Polarization descriptions of quantized fields

Anita Sehat, Jonas Söderholm, Gunnar BjörkRoyal Institute of Technology

Stockholm, Sweden

Pedro Espinoza, Andrei B. KlimovUniversidad de Guadalajara, Jalisco, Mexico

Luis L. Sánchez-SotoUniversidad Complutense, Madrid, Spain

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

• 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

In 1852, G. G. Stokes introduced operational parameters to classify the polarization state of light

thenreal, are and ,,, whereˆeˆe If 2121)ti(

2)ti(

121 aaeaeaE yx

2122

210

22

21

21

21

where,

sin2

cos2

aaS

aaS

aaS

aaS

z

y

x

onpolarizati of degree the

0

2/1222

S

SSSP zyx

C

If P=0, then the light is (classically) unpolarized

tests + linear polarization

tests x linear polarization

tests circular polarization

The Stokes operators

E. Collett, 1970:

bbaaNS

bbaasSS

ibabasSS

babasSS

zz

yy

xx

ˆˆˆˆˆ

ˆˆˆˆˆ

ˆˆˆˆˆ

ˆˆˆˆˆ

0

0

0

0

onpolarizati of degree quantum"" a

ˆ

ˆˆˆ2/1222

N

sssP

zyx

SC

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

10 SCPTwo-mode thermal state

Any two-mode coherent state

A problem with PSC

.1 has , statecoherent mode-Any two SCP

1 0, lim 0 SCP

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

ysi ˆ expby realized is angle by therotation lgeometricaA

zsi ˆ expby realized is by shift phase-aldifferentiA

zyz sisisiU ˆ expˆ expˆ exp,,ˆ

operator by the realized becan SU(2) group theoftion representaunitary A

Only waveplates, rotating optics holders, and polarizers needed for all SU(2) transformations and measurements.

Another problem: Unpolarized light – hidden polarization

PSC = 0 => Is the corresponding state is unpolarized?

1,1

Frequencydoubled pulsed

Ti:Sapphirelaser

=780 nm =390 nm

BBOType II

PBS Detector

Counter

HWP

11 state heConsider t ,

Experimental results

0 30 60 90 120 150 180 210 240 270 300 330 360

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

11000

HWP@ QWP@ HWP@22.5+QWP@ light off

Sin

gle

cou

nts

per

10 s

ec

Phase plate rotation , deg

xs

ys

zs

xszs

ys

HWP

QWP

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).

A coincidence count experiment

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.

0 15 30 45 60 75 90 105 120 135 150 165 180 195

0

100

200

300

400

500

600

700

800 HWP@ curve fit

Coi

ncid

ence

cou

nts

per

10

sec

Half-wave plate rotation , deg

BBOType II

1,1

PBS

Detector

Detector

Coincidencecounter

HWP

D. M. Klyshko, Phys. Lett. A 163, 349 (1992).

States invariant to differential phase shifts -“Linearly” polarized quantum states

0,2

2,0

1,1

Vertical

Classical polarization

Horizontal

Unpolarized!

Vertical

Quantumpolarization

Horizontal

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.

:ˆ tosEigenstate zs

Rotationally invariant states - Circularly polarized quantum states

0,21,122,02

12,0 i

0,21,122,02

10,2 i

0,22,02

11,1

Left handed

Classical polarization

Right handed

Unpolarized

Left handed

Quantumpolarization

Right handed

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

0,26

22,0

6

20,21,12,0

3

10002

iii

0323121

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

0

3/200

3/23 0,21,12,0

3

1 ii eie

0

3/200

3/21 0,21,12,0

3

1 ii eie

2

Consider:

Experimental demonstration

Polarization rotation angle (deg)

Coi

ncid

ence

cou

nts

per

500

s

0 180-1800

2500

Back-groundlevel

1 3

2

13

22000

1500

1000

500

-120 -60 60 120

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.

2

2

ˆ

2 ySie

Measured data (dots) and curve fit for the overlap

Existing proposals for quantum polarization quantification

NsΔsΔsΔ zyxˆ2ˆˆˆ :relationy uncertaintoperator Stokes 222

mNme

m

NN immmN

N

m

,2cos2sin,, :statescoherent SU(2)0

,,ˆ,,4

1, :function- SU(2)

0

NNN

QQN

2)2(,4

11 :onpolarizati of Degree

QdPSU

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

1

:statecoherent SU(2)2

SU(2)

N

NP

That is, the vacuum state is unpolarized and highly excited states are polarizedNote that:

Nm

N

m

N

N

NPmNN

1 2

2

1

121 :, statenumber mode-Two

2

2SU(2)

N Max{PSU(2)}

1 1/4

2 4/9

4 16/25

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

rmationon transfopolarizatiunitary a is ˆ where,ˆMin12

UUP UQ

Original state

Transformed 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

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 => No ”hidden” polarization.

0 state vacuumFor the QP

2222exp1 : where, statecoherent mode- twoaFor QPNβ α

Orbits

on.polarizati quantum of degree same thehave states two

then theˆ if that followsit of definition theFrom UPQ

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

integer)nearest the todownwards (rounded 12 orbits ofNumber

, states generating orbits, Discrete :1 typeOrbits

N

nNN

.etc,3for

sincose

sinsin

cossin

sincose

2for

sin

0

cos

2,0sin0,2cos states Generating :2 typeOrbits

i

i

N

N

Orbit generating states where the orbit spans the whole Hilbert space

states). (3set basis wholea generatecan 22,00,2 state theseen, have weAs

Moreover, to generate the basis set we need only make geometrical rotations or differential phase shifts.

.4for set basis wholea generatecan 23,00,3 state theshown that becan It N

. 22,11,2 is 4for state generating basisAnother N

In higher excitation manifolds it is not known if it is possible to find complete-basis generating orbits, but it seems unlikely.

Such orbits are of particular interest for experimentalists to implement 3-dimensional quantum information protocols, and to demonstrate effects of two-photon interference.

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.

Schematic experimental setup

Phase shift HWP PBSBBOType II

4

Phase shift

4

24

1,1

0,26

22,0

6

22

ii

Generated state:

Projection onto the state .(This state causes coincidence counts.)

2

Detector

Detector

Coincidence

32arccos