Nonclassical Properties of Even and Odd Semi-Coherent States

IOSR Journal of Applied Physics (IOSR-JAP)
e-ISSN: 2278-4861.Volume 9, Issue 1 Ver. III (Jan. – Feb. 2017), PP 59-67
www.iosrjournals.org
DOI: 10.9790/4861-0901035967 www.iosrjournals.org 59 | Page
Nonclassical Properties of Even and Odd Semi-Coherent States
H. H. Salah
Department of Physics, Faculty of Science (Girls Branch), Al-Azher University, Nasr City, Cairo, Egypt
Abstract: Even and odd semi-coherent states have been introduced. Some of the nonclasscial properties of the
states are studied in terms of the quadrature squeezing as well as sub-Poissonian photon statistics. The Husimi–
Kano Q-function and the phase distribution in the framework of Pegg and Barnett formalism, are also
discussed.
Keywords: Semi-coherent states, quadrature squeezing, sub-Poissonian statistics, quasi-probability
distribution function (Q function), Phase distribution.
I. Introduction
Over the past four decades, various works were devoted to the study of coherent states (CSs) and their
application in different branches of physics from both theoretical and experimental points of view. The concept
of CSs was first introduced by Schrodinger as Gaussian wave-function that could be constructed from a
particular superposition of the wave functions corresponding to the discrete eigenvalues of the harmonic
oscillator [1]. These states prove a close connection between quantum and classical formulations of a given
physical system. The importance of CSs was put forward by Glauber [2, 3] in quantum optics as eigenstates of
the annihilation operator zzza ˆ , which he realized that these states have interesting property of
minimizing the Heisenberg uncertainty relation, since the product of their uncertainties in the position and
momentum saturates the Hiesenberg uncertainty relation with equal values in each quadrature. Moreover CSs
were reintroduced by Klauder in [4, 5] by action of the Glauber displacement operator,   azaz
ezD ˆˆ 
 , on the
vacuum states, where z is a complex variable. These states can be served as a starting point to introduce non-
classical states that have attracted considerable attention in quantum optics over the last decades. Over the last
four decades there have been several experimental demonstrations of non-classical effects, such as photon anti-
bunching [6], sub-Poissonian statistics [7] and squeezing [8]. Also, significant progress was achieved in the
development of practical realizable schemes of nonlinear quantum states, such as f-CS or nonlinear CS [9–15],
even and odd CS [8, 16] and semi CS [17]. These states have attracted much attention in recent decades due to
their helpful applications in various fields of research [10, 18–22].
For instance, the concept of the semi-CSs was introduced in [23] as the states of a harmonic oscillator
which possess time-independent values of the quadrature variances x and p , different from the vacuum (or
coherent state) values. These states are different from the CSs in not having the minimum value 2
1
for
the uncertainty product px  . . They have shown that the necessary and sufficient condition for such states is
22
ˆˆ aa  , (1)
Where   2/ˆˆˆ pixa  is the usual bosonic annihilation operator (using the units 1 m ).
Condition (1) is obviously satisfied for the usual coherent states , for which  aˆ . Another trivial
example is the Fock state n , for which 0ˆ2
nan .
An immediate consequence of equation (1) is the equality of the quadrature variances, x = p , and zero value
of the covariance, 0ˆˆˆˆˆˆ2
1
 pxxppx . Thus ‘semi-coherent’ states cannot exhibit usual (second-order)
squeezing.
A nontrivial example of ‘semi-coherent’ states, only briefly discussed in [23], is a normalized superposition of
two coherent states of the form
 2
1
2
1 




 . (2)

