Entangled state generation in a linear coupling coupler

ators excited by external 
 coherent fields, is investigated. We show that evolution of the nonlinear coupler is 
 possible closed in a finite set of n-photon Fock states and can create Bell-like states. 
 Especially, the entropy of entanglement and the Bell-like states vary dramatically with 
 the different initial conditions are discussed. These results are compared with that 
 obtained previously in the literature. 
 Keywords: Kerr nonlinear coupler; Bell-like state; entropy of entanglement. 
 1. Introduction 
 Scientists are interested in two-mode nonlinear couplers, which are introduced by 
Jensen [1] because of their wide applicability. The nonlinear optical couplers, which rely 
on Kerr effect, have drawn exceptional care about both classical [1] and quantum [2] 
systems. The Kerr nonlinear couplers can display changes of effects as self-switching 
and self-trapping. For quantum fields, they are able to also create squeezed light and sub-
Poissonian [3]. It is also researched on the probabilities of creating entangled states in 
Kerr nonlinear couplers [4]. Kerr nonlinear couplers involve two nonlinear oscillators 
interacting linear [5] and nonlinear [6] together. The models are advance in couplers with 
three nonlinear oscillators [7], the three-qubit models in phenomena of quantum steering 
[8], the model of three interacting qubits [9]. 
 In this paper, we investigate Kerr nonlinear couplers including two quantum 
nonlinear oscillators linearly coupled together in which one or two of these oscillators 
excited by external classical fields and extend the consideration for all initial conditions 
of the motion equations of complex probability amplitudes. We show that the Bell-like 
states can be created in the Kerr nonlinear couplers under suitable conditions. We also 
compare the abilities to create Bell-like states by the nonlinear couplers pumped in one 
and two modes for different initial conditions of the motion equations. 
 2. The Kerr nonlinear coupler 
 2.1. The Kerr nonlinear coupler pumped in one mode 
Email: khoa_dqspqt@yahoo.com (Đ. Q. Khoa) 
 38 
Trường Đại học Vinh Tạp chí khoa học, Tập 49 - Số 2A/2020, tr. 38-46 
 The Kerr nonlinear coupler, which involves two nonlinear oscillators linearly 
interacted together, and one of these oscillators linearly interacts to an external coherent 
field, is studied. Therefore, the system might be depicted by the Hamiltonian [10] with 
the form as 
 2
 ˆ ˆ ˆ a 2 2 b ˆ ˆ2 ˆ * ˆ *
 H aaˆ aˆ bb b aˆ aˆ b b aˆ b  aˆb aˆ aˆ , (1) 
 2 2
here aˆ bˆ and abˆ ˆ are bosonic creation and annihilation operators, corresponding to 
the a (b) mode of the nonlinear oscillators, respectively; a ( b ) is Kerr nonlinearity of 
the mode a (b); the parameters and  are the external coherent field for the mode a 
and the oscillator-oscillator coupling strength, respectively. 
 The evolution of our system without damping processes can be represented in the 
n-photon Fock basis states with the following form 
 , (2) 
  (t) cmn (t) mn
 m,n 0
in which cmn (t) are the complex probability amplitudes of the system. 
 By using the formalism of the nonlinear quantum scissors discussed in [5], we 
show that the time-dependent wave function of our system can be truncated into the 
simple form as 
  (t) c ij (t) 00 c ij (t) 01 c ij (t) 10 c ij (t) 11
 cut 00 01 10 11 , (3) 
i, j 0,1 are the sign of oscillator modes, which are initially in states ij . 
 Using the Schrödinger equation, the motion equations of the complex probability 
amplitudes can be depicted by the equations as 
 d
 i c ij (t) *c ij (t),
 dt 00 10
 d
 i c ij (t)  *c ij (t) *c ij (t),
 dt 01 10 11
 d (4) 
 i c ij (t) c ij (t) c ij (t),
 dt 10 01 00
 d
 i c ij (t) c ij (t).
 dt 11 01
By supposing that and  are real and for the time t = 0, both modes are originally in 
vacuum states (  (t 0) 00 ), then the solutions of Eqs. (4) grow into exactly the same 
 cut
as those in [5]: 
 39 
L. T. T. Oanh, C. V. Lanh, N. T. Mạnh, Đ. Q. Khoa / Entangled state generation in a linear coupling coupler 
 00 1 1t 2t 
 c00 (t)   cos   cos ,
 2 2 2 
 00 1t 2t 
 c01 (t) cos cos ,
  2 2 
 (5) 
 00 i   2 1t 2t 
 c10 (t) sin sin ,
 4  2 2 
 00 i 1t 2t 
 c11 (t) 2 sin 1 sin .
