Statistical properties of the nonlinear negative binomial state

Optics Communications 274 (2007) 372–383 www.elsevier.com/locate/optcom

Statistical properties of the nonlinear negative binomial state M. Sebawe Abdalla a

a,*

, A.-S.F. Obada b, M. Darwish

c

Mathematics Department, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia b Department Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt c Department of Physics, Faculty of Education, Suez Canal University, Al-Arish, Egypt Received 28 December 2006; received in revised form 4 February 2007; accepted 16 February 2007

Abstract In this context, we introduce and investigate the properties of the nonlinear negative binomial state (the state which interpolates between the nonlinear coherent and the number states). Mainly we concentrate on the statistical properties for such state where we have discussed two different cases of squeezing phenomenon. The first case is the normal squeezing while the second is the amplitude squared squeezing, further the second order correlation function is also considered. Our discussion have been extended to include the quasi-probability distribution functions (W-Wigner and Q-functions). The quadrature distribution and the phase properties in Pegg–Barnett formalism besides the phase variances are considered. Examination of the resonance fluorescence against the present state is given (single atom and thermodynamic limit). It has been shown that the atomic inversion is sensitive to any variation in the nonlinear negative binomial number m. Ó 2007 Elsevier B.V. All rights reserved.

1. Introduction The idea of considering and generalizing quantum states of light have been accumulated in the last decades. In this sense many authors have considered and generalized some fundamental states in different directions. For instance, the SU(2) generalized coherent states are an especially nice example of a successful extension of the coherent state idea. More challenging has been the objective of obtaining generalized coherent states for the Coulomb potential problem, as was originally proposed by Schro¨dinger [1]. In the meantime, it is noted that the coherent state is a linear combination of a pure photon number state jni with coefficients chosen such that the photon counting distribution is Poissonian. The state is widely used as a basis for representation of the radiation field and it can be produced by acting on the vacuum state j0i using the Glauber displace^ ment operator DðaÞ such that [2]

*

Corresponding author. E-mail address: [email protected] (M.S. Abdalla).

0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.02.051

^ jai ¼ DðaÞj0i ¼ expða^ay  a ^aÞj0i   1 1 2 X an pffiffiffiffi jni: ¼ exp  jaj 2 n! n¼0

ð1:1Þ

This in fact leads to the idea of thinking to construct intermediate states with nonclassical properties of light. Therefore, we have seen a considerable effort to generate new quantum state besides the number state and coherent state. Most of these states interpolate between the coherent state and the number state or between the thermal state and the number state. Here we may point out to another extension of the coherent state that is the SU(1,1) generalized coherent states. This state can be produced from the action of ^ the usual squeeze operator SðfÞ (say) on the vacuum state j0,ki. In this case the generalized coherent state is given ^ by jf; ki ¼ SðfÞj0; ki, where k is Bargmann number, for more details see Ref. [3]. In fact the generalized coherent state is essentially known as negative binomial state and can be regarded as one of the intermediate states as we shall see later. Furthermore, we can see the generalized geometric state interpolates between the number state and chaotic state [4–6]. Also one can find the binomial state interpolates

M.S. Abdalla et al. / Optics Communications 274 (2007) 372–383

between the number state and coherent state [7], while the negative binomial state tends for some limiting cases to the coherent state, or to the pure thermal states [8–11], depending upon the value of the parameters. In the meantime, it is easy to prove the negative binomial state tends to the logarithmic state if one removes the vacuum state j0i, [12], and also to show it has some nonclassical effects. Thus from the above we come to conclusion that is, the coherent and the number states would play a great rules to generate and produce a new states with nonclassical properties of light. For more details one may consult Ref. [13] and the references therein. On the other hand, it is well known that to produce a basic photon number state jni we have to act on the vacuum state j0i by the creation operator aˆ  of power n. This is because the operator aˆ  raises the energy of the optical cavity. Using this fact one would be able to produce a new interesting class of nonclassical states of light. For example, added photon states jw,mi = Nmaˆ mjwi, where jwi can be an arbitrary state, m is a positive integer (the number of add quanta) [14–18]. The same concept has been used to introduce the nonlinear coherent states or the f-coherent states based on algebraic aspects, which is the eigenstates ja, fi of the non-boson non-Hermitian operator ^ af ð^ nÞ, for more details see Refs. [19–21]. The concept of nonlinear coherent states is given individually in Refs. [18,22]. However, the authors of Ref. [22] managed to apply it to photon distribution function as well as to the quasi-probability distribution function. Moreover, they also managed to examine its effect on the Planck distribution formula. In the present paper the same process will be employed to introduce and to study the nonlinear negative binomial state. This can be achieved if one managed to modify the negative binomial state given by [13] 12 1  X ðn þ mÞ! 2n 2 mþ1 jm; fi ¼ f ð1  jfj Þ jni; ð1:2Þ n!m! n¼0 where m > 0 (i.e., m in general is any real positive number), 0 6 jfj2 6 1. It should be noted that the negative binomial state as we have mentioned above can be produced from the action of a single mode squeeze operator on the state j0, fi. In this case Eq. (1.2) can be re-written in the form rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 X mþ1 ^ ayn ðm þ nÞ! n jm; fi ¼ f ð1  jfj2 Þ 2 j0i: ð1:3Þ m! n! n¼0 Since our aim of the present work is to study the nonlinear negative binomial state then it would be interesting to introduce the non-commuting operators ^¼^ A af ð^ nÞ ¼ f ð^ n þ 1Þ^ a;

