基于退相干环境克尔介质的贝尔型纠缠相干.doc

精品论文基于退相干环境克尔介质的贝尔型纠缠相干态的产生宋思谕1,2, 王书浩1,2, 许国富1,2 , 龙桂鲁1,21 清华大学物理学院低维量子物理国家重点实验室，北京 1000842 清华大学清华信息科学技术国家实验室，北京 100084 摘要：我们提出了两个产生贝尔型纠缠相干态的方案。一个是利用单光子，另一个是利用纠缠 光子对。对比与以前的方案，方案仅应用线性光学器件，单光子探测器，和克尔介质来完成 的，并且整个的产生过程考虑了克尔介质的退相干效应的影响。代替相干叠加态，我们选择相 干态作为输入态。文中进行了产生的贝尔纠缠相干态的保真度与退相干效应的作用的数值研 究，并且讨论了光子的损耗对产生的成功率和保真度的影响。我们的方案也适用于多模 GHZ 态的产生关键词：贝尔型纠缠相干态，态产生，克尔介质中图分类号： O43Generation of Bell-type entangled coherent states with realistic cross-Kerr nonlinearitySi Yu Song1,2 , Shuhao Wang1,2 , Guo Fu Xu1,2 , Gui Lu Long1,21 State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University, Beijing, 1000842 Tsinghua National Laboratory For Information Science and Technology, TsinghuaUniversity, Beijing, 100084Abstract: We propose two protocols for generating Bell-type entangled coherent states (ECSs). One uses a single photon; the other one uses an entangled photon pair. In contrast to previous works, the protocols are completed merely using linear optical elements,single-photon detectors, and the cross-Kerr nonlinearity, where the decoherence eect of the cross-Kerr nonlinearity is imposed onto the generation process. Considering the diculty in generating a coherent super-positioned state with large amplitude, the input state is taken as the coherent state. The delity of the generated ECSs in the presence of decoherence of weak nonlinearity is numerical studied. Photon loss is found to aect only the probability, and not the delity, of the generation process. The protocol is also proved to be easily applicable forthe generation of multi-mode GHZ-type ECSs.基金项目： National Natural Science Foundation (11175094)，National Basic Research Program (2009CB929402), Na- tional Basic Research Program (2011CB9216002)作者简介： Song Si Yu(1986-)，female，PHD，major research direction：quantum information. Correspondence author：Long Gui Lu (1962-)，male，professor，major research direction：quantum information.- 20 -Key words: Bell-type entangled coherent states, state generation, cross-Kerr nonlinearityEntanglement is a signicant ingredient of quantum systems dierent from classical sys- tems, and is thus important in studying non-local and non-classical behaviors. Using entangled states, many schemes have been proposed to implement the tasks in quantum information processing (QIP) 1. One of the most important states exhibiting quantum entanglement is the entangled coherent state (ECS). The ECS was rst shown in Ref. 2, and the theoretical generation process of ECS was subsequently proposed through the optical system 3. Soon after, ECSs have been theoretically achieved by trapped ions 4, nanomechanical systems 5, quantum dots 6, microwave cavity quantum electrodynamics 7, Bose-Einstein condensation systems 8, and superconducting systems 9. ECSs exhibit several advantages, such as ro- bustness against absorption and decoherence of the environment 10, which leads to the better performance of small-amplitude ECSs than photon systems in teleportation 11. Meanwhile, the four Bell-type ECSs can be simply discriminated by a beam splitter (BS) and two photon- number-resolving detectors 12, which is a great advantage in designing quantum computing protocols 11. ECSs are widely applied in quantum communication and quantum computation 3, 13, 14, 15, 16, 17, 18, 19, 20, 21, such as quantum teleportation 15, 21, 22, entanglement purication 23, quantum error corrections 24, 25, etc.The generation of a two-mode ECS in an optical system can be simply realized by injecting a coherent superposition state (CSS) into one port of a BS 22, 26, 27. The explicit form ofthe CSS is|CSS± = N± (| ± | ), (1)where the subscripts “+” and “” correspond to even and odd CSSs, respectively, and N± =212(1 ± e2| ) 2 are the normalization constants. In the following context, for simplicity, thestates are not necessarily normalized, and “” is used instead of “=” to denote non-normalized±states. The output state is the Bell-type ECS |Bell |, ± | , . Similarly, theGreenberger-Horne-Zeilinger (GHZ)-type ECS of N modes can be obtained by passing a CSS of the form of N± (|N ± | N ) through a sequence of N 1 BSs with reectivityr1 = 