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)†, 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). †Corresponding author. E-mail: [email protected] © 2012 Chinese Physical Society and IOP Publishing Ltd http://iopscience.iop.org/cpb http://cpb.iphy.ac.cn

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2. Brief review of SCS whose overlap is

As is commonly known in quantum optics, single- ⟨z′′| z′⟩ ∫g g mode coherent squeezed states are constructed either d2z = ⟨z′′|z⟩⟨z| z′⟩ by[18] π g g [ ( )] [ 1 (fz′ + gz′∗) |z, λ⟩ = D (z) S (λ) |0⟩ = exp − |z′|2 + |z′′|2 exp [ 2 1 − 4 |fg|2 1 2 † = sech1/2 λ exp − |z| + za ∗ ′′ ∗ ′′∗ ∗ ∗ ′ ′∗ 2 2 × (f z + g z ) − f g (fz + gz ) ] 2 2 1 ( ) 1 − 4 |fg| 1 − 4 |fg| + a† − z∗ tanh λ |0⟩ , (1) ] 2 fg (f ∗z′′ + g∗z′′∗)2 − , (6) − | |2 or by 1 4 fg

| ⟩ | ⟩ where the overcompleteness relation of coherent state λ, z = S (λ) D (z) 0[ 1 ∫ = sech1/2 λ exp − |z|2 + za† sech λ d2z 2 |z⟩⟨z| = 1 (7) ] π 1 ( ) + tanh λ a†2 − z2 |0⟩ , (2) 2 has been used. Equation (6) indicates that the SCS is non-orthogonal. Especially, when z′′ = z′ and where a† and a are Bose creation and annihilation ⟨z′| z′⟩ = 1, Eq. (6) proves that the SCS is capable operators, respectively, [a, a†] = 1, |0⟩ 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: ||z⟩ = exp − |z|2 + (fz + gz∗) a† − fga†2 |0⟩ , (3) g 2 †m |z⟩ = Ng,ma |z⟩ , (8) where z is the displacement parameter, and f = g,m g |f| exp(iϕ ) and g = |g| exp(iϕ ) are complex param- f g where Ng,m is the normalization factor. Next, we shall eters satisfying |f|2 + |g|2 = 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 | ⟩ 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 | ⟩⟨ | −a†a projector 0 0 =: e : , the overcompleteness can normalization condition, we have be easily proved as follows: ∫ ∫ 2 2 1 = g,m⟨z|z⟩g,m d z d z 2 ∫ [ ||z⟩g g⟨z|| = : exp[−|z| 2 π π 2 d α 1 2 ( ) ( ) = N 2 (1 − 4 |fg| )1/2 ⟨0| exp − |z| † ∗ † ∗ g,m π 2 + fa + ga z + f a + ga z ] † ∗ ∗ † − fga 2 − f g a2 − a a]: + (f ∗z∗ + g∗z) a − f ∗g∗a2 am |α⟩⟨α| a†m = 1. (4) [ ] 1 × exp − |z|2 + (fz + gz∗) a† − fga†2 |0⟩ || ⟩ 2 Moreover, z g can be normalized as follows: ∫ 2m 2 [ 2 d d α 1 = N 2 (1 − 4 |fg| )1/2 |z⟩ = (1 − 4 |fg|2)1/4 exp − |z|2 g,m djm dkm π g 2 ] × exp[− |α|2 + (f ∗z∗ + g∗z + j) α − f ∗g∗α2 ∗ † †2 + (fz + gz ) a − fga |0⟩ , (5) ∗ ∗ ∗2 2 + (fz + gz + k) α − fgα − |z| ]|j=k=0. (9)