^ y ¼ f y ð^ A nÞ^ ay ¼ ^ ay f y ð^ n þ 1Þ;

y

^ y ¼ f y ð^nÞ^ay or by its canonical conjugate operator ator A y 1 y ^ ^a , then the above equation becomes B ¼ f ðnÞ  1  X mþn 2 1

jm; fif ¼ k

n¼0

n

2

fn ð1  jfj Þ

mþ1 2

f y ðnÞ!jni:

ð1:6Þ

^ and B ^ are the It should be noted that the operators A canonical conjugate of each other and satisfy the commu^ B ^ y , and ½A; ^ B ^ y  ¼ 1 ¼ ½B; ^ A ^ 6¼ 0. Protation relation ½A; pffiffiffiffi y vided we have taken the case in which f ðnÞ! ¼ 1= n! for pffiffiffiffi ^ or f ðnÞ! ¼ n! for the operator B, ^ then the operator A Eq. (1.6) takes the form rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 X ðm þ nÞ! n 2 mþ1 1 f ð1  jfj Þ 2 jni: ð1:7Þ jm; fif ¼ k m! n! n¼0 and hence the normalization constant k has the expression 2 ðmþ1Þ

2

jkj ¼

ð1  jfj Þ

2

expðjfj Þ 2

Lm ðjfj Þ

:

ð1:8Þ

where Lm(z) being the Laguerre polynomial defined as m X m!zr r : ð1:9Þ ðÞ Lm ðzÞ ¼ 2 ðr!Þ ðm  rÞ! r¼0 Before we go further we should emphasis that in obtaining the state given by Eq. (1.6) it is not just a matter of replacing certain operator by another one has different properties. In fact we have been used the SU(1,1) realization to derive such state. This will be seen in the following subsection. 1.1. SU(1,1) realization of the nonlinear negative binomial state In this subsection we shall give in details the full derivation for the state given by Eq. (1.6). To do so we have to look for an SU(1,1) group realization of the nonlinear negative binomial state. For this reason we assume that f ð^ nÞ is an operator valued unitary function i.e. f y ð^nÞ ¼ f 1 ð^ nÞ. Thus we can define the generators of SU(1,1) in this case as follows: pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi ^ þ ¼ ^ay f y ð^nÞ m þ ^n; K ^  ¼ m þ ^nf ð^nÞ^a; K ^0 ¼ m þ ^ K n; 2 ð1:10Þ ^  , and K ^ 0 satisfy the SU(1,1) commutation where K relations ^ 0; K ^   ¼ K ^ ; ½K

^ þ; K ^ 0  ¼ 2K ^ 0: ½K

ð1:11Þ

ð1:4Þ

Consequently, we can define a unitary evolution operator ^ II ðgÞ ¼ expðnK ^  Þ; ^ þ  n K D ð1:12Þ

ð1:5Þ

which can be disentangled into m ^ II ðgÞ ¼ expðnK ^  Þ; ^ þ Þð1  jnj2 Þ 2 expðn K D g tanh jgj: where; n ¼ jgj

with the property ^ A ^ y  ¼ ð^ ½A; n þ 1Þf 2 ð^n þ 1Þ  ^ nf 2 ð^ nÞ;

373

where ^ n¼^ a^ a is the photon number operator. Therefore if we replace the creation operator aˆ  in Eq. (1.2) by the oper-

ð1:13Þ

374

M.S. Abdalla et al. / Optics Communications 274 (2007) 372–383

When this unitary operator is applied on the vacuum state j0i we obtain 1 n m X n ðK þ Þn j0i jm; n; if ¼ ð1  jnj2 Þ 2 n! n¼0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 X ðm þ nÞ!  2 mþ1 n ðf ðnÞÞ!jni; ð1:14Þ ¼ ð1  jnj Þ 2 n n! n¼0 which is the state given by Eq. (1.6) but here f (n) is a unitary complex function such that f*f = 1. For the non-unitary case we can also use a similar expansions, however, we shall employ the canonical conju^ gate operators. For this reason we define the operators B ^ to be the canonical conjugate of the operators K  such that 1 ^þ ¼ ^ B ay pffiffiffiffiffiffiffiffiffiffiffiffi ; mþ^ nf ð^ nÞ