The notation  means that the state (2) is orthogonal to the state  :
0 .
Therefore, the state  can be considered as an orthogonal projection of the coherent state  to another
coherent state  . In the state (2) one has [23]
2
22
2
1
1 




 px , (3)
so that, indeed, x = 2
1
p .
Later Another investigation have demonstrated that firstly, state (2) has the same average value of the
annihilation operator aˆ as the coherent state   aˆ, .for any value of  , secondly, a
possibility of introducing a large family of ‘generalized semi-coherent states’, which satisfy property
nnn
aa  ˆˆ for an arbitrary operator Aˆ ( All such states have the same structure as the superposition (2),
where the components  and  should be understood as eigenstates of the operator Aˆ and thirdly semi-
CSs possess other nonclassical properties, in particular, the possibility of the fourth-order squeezing and sub-
Poissonian statistics [17]. Recently, it has been demonstrated that Semi-CSs as superposition of two coherent
states, are suasible states for considering upper bounds for the relative energy difference with a fixed fidelity
between them [24]. Also a new family of semi-CSs for a charged particle moving in a constant magnetic field
have been introduced [25]. These states interpreted as a nonlinear coherent states (NLCSs) with a special
nonlinearity function. Another investigation demonstrated that, the semi-CSs can be considered as a special case
of nonlinear semi-CSs when f (n) = 1. Then, the nonclassical properties such as quadrature squeezing,
amplitude-squared squeezing, fourth-order squeezing, number and phase squeezing and sub-Poissonian statistics
for the introduced states in addition to their Pegg–Barnett phase distribution are generally studied [26].
The main goal of this paper is the introducing of even and odd semi-CSs. Our motivation is to search
for more nonclassical properties in the obtained states, since, as reported in [17, 23], the semi-CSs only possess
sub-Poissonian statistics and fourth-order squeezing. The paper is organized as follows. In section II, we
introduce the even and odd semi-CSs. Certain nonclassical properties associated with these states are
deduced, in section III, such as the quadrature squeezing, sub-Poissonian distribution, the Husimi-Kano Q-
function and Phase distribution. Finally the conclusion is presented in section IV.
II. Introducing Even and Odd Semi-CSs (ESemi-CSs and OSemi-CSs)
The discrete energy values corresponding to motion of a charged particle on the infinite flat surface in
the presence of a uniform external magnetic field perpendicular to this plane are called Landau levels [27]. The
exact solvability of the symmetric-gauge Landau Hamiltonian,
    32
1
2
1
LbbaaH   
 with  /3 iL
have been considered. It includes an infinite-fold degeneracy on the Landau levels [28], i.e.
  mnnmnH ,, 2
1
  , (4)
Where Landau cyclotron frequency, ω, is expressed in terms of the value of the electron charge, its mass, the
magnetic field strength B and also the velocity of light as MceB / . The number m is an integer and n is a
non-negative one together with mn  limitation. Each pair of creation and annihilation operators of the
system,  
aa, and  
bb, , form two separate copies of Weyl-Heisenberg algebra,
1],[ 
aa , 1],[ 
bb , 0],[],[],[],[  
babababa , (5)
With the unitary representations as
1,1,  mnnmna , mnnmna ,1,1 
, (6)
1,,  mnmnmnb , mnmnmnb ,1, 
. (7)
The creation and annihilation operators of the system are used to express the coordinates x and y and their
conjugate momenta, i.e., xp and yp ,

 
 aabb
M
x
2

,  
 aabb
Mi
px
22

, (8)
 
 aabb
M
iy
2

,  
 aabb
M
py
22
1 
, (9)
Using the constructed the minimum uncertainty two variable CSs, , for a charged particle in the magnetic
field as [28],
b
n
n
n
baa
n
ee 










0
2
0
!
,
2
(10)
where they satisfy following eigenvalue equations
 ,, a ,
 ,, b . (11)
Also,
b
n
 are the normalized Glauber CSs, which are obtained by using the displacement operator
  bb
eD 
 
 acting on the Landau level nn , :
 
 
mn
mn
ennD
nm
mn
b
n
,
!
, 2
2










(12)
By using the orthonormall relation of Landau levels, i.e., mmnnmnmn  ,, , one can deduce that
coherent states
b
n
 and
a
m
 form two separate complete bases, that is, nn
b
n
b
n 
  and
mm
a
m
a
m 
  . As mentioned in previous section, , satisfy the condition (1) and thus can be regarded
as semi-CSs, which can be introduced by substituting state
b
n
 (12) into (10) the state  can be
rewritten in the Fock basis as
n
n
ee
N n
nn







