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Superposition of two coherent states π out of phase with average photon number as relative phase
ARTICLE IN PRESS
Optik
Optics
Optik 120 (2009) 68–73 www.elsevier.de/ijleo
Superposition of two coherent states p out of phase with average photon number as relative phase Muhammad Ashfaq Ahmada,b,, Shu-Tian Liub a
Department of Physics, COMSATS Institute of Information Technology, Lahore 54000, Pakistan Department of Physics, Harbin Institute of Technology, Harbin 150001, China
b
Received 22 September 2006; accepted 19 February 2007
Abstract This paper discusses nonclassical features of the superposition of two coherent states p out of phase with each other and parameterized solely by amplitude a, that is, when the relative phase has a relationship with average photon number jaj2. In particular, it discusses oscillatory and sub-Poissonian photon statistics, degree of quadrature squeezing, and quasiprobability distribution functions. We examine that some nonclassical properties of these states are different from those of the even and odd coherent states. r 2007 Elsevier GmbH. All rights reserved. Keywords: Nonclassical states of light; Superposition of coherent states; Photon statistics; Squeezing properties
1. Introduction The superposition principle is one of the most fundamental kinematical concepts of quantum theory. In particular, a quantum mechanical superposition of two coherent states of light results in a new state which displays various nonclassical features. There has been considerable interest in the properties of quantum mechanical superposition of coherent states. The main focus of the present work is to study the nonclassical properties of the superposition of two coherent states p out of phase with each other and parameterized solely by amplitude parameter a. Studies on nonclassical properties of the quantum superposition of coherent states are of great interest because of their applications in quantum information Corresponding author. Department of Physics, COMSATS Institute of Information Technology, Lahore 54000, Pakistan. Tel.:+92 42 9203101x235; fax: +92 42 92 3100. E-mail address: [email protected] (M.A. Ahmad).
0030-4026/$ - see front matter r 2007 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2007.02.012
theory such as communication [1], quantum teleportation [2], dense coding [3], and quantum cryptography [4]. Kim et al. [5] demonstrated that nonclassicality is the necessary input for entangled states. Thus, the nonclassicality of the state would be worthily analyzed for quantum information theory. States with certain features that cannot be understood by the classical theory are called nonclassical states. The nonclassical nature of a quantum state can manifest itself in different ways. It has been shown that photon antibunching [6], sub-Poissonian distribution of photon numbers [7], the degree of quadrature squeezing [8], and oscillations of the photon number distribution [9,10] are manifestations of the nonclassical states of light. Moreover, an analysis of nonclassicality using Wigner function was also carried out by Lu¨tkenhaus and Barnnett [11]. Nonclassical properties of the quantum superposition of coherent states of certain forms have already been discussed. Various schemes have been suggested to produce such states extensively [12,13], and their
ARTICLE IN PRESS M.A. Ahmad, S.-T. Liu / Optik 120 (2009) 68–73
applications in quantum computation [14] have been discussed. A familiar example of such states is the superposition of two classical-like coherent states of same amplitude but with a phase shift of 1801: jci ¼ Nðjai þ eif j aiÞ.
(1)
For f ¼ 0, Eq. (1) describes the even coherent state, while for f ¼ p, it describes the odd coherent state. Recently, it has been shown that the superpositions of two coherent states p/2 out of phase [15] and 3p/2 out of phase [16,17] have some different nonclassical features than those of even–odd coherent states [18] when the relative phase f has a relationship with average photon number jaj2. Here we consider the superposition state of two coherent states shifted in phase by p dependent on average photon number jaj2 and describe in detail the nonclassical properties of this state. By fixing f ¼ jaj2 and f ¼ jaj2p, the states jcS+ and jcS, respectively, are given as 2
jci ¼ N ðjai eijaj j aiÞ,
(2)
The Wigner function may be defined as the Fourier transform of the symmetrically ordered characteristic function w(Z) [21]: Z 1 exp½Z b Zb wðZÞ d2 Z p2 Z 1 ¼ 2 exp½Z b Zb Trfr exp½Zay Z ag d2 Z. p
W ðbÞ ¼
ð3Þ
Inserting equation Eq. (2) into Eq. (3), after minor algebra we arrive at W ðbÞ ¼ N 2 fW jai ðbÞ þ W jai ðbÞ þ W int ðbÞg.
(4)
Here WjaS(b) or WjaS(b) denotes the Wigner function of the single coherent states. Wint(b) is due to the quantum interference between the two states and is responsible for nonclassicality. Finally, the Wigner function for the state jcS+ takes the form W þ ðbÞ ¼
2
where N 2 ¼ ½2 2e2jaj cosðjaj2 Þ1 is the normalization constant. The paper is organized in the following sequence: In Section 2, we discuss the quasi-probability distribution functions. The oscillatory behavior of the photon number distribution, sub-Poissonian photon statistics, and photon anti-bunching are presented in Section 3, while the degree of quadrature squeezing is discussed in Section 4. The final section presents a short discussion of the results.