1 ^  ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ^ B a; mþ^ nf ð^ nÞ

ð1:15Þ

where we have ^ ; B ^ þ  ¼ 1; ½K

^; K ^ þ  ¼ 1: ½B

ð1:16Þ

Now suppose we define the operator ^ 1 ðgÞ ¼ expðnK ^  Þ; ^ þ  n B D

ð1:17Þ

which is not a unitary operator. Yet we can decompose this operator into the following ^  Þ: ^ 1 ðgÞ ¼ expðjnj2 Þ expðnK ^ þ Þ expðn B D

ð1:18Þ

The effect of this operator on the vacuum state j0i leads to the nonlinear negative binomial state (1.6) apart from a normalization factor. Furthermore we may also define ^ þ by using another state associated with the operator B another operator ^  Þ: ^ 2 ðgÞ ¼ expðnB ^ þ  n K D

ð1:19Þ

But in this case the state looks different from that of (1.6) because  s 1 X ns y 1 2 ^ pffiffiffiffiffiffiffiffiffiffiffiffi j0i ð1:20Þ ^ D2 ðgÞj0i ¼ expðjnj Þ a s! f ð^ nÞ m þ ^ n s¼0 apart from a normalization constant k it defines another set of non-linear states of the form 1 X nn pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jni: ð1:21Þ jniB ¼ k n!ðf ðnÞÞ!ðn þ mÞ! n¼0 As it has been mentioned above, these states belong to the class of nonlinear coherent states. The physical meaning of such states was elucidated in [23] where it was shown that they may appear as stationary states of the centreof-mass motion of trapped ions. Furthermore, they may be related to nonlinear processes in high intensity photon beams [22,24]. Having constructed the nonlinear negative binomial state, we are therefore in position to discuss some of the non-classical properties related to this state. This will be considered partially in the forthcoming section. For this reason we organize the paper as follows: In Section 2 we consider the normal squeezing as well as the amplitude

squared squeezing, further we discuss the behavior of the correlation function. In Section 3 we turn our attention to the quasi-probability distribution functions, more precisely W-Wigner and Q-Husimi functions. Section 4 is devoted to discuss the photon number distribution and phase distribution besides its quadrature variances. While in Section 5 as an application we examined the behavior of the resonance fluorescence for a single atom and many cooperative atoms against the present state. Finally, our conclusion is given in Section 6. 2. Nonclassical properties We devote this section to discuss the nonclassical properties of light for the nonlinear negative binomial state. For this reason we consider two different kinds of phenomena; squeezing phenomenon and Poissonian behavior. To examine squeezing phenomenon we shall pay attention to two different types of squeezing; normal squeezing, and amplitude squared squeezing. This can be achieved by examining the variation in the quadrature variances for each case separately. Also we extend our investigation to include the normalized second-order correlation function that to see the effect of the nonlinearity on the Poissonian behavior. In fact these two quantities are purely independent, however both of them are a measure nonclassical properties of light. To carry out our investigations we have to calculate the expectation values of the mean photon number nf ¼ h^ay ^aif as well as the expectation values of the higher order of the creation and annihilation operators haˆ rif and haˆrif. In this case if one uses Eq. (1.6) then we have for haˆrif the expression 1  1 1  X m þ k þ r 2 m þ k 2 jfj2k r 2 2 mþ1 r f; h^a if ¼ jkj ð1  jfj Þ k! m m k¼0 ð2:1Þ on the other hand we can obtain expression for haˆ raˆrif in the form  2 1  expðjfj Þ X Cðm þ k þ 1Þ  jfj2k h^ayr ^ar if ¼  Lm jfj2 k¼0 m!k!Cðk  r þ 1Þ ¼

LmðrÞ ðjfj2 Þ 2

Lm ðjfj Þ

2r

ð2:2Þ

jfj ;

where LmðrÞ ðxÞ is the associated Laguerre polynomial given in terms of the hypergeometric function as LmðrÞ ðxÞ ¼

ðr þ 1Þm F 1 ðm; r þ 1; xÞ: m! 1

ð2:3Þ

Therefore, we can deduce the expectation value of the photon number to take the form 2

nf ¼ h^ay ^aif ¼

Lmð1Þ ðjfj Þ Lm ðjfj2 Þ

2

jfj ;

while the second moment can be written as

ð2:4Þ

M.S. Abdalla et al. / Optics Communications 274 (2007) 372–383 2  ay ^ n2f ¼ hð^ aÞ i f

jfj

h 2

Lm ðjfj Þ

i 2 2 2 Lmð1Þ ðjfj Þ þ jfj Lð2Þ m ðjfj Þ :

