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Higher-order nonclassical properties of nonlinear charge pair cat states
To cite this article before publication: Duc Minh Truong et al 2019 J. Phys. B: At. Mol. Opt. Phys. in press https://doi.org/10.1088/1361- 6455/ab51f7
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Higher-order nonclassical properties of nonlinear charge pair cat states
Truong Minh Duc1, Dang Huu Dinh2, and Tran Quang Dat1,3
1Center for Theoretical and Computational Physics, University of Education, Hue University, Vietnam
2Industrial University of Ho Chi Minh City, 12 Nguyen Van Bao, Ward 4, Go Vap District, Ho Chi Minh City, Vietnam
3University of Transport and Communications - Campus in Ho Chi Minh City, 450-451 Le Van Viet Street, Tang Nhon Phu A Ward, District 9, Ho Chi Minh City, Vietnam
Abstract. The concept of nonlinear charge pair cat states is introduced and their higher-order nonclassical properties are studied. By comparing the nonlinear cases of photon-added states f2(n) and a laser-driven trapped ion f3(n) with the linear case f1(n), it is shown that these states exhibit two-mode higher-order antibunching to all orders and the degree of antibunching depends on the suitable variables, especially on the nonlinear functions. We also show that in such states, the higher-order squeezing appears only in the even orders and the squeezing behaviours for the nonlinear cases f2(n) and f3(n) are quite opposite. These states are two-mode genuinely entangled regardless of the nonlinear functions and their entanglement properties are more obvious in the photon-added statesf2(n) case.
1. Introduction
Historically, the conceptual introduction of coherent states dated back to 1926 and was credited to Schr¨odinger [1] who was looking for classical-like states that would best satisfy what is now known as the minimum uncertainty condition. Then, these states became widely recognized thanks to the fundamental works in 1963 by Glauber [2], Klauder [3] and Sudarshan [4]. Mathematically, a coherent state results from the action of a displacement operator on the vacuum state. It can also be defined as the eigenstate of the annihilation operator of the electromagnetic field. Although coherent states are classical states, the superpositions of them turn out to be nonclassical ones.
The nonclassical states and their nonclassical properties have been applied in quantum systems such as detection of gravitational waves [5], application in laser interferometers [6], quantum state engineering [7, 8, 9, 10], quantum teleportation with continuous variables [11, 12], quantum key distribution [13] and implementation of many other tasks in quantum information processing [14].
The nonlinear coherent states were first introduced in Refs. [15, 16] as the states resulting from the action of the nonlinear function of number operator f(n) on the 4
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coherent state and form a class of the nonclassical states. Up to now, many nonlinear coherent states have been introduced and their nonclassical properties have been studied [17, 18, 19, 20, 21, 22, 23, 24, 25]. Although some nonclassical states are not expressed in terms of the nonlinear function f(n), those states are shown to belong to the nonlinear coherent states [26]. For example, the photon-added coherent state [27, 28] and the q-deformed oscillator [29, 30] were known as the classes of the nonlinear coherent states with the different nonlinear functions f(n) [15, 16, 31].
The even and odd coherent states that are the superposition of two different single- mode coherent states were introduced [32]. The generalization of these states as the even and odd nonlinear coherent states have been studied [17, 18]. The two-mode coherent states as pair coherent states were introduced by Agarwal [33, 34] and the extension of these states as pair cat states [35] or even and odd charge coherent states [36] were proposed and their nonclassical properties have been studied [37]. In this paper, as an extension of the pair cat states and the even and odd charge coherent states [35, 36], we introduce nonlinear charge pair cat states in section 2 and study their nonclassical properties. We derive the explicit expressions for higher-order antibunching and squeezing as well as higher-order entanglement. We pay attention not only to the usual but also to the higher-order antibunching and squeezing of these states with some specific nonlinear functions f(n). In section 3, it is shown that these states can exhibit two-mode antibunching in any orders and the degree of antibunching depends not only on the orders but also on the nonlinear functions f(n). In section 4, the two-mode squeezing appears only in the higher order but not in the usual. We also show in section 5 that these states are two-mode entangled in difference of the nonlinear functionsf(n).