0
2
!
1
2
2
2




. (13)
2 (a) ESemi-CSs:
From equation (13) ESemi-CSs is given as
nC
N n
n
e
e
2
1
0
2


  (14)
Where,
!2
1
2222
2
22
n
ee
N
C
nn
e
n




 (15)
and the normalization constant eN is given by
    2
1
2222
22
]]
[
!2
1
[
22
1
2*22
1
2*
244
0
nnnn
nn
n
e
ee
ee
n
N









 
(16)
Where
22 
 
 e

2 (b) OSemi-CSs:
From equation (13) OSemi-CSs is given as
12
1
0
12  


 nC
N n
n
o
o
 (17)
Where,
 !12
1
122122
12
22








n
ee
N
C
nn
o
n


(18)
The normalization constant oN is also given by
 
    2
1
2222
22
]]
[
!12
1
[
122
1
12*122
1
12*
22424
0











 
nnnn
nn
n
o
ee
ee
n
N




(19)
III. Nonclassical Properties of Even and Odd Semi-CSs
Nonclassical properties may be established through investigating quadrature squeezing, Sub-Poissonian
distributions, The Husimi Q-function and phase distribution of the quantum states. In this section, we express
these quantities with respect to even and odd semi-CSs which have been introduced in (14) and (17),
respectively. Recall that a common feature of all of the mentioned criteria is that the Glauber–Sudarshan P-
function [29] of the corresponding field state is not positive definite, i.e. it cannot be illustrated as a (classical)
probability distribution. But, as a well-known fact, we would like to imply that finding this function is ordinarily
a hard task. Altogether, each of the above signs, which will be considered in this paper, is indeed sufficient (not
necessary) for a state to belong to nonclassical states. It should be mentioned that the necessary and sufficient
criterion for the nonclassicality of a state is the subject of a recent research work [30].
3.1. Quadrature Squeezing
In order to study quadrature squeezing [31], we consider the following quadrature operators:
  2/ˆˆˆ 
 aax and   iaap 2/ˆˆˆ 
 . (20)
From which we have
  Cipx ˆˆ,ˆ  , with
2
1ˆ C
Then the uncertainty relation holds    
2
22 ˆ
4
1
ˆˆ Cpx  where   222
ˆˆˆ iii xxx  , xxi
ˆˆ  or pˆ
. A state is squeezed if any of the following conditions holds:
 
4
1
ˆ 2
x or  
4
1
ˆ 2
p . (21)
Now by using Eqs. (20) and (21) the quadrature variances are given in terms of annihilation and creation
operators expectation values by
   nax ˆˆRe
2
1
4
1
ˆ 22
 ,
   nap ˆˆRe
2
1
4
1
ˆ 22
 (22)
The state would possess xˆ or pˆ quadrature frequency sum squeezing if the S -factor defined by
 
C
Cx
SX
ˆ
ˆˆ2
2

 , or
 
C
Cp
Sp
ˆ
ˆˆ2
2

 (23)

Satisfies the inequalities 01  XS or 01  pS . For even or odd semi-CSs 0ˆˆ  
aa . Also
the expectation values of the square annihilation operator and the mean photon number for even and odd semi-
CSs are respectively given by
 122ˆ
0
2
2
2
 


