- Email: [email protected]
Non-classical properties of superposition of two coherent states having phase difference ϕ
Optik 122 (2011) 1058–1060
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Non-classical properties of superposition of two coherent states having phase difference Hari Prakash a , Pankaj Kumar b,∗ a b
Department of Physics, University of Allahabad, Allahabad (U.P) 211002, India Department of Physics, Bhavan’s Mehta Mahavidyalaya (V. S. Mehta College of Science), Bharwari, Kaushambi (U.P) 212201, India
a r t i c l e
i n f o
Article history: Received 24 March 2010 Accepted 4 July 2010
PACS: 42.50.Dv Keywords: Non-classical features of light Coherent state Squeezing Sub-Poissonian photon statistics Displacement operator Phase shifting operator
a b s t r a c t Recently Ahmad et al. [Optik 2009;120;68; Optics Commun. 2007;271:162; Chin. Phys. Lett. 2006;23:2438] have non-classical properties of superposition of two-coherent states of the form, studied = K[|˛ + ei ˛ei ] for the special cases with values = /2, 3/2, and , and for arbitrarily fixed values of . We point out that some of their results are special cases of our recently published work [Physica A 319, 305 (2003); Physica A 341, 201(2004)] on the most general superposition of two arbi trary coherent states of the form ∼(Z1 |˛ + Z2 ˇ ), where X1,2 , ˛ and ˇ are arbitrary and only restriction on these is the normalization condition for the superposed state. To make our point we first obtain results for (i) squeezing of the most general Hermitian operator X = X1 cos + X2 sin , with X1 + iX2 = a, is the annihilation operator, and (ii) sub-Poissonian photon statistics, for the superposed state with a general and, then obtain results of Ahmad et al. for = /2, 3/2, and and for = 0 and /2. It is interesting to note that the arbitrarily fixed values = |˛|2 and −|˛|2 for = /2 and 3/2, respectively by Ahmad et al. are the values at which we get maximum squeezing working in a rigorous way. © 2010 Elsevier GmbH. All rights reserved.
1. Introduction States of light, properties of which cannot be explained on the basis of classical theory are called non-classical states [1]. The nonclassical nature of a state can be manifested in different ways like antibunching, sub-Poissonian photon statistics and various kinds of squeezing, etc. Earlier study of such non-classical effects was largely in academic interest [2], but now their applications in quantum information theory such as communication [3], quantum teleportation [4], dense coding [5] and quantum cryptography [6] are well realized. It has been demonstrated that non-classicality is the necessary input for entangled state [7]. A coherent state [8] defined as the eigenstate of annihilation operator, i.e., a |˛ = ˛ |˛, does not exhibit non-classical effects but a superposition of coherent states exhibit [9–16] various non-classical effects such as squeezing, higher-order squeezing, sub-Poissonian statistics and higher-order sub-Poissonian statistics. Buzek et al. [9] and Xia et al. [12] studied such effects in the superposition of two-coherent states |˛ and |−˛ and reported that the even coherent state exhibits squeezing but not sub-Poissonian statistics while the odd coherent state exhibits sub-
∗ Corresponding author. E-mail addresses: prakash [email protected] (H. Prakash), pankaj [email protected] (P. Kumar). 0030-4026/$ – see front matter © 2010 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2010.07.004
Poissonian statistics but not squeezing. Xia et al. also studied [12] such effects in the displaced even and odd coherent state. Schleich et al. [10] studied such effects in the superposition of two-coherent states, |˛ and |˛∗ , of identical mean photon number but different phases and reported that such superposition can exhibit both squeezing and sub-Poissonian statistics when |˛|2 1. Recently, we studied maximum squeezing in the most general superposed coherent state [13]. Later, we used this result to study [14] maximum simultaneous squeezing and sub-Poissonian statistics in superposed coherent states. We also studied [15] maximum fourthorder squeezing in superposition of two arbitrary coherent states. In practice, the superposition of coherent states can be generated in interaction of coherent state with nonlinear media [16] and in quantum nondemolition techniques [17]. Recently Ahmad et al. [18–20] studied the non-classical properties of the superposition with equal weights of pairs of coherent states (|˛ and i˛ ), (|˛ and −i˛ ) and (|˛ and |−˛). We point out that the results obtained by Ahmad et al. [18–20] are special cases of our recently published work [13,14] for the most general superposition of two-coherent states of the form =