2. Nonlinear charge pair cat states
Letaandbbe the bosonic annihilation operators of two independent modes. Completely similar to the nonlinear charge coherent states introduced in [20, 21], in terms of the nonlinear functions f(n), we define the nonlinear charge pair coherent states |ξ, q, fi as the eigenstates of two operators A = af(n), B = bf(n) and the charge operator Q=a+a−b+b as follows
AB|ξ, q, fi=ξ|ξ, q, fi, Q|ξ, q, fi=q|ξ, q, fi, (1) where ξ = |ξ|eiθ is complex number and q is referred to as charge. The nonlinear function f(n) in Eq. (1) is not deformed as in the deformed nonlinear functions in [20, 21]. Assuming the charge number q≥0 implies that the photon number in modea is larger than the photon number in mode b. Then the nonlinear charge pair coherent states can be expressed in the Fock-state (i.e., number-state) representation as
|ξ, q, fi=Nq
∞
X
n=0
ξn
[n!(n+q)!]1/2f(n)!f(n+q)!|n+q, ni, (2) where |m, ni is two-mode Fock state, f(n)! = f(n)f(n−1)...f(1), f(0)! = 1 and the 4
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normalization factor is Nq =
∞
X
n=0
|ξ|2n
n!(n+q)![f(n)!f(n+q)!]2
−1/2
. (3)
Now, we introduce the new states called nonlinear charge pair cat states via superposition of nonlinear charge pair coherent states introduced in Eq. (1) as
|ξ, q, f, φi=Nφ(|ξ, q, fi+eiφ|−ξ, q, fi), (4) where the normalization factor is
Nφ = 1
√2
1 +Nq2cos(φ)
∞
X
n=0
(−1)n|ξ|2n
n!(n+q)![f(n)!f(n+q)!]2
−1/2
, (5)
with 0 ≤ φ ≤ 2π. Noting that when φ = 0 or π, the nonlinear charge pair cat states reduce to the even or odd nonlinear charge pair coherent states, respectively. In terms of the Fock states, the nonlinear charge pair cat states are given by
|ξ, q, f, φi=Nφ,q,f
∞
X
n=0
ξn[1 + (−1)neiφ]
[n!(n+q)!]1/2f(n)!f(n+q)!|n+q, ni, (6) with Nφ,q,f is the normalization factor of the form
Nφ,q,f−2 =
∞
X
n=0
2|ξ|2n[1 + (−1)ncos(φ)]
n!(n+q)![f(n)!f(n+q)!]2. (7) 3. Two-mode higher-order antibunching
In the recent years, some experiment schemes have been proposed to detect higher- order antibunching by the time multiplexing and by means of hybrid photodetectors [38, 39, 40]. As in the theoretical field, the criteria of single-mode and two-mode higher- order antibunching were first introduced by Lee [41, 42] and were considered further by several authors [43, 44, 45]. Up to now, these criteria have been applied to detect and measure the higher-order antibunching properties of many nonclassical states and many different optical systems [33, 46, 47, 48, 49, 50, 51, 52]. According to Lee [41], the two-mode higher-order antibunching criterion for two arbitrary modes a and b was defined in the form of antibunching coefficient Aab(l, m) as
Aab(l, m)≡ hn(l+1)a n(m−1)b i+hn(m−1)a n(l+1)b i
hn(l)a n(m)b i+hn(m)a n(l)b i −1, (8) where nx = x+x is the number operator of mode x with x = a, b, and hn(i)x i = hnx(nx−1)...(nx−i+ 1)i=hx+ixiidenotes the ith factorial moment, while the integers l and m satisfy the conditionsl ≥ m ≥1. The two-mode higher-order antibunching in states occurs only ifAab(l, m)<0 and the usual antibunching corresponds tol =m = 1.
Based on the Eqs. (1)-(7), after calculating the average values of the nonlinear charge pair cat states in Eq. (8), we have obtained
Aab(l, m) = Cm−1,l+1,0+Cl+1,m−1,0
Cm,l,0+Cl,m,0 −1, (9)
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with
Cl,k,h =
∞
X
n=max(l,k−q)
[1 + (−1)ncos(φ)]|ξ|2n
(n−l)!(n+q−k)!f(n)!f(n+q)!f(n+h)!f(n+h+q)!. (10) We study the two-mode higher-order antibunching in the nonlinear charge pair cat states by using Eqs. (9)-(10) with some special functions of f(n) from f1(n) to f3(n).