ð2:5Þ

Having obtained the above expressions we are therefore in position to discuss the above mentioned physical phenomena. This will be seen in the following subsections. 2.1. Squeezing phenomenon It is well known that squeezing phenomenon represents one of the interesting phenomena in the field of quantum optics, which is a direct quantum effect of the Heisenberg uncertainty principle. It reflects the reduced quantum fluctuations in one of the field quadratures at the expense of the other corresponding stretched quadrature. In fact the squeezed light is related to several applications in optical communication networks [25], to interferometric techniques [26], and to optical waveguide tap [27]. There are two different types of squeezing we are concern to discuss in this subsection; normal squeezing [28], and amplitude squared squeezing [28–30]. The investigation of normal squeezing is based on defining two field quadrature operators: 1 1 X^ ¼ ½^ aþ^ ay ; and Y^ ¼ ½^ a^ ay : ð2:6Þ 2 2i In fact these quadratures are strongly related to the con^ and H ^ (say) jugate electric and magnetic fields operators E in the optical cavity of electromagnetic and satisfy the commutation relation ½X^ ; Y^  ¼ i=2 and the uncertainty relation has the form hðDX^ Þ2 ihðDY^ Þ2 i P 161 . The system would acquires squeezing if one of the quadratures satisfies the inequality S x ¼ 4hðDX^ Þ2 i  1 < 0

or

S y ¼ 4hðDY^ Þ2 i  1 < 0: ð2:7Þ

2 To calculate the quadrature variances hðDX^ Þ i and 2 ^ hðDY Þ i we have to use Eq. (2.1) together with Eq. (2.6). In this case we have for the first quadrature the expression

Sx ¼ 2

Lmð1Þ ðjfj2 Þ 2

2

j1j þ 2

expðjfj2 Þ 2

cos 2/

the squeezing for the present state is too sensitive to any variation in the value of the parameter m. As a second example of the squeezing phenomenon we shall discuss that is the amplitude squared squeezing. This kind of squeezing arises in a natural way in the second harmonic generation, and its quadratures component defined as 1 d^1 ¼ ð^a2 þ ^ay2 Þ; 2

ð2:9Þ

The nonlinear negative binomial state is said to be amplitude squared squeezing if one of the quadratures D2 d^1 or D2 d^2 6 ðh^nif þ 12Þ. This means that 2

D2 d^1 ¼ Reh^a4 if þ h^n2 if  h^nif  2Reðh^a2 if Þ < 0; 2

D2 d^2 ¼ h^n2 if  h^nif  Reh^a4 if þ 2Imðh^a2 if Þ < 0:

ð2:11Þ

In Fig. 2 we have plotted the quadrature variances D2 d^2 against the parameter f for different values of m. In this figure, similar behavior to that of the normal squeezing can be seen, however the state in the present case exhibits strong amplitude square squeezing for f > 0. Increasing the value of the parameter m results in stronger squeezing for all values of f > 0. For instance in the case when m = 40 we can observe a strong as well as faster squeezing compared with the case in which m = 5. 2.2. Correlation function Antibunching light is another example of nonclassical light and can be determined from a photocounting-correlation measurement. In practice, the measurement can be performed in an experiment of Hanbury Brown–Twiss type. The measure for bunching (classical phenomenon) and antibunching (nonclassical phenomenon) of photons

1 X ðm þ kÞ!

0.00

2

Lm ðjfj Þ Lm ðjfj Þ k¼0 m!ðk!Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ffi 2 m m 2kþ2 2 expð2jfj Þ  1þ  4j1j 1þ jfj 2 2 kþ1 kþ2 ½Lm ðjfj Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi " #2   1 X ðm þ kÞ! m jfj2k ; 1þ  cos2 / ð2:8Þ 2 k þ 1 k¼0 m!ðk!Þ

1 d^2 ¼ ð^a2  ^ay2 Þ; 2i

while their quadrature variances satisfy the uncertainty relation  2 1 : ð2:10Þ D2 d^1 :D2 d^2 P h^nif þ 2

Sx for m=5

-0.02

Sx for m=10

-0.04

NS

¼

2

375

Sx for m=15

-0.06

Sx for m=25

-0.08

Sx for m=40

-0.10 -0.12 -0.14

where / is a phase angle. To discuss the normal squeezing we have plotted the quadrature variances Sx against the parameter f for different values of the number m. This can be seen in Fig. 1. In this figure, the state shows squeezing in the first quadrature for all values of f > 0. Increasing the values of m more squeezing can be seen (even stronger). This means that

-0.16 -0.18 -0.20 0.0

0.2

0.4

ξ

0.6

0.8

Fig. 1. Normal squeezing for different values of m.

1.0

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M.S. Abdalla et al. / Optics Communications 274 (2007) 372–383

at f = 0 and then its value starts to increase as long as we increase the value of the parameter f. However, it is always less than unity for all values 0 < f < 1. Increasing m just changes the beginning to lower values at f = 0 but at f increases g(2)(0) attains higher values for higher values of m for f = 0.2.