(Z1 |˛ + Z2 ˇ ), where Z1,2 , ˛ and ˇ are arbitrary and only restriction on these is the normalization condition for the superposed coherent state. To make our point, we consider the more general superposed state,
= K[|˛ + ei ˛ei ],
(1)
H. Prakash, P. Kumar / Optik 122 (2011) 1058–1060
of two-coherent states |˛ and ˛ei , and study squeezing of the most general Hermitian Operator X defined by X = X1 cos + X2 sin ,
(2)
and sub-Poissonian photon statistics in the superposed coherent using recent results [13,14]. Here Hermitian operators state X1,2 are defined by X1 + iX2 = a, the annihilation operator, and complex number ˛ = |˛| ei˛ , angles and are completely arbitrary with the only restriction being the normalization condition for . We show that the results of Ahmad et al. are special cases = /2, 3/2, and . Also we show that the arbitrarily fixed values = |˛|2 and −|˛|2 for = /2 and 3/2, respectively by Ahmad et al. are, as a matter of fact, the values obtained for maximum squeezing, working in a rigorous way.
2. Squeezing in superposed coherent state Using the Hesenberg’s uncertainty relation, the condition for squeezing of X in the any state can be defined as
(X )2 < 1/4, where X = X − X . Recently we studied [14] squeezing of the operator hermitian X in the most general superposition state = Z1 |˛ + Z2 ˇ . Using the fact
that displacement and phase shift does not change the amount of squeezing in any state we related to superposition of coher-
˛+ˇ
ent states
2
and −
˛+ˇ 2
and showed [14] that the maximum
with the absolute mini 2 mum value 0.11077 of variance (X ) occurs for infinite squeezing in the superposed state
numbers of combinations with ˛ − ˇ = 1.59912 exp[± i
2
+ i],
Z1 1 = exp (ˇ˛∗ − ˇ∗ ˛) , Z2 2
(3)
and with arbitrary (˛ + ˇ) and . For the state , we can obtain results for maximum squeezing by using Eq. (3). For ˛ = |˛| ei˛ , this gives maximumsqueezing with (X )2 the absolute minimum value 0.11077 of variance occurring for infinite numbers of combinations with 2 |˛| sin
= 1.59912; 2
for = D (˛ + ˇ)/2
4
Using the results [13], i.e., Eq. (3) we find that the state exhibits maximum simultaneous and squeezing antibunching with (X )2 ˙ and minimum value minimum value 0.11077 of
and the condition ˛ + ˇ ˛ − ˇ.
−0.55692 of Mandel’s Q parameter for an infinite number of conditions in Eq. (3) and = arg(˛ + ˇ), for the as mentioned case ˛ + ˇ ˛ − ˇ. Using these results we report that the
superposed coherent state exhibits maximum simultaneous squeezing and antibunching with minimum value 0.11077 (X )2 of and minimum value −0.55692 of Mandel’s Q parameter for an infinite combination with 2 |˛| sin /2 = 1.59912, = ˛ + /2, = −|˛|2 sin under the consideration |˛| 1. 4. Conclusion In this paper we consider a more general superposition of = K[|˛ + ei ˛ei ]; ˛ = |˛| ei˛ and studcoherent states, ied photon statistics and ordinary squeezing of the most general Hermitian operator X defined by X = X1 cos + X2 sin , in this using our recently reported results superposed coherent state [13,14]. We conclude that maximum squeezing of X in the state occurs with the absolute minimum value 0.11077 of variance
(X )2 for infinite numbers of combinations with,
2 |˛| sin /2 = 1.59912, = ˛ + /2 and = −|˛|2 sin . Using these results and the properties of displacement operator we also conclude that this superposed coherent states also exhibits subPoissonian photon statistics for infinite numbers of combinations with, 2 |˛| sin /2 = 1.59912, = ˛ + /2 and = −|˛|2 sin when |˛| 1. The results of Ahmad et al. [18–20] are special cases ( = /2, 3/2 and ) of this general study. Acknowledgement We would like to thank Professors N. Chandra and R. Prakash for their interest and some critical comments. References
= −|˛|2 sin ,
(4)
and = ˛ + /2. We can obtain the results of Ahmad et al. for squeezing by putting the values = /2, 3/2, and which gives |˛| = 1.1307, 1.1307, 0.7996 and = −|˛|2 , |˛|2 and 0, respectively. This gives a rigorous derivation of values |˛|2 and –|˛|2 for for the cases = /2 and 3/2, respectively, which have been fixed arbitrary by Ahmad et al.
