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Maximum simultaneous squeezing and antibunching in superposed coherent states
Physica A 341 (2004) 201 – 207
www.elsevier.com/locate/physa
Maximum simultaneous squeezing and antibunching in superposed coherent states Hari Prakasha; b;∗ , Pankaj Kumara b M.N.
a Department of Physics, University of Allahabad, Allahabad 211002, India Saha Centre of Space Studies, Institute of Interdisciplinary Studies, University of Allahabad, Allahabad 211002, India
Received 2 February 2004
Abstract Maximum simultaneous squeezing and antibunching in the superposition states, | = Z1 | + Z2 |, of two coherent states | and |, where Z1 ; Z2 ; ; are complex numbers, is studied for the case | + || − |. We show that the maximum squeezing for the operator X = X1 cos + X2 sin , where Hermitian operator X1; 2 are de1ned by X1 + iX2 = a, the annihilation operator and is the argument of ( + ), with the minimum value 0.11077 of |(2X )2 | and maximum antibunching with the minimum value −0:55692 of Mandel’s Q parameter occur for an in1nite combinations with − = 1:59912 exp[ ± i( =2) + i], = arg( + ) and Z1 =Z2 = exp(∗ − ∗ ) and with | + || − |. c 2004 Elsevier B.V. All rights reserved. PACS: 42.50.Dv Keywords: Quantum features of light; Coherent state; Squeezing; Sub-Poissonian photon statistics; Displacement operator; Phase shift operator
In quantum optics, much attention is being paid to non-classical phenomena (for excellent review on non-classical phenomena see Ref. [1]), which cannot be explained on the basis of classical probability concepts. Earlier, study of such non-classical eCects were regarded as being of interest academically only (see, e.g. Ref. [2]), but now their applications in communications (see, e.g. Refs. [3,4]), quantum teleportation (see, e.g. Ref. [5]), dense coding [6], quantum cryptography (see, e.g. Ref. [7]) and detection of gravitational waves [8] are well understood. In general, there is no connection between ∗
Corresponding author. Department of Physics, University of Allahabad, Allahabad 211002, India. E-mail addresses: prakash [email protected] (H. Prakash), pankaj [email protected] (P. Kumar).
c 2004 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2004.04.119
202
H. Prakash, P. Kumar / Physica A 341 (2004) 201 – 207
the two important non-classical eCects, squeezing and antibunching (or sub-Poissonian photon statistics), i.e., some states exist that exhibit the 1rst but not the second and vice versa. The phenomenon of resonance Juorescence [9] from an atom is one of such distinguished examples where light exhibits both squeezing and antibunching at the same time. The coherent state [10] does not exhibit non-classical eCects but a superposition of coherent states can exhibit [11–14] various non-classical eCects, such as, squeezing, higher-order squeezing and antibunching. Buzek et al. [11] and Yunji et al. [14] studied such eCects in the superposition of two coherent states | and | − . The authors reported that the even coherent states exhibit squeezing but not sub-Poissonian statistics while the odd coherent states exhibit sub-Poissonian statistics but not squeezing. Schleich et al. [12] studied such eCects in the superposition of two coherent states, |ei’=2 and |e−i’=2 ( being a real number), of identical mean photon number but diCerent phases and reported that such superposition can exhibit both squeezing and antibunching when 2 1. Yunji et al. [14] studied such eCects in even coherent states, odd coherent states and the states obtained by their displacement. The authors showed that the displaced even coherent states can exhibit both squeezing and antibunching while the displaced odd coherent states can preserve antibunching. Mandel [15] has shown much earlier that the squeezing in a 1eld gives rise to antibunching in the displaced 1eld if the displacement is very large. In this paper, we consider simultaneous squeezing and antibunching in superposed coherent states and study the condition when both eCects are maximum. We make use of Mandel’s results [15] and our own recent results [16] for maximum squeezing in superposed coherent states. Superposition of coherent states can be generated in interaction of coherent state with non-linear media (see, e.g. Ref. [17]) and in quantum nondemolition techniques (see, e.g. Ref. [18]). Coherent state [10] de1ned by a| = |, is given by ∞ √ 1 | = exp − ||2 (n = n!)|n = D()|0 ; 2
(1)
n=0
where |n is the occupation number state and D()=exp(a+ −∗ a) is the displacement operator. Let us consider superposition of two coherent states | and |, given by | = Z1 | + Z2 | ;
(2)
where Z1 ; Z2 ; ; are complex numbers, and investigate the conditions for maximum simultaneous squeezing and antibunching in the state | . We consider the Mandel’s Q parameter [15], Q=
|(2N )2 | − |N | ; |N |
(3)
where |(2N )2 | = |N 2 | − |N | 2 and N = a+ a, which characterizes the departure from Poissonian photon statistics. For classical light Q ¿ 0. When Q ¡ 0, the photon statistics is called sub-Poissonian and Q has values between 0 and −1.
