
Fn ( z, t )  f n ( z  a  ct )  f n ( z  a  ct ).
1D ultrafast pulse
l0/2≈1
real part
(localizable)
pulse propagation (k>0 only)
peak at z=-a+ct
imaginary part (tails
go to 0 as 1/Dz)
Causality paradox in 1D: photon at a=0 time t=0 can immediately
be found anywhere in space (dark blue imaginary part).
Resolution of the “causality paradox” in the recent literature is
localizable states are not physically realizable, but is this the case?
nonlocalizable PVs cancel (interfere
destructively) when coincident.
localizable (d-function)
nonlocalizable PV~1/Dz
At any t ≠ 0 the probability to find the photon anywhere is
space in nonzero. Due to interference there exists a single
instant when QM says that the photon can be detected at
only one place. But this is just familiar spooky quantum
mechanics, and I think the effect is physically real.
Let’s have a closer look as the pulses collide.
nonlocalizable PV tails
→0 as pulses collide, a
QM interference effect
Have total destructive interference of nonlocalizable part
when counter propagating pulses peaks are coincident.
Back to 3D (or 2D beam) causality paradox:
wave fronts
red particle localized at r=0 at
t=0 can be found any where in
space at all other times.
propagation direction
.
Assume  0,l (0)  0, l at t  0,
rl |  0,l (t )    d 3 keik r e ikct
i
1 
3
ik ( r  ct )
ik ( r  ct )
~
d
k
e

e
r r 
1   1
1 
~



r r  r  ct  r  ct 


There is an outgoing plus an incoming wave.
In 3D have sum of incoming and outgoing spherical pulses:
e
ik ( r  ct )
e
e
i ( k r  kct )
 ik ( r  ct )
Conclusion 1
A single quantum mechanical pulse is not
localizable. For a pair of counter propagating
pulses the probability to detect the photon can
be exactly localized at the instant when their
peaks collide. This gives a physical
interpretation to photon localizability, it
implies that we don’t know whether the photon
is arriving or departing.
2) Fields versus probability amplitudes
Recall that pulses were described by

f n (Dz )   k exp[ ikDz  e (k  k0 )]dk
n
0
and a pair of pulses initially at –a travelling to the
right (k's>0) and at a travelling to the left (k's<0) is
Fn ( z, t )  f n ( z  a  ct )  f n ( z  a  ct ).
If n=0 (an integer in general) we get localizability.
For a monochromatic wave
energy density
number density 
w
but this is ambiguous for a localized pulse that
incorporates all frequencies, for which number and
energy density have a different functional form.
f1/ 2 (Dz)   / e  iDz 
1/ 2
f 0 (Dz )  1 / e  iDz 
 iPV 1 / Dz   d Dz 
e 0
I.Based on photodetection theory,the photon wave
function is sometimes defined as the expectation
value of the +ve energy field operator as below
where |> is a 1-photon state and |0> the vacuum:
0 E (  ) ( z , t )   F1/ 2 ( z, t ) if k ' s have equal weight
 F0 ( z , t ) if weights go as k 1/ 2
II.If we consider instead the probability amplitude
to find a photon at z the interpretation is:
position prob. ampl.  z  t   F0 ( z , t ) for equal weights.
The important role of the position operator is to define
the z basis and the prob ampl to detect a photon.
Energy density
If E(z±ct) is LP along x, dtBy=-dzEx the magnetic
field has the opposite sign for pulses travelling in
the positive and negative directions. Thus if the
nonlocalizable (PV) part of the E contributions
cancel, the nonlocalizable contributions to B add. In
a QM description, the photon energy density is not
localizable.
We have a localizable position probability amplitude if
k's equally weighted, electric field if weighted as k -1/2.
What “wave function” should we consider?
The important thing is what can be produced
and detected. And does a photodetector see just
the electric field?
I don’t know, really, but consider the E-field
due to a planar current source localized in z
and approximately localized in t.
The source is localized in space but can’t be current
exactlysource
localized
in time since w>0. QED is required to do a proper job.
 2z A  c12  t2 A  d z  f t 
In kw - space  k A  w A  
2
2
1
e -w
2 2
1
 iw
-it  w ikz
E ( z, t )   t A 
dkd
w
e
e
2
2 
2
2
kc  w
the residue contributi on is
1
1
E 