nnCa
n
ne
, (24)
 nCn
n
ne
2ˆ
0
2
2


 , (25)
 122ˆ
0
2
12
2
 


 nnCa
n
no
, (26)
 12ˆ
0
2
12  


 nCn
n
no
. (27)
In Fig. 1, the squeezing function corresponding to states
e
 and
o
 versus the parameters α and β
have been plotted. These figures show that for both states, xS are always negative, and pS are always positive
for all values of α and β, see Fig. 1 (a, c) and (b, d) respectively. This means that squeezing occurs, for both
states, only in the x quadrature and so nonclassicality feature of these states is revealed.
( a ) ( b )
( c ) ( d )
Fig. 1 The squeezing function xS and pS as a function of the parameters α and β (a and b) for the even semi-
CSs and (c and d) for the odd semi-CSs.
3.2. Sub-Poissonian Distributions
In this subsection we shall investigate the sub-Poissonian distributions for the even and odd semi-CSs. The
distribution can be measured by using photon detectors based on photoelectric effect [32; 33]. The second-order
correlation function, denoted  02
g , takes the following simple form [34],
  2
22
2
ˆ
0
aa
aa
g



This expression is easily rewritten in terms of the photon number operator and can then be evaluated for even
and odd semi-CSs using Eqs. (25,) and (27) as

  2
2
2
ˆ
ˆˆ
0
n
nn
g

 . (28)
If
 
   1102
g , then the light field exhibits sub(super)-Poissonian distribution; if
 
  102
g the
distribution is called Poissonian; if
 
   2202
g the distribution is referred to as thermal (super-thermal).
Fig. 2 (a, b) represents the second order correlation function for the even and odd semi-CSs
respectively. As shown in this figure, for even semi-CSs  02
g ˃ 2 in a specific regions of α and β indicating
super-thermal-Poissonian distribution then with increasing both α and β the function becomes super-Poissonian.
In case of the odd semi-CSs,  02
g > 1, in a specific regions of α and β, this means that it shows super-
Poissonian distribution. With increasing both α and β parameters the function changes gradually to sub-
Poissonian distribution. This means that both states has nonclassicality feature.
( a ) ( b )
Fig. 2 Plot of the second order correlation function as a function of α and β, (a) for the even semi-CSs and (b)
for the odd semi-CSs.
3.3. The Husimi – Kano Q-function
In this subsection we shall consider the Husimi - Kano Q-function associated with the even and odd semi-CSs.
The Q-function is positive-deﬁnite at any point in the phase space for any quantum state and can be also
calculated in terms of the following formula [35]
   1
Q , (29)
where  is the density matrix for the even and odd semi-CSs and  is the usual coherent state.[2] Thus the Q-
function for light ﬁelds in the even and odd semi-CSs will be given respectively, by
3.3 (a) For the ESemi-CSs
 
2
21 
 
 eGQ ee (30)
where
n
n
n
e C
n
G 2
0
2*
!2





( a ) ( b )

( c ) ( d )
Fig. 3 The eQ - function against γ for even semi–CSs at fixed β = 2 (a) α = 0, (b) α = 1, (c) α = 2 and (d) α = 3.
Since the maximization or minimization of the  eQ -function depends on the parameter α.
Therefore, our main task is to examine the behaviour of the  eQ -function due to the variation in the α-
parameter. For this reason we plot  eQ in Fig. (3) for different values of the α-parameter keeping β parameter
unchanged. For ﬁxed β = 2 and for instance when we consider α = 0, the function exhibits one peak but with
squeezing apparent on the contours of the base, see Fig.3 (a). For α = 1, we note that more squeezing occur on
the contours of the peak base where it is elliptically shaped, see Fig.3 (b). For α = 2, we note that the peak splits
into two peaks but they are joined near of the base and each base has a crescent like shape as shown in Fig.3 (c).
More increase in the value of α = 3 leads to a complete separation of the two peaks, see Fig. 3 (d). It is also
noted that there is a slight difference between the heights of the peaks. This means that there is a slight
asymmetry in the function shape which reﬂects the effect of the squeezing.
3.3 (b) For the OSemi-CSs
 
2
21 
 
 eGQ oo (31)
where
  12
0
12*
!12




 
 n
n
n
o C
n
G

Where nC2 and 12 nC are given by Eqs.(15) and (18), respectively.
( a ) ( b )
( c ) ( d )
Fig. 4 The  oQ - function against γ for odd semi–CSs at fixed β = 2 (a) α = 0, (b) α = 1, (c) α = 2 and (d)
α = 3.