3. Simultaneous squeezing and sub-Poissonian photon statistics in superposed coherent state For characterization of the photon statistics of optical field in the state , Mandel [21] introduced a parameter based of the field defined fluctuations on 2intensity by Q = N ] − 1, where N = N − N and / [ (N) N = a+ a. When 0 < Q ≤ − 1, the photon statistics is called subPoissonian and the field is called antibunched. Recently we studied [13] simultaneous occurrence of antibunchingand squeezing of the hermitian operator X in the superposed state of two-coherent
1059
ing and antibunching in superposed state and found a relation (X )2 , between Mandel’s Q parameter and variance 1 2 (X ) − . (10) Q ∼ =4
states |˛ and ˇ . We investigated the connection between squeez-
[1] For excellent review on non-classical phenomena see, D.F. Walls, Squeezed states of light, Nature 306 (1983) 141–146; R. Loudon, P.L. Knight, Squeezed light, J. Mod. Opt. 34 (1987) 709–759; V.V. Dodonov, ‘Nonclassical’ states in quantum optics: a ‘squeezed’ review of the first 75 years, J. Opt. B 4 (2002) R1–R33. [2] See, e.g., B.R. Mollow, R.J. Glauber, Quantum theory of parametric amplification. I, Phys. Rev. 160 (1967) 1076; N. Chandra, H. Prakash, Quantum statistics of photon amplifiers, Indian J. Pure Appl. Phys. 2 (1971), 677, 688, 767; N. Chandra, H. Prakash, Anticorrelation in two-photon attenuated laser beam, Phys. Rev. A 1 (1970) 1696–1698. [3] C.H. Bennett, P.W. Shor, J.A. Smolin, A.V. Thapliyal, Entangled-assisted classical capacity of noisy quantum channels, Phys. Rev. Lett. 83 (1999) 3081–3084. [4] S.L. Braunstein, G.M. D’ Ariano, G.J. Milburn, M.F. Sacchi, Universal teleportation with a twist, Phys. Rev. Lett. 84 (2000) 3486–3489. [5] S.L. Braunstein, H.J. Kimble, Dense coding for continuous variables, Phys. Rev. A 61 (2000) 042302–042305. [6] C.H. Bennette, G. Brassard, N.D. Mermin, Quantum cryptography without Bells theorem, Phys. Rev. Lett. 68 (1992) 557–559. [7] M.S. Kim, W. Son, V. Buzek, P.L. Knight, Entanglement by a beam splitter: non-classicality as a prerequisite for entanglement, Phys. Rev. A 65 (2002) 032323–032329. [8] R.J. Glauber, Coherent and incoherent states of the radiation field, Phys. Rev. 131 (1963) 2766–2788. [9] V. Buˇzek, A. Vidiella-Barranco, P.L. Knight, Superpositions of coherent states: squeezing and dissipation, Phys. Rev. A 45 (1992) 6570–6585. [10] W. Schleich, M. Pernigo, F.L. Kien, Phys. “Nonclassical state from two pseudoclassical states, Phys. Rev. A 44 (1991) 2172–2187.
