Polarization descriptions of quantized fields

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Polarization descriptions of quantized fields
Anita Sehat, Jonas Söderholm, Gunnar Björk Royal
Institute of Technology Stockholm, Sweden Pedro
Espinoza, Andrei B. Klimov Universidad de
Guadalajara, Jalisco, Mexico Luis L.
Sánchez-Soto Universidad Complutense, Madrid,
Spain
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Outline

• Motivation
• Stokes parameters and Stokes operators
• Unpolarized light hidden polarization
• Quantification of polarization for quantized
  fields
• Generalized visibility
• Polarization of pure N-photon states
• Orbits and generating states
• Arbitrary pure states
• Summary

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Motivation

• The polarization state of a propagating
  electromagnetic field is relatively robust
• The polarization state is relatively simple to
  transform
• Transformation of the polarization state
  introduces only marginal losses
• The polarization state can easily and relatively
  efficiently be measured
• The polarization is an often used property to
  encode quantum information
• Typically, photon counting detectors are used to
  measure the polarization
• gt The post-selected polarization states are
  number states
• A (semi)classical description of polarization is
  insufficient.

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The Stokes parameters
In 1852, G. G. Stokes introduced operational
parameters to classify the polarization state of
light
tests x linear polarization
tests circular polarization
tests linear polarization
If P0, then the light is (classically)
unpolarized
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The Stokes operators
E. Collett, 1970
Two-mode thermal state
Any two-mode coherent state
E. Collett, Am. J. Phys. 38, 563 (1970).
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A problem with PSC
A two-mode coherent state, arbitrarily close to
the vacuum state is fully polarized according to
the semiclassical definition!
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SU(2) transformations realized by geometrical
rotations and differential-phase shifts
Only waveplates, rotating optics holders, and
polarizers needed for all SU(2) transformations
and measurements.
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Another problem Unpolarized light hidden
polarization
PSC 0 gt Is the corresponding state is
unpolarized?
Counter
Frequency doubled pulsed TiSapphire laser ?780
nm ? ?390 nm
BBO Type II
PBS
Detector
HWP
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Experimental results
HWP

HWP_at_
J
Single counts per 10 sec

QWP_at_
J

HWP_at_22.5QWP_at_
J
light off
QWP
J
Phase plate rotation
, deg
The state is unpolarized according to the
classical definition
P. Usachev, J. Söderholm, G. Björk, and A.
Trifonov, Opt. Commun. 193, 161 (2001).
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Unpolarized light in the quantum world
A quantum state which is invariant under any
combination of geometrical rotations (around its
axis of propagation) and differential
phase-shifts is unpolarized. H. Prakash and N.
Chandra, Phys. Rev. A 4, 796 (1971). G. S.
Agarwal, Lett. Nuovo Cimento 1, 53 (1971). J.
Lehner, U. Leonhardt, and H. Paul, Phys. Rev. A
53, 2727 (1996).
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A coincidence count experiment
Coincidence counter

HWP_at_
J
curve fit
Detector
Coincidence counts per 10 sec
BBO Type II
PBS
Detector
HWP
J
Half-wave plate rotation
, deg
Since the state is not invariant under
geometrical rotation, it is not unpolarized. The
raw data coincidence count visibility is 76,
so the state has a rather high degree of
(quantum) polarization although by the classical
definition the state is unpolarized. This is
referred to as hidden polarization.
D. M. Klyshko, Phys. Lett. A 163, 349 (1992).
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States invariant to differential phase shifts
-Linearly polarized quantum states
Classical polarization
Quantum polarization
Vertical
Vertical
Horizontal
Horizontal
Unpolarized!
Neutral, but fully polarized?
The linear neutrally polarized state lacks
polarization direction (it is symmetric with
respect to permutation of the vertical and
horizontal directions). It has no classical
counterpart. For all even total photon numbers
such states exist.
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Rotationally invariant states - Circularly
polarized quantum states
Classical polarization
Quantum polarization
Left handed
Left handed
Right handed
Right handed
Unpolarized
Neutral, but fully polarized?
The circular neutrally polarized state is
rotationally invariant but lacks chirality. It
has no classical counterpart. For all even total
photon numbers such states exist.
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States with quantum resolution of geometric
rotations
Consider
A geometrical rotation of this state by ?/3 (60
degrees) will yield the state
A rotation of by 2 ? /3 (120 degrees) or
by - ? /3 will yield the state
? Complete set of orthogonal two-mode two photon
states. There states are not the linearly
polarized quantum states PSC 0 for these
states gt Semiclassically unpolarized, hidden
polarization
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Experimental demonstration
2500
2000
Measured data (dots) and curve fit for the overlap
1500
Coincidence counts per 500 s
1000
Back- ground level
500
0
0
180
-180
-120
-60
60
120
Polarization rotation angle (deg)
T. Tsegaye, J. Söderholm, M. Atatüre, A.
Trifonov, G. Björk, A.V. Sergienko, B. E. A.
Saleh,
and M. C. Teich, Phys. Rev. Lett., vol. 85, pp.
5013
-
5016, 2000.

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Existing proposals for quantum polarization
quantification
The measures quantify to what extent the states
SU(2) Q-function is spread out over the spherical
coordinates. That is, how far is it from being a
Stokes operator minimum uncertainty state?
A. Luis, Phys. Rev. A 66, 013806 (2002).
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Examples
That is, the vacuum state is unpolarized and
highly excited states are polarized Note that
N MaxPSU(2)
1 1/4
2 4/9
4 16/25
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Degree of polarization based on distance to
unpolarized state
Another proposal is to define the degree of
polarization as the distance (the
distinguishability) to a proximal unpolarized
state. Will be covered in L. Sánchez-Sotos talk.
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Proposal for quantification of polarization
Generalized visibility
Transformed state
Original state
How orthogonal (distinguishable) can the original
and a transformed state become under any
polarization transformation?
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All pure, two-mode N-photon states are polarized
One can show that all pure, two-mode N-photon
states with N 1 have unit degree of
polarization using this definition, even those
states that are semiclassically unpolarized gt No
hidden polarization.
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Orbits
The set of all such states define an orbit. If
one state in an orbit can be generated, then we
can experimentally generate all states in the
orbit.
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Orbit generating states
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Orbit generating states where the orbit spans the
whole Hilbert space
Moreover, to generate the basis set we need only
make geometrical rotations or differential phase
shifts.
Such orbits are of particular interest for
experimentalists to implement 3-dimensional
quantum information protocols, and to demonstrate
effects of two-photon interference.
In higher excitation manifolds it is not known if
it is possible to find complete-basis generating
orbits, but it seems unlikely.
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Summary
Polarization is a useful and often used
characteristic for coding of quantum info. The
classical, and semiclassical description of
polarization is unsatisfactory for quantum
states. Other proposed measures have been
discussed and compared. We have proposed to use
the generalized visibility under (linear)
polarization transformations as a quantitative
polarization measure. Polarization orbits
naturally appears under this quantitative
measure. Orbits spanning the complete N-photon
space have special significance and interest for
experiments and applications.
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Schematic experimental setup
Generated state
Phase shift
HWP
PBS
BBO Type II
Phase shift
Projection onto the state . (This state
causes coincidence counts.)