Generation of maximally entangled states in pumped nonlinear couplers induced by broadband laser light

Physics 
N.T.H. Sang, L.T.T. Oanh,, “Generation of maximally entangled states” 122 
GENERATION OF MAXIMALLY ENTANGLED STATES 
IN PUMPED NONLINEAR COUPLERS INDUCED 
BY BROADBAND LASER LIGHT 
Nguyen Thi Hong Sang1, Luong Thi Tu Oanh2, Chu Van Lanh1, 
Doan Quoc Tuan3, Le Thi Thuy4, Le Thi Thanh Binh5, Doan Quoc Khoa5,* 
Abstract: In this paper, model of optical state truncation of two cavity modes are 
analysed. The Kerr nonlinear coupler with two of them pumped by external 
classical fields, which is assumed to be decomposed into two parts: a coherent part 
and a white noise. We can see that the quantum evolution of the pumped couplers is 
closed in a Hilbert space of two-qubit spanned by single-photon and vacuum states 
only. Hence, the pumped couplers can treat as a system of two-qubit. Analysis of 
time evolution of the quantum entanglement shows that maximally entangled states 
can be generated and compare these results with that obtained previously in the 
literature. 
Keywords: Kerr nonlinear coupler, Quantum entanglement, Bell states, White noise. 
1. INTRODUCTION 
An important research area in quantum optics is methods for manipulation of 
nonclassical states of light, especially in relation to possible optical implementations of 
systems for quantum communication and quantum computers and quantum cryptography 
[1]. Among the various schemes for the generation of optical-qubit, the device of quantum 
scissors of Pegg et al. [2] produces a superposition of single-photon and vacuum states, by 
optical-state truncation of an input single-mode coherent light. The quantum scissors 
device was considered in numerous papers [3–8]. In all above-mentioned schemes are 
restricted to the single-mode optical truncation and have supposed that the laser light is 
perfectly monochrome. Nevertheless, a real laser is never completely monochrome and it 
is usually modeled by Gaussian process. Furthermore, exactly analytical averaging of 
stochastic equations with Gaussian process is a not easy task. Nearly only the extreme case 
of white noise has been well studied. Even for the case of white noise, we can achieve 
some interesting results [9-12]. Here we should expand the formalism given in [13] to the 
more realistic case, when the laser width should be taken into account. By generalising 
former scheme in [14], we display a realization of nonlinear quantum scissors for optical-
state truncation of two cavity modes by means of a pumped nonlinear coupler. We analyze 
Kerr-like nonlinear couplers that can be modelled by systems composed of two quantum 
nonlinear oscillators linearly coupled to each other and these oscillators are excited by the 
external fields, which are assumed to be decomposed into two parts: a coherent part and 
white noise. We consider scheme based on the coupler with an external excitation of the 
coupler with two modes pumped. We demonstrate that the states generated in the excited 
nonlinear couplers under appropriate conditions can be limited to a superposition of only 
single-photon and vacuum states. We compare the possibilities of generation of Bell states 
by the couplers excited in two modes when chaotic parameter is present and absent. 
2. COUPLER PUMPED IN TWO MODES 
 This section is devoted to the general scheme of Kerr-like nonlinear coupler, which 
includes two nonlinear oscillators linearly coupled to each other and linearly coupled to 
external excitations. We suppose here that both modes of the coupler are excited by 
Research 
Journal of Military Science and Technology, Special Issue, No. 48A, 5 - 2017 123
external classical fields, whereas for the case considered previously we supposed that only 
one of the modes was coupled to the external classical field [12]. The Hamiltonian 
describing such system is of the form 
2 2
2 2 * * *ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
2 2
a b
a bH a a b b a a b b a b ab a a b b
 
      , (1) 
where ˆaˆ b and ˆaˆ b
 are boson annihilation and creation operators, respectively; the 
parameters a and b are constants of the nonlinearity of the oscillators a and b , 
respectively;  is the strength of the oscillator-oscillator coupling; and  are the 
strength of the external excitation of the oscillators a and b, respectively. 
 We assume that amplitude of the external classical field includes two parts: a 
deterministic coherent part and a randomly fluctuating chaotic part (white noise) and shall 
restrict ourselves to the case of real   . In addition, we also presume that for the 
time 0 t , two oscillators are in vacuum states, so we can write the wave function of our 
model as 
babababa
tctctctct 11)(10)(01)(00)()( 11011000
)00(  . (2) 
Hence, we obtain the solutions of stochastic averages equations of the variables for the 
probability amplitudes mnc t m, n 0,1 as (we shall focus here only on physical 
aspects of the problem because the volume of this paper is limited, whereas the 
mathematical details of this procedure will be discussed elsewhere [15]): 
0 0 0 0
0 0 0 0
3 2 7 2
0 0 0 01 1 2 24 4
00
1 2
3 2 7 2
0 00 1 24 4
01
1 2
2 3 21 1
( ) sin os sin os ,
2 4 4 2 4 4
4
( ) sin sin
4 4
i a t i a t
i a t i a t
i a i at t t t
c t e c e c
aa t t
c t i e e
    
 
  