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Further using the following integral formula:[20] of the PASCS in terms of Mandel’s Q parameter ∫ 2 ⟨ ⟩ d α 2 ∗ 2 ∗2 †2 2 ⟨ ⟩ exp(ζ |α| + ξα + ηα + fα + gα ) a a † π Q = − a a , (16) ( ) ⟨a†a⟩ 1 −ζξη + ξ2g + η2f = √ 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 ⟨z| z⟩ 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 + | is classical because a state may be nonclassical even 1 2 − | |2 j=k=0 1 4 fg though Q is positive, as pointed out in Ref. [21]. 2m ∑∞ l l 2 d k j To calculate Mandel’s Q parameter, we begin = Ng,m 2 djm dkm − | | l † †2 2 l=0 (1 4 fg ) l! with converting operators a a and a a into its anti- × − 2 ∗ − ∗ 2 | exp(A1k A2k + A1j A2j ) j=k=0 normal ordering form ∑∞ | |m 2 A2 = N a†a = aa† − 1, g,m − | |2 l l=0 (1 4 fg ) l! ( ) a†2a2 = a2a†2 − 4aa† + 2. (17) l 2 × ∂ √A1 l Hm , (11) ∂A1 2 A2 One can easily obtain where ⟨ ⟩ † † ∗ ∗ ∗ ∗ 2 ∗ ∗ 2 2⟨ | m+1 m+1 | ⟩ − g z + f z − 2g z |f| − 2f z |g| a a = Ng,m g z a a z g 1 A = , 1 2 N 2 1 − 4 |fg| g,m − ∗ ∗ = 2 1, (18) f g Ng,m+1 A2 = . (12) 1 − 4 |fg|2 and In the last step, we have used the generating function ⟨ ⟩ ( ) of single-variable Hermite polynomials †2 2 2 ⟨ | m 2 †2 − † †m | ⟩ 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 |t=0. (13) − ∂tn = 2 4 2 + 2. (19) Ng,m+2 Ng,m+1 Noticing the relation ∂l 2ln! Substituting Eqs. (18) and (19) into Eq. (16), we have Hn (x) = Hn−l (x) , (14) ∂xl (n − l)! − − N 2 N 2 − 4N 2 N 2 + 2 N 2 Q = g,m g,m+2 g,m g,m+1 − g,m + 1 the compact form of Ng,m is found to be N 2 N −2 − 1 N 2 ( ) g,m g,m+1 g,m+1 ∑m (m!)2 |A |m A 2 2 2 − 2 2 2 2 −2 2 √1 Ng,mNg,m+1 4Ng,mNg,m+2 + 2Ng,m+1Ng,m+2 Ng,m = Hm−l , (15) = − 2 | |l 2 A 2 2 − 2 2 l=0 l![(m l)!] fg 2 Ng,mNg,m+2 Ng,m+1Ng,m+2 N 2 which is related to a single-variable Hermite polyno- − g,m 2 + 1, (20) mial. Especially when m = 0, the normalization con- Ng,m+1

stant Ng,0 = 1, as expected. which implies that Mandel’s Q parameter of the PASCS is related to the diﬀerence values of the phase

4. Mandel’s Q parameter of the angles ϕf and ϕg. From Fig. 1, one can clearly see that PASCS all Mandel’s Q parameters become positive when the modulus of f is larger than a certain threshold value,

By virtue of the analytical expression of Ng,m, it and the threshold value achieves the maximum when

is easy for us to examine the sub-Poissonian statistics the values of the phase angles ϕf and ϕg are π/2.

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3 4 m/ (a) m/ (b) m/ m/ 3 m/ 2 m/ m/ m/ 2 m/ Q m/

Q 1 1 0 0

-1 -1 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 f f

2.0 m/ m/ (c) 4 (d) m/ m/ m/ m/ 3 1.0 m/ m/ m/ 2 m/ Q Q

0 1 0

-1.0 1 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 f f √ Fig. 1. (colour online) Mandel’s Q parameter of the PASCS as a function of r with z = (1 + i)/ 2 and m = 0, 1, 5,

10, and 30 for (a) ϕg − ϕf = 0, π; (b) ϕg − ϕf = π/4, 3π/4; (c) ϕg − ϕf = π/2; (d) ϕg − ϕf = 3π/2.

( ) ∗ 2 5. Photon-number distribution of fz + gz × Hn √ , (22) PASCS 2 fg which is exactly the PND of SCS. From Eqs. (21) and Next, we discuss the photon-number distribution (22), it is easy to see that, when z, |f|, and |g| are of the PASCS. Using the un-normalized coherent state fixed, the value of P(n) or P(n) is only related to ( ) dn √ m=0 |α⟩ = exp αa† |0⟩, we have |n⟩ = |α⟩ | / n! the difference values of the phase angles ϕ and ϕ . dαn α=0 f g and ⟨β|α⟩ = exp(αβ∗). Based on Eq. (8), the proba- Now we firstly analyze the changes of the PND of bility of finding m photons in the field is given by SCS with the difference values of the phase angles ϕf and ϕg for given z, |f|, and |g|. Using Eq. (22), the P 2 ⟨ | †m| ⟩ ⟨ | m | ⟩ (n) = Ng,m n a z g g z a n change of the PND of SCS with (ϕg −ϕf ) in the range 2 2 N 2 (1 − 4 |fg| )1/2 exp(− |z| ) d2n of [0, 2π] is shown in Fig. 2, from which we can see that = g,m n! dβ∗n dαn the PND is extremely sensitive to the difference values ∗m m ∗ ∗ ∗2 × [β α exp (fz + gz ) β − fgβ of phase angles ϕf and ϕg. When (ϕg − ϕf ) ∈ [0, π],