The f1(n) = 1 corresponds to the linear case, in which the nonlinear charge pair cat states reduce to the pair cat states [35]. The f2(n) = 1 − s/(1 + n), with s being the number of photons added to mode, corresponds to the photon-added states.
The f3(n) = L(1)n (η2)/[(n + 1)L(0)n (η2)] is associated with the vibrational motion of a laser-driven trapped ion with η being the Lamb–Dicke parameter and Lpn(η2) the nth generalized Laguerre polynomial in η for a parameter p. For the two-mode usual antibunching l = m = 1, we plot in Figure 1 the dependence of the antibunching coefficient Aab(1,1) as a function of |ξ| in the nonlinear charge pair cat states as the even nonlinear charge pair coherent states (a) for φ = 0 and the odd nonlinear charge pair coherent states (b) for φ = π with the same values q = 0, η = 0.15, s = 1 and difference of the nonlinear functions from f1(n) to f3(n) respectively. The two- mode usual antibunching always exists with any value of f(n) and |ξ|. In the small values of |ξ|, the degree of antibunching tends to −0.5 and −1 in the even and odd nonlinear charge pair coherent states respectively. In both states, if |ξ| increases, the coefficient Aab(l, m) becomes less negative, corresponding to a decrease in the degree of antibunching and tends to 0 in the large values of |ξ|. With the same values of
|ξ|, the degree of antibunching depends on the nonlinear functions f(n). Compared with the linear case f1(n) = 1, it becomes smaller than that in the photon-added case f2(n) = 1−s/(1 +n), but bigger than that in the vibrational motion of a laser-driven trapped ion case f3(n) = L(1)n (η2)/[(n + 1)L(0)n (η2)]. For the two-mode higher-order antibunching with l = m = 2, we plot in Figure 2 the dependence of the higher- order antibunching coefficient Aab(2,2) as a function of |ξ| in the nonlinear charge pair cat states as the even nonlinear charge pair coherent states (a) for φ = 0 and the odd nonlinear charge pair coherent states (b) for φ = π with the same values q = 0, η = 0.15, s = 1 and different nonlinear functions f1(n), f2(n) and f3(n). Note that, in the linear casef1(n) = 1, the results given in Figures 1 and 2 completely coincide with the previous results in [37]. As same as the two-mode usual antibunching case, the two-mode higher-order antibunching always exists with any values of f(n) and |ξ|, and it becomes smaller than that in the photon-added casef2(n), but bigger than that in the vibrational motion of a laser-driven trapped ion casef3(n) in comparison with the linear casef1(n) = 1. However, the degree of antibunching in the higher-order case is opposite in the even and the odd nonlinear charge pair coherent states in the usual case. In case q≥1 and arbitrary φ, our numerical calculations show that the two-mode higher-order antibunching also occurs for any values of l, m, |ξ| and the nonlinear functions from f1(n) to f3(n).
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f1HnL f2HnL f3HnL (a)
0 5 10 15 20
-0.5 -0.4 -0.3 -0.2 -0.1 0.0
ÈΞÈ AabH1,1L
f1HnL f2HnL f3HnL (b)
0 5 10 15 20
-1.0 -0.8 -0.6 -0.4 -0.2 0.0
ÈΞÈ AabH1,1L
Figure 1. The antibunching coefficient Aab(1,1) as a function of |ξ| in the nonlinear charge pair cat states as the even nonlinear charge pair coherent states (a) for φ= 0 and the odd nonlinear charge pair coherent states (b) for φ= π with q= 0, l=m= 1, η= 0.15, s= 1 and different nonlinear functions f(n) (from top to the bottom f2(n) = 1−s/(1 +n), f1(n) = 1 and f3(n) =L(1)n (η2)/[(n+ 1)L(0)n (η2)]
respectively).