 2 2 2 
On the other hand, by assuming that for the time t = 0, one mode is originally in vacuum 
state and other mode is in single-photon Fock state (  (t 0) 01 ), we get the solutions 
 cut
 ij 
of Eqs. (4) for cmn , m,n 0,1 in the form as 
 01 00 
 c00 (t) c01 (t),
 01 1 1t 2t 
 c01 (t)   cos   cos ,
 2 2 2 
 (6) 
 01 i 1t 2t 
 c10 (t) 2 sin 1 sin ,
 2 2 2 
 01 00 
 c11 (t) c10 (t),
 2 2 2 2 2 2
where 1 2[2  ] , 2 2[2  ] ,  4  . 
 We now examine the evolution of our system for the cases when the modes are 
primarily in states 10 and 11 . Therefore, the evolution of the system for these initial 
states has the form as 
  10 t c 00 00 c 01 01 c 01 10 c 00 11 , (7) 
 cut 10 10 01 01
and 
  11 t c 00 00 c 00 01 c 00 10 c 00 11 , (8) 
 cut 11 10 01 00
and the entropies of entanglement are also easily obtained as 
 E 11 t E 00 t , E 10 t E 01 t . (9) 
 2.2. The Kerr nonlinear coupler pumped in two modes 
 The Kerr nonlinear coupler pumped in two modes is similar to the coupler 
pumped in single mode, except both modes of this coupler are coupled by external 
coherent fields. Hence, the Hamiltonian depicting such system has the following form 
  2  2
 Hˆ a aˆ aˆ 2 b bˆ bˆ2 aˆ bˆ  *aˆbˆ aˆ *aˆ bˆ  *bˆ . (10) 
 2 2
 40 
Trường Đại học Vinh Tạp chí khoa học, Tập 49 - Số 2A/2020, tr. 38-46 
This Hamiltonian is similar to the one defined by (1), except for the term bˆ  *bˆ , in 
which  is the coupling strength of the mode b with an external coherent field. 
 In this case, we also use the Schrödinger equation and obtain the motion 
equations of the complex probability amplitudes in the form 
 d
 i c ij (t) *c ij (t)  *c ij (t),
 dt 00 10 01
 d
 i c ij (t)  *c ij (t) *c ij (t) c ij (t),
 dt 01 10 11 00
 (11) 
 d
 i c ij (t) c ij (t) c ij (t)  *c ij (t),
 dt 10 01 00 11
 d
 i c ij (t) c ij (t) c ij (t).
 dt 11 01 10
By solving these equations for all initial states of the modes, we shall obtain their 
solutions similar to those for coupler pumped in single mode. Because of the limitation 
of the volume in this work scale, we focus only on studying the generation of Bell-like 
states in the next section, whereas their mathematical details will not be presented. 
 3. The generation of Bell-like states in the Kerr nonlinear coupler 
 The entropy of entanglement of our system is defined as in [5]: 
 ij 
 E (t) .log2  (1 ).log2 (1 ) , (12) 
 2
 1 1 C ij ij ij ij ij ij 
where  and C 2c00 (t)c11 (t) c01 (t)c10 (t) . 
 2
 The truncation state in (3) can be represented in the Bell basis states in the form: 
 4
 ij ij 
  (t) bl (t) Bl , (13) 
 cut 
 l 1
where Bell states are expanded by the Bell-like states with the form as 
 1 ij 1 ij 
 B ij 00 i 11 , B ij 11 i 00 ,
 1 2 2 2
 (14) 
 1 ij 1 ij 
 B ij 01 i 10 , B ij 10 i 01 .
 3 2 4 2
 ij 
By using (3) and (13), the coefficients bl can be achieved in the following form 
 1 1
 b ij c ij (t) ic ij (t) , b ij c ij (t) ic ij (t) ,
 1 2 00 11 2 2 11 00
 (15) 
 1 1
 b ij c ij (t) ic ij (t) , b ij c ij (t) ic ij (t) .