In order to visualize the behaviour of  oQ -function we plot the  oQ -function in Fig. (4) for the
same values of the α and β –parameters as in  eQ . The function exhibits two peaks but with squeezing
apparent on the contours of the base at α = 0, see Fig.4 (a). In Fig.4 (b), for α = 1, we note that the function
shows a two-peak squeezed structure with pattern of interference at the base. From Fig. 4 (c), it is seen that, for
α = 2, the function has the same behavior as in the case α = 0 with a decrease in the peak height. More increase
in the value of α = 3 leads to a complete separation of the two peaks, see Fig. 4 (d). Generally, a slight
difference between the heights of the peaks has been observed with changing α value. Also, the slight
asymmetry in the function shape may be due to the effect of the squeezing.
3.4. Phase distribution
Phase distribution of a quantum state of the electromagnetic field provides information about
uncertainty in the phase of the field. This, in turn, imposes a lower limit on the fluctuations in the number of
photons. A useful approach to define phase distribution is the Pegg-Barnett formulation [36]. Phase distribution
of an arbitrary state of the field is obtained by the overlap of the state with the phase state. Pegg and Barnett
defined a complete set of s + 1 orthonormal phase states p by
 nin
s
s
n
pp 

0
exp
1
1
 , (32)
Where
1
2
0


s
p
p

 with sp ,.....,1,0 and an arbitrary value of 0 based on these phase states, the
Hermitian phase operator is defined as
pp
s
p
p  

0
ˆ (33)
Henceforth, the Pegg–Barnett phase probability distribution of an arbitrary state  may be defined as
 
2
2
1
lim 

 p
s
s
p



. (34)
Now inserting the introduced states (14) and (17) into (34), the respective phase distributions for the states
e
 and
o
 are,
     






 





0 0
*
22
1
22cos212
n m
mne mnCCP  (35)
     






 






0 0
*
1212
1
22cos212
n m
mno mnCCP  (36)
Fig. 5 The phase distribution  PBP against θ for a fixed value β= 1 and different values of the parameter α =
1, 2 and 3, for even semi-CSs.
In Fig. 5, we have plotted the Pegg–Barnett phase distribution for even semi-CSs for various values of
α = 1, 2 and 3 keeping β fixed at 1. Generally, for even semi-CSs, the figure show that all of the graphs have
single peaks at 0 and the distribution is symmetric around this peaks with two lateral wings at  

developing. As α increase, the distribution  PBP peaks become sharper and higher. In fact, the same results
and behavior have been seen also for odd semi-CSs.
IV. Conclusion
In this article we have introduced and discussed some nonclassical properties of even and odd semi-
coherent states. The quadrature squeezing is investigated. It shows that both even and odd semi-CSs exhibit
quadrature squeezing in x component. The second order correlation function for even and odd semi-CSs is
studied. It shows that even and odd semi-CSs exhibits super-thermal-Poissonian and super-Poissonian
distribution in a specific regions of α and β respectively. On the other hand even and odd semi-CSs with
increasing α and β parameters changes to super and sub-Poissonian distribution respectively. The Husimi - Kano
Q-function associated with the even and odd semi-CSs reveals that Q-function for both states are positive-
definite at any point in the phase space with a slight asymmetry in the function shape which reflects the effect of
the squeezing. Also Q-function for both states tends to split up into two peaks as α-parameter increases. Finally
the phase properties using the Pegg–Barnett formalism are studied for even and odd semi-CSs. For both states
the phase distribution function has one central peak at 0 with two lateral wings at   developing and
as α increase, the distribution  PBP peaks become sharper and higher.
Acknowledgments
The auther would like to thank Mrs. S. Sanad of the Mathemtics Depat-ment, Faculty of Science, Al-Azhar
University (Girls Branch) for helpingin the programs for the figures.
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