0

ASS

-1

-2

F2 for m=5

3. Quasi-probability distribution functions

F2 for m=10 F2 for m=15

0.75

Since the quantum properties of the radiation field can be investigated under different points of view. Therefore, we continue our progress and devote the present section to consider and discuss the quasi-probability distribution functions. There are three well known types of these functions: Glauber-Sudarshan P, Wigner-Moyal W, and Husimi-Kano Q functions corresponding to normally ordered, symmetric, and anti-normally ordered, respectively. In fact these functions are important tools to give insight in the statistical description of a quantum mechanical system. Experimentally these functions can be measured via homodyne tomography [31]. To obtain one of these functions we have to calculate the characteristic function Cp ðb; fÞ which is defined by

0.70

qðfÞ expðb^ay Þ expðb ^aÞ; Cp ðb; fÞ ¼ Tr½^

-3

F2 for m=25 F2 for m=40

-4 0.0

0.2

0.4

0.6

ξ

0.8

1.0

Fig. 2. Amplitude squar squeezing for different values of m.

0.95

m=5 m=10 m=15 m=25 m=40

0.90 0.85

(2)

g (0)

0.80

0.65 0.60 0.55 0.50 0.0

0.2

0.4

ξ

0.6

0.8

1.0

Fig. 3. g(2)(0) for different values of m.

in the system was given by normalized normal secondorder correlation function which defined as gð2Þ ð0Þ ¼

2 ai h^ ay ^

¼1þ

hðD^ nÞ i  h^ ni h^ ni

2

;

ð2:12Þ

n2 i  h^ ni2 , is the photon-number variwhere hðD^ nÞ2 i ¼ h^ y ance with ^ n¼^ a^ a as we have mentioned before. Now if one uses Eq. (2.2) together with Eq. (2.12), then after straightforward calculations the correlation takes the form gð2Þ ð0Þ ¼

^ðfÞ is the density matrix such that q ^ðfÞ ¼ where q jm; fihm; fj. Now if we use Eqs. (1.7) and (3.1) then the off-diagonal form of the characteristic function can be written as "  1  m þ n jfj2n expðjfj2 Þ X 2 Cp ðb; fÞ ¼ Ln ðjbj Þ 2 n! m Lm ðjfj Þ n¼0   1 1 r X X mþn m þ r 2 jfj2n þ

r! n r r¼0 n¼0 nr

 ðf



ðrnÞ ðrnÞ 2 bÞ Ln ðjbj Þ

# :

ð3:2Þ

2

y2 2

h^ a ^ ai

ð3:1Þ

Lmð2Þ ðjfj2 ÞLm ðjfj2 Þ 2 2 ½Lmð1Þ ðjfj Þ

;

ð2:13Þ

it can be used as a measure of sub-Poissonian behavior of light. It holds that g(2)(0) < 1 for sub-Poissonian distribution (nonclassical light), g(2)(0) > 1 for super-Poissonian distribution (chaotic behavior), while the coherent light occurs when g(2)(0) = 1. To demonstrate these phenomena we have plotted the function against the binomial parameter f. From Fig. 3, we can observe that the state is always subPoissonian and it starts from gð2Þ ð0Þ ¼ 12ðm þ 2Þ=ðm þ 1Þ

Since the quasi-probability distribution function is the Fourier transform of the characteristic function, therefore the W-Wigner function can be obtained from the direct evaluation of the integral   Z 1 1 1 W ða; fÞ ¼ 2 d2 bCp ðb; fÞ expðab  ba Þ exp  jbj2 : p 1 2 ð3:3Þ Straightforward calculations lead to the expression " 1 X 2 2 n 2 ð1Þ qn;n ðfÞLn ð4jaj Þ W ða; fÞ ¼ expð2jaj Þ p n¼0 rffiffiffiffi 1 X r1 X n! ðrnÞ n 2 ðrnÞ Ln ð4jaj Þ½ð2a Þ þ ð1Þ qr;n ðfÞ r! r¼1 n¼0 # ðrnÞ

þ ð2aÞ

 ;

ð3:4Þ

M.S. Abdalla et al. / Optics Communications 274 (2007) 372–383

377

Fig. 4. (a)–(c) Wigner function as a function of a.

where Lmn ð Þ is the associated Laguerre polynomial defined by Eq. (2.3), while qr,n(f) is the off-diagonal element of the density operator given by:

2

qr;n ðfÞ ¼

expðjfj Þ 2

Lm ðjfj Þ

Fig. 5. (a) and (b) Q-function as a function of a.



mþn m



mþr m

12

nn nr pffiffiffiffiffiffiffi ¼ q2n;2r : n!r! ð3:5Þ

378

M.S. Abdalla et al. / Optics Communications 274 (2007) 372–383

According to (3.4) we have plotted in Figs. 4, the Wigner function W(a, f) against the parameter a = x + iy for different values of m and f. For small f (f = 0.2) we considered in Fig. 4a the case in which m = 5, where we can see the Wigner function is positive with a very little structure around the base. As the value of the parameter f increases to f = 0.5, then we observe the negative values for the Wigner function are attained, see Fig. 4b. This in fact is a signature of the non-classical effect. Moreover, asymmetry is also shown in the figure. It should be noted that there is a different between the figure in the present case and in the usual negative binomial state case [11]. Increasing the value of the parameter m shows pronounced structure for the function even when n = 0.2 as shown in Fig. 4c where m = 25. As another example for the quasi-probability distribution function which is worth to be considered is the anti-normal ordered Q-function. To discuss the behavior of this function against the non-linear negative binomial state we have to use as usual the characteristic function given by Eq. (3.2) to evaluate the integral Z 1 1 Qða; fÞ ¼ 2 d2 bCp ðb; fÞ p 1 2

 exp ðab  ba Þ expðjbj Þ;