 
0 0 0 0
0 0 0 0
3 2 7 2
0 00 1 24 4
10
1 2
3 2 7 2
0 0 0 01 1 2 24 4
11
1 2
,
4
( ) sin sin ,
4 4
2 3 21 1
( ) sin os sin os
2 4 4 2 4 4
i a t i a t
i a t i a t
aa t t
c t i e e
i a i at t t t
c t e c e c
  
 
    
 
.
 (3) 
Where: 0 is a coherent component of the external classical field, 0a is chaotic parameter, 
2 2
1 0 0 0 05 4 4 ,a a and 
2 2
2 0 0 0 013 44 68a a . 
 We are going to express the derived wave function in the Bell basis 
44332211 BbBbBbBb  , (4) 
Where: 
iB , 4,3,2,1 i are Bell-like states, which can be expressed as functions of the 
n -photon states discussed here: 
1 2 3 4
11 00 00 11 01 10 10 01
, , ,
2 2 2 2
i i i i
B B B B
 , (5) 
Physics 
N.T.H. Sang, L.T.T. Oanh,, “Generation of maximally entangled states” 124 
these states are maximally entangled states. 
 The entanglement degree of the system is defined as in [16]: 
)1(log).1(log.)( 22 pppptE , (6) 
in which 
2
11 2C
p
 and )()()()(2 10011100 tctctctcC . These results show that in the 
absence of chaotic component, our result becomes exactly the same as that obtained by 
Miranowicz et al. [13]. The entanglement degree of states (2) is shown in figure 1. For the 
case 
0 0a , entanglement degree of states (2) varies over period of time. The maximum 
values of them gradually reduce after each period 2 /T . The entanglement can 
maximally achieve at the time of ( ) 1 / 2t n n T . At the time of (1) 1 / 2t T , the 
entanglement can maximally achieve 0.995 ebits. When the chaotic component is present, 
entanglement degree of states (2) also varies over period of time but the maximum values of 
them gradually increase and the entanglement can also maximally achieves 0.995 ebits but 
the maximum position change in comparison to the case when 
0 0a . 
 When we express the states (2) in the basis of the Bell states, the probabilities of finding 
the system in these states are shown in figure 2. We can see that when 
0 0a , the 
probabilities to the system exists in Bell states B1 and B2 vary with time corresponds to the 
modulation of the oscillations with frequencies greater according to the oscillations with 
frequencies smaller. Maximum entanglement of the states B1 achieves 0.992 ebits and B2 
achieves 0.997 ebits. When 
0 0a , the probabilities to the system exists in Bell states B1 
and B2 also vary with time but the maximum position of them change and maximum 
entanglement of these states decrease in comparison to the case when the chaotic 
component is absent. 
On the other hand, when 
0 0a , the probabilities to the system exists in Bell states B3 and 
B4 are equal. However, the maximum values of these states are only about 0.235 ebits. 
Especially when 
0 0a , the probabilities to system exists in the Bell states B3 and B4 are 
different on the maximum position and have much larger value than when 
0 0a . 
Figure 1. Entanglements degree 
of the generated states by the 
coupler pumped in two modes 
with 5
0 5 10 rad/s. Solid line: 
0 0a . Dotted line: 
4
0 5 10a 
rad/s. Dashed-dotted line: 
5
0 5 10a rad/s. The time unit is 
1 /  . 
Research 
Journal of Military Science and Technology, Special Issue, No. 48A, 5 - 2017 125
Figure 2. Probabilities 2ib to the system exist in the Bell states with 
5
0 5 10 rad/s. 
Solid line:
0 0a . Dotted line: 
4
0 5 10a rad/s. Dashed-dotted line: 
5
0 5 10a rad/s. 
 The time unit is 1 /  . 
3. CONCLUSION 
 In this paper, we discussed the model of Kerr-like nonlinear coupler comprising 
two nonlinear oscillators linearly coupled to each other. These two nonlinear oscillators 
are coupled to external classical fields, which are modelled by chaotic processes. 
Furthermore, we studied the dependence of maximally entangled states on noise parameter 
and compared maximally entangled states when chaotic parameter is present and absent. 
We found that in the presence of chaotic part, the location and the magnitude of the 
maximum change, especially maximally entangled of Bell states B3 and B4 increase in 
comparison to the case when chaotic component is absent. Consequently, the parameter 
0a 
related to the chaotic component is an important parameter that controls the maxima 
entanglement of Bell states. 
ACKNOWLEDGMENT: This research is funded by Vietnam National Foundation for 
Science and Technology Development (NAFOSTED) under grant number 103.03-2017.28 