1/N , r2 = 1/N 1, · · · , and rN = 1/2 20. However, the protocols introducedabove can only obtain ECSs with small amplitudes, which is mainly due to the diculty ofgenerating CSSs with large amplitudes. Although considerable eorts have been devoted to the enlargement of the amplitudes of CSSs and improving the delity in the generation process 26, 28, 29, 30, 31, 32, the generation of CSSs with large amplitudes is still an open question.The scheme in Ref. 30 can produce CSSs with 2.2 using a squeezed single-photon state,and then is improved to be close to 3.4 with high delity using the cavity-assisted interaction31. In Ref. 32, they obtained CSSs with high delity ( 4), where double cross-phasemodulations are applied. The amplitude of ECSs obtained by BSs and a vacuum state is atmost CSS , which still cannot satisfy the requirements of QIP.2 !"!" !"!"!图 1: Schematic of the generation of Bell-type ECSs with realistic cross-Kerr nonlinearity. BSand PBS denote beam splitter and polarization splitter, respectively. D1 and D2 denote the single-photon detectors. The four cross-Kerr nonlinearities give one mode of coherent state a phase shift .In this paper, we simply generate two-mode ECSs using the coherent state assisted by the photon states without using CSSs. Our protocols are completed by BSs, a polarization beam splitter (PBS), single-photon detectors, and the cross-Kerr nonlinearity 33. The cross- Kerr nonlinearity has been used to generate entanglement and CSSs 26, 27, 34, 35, 36, 37; it has also been used in quantum computation 28, 38, 39, 40, 41. Experimentally, a strong Kerr nonlinearity is extremely dicult to obtain because the nonlinear eects in a nonlinear mediumare suciently weak (3 1022 m2 V 2 ) 43, 42, even with the method of electromagneticallyinduced transparency 29, 44. Thus far, the achieved phase shift at the single-photon levelis only in the order of 105 43. In experiments, the Kerr interaction is not ideal because of the decoherence existing in Kerr nonlinear materials. Decoherence leads to a phase shift and energy loss of the coherent state; therefore, the Kerr interaction cannot be describedby the ideal Hamiltonian H = a a1 a a2 . The remainder of this paper is organized as1 2follows. First, we describe the two protocols for generating the Bell-type ECSs with realistic Kerr nonlinearity. Next, the delity and entanglement of the generated Bell-type ECSs are numerically investigated with the decoherence eect. Subsequently, the scheme is applied to the generation of GHZ-type ECSs. And the eect of photon loss on the generation is investigated. The advantages of the two generation protocols are then enumerated. Finally, the conclusions are drawn.The scheme for the generation of the Bell-type ECSs with realistic cross-Kerr nonlinearityis shown in Fig. 1. Our input single-photon state is 21/2 (|H + |V ), where |H and |V represent the horizontal and vertical polarization states, respectively. The input state of thewhole system is1 |in = 2 (|H + |V )|2|2. (2)By injecting the input state into two balanced BSs and a PBS, the whole system becomes1| = 2 (|H, 0 + |0, V )|, , , , (3)where |0 is the vacuum state. Then, the single-photon state is coupled with the coherent stateswithin the cross-Kerr medium, and the evolution of the system is determined by the masterequation(t)= i a ai a aj , (t)t ij i,j +2ai (t)a a ai (t) (t)a ai , (4)2iii iwhere ai (a ) and aj (a ) are the annihilation operators (creation operator) of the i and j modes,ijrespectively, and are the nonlinear strength and damping rate of the coupling optical modes, which are decided by the real and imaginary parts of the third-order susceptibility (3) of the nonlinear material, respectively.For clarication, we divide the Kerr interaction, i.e., the right side of Eq. (4), in-to two parts: the phase shift item i i,j a ai a aj , (t) and the decoherence eect itemij 2 i 2ai (t)a a ai (t) (t)a ai . Using the following super operators,iiiJi = ai a ,i Li = 2 (ai ai + ai ai ), (5)and dividing the interaction time into innite small periods, the state of the whole system attime t can be expressed as(t) = limN 1 exp(Ji + Li )t + Ki,j t(t0 ), (6)N k=1 ii,jwhere Kij = ia ai a aj , , t = (t t0 )/N , and tk = t0 + kt. Ji and Li act on allij2the involved coherent modes, and Ki,j acts on each coupled modes of the single photon and coherent mode. To obtain (t1 ) for (t0 ) = 1 (|H, 0 + |0, V )|, , , , we rst calculatee(Ji +Li )t |. Supposing t 0, then 45e(Ji +Li )t | = |1exp(t) |et/2 et/2 |. (7)Thus, the density matrix of the whole system at time t1 is1 (t1 ) = e i (Ji +Li )t e2i,j Ki,j t(|H + |V )(H | + V |)1| | | |2=ei (Ji +Li )t (|H |ei1 |ei1 | + |V |ei1 |ei1 )(H |ei1 |ei1 | + V |ei1 |ei1 |) (|H H | |A1 ei1 , A1 , A1 ei1 , A1 A1 ei1 , A1 , A1 ei1 , A1 |+ |V V | |A1 , A1 ei1 , A1 , A1 ei1 A1 , A1 ei1 , A1 , A1 ei1 |+ C1 |H V | |A1 ei1 , A1 , A1 ei1 , A1 A1 , A1 ei1 , A1 , A1 ei1 |+ C1 |V H | |A1 , A1 ei1 , A1 , A1 ei1 A1 ei1 , A1 , A1 ei1 , A1 |), (8)where 1 = t is the phase factor, C1 = exp2(1 et )|ei1 |2 , and A1 = et/2 .After N steps of evolution, the density matrix of the whole system is(t) |H H | |Aei , A, Aei , AAei , A, Aei , A|+ |V V | |A, Aei , A, Aei A, Aei , A, Aei |+ C (t)|H V | |Aei , A, Aei , AA, Aei , A, Aei |+ C (t)|V H | |A, Aei , A, Aei Aei , A, Aei , A|, (9)where = t, A = et/2 , and the coherence parameter C (t) isC (t) =limN C1 C2 · · · CN 12 | |= exp 4 2 ( 2 + 22 et+2 + 2et cost 2 + 2et sint).(10)We transform the system of Eq. (9) using three BSs, and then detect with the single- photon detectors D1 and D2 . When one detector respond, the density matrix of modes 1, 2, 3, and 4 is |, , , , , , |+ |, , , , , , |+ C (t)|, , , , , , |+ C (t)|, , , , , , |, (11)where = A(e21)= ei t/2(ei21) , and = ei +1 . The state of modes 2 and 4 can beei 11.00.8F0.60.40.2 =100, G=0.5 =100, G=1 =200, G=10.00.00 0.01 0.02 0.03 0.04 0.05 0.060.07图 2: Fidelity versus the phase shift of the obtained Bell-type ECS. The solid, dashed, anddot-dashed lines represent the cases in which the initial amplitude | = 100 with = 0.5,| = 100 with = 1, and | = 200 with = 1, respectively, where = /.decomposed as1 + C (t)Bell 2,4 Bell |2,42N 2 |+N1 C (t)+|Bell 2,4 Bell |,(12)4|2 122 where N± = 2(1 ± e) 2 are the normalization constants of the Bell-type ECSs. Con-sequently, the generated state of modes 2 and 4 is the mixture of |Bell and |Bell , and itsamplitude is+ t| =2et/2 |sin2 |.(13)The output state of modes 1 and 3 in Fig. 1 is a pure coherent state | e t/2(ei+1) , e t/2(ei+1).1,3The delity is2 21 + C (t)which can be calculated as 46=, (14)F2+by taking 2 = , |1 = |Bell .F = 1 |2 |1 (15)When the coherence parameter C (t) = 1, the output state of modes 2 and 4 is a pure state|Bell , which corresponds to the case in which no decoherence exists. In the case of C (t) = 0,+the output state of modes 2 and 4 is the mixture of |Bell and |Bell with half probability.+ The delity F as a function of the phase shift (Eq. (14) is illustrated in Fig. 2. Strongerdamping rates are found to reduce C (t) and the delity; stronger input coherent states makedecoherence occur easier, and reduces C (t) as well as the delity. The delity decreases with increased phase shift. Using Eq. (13), we see that the amplitude of the obtained Bell-typeECSs reaches 4 with good delity as the input coherent state with | = 100. Fig. 3 shows1.00.8CHtL0.60.40.20.0 =100, G=0.5=100, G=1 =400, G=10 1 2 3 4 5 6 7 图 3: Coherence parameter C (t) versus the amplitude | of the obtained Bell-type ECSs. The solid, dashed, and dot-dashed lines represent the cases in which the initial amplitude | = 100 with = 0.5, | = 100 with = 1, and | = 200 with = 1, respectively.0.950.90CHtL0.850.800.750.700 500 1000 15002000图 4: Coherence parameter C (t) versus the amplitude of the initial state.the change in the coherence parameter C (t) with the amplitude | of the obtained Bell-typeECSs. A larger required separation results in a lower coherence parameter C (t). Fig. 4 showsthat under the same separation, the initial state with a larger amplitude achieves a higher C (t). Fig. 5 shows the maximum value of the amplitude as a function of the phase shift (Eq. (13).The maximum amplitudes |max are 72.719 at = 2.214 when | = 100 and = 0.5, 45.594 at = 1.571 when | = 100 and = 1, as w