1060
H. Prakash, P. Kumar / Optik 122 (2011) 1058–1060
[11] J. Janszky, A.V. Vinogradov, Squeezing via one-dimensional distribution of coherent states, Phys. Rev. Lett. 64 (1990) 2771–2774. [12] X.I.A. Yunjie, G.U.O. Guangcan, Nonclassical properties of even and odd coherent states, Phys. Lett. A 136 (1989) 281–283. [13] H. Prakash, P. Kumar, Maximum simultaneous squeezing and antibunching in superposed coherent states, Physica A 341 (2004) 201–207. [14] H. Prakash, P. Kumar, Squeezing in superposed coherent states, Physica A 319 (2003) 305–310. [15] H. Prakash, P. Kumar, Fourth-order squeezing in superposed coherent states, Acta Phys. Polon. B 34 (2003) 2769–2774. [16] See, e.g., B. Yurke, D. Stoler, Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion, Phys. Rev. Lett. 57 (1986) 13–16; G.J. Milburn, C.A. Holmes, Dissipative quantum and classical Liouville mechanics of the anharmonic oscillator, Phys. Rev. Lett. 56 (1986) 2237–2240.
[17] See, e.g., C.M. Caves, B. Yurke, Generation of superpositions of classically distinguishable quantum states from optical back-action evasion, Phys. Rev. A 41 (1990) 5261–5264; B. Yurke, W. Schleich, D.F. Walls, Quantum superpositions generated by quantum nondemolition measurements, Phys. Rev. A 42 (1990) 1703–1711. [18] M.A. Ahmad, R. Zeng, S.-T. Liu, Nonclassical properties of superposition of two coherent states shifted in phase by 3/2, Chin. Phys. Lett. 23 (2006) 2438–2441. [19] R. Zeng, M.A. Ahmad, S. Liu, Nonclassical state via superposition of two coherent states (/2 out of phase) and related entangled states, Optics Commun. 271 (2007) 162–168. [20] M. Ashfaq, S.-T. Liu, Superposition of two coherent states out of phase with average photon number as relative phase, Optik 120 (2009) 68–73. [21] L. Mandel, Squeezed states and sub-Poissonian photon statistics, Phys. Rev. Lett. 49 (1982) 136.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Non-classical properties of superposition of two coherent states having phase difference Hari Prakash a , Pankaj Kumar b,∗ a b
Department of Physics, University of Allahabad, Allahabad (U.P) 211002, India Department of Physics, Bhavan’s Mehta Mahavidyalaya (V. S. Mehta College of Science), Bharwari, Kaushambi (U.P) 212201, India
a r t i c l e
i n f o
Article history: Received 24 March 2010 Accepted 4 July 2010
PACS: 42.50.Dv Keywords: Non-classical features of light Coherent state Squeezing Sub-Poissonian photon statistics Displacement operator Phase shifting operator
a b s t r a c t Recently Ahmad et al. [Optik 2009;120;68; Optics Commun. 2007;271:162; Chin. Phys. Lett. 2006;23:2438] have non-classical properties of superposition of two-coherent states of the form, studied = K[|˛ + ei ˛ei ] for the special cases with values = /2, 3/2, and , and for arbitrarily fixed values of . We point out that some of their results are special cases of our recently published work [Physica A 319, 305 (2003); Physica A 341, 201(2004)] on the most general superposition of two arbi trary coherent states of the form ∼(Z1 |˛ + Z2 ˇ ), where X1,2 , ˛ and ˇ are arbitrary and only restriction on these is the normalization condition for the superposed state. To make our point we first obtain results for (i) squeezing of the most general Hermitian operator X = X1 cos + X2 sin , with X1 + iX2 = a, is the annihilation operator, and (ii) sub-Poissonian photon statistics, for the superposed state with a general and, then obtain results of Ahmad et al. for = /2, 3/2, and and for = 0 and /2. It is interesting to note that the arbitrarily fixed values = |˛|2 and −|˛|2 for = /2 and 3/2, respectively by Ahmad et al. are the values at which we get maximum squeezing working in a rigorous way. © 2010 Elsevier GmbH. All rights reserved.