REFERENCES 
[1]. M. A. Nielsen, and I. L. Chuang, “Quantum Computation and Quantum 
Information,” Cambridge University Press (2000). 
Physics 
N.T.H. Sang, L.T.T. Oanh,, “Generation of maximally entangled states” 126 
[2]. D. T. Pegg, L. S. Phillips, and S. M. Barnett, “Optical State Truncation by Projection 
Synthesis,” Physical Review Letters, Vol. 81, pp. 1604-1606, (1998). 
[3]. M. Koniorczyk, Z. Kurucz, A. Gabris, and J. Janszky, “General optical state 
truncation and its teleportation,” Physical Review A, Vol. 62, pp. 1-8, (2000). 
[4]. M. G. A. Paris, “Optical qubit by conditional interferometry,” Physical Review A, 
Vol. 62, pp. 1-8, (2000). 
[5]. S. K. Ozdemir, A. Miranowicz, M. Koashi, and N. Imoto, “Quantum-scissors device 
for optical state truncation: A proposal for practical realization,” Physical Review A, 
Vol. 66, pp. 1-10, (2001). 
[6]. C. J. Villas-Boas, Y. Guimaraes, M. H. Y. Moussa, and B. Baseia, “Recurrence 
formula for generalized optical state truncation by projection synthesis,” Physical 
Review A, Vol. 63, pp. 1-4, (2001). 
[7]. A. Miranowicz, and W. Leoński, “Dissipation in systems of linear and nonlinear 
quantum scissors,” Journal of Optics B: Quantum Semiclassical Optics, Vol. 6, pp. 
S43-S46, (2004). 
[8]. A. Miranowicz, “Optical-state truncation and teleportation of qudits by conditional 
eight-port interferometry,” Journal of Optics B: Quantum Semiclassical Optics, Vol. 
7, pp. 142-150, (2005). 
[9]. K. Doan Quoc, V. Cao Long, W. Leoński, “Electromagnetically induced 
transparency for Λ-like systems with a structured continuum and broad-band 
coupling laser,” Physica Scripta T, Vol. 147, pp. 1-5, (2012). 
[10]. K. Doan Quoc, V. Cao Long, W. Leoński, “A broad-band laser-driven double Fano 
system-photoelectron spectra,” Physica Scripta Vol. 86, pp. 1-6, (2012). 
[11]. K. Doan Quoc, V. Cao Long, L. Chu Van, P. Huynh Vinh, “Electromagnetically 
induced transparency for Λ-like systems with degenerate autoionizing levels and a 
broadband coupling laser,” Optica Applicata, Vol. 46, pp. 93-102, (2016). 
[12]. K. Doan Quoc, V. Cao Long, L. Chu Van, V. Nguyen Thanh, H. Tran Thi, Q. Ha 
Kim, and S. Nguyen Thi Hong, “Kerr nonlinear coupler and entanglement induced 
by broadband laser light,” Photonics Letters of Poland, Vol. 8, pp. 64-66, (2016). 
[13]. A. Miranowicz, and W. Leoński, “Two-mode optical state truncation and 
generation of maximally entangled states in pumped nonlinear couplers,” Journal of 
Physics B: Atomic, Molecular and Optical Physics, Vol. 39, pp. 1683-1700, (2006). 
[14]. W. Leoński, A. Miranowicz, “Kerr nonlinear coupler and entanglement,” Journal of 
Optics B: Quantum Semiclassical Optics, Vol. 6, pp. S37-S42, (2004). 
[15]. K. Doan Quoc, to be Published. 
[16]. C. Christopher, P. L. Knight, “Introductory Quantum Optics,” Cambridge University 
Press, (2010). 
TÓM TẮT 
SỰ TẠO RA TRẠNG THÁI ĐAN RỐI CỰC ĐẠI TRONG BỘ NỐI PHI TUYẾN 
ĐƯỢC BƠM BỞI ÁNH SÁNG LASER BĂNG RỘNG 
Trong bài báo này, mô hình cắt trạng thái quang học của buồng cộng hưởng hai 
mode được phân tích. Bộ nối phi tuyến kiểu Kerr với hai mode được kích thích bởi 
trường cổ điển ngoài được giả thiết tách thành hai phần: kết hợp và nhiễu trắng. 
Chúng ta có thể thấy rằng sự tiến triển lượng tử của các bộ nối khép kín trong 
không gian Hilbert hai qubit chỉ mở rộng ra bởi các trạng thái đơn photon và chân 
không. Do đó, các bộ nối có thể nghiên cứu như hệ hai qubit. Phân tích sự tiến triển 
theo thời gian của sự đan rối lượng tử chỉ ra rằng các trạng thái đan rối cực đại có 
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Journal of Military Science and Technology, Special Issue, No. 48A, 5 - 2017 127
thể được tạo ra và so sánh những kết quả này với những kết quả tìm được trong các 
tài liệu trước đó. 
Từ khóa: Bộ nối phi tuyến Kerr, Sự đan rối lượng tử, Các trạng thái Bell, Nhiễu trắng. 
Received date, 1st March, 2016 
Revised manuscript, 10th April 2017 
Published on 26th April 2017 
Author affiliations: 
1 Vinh University, 182 Le Duan, Vinh, Viet Nam; 
2 Nghe An College of Education, Nghe An, Viet Nam; 
 3 Department of Metrology, Ministry of Defense, Hanoi, Viet Nam; 
4 Hong Duc University, 565 Quang Trung, Thanh Hoa, Viet Nam; 
5 Quang Tri Teacher Training College, Quang Tri, Viet Nam; 
Corresponding author: khoa_dqspqt@yahoo.com.