69
2 2 N fexp½2ðy2 x2 Þ2 2ðy1 x1 Þ2 p þ þ exp½2ðy2 þ x2 Þ2 2ðy1 þ x1 Þ2 þ 2 cosðjaj2 þ 4DÞ exp½2jbj2 g,
ð5Þ
and for the state jcS, it looks like W ðbÞ ¼
2 2 N fexp½2ðy2 x2 Þ2 2ðy1 x1 Þ2 p þ exp½2ðy2 þ x2 Þ2 2ðy1 þ x1 Þ2 2 cosðjaj2 þ 4DÞ exp½2jbj2 g,
ð6Þ
where we have introduced D ¼ x 1 y 2 x2 y 1 .
2. Quasiprobability distribution functions Quasiprobability distribution functions are very important to study the quantum features of the state under control. Three types of these functions are GlauberSudarshan P-function, Q function, and Wigner function. As for the quasiprobability distribution, the P function is highly singular, involving an infinite sum of higherorder derivatives of a delta function. The Q function is always non-negative and does not exhibit a clear signal for nonclassicality. But the Wigner function, when it takes on negative values, does exhibit a clear signal for nonclassicality. A sophisticated approach to discuss the nonclassical features of the superposition state of two coherent states is Wigner distribution, and the quantum interference structure of Wigner function gives rise to nonclassical features of the superposition state [18]. Moreover, negative regions in the Wigner function of a given state can be seen as signatures of non-classical behavior [11,19,20].
The Wigner functions for Eqs. (5) and (6) are plotted in Fig. 1 for several values of a. The Wigner function of jcS+ for a ¼ 0 represents the state of vacuum because no negative regions are located in phase space (Fig. 1a). The Wigner functions of both states jcS+ and jcS show strong quantum interference between states jaS and jaS. The two Gaussian bells in Fig. 1b represent the two coherent states, and the structure halfway between them is the result of the quantum interference. The two bells and the interference structure gradually separate with the increase in a (Fig. 1b and c). As cited before, the quantum interference is responsible for nonclassical features and that results in the appearance of negative values of the Wigner distribution in interference structure, which is the sign of nonclassical features. Moreover, if the coherent states are far away, the interference structure has many well-pronounced peaks situated near to each other. On the contrary, if the coherent states are near enough, the interference structure has only several peaks, which, partly merging with the bells of the coherent states, decrease the uncertainty of one of the quadrature below the vacuum
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With the help of Eq. (7), that is, jaj2 an pffiffiffiffi , hnjai ¼ exp 2 n! jaj2 ð1Þn an pffiffiffiffi , hnj ai ¼ exp 2 n! the photon count probability for jcS+ takes the form 2
2 Pþ n ¼ Nþ
2 ejaj jaj2n j1 þ ð1Þn eijaj j2 , n!
(9)
and for the state jcS 2
2 P n ¼ N
Fig. 1. Wigner function (a) for the state jcS+ with a ¼ 0 representing the vacuum state, (b) for the state jcS+ with a ¼ 2. Structure can be seen emerging from the quantum interference between the two Gaussian bells of the individual coherent states, (c) for the state jcS+ with a ¼ 5. Here the interference structure and two Gaussian bells are separated with the increase in a, (d) for the state jcS with a ¼ 2, and (e) for the state jcS with a ¼ 5.
level. That is, the closer the two bells get to each other, the lesser the number of peaks that are significant in the interference structure. In other words, the wavelength of the interference structure is reciprocally proportional to the value of a.
3. Oscillatory and sub-Poissonian photon statistics A coherent state jaS which can be regarded as a classical-like state in terms of number (Fock) states jnS is given by jaj2 X an pffiffiffiffi jni: jai ¼ exp (7) 2 n! n The probability P(n) to find n photons in the given state jcS is given by PðnÞ ¼ jhnjcij2 ¼ N 2 jhnjai þ eif hnj aij2 .
(8)
2 ejaj jaj2n j1 ð1Þn eijaj j2 . n!