ð3:6Þ

therefore, if we insert Eq. (3.2) into the above equation and preform the integration one can find 2  1 1  m þ n 2 ða nÞn  1 exp½ðjaj2 þ jfj2 Þ X Qða; fÞ ¼ :  2  n¼0 p n!  n Lm ðjfj Þ ð3:7Þ Since we have obtained the Q-function then it is possible to discuss its behavior related to the non-linear negative binomial state. In Figs. 5, we have plotted the Q-function against Re a and Im a according to Eq. (3.7) for different values of m and f. For example, in Fig. 5a we can observe that the graph for the Q-function shows a Gaussian shape centered at ðx; 0Þ. However, the center is shifted towards higher x as f increases. Also we may report that the squeez-

ing is exhibited in the form of the Gaussian by slight asymmetry in the shape, see Fig. 5b. Finally it is remarked that change in the parameter m does not change the behavior of the function. Before we close this section let us investigate the quadrature (or position) distribution P(x, f),which can be measured in the homodyne detector [31]. The quadrature distribution can be evaluated via the W-Wigner function through the relation Z 1 W ðx þ iy; fÞ dy; ð3:8Þ P ðx; fÞ ¼ 1

in this case if one inserts Eq. (3.4) into the above equation and evaluates the integral, then we have rffiffiffi 2 2 exp½ðjfj þ 2x2 Þ P ðx; fÞ ¼ 2 p Lm ðjfj Þ 1 X pffiffiffi i2 ð1Þn ðm þ nÞ!f2n h ð 2xÞ ; ð3:9Þ H  n 3 2n ðn!Þ m! n¼0 where Hm(z) is the Hermite polynomial of degree n defined by H k ðzÞ ¼

½k=2 ðk2sÞ X s k!ð2zÞ : ð1Þ s!ðk  2sÞ! s¼0

ð3:10Þ

To demonstrate the behavior of the distribution function P(x, f) we have plotted in Figs. 6 the function against the variable x. For example in Fig. 6a we have taken different values of the parameter f and fixed value of m such that m = 5. In this case the quadrature distribution function P(x, f) shows in general that the function takes an almost Gaussian shape around x = 0 and decreases rapidly as x increases. The maximum value at x = 0 reduces as x increases for the same value of m. Moreover, it is also observed that as we decrease the value of the parameter f, the function increases its maximum and vise verse. Further we have plotted the same function against the parameter f in Fig. 6b, however, for different values of m. In this case we realize that the maximum value of P(x = 0, f) reduces as m increases for the same value of f.

Fig. 6. (a) Quadrature distribution (at m = 5) for different values of f. (b) Quadrature distribution (at x = 0) for different values of m.

M.S. Abdalla et al. / Optics Communications 274 (2007) 372–383

4. More statistical properties As it is known the nonclassical properties of electromagnetic waves are progressively destroyed by the presence of noise and losses. This in fact leads to attract a little attention to the effect of thermal noise on quantum phase measurement, especially of nonclassical light. In the present section we turn our attention to this specific point and consider the phase distribution as well as its variances. To do so there are different techniques for phase description [19,20] based on a Hermitian quantum phase operator or associated with quasi-probability distribution functions in a phase space. However, most of these approaches have advantages and disadvantages. In the present communication we shall restrict ourself with that one which adopted by Barnett and Pegg. They have defined a Hermitian phase operator in a finite dimensional state space [32,33] . This phase operator is defined as the projection operator on the particular phase state multiplied by the corresponding value of the phase. The Pegg–Barnett phase is given through the probability phase PPB(f, h) by 1 X P PB ðf; hÞ ¼ q ðfÞ exp½iðr  sÞðh  h0 Þ; ð4:1Þ 2p r;s¼0 rs where qrs(f) is the density matrix operator which can be evaluated if one uses the state (1.7), while h0 is the phase reference angle. The phase distribution for the present state can be then written as 1 X n1 X 1 2 ðmþ1Þ 2 1 þ 2ð1  jfj Þ jkj 2p n¼1 r¼0 !   12 mþn mþr fn fr   pffiffiffiffiffiffiffi cos½ðn  rÞh ; m m n!r!