∗ ∗ ∗ ∗ ∗ 2 the value of P(n)m=0 is symmetrical with respect to + (f z + g z) α − f g α ]|α=β∗=0 − 2 − | |2 − | |2 1/2 | |n−m (ϕg ϕf ) = π/2, as shown in Figs. 2(a) and 2(b). Ng,mn! exp( z )(1 4 fg ) fg = However, if (ϕ − ϕ ) ∈ [π, 2π], the value of P(n) [(n − m)!2] g f m=0 ( ) − ∗ 2 is symmetrical with respect to (ϕg ϕf ) = 3π/2, as fz + gz × Hn−m √ , (21) shown in Fig. 2(d). Especially, if (ϕg − ϕf ) = π/2, the 2 fg PND of SCS obviously shows squeezed vacuum os- which is a compact result of a single-variable Hermite cillations for strong squeezing in Fig. 2(c), or else the polynomial. In particular, when m = 0 (Ng,m = 1), PND of SCS shows Schleich–Wheeler oscillations[22,23] Eq. (21) is reduced to (nonclassical intense oscillations) which reﬂect two different oscillating behaviors for even and odd photon exp(− |z|2)(1 − 4 |fg|2)1/2 |fg|n P(n) = m=0 n! numbers of the PND of squeezed states.

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0.10 0.10 0.10 (a) (b) (c) 0.08 0.08 0.08 0.06 0.06 0.06 0.04 0.04 0.04 0.02 0.02 0.02

0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 n n n

0.20 0.15 (d) (e) 0.15 0.10 0.10

0.05 0.05

0 20 40 60 80 100 0 20 40 60 80 100 n n √ Fig. 2. (colour online) Photon-number distributions of the SCS with |f| = 0.6, |g| = 0.8, z = (1 + i)/ 2, and m = 0

for (a) ϕf − ϕg = 0, π; (b) ϕf − ϕg = π/4, 3π/4; (c) ϕf − ϕg = π/2; (d) ϕg − ϕf = 5π/4, 7π/4; (e) ϕg − ϕf = 3π/2.

Next we plot the PND of PASCS as a function wide. of m for given z, |f|, |g|, and (ϕg − ϕf ) in Fig. 3. Finally, we plot the nonclassical oscillations of the

From Figs. 2(a) and 3(a)–3(d), we can see that as m PND of PASCS at various values of ϕg −ϕf for given z, increases, the nonclassical intense oscillations of the |f|, |g|, and m in Fig. 4. It is easy to see that the sym-

PND are more and more unconspicuous, and almost metry of the PND with respect to ϕg − ϕf is similar disappear in Fig. 3(d). In addition, all peaks of the to that in Fig. 2, but the PND exhibits squeezed vac-

PND move toward a larger photon-number region, and uum oscillations when ϕg − ϕf = 3π/2. These results the curves of the PND become more and more ﬂat and indicate distinctly the nonclassicality of PASCS.

0.020 0.010 (a) (b) 0.015 0.008 0.006 0.010 0.004 0.005 0.002

0 0 0 50 100 150 200 0 50 100 150 200 n n 0.008 0.005 (c) (d) 0.006 0.004 0.003 0.004 0.002 0.002 0.001

0 0 100 200 300 400 0 0 200 400 600 n n √ Fig. 3. (colour online) Photon-number distributions of the SCS with |f| = 0.6, |g| = 0.8, z = (1 + i)/ 2, and

ϕg − ϕf = 0 for (a) m = 1; (b) m = 3; (c) m = 5; (d) m = 9. Blue solid lines represent the even-photon-number distribution, while red dotted lines denote odd-photon-number distribution.