H. Prakash, P. Kumar / Physica A 341 (2004) 201 – 207
203
We can write the state | as a state obtained by displacement of a superposition of even and odd coherent state | = D()| 1 ;
=
+ = ||ei ; 2
(4)
with | 1 = Z1 | + Z2 | − ; Z1; 2 = Z1; 2 exp[ ± 12 (∗ − ∗ )], using the property [10] D()D() = exp(∗ − ∗ )D( + )
(5)
of the displacement operator. Since D+ ()aD() = a + , we have D+ ()X D() = X + ||;
X = X1 cos + X2 sin ;
(6)
where Hermitian operator X1; 2 are de1ned by X1 + iX2 = a and a is the annihilation operator. Variance of X in any state | 1 de1ned by 1 |(2X )2 | 1 = 1 |X2 | 1 − 1 |X | 1 2 , therefore, does not change on operation of D() on the state | 1 , and we have |(2X )2 | = | : (2X )2 : | + 0:25 = 1 | : (2X )2 : | 1 + 0:25 = 1 |(2X )2 | 1 ;
(7)
where : : denotes the normal form of the operator within colons. Using Eq. (4) and the properties of displacement operator, we can also show that (see also Ref. [15]), |(2N )2 | − |N | = 1 |(2N )2 | 1 − 1 |N | 1 +4||2 1 | : (2X)2 : | 1 +4||cos 1 |(2a+)(2N)| 1 ; (8) where 2a+ = a+ − 1 |a+ | 1 and 2N = N − 1 |N | 1 . When we consider the case ||||, then the terms involving || in Eq. (8) can be made to dominate. Hence, under this consideration and using Eqs. (8) and (7), we have |(2N )2 | − |N | ∼ = 4||2 1 | : (2X )2 : | 1 = 4||2 | : (2X )2 : | : (9) We also have |N | ∼ = ||2 for the case ||||. Hence, we 1nally get Q∼ = 4 | : (2X )2 : | :
(10)
This expression shows that squeezing of X in the state | (i.e., | : (2X )2 : | ¡ 0) always gives rise to antibunching for the case ||||. In Ref. [16] we considered |(2X )2 | for the state | given by Eq. (2) and an arbitrary and reported the minimum value 0.11077 for an in1nite number of combinations with − = 1:59912 exp[ ± i( =2) + i], Z1 =Z2 = exp(∗ − ∗ ) with arbitrary values of + and . Hence we conclude that the Mandel’s Q parameter has minimum value −0:55692 for an in1nite combinations with − = 1:59912 exp[ ± i( =2) + i], = arg( + ), (see Eq. (4)) and Z1 =Z2 = exp(∗ − ∗ ) for the case | + || − |.
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H. Prakash, P. Kumar / Physica A 341 (2004) 201 – 207
Fig. 1. (a) Variation of Q in the state | with x at = 4:00582 × 10−3 ; (b) variation of Q in the state | with at x = 400.
