Download 1 simultaneous polarization squeezing of all three stokes

ICOP 2009-International Conference on Optics and Photonics
CSIO, Chandigarh, India, 30 Oct.-1 Nov. 2009
SIMULTANEOUS POLARIZATION SQUEEZING OF ALL THREE STOKES
PARAMETERS IN PURE PHOTON NUMBER STATES
Ranjana Prakash1,2* & Namrata Shukla1**
1
2
Physics Department, University of Allahabad, Allahabad 211002, INDIA
M.N. Saha Centre of Space Studies, Institute of Inter-Disciplinary Studies, University of
Allahabad, Allahabad 211002, INDIA
[email protected] [email protected]
Abstract: We study Polarization squeezing of a pure photon number state which is obviously polarized but if the
basis of linear polarization is adopted, simultaneous polarization squeezing in all the stokes parameters is
observed. We use our definition of Polarization squeezing, which is similar to the recent definition of Prakash
and Kumar used for Atomic Squeezing and formulated by uncertainty relations taken into account.
1.INTRODUCTION
Products of quantum fluctuations in two noncommuting observables satisfy the uncertainty
relation but the individual fluctuations can be
reduced and this gives the well known concept of
squeezing. In classical optics the polarization state
of light beam can be visualized as direction of a
Stokes vector in Poincare sphere and is determined
by the four Stokes parameters [1,2] S0 , S1 , S 2 , and
S3 . Quantum nature of Polarization of light [3-8]
has received more attention in a number of papers
because of more chances in the same for Quantum
information communication. The quantum Stokes
parameter operators [9-14] and associated Poincare
sphere describe the quantum optical polarization
properties of light, that can also apply to non
classical light [10,12,15-18].
Operators, S0 , S1 , S 2 , and S3 follow commutation
relations and their variances are restricted by
quantum
mechanical
uncertainty
relations.
Polarization squeezing is a non classical property
which can be mathematically defined with a
criterion developed using the uncertainty relations
under consideration. Polarization squeezing was
firstly discussed by Chirkin [9] and the quantum
Stokes parameters and polarization squeezing is
quite widely studied [9-11, 19-23]. Some
definitions for polarization squeezing have been
given and formation and conversion of polarization
squeezed states has been done [24-26]. It has also
been defined in such a way that uncertainty
relations as well as MUS limit are satisfied [23, 27]
but using this definition we can investigate
squeezing only along the coordinate axes formed
∧
∧
∧
by stokes parameter operators S 1 , S 2 , S 3 .
The main outline of this paper is as follows: We
give a general definition of Polarization squeezing
and using that definition, we investigate simultaneous
squeezing in all the three components of stokes
parameter operators.of a pure photon number state
which is polarized, by adopting the linear basis of
polarization.
2. CRITERION FOR POLARIZATION
SQUEEZING
The Hermitian Stokes operators can be defined as
quantum versions of their classical counterparts [1, 2],
by
∧
∧
∧
∧
∧
∧
∧
∧
∧
∧
S 0 = a x† a x + a y† a y , S1 = a x† a x − a y† a y ,
∧
∧
∧
∧
∧
∧
 ∧ ∧ ∧ ∧  (1)
S 2 = a x † a y + a y † a x , S 3 = i a y † a x − a x † a y ,


∧
∧
∧
∧
where a x , a y , a x † , a y † refer to the photon annihilation
and creation operators respectively of two orthogonal
polarization modes x and y satisfying the commutation
∧ ∧ 
relations, a j , a k  = δ jk for j , k = x, y . The operator


∧
S 0 is equal to the total photon number operator and
corresponds to the beam intensity. Polarization state is
∧
∧
described by the four operators S 0 and S (given by
∧
∧
∧
S 1 , S 2 and S 3 ). Commutation relations for these
stokes parameters are,
∧
∧ ∧ 
∧ ∧ 
 S 0 , S i  = 0 , i = 1,2,3 ,  S 1 , S 2  = 2i S 3 ,




∧
∧
∧ ∧ 
∧ ∧ 
(2)
 S 2 , S 3  = 2i S 1 ,  S 3 , S 1  = 2i S 2 .