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0.04 0.04 0.020 (a) (b) (c)

0.02 0.02 0.010

0 0 0 0 100 200 0 100 200 0 100 200 n n n

0.020 0.020 (d) (e)

0.010 0.010

0 0 0 100 200 0 100 200 n n √ Fig. 4. (colour online) Photon-number distributions of the SCS with |f| = 0.6, |g| = 0.8, z = (1 + i)/ 2, and m = 1 for

(a) ϕg −ϕf = π/4, 3π/4; (b) ϕg −ϕf = π/2; (c) ϕg −ϕf = 0, π; (d) ϕg −ϕf = 5π/4, 7π/4; (e) ϕg −ϕf = 3π/2. Blue solid lines represents the even-photon-number distribution, while red dotted lines denote odd-photon-number distribution.

6. Wigner function of PASCS − fgα∗2 − |z|2] ∞ 2 B1 2m ∑ p p Ng,m e d r s The WF[24] is a powerful tool to investigate the = π drmdsm (4 |fg|2 − 1)pp! nonclassicality of optical ﬁelds. Its partial negativity ( p=0 ) × exp B r + B∗s − B r2 − B∗s2 | implies the highly nonclassical properties of quantum 2 2 3 3 r=s=0 m N 2 eB1 ∑ (−1)p (m!)2 |B |m states and is often used to describe the decoherence = g,m 3 π [(m − p)!]2p! |fg|p of quantum states. In this section, we derive the ana- p=0 ( ) 2 lytical expression of the WF for the PASCS. For this B2 × Hm−p √ , (26) purpose, we ﬁrst recall the Wigner operator in the 2 B3 coherent state |z⟩ representation[25,26] where ∫ 2 ∗ ∗ 2 d α ∗ ∗ − | |2 − | |2 − 2 − ∆ (β) = e2|β| |α⟩ ⟨−α| e−2(αβ −α β), (23) B1 = [ 2 z 2 β 4Re(fg z βg z π2 − fβ∗z − 2fβ∗|g|2z − 2βg∗|f|2z √ where β = (q + ip) / 2, the expression of the WF + 2fgβ∗2)]/(1 − 4|fg|2), B = (2β − gz∗ − fz + 4fgβ∗ − 2gz∗|f|2 W (β) = tr[ρ∆ (β)], (24) 2 − 2fz|g|2)/(1 − 4|fg|2), as well as the density operator of the PASCS fg B = . (27) 3 1 − 4|fg|2 ρ = N 2 a†m |z⟩ ⟨z| am. (25) g,m gg Equation (25) seems to be a new result related to Substituting Eqs. (23) and (25) into Eq. (24), we have single-variable Hermite polynomials. In particular, ∫ when the photon-added number m = 0, Eq. (25) is 2 2 2|β|2 d α ⟨− | †m | ⟩ ⟨ | reduced to W (β) = Ng,m e 2 α a z gg z [ π 1 −2|z|2 − 2|β|2 m −2(αβ∗−α∗β) W (β) = exp × a |α⟩ e m=0 2 ∫ π 1 − 4|fg| 2 ∗ ∗ ∗ ∗ m 2 | |2 d α − 2 − − − = (−1) N 2 (1 − 4 |fg| )1/2 e2 β 4Re(fg z βg z fβ z fβ z g,m π2 − 2fβ∗|g|2z − 2βg∗|f|2z × α∗mαm exp[− |α|2 + (f ∗z∗ + g∗z ] ∗2 − | |2 − 2β∗)α − (fz + gz∗ − 2β) α∗ − f ∗g∗α2 + 2fgβ )/(1 4 fg ) . (28)

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Now we shall analyze the changes of the Wigner play that the peak will rotate clockwise when either function of PASCS as we change the complex squeez- ϕg or ϕf decreases. ing parameters f and g and the photon-added number In addition, the Wigner functions are depicted m. The complex squeezing parameters f and g are in Fig. 6 for different values of m with |g| = 0.1, not independent of each other and can be adjusted to √ ϕg = ϕf = 0, and z = (1 + i)/ 2. It is easy to achieve different degrees of squeezing. Using Eq. (26), see that the WF is non-Gaussian in phase space when we depict the WFs of PASCS in Fig. 5 for several dif- m ≠ 0 . As evidence of the nonclassicality of the ferent values of complex squeezing√ parameters f and state, some negative regions of the WF are clearly seen g with m = 0 and z = (1 + i)/ 2. From Figs. 5(a) and 1(b), we can see that the WF with m = 0 is Gaus- in Figs. 6(a)–6(c), and they are gradually enlarged sian in phase space and has one positive peak which but their numerical fluctuations become inconspicu- is squeezed along the q direction and expanded along ous gradually with the increasing number of photon the p direction as |g| increases. Figures 5(c)–5(f) dis- addition.

