Download polarized quantum states

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
• 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
If E  a1ei(t 1 )eˆx  a2ei(t 2 )eˆy where a1 , a2 ,1 , and 2 are real, then
S x  2a1a2 cos 
tests x linear polarization
S y  2a1a2 sin 
tests circular polarization
S z  a12  a22
tests + linear polarization
S 0  a12  a22 , where   1  2
PC

S

2
x
 S y2  S z2
S0

1/ 2
 the degree of polarizati on
If P=0, then the light is (classically) unpolarized
The Stokes operators
S x S 0  sˆx  aˆ  bˆ  aˆbˆ 
S y S 0  sˆ y  aˆ  bˆ  aˆbˆ  i
E. Collett, 1970:


S z S 0  sˆz  aˆ  aˆ  bˆ  bˆ
S
 Nˆ  aˆ  aˆ  bˆ  bˆ
0
PSC

sˆ

2
x
 sˆ y
Two-mode
thermal state
2
Nˆ
 sˆz
2

1/ 2
 a " quantum" degree of polarizati on
0  PSC  1
E. Collett, Am. J. Phys. 38, 563 (1970).
Any two-mode
coherent state
A problem with PSC
Any two - mode coherent state  ,  has PSC  1.
 lim  0 PSC   ,0   1
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
A geometrica l rotation by the angle  is realized by exp i sˆ y 
A differenti al - phase shift by  is realized by exp i sˆz 
A unitary representa tion of the group SU(2) can be realized by the operator
Uˆ  ,  ,    exp i sˆ exp i sˆ exp i sˆ 
z
y
z
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?
Consider t he state 1,1 
Counter
1,1 
Frequency
BBO
doubled pulsed
Type II
Ti:Sapphire
laser
=780 nm  =390 nm
HWP

PBS
Detector
Experimental results
sy
HWP
sx
sz
sy
QWP
Single counts per 10 sec
11000
10000
9000
8000
HWP@ 
QWP@ 
[email protected]+QWP@ 
light off
7000
6000
5000
4000
3000
2000
1000
sx
sz
0
0
30
60
90
120
150
180
210
240
270
300
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).
330
360
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).
1,1 
Coincidence
counter
Detector
BBO HWP PBS Detector
Type II

Coincidence counts per 10 sec
A coincidence count experiment
800
HWP@ 
700
curve fit
600
500
400
300
200
100
0
0
15
30
45
60
75
90
105 120 135 150 165 180 195
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
Eigenstate s to sˆz :
Classical
polarization
Quantum
polarization
2,0

Vertical
Vertical
0,2

Horizontal
Horizontal
Unpolarized!
Neutral,
but fully
polarized?
1,1 
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
0,2  
1
 0,2
2
2,0  
1
 0,2
2
1,1  
1
 0,2   2,0
2
Classical
polarization
Quantum
polarization

 i 2 1,1   2,0


Left handed
Left handed

 i 2 1,1   2,0


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:
2 
1

0,2 0  i 1,1 0  2,0
3

0
2 i
2 i
0,2  
2,0
6
6

A geometrical rotation of this state by /3 (60 degrees) will yield the state:
1 i 2 / 3
3 
e
0,2 0  i 1,1 0  ei 2 / 3 2,0 0
3
A rotation of  2 by 2  /3 (120 degrees) or by -  /3 will yield the state:
1 i 2 / 3
1 
e
0,2 0  i 1,1 0  ei 2 / 3 2,0 0
3




1  2  1  3   2  3  0
 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
Measured data (dots)
and curve fit for the
overlap  e iSˆ y 
2
2
2
Coincidence counts per 500 s
2500
2
2
2000
1500
1000
3
1
3
1
Background
level
500
0
-180
-120
-60
0
60
120
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
Stokes operator uncertaint y relation : Δsˆx2  Δsˆ y2  Δsˆz2  2 Nˆ
 N  N m
 2cos m  2eim m, N  m
SU(2) coherent states : N , ,      sin
m 0  m 

N 1
SU(2) Q - function : Q ,    
N ,  ,  ˆ N , , 
N  0 4
N
Degree of polarizati on : PSU ( 2)  1 
1
4  dQ ,  
2
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
SU(2) coherent state : PSU(2)
 N 


 N 1
2
That is, the vacuum state is unpolarized and highly excited states are polarized
Note that:
N
Max{PSU(2)}
1
1/4
2
4/9
4
16/25
Two - mode number state N , N  m : PSU(2)
2N 1  2N 


 1
2 
N  1  2m 
2
N

   1 
N
m
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
PQ  1  Min U
Original
state
 Uˆ 
2
, where Uˆ is a unitary polarizati on transfo rmation
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.
For the vacuum state PQ  0
 
For a two - mode coherent state  ,  where α  β  N : PQ  1  exp 2 
2
2
2
Orbits
From the definition of PQ it follows that if   Uˆ  then the
two states have the same degree of quantum polarizati on.
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
Orbits type 1 : Discrete orbits, generating states N , N  n
Number of orbits N 2  1 (rounded downwards to the nearest integer)
 cos  


Orbits type 2 : Generating states cos  2,0  sin  0,2   0  for N  2
 sin  


 ei cos  sin  


 sin  cos  

 for N  3, etc.
  sin  sin  
 ei cos  sin  


Orbit generating states where the orbit spans
the whole Hilbert space
As we have seen, the state  2,0  0,2

2 can generate a whole basis set (3 states).
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.
It can be shown that the state  3,0  0,3

2 can generate a whole basis set for N  4.
Another basis generating state for N  4 is  2,1  1,2

2.
In higher excitation manifolds it is not known if it is possible to find completebasis generating orbits, but it seems unlikely.
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
2
Generated state:
2 i
2 i

0,2  
2,0
6
6
1,1
BBO
Type II

4
Phase shift
  arccos 23

4


2
HWP

Coincidence
Detector

4
Phase shift

Projection onto the state  .2
(This state causes coincidence counts.)
PBS
Detector