1. Introduction States of light, properties of which cannot be explained on the basis of classical theory are called non-classical states [1]. The nonclassical nature of a state can be manifested in different ways like antibunching, sub-Poissonian photon statistics and various kinds of squeezing, etc. Earlier study of such non-classical effects was largely in academic interest [2], but now their applications in quantum information theory such as communication [3], quantum teleportation [4], dense coding [5] and quantum cryptography [6] are well realized. It has been demonstrated that non-classicality is the necessary input for entangled state [7]. A coherent state [8] defined as the eigenstate of annihilation operator, i.e., a |˛ = ˛ |˛, does not exhibit non-classical effects but a superposition of coherent states exhibit [9–16] various non-classical effects such as squeezing, higher-order squeezing, sub-Poissonian statistics and higher-order sub-Poissonian statistics. Buzek et al. [9] and Xia et al. [12] studied such effects in the superposition of two-coherent states |˛ and |−˛ and reported that the even coherent state exhibits squeezing but not sub-Poissonian statistics while the odd coherent state exhibits sub-
∗ Corresponding author. E-mail addresses: prakash [email protected] (H. Prakash), pankaj [email protected] (P. Kumar). 0030-4026/$ – see front matter © 2010 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2010.07.004
Poissonian statistics but not squeezing. Xia et al. also studied [12] such effects in the displaced even and odd coherent state. Schleich et al. [10] studied such effects in the superposition of two-coherent states, |˛ and |˛∗ , of identical mean photon number but different phases and reported that such superposition can exhibit both squeezing and sub-Poissonian statistics when |˛|2 1. Recently, we studied maximum squeezing in the most general superposed coherent state [13]. Later, we used this result to study [14] maximum simultaneous squeezing and sub-Poissonian statistics in superposed coherent states. We also studied [15] maximum fourthorder squeezing in superposition of two arbitrary coherent states. In practice, the superposition of coherent states can be generated in interaction of coherent state with nonlinear media [16] and in quantum nondemolition techniques [17]. Recently Ahmad et al. [18–20] studied the non-classical properties of the superposition with equal weights of pairs of coherent states (|˛ and i˛ ), (|˛ and −i˛ ) and (|˛ and |−˛). We point out that the results obtained by Ahmad et al. [18–20] are special cases of our recently published work [13,14] for the most general superposition of two-coherent states of the form =
(Z1 |˛ + Z2 ˇ ), where Z1,2 , ˛ and ˇ are arbitrary and only restriction on these is the normalization condition for the superposed coherent state. To make our point, we consider the more general superposed state,
= K[|˛ + ei ˛ei ],
(1)
H. Prakash, P. Kumar / Optik 122 (2011) 1058–1060
of two-coherent states |˛ and ˛ei , and study squeezing of the most general Hermitian Operator X defined by X = X1 cos + X2 sin ,
(2)
and sub-Poissonian photon statistics in the superposed coherent using recent results [13,14]. Here Hermitian operators state X1,2 are defined by X1 + iX2 = a, the annihilation operator, and complex number ˛ = |˛| ei˛ , angles and are completely arbitrary with the only restriction being the normalization condition for . We show that the results of Ahmad et al. are special cases = /2, 3/2, and . Also we show that the arbitrarily fixed values = |˛|2 and −|˛|2 for = /2 and 3/2, respectively by Ahmad et al. are, as a matter of fact, the values obtained for maximum squeezing, working in a rigorous way.
2. Squeezing in superposed coherent state Using the Hesenberg’s uncertainty relation, the condition for squeezing of X in the any state can be defined as
(X )2 < 1/4, where X = X − X . Recently we studied [14] squeezing of the operator hermitian X in the most general superposition state = Z1 |˛ + Z2 ˇ . Using the fact
that displacement and phase shift does not change the amount of squeezing in any state we related to superposition of coher-
˛+ˇ
ent states
2
and −
˛+ˇ 2
and showed [14] that the maximum
with the absolute mini 2 mum value 0.11077 of variance (X ) occurs for infinite squeezing in the superposed state
numbers of combinations with ˛ − ˇ = 1.59912 exp[± i
2
+ i],
Z1 1 = exp (ˇ˛∗ − ˇ∗ ˛) , Z2 2
(3)
and with arbitrary (˛ + ˇ) and . For the state , we can obtain results for maximum squeezing by using Eq. (3). For ˛ = |˛| ei˛ , this gives maximumsqueezing with (X )2 the absolute minimum value 0.11077 of variance occurring for infinite numbers of combinations with 2 |˛| sin
= 1.59912; 2
for = D (˛ + ˇ)/2
4
Using the results [13], i.e., Eq. (3) we find that the state exhibits maximum simultaneous and squeezing antibunching with (X )2 ˙ and minimum value minimum value 0.11077 of
and the condition ˛ + ˇ ˛ − ˇ.
