Entangled state generation by a Kerr-like nonlinear coupling coupler

This system can be generated maximally entangled states. Specially, we can show that the entropy of entanglement and the maximally entangled states change dramatically when the initial conditions are different. The reasonableness obtained results is affirmed by the comparison to that of previous works.

Vật lý ENTANGLED STATE GENERATION BY A KERR-LIKE NONLINEAR COUPLING COUPLER Luong Thi Tu Oanh1,2, Do Hong Son3, Chu Van Lanh1, Ho Quang Quy4, Doan Quoc Khoa5* Abstract: We study a Kerr-like nonlinear coupler including two nonlinear oscillators coupled to each other by a nonlinear interaction One of these oscillators is pumped by an external coherent field We show that evolution of the system can be closed within a finite set of Fock states This system can be generated maximally entangled states Specially, we can show that the entropy of entanglement and the maximally entangled states change dramatically when the initial conditions are different The reasonableness obtained results is affirmed by the comparison to that of previous works Keywords: Kerr-like nonlinear coupler; Maximally entangled state; Entropy of entanglement INTRODUCTION In recent decades, scientists have been interested in the generative possibility of nonclassical states, which is indeed important for the implementation of models in the quantum teleportation, quantum cryptography and quantum entanglement In practice, the particular interests are the systems can be closed in a finite set of Fock states, as the capabilities of implementations of the models are described for quantum computing systems Models that lead to the creation of finite-dimensional state are seen as linear [1,2] or nonlinear [3,4] quantum scissors The models include two Kerr-like oscillators that interact linearly [5-7] and nonlinearly [8,9] with each other and one or two of them pumped by external coherent fields were discussed The systems were developed for the case of three Kerr-like nonlinear oscillators [10], the three-qubit systems in quantum steering phenomena [11], the system of three interacting qubits [12] In this work, we study in detail the Kerr-like nonlinear coupler consisting of two nonlinear oscillators that couple nonlinearly to each other and one of these oscillations is pumped by the external coherent field in which the initial conditions for differential equation of probability amplitudes involved in the problem are arbitrarily chosen What is mainly interesting from our opinion is that for different initial conditions, the system can change the creation of the maximally entangled states (Bell-like states) We shall show that selecting the appropriate initial conditions will significantly increase the Bell-like states generation THE MODEL AND THE MAXIMALLY ENTANGLED STATES CREATION The model of a Kerr-like nonlinear coupler studied here is constructed by two nonlinear oscillators, corresponding to the two field modes a and b, respectively These oscillators are coupled with each other by the nonlinear interaction Furthermore, the field mode corresponding to mode a is linearly coupled to an external coherent field Hence, in the interaction picture, the effective Hamiltonian to describe this system can has the following form:  Hˆ   a aˆ  aˆ   b bˆ  bˆ  a aˆ    aˆ where  aˆ  bˆ  2  b ˆ ˆ2 2 b b   aˆ   bˆ   * aˆ  bˆ         aˆ    * aˆ (1) ( aˆ  bˆ  ) are bosonic creation (annihilation) operators, corresponding to the a and b modes of the nonlinear oscillators, respectively; the parameters  a and b 170 L T T Oanh, …, D Q Khoa, “Entangled state generation … nonlinear coupling coupler.” Nghiên cứu khoa học công nghệ are nonlinearity constants of the nonlinear oscillators a and b , respectively;  and  is the strength of the external excitation field for the oscillators a and the oscillatoroscillator coupling, respectively We will depict the evolution of