f1HnL f2HnL f3HnL (a)
0 5 10 15 20
-1.0 -0.8 -0.6 -0.4 -0.2 0.0
ÈΞÈ AabH2,2L
f1HnL f2HnL f3HnL (b)
0 5 10 15 20
-0.5 -0.4 -0.3 -0.2 -0.1 0.0
ÈΞÈ AabH2,2L
Figure 2. The antibunching coefficientAab(2,2) as a function of|ξ|in the nonlinear charge pair cat states as the even nonlinear charge pair coherent states (a) forφ= 0 and the odd nonlinear charge pair coherent states (b) for φ = π, q = 0, l = m = 2, η = 0.15, s = 1 and different nonlinear functions f(n) (from top to the bottom f2(n) = 1−s/(1 +n), f1(n) = 1 andf3(n) =L(1)n (η2)/[(n+ 1)L(0)n (η2)] respectively).
4. Two-mode higher-order squeezing
The higher-order squeezing criteria for single-mode states of a quantum field were introduced by Hong-Mandel [53] and Hillery [54] and have been applied to several nonclassical states and systems [43, 50, 55, 56, 57]. For two-mode states, there exist several higher-order squeezing criteria known as sum- and difference-squeezing [58] as well as two-mode squeezing [59]. The criterion for the two-mode squeezing in [59] was extended by An in [47] to the higher-order case, in which the two-mode higher-order squeezing is associated with the operator Qab(N, ϕ) which, for two arbitrary modes a and b, has the following form
Qab(N, ϕ) = 1 2√
2
(a++b+)Neiϕ+ (a+b)Ne−iϕ, (11) 4
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where N is positive integers andϕ is a phase determining the direction of squeezing in the complex plane. According to [47], the squeezing coefficient Sab(N, ϕ) is of the form
Sab(N, ϕ) = 1
4{<[h(a+b)2Nie2iϕ]+h(a++b+)N(a+b)Ni−2<2[h(a+b)Nieiϕ]}.(12) A state is said to be two-mode higher-order squeezed in an orderN >1 ifSab(N, ϕ)<0.
WhenN = 1 andϕ=kπ, the two-mode higher-order squeezing reproduces to the usual two-mode squeezing introduced by Loudon and Knight [59]. By using the Eqs. (4)-(6) for calculating the averaged values in the nonlinear charge pair cat states in Eq. (12), we can obtain the analytical forms for the squeezing coefficient Sab(N, ϕ) in any orders of squeezing. If the order N is odd, theSab(N, ϕ) can be expressed as
Sab(N, ϕ) = 2Nφ,q,f2
N X
n=0
N! 2n!(N −n)!
2
CN−n,n,0
+ (2N)!
4(N!)2|ξ|Nsin(N θ+ 2ϕ) sin(φ)BN
, (13)
with Cl,k,h given in Eq. (10), and BN =
∞
X
m=0
(−1)m|ξ|2m
m!(m+q)!f(m)!f(m+q)!f(m+N)!f(m+N +q)!. (14) In our numerical calculations, the Sab(N, ϕ) in Eq. (13) is never less than 0. It means that when the order N is odd, there is no higher-order two-mode squeezing for the nonlinear charge pair cat states.
When the order N = 2k with k is even, the Sab(N, ϕ) can be expressed in the explicit forms of the nonlinear charge pair cat states as
Sab(N, ϕ) = 2Nφ,q,f2
N
X
n=0
N! 2n!(N −n)!
2
CN−n,n,0
+ (2N)!
4(N!)2|ξ|Ncos(N θ+ 2ϕ)C0,0,N
−
√2N!
[(N2)!]2|ξ|N2 cosN θ
2 +ϕNφ,q,f2 C0,0,N
2
2
. (15)
If k is odd, we have
Sab(N, ϕ) = 2Nφ,q,f2
N X
n=0
N! 2n!(N −n)!
2
CN−n,n,0
+ (2N)!
4(N!)2|ξ|Ncos(N θ+ 2ϕ)C0,0,N
−
√2N!
[(N2)!]2|ξ|N2 sinN θ
2 +ϕsin(φ)Nφ,q,f2 BN
2
2
. (16)
Figure 3 plots the dependence ofSab(2, ϕ) on |ξ| in the nonlinear charge pair cat states as the even nonlinear charge pair coherent states (a) for φ = 0 and the odd nonlinear 4
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charge pair coherent states (b) for φ = π with the same values q = 0, η= 0.15, s = 1, cos[2(θ+ϕ)] = −1 and different nonlinear functions fromf1(n) tof3(n) as given above.
