A New Kind of Even and Odd Nonlinear Coherent States and Their Quantum Statistical Properties

In the paper, a new kind of even and odd nonlinear coherent states is constructed in the Fock space. By using numerical method, their quantum statistical properties such as amplitude-squeezing and anti-bunching are investigated. These results show that in the range of  , the new kind of even and odd coherent states appear amplitude-squeezing effect in the direction X2 and anti-bunching effect.


Introduction
The concept of coherent states was introduced by Glauber in 1963.The coherent states have attained an important position in optics quantum because they not only have physical substance but also yield useful representation (Fan H. Y., 1987).As is well known, the coherent states are eigenstates of annihilation operator a of harmonic oscillator, i.e.

 
a    .On the basis of the concept of the coherent states, the even and odd coherent states were introduced (S.Sivakumar, 1998).They are eigenstates of the operator 2 a , i.e. 2 2 , ,     a    , and they are the symmetric and asymmetric combination of coherent states, respectively.They have two kinds of nonclassical effect: the event coherent states are possessed of squeezing but on antibunching effect, however, odd coherent states has antibuching but on squeezing effect (Wu W., et al., 2008).
The concept of nonlinear coherent states was introduced by de Mators Filho R. L. and Vogel W. in 1996.They are eigenstates of a kind of nonlinear operator ( )  af N , which is called as f type harmonic oscillator annihilation operator, such as ( )  was defined by Stefano and Mancini (1997), i.e.
In 1992, the concept of inverse operators about harmonic oscillator is introduced by Metha C. L. and Roy A. K. (1992): Apparently, 1  a is the same as a an annihilation operator, they both let optical number decrease; 1  a is the same as  a a creation operator, they make the optical number increase.Their properties have investigated.After they act on coherent state and squeezing state, lot of new states are derived (Fan H. Y., 1993;Fan H. Y. & Fan T. F., 1994).The method is useful in quantum optics.Therefore, in the article, a kind of nonlinear operator 1 ( )   A a f N is introduced under the influence of paper (De Mators Filho R. L. & Vogel W., 1996;Roy B. & Roy P., 2000;Stefano & Mancini, 1997), and then its nonlinear coherent states, nonlinear even and odd coherent states were obtained, non-classical effects of the new even and odd coherent states are discussed.

New Even and Odd Nonlinear Coherent States
For convenience, a kind of annihilation and creation operator about f type harmonic oscillator is defined below where a and  a are annihilation and creation operator, respectively, is the inverse operator of  a , f is the non-negative function of the number operator.By means of the equations (1), (2), the relations below are derived: where . According to the expression (6), even ( )   and odd ( )  f -coherent states may be defined in straightforward manner as where , ,

Statistical Properties of Even and Odd f -coherent States
According to the expression (7), the non-classical properties of squeezing and anti-bunching are discussed in the paper.

Amplitude Squeezing Effect
At first, two hermite normal operators of complex amplitude operator are defined below: , we called that optical field exists two order squeezing effect in the component i X .In view of indicating magnitude of squeezing in i X , the squeezing degree is introduced, in order to characterize the compression degree, can be defined as the compression degree (Wang J. S., 2002).
, the squeezing effect exists in the component i X .The magnitude of (1) i D describes the squeezing degree.
(1) 1 remarks that squeezing degree is 100% in the component i X .
Under the expression (7), the below relations are obtain:  is real number.
taking ( 13)~( 16) into ( 11) and ( 12) respectively, these expression are obtained The squeezing properties of even and odd coherent state in ( 7) are researched under the conduction that Lamb-Dicke parameter  obtains certain constant.By means of numerical calculation technique, some of curves are obtained, which show squeeze degree varying with complex parameter  .Dot line and solid line denote even and odd coherent state squeezing degree respectively.In the Figure 1, 0.25   , the argument of  is zero.
The Figure 1 shows that in range of  and at given lamb-Dicke parameter  , the new kind of even and odd coherent stastes appear squeezing effect in the direction 2 X .For example, under 0.25   , even coherent state appears amplitude-squeezing effect in range of 0 0 .7 6    in the direction 2 X ; Odd coherent state appears squeezing effect in range of 0.57 0.72    in the direction 2 X ; Specially, in range of 0.57 0.72    both even and odd coherent states appear amplitude-squeezing effect in the direction 2 X .As is well known that usual even coherent state always appear squeezing effect and usual odd coherent state doesn't appear effect.Hence (7) define the even and odd coherent states appearing different properties with usual even and odd coherent state in amplitude squeezing effect.

Photon Anti-bunching Effect
As to single mode field, two order correlation degree may be defined , photon shows anti-bunching effect, i.e. optical field appears non-classical effect.According to the definition of ( 7), ( 18) is put into (19), two order degree of even and odd coherent states are obtained below By means of numerical calculation technique, some of curves is obtained, which show two order correlation degree varying with complex parameter  when Lamb-Dicke parameter is given.Dot line and solid line denote two order correlation degree of even and odd even respectively.In the Figure 1, 0.3   and the argument of  is zero.
The Figure 2 shows that new even and odd coherent state appear anti-bunching effect for given Lamb-Dicke parameter  in range of  .For example, under the condition


. As is well known that general odd coherent state always appear anti-bunching effect and odd coherent state doesn't appear anti-bunching effect.Hence (7) define the even and odd coherent states appearing different properties with usual even and odd coherent state in anti-bunching effect.

Conclusions
The kind of new even and odd coherent states are introduced in the paper.Their properties of squeezing and anti-bunching effect have studied.The results show that these even and odd coherent states have rather different statistical properties from those of the usual even and odd coherent states in non-classical effect.As to given Lamb-Dicke parameter  , new even and odd coherent states appear squeezing effect meantime in range of  ; New even and odd coherent states may appear anti-bunching effect.

Figure 2 .Figure 3 .
Figure 1.The squeezing degree curves of even coherent (solid line) and odd coherent (dot line) vary with  at 0.25  