To illustrate this point, let us consider a superposition state in the form | = K (|x + |xei ); K = 1= {2[1 + e−x2 (1−cos ) cos(x2 sin )]} :
(11)
H. Prakash, P. Kumar / Physica A 341 (2004) 201 – 207
205
Fig. 2. (a) Variation of |(2X )2 | with x at = 4:00582 × 10−3 and = =2; (b) variation of |(2X )2 | at = 2:00291 × 10−3 and x = 400.
We get Mandel’s parameter, −x2 (1−cos ) 2 −x2 (1−cos ) 2 {1 + e {1 + e cos(2 + x sin )} cos( + x sin )} − ; Q = x2 {1 + e−x2 (1−cos ) cos( + x2 sin )} {1 + e−x2 (1−cos ) cos(x2 sin )} (12)
206
H. Prakash, P. Kumar / Physica A 341 (2004) 201 – 207
and the variance of X in the state | , |(2X )2 |
2 1 K 2 x2 = + [2 + cos 2 + cos(2 − 2) + e−x (1−cos ) {cos(2 − 2 + x2 sin ) 4 2
+ cos(2 + x2 sin ) + 2cos( + x2 sin )}] − [K 2 x{cos + cos( − ) + e−x
2
(1−cos )
(cos( − + x2 sin ) + cos( + x2 sin ))}]2 :
(13)
The state | may have maximum simultaneous antibunching and squeezing with the minimum value −0:55692 of Q and minimum value 0.11077 of |(2X )2 | , if 2x cos (=2)1:59912 and x(1 − ei ) = 1:59912ei(±( =2)+) :
(14)
This is satis1ed for an in1nite combinations of x; , and with 2x sin(=2) = 1:59912 and = =2. For example, if we consider a large value 400 of x, we 1nd the absolute minimum value −0:5569 of Q for the state | and absolute minimum value 0.11077 of |(2X )2 | at = 4:00582 × 10−3 and = =2. Variations of Q and of |(2X )2 | with x and near the absolute minimum are shown in Figs. 1(a)– (b) and Figs. 2(a)–(b), respectively. We note that the variations of Q and also of |(2X )2 | with x and are quite fast, and hence for observation of such eCects a sensitive setting of the parameters may be required. We would like to thank Prof. N. Chandra and Prof. R. Prakash for their interest and some critical comments and Dr. R.S. Singh, Mr. D.K. Singh, Mr. Rakesh Kumar, Mr. D.K. Mishra, Mr. A. Dixit, Miss. P. Shukla and Miss. Shivani for helpful and stimulating discussions. One of the author (HP) is grateful to Indian Space Research Organization, Bangalore, and the author (PK) is grateful to Council of Scienti1c and Industrial Research, New Delhi for 1nancial supports. References [1] D.F. Walls, Nature 306 (1983) 141; R. Loudon, P.L. Knight, J. Mod. Opt. 34 (1987) 709; V.V. Dodonov, J. Opt. B 4 (2002) R1. [2] B.R. Mollow, R.J. Glauber, Phys. Rev. 160 (1967) 1076; N. Chandra, H. Prakash, Indian J. Pure Appl. Phys. 2 (1971) 677,688,767. N. Chandra, H. Prakash, Lett. Nuovo. Cimento 4 (1970) 1196. [3] H.P. Yuen, J.H. Shapiro, IEEE Trans. Inform. Theory IT 24 (1978) 657; H.P. Yuen, J.H. Shapiro, IEEE Trans. Inform. Theory IT 26 (1980) 78; J.H. Shapiro, H.P. Yuen, J.A. Machado Mata, IEEE Trans. Inform. Theory IT 25 (1979) 179. [4] C.H. Bennett, P.W. Shor, J.A. Smolin, A.V. Thapliyal, Phys. Rev. Lett. 83 (1999) 3081; B. Schumacher, Phys. Rev. A 54 (1996) 2614; B. Schumacher, M.A. Nielsen, Phys. Rev. A 54 (1996) 2629. [5] S.L. Braunstein, G.M.D’ Ariano, G.J. Millburn, M.F. Sacchi, Phys. Rev. Lett. 84 (2000) 3486; S.L. Braunstein, H.J. Kimble, Phys. Rev. Lett. 80 (1998) 869; C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W.K. Wootters, Phys. Rev. Lett. 70 (1993) 1895.
