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A New Type of Photon-Added Squeezed Coherent State and Its Statistical Properties∗

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A New Type of Photon-Added Squeezed Coherent State and Its Statistical Properties∗ Chin. Phys. B Vol. 21, No. 7 (2012) 070301 A new type of photon-added squeezed coherent state and its statistical properties∗ Zhou Jun(周 军)a)b)y, Fan Hong-Yi(范洪义)b), and Song Jun(宋 军)a)b) a)Department of Material and Chemical Engineering, West Anhui University, Liu'an 237012, China b)Department of Material Science and Engineering, University of Science and Technology of China, Hefei 230026, China (Received 21 December 2011; revised manuscript received 7 January 2012) We construct a new type of photon-added squeezed coherent state generated by repeatedly operating the bosonic creation operator on a new type of squeezed coherent state [Fan H Y and Xiao M 1996 Phys. Lett.A 220 81]. We find that its normalization factor is related to single-variable Hermite polynomials. Furthermore, we investigate its statistical properties, such as Mandel's Q parameter, photon-number distribution, and Wigner function. The nonclassicality is displayed in terms of the intense oscillation of photon-number distribution and the negativity of the Wigner function. Keywords: non-Gaussian state, Hermite polynomial, squeezed coherent state, Wigner function PACS: 03.65.{w, 42.50.{p DOI: 10.1088/1674-1056/21/7/070301 1. Introduction of squeezed coherent state (SCS), which differs from conventional squeezed states whose displacement pa- As the most common Gaussian states of a radi- rameter depends on its squeezing parameter and is ation field, coherent states, and squeezed states are important topics in quantum optics and play signif- more general. For instance, the SCS can be reduced icant roles in quantum information processing with to the simple coherent state, the coordinate eigen- continuous variables, such as quantum teleportation, state, or the momentum eigenstate. Then, a ques- [1−3] dense coding, and quantum cloning. Experimen- tion of what properties the new type photon-added tally, Gaussian states, such as Fock states, coherent squeezed coherent state (PASCS) has is raised. In this states, and squeezed states, have been generated, but there are some limitations in using them for quan- paper, we will introduce the PASCS and investigate tum information processing.[4] Non-Gaussian coher- its statistical properties, such as Mandel's Q parame- ent states and non-Gaussian squeezed states with ex- ter, photon-number distribution (PND), as well as the teme nonclassical properties may constitute powerful Wigner function (WF). This work is arranged as fol- resources for the efficient implementation of quantum lows. In Section 2, we give a brief review of the SCS. communication and computation.[5;6] It is well known In Section 3, we introduce the PASCS and derive its that operating-photon addition or subtraction of a given Gaussian state is an effective method to generate normalized constant, which turns out to be related to non-Gaussian states.[7;8] Thus, subtracting or adding a single-variable Hermite polynomial with a remark- photon states, which have been successfully demon- able result. In Section 4, Mandel's Q parameter of the strated experimentally by Parigi et al.,[9] has received PASCS is calculated analytically. The PND and WF increasing attention from both experimentalists and are discussed in Sections 5 and 6, respectively, and theoreticians recently[10−14] because it is possible to the results display the nonclassicality of the PASCS. generate and manipulate various nonclassical optical fields by subtracting/adding photons from/to tradi- Section 7 is devoted to projecting a scheme to prod- tional quantum states.[13;15;16] uct the PASCS. The main results are summarized in In Ref. [17], Fan and Xiao introduced a new type Section 8. ∗Project supported by the National Natural Science Foundation of China (Grant Nos. 10905015 and 41104111) and the Excellent Young Talents Fund of Higher Schools in Anhui Province, China (Grant No. 2011SQRL147). yCorresponding author. E-mail: [email protected] © 2012 Chinese Physical Society and IOP Publishing Ltd http://iopscience.iop.org/cpb http://cpb.iphy.ac.cn 070301-1 Chin. Phys. B Vol. 21, No. 7 (2012) 070301 2. Brief review of SCS whose overlap is As is commonly known in quantum optics, single- hz00j z0i Zg g mode coherent squeezed states are constructed either d2z = hz00jzi hzj z0i by[18] π g g [ ( )] [ 1 (fz0 + gz′∗) jz; λi = D (z) S (λ) j0i = exp − jz0j2 + jz00j2 exp [ 2 1 − 4 jfgj2 1 2 y = sech1=2 λ exp − jzj + za ∗ 00 ∗ ′′∗ ∗ ∗ 0 ′∗ 2 2 × (f z + g