the system in terms of the time-dependent wave function with the model without damping processes It can be expressed in the n-photon Fock basis states with form as   (t )  c mn (t ) m a n (2) b m,n 0 where cmn (t ) are the amplitudes of complex probability of finding the system in the mphoton state for mode a and the n-photon state for mode b We now assume the weak external pumping and constant amplitudes of the couplings to use the method of the nonlinear quantum scissors extensively discussed in [3] In corollary, the wave function (2) is able to truncate into the wave function describing only the evolution of the resonant states Hence, the wave function receives the form as  pq   pq   ( t ) cut  c 20 ( t ) a b  c12 pq  ( t ) a b  c 02 (t ) a b (3) in with p, q  0,2 are the symbol of oscillator modes that are initially in states p a q b Our considerations here have limited to a finite set of the states By applying the Schrödinger equation, the set of equations of motion for three probability amplitudes in the closed form can be written as follows d  pq   pq  c 20 ( t )   c 02 ( t ), dt d  pq   pq  i c12 ( t )   c 02 ( t ), dt d  pq   pq  i c 02 ( t )   * c 20 ( t )   * c12 pq  ( t ) dt i (4) Supposing that for the time t = 0, both two photons are in mode a and no photon is in mode b, namely c 2020  (0)  and c1220  (0)  c0220  (0)  (  (t  0) cut  a b ), then its solutions become exactly the same as those in [8]:  20  c 20 (t )     cos(  t ) 2 2 cos(  t )  1 c1220  ( t )  , 2 sin(  t )  20  c 02 (t )   i ,  , (5) On the other hand, for the time t  , no photons is in mode a and both two photon are in mode b, i.e., c2002  (0)  c1202  (0)  and c0202  (0)  (  (t  0) cut  a b ), we obtain the  pq  solutions for cmn , m, n  0,2 with the following form: sin(  t ) ,  sin(  t ) c1202  ( t )   i  ,  02  c 02 ( t )  cos(  t ), 02  c 20 (t )   i Tạp chí Nghiên cứu KH&CN quân sự, Số 61, - 2019 (6) 171 Vật lý where     4 From equations (5) and (6), we can see that c0220  (0)  c2002  (0) We can express the state (3) in the Bell basis states with the following form:  (t ) cut   bi pq  (t ) Bi pq  (7) i 1 where Bell states Bi pq  , which are maximally entangled states The entropy of entanglement of the system is defined as in [15]: E  pq (t )  1 log2 1  2 log2 2  pq  20  pq  02 in which 1  c and 2  c  pq  12 c (8) The probabilities of the states in our system for  (t  0) cut  a b and  (t  0) cut  a b are shown in figure The probabilities of the states in figure 1a become exactly the same as those in [8] We can see that the probabilities of the state pairs a b and a b , and a b (figure 1a) as well as a b and a b (figure 1b) intersect for the values close to 0.5 This means that the Bell-like states could be created in our system Hence we shall concentrate on the states in the following form 1a2 b B1 pq   B 3 pq   2 2 1 a a b b i i a a 2 b b   , B 2 pq   , B 4 pq   pq  pq  2 2 1 a a b b i i a a 2 b b   pq  pq  , (9) The probability for the system to exist in Bell-like states is represented by the formula as: P( Bi pq  )  Bi pq   cut (t ) (10) Thus, we can obtain the coefficients bi pq  as b1 pq   b3 pq   (a) 2 c   pq  (t )  ic02 (t ) ,  b2 pq   c   pq  (t )  ic02 (t ) ,  b4 pq   pq  20 pq  12 2 c   pq  (t )  ic02 (t ) , c   pq  (t )  ic02 (t ) pq  20 pq  12   (11) (b) Figure The probabilities to the system exist in the states a b (solid line), a b (dashed line), and a b (dashed-dotted line) with (a)      / 25 ,  (t  0) cut  a b and (b)    / 25,    / ,  (t  0) cut  a b The time is scaled in /  units The probabilities to the system exist in the Bell-like states with two initial conditions  (t  0) cut  a b and  (t  0) cut  a b are showed in figures to The values of the 172 L T T Oanh, …, D Q Khoa, “Entangled state generation … nonlinear coupling coupler.” Nghiên cứu khoa học công nghệ highest maxima of the probabilities corresponding to the Bell-like states B120 (Fig 2a), B220  (Fig 3a), B320  (Fig4a) and B420  (Fig 4b) with      / 25 are approximately equal to unity That is, the Bell-like states are generated while the Bell-like states Bi pq  in these figures are not created in our system For    / 25,    / , The values of all maximum of the probabilities of the Bell-like states B102 (Fig 2b), B202 (Fig 3b) are close to 1, i.e the Bell-like states are created with a high accuracy Thus, the system can generate Bell-like states with high fidelity for appropriate values of  and  (a) (b) Figure The probabilities to the system exist in the Bell-like states B120 (solid line) and B102 (dashed line) with (a)      / 25 and (b)    / 25,    / The time is scaled in /  units (a) (b) Figure The probabilities to the system exist in the Bell-like states B220 (solid line) and B202 (dashed line) with (a)      / 25 and (b)    / 25,    / The time is scaled in /  units (a) (b) Figure The probabilities to the system exist in the Bell-like states (a) B320  (solid line), B302  (dashed line) and (b) B420 (solid line), B402 (dashed line) with      / 25 The time is scaled in /  units Tạp chí Nghiên cứu KH&CN quân sự, Số 61, - 2019 173 Vật lý The entropy of entanglement is shown in figure The result of E 20 in figure 5a is similar to those in [8] The entropy of entanglement varies over period of time and is equal to ebit for Bell-like states, while for separable states it is equal to zero Except for the second maximum of E 20 in figure 5a, the values of all the maxima are almost equal to unity i.e this system can be treated as a source of Bell-like states The values of odd minima of the entangled entropy E 02 are always greater than zero It means that in this case, our system is unfailingly in the tangled states As a consequence, the entropy of entanglement and the Bell-like states change significantly when there are differences between the initial conditions (a) (b) Figure The entropies of entanglement (ebits unit) E 20 (solid line) and E 02 (dashed line) with (a)      / 25 and (b)    / 25,    / The time is scaled in /  units CONCLUSIONS In this paper, we have researched the Kerr-like nonlinear coupler comprising two nonlinear oscillators nonlinearly coupled to together and one of them interacts linearly with external coherent field By using the nonlinear scissors formalism, we have obtained the evolution of system grow closed inside a set of three states a b , a b and a b We have shown that our system can be generated the maximally entangled states for both different initial conditions Moreover, the entangled entropy and the Bell-like states changed dramatically these initial conditions Acknowledgment: This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103.03-2017.28 REFERENCES [1] 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) [2] S M Barnett and D T Pegg, “Optical state truncation,” Phys Rev A, Vol 60, pp 4965-4973, (1999) [3] W Leoński, 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dao động bơm trường kết hợp Chúng tiến triển hệ đóng tập hợp hữu hạn trạng thái Fock Hệ tạo trạng thái đan rối cực đại Đặc biệt, chúng tơi trạng thái đan rối đan rối cực đại thay đổi cách đáng kể điều kiện đầu vào khác Tính hợp lý kết khẳng định so sánh với kết tìm cơng trình trước Từ khóa: Bộ liên kết phi tuyến kiểu Kerr; Trạng thái đan rối cực đại; Entropy đan rối Nhận ngày 16 tháng năm 2019 Hoàn thiện ngày 07 tháng năm 2019 Chấp nhận đăng ngày 17 tháng năm 2019 Địa chỉ: Đại học Vinh; Cao đẳng Sư phạm Nghệ An; Đại học Hồng Đức; Viện Khoa học Công nghệ quân sự; Cao đẳng Sư phạm Quảng Trị * Email: khoa_dqspqt@yahoo.com Tạp chí Nghiên cứu KH&CN quân sự, Số 61, - 2019 175 ... [6] 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... 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... become exactly the same as those in [8] We can see that the probabilities of the state pairs a b and a b , and a b (figure 1a) as well as a b and a b (figure 1b) intersect for the values close