These relations are parallel to the commutation
relations for components of the angular momentum
operator [28-31, see also 32]. These non zero
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ICOP 2009-International Conference on Optics and Photonics
CSIO, Chandigarh, India, 30 Oct.-1 Nov. 2009
commutators show that simultaneous exact
measurement of the quantities represented by these
Stokes operators are impossible and the following
uncertainty
relations
hold,
2
∧
2
∧
, V2V3 ≥ S 1
V1V2 ≥ S 3
The criterion for squeezing of the component.V1 is
then
∧
, V3V1 ≥ S 2
∧
.
∧
of
∧
2
∧
∧
2
∧
i.e. variance of stokes operator S j should be less
than the corresponding variance in the coherent
mode [331] . Any polarization state which follows
equation (6) will be a non-classical state. This
definition of polarization squeezing actually does
not contain minimum uncertainty state (MUS)
limits of equation (3) and hence may show only
quadrature squeezing observed by calculating
stokes parameters.
According to the improved definition of
polarization squeezed states containing MUS limit,
polarization squeezed state [23, 27] is one in which
one of Stokes variances lies not only below the
coherent limit but also below respective MUS limit.
i.e.. light is said to be polarization squeezed [34-36]
if,
∧
V j < S l < Vk , j, k , l = 1,2,3 , j ≠ k ≠ l ≠ j
(6)
.We now make more general considerations and
define polarization squeezed states with squeezing
any general direction in the coordinate axes system.
If one considers a set of three mutually
perpendicular unit vectors ( n̂ , nˆ ⊥1 , nˆ ⊥ 2 ), for
of
^
^
S n ⊥1 = n ⊥1 . S
∧
S
and
^
along these,
^
^
S n⊥2 = n ⊥2 . S
=
(n.S),
V = (∆S nˆ ) ≤ 1 (Snˆ ⊥1 or/and 1 (Snˆ ⊥2 .. (7)
2
2
Since the maximum value of the right hand sides is
2
− S nˆ
2
.
(10)
D = 1- S
(11)
3.PHOTON NUMBER STATE AND
POLARIZATION SQUEEZING
Pure state with N photons polarized along complex
unit vector,
∧ ∧
∧
(12)
ε = e x cos (θ / 2 ) + e y sin (θ / 2 )e iφ
can be written as,
ψ = (N! ) −1/2 (ε x a x † + ε y a y † ) N 0,0 x,y
= 0,0
, the most general criterion for
polarization squeezing is [4-7], is
V = 〈∆Sn2〉 < [〈S〉 2 − 〈Sn〉 2] ½.
(8)
(13)
ε,ε⊥
in (ε , ε ⊥ ) basis with,
∧
∧
∧
∧
∧
∧
*
*
a ε = ε x a x + ε y a y , a ε ⊥ = −ε y a x + ε x a y
(14)
If we consider the same state in ( x , y ) basis of
polarization, creation and annihilation operators in
both the modes can be written as,
∧
∧
∧
∧
∧
∧
∗
*
a x = ε x aε − ε y aε ⊥ , a y = ε y aε + ε x aε ⊥
(15)
Initially, the state of polarization is N ,0 and
ε x = cos(θ / 2), ε y = e iφ sin(θ / 2)
(16)
Straight forward calculations for expectation values
of the Stokes parameters and their variances for the
state (13) give
∧
∧
S 0 = N , S 1 = N cos θ = n x
∧
S 2 = N sin θ cos φ = n y ,
^
2
2
2
The Degree of Squeezing is often defined as
S n = n. S ,
uncertainty relations give squeezing if
S − Snˆ
S
∧
2
(4)
S2 + S3 = S0 + 2S0 ,
and it is fuzzy and ill defined because of the
inherent uncertainties.
. According to the definition [9-10] a state is said to
be polarization squeezed if,
V j < V j (coh), j = 1,2,3
(5)
component
V
S =
the quantum stokes parameter S 1 and so on.
The mean value of the radius of quantum Poincare
sphere is given by square root of expectation value
of either side of the equation,
S 12 +
(9)
We also define squeezing factor, as the ratio of the
left and right hand sides of Eq. (8)
∧
∧
2
∧
+ S3
S2
(3)
2
S 12 − S 1
Here V1 stands for the variance
2
∧
V1 <
∧
(17a,b,c)
S 3 = N sin θ sin φ = nz ,
and,
(
)
= N (1 − sin θ cos φ ) = N (1 − n
= N (1 − sin θ sin φ ) = N (1 − n
V1 = N 1 − cos 2 θ = N (1 − n x 2 )
V2
2
2
2
2
V3
where
n = nx e x + ny e y + nz e z
y
z
2
2
) , (18a,b,c)
)
= cos θ e x + sinθ (cosφ e y + sinφ e z )
(20)
2
ICOP 2009-International Conference on Optics and Photonics
CSIO, Chandigarh, India, 30 Oct.-1 Nov. 2009
Is a unit vector in the direction specified by the
angles θ and φ .
∧
As per the criterion (9), S 1 would be squeezed if
2
∧
V1 <
2
∧
. Eqs. (18a) and (17b,c)
+ S3
S2
then give, (1-nx2) < (1-nx2)1/2, which is always true
except when nx is 0 or 1. Similary conditions for
squeezing
are V 2 <
∧
of
∧
S3
2
∧
+ S1
∧
and
S2
2
, V3 <
∧
S1
S3
2
∧
2
+ S2
, which lead to (1-ny2) < (1-ny2)1/2 and (1-nz2) < (1nz2)1/2 , respectively, which are always true unless
the component of n under consideration has the
value 0 or 1.
Thus we find that all Stokes parameters (all
∧
Components of S ) are squeezed in general. To
study the exceptions, we may consider the
squeezing factors, given by Eq. (10).
Squeezing factors corresponding to all these
operators are as,
S1 = 1− cos 2 θ = , (1-nx2),
S2 = 1 − sin 2 θ cos 2 φ = , (1-ny2)
S3.= 1 − sin 2 θ sin 2 φ
2
= , (1-nz )
This shows that any Stokes parameter which has a
nonzero mean value, i.e., is in a direction
perpendicular to n̂ , is squeezed.
4. RESULT AND DISCUSSION
Starting with a pure number state, polarization
squeezed light can be achieved and that is possible
just by changing the basis of polarization of pure
photon number state. Squeezing factors show that
all the stokes parameter operators visualizing
polarization of state are found to be squeezed.
ACKNOWLEDGEMENT
We would like to thank Prof. H. Prakash and
Prof. N. Chandra for their support, interest and
critical comments.
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CSIO, Chandigarh, India, 30 Oct.-1 Nov. 2009
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