(10)
The quantum interference between the two states, two contributors of interfering probability amplitudes /njaS and /njaS, creates the interference term ½1 2 ð1Þn eijaj ; which modulates the familiar Poissonian photon statistics of a single coherent state. As it has been shown [10], the oscillatory behavior of the interference part of the Wigner function not only results in the oscillation of the photon number distribution but also gives rise to quadrature squeezing, which is discussed in the next section. We analyzed the consequences of this quantum interference in more detail from the curves of probability P(n) of Eqs. (9) and (10) as a function of n and a depicted in Fig. (2). From this figure, it is seen that for certain values of n probability P(n) falls to zero, which can be seen as a signature of nonclassical behavior [9]. It is also evident that the oscillatory behavior of the interference part of the Wigner function of both states is closely related to oscillations of the photon number distribution. To characterize sub-Poissonian statistics, following Mandel [7], we consider Q parameter 2
Q¼
hðDnÞ2 i hni hay a2 i hay ai2 ¼ , hni hay ai
(11)
which characterizes the departure from Poissonian photon statistics. For classical light, Q40. When Q has some value between 0 and 1 (1oQo0), then the photon statistics is called sub-Poissonian. The Q parameter for the state jcS+ is 2
Q ¼
4jaj2 e2jaj cosðjaj2 Þ , 2 1 e4jaj cos2 ðjaj2 Þ
(12)
indicating sub-Poissonian photon statistics for large values of jaj2 in contrast to the even coherent states (Fig. 3a). The Q parameter for the state c is 2
Q ¼
4jaj2 e2jaj cosðjaj2 Þ , 2 1 e4jaj cos2 ðjaj2 Þ
(13)
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4. Degree of quadrature squeezing In order to study the quadrature squeezing, we now introduce the so-called quadrature operators corresponding to creation (ay) and annihilation (a) operators: X 1 ¼ 12ða þ ay Þ; Fig. 2. Photon number distribution (a) of the state jcS+ and (b) of the state jcS with a ¼ 3.
X 2 ¼ 12ða ay Þ.
Note that X1 and X2, essentially the position and momentum operators but scaled to be dimensionless, satisfy the commutation relation [X1,X2] ¼ i/2 and, as a result, the uncertainty relation for these operators is 1 . hðDX 1 Þ2 ihðDX 2 Þ2 iX16
Quadrature squeezing exists whenever the degree of squeezing defined by [8] Dj ¼ 4hðDX j Þ2 i 1 ¼ 4ðhX 2j i hX j i2 Þ 1;
j ¼ 1; 2 (17)
satisfies the condition 1oDjo0, j ¼ 1,2. With the help of the equations 2
hX 21 i ¼ 14ha2 þ ay þ 2ay a þ 1i Fig. 3. The Q parameter vs. jaj2 (a) for the state jcS+ and (b) for the state jcS. The maximum degree of sub-Poissonian photon statistics (Q-1) is obtained for jaj2-0.
2
¼ 14f1 þ a2 þ a2 þ 2N 2 jaj2 ½2 2e2jaj cosðjaj2 Þg, ð18Þ 2
hX 22 i ¼ 14ha2 ay þ 2ay a þ 1i indicating sub-Poissonian photon statistics for small values of jaj2 (Fig. 3b). It is interesting to note that the state jcS has the maximum degree of sub-Poissonian photon statistics for small values of jaj2, i.e., Q-1 when jaj2-0. To study the effect of photon antibunching, the second-order quantum coherence function can be written as [21] hðDnÞ2 i hni hay2 a2 i g ð0Þ ¼ 1 þ ¼ y 2. hni2 ha ai ð2Þ
(14)
2
½1 þ e2jaj cosðjaj2 Þ2 , 2 ½1 e2jaj cosðjaj2 Þ2
ð19Þ hX 1 i2 ¼ 14½hai þ hay i2 2
¼ N 4 e4jaj sinðjaj2 Þ½a2 þ a2 2jaj2 ,
¼ N 4 e4jaj sinðjaj2 Þ½a2 þ a2 þ 2jaj2 ,
ð21Þ
we find the degree of quadrature squeezing for the state jcS+ as 2
Dþ 1 ¼
2
ða2 þ a2 Þ½1 þ 2e2jaj cosðjaj2 Þ þ ða a Þ2 e4jaj þ 2jaj2 , 2 ½1 þ e2jaj cosðjaj2 Þ2
(15)
which shows that photon antibunching exists for large values of jaj2, and for the state jcS
(22a) 2
Dþ 2 ¼
2
ða2 þ a2 Þ½1 þ 2e2jaj cosðjaj2 Þ þ ða þ a Þ2 e4jaj 2jaj2 , 2 ½1 þ e2jaj cosðjaj2 Þ2
(22b)
2
gð2Þ ð0Þ ¼
½1 e2jaj cosðjaj2 Þ2 . 2 ½1 þ e2jaj cosðjaj2 Þ2
ð20Þ
hX 2 i2 ¼ 14½hai hay i2 2
The second-order quantum coherence function for the state jcS+ is gð2Þ ð0Þ ¼
2
¼ 14f1 a2 a2 þ 2N 2 jaj2 ½2 2e2jaj cosðjaj2 Þg,
(16)
and for the state jcS
2
This expression shows that the state gives rise to photon antibunching for small values of jaj2. We conclude this section by emphasizing that although a single coherent state does not exhibit sub-Poissonian photon statistics, a state built out of the superposition of two coherent states can exhibit sub-Poissonian photon statistics and photon antibunching.