P PB ðf; hÞ ¼

ð4:2Þ where we have dropped the phase reference angle h0 assuming its value to be zero. It should be noted that the phase distribution may be calculated using the W-Wigner function given by Eq. (3.4). However, in this case it

would leads to negative values in contrast to that associated with Pegg–Barnett phase distribution. To illustrate the behavior of the phase distribution related to the state (1.7) we have plotted in Fig. 7 the function PPB(f, h) against the parameters f and h, for fixed value of m (m = 25). Information about the phase is gained with the increase of f. For example, when we take f = 0, the vacuum state is present and no phase information can be seen. However, as the value of f increases and more Fock states are added the phase starts to build up as a single peak around h = 0. The peak gets sharper by increasing the total number of states m. In the meantime we observe that the amplitude gets larger as long as the value of f increases. However it drops sharply when f approaches the value 1 where no state present and again the phase infor1 mation is lost where P PB ð1; hÞ ¼ 2p ¼ P PB ð0; hÞ. Numerical calculations show also that increasing the value of m results in increasing the amplitude. Before we close the present section let us turn our attention to consider the second task that is the phase variance which is defined by hðDH1 Þ2 i ¼ hH21 i  hH1 i2 :

p

After minor algebra the expression for second moment hH21 i can be written in the form hH21 i ¼

2 ðnrÞ n r 1 X n1 p2 expðjfj Þ X ð1Þ ff p ffiffiffiffiffiffiffi þ4 2 3 Lm ðjfj Þ n¼1 rr¼0 n!r!ðn  rÞ2   1 mþn mþr 2  ; m m

m=25

P PB(ξ,θ)

1.0 0.8 0.6 0.4 0.2 3

0.0

2

0.2

1 0.4

ξ

0 -1

0.6 -2

0.8 1.0

θ

-3

Fig. 7. The phase distribution function at m = 25.

ð4:3Þ

Since we have employed the Pegg–Barnett phase to examine the present state, therefore we shall use the same distribution to calculate its variances. To do so we use Eq. (4.2) to find the lth moment of the phase distribution which can be evaluated through the relation Z p hHl1 i ¼ hl P ðf; hÞ dh: ð4:4Þ

1.4 1.2

379

Fig. 8. The phase variance for difference values of m.

ð4:5Þ

380

M.S. Abdalla et al. / Optics Communications 274 (2007) 372–383

while the first moment in this case is zero. This in fact is on contrary to the case associated with the W-Wigner function where we can find expression for both first and second mo-

ment. Therefore, the second moment is essentially the variance of the phase distribution. In Fig. 8 we have plotted the quadrature variance against the parameter f, for differ-

Fig. 9. (a) The atomic inversion for different values of m, and c = 0.5, g = 0.2. (b) The atomic inversion for different values of c, and m = 10, g = 0.2. (c) The atomic inversion for different values of g, and m = 10, c = 0.5. (d) The atomic inversion for different values of g, and m = 1, c = 0.5. (e) The atomic inversion for different values of c, and m = 1, g = 0.2.

M.S. Abdalla et al. / Optics Communications 274 (2007) 372–383

ent value of m. It is easy to observe that although the function shows in general similar behavior for all cases, such that start with a decrease in its value, however, it is faster for large value of m (m = 25). This behavior is emphasized for the other cases. For instance, when we considered m = 5 (solid line) the function reaches its minimum after considerable value of f. In the meantime there is no negative values for the quadrature variance can be observed, however we realize there is an asymptotes behavior for all cases. 5. Resonance fluorescence As an application to the present state we shall consider the resonance fluorescence. In fact the phenomenon of resonance fluorescence provides an interesting manifestation of the quantum theory of light. The phenomenon is related to a radioactively decaying two-level atomic system coupled to an external radiation field in free space. In what follows we shall restrict ourself with the steady-state regime (i.e, t ! 1) in two different cases: the single atom (N = 1), and the thermodynamic limit in a cooperative many-atom system where N ! 1. To do so we shall examine the mean atomic inversion against the non-linear negative binomial state. This will be done in the following subsection. 5.1. A single atom It is well known that in the steady-state case the mean atomic inversion for a single two-level atom in interaction with an external field at exact resonance is given by [34,35] pffiffiffi !2n  2 2 1 2 2g c h hS^z ð1Þin ¼  Lðn1Þ  2 ; ð5:1Þ n 2 c h 8g where c is the Einstein coefficient, and g is the coupling constant while LnðaÞ ðxÞ is the generalized Laguerre polynomial. Using the non-linear negative binomial state defined by Eq. (1.7), we get the following expression for the mean atomic inversion: hS^z ð1Þif ¼

1 X

P n ðm; fÞhS^z ð1Þin ;

ð5:2Þ

n¼0

where Pn(m, f) = jhnjm, fifj2, in this case the atomic inversion can be written as pffiffiffi !2n 2 X 1 1 expðjfj Þ 2 2g ðm þ nÞ! hS^z ð1Þif ¼  2 2 Lm ðjfj Þ n¼0 ch m!   2n jfj c2  h2  Lðn1Þ  2 : ð5:3Þ 2 n 8g ðn!Þ We have examined the variation of the atomic inversion against the parameter f for three different cases. For example, in Fig. 9a we have plotted the atomic inversion for dif-