For the nonlinear functions f1(n) and f3(n), shape of two graph lines has a similar posture. In the domain of small values of |ξ|, there is two-mode higher-order squeezing in the even nonlinear charge pair coherent states, but in the odd nonlinear charge pair coherent states. If the value of |ξ| is large enough, there is two-mode higher-order squeezing in both states. For the nonlinear function f2(n), the shape of graph line looks quite different from f1(n) and f3(n) cases. In the domain of small values of |ξ|, there is no two-mode higher-order squeezing in both states. The two-mode higher-order squeezing only appears in a small region of|ξ|in the odd nonlinear charge pair coherent states. The two-mode higher-order squeezing appears and shows more and more obvious in both states if |ξ| is large. Figure 4 plots the dependence of Sab(4, ϕ) on |ξ| in the nonlinear charge pair cat states as the even nonlinear charge pair coherent states (a) for φ = 0 and the odd nonlinear charge pair coherent states (b) for φ = π with the same values q = 0, η = 0.15, s = 1, cos[2(2θ +ϕ)] = −1 and different nonlinear functions from f1(n) to f3(n) as given above. In the linear case f1(n) = 1, the two-mode higher- order squeezing in both states always appears for any value of |ξ|. It is more and more obvious if |ξ| is increased. There are quite opposite behaviours between nonlinearf2(n) and f3(n) cases. The higher-order squeezing appears in the vibrational motion of a laser-driven trapped ion case f3(n) but it does not appear in the photon-added case f2(n) in the small region of |ξ|. In contrary, in the big value of |ξ|, the higher-order squeezing does not appear in the vibrational motion of a laser-driven trapped ion case f3(n) but it appears more and more obvious in the photon-added case f2(n).
f1HnL f2HnL f3HnL (a)
0 1 2 3 4 5 6
-3 -2 -1 0 1
ÈΞÈ SabH2,jL
f1HnL f2HnL f3HnL (b)
0 1 2 3 4 5 6
-3 -2 -1 0 1 2 3
ÈΞÈ SabH2,jL
Figure 3. The squeezing coefficientSab(2, ϕ) as a function of|ξ|in the even nonlinear charge pair coherent states (a) forφ= 0 and the odd nonlinear charge pair coherent states (b) forφ=πwithq= 0, η= 0.15, s= 1 and different nonlinear functionsf(n) (the solid line forf1(n) = 1, the dashed line forf2(n) = 1−s/(1+n), the dotted-dashed line forf3(n) =L(1)n (η2)/[(n+ 1)L(0)n (η2)].
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f1HnL f2HnL f3HnL (a)
0 2 4 6 8 10 12
-2000 -1000 0 1000 2000
ÈΞÈ SabH4,jL
f1HnL f2HnL f3HnL (b)
0 2 4 6 8 10 12
-2000 -1000 0 1000 2000
ÈΞÈ SabH4,jL
Figure 4. The squeezing coefficientSab(4, ϕ) as a function of|ξ|in the even nonlinear charge pair coherent states (a) forφ= 0 and the odd nonlinear charge pair coherent states (b) forφ=πwithq= 0, η= 0.15, s= 1 and different nonlinear functionsf(n) (the solid line forf1(n) = 1, the dashed line forf2(n) = 1−s/(1+n), the dotted-dashed line forf3(n) =L(1)n (η2)/[(n+ 1)L(0)n (η2)].