D 1 ¼
2
ða2 þ a2 Þ½1 2e2jaj cosðjaj2 Þ þ ða a Þ2 e4jaj þ 2jaj2 , 2 ½1 e2jaj cosðjaj2 Þ2
(23a) 2
D 2 ¼
2
ða2 þ a2 Þ½1 2e2jaj cosðjaj2 Þ þ ða þ a Þ2 e4jaj 2jaj2 . 2 ½1 e2jaj cosðjaj2 Þ2
(23b)
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of entangled states based on such states need to be discussed for quantum information research.
References
Fig. 4. Degree of squeezing (a) for the state jcS+ vs. the parameter a. We see that the highest degree of squeezing can be observed for small intensities of the light field, that is, for small values of the parameter a and (b) for the state jcS.
From the above equations, it is evident that both states jcS+ and jcS do exhibit squeezing in X2 quadrature. The squeezing effects for both states are plotted in Fig. 4. From Fig. 4a we find that the maximum degree of squeezing in the case of state jcS+ appears for small values of a. On the other hand, the state jcS exhibits squeezing for large values of a in contrast to the odd coherent states discussed in [17], which do not exhibit squeezing. But the degree of quadrature squeezing in the state jcS is significantly smaller than that for the state jcS+. We should underline here that, in the case of the state jcS+, there is a close relation between the presence of squeezing and the shape of the Wigner function. As seen from Fig. 1b, the Wigner function itself is ‘‘squeezed’’ in phase space in the y2-direction corresponding to the reduction of fluctuations in the X2 quadrature. Hence, we conclude that the present superposition state is a nonclassical state.
5. Conclusion In this paper, we have discussed the nonclassical features of the superposition state of two coherent states shifted in phase by p, parameterized solely with respect to a. It was shown that the proposed states are nonclassical and some nonclassical features of these states are different from the famous Schro¨dinger cat states, namely, even and odd coherent states discussed in [17]. The present state jcS+ exhibits sub-Poissonian photon statistics for large values of jaj2 in contrast to the even coherent states, which do not exhibit sub-Poissonian photon statistics. The state jcS does exhibit maximum degree of sub-Poissonian photon statistics for small values of jaj2 similar to the odd coherent states, that is, (Q-1) is obtained when jaj2-0. Similarly, the state jcS+ exhibits maximum degree of squeezing for small values of jaj2, but the state jcS does exhibit squeezing for large values of jaj2 in contrast to the odd coherent states. Further investigations of the properties
[1] C.H. Bennett, P.W. Shor, J.A. Smolin, Thapliyal AV: entanglement-assisted classical capacity of noisy quantum channels, Phys. Rev. Lett. 83 (1999) 3081–3084. [2] S.L. Braunstein, G.M.D. Ariano, G.J. Millburn, M.F. Sacchi, Universal teleportation with a twist, Phys. Rev. Lett. 84 (2000) 3486–3489. [3] S.L. Braunstein, H.J. Kimble, Dense coding for continuous variables, Phys. Rev. A 61 (2000) 042302–042305. [4] C.H. Bennett, G. Brassard, N.D. Mermin, Quantum cryptography without Bells theorem, Phys. Rev. Lett. 68 (1992) 557–559. [5] M.S. Kim, W. Son, V. Buzˇek, P.L. Knight, Entanglement by a beam splitter: nonclassicality as a prerequisite for entanglement, Phys. Rev. A 65 (2002) 032323–032329. [6] M.C. Teich, B.E.A. Saleh, Squeezed and antibunched light, Phys. Today 43 (6) (1990) 26–34. [7] L. Mandel, Sub-Poissonian photon statistics in resonance fluorescence, Opt. Lett. 4 (1979) 205–207; L. Mandel, Non-classical states of the electromagnetic field, Phys. Scr. T 12 (1986) 34–42. [8] V. Buzˇek, I. Jex, Squeezing properties of coupled nonlinear oscillators, Int. J. Mod. Phys. B 4 (1990) 659–676. [9] R. Simon, M. Selvadoray, Arvind, N. Mukunda, Necessary and sufficient classicality conditions on photon number distributions, e-print quant-ph/9709030. [10] W. Schleich, M. Pernigo, F.L. Kien, Nonclassical state from two pseudoclassical states, Phys. Rev. A 44 (1991) 2172–2187. [11] N. Lu¨tkenhaus, S.M. Barnnett, Nonclassical effects in phase space, Phys. Rev. A 51 (1995) 3340–3342. [12] M. Brune, E. Hagley, J. Dreyer, X. Maitre, C. Wunderlich, J.M. Raimond, S. Haroche, Observing the progressive decoherence of the Meter in a quantum measurement, Phys. Rev. Lett. 77 (1996) 4887–4890. [13] J.M. Raimond, M. Brune, S. Haroche, Reversible decoherence of a mesoscopic superposition of field states, Phys. Rev. Lett. 79 (1997) 1964–1967. [14] T.C. Ralph, A. Gilchrist, G.J. Milburn, W.J. Munro, S. Glancy, Quantum computation with optical coherent states, Phys. Rev. A 68 (2003) 042319–042329. [15] R. Zeng, M.A. Ahmad, S. Liu, Nonclassical state via superposition of two coherent states (p/2 out of phase) and related entangled states, Opt. Commun. 271 (1) (2007) 162–168. [16] M.A. Ahmad, R. Zeng, S. Liu, Nonclassical properties of superposition of two coherent states shifted in phase by 3p/2, Chin. Phys. Lett. 23 (9) (2006) 2438–2441. [17] M.A. Ahmad, S. Liu, Entangled states based on two coherent states 3p/2 out of phase, Chin. Phys. Lett. 23 (11) (2006) 2964–2967. [18] V. Buzˇek, A. Vidiella-Barranco, P.L. Knight, Superpositions of coherent states: squeezing and dissipation, Phys. Rev. A 45 (1992) 570–6585; M. Hillery, R.F. O’Connell, M.O. Scully, E.P. Wigner,