 a  ct  z  i 
far field  a  ct  z  i 
This gives the same simple solution in the far field that I have
been plotting and has a localized E-field.
Plots with near field + far field
t
60
2
50
40
30
20
10
-10
-5
5
-10
-20
source
10
Plots with near field + far field
t 2
60
50
40
30
20
10
-10
-5
5
-10
-20
10
In far field, get propagating pulses i/Dz as
previously plotted.
t 4
60
50
40
30
20
10
-10
-5
5
-10
-20
10
detector
propagating
free photon
source
emission/absorption
should 2nd quantize
detector
Conclusion 2
Photon position probability amplitude and fields
are not simultaneously exactly localizable.
Exponential localization of both is possible, but
what matters is the field/probability amplitude that
can be produced and detected. A localized current
source in 1D produces a localizable E in the far
field. Photon energy density is not localizable.
3) Transverse fields in 3D
It has long been claimed that there is no hermitian photon
position operator with commuting components, and hence
there is not a basis of localized eigenvectors. However,
we have recently published papers where it is
demonstrated that a family of position operators exists.
Since a sum over all k’s is required, we need to define 2
transverse directions for each k. One choice is the
spherical polar unit vectors in k-space.
ẑ
kz or z
q
f
kx
φˆ ~ zˆ  kˆ ; θˆ ~ φˆ  kˆ
Use CP basis vectors :
k̂
θ̂
ky
φ̂
el 
(0)
1
2
θˆ  ilφˆ 
e ( 0 )  kˆ
 is the k - space gradient
More generally can use any Euler angle basis
De
kz
φ̂
q
k
f
kx
θ̂

ky
 iS p 
e
 iS zf
e
 iS yq
O  DOD 1
F  DF
i  r
( )
 D i  D
1
Position operator with commuting components
r
( )
kˆ  S  cos q
ˆ
 ik 

  k  S
k
 k sin q

A unique direction in space and jz is specified by the
operator so it is rather complicated. It does not
transform like a vector and nonexistence proofs in the
literature do not apply.
e
( )
l
e
is rotation by l about pˆ :
θˆ  ilφˆ   cos f  il sin f 
 cos fθˆ  sin fφˆ 
ˆ
 il  sin f θˆ  cos f φ
e(lf ) 

 e
 il (0)
l
1
2
1
2
q
1
2
Rotated about zˆ by
 f  pˆ .zˆ  f cos q
 f at q =0, =f at q =
φ̂
f
q0
θ̂
Topology: You can’t comb the hair on a fuzz
ball without creating a screw dislocation.
Phase discontinuity at
origin gives d-function
string when differentiated.
Is the physics -dependent?
Localized basis states depend on choice of , e.g.
el(0) or el(-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.
Optical angular momentum (AM)
Helicity l: e(l ) 
Spin sz : e(l ) ~
1
2

xˆ  isz yˆ 

2
1
Usual orbital AM: Lz  i
If coefficient f  p  ~ e



ˆ e  il
θˆ  il φ
p   z

 i
f
ilzf
Lz eilzf  lz eilzf and lz is OAM
Interpretation for helicity l1, single
valued, dislocation -ve z-axis, =-f
( f )
1
e
cos q  1  xˆ  iyˆ 


2
sz=1, lz= 0
2
cos q  1  xˆ  iyˆ 


exp
2
2


1
  2 sin q  zˆ exp  if 


 2if 
sz= -1, lz= 2
sz=0, lz= 1
Basis has uncertain spin and orbital AM, definite jz=1.
Position space
e

ipr /
2
0
imf
 ;l
 pr 
 4  i Yl  ,   Yl q , f  jl  
 
l  0;n  l
l
n
n*
Yl n* q , f  eimf df ~ d n,m eim
im
e dependence in p-space  e 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
Fl  r, t   
3
d p
 2 3
f  p  el e
(  ) i  p.r  pct  /
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 q 0 , azimuthal and radial (jz =0):
Volke-Sepulveda et al, J. Opt. B 4 S82 (2002).
A has   zˆ and     zˆ terms.
e1(0)  e(0)
1
φˆ 
i 2
xˆ  iyˆ if 1 xˆ  iyˆ if
1
i 2
e i 2
e
2
2
ˆθ  1 cos q xˆ  iyˆ eif  1 cos q xˆ  iyˆ e  if  sin q zˆ
2
2
2
2
The basis vectors contribute orbital AM.
e1( f ) and e(f1) have same lz  1
Nonparaxial optical beams
Barnett&Allen, Opt. Comm. 110, 670 (1994) get
xˆ  iyˆ
1 zˆ sin q eif
co
s
q

2
2
 1 2if ( f )
( f )
cos
q

1
cos
q

e1 +
e e 1
2
2
Elimination of e2if term requires linear combination of
RH and LH helicity basis states.
Conclusion 3
A transverse basis is required for the general
.
description
of pulses and beams, for example
spherical polars. This necessarily singles out some
direction in space, call it z. The transverse vectors
form a screw dislocation with an associated
definite total angular momentum, jz, which can’t
in general be separated into spin and orbital AM.
Summary
• Unidirectional pulses are not localizable, but
counter propagating pulses can be constructed
such that when they collide the particle can be
detected in only one place.
• Relevance of field or energy density or
probability amplitude depends on the experiment.
• Localized photons are not just fuzzy balls, they
contain a screw phase dislocation. This applies
quite generally, e.g. to optical beam AM.