5. Intermodal entanglement
In the recent years, the criteria in the form of class of the inequalities to detect the entanglement of bipartite systems, especially of the two-mode nonclassical states were introduced [60, 61, 62, 63, 64, 65]. These criteria have been applied to detect the entanglement properties of some two-mode nonclassical states [66, 67, 68, 69]. One of these criteria, the higher-order criterion was given by Hillery-Zubairy [63] in 2006 in the following inequality
ha+mamihb+nbni ≥ |hambni|2, (17)
in which a state is entangled if the above inequality is violated. Based on the inequality in Eq. (18), a state is entangled if the entanglement coefficient E in the form
E =ha+mamihb+nbni − |hambni|2 <0 (18) is satisfied with any integers m, n≥1. For the case m=n = 2k with k= 1,2,3, ..., by using the Eqs. (2)-(4) and Eq. (18), in the nonlinear charge pair cat states, we have
E = 4Nφ,q,f4 [C0,2k,0C2k,0,0−(|ξ|2kC0,0,2k)2], (19) with Cl,k,h given in Eq. (10). Based on the Eq. (19), the states are intermodal entanglement ifE <0. In Figure 5 we plot the entanglement coefficientE as a function of |ξ| in the even nonlinear charge pair coherent states for φ = 0 (a) and in the odd nonlinear charge pair coherent states for φ = π (b) with q = 0, k = 2, η = 0.15, s = 1 and different nonlinear functions from f1(n) to f3(n) as given above. It is clear that for both states, the entanglement coefficient E is always negative for any values of |ξ|
and f(n), and it becomes more and more negative when |ξ| is increased. For arbitrary φ, our numerical calculations show that the entanglement coefficient E is also negative regardless of any values of the nonlinear functions from f1(n) to f3(n) and |ξ|. It indicates that the nonlinear charge pair cat states are two-mode genuinely entangled.
In the Figure 5(a) and Figure 5(b), with the same value of |ξ|, the dashed line for 4
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f2(n) = 1−s/(1+n) corresponds to the photon-added case where the value of coefficient E is smallest. It means that the entanglement behaviours of nonlinear charge pair cat states in the photon-added case are more pronounced compared with the cases of linear and a laser-driven trapped ion.
(a)
f1HnL f2HnL f3HnL
0.0 0.2 0.4 0.6 0.8 1.0 -1.0
-0.8 -0.6 -0.4 -0.2 0.0
ÈΞÈ
E f1HnL
f2HnL f3HnL (b)
0.0 0.2 0.4 0.6 0.8 1.0 -1.0
-0.8 -0.6 -0.4 -0.2 0.0
ÈΞÈ
E
Figure 5. The entanglement coefficientE as a function of |ξ| in the even nonlinear charge pair coherent states (a) forφ= 0 and the odd nonlinear charge pair coherent states (b) for φ = π with q = 0, k = 2, η = 0.15, s = 1 and different nonlinear functionsf(n) (the solid line forf1(n) = 1, the dashed line forf2(n) = 1−s/(1 +n), the dotted-dashed line forf3(n) =L(1)n (η2)/[(n+ 1)L(0)n (η2)].
6. Conclusion
In summary, we have introduced the nonclassical states, which are referred to as nonlinear charge pair cat states and studied the nonclassical properties of these states.
It is found that these states exhibit two-mode higher-order antibunching in any orders.
For given the order of l and m, the charge number q and |ξ|, the degree of higher- order antibunching becomes bigger and bigger as the nonlinear function changes from f2(n) to f1(n) and then tof3(n) respectively. It means that the degree of higher-order antibunching is the biggest in the vibrational motion of a laser-driven trapped ion case.
For the two-mode higher-order squeezing, it shows that when the order N is odd, there is no higher-order squeezing in the nonlinear charge pair cat states. The higher-order squeezing appears only if the order N is even. For the orders N = 2,4 with the linear casef1(n) = 1, these states exhibit the higher-order squeezing and the squeezing appears more and more obvious when|ξ|is increased. However, in the nonlinear casesf2(n) and f3(n) with N = 4, there are quite opposite squeezing behaviours between them. The higher-order squeezing appears in the vibrational motion of a laser-driven trapped ion case f3(n) but it does not appear in the photon-added states case f2(n) in the small region of |ξ| and vice versa in the big region of |ξ|. It is also found that the nonlinear charge pair cat states are two-mode genuinely entangled according to Hillery-Zubairy criterion and the entanglement behaviours are more obvious in the photon-added states f2(n) case. That makes us think about whether to change the nonlinear functionsf(n) to control the entanglement degree of the nonlinear charge pair cat states in order to 4
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use these states as a two-mode entangled resource with continuous variables to perform quantum teleportation process.
Acknowledgments
This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103.01-2018.361.
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