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Distribution functions in physics: fundamentals, Phys. Rep. 106 (3) (1984) 121–167. [19] P. Fo¨ldi, A. Czirja´k, B. Molna´r, M.G. Benedict, Formation of Schro¨dinger-cat states in the morse potential: Wigner function picture, Opt. Express 10 (2002) 376–381.
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˙ yczkowski, Negativity of the Wigner [20] A. Kenfack, K. Z function as an indicator of nonclassicality, J. Opt. B Quantum Semiclass. Opt. 6 (2004) 396–404. [21] D.F. Walls, G.J. Milburn, Quantum Optics, Springer, Berlin, 1994.
Optik
Optics
Optik 120 (2009) 68–73 www.elsevier.de/ijleo
Superposition of two coherent states p out of phase with average photon number as relative phase Muhammad Ashfaq Ahmada,b,, Shu-Tian Liub a
Department of Physics, COMSATS Institute of Information Technology, Lahore 54000, Pakistan Department of Physics, Harbin Institute of Technology, Harbin 150001, China
b
Received 22 September 2006; accepted 19 February 2007
Abstract This paper discusses nonclassical features of the superposition of two coherent states p out of phase with each other and parameterized solely by amplitude a, that is, when the relative phase has a relationship with average photon number jaj2. In particular, it discusses oscillatory and sub-Poissonian photon statistics, degree of quadrature squeezing, and quasiprobability distribution functions. We examine that some nonclassical properties of these states are different from those of the even and odd coherent states. r 2007 Elsevier GmbH. All rights reserved. Keywords: Nonclassical states of light; Superposition of coherent states; Photon statistics; Squeezing properties
1. Introduction The superposition principle is one of the most fundamental kinematical concepts of quantum theory. In particular, a quantum mechanical superposition of two coherent states of light results in a new state which displays various nonclassical features. There has been considerable interest in the properties of quantum mechanical superposition of coherent states. The main focus of the present work is to study the nonclassical properties of the superposition of two coherent states p out of phase with each other and parameterized solely by amplitude parameter a. Studies on nonclassical properties of the quantum superposition of coherent states are of great interest because of their applications in quantum information Corresponding author. Department of Physics, COMSATS Institute of Information Technology, Lahore 54000, Pakistan. Tel.:+92 42 9203101x235; fax: +92 42 92 3100. E-mail address: [email protected] (M.A. Ahmad).
0030-4026/$ - see front matter r 2007 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2007.02.012
theory such as communication [1], quantum teleportation [2], dense coding [3], and quantum cryptography [4]. Kim et al. [5] demonstrated that nonclassicality is the necessary input for entangled states. Thus, the nonclassicality of the state would be worthily analyzed for quantum information theory. States with certain features that cannot be understood by the classical theory are called nonclassical states. The nonclassical nature of a quantum state can manifest itself in different ways. It has been shown that photon antibunching [6], sub-Poissonian distribution of photon numbers [7], the degree of quadrature squeezing [8], and oscillations of the photon number distribution [9,10] are manifestations of the nonclassical states of light. Moreover, an analysis of nonclassicality using Wigner function was also carried out by Lu¨tkenhaus and Barnnett [11]. Nonclassical properties of the quantum superposition of coherent states of certain forms have already been discussed. Various schemes have been suggested to produce such states extensively [12,13], and their
ARTICLE IN PRESS M.A. Ahmad, S.-T. Liu / Optik 120 (2009) 68–73
applications in quantum computation [14] have been discussed. A familiar example of such states is the superposition of two classical-like coherent states of same amplitude but with a phase shift of 1801: jci ¼ Nðjai þ eif j aiÞ.