z ) − f g (fz + gz ) ] 2 2 1 ( ) 1 − 4 jfgj 1 − 4 jfgj + ay − z∗ tanh λ j0i ; (1) ] 2 fg (f ∗z00 + g∗z′′∗)2 − ; (6) − j j2 or by 1 4 fg j i j i where the overcompleteness relation of coherent state λ, z = S (λ) D (z) 0[ 1 Z = sech1=2 λ exp − jzj2 + zay sech λ d2z 2 jzi hzj = 1 (7) ] π 1 ( ) + tanh λ ay2 − z2 j0i ; (2) 2 has been used. Equation (6) indicates that the SCS is non-orthogonal. Especially, when z00 = z0 and where ay and a are Bose creation and annihilation hz0j z0i = 1, Eq. (6) proves that the SCS is capable operators, respectively, [a; ay] = 1, j0i is the vac- g g of making up a new quantum mechanical representa- uum state, and λ and z are the single-mode squeezing tion. and displacement parameters, respectively. In these two kinds of states, λ and z are independent of each other. By constrast, in Ref. [17], Fan and Xiao in- 3. PASCS and its normalization troduced a special kind of unnormalized single-mode squeezed coherent state whose displacement is related Enlightened by the above studies, we introduce a to its squeezing, and the state also spans an overcom- new type of PASCS. Theoretically, the PASCS can be pleteness space obtained by repeatedly operating the photon creation [ ] 1 operator on a SCS as follows: jjzi = exp − jzj2 + (fz + gz∗) ay − fgay2 j0i ; (3) g 2 ym jzi = Ng;ma jzi ; (8) where z is the displacement parameter, and f = g;m g jfj exp(iϕ ) and g = jgj exp(iϕ ) are complex param- f g where Ng;m is the normalization factor. Next, we shall eters satisfying jfj2 + jgj2 = 1, where ϕ and ϕ are f g determine the normalization constant Ng;m, which is phase angles. Particularly, when g = 0 or f = 0, the key to analyze the quantum statistical properties j i z g will be reduced to a normalized coherent state. of the PASCS. For this purpose, using the overcom- [19] By virtue of the IWOP technique and the vacuum pleteness relation of coherent state (Eq. (7)) and the j i h j −aya projector 0 0 =: e : , the overcompleteness can normalization condition, we have be easily proved as follows: Z Z 2 2 1 = g;mhzjzig;m d z d z 2 Z [ jjzig ghzjj = : exp[−|zj 2 π π 2 d α 1 2 ( ) ( ) = N 2 (1 − 4 jfgj )1=2 h0j exp − jzj y ∗ y ∗ g;m π 2 + fa + ga z + f a + ga z ] y ∗ ∗ y − fga 2 − f g a2 − a a]: + (f ∗z∗ + g∗z) a − f ∗g∗a2 am jαi hαj aym = 1: (4) [ ] 1 × exp − jzj2 + (fz + gz∗) ay − fgay2 j0i jj i 2 Moreover, z g can be normalized as follows: Z 2m 2 [ 2 d d α 1 = N 2 (1 − 4 jfgj )1=2 jzi = (1 − 4 jfgj2)1=4 exp − jzj2 g;m djm dkm π g 2 ] × exp[− jαj2 + (f ∗z∗ + g∗z + j) α − f ∗g∗α2 ∗ y y2 + (fz + gz ) a − fga j0i ; (5) ∗ ∗ ∗2 2 + (fz + gz + k) α − fgα − jzj ]jj=k=0: (9) 070301-2 Chin. Phys. B Vol. 21, No. 7 (2012) 070301 Further using the following integral formula:[20] of the PASCS in terms of Mandel's Q parameter Z 2 ⟨ ⟩ d α 2 ∗ 2 ∗2 y2 2 ⟨ ⟩ exp(ζ jαj + ξα + ηα + fα + gα ) a a y π Q = − a a ; (16) ( ) hayai 1 −ζξη + ξ2g + η2f = p exp ; (10) ζ2 − 4fg ζ2 − 4fg which measures the deviation of the variance of the whose convergent conditions are Re (ζ ± f ± g) < 0 photon number distribution of the field state under and Re((ζ2 −4fg)=(ζ ±f ±g)) < 0 (the conditions are consideration from the Poissonian distribution of the physical and always satisfied), Eq. (9) can be rewritten coherent state. It is well known that if Q = 0, the field as follows: has Poissonian photon statistics, if Q > 0 (Q < 0), the field has super- (sub-)Poissonian photon statistics. In hzj zi g;m g;m [ addition, the case of Q < 0 means that a state is non- d2m = N 2 exp A k − A k2 classical. However, the positive value of Mandel's Q g;m djm dkm 1 2 ] parameter cannot be used to conclude that a state kj + A∗j − A∗j2 + j is classical because a state may be nonclassical even 1 2 − j j2 j=k=0 1 4 fg though Q is positive, as pointed out in Ref. [21]. 2m X1 l l 2 d k j To calculate Mandel's Q parameter, we begin = Ng;m 2 djm dkm − j j l y y2 2 l=0 (1 4 fg ) l! with converting operators a a and a a into its anti- × − 2 ∗ − ∗ 2 j exp(A1k A2k + A1j A2j ) j=k=0 normal ordering form X1 j jm 2 A2 = N aya = aay − 1; g;m − j j2 l l=0 (1 4 fg ) l! ( ) ay2a2 = a2ay2 − 4aay + 2: (17) l 2 × @ pA1 l Hm ; (11) @A1 2 A2 One can easily obtain where ⟨ ⟩ y y ∗ ∗ ∗ ∗ 2 ∗ ∗ 2 2h j m+1 m+1 j i − g z + f z − 2g z jfj − 2f z jgj a a = Ng;m g z a a z g 1 A = ; 1 2 N 2 1 − 4 jfgj g;m − ∗ ∗ = 2 1; (18) f g Ng;m+1 A2 = : (12) 1 − 4 jfgj2 and In the last step, we have used the generating function ⟨ ⟩ ( ) of single-variable Hermite polynomials y2 2 2 h j m 2 y2 − y ym j i a a = Ng;mg z a a a 4aa + 2 a z g @n ( ) 2 2 2 Ng;m Ng;m Hn (x) = exp 2xt − t jt=0: (13) − @tn = 2 4 2 + 2: (19) Ng;m+2 Ng;m+1 Noticing the relation @l 2ln! Substituting Eqs.
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