−0.55692 of Mandel’s Q parameter for an infinite number of conditions in Eq. (3) and = arg(˛ + ˇ), for the as mentioned case ˛ + ˇ ˛ − ˇ. Using these results we report that the
superposed coherent state exhibits maximum simultaneous squeezing and antibunching with minimum value 0.11077 (X )2 of and minimum value −0.55692 of Mandel’s Q parameter for an infinite combination with 2 |˛| sin /2 = 1.59912, = ˛ + /2, = −|˛|2 sin under the consideration |˛| 1. 4. Conclusion In this paper we consider a more general superposition of = K[|˛ + ei ˛ei ]; ˛ = |˛| ei˛ and studcoherent states, ied photon statistics and ordinary squeezing of the most general Hermitian operator X defined by X = X1 cos + X2 sin , in this using our recently reported results superposed coherent state [13,14]. We conclude that maximum squeezing of X in the state occurs with the absolute minimum value 0.11077 of variance
(X )2 for infinite numbers of combinations with,
2 |˛| sin /2 = 1.59912, = ˛ + /2 and = −|˛|2 sin . Using these results and the properties of displacement operator we also conclude that this superposed coherent states also exhibits subPoissonian photon statistics for infinite numbers of combinations with, 2 |˛| sin /2 = 1.59912, = ˛ + /2 and = −|˛|2 sin when |˛| 1. The results of Ahmad et al. [18–20] are special cases ( = /2, 3/2 and ) of this general study. Acknowledgement We would like to thank Professors N. Chandra and R. Prakash for their interest and some critical comments. References
= −|˛|2 sin ,
(4)
and = ˛ + /2. We can obtain the results of Ahmad et al. for squeezing by putting the values = /2, 3/2, and which gives |˛| = 1.1307, 1.1307, 0.7996 and = −|˛|2 , |˛|2 and 0, respectively. This gives a rigorous derivation of values |˛|2 and –|˛|2 for for the cases = /2 and 3/2, respectively, which have been fixed arbitrary by Ahmad et al.
3. Simultaneous squeezing and sub-Poissonian photon statistics in superposed coherent state For characterization of the photon statistics of optical field in the state , Mandel [21] introduced a parameter based of the field defined fluctuations on 2intensity by Q = N ] − 1, where N = N − N and / [ (N) N = a+ a. When 0 < Q ≤ − 1, the photon statistics is called subPoissonian and the field is called antibunched. Recently we studied [13] simultaneous occurrence of antibunchingand squeezing of the hermitian operator X in the superposed state of two-coherent
1059
ing and antibunching in superposed state and found a relation (X )2 , between Mandel’s Q parameter and variance 1 2 (X ) − . (10) Q ∼ =4
states |˛ and ˇ . We investigated the connection between squeez-
[1] For excellent review on non-classical phenomena see, D.F. Walls, Squeezed states of light, Nature 306 (1983) 141–146; R. Loudon, P.L. Knight, Squeezed light, J. Mod. Opt. 34 (1987) 709–759; V.V. Dodonov, ‘Nonclassical’ states in quantum optics: a ‘squeezed’ review of the first 75 years, J. Opt. B 4 (2002) R1–R33. [2] See, e.g., B.R. Mollow, R.J. Glauber, Quantum theory of parametric amplification. I, Phys. Rev. 160 (1967) 1076; N. Chandra, H. Prakash, Quantum statistics of photon amplifiers, Indian J. Pure Appl. Phys. 