(1)
For f ¼ 0, Eq. (1) describes the even coherent state, while for f ¼ p, it describes the odd coherent state. Recently, it has been shown that the superpositions of two coherent states p/2 out of phase [15] and 3p/2 out of phase [16,17] have some different nonclassical features than those of even–odd coherent states [18] when the relative phase f has a relationship with average photon number jaj2. Here we consider the superposition state of two coherent states shifted in phase by p dependent on average photon number jaj2 and describe in detail the nonclassical properties of this state. By fixing f ¼ jaj2 and f ¼ jaj2p, the states jcS+ and jcS, respectively, are given as 2
jci ¼ N ðjai eijaj j aiÞ,
(2)
The Wigner function may be defined as the Fourier transform of the symmetrically ordered characteristic function w(Z) [21]: Z 1 exp½Z b Zb wðZÞ d2 Z p2 Z 1 ¼ 2 exp½Z b Zb Trfr exp½Zay Z ag d2 Z. p
W ðbÞ ¼
ð3Þ
Inserting equation Eq. (2) into Eq. (3), after minor algebra we arrive at W ðbÞ ¼ N 2 fW jai ðbÞ þ W jai ðbÞ þ W int ðbÞg.
(4)
Here WjaS(b) or WjaS(b) denotes the Wigner function of the single coherent states. Wint(b) is due to the quantum interference between the two states and is responsible for nonclassicality. Finally, the Wigner function for the state jcS+ takes the form W þ ðbÞ ¼
2
where N 2 ¼ ½2 2e2jaj cosðjaj2 Þ1 is the normalization constant. The paper is organized in the following sequence: In Section 2, we discuss the quasi-probability distribution functions. The oscillatory behavior of the photon number distribution, sub-Poissonian photon statistics, and photon anti-bunching are presented in Section 3, while the degree of quadrature squeezing is discussed in Section 4. The final section presents a short discussion of the results.
69
2 2 N fexp½2ðy2 x2 Þ2 2ðy1 x1 Þ2 p þ þ exp½2ðy2 þ x2 Þ2 2ðy1 þ x1 Þ2 þ 2 cosðjaj2 þ 4DÞ exp½2jbj2 g,
ð5Þ
and for the state jcS, it looks like W ðbÞ ¼
2 2 N fexp½2ðy2 x2 Þ2 2ðy1 x1 Þ2 p þ exp½2ðy2 þ x2 Þ2 2ðy1 þ x1 Þ2 2 cosðjaj2 þ 4DÞ exp½2jbj2 g,
ð6Þ
where we have introduced D ¼ x 1 y 2 x2 y 1 .
2. Quasiprobability distribution functions Quasiprobability distribution functions are very important to study the quantum features of the state under control. Three types of these functions are GlauberSudarshan P-function, Q function, and Wigner function. As for the quasiprobability distribution, the P function is highly singular, involving an infinite sum of higherorder derivatives of a delta function. The Q function is always non-negative and does not exhibit a clear signal for nonclassicality. But the Wigner function, when it takes on negative values, does exhibit a clear signal for nonclassicality. A sophisticated approach to discuss the nonclassical features of the superposition state of two coherent states is Wigner distribution, and the quantum interference structure of Wigner function gives rise to nonclassical features of the superposition state [18]. Moreover, negative regions in the Wigner function of a given state can be seen as signatures of non-classical behavior [11,19,20].
The Wigner functions for Eqs. (5) and (6) are plotted in Fig. 1 for several values of a. The Wigner function of jcS+ for a ¼ 0 represents the state of vacuum because no negative regions are located in phase space (Fig. 1a). The Wigner functions of both states jcS+ and jcS show strong quantum interference between states jaS and jaS. The two Gaussian bells in Fig. 1b represent the two coherent states, and the structure halfway between them is the result of the quantum interference. The two bells and the interference structure gradually separate with the increase in a (Fig. 1b and c). As cited before, the quantum interference is responsible for nonclassical features and that results in the appearance of negative values of the Wigner distribution in interference structure, which is the sign of nonclassical features. Moreover, if the coherent states are far away, the interference structure has many well-pronounced peaks situated near to each other. On the contrary, if the coherent states are near enough, the interference structure has only several peaks, which, partly merging with the bells of the coherent states, decrease the uncertainty of one of the quadrature below the vacuum
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With the help of Eq. (7), that is, jaj2 an pffiffiffiffi , hnjai ¼ exp 2 n! jaj2 ð1Þn an pffiffiffiffi , hnj ai ¼ exp 2 n! the photon count probability for jcS+ takes the form 2
2 Pþ n ¼ Nþ
2 ejaj jaj2n j1 þ ð1Þn eijaj j2 , n!