 3 2 01 10 4 2 10 01
 41 
L. T. T. Oanh, C. V. Lanh, N. T. Mạnh, Đ. Q. Khoa / Entangled state generation in a linear coupling coupler 
 00 00 
 Figure 1: The probabilities to the system exist in the Bell-like states B1 and B2 
 for the coupler pumped in one mode with  106 rad/s,  0 (solid line) 
 and in two modes with   106 rad/s (dashed line) 
 and   /2 106 rad/s (dashed dotted line). 
 00 00 
 Figure 2: The probabilities to the system exist in the Bell-like states B3 and B4 
 for the coupler pumped in one mode with  106 rad/s,  0 (solid line) 
 and in two modes with   106 rad/s (dashed line) 
 and   /2 106 rad/s (dashed dotted line). 
 Here, the figures of probabilities, which maintain the system in the Bell-like 
 01 01 01 2 00 2
states B1 and B2 is not presented, as we have already obtained b1 b4 and 
 42 
Trường Đại học Vinh Tạp chí khoa học, Tập 49 - Số 2A/2020, tr. 38-46 
 01 2 00 2
b2 b3 . The probabilities to the system exist in the Bell-like states in which the 
modes are originally in states 00 and 01 are presented in figures from 1 to 3. When the 
coupler pumped in one mode (  0), for the modes are originally in states 00 , we get 
the same results as the ones in [5] (Figs. 1 and 2). For the modes are primarily in states 
01 , the probabilities for the creation of the maximally entangled states as well as a 
function of time for the single-mode control couplers and the system can be also 
 01 01 
generated Bell-like states for the states B3 and B4 (Fig. 3). When the coupler 
pumped in two modes, the system can be generated the maximally entangled states for 
 00 00 01 01 
the states B1 , B2 (Fig. 1) and B3 , B4 (Fig. 3), but it cannot be created the 
 00 00 
maximally entangled states for the states B3 and B4 (Fig. 2). Especially, when 
 00 00 
 , the maximum values of the probabilities are the greatest for states B1 , B2 
 01 00 00 
and B4 , whereas they are the smallest for states B3 and B4 . Moreover, when the 
 00 00 
parameter  , the probabilities for the existence of the system in states B1 , B2 
 01 01 00 00 
and B3 , B4 decrease, while the probabilities B3 and B4 increase. 
 01 01 
 Figure 3: The probabilities to the system exist in the Bell-like states B3 and B4 
 for the coupler pumped in one mode with  106 rad/s,  0 (solid line) 
 and in two modes with   106 rad/s (dashed line) 
 and   /2 106 rad/s (dashed dotted line). 
 The entropies of entanglement of the system are shown in figure 4. The results of 
E 00 for the coupler pumped in single mode (  0) and in two modes (  ) are the 
same as those in [5]. The entangled entropies E 00 and E 01 are progressing in cycles of 
time and they approximately are equal to 1 ebit for maximally entangled states, whereas 
they are equal to zero for separable states. For  , the maximum values of the E 00 
 43 
L. T. T. Oanh, C. V. Lanh, N. T. Mạnh, Đ. Q. Khoa / Entangled state generation in a linear coupling coupler 
and E 01 are the highest while they are the lowest for  . Furthermore, the entropy of 
entanglement E 01 has more maxima than E 00 , which means that E 01 oscillates faster 
than E 00 . Consequently, the maximally entangled states and the entropy of entanglement 
vary considerably for the modes, which are initially in different states. 
 Figure 4: Evolution of the entropies of entanglement E 00 and E 01 
 for the coupler pumped in one mode with  106 rad/s,  0 (solid line) 
 and in two modes with   106 rad/s (dashed line) 
 and   /2 106 rad/s (dashed dotted line). 
 For brevity, we do not present the figures of the probabilities for the system to 
exist in Bell-like states, and the entropies of entanglement for the modes in states 10 
and 11 because they are shown in figures from 1 to 4 for the modes are initially in states 
00 and 01 . 