H. Prakash, P. Kumar / Physica A 341 (2004) 201 – 207 [6] S.L. Braunstein, H.J. Kimble, Phys. Rev. A 61 (2000) 042302. [7] C.H. Bennett, G. Brassard, N.D. Mermin, Phys. Rev. Lett. 68 (1992) 557; J. Kempe, Phys. Rev. A 60 (1999) 910. [8] C.M. Caves, Phys. Rev. D 23 (1981) 1693. [9] D.F. Walls, P. Zoller, Phys. Rev. Lett. 47 (1981) 709. [10] R.J. Glauber, Phys. Rev. 131 (1963) 2766. [11] V. Buzek, A. Vidiella-Barranco, P.L. Knight, Phys. Rev. A 45 (1992) 6570. [12] W. Schleich, M. Pernigo, Fam Le Kien, Phys. Rev. A 44 (1991) 2172. [13] J. Janszky, An.V. Vinogradov, Phys. Rev. Lett. 64 (1990) 2771. [14] Yunjie XIA, Guangcan GUO, Phys. Lett. A 136 (1989) 281. [15] L. Mandel, Phys. Rev. Lett. 49 (1982) 136. [16] H. Prakash, P. Kumar, Physica A 319 (2003) 305. [17] B. Yurke, D. Stoler, Phys. Rev. Lett. 57 (1986) 13; G.J. Milburn, C.A. Holmes, Phys. Rev. Lett. 56 (1986) 2237; P. Tombesi, A. Mecozzi, J. Opt. Soc. Amer. B 4 (1987) 1700. [18] C.M. Caves, B. Yurke, Phys. Rev. A 41 (1990) 5261; B. Yurke, W. Schleich, D.F. Walls, Phys. Rev. A 42 (1990) 1703.
207
www.elsevier.com/locate/physa
Maximum simultaneous squeezing and antibunching in superposed coherent states Hari Prakasha; b;∗ , Pankaj Kumara b M.N.
a Department of Physics, University of Allahabad, Allahabad 211002, India Saha Centre of Space Studies, Institute of Interdisciplinary Studies, University of Allahabad, Allahabad 211002, India
Received 2 February 2004
Abstract Maximum simultaneous squeezing and antibunching in the superposition states, | = Z1 | + Z2 |, of two coherent states | and |, where Z1 ; Z2 ; ; are complex numbers, is studied for the case | + || − |. We show that the maximum squeezing for the operator X = X1 cos + X2 sin , where Hermitian operator X1; 2 are de1ned by X1 + iX2 = a, the annihilation operator and is the argument of ( + ), with the minimum value 0.11077 of |(2X )2 | and maximum antibunching with the minimum value −0:55692 of Mandel’s Q parameter occur for an in1nite combinations with − = 1:59912 exp[ ± i( =2) + i], = arg( + ) and Z1 =Z2 = exp(∗ − ∗ ) and with | + || − |. c 2004 Elsevier B.V. All rights reserved. PACS: 42.50.Dv Keywords: Quantum features of light; Coherent state; Squeezing; Sub-Poissonian photon statistics; Displacement operator; Phase shift operator
In quantum optics, much attention is being paid to non-classical phenomena (for excellent review on non-classical phenomena see Ref. [1]), which cannot be explained on the basis of classical probability concepts. Earlier, study of such non-classical eCects were regarded as being of interest academically only (see, e.g. Ref. [2]), but now their applications in communications (see, e.g. Refs. [3,4]), quantum teleportation (see, e.g. Ref. [5]), dense coding [6], quantum cryptography (see, e.g. Ref. [7]) and detection of gravitational waves [8] are well understood. In general, there is no connection between ∗
Corresponding author. Department of Physics, University of Allahabad, Allahabad 211002, India. E-mail addresses: prakash [email protected] (H. Prakash), pankaj [email protected] (P. Kumar).