(9)
and for the state jcS 2
2 P n ¼ N
Fig. 1. Wigner function (a) for the state jcS+ with a ¼ 0 representing the vacuum state, (b) for the state jcS+ with a ¼ 2. Structure can be seen emerging from the quantum interference between the two Gaussian bells of the individual coherent states, (c) for the state jcS+ with a ¼ 5. Here the interference structure and two Gaussian bells are separated with the increase in a, (d) for the state jcS with a ¼ 2, and (e) for the state jcS with a ¼ 5.
level. That is, the closer the two bells get to each other, the lesser the number of peaks that are significant in the interference structure. In other words, the wavelength of the interference structure is reciprocally proportional to the value of a.
3. Oscillatory and sub-Poissonian photon statistics A coherent state jaS which can be regarded as a classical-like state in terms of number (Fock) states jnS is given by jaj2 X an pffiffiffiffi jni: jai ¼ exp (7) 2 n! n The probability P(n) to find n photons in the given state jcS is given by PðnÞ ¼ jhnjcij2 ¼ N 2 jhnjai þ eif hnj aij2 .
(8)
2 ejaj jaj2n j1 ð1Þn eijaj j2 . n!
(10)
The quantum interference between the two states, two contributors of interfering probability amplitudes /njaS and /njaS, creates the interference term ½1 2 ð1Þn eijaj ; which modulates the familiar Poissonian photon statistics of a single coherent state. As it has been shown [10], the oscillatory behavior of the interference part of the Wigner function not only results in the oscillation of the photon number distribution but also gives rise to quadrature squeezing, which is discussed in the next section. We analyzed the consequences of this quantum interference in more detail from the curves of probability P(n) of Eqs. (9) and (10) as a function of n and a depicted in Fig. (2). From this figure, it is seen that for certain values of n probability P(n) falls to zero, which can be seen as a signature of nonclassical behavior [9]. It is also evident that the oscillatory behavior of the interference part of the Wigner function of both states is closely related to oscillations of the photon number distribution. To characterize sub-Poissonian statistics, following Mandel [7], we consider Q parameter 2
Q¼
hðDnÞ2 i hni hay a2 i hay ai2 ¼ , hni hay ai
(11)
which characterizes the departure from Poissonian photon statistics. For classical light, Q40. When Q has some value between 0 and 1 (1oQo0), then the photon statistics is called sub-Poissonian. The Q parameter for the state jcS+ is 2
Q ¼
4jaj2 e2jaj cosðjaj2 Þ , 2 1 e4jaj cos2 ðjaj2 Þ
(12)
indicating sub-Poissonian photon statistics for large values of jaj2 in contrast to the even coherent states (Fig. 3a). The Q parameter for the state c is 2
Q ¼
4jaj2 e2jaj cosðjaj2 Þ , 2 1 e4jaj cos2 ðjaj2 Þ
(13)
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4. Degree of quadrature squeezing In order to study the quadrature squeezing, we now introduce the so-called quadrature operators corresponding to creation (ay) and annihilation (a) operators: X 1 ¼ 12ða þ ay Þ; Fig. 2. Photon number distribution (a) of the state jcS+ and (b) of the state jcS with a ¼ 3.
X 2 ¼ 12ða ay Þ.
Note that X1 and X2, essentially the position and momentum operators but scaled to be dimensionless, satisfy the commutation relation [X1,X2] ¼ i/2 and, as a result, the uncertainty relation for these operators is 1 . hðDX 1 Þ2 ihðDX 2 Þ2 iX16
Quadrature squeezing exists whenever the degree of squeezing defined by [8] Dj ¼ 4hðDX j Þ2 i 1 ¼ 4ðhX 2j i hX j i2 Þ 1;
j ¼ 1; 2 (17)
satisfies the condition 1oDjo0, j ¼ 1,2. With the help of the equations 2
hX 21 i ¼ 14ha2 þ ay þ 2ay a þ 1i Fig. 3. The Q parameter vs. jaj2 (a) for the state jcS+ and (b) for the state jcS. The maximum degree of sub-Poissonian photon statistics (Q-1) is obtained for jaj2-0.