2 (1971), 677, 688, 767; N. Chandra, H. Prakash, Anticorrelation in two-photon attenuated laser beam, Phys. Rev. A 1 (1970) 1696–1698. [3] C.H. Bennett, P.W. Shor, J.A. Smolin, A.V. Thapliyal, Entangled-assisted classical capacity of noisy quantum channels, Phys. Rev. Lett. 83 (1999) 3081–3084. [4] S.L. Braunstein, G.M. D’ Ariano, G.J. Milburn, M.F. Sacchi, Universal teleportation with a twist, Phys. Rev. Lett. 84 (2000) 3486–3489. [5] S.L. Braunstein, H.J. Kimble, Dense coding for continuous variables, Phys. Rev. A 61 (2000) 042302–042305. [6] C.H. Bennette, G. Brassard, N.D. Mermin, Quantum cryptography without Bells theorem, Phys. Rev. Lett. 68 (1992) 557–559. [7] M.S. Kim, W. Son, V. Buzek, P.L. Knight, Entanglement by a beam splitter: non-classicality as a prerequisite for entanglement, Phys. Rev. A 65 (2002) 032323–032329. [8] R.J. Glauber, Coherent and incoherent states of the radiation field, Phys. Rev. 131 (1963) 2766–2788. [9] V. Buˇzek, A. Vidiella-Barranco, P.L. Knight, Superpositions of coherent states: squeezing and dissipation, Phys. Rev. A 45 (1992) 6570–6585. [10] W. Schleich, M. Pernigo, F.L. Kien, Phys. “Nonclassical state from two pseudoclassical states, Phys. Rev. A 44 (1991) 2172–2187.
1060
H. Prakash, P. Kumar / Optik 122 (2011) 1058–1060
[11] J. Janszky, A.V. Vinogradov, Squeezing via one-dimensional distribution of coherent states, Phys. Rev. Lett. 64 (1990) 2771–2774. [12] X.I.A. Yunjie, G.U.O. Guangcan, Nonclassical properties of even and odd coherent states, Phys. Lett. A 136 (1989) 281–283. [13] H. Prakash, P. Kumar, Maximum simultaneous squeezing and antibunching in superposed coherent states, Physica A 341 (2004) 201–207. [14] H. Prakash, P. Kumar, Squeezing in superposed coherent states, Physica A 319 (2003) 305–310. [15] H. Prakash, P. Kumar, Fourth-order squeezing in superposed coherent states, Acta Phys. Polon. B 34 (2003) 2769–2774. [16] See, e.g., B. Yurke, D. Stoler, Generating quantum mechanical superpositions of macroscopically distinguishable states via amplitude dispersion, Phys. Rev. Lett. 57 (1986) 13–16; G.J. Milburn, C.A. Holmes, Dissipative quantum and classical Liouville mechanics of the anharmonic oscillator, Phys. Rev. Lett. 56 (1986) 2237–2240.
[17] See, e.g., C.M. Caves, B. Yurke, Generation of superpositions of classically distinguishable quantum states from optical back-action evasion, Phys. Rev. A 41 (1990) 5261–5264; B. Yurke, W. Schleich, D.F. Walls, Quantum superpositions generated by quantum nondemolition measurements, Phys. Rev. A 42 (1990) 1703–1711. [18] M.A. Ahmad, R. Zeng, S.-T. Liu, Nonclassical properties of superposition of two coherent states shifted in phase by 3/2, Chin. Phys. Lett. 23 (2006) 2438–2441. [19] R. Zeng, M.A. Ahmad, S. Liu, Nonclassical state via superposition of two coherent states (/2 out of phase) and related entangled states, Optics Commun. 271 (2007) 162–168. [20] M. Ashfaq, S.-T. Liu, Superposition of two coherent states out of phase with average photon number as relative phase, Optik 120 (2009) 68–73. [21] L. Mandel, Squeezed states and sub-Poissonian photon statistics, Phys. Rev. Lett. 49 (1982) 136.