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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. Outline.

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Polarization descriptions of quantized fields
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


Outline
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


Motivation
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

    => The post-selected polarization states are number states

    A (semi)classical description of polarization is insufficient.


The stokes parameters
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 P=0, then the light is (classically) unpolarized


The Stokes operators

E. Collett, 1970:

Two-mode thermal state

Any two-mode coherent state

E. Collett, Am. J. Phys. 38, 563 (1970).


A problem with p sc
A problem with PSC

A two-mode coherent state, arbitrarily close to the vacuum state is fully polarized according to the semiclassical definition!


Su 2 transformations realized by geometrical rotations and differential phase shifts
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.


Another problem unpolarized light hidden polarization
Another problem: Unpolarized light – hidden polarization

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


11000

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1000

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0

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Experimental results

HWP

[email protected]

J

Single counts per 10 sec

[email protected]

J

[email protected][email protected]

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).


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

J

[email protected]

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).


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 two-mode two photon states.

There states are not the “linearly” polarized quantum states

PSC = 0 for these states => Semiclassically unpolarized, “hidden” polarization


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

-120

-60

60

120

-180

0

180

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.


Existing proposals for quantum polarization quantification
Existing proposals for quantum polarization quantification

The measures quantify to what extent the state’s 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).


Examples
Examples

That is, the vacuum state is unpolarized and highly excited states are polarized

Note that:


Degree of polarization based on distance to unpolarized state
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-Soto’s talk.


Proposal for quantification of polarization generalized visibility
Proposal for quantification of polarization state– Generalized visibility

Transformed state

Original state

How orthogonal (distinguishable) can the original and a transformed state become under any polarization transformation?


All pure two mode n photon states are polarized
All pure, two-mode stateN-photon states are polarized

One can show that all pure, two-mode N-photon states with N ≥ 1have unit degree of polarization using this definition, even those states that are semiclassically unpolarized => No ”hidden” polarization.


Orbits
Orbits state

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.



Orbit generating states where the orbit spans the whole hilbert space
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.


Summary
Summary Hilbert space

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.


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.)


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