c 2004 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/j.physa.2004.04.119
202
H. Prakash, P. Kumar / Physica A 341 (2004) 201 – 207
the two important non-classical eCects, squeezing and antibunching (or sub-Poissonian photon statistics), i.e., some states exist that exhibit the 1rst but not the second and vice versa. The phenomenon of resonance Juorescence [9] from an atom is one of such distinguished examples where light exhibits both squeezing and antibunching at the same time. The coherent state [10] does not exhibit non-classical eCects but a superposition of coherent states can exhibit [11–14] various non-classical eCects, such as, squeezing, higher-order squeezing and antibunching. Buzek et al. [11] and Yunji et al. [14] studied such eCects in the superposition of two coherent states | and | − . The authors reported that the even coherent states exhibit squeezing but not sub-Poissonian statistics while the odd coherent states exhibit sub-Poissonian statistics but not squeezing. Schleich et al. [12] studied such eCects in the superposition of two coherent states, |ei’=2 and |e−i’=2 ( being a real number), of identical mean photon number but diCerent phases and reported that such superposition can exhibit both squeezing and antibunching when 2 1. Yunji et al. [14] studied such eCects in even coherent states, odd coherent states and the states obtained by their displacement. The authors showed that the displaced even coherent states can exhibit both squeezing and antibunching while the displaced odd coherent states can preserve antibunching. Mandel [15] has shown much earlier that the squeezing in a 1eld gives rise to antibunching in the displaced 1eld if the displacement is very large. In this paper, we consider simultaneous squeezing and antibunching in superposed coherent states and study the condition when both eCects are maximum. We make use of Mandel’s results [15] and our own recent results [16] for maximum squeezing in superposed coherent states. Superposition of coherent states can be generated in interaction of coherent state with non-linear media (see, e.g. Ref. [17]) and in quantum nondemolition techniques (see, e.g. Ref. [18]). Coherent state [10] de1ned by a| = |, is given by ∞ √ 1 | = exp − ||2 (n = n!)|n = D()|0 ; 2
(1)
n=0
where |n is the occupation number state and D()=exp(a+ −∗ a) is the displacement operator. Let us consider superposition of two coherent states | and |, given by | = Z1 | + Z2 | ;
(2)
where Z1 ; Z2 ; ; are complex numbers, and investigate the conditions for maximum simultaneous squeezing and antibunching in the state | . We consider the Mandel’s Q parameter [15], Q=
|(2N )2 | − |N | ; |N |
(3)
where |(2N )2 | = |N 2 | − |N | 2 and N = a+ a, which characterizes the departure from Poissonian photon statistics. For classical light Q ¿ 0. When Q ¡ 0, the photon statistics is called sub-Poissonian and Q has values between 0 and −1.