381

ferent values of the parameter m however, for fixed values of c = 0.5 and g = 0.2. In this case we observe that the value of the atomic inversion decreases faster as long as we increase the value of m. However, it backs again to increase its value with nearly the same ratio or even faster. For example, the function reaches its minimum at f ’ 0.3, and then it changed its direction to increase above 0.5, see for instance (m = 40). This cannot be seen for the same interval of f when m = 5. Further, we may report that for each value of m the function has different critical point (the point in which the function changes its direction). Changing the direction of the atomic inversion refers to the excitation and de-excitation in the atomic system which would occur during the interaction between the atom and the field. It is quite obvious the function gets nearby the ground state showing de-excitation for large value of m faster than for small value of m provided f is small. This means that the atomic inversion is very sensitive to the variation in the parameter m. On the other hand, we have examined the variation of the atomic inversion for fixed value of both g = 0.2 and m = 10, but with variation in the Einstein coefficient c. In this case we realize that the function starts to decrease its value with different ratio depends on the value of the parameter c. However, it backs again to show a slight increment in its value for each case. It should be noted that the decreasing in the function value for the case of c = 0.4 is significant compare with the other two cases c = 0.5, and 0.6, see Fig. 9b. In addition to the effect of the nonlinearity this may due to the existence of the c factor in the argument of the Laguerre polynomial, Eq. (5.3). To examine the variation in the function resultant of changing the value of the coupling parameter g we have plotted Fig. 9c. In this case the situation is different compare with the previous cases. For example, we can see a slight increasing in the atomic inversion for g = 0.1. This is not the case when we take g = 0.2, where the function starts to decrease its value for certain interval of f, and then it backs again to show slight increasing. Drastic decreasing occurred in function value as soon as we consider the case in which g = 0.3. This means that the system gets deexcited where the atom nearby the ground state. Finally let us consider the case in which the value of m is small (m = 1) while the value of c is fixed (c = 0.5). In this case the function decreases its value as g increases, however, the amount of increment is nearly three times the case in which m = 10, see Fig. 9d. On the other hand, when we consider the same case where m = 1 but g = 0.2, the function increases its value as c increases. In this case the value of increment of the function is nearly twice the case in which m = 10, see Fig. 9e. 5.2. The thermodynamic limit Now we turn our attention to the case of the N-cooperative-atom resonance fluorescence and in the limit N ! 1, the scaled atomic inversion at exact resonance in the number state field is given by

382

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tical properties mainly the squeezing and the correlation function. We extended our discussion to include the quasi-probability distribution functions W-Wigner and Q-function. Furthermore we have considered the phase distribution in Pegg–Barnett formalism beside its variance. Finally, an application to the resonance fluorescence is given. For a fixed value of the parameter g and the Einstein coefficient c it has been shown that in a single atom case the excitation in the atomic inversion occurs for large value of both m and f and the de-excitation occurs for small value of f, with large value of m. For thermodynamic limit case the increasing in the atomic inversion is found to be pronounced for all values of the nonlinear negative binomial state m. Acknowledgment Fig. 10. The scaled atomic inversion for different values of m, and g = 0.5, c = 0.5.

"

#   hS^z ð1Þin 1 1 lim ¼  Cn ; X 2 N !1 2 2 N ¼

1 2

n X s¼0

n! cos pðs þ 1Þ ðn  sÞ!X 2s



 1=2 ; s

One of us (M.S.A.) is grateful for the financial support from the project Math 2005/32 of the Research Centre, College of Science, King Saud University. References

ð5:4Þ

where X 2 ¼  h2 cN =ð2g2 Þ, and Cn are the Poisson–Charlier polynomials. For the case of the non-linear negative binomial state we have the expression " # 1 hS^z ð1Þif 1 expðjfj2 Þ X ðm þ nÞ! lim ¼ 2 N !1 2 Lm ðjfj Þ n¼0 m! N   2n jfj 1 2 ;X :  C ð5:5Þ 2 n 2 ðn!Þ To illustrate the behavior of the atomic inversion in the thermodynamic limit case we have plotted the function against the parameter f for different values of m and fixed of g and c. On contrary to the single atom case we realize that the atomic inversion increases its value (showing excitation), however, it is more rapid for large value of m (m = 40). This means that the function in the present case shows fast and strong interaction as long as we increase the value of the nonlinear negative binomial state parameter m. There is also a slight decreasing in the function value, however it is not significant, see Fig. 10. Finally, we can say that the existence of the nonlinearity in the present state is in general response of the drastic change in the function behavior. This leads to the observation of the excitation and de-excitation phenomenon which can be regarded as a new result in the present communication. 6. Conclusion In the previous sections of the present paper we have discussed in detail the behavior of nonlinear negative binomial state. We concentrated on the discussion of the statis-

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