2
¼ 14f1 þ a2 þ a2 þ 2N 2 jaj2 ½2 2e2jaj cosðjaj2 Þg, ð18Þ 2
hX 22 i ¼ 14ha2 ay þ 2ay a þ 1i indicating sub-Poissonian photon statistics for small values of jaj2 (Fig. 3b). It is interesting to note that the state jcS has the maximum degree of sub-Poissonian photon statistics for small values of jaj2, i.e., Q-1 when jaj2-0. To study the effect of photon antibunching, the second-order quantum coherence function can be written as [21] hðDnÞ2 i hni hay2 a2 i g ð0Þ ¼ 1 þ ¼ y 2. hni2 ha ai ð2Þ
(14)
2
½1 þ e2jaj cosðjaj2 Þ2 , 2 ½1 e2jaj cosðjaj2 Þ2
ð19Þ hX 1 i2 ¼ 14½hai þ hay i2 2
¼ N 4 e4jaj sinðjaj2 Þ½a2 þ a2 2jaj2 ,
¼ N 4 e4jaj sinðjaj2 Þ½a2 þ a2 þ 2jaj2 ,
ð21Þ
we find the degree of quadrature squeezing for the state jcS+ as 2
Dþ 1 ¼
2
ða2 þ a2 Þ½1 þ 2e2jaj cosðjaj2 Þ þ ða a Þ2 e4jaj þ 2jaj2 , 2 ½1 þ e2jaj cosðjaj2 Þ2
(15)
which shows that photon antibunching exists for large values of jaj2, and for the state jcS
(22a) 2
Dþ 2 ¼
2
ða2 þ a2 Þ½1 þ 2e2jaj cosðjaj2 Þ þ ða þ a Þ2 e4jaj 2jaj2 , 2 ½1 þ e2jaj cosðjaj2 Þ2
(22b)
2
gð2Þ ð0Þ ¼
½1 e2jaj cosðjaj2 Þ2 . 2 ½1 þ e2jaj cosðjaj2 Þ2
ð20Þ
hX 2 i2 ¼ 14½hai hay i2 2
The second-order quantum coherence function for the state jcS+ is gð2Þ ð0Þ ¼
2
¼ 14f1 a2 a2 þ 2N 2 jaj2 ½2 2e2jaj cosðjaj2 Þg,
(16)
and for the state jcS
2
This expression shows that the state gives rise to photon antibunching for small values of jaj2. We conclude this section by emphasizing that although a single coherent state does not exhibit sub-Poissonian photon statistics, a state built out of the superposition of two coherent states can exhibit sub-Poissonian photon statistics and photon antibunching.
D 1 ¼
2
ða2 þ a2 Þ½1 2e2jaj cosðjaj2 Þ þ ða a Þ2 e4jaj þ 2jaj2 , 2 ½1 e2jaj cosðjaj2 Þ2
(23a) 2
D 2 ¼
2
ða2 þ a2 Þ½1 2e2jaj cosðjaj2 Þ þ ða þ a Þ2 e4jaj 2jaj2 . 2 ½1 e2jaj cosðjaj2 Þ2
(23b)
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of entangled states based on such states need to be discussed for quantum information research.
References
Fig. 4. Degree of squeezing (a) for the state jcS+ vs. the parameter a. We see that the highest degree of squeezing can be observed for small intensities of the light field, that is, for small values of the parameter a and (b) for the state jcS.
From the above equations, it is evident that both states jcS+ and jcS do exhibit squeezing in X2 quadrature. The squeezing effects for both states are plotted in Fig. 4. From Fig. 4a we find that the maximum degree of squeezing in the case of state jcS+ appears for small values of a. On the other hand, the state jcS exhibits squeezing for large values of a in contrast to the odd coherent states discussed in [17], which do not exhibit squeezing. But the degree of quadrature squeezing in the state jcS is significantly smaller than that for the state jcS+. We should underline here that, in the case of the state jcS+, there is a close relation between the presence of squeezing and the shape of the Wigner function. As seen from Fig. 1b, the Wigner function itself is ‘‘squeezed’’ in phase space in the y2-direction corresponding to the reduction of fluctuations in the X2 quadrature. Hence, we conclude that the present superposition state is a nonclassical state.
5. Conclusion In this paper, we have discussed the nonclassical features of the superposition state of two coherent states shifted in phase by p, parameterized solely with respect to a. It was shown that the proposed states are nonclassical and some nonclassical features of these states are different from the famous Schro¨dinger cat states, namely, even and odd coherent states discussed in [17]. The present state jcS+ exhibits sub-Poissonian photon statistics for large values of jaj2 in contrast to the even coherent states, which do not exhibit sub-Poissonian photon statistics. The state jcS does exhibit maximum degree of sub-Poissonian photon statistics for small values of jaj2 similar to the odd coherent states, that is, (Q-1) is obtained when jaj2-0. Similarly, the state jcS+ exhibits maximum degree of squeezing for small values of jaj2, but the state jcS does exhibit squeezing for large values of jaj2 in contrast to the odd coherent states. Further investigations of the properties
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