 4. Conclusion 
 In this work, we have investigated the model of the Kerr nonlinear coupler 
consisting of two nonlinear oscillators linearly coupled at one another and one or two of 
these oscillators are linear interaction with external classical fields. By using the method 
of nonlinear quantum scissors, we have achieved the probabilities for the existence of the 
system in the maximally entangled states and the entropies of entanglement for the 
original modes in four states 00 , 01 , 10 and 11 . We have also shown that the Kerr 
nonlinear coupler creates the Bell-like states for the primary modes in all these states. 
Furthermore, the entangled entropy and the Bell-like states potentially vary for the modes 
in different states. 
 44 
Trường Đại học Vinh Tạp chí khoa học, Tập 49 - Số 2A/2020, tr. 38-46 
 REFERENCES 
[1] S. M. Jensen, “The nonlinear coherent coupler”, IEEE Journal of Quantum 
 Electronics, Vol. QE-18, No. 10, pp. 1580-1583, 1982. 
[2] F. A. A. El-Orany, M. Sebawe Abdalla and J. Peřina, “Quantum properties of the 
 codirectional three-mode Kerr nonlinear coupler”, The European Physical Journal 
 D, Vol. 33, No. 3, pp. 453-463, 2005. 
[3] A. B. M. A. Ibrahim, B. A. Umarov and M. R. B. Wahiddin, “Squeezing in the Kerr 
 nonlinear coupler via phase-spacerepresentation”, Phys. Rev. A, Vol. 61, No. 4, pp. 
 043804(1-6), 2000. 
[4] L. Sanz, R. M. Angelo and K. Furuya, “Entanglement dynamics in a two-mode 
 nonlinear bosonic Hamiltonian”, Journal of Physics A: Mathematical and General, 
 Vol. 36, No. 37, pp. 9737-9754, 2003. 
[5] 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, No. 7, pp. 1683-1700, 2006. 
[6] A. Kowalewska-Kudłaszyk and W. Leoński, “Finite-dimensional states and 
 entanglement generation for a nonlinear coupler”, Physical Review A, Vol. 73, No. 
 4, pp. 042318(1-8), 2006. 
[7] J. K. Kalaga, A. Kowalewska-Kudłaszyk, W. Leoński and A. Barasiński, Quantum 
 correlations and entanglement in a model comprised of a short chain of nonlinear 
 oscillators, Physical Review A, Vol. 94, No. 3, pp. 032304(1-12), 2016. 
[8] J. K. Kalaga and W. Leoński, “Quantum steering borders in three-qubit systems”, 
 Quantum Information Processing, Vol. 16, No. 7, 175(1-23), 2017. 
[9 J. K. Kalaga, W. Leoński and J. Peřina, Jr., “Einstein-Podolsky-Rosen steering and 
 coherence in the family of entangled three-qubit states”, Physical Review A, Vol. 97, 
 No. 4, pp. 042110(1-12), 2018. 
[10] W. Leoński and A. Miranowicz, “Kerr nonlinear coupler and entanglement”, 
 Journal of Optics B: Quantum Semiclassical Optics, Vol. 6, No. 3, pp. S37-S42, 
 2004. 
 45 
L. T. T. Oanh, C. V. Lanh, N. T. Mạnh, Đ. Q. Khoa / Entangled state generation in a linear coupling coupler 
 TÓM TẮT 
 SỰ SINH TRẠNG THÁI ĐAN RỐI 
 TRONG BỘ NỐI LIÊN KẾT TUYẾN TÍNH 
 Bộ nối phi tuyến gồm hai dao động tử phi tuyến liên kết tuyến tính với nhau và 
một hoặc hai dao động tử này được kích thích bởi các trường kết hợp ngoài được nghiên 
cứu một cách chi tiết. Chúng tôi chỉ ra rằng sự tiến triển của bộ nối phi tuyến này có thể 
được đóng trong một tập hợp hữu hạn các trạng thái Fock n-photon và có thể tạo ra các 
trạng thái kiểu Bell. Đặc biệt, entropy đan rối và các trạng thái kiểu Bell thay đổi một 
cách đáng kể với các điều kiện đầu khác nhau sẽ được thảo luận. Các kết quả này sẽ được 
so sánh với những kết quả tìm được trong các công trình trước đó. 
 Từ khóa: Bộ nối phi tuyến Kerr; trạng thái kiểu Bell; entropy đan rối. 
 46