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. SánchezSoto Universidad Complutense, Madrid, Spain. Outline.
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Anita Sehat, Jonas Söderholm, Gunnar Björk
Royal Institute of Technology
Stockholm, Sweden
Pedro Espinoza, Andrei B. Klimov
Universidad de Guadalajara, Jalisco, Mexico
Luis L. SánchezSoto
Universidad Complutense, Madrid, Spain
Typically, photon counting detectors are used to measure the polarization
=> The postselected polarization states are number states
A (semi)classical description of polarization is insufficient.
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 P=0, then the light is (classically) unpolarized
E. Collett, 1970:
Twomode thermal state
Any twomode coherent state
E. Collett, Am. J. Phys. 38, 563 (1970).
A twomode coherent state, arbitrarily close to the vacuum state is fully polarized according to the semiclassical definition!
Only waveplates, rotating optics holders, and polarizers needed for all SU(2) transformations and measurements.
PSC = 0 => Is the corresponding state is unpolarized?
Counter
Frequency
doubled pulsed
Ti:Sapphire
laser
=780 nm =390 nm
BBO
Type II
PBS
Detector
HWP
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Experimental results
HWP
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Single counts per 10 sec
J
[email protected][email protected]
J
light off
QWP
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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).
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 phaseshifts 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
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curve fit
Detector
Coincidence counts per 10 sec
BBO
Type II
PBS
Detector
HWP
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Halfwave 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).
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.
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.
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 twomode two photon states.
There states are not the “linearly” polarized quantum states
PSC = 0 for these states => Semiclassically unpolarized, “hidden” polarization
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Measured data (dots) and curve fit for the overlap
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Coincidence counts per 500 s
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Back
ground
level
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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.
The measures quantify to what extent the state’s SU(2) Qfunction 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).
That is, the vacuum state is unpolarized and highly excited states are polarized
Note that:
Another proposal is to define the degree of polarization as the distance (the distinguishability) to a proximal unpolarized state.
Will be covered in L. SánchezSoto’s talk.
Transformed state
Original state
How orthogonal (distinguishable) can the original and a transformed state become under any polarization transformation?
One can show that all pure, twomode Nphoton states with N ≥ 1have unit degree of polarization using this definition, even those states that are semiclassically unpolarized => No ”hidden” polarization.
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.
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 3dimensional quantum information protocols, and to demonstrate effects of twophoton interference.
In higher excitation manifolds it is not known if it is possible to find completebasis generating orbits, but it seems unlikely.
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 Nphoton space have special significance and interest for experiments and applications.
Coincidence Hilbert space
Detector
Detector
Schematic experimental setup
Generated state:
BBO
Type II
Phase shift
HWP
Phase shift
PBS
Projection onto the state .
(This state causes coincidence counts.)