H. Prakash, P. Kumar / Physica A 341 (2004) 201 – 207
203
We can write the state | as a state obtained by displacement of a superposition of even and odd coherent state | = D()| 1 ;
=
+ = ||ei ; 2
(4)
with | 1 = Z1 | + Z2 | − ; Z1; 2 = Z1; 2 exp[ ± 12 (∗ − ∗ )], using the property [10] D()D() = exp(∗ − ∗ )D( + )
(5)
of the displacement operator. Since D+ ()aD() = a + , we have D+ ()X D() = X + ||;
X = X1 cos + X2 sin ;
(6)
where Hermitian operator X1; 2 are de1ned by X1 + iX2 = a and a is the annihilation operator. Variance of X in any state | 1 de1ned by 1 |(2X )2 | 1 = 1 |X2 | 1 − 1 |X | 1 2 , therefore, does not change on operation of D() on the state | 1 , and we have |(2X )2 | = | : (2X )2 : | + 0:25 = 1 | : (2X )2 : | 1 + 0:25 = 1 |(2X )2 | 1 ;
(7)
where : : denotes the normal form of the operator within colons. Using Eq. (4) and the properties of displacement operator, we can also show that (see also Ref. [15]), |(2N )2 | − |N | = 1 |(2N )2 | 1 − 1 |N | 1 +4||2 1 | : (2X)2 : | 1 +4||cos 1 |(2a+)(2N)| 1 ; (8) where 2a+ = a+ − 1 |a+ | 1 and 2N = N − 1 |N | 1 . When we consider the case ||||, then the terms involving || in Eq. (8) can be made to dominate. Hence, under this consideration and using Eqs. (8) and (7), we have |(2N )2 | − |N | ∼ = 4||2 1 | : (2X )2 : | 1 = 4||2 | : (2X )2 : | : (9) We also have |N | ∼ = ||2 for the case ||||. Hence, we 1nally get Q∼ = 4 | : (2X )2 : | :
(10)
This expression shows that squeezing of X in the state | (i.e., | : (2X )2 : | ¡ 0) always gives rise to antibunching for the case ||||. In Ref. [16] we considered |(2X )2 | for the state | given by Eq. (2) and an arbitrary and reported the minimum value 0.11077 for an in1nite number of combinations with − = 1:59912 exp[ ± i( =2) + i], Z1 =Z2 = exp(∗ − ∗ ) with arbitrary values of + and . Hence we conclude that the Mandel’s Q parameter has minimum value −0:55692 for an in1nite combinations with − = 1:59912 exp[ ± i( =2) + i], = arg( + ), (see Eq. (4)) and Z1 =Z2 = exp(∗ − ∗ ) for the case | + || − |.
204
H. Prakash, P. Kumar / Physica A 341 (2004) 201 – 207
Fig. 1. (a) Variation of Q in the state | with x at = 4:00582 × 10−3 ; (b) variation of Q in the state | with at x = 400.
To illustrate this point, let us consider a superposition state in the form | = K (|x + |xei ); K = 1= {2[1 + e−x2 (1−cos ) cos(x2 sin )]} :
(11)
H. Prakash, P. Kumar / Physica A 341 (2004) 201 – 207
205
Fig. 2. (a) Variation of |(2X )2 | with x at = 4:00582 × 10−3 and = =2; (b) variation of |(2X )2 | at = 2:00291 × 10−3 and x = 400.
We get Mandel’s parameter, −x2 (1−cos ) 2 −x2 (1−cos ) 2 {1 + e {1 + e cos(2 + x sin )} cos( + x sin )} − ; Q = x2 {1 + e−x2 (1−cos ) cos( + x2 sin )} {1 + e−x2 (1−cos ) cos(x2 sin )} (12)
206
H. Prakash, P. Kumar / Physica A 341 (2004) 201 – 207
and the variance of X in the state | , |(2X )2 |
2 1 K 2 x2 = + [2 + cos 2 + cos(2 − 2) + e−x (1−cos ) {cos(2 − 2 + x2 sin ) 4 2
+ cos(2 + x2 sin ) + 2cos( + x2 sin )}] − [K 2 x{cos + cos( − ) + e−x
2
(1−cos )
(cos( − + x2 sin ) + cos( + x2 sin ))}]2 :
(13)
The state | may have maximum simultaneous antibunching and squeezing with the minimum value −0:55692 of Q and minimum value 0.11077 of |(2X )2 | , if 2x cos (=2)1:59912 and x(1 − ei ) = 1:59912ei(±( =2)+) :
