Localized Photon States Here be dragons Margaret Hawton
Lakehead University Thunder Bay, Canada
Slide 2
Introduction In standard quantum mechanics a measurement is
associated with an operator and collapse to one of its
eigenvectors. For the position observable this requires a position
operator and collapse to a localized state. The generalized theory
of observables only requires a partition of the identity operator,
i.e. a positive operator valued measure (POVM). I will show here
that the elements the POVM of a photon counting array detector are
projectors onto localized photon states.
Slide 3
Since the early days of quantum mechanics it has been believed
that there is no photon position operator with localized
eigenvectors. The emergence of localized photon states as the POVM
of a photon counting array sheds light on this long standing
problem. By using the generalized theory of observables many of the
theoretical difficulties are avoided. Recently Tsang [Phys. Rev.
Lett. 102, 253601 (2009)] defined a similar photon position POVM
consisting of projectors onto localized states and applied it to a
new quantum imaging method.
Slide 4
Outline Localized electron and photon states POVM of a photon
counting array Hegerfeldt theorem and scattering Conclude
Slide 5
is a -function in coordinate space. This energy fiction
describes localization in a small region. Localized basis for
nonrelativistic electron An electron has spin AM s parallel to the
arbitrary z- axis where s= . The position/spin basis The position
eigenvector,
Slide 6
The following has been proved regarding photon position: (1)
The relationship between the electric/magnetic field and photon
number amplitude is nonlocal in r-space. (2) There are no definite
s, l=0 localized photon states (Newton and Wigner 1949) and no
photon position operator with localized eigenvectors that
transforms like a vector. (3) If a relativistic particle is
localized for an instant, at all other times it is not confined to
any bounded region (Hegerfeldt 1974).
Slide 7
This does not exclude localized photon states with the
following properties: (1) While the probability of absorption by an
atom , I will show here that photon counting by a thick detector is
described by photon number density. (2) Helicity and total AM can
have definite values, but spin cannot. Their nonintegrable AM leads
to the Berry phase observed in helically wound fibers. (3) Incoming
and outgoing waves are equally likely. Localization is due to
destructive interference of these counterpropagating waves. ^
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annihilates a photon at r. a (r,t) creates a localized photon
while E (r,t) creates its nonlocal field. Photon annihilation and
creation operators ^^ In the IP the positive energy QED electric
field operator annihilates the field due to a photon at r while the
photon number amplitude operator ^ (-)
Slide 9
The localized states, are orthonormal, i.e. and form a
partition of the identity operator, Photon number amplitude and
field operators differ by k constant. Both require transverse unit
vectors e k, for helicities = 1 and all k. Localized photon
states
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kxkx kzkz kyky k (2) The transverse unit vectors where is the
Euler angle and helicity is internal AM parallel to k. These unit
vectors have definite helicity and total AM but no definite spin.
The choice =- gives j= and the total AM is . The ks close to the
+z-direction needed to describe a paraxial beam have spin s= and
l=0 so all orbital AM is in | >, not the basis. in k-space
spherical polar coordinates are
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The sketch shows =+1 E and B and wave fronts for k- components
of a localized state close to +z and z. For the latter, s=-1 so l=2
is required to give j=1.
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(3) In+Out: We dont know if a component plane wave is
approaching or leaving the point of localization. The t=0 localized
state is a sum of incoming and outgoing spherical shells. More
details later in II.
Slide 13
^ z I. The POVM of a photon counting array detector and its
relationship to photon density. II. The Hegerfeldt theorem,
counterpropagating waves, and scattering of a photon by a
nanoparticle. Position measurement will be discussed from two
perspectives:
Slide 14
I. POVM of a photon counting array When measuring photon
position using a photon counting array the photons are the
particles of interest and the detector atoms form an ancillary
Hilbert subspace. The POVM is the partial trace over the atom
subspace. The measurement consists of counting the e-h pairs
created in each pixel. For simplicity it will be assumed here that
1 photon is counted, but 2 photons is treated in [1]: Hawton, Phys.
Rev. A 82, 012117 (2010).
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It will be assumed that each atom has a ground state |g> and
3 mutually orthogonal excited states, |e r,p >. The IP or SP
1-photon counting operator checks for an excited atom in the n th
pixel by projecting onto one of these excited states using where D
n denotes the n th pixel of the array.
Slide 16
To incorporate the dynamics into the counting operator we can
transform to the HP using the usual E interaction Hamiltonian, This
operator promotes an atom from the ground state to an excited state
while annihilating a photon. We can choose p for normal incidence
in the paraxial approximation
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Transformation requires the unitary operator gives the 1-photon
counting operator where the first order term
Slide 18
the (Glauber) counting operator becomes Writing out the unitary
and counting operators for a measurement performed between t 0 and
t 0 + t
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A photon counting detector should be thick enough to absorb all
photons incident on it. We can change the sum over atoms to an
integral using For a single mode with k z =nk+i k the absorptivity
is Integration of exp(-2 k z) over detector thickness gives 1/2 k
that eliminates the ks in E (-) E (+). In [1], following Bondurant
1985, this was proved for a sum over modes to first order in
Slide 20
The trace over the ancillary atom states eliminates the factor
|g>
Position information is obtained by projection of the QED state
vector onto the localized states, and the probability to count a
photon equals an integral over its absolute square, i.e. Only modes
present in | > contribute to (r,t).
Slide 22
The Hegerfeldt theorem doesnt apply to photon counting since no
localized photon state is created, but a scattered photon is not
annihilated, at least not permanently, so it might apply to
scattering. Next Ill consider how this relates to exactly localized
states, II. Hegerfeldt theorem and scattering ct If a particle is
confined to the red region at t=0 it is not confined to a sphere of
radius ct at other time t. This could lead to causality
violations.
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Exact ( -function) localization: The real and imaginary parts
of cant be localized simultaneously for arbitrary t since the sum
over cos[k(r ct)] gives - functions but isin[k(r ct)] gives i/(r
ct) tails. Nonlocal tail -function r r=0 t0 cc
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Only at t=0 does destructive interference completely eliminate
the nonlocal tails leaving just a -function. Hegerfeldt proved that
generalization to localization in a finite region doesnt change
this property. Destructive interference explains the physics behind
his theorem. r
Slide 25
Scattering: In Celebrano et al [Optics Express 18, 13829
(2010)] a one-photon pulse is focused onto a nanoparticle and
scattered onto a detector. Photon localization is achieved by
focusing with a microscope objective. nanoparticle In+Out pulses
detector 1-photon input,
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Zumofen et al [Phys. Rev. Lett. 101, 180404 (2008)] predicts up
to 100% reflection and 55% was achieved in the experiment.
Wavelength is 589nm and the nanoparticles are smaller, 46x94nm.
Many wave vectors, both in and out, are present. This approaches a
physical realization of a localized state.
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Conclusion Localized photon states with definite total AM exist
due to destructive interference of in/out pulses. A photon counting
array measures coarse grained photon number density but collapse is
to the vacuum state. This photon number density equals the absolute
square of the projection of the photon state vector onto the
localized basis. Localized states can exist in a scattering
experiment.