(a) (b)

0.3 0.4 6 0.2 6 0.2 4 4 0.1 2 p 0 0 2 p 0 -2 -2 0 0 0 2 -2 2 -2 q 4 q 4

(d) (c)

0.4 0.3 6 6 0.2 4 0.2 4 0.1 2 p 2 p 0 0 0 0 2 -2 0 -2 0 2 2 -2 q 4 q 4

(e) (f)

0.6 0.6 6 6 0.4 0.4 4 4 0.2 0.2 2 p 2 p 0 0 0 -2 0 -4 0 2 -2 2 -2 0 q 4 q 2

Fig. 5. (colour online) Wigner-function distributions of the PASCS for diﬀerent f and g. (a) |g| = 0.1,

ϕf = ϕg = 0; (b) |g| = 0.2, ϕf = ϕg = 0; (c) |g| = 0.2, ϕg = π/4, ϕf = 0; (d) |g| = 0.2, ϕg = π/2, ϕf = 0; (e) |g| = 0.2, ϕg = 0, ϕf = π/4; (f) |g| = 0.2, ϕg = 0, ϕf = π/2.

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adjusting all the experimental parameters, the con- (a) dition |f|2 + |g|2 = 1 will be satisfied and the light | ⟩ representing z g is produced. The second step is the 0.2 process of m-photon excitation on this light, letting 4 0 this beam inject into a cavity including an atom with 2 two levels, the interaction between them is described -2 0 p 0 by the effective Harmiltonian 2 q 4 -2 ( ) †m − ∗ † m Heff = ~ ga s + g s a , (29)

(b) where g is the coupling factor between m photons and the atom, s− is the operator which represents the ab-

0.2 sorption of the atom. To lower the order of the cou- 4 pling constant, the wave function of the system at time 0 2 t is

0 p -2 †m − 0 |ψ (t)⟩ = |ψR⟩ |G⟩ − it(ga s 2 -2 q 4 † ∗ m + s g a ) |ψR⟩ |G⟩ , (30)

where |ψR⟩ (|G⟩) is the initial state of the ﬁeld (atom). (c) 0.6 Suppose that at time t, the atom is measured to be

0.4 in the absorption state, then the state of the ﬁeld is reduced to 0.2 4 †m m 0 2 ρﬁeld ∝ a |ψ (t)⟩⟨ψ (t)| a , (31)

0 p -2 0 2 which is apart from a normalization constant. Thus, 4 -2 q | ⟩ if the initial state of the radiation field is z g, Eq. (31) Fig. 6. (colour online) Wigner-function distributions of becomes the PASCS for different values of m: (a) 1; (b) 3; (c) 5. ∝ †m | ⟩ ⟨ | m ρfield a z gg z a . (32) 7. Production of the PASCS Thus, the photon-added squeezed coherent state of the We finally consider how the PASCS can be pro- field we need is obtained. duced in optical processes. The whole process may be divided into two steps, and the first one is to produce 8. Conclusion | ⟩ the z g state. In order to achieve this aim, we propose a scheme that an optical parametric down-conversion In summary, we introduce a new type of photon- process coexists with a process of generating a coher- added squeezed coherent state, which is obtained by ent state. For instance, one may consider a composite repeatedly exerting the photon creation operator on a system consisting of a gain medium and a parametric new type of squeezed coherent state. We first derive optical parametric oscillator in the same optical cav- the normalization factor of PASCS, which is related ity. Then, the laser mode (through a nonlinear crys- to single-variable Hermite polynomials. Based on the tal) will generate a second harmonic field (ω → 2ω), technique of integration within an ordered product which will, in turn, generate (through another nonlin- of operators, we investigate its statistical properties ear crystal) a sub-harmonic mode (ω → 2ω). If the such as Mandel’s Q parameter, photon-number distri- variables for the second harmonic mode can be adi- bution, and the Wigner function, whose changes with abatically eliminated, the system Harmiltonian will the phase difference are analyzed graphically. Fur- | ⟩ have the form that may produce the z g state given thermore, the nonclassicality is discussed in terms of by Eq. (5). The parameter f depends on the las- intense oscillations of the PND and the negativity of ing gain and the parameter g is determined by the the Wigner function after deriving the explicit expres- pumping-field strength and the second-order suscep- sions of the PND and WF, and the results demonstrate tibility of the down-conversion process. By carefully explicitly the nonclassical properties of the PASCS.

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