(14)
This is satis1ed for an in1nite combinations of x; , and with 2x sin(=2) = 1:59912 and = =2. For example, if we consider a large value 400 of x, we 1nd the absolute minimum value −0:5569 of Q for the state | and absolute minimum value 0.11077 of |(2X )2 | at = 4:00582 × 10−3 and = =2. Variations of Q and of |(2X )2 | with x and near the absolute minimum are shown in Figs. 1(a)– (b) and Figs. 2(a)–(b), respectively. We note that the variations of Q and also of |(2X )2 | with x and are quite fast, and hence for observation of such eCects a sensitive setting of the parameters may be required. We would like to thank Prof. N. Chandra and Prof. R. Prakash for their interest and some critical comments and Dr. R.S. Singh, Mr. D.K. Singh, Mr. Rakesh Kumar, Mr. D.K. Mishra, Mr. A. Dixit, Miss. P. Shukla and Miss. Shivani for helpful and stimulating discussions. One of the author (HP) is grateful to Indian Space Research Organization, Bangalore, and the author (PK) is grateful to Council of Scienti1c and Industrial Research, New Delhi for 1nancial supports. References [1] D.F. Walls, Nature 306 (1983) 141; R. Loudon, P.L. Knight, J. Mod. Opt. 34 (1987) 709; V.V. Dodonov, J. Opt. B 4 (2002) R1. [2] B.R. Mollow, R.J. Glauber, Phys. Rev. 160 (1967) 1076; N. Chandra, H. Prakash, Indian J. Pure Appl. Phys. 2 (1971) 677,688,767. N. Chandra, H. Prakash, Lett. Nuovo. Cimento 4 (1970) 1196. [3] H.P. Yuen, J.H. Shapiro, IEEE Trans. Inform. Theory IT 24 (1978) 657; H.P. Yuen, J.H. Shapiro, IEEE Trans. Inform. Theory IT 26 (1980) 78; J.H. Shapiro, H.P. Yuen, J.A. Machado Mata, IEEE Trans. Inform. Theory IT 25 (1979) 179. [4] C.H. Bennett, P.W. Shor, J.A. Smolin, A.V. Thapliyal, Phys. Rev. Lett. 83 (1999) 3081; B. Schumacher, Phys. Rev. A 54 (1996) 2614; B. Schumacher, M.A. Nielsen, Phys. Rev. A 54 (1996) 2629. [5] S.L. Braunstein, G.M.D’ Ariano, G.J. Millburn, M.F. Sacchi, Phys. Rev. Lett. 84 (2000) 3486; S.L. Braunstein, H.J. Kimble, Phys. Rev. Lett. 80 (1998) 869; C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W.K. Wootters, Phys. Rev. Lett. 70 (1993) 1895.
H. Prakash, P. Kumar / Physica A 341 (2004) 201 – 207 [6] S.L. Braunstein, H.J. Kimble, Phys. Rev. A 61 (2000) 042302. [7] C.H. Bennett, G. Brassard, N.D. Mermin, Phys. Rev. Lett. 68 (1992) 557; J. Kempe, Phys. Rev. A 60 (1999) 910. [8] C.M. Caves, Phys. Rev. D 23 (1981) 1693. [9] D.F. Walls, P. Zoller, Phys. Rev. Lett. 47 (1981) 709. [10] R.J. Glauber, Phys. Rev. 131 (1963) 2766. [11] V. Buzek, A. Vidiella-Barranco, P.L. Knight, Phys. Rev. A 45 (1992) 6570. [12] W. Schleich, M. Pernigo, Fam Le Kien, Phys. Rev. A 44 (1991) 2172. [13] J. Janszky, An.V. Vinogradov, Phys. Rev. Lett. 64 (1990) 2771. [14] Yunjie XIA, Guangcan GUO, Phys. Lett. A 136 (1989) 281. [15] L. Mandel, Phys. Rev. Lett. 49 (1982) 136. [16] H. Prakash, P. Kumar, Physica A 319 (2003) 305. [17] B. Yurke, D. Stoler, Phys. Rev. Lett. 57 (1986) 13; G.J. Milburn, C.A. Holmes, Phys. Rev. Lett. 56 (1986) 2237; P. Tombesi, A. Mecozzi, J. Opt. Soc. Amer. B 4 (1987) 1700. [18] C.M. Caves, B. Yurke, Phys. Rev. A 41 (1990) 5261; B. Yurke, W. Schleich, D.F. Walls, Phys. Rev. A 42 (1990) 1703.
207