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Beam splitter,as a kind of linear optics instruments,has many applications such as in quantum optics and quantum information,including the preparation of nonclassical quantum states and entangled state representation.In Heisenberg picture,on the one hand,the relation of input-output of beam splitter can be easily obtained.Especially for the multicascaded beam-splitters,the input-output relation can also be directly obtained by the input-output relation of single beam splitter.On the other hand,we often need to calculate the probabilities of detecting photon number in many cases,thus we need to turn into Schrdinger picture for simplifying our calculation.Based on the equivalence between both pictures,the relation between transformation matrixes connecting these two pictures is derived.That is to say, the transform matrix corresponding to the Schrdinger picture can be obtained by transposing the transform matrix in Heisenberg picture.This concise relation constructs a bridge connecting two pictures and simplifies our calculation in the Schrdinger picture rather than step by step.Using the relation between transform matrixes of both pictures and combining the technique of integration within ordered product of operator,we further consider the coordination representation,normally ordering form and exponential expression of single beam-splitter.Then we further examine the coordination representation,normally ordering form and exponential expression of two-cascaded beam-splitters.As a generalization,the method is extended to the case of multi-cascaded beam-splitters.These investigations provide an effective way to prepare multi-mode entangled states and qubit states.In addition,a general method is shown of obtaining the total operator and its normally ordering form as well as Schmidt decomposition of the linear systems consisting of beam-splitters.As applications,2-cascaded beam-splitters is used to generate a new quantum mechanics representation and prepare the qubit states with the help of conditional measurement.The Schmidt decomposition of three-mode entangled state representation can be directly obtained by the coordination representation of 2-cascaded beam-splitters,which shows the property of entanglement.In addition,based on this representation we can clearly see that when the input states of first beam splitter are two coordinate states,the output states cannot be entangled.This implies that although the coordinate states are nonclassical,the entangled state can not be prepared either.The new proposed quantum mechanics representation will be further used to investigate the optical transformations,including wavelet transformation,Fourier transform,fractional Fourier transform,et al.Therelevant discussion will be our aim in the future research.
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Keywords:
- multi-cascaded beam-splitter /
- entangled state representation /
- technique of integration within ordered product of operator /
- Schmidt decomposition
[1] Fan H Y, Klauder J R 1994 Phys. Rev. A 49 704
[2] Fan H Y, Chen J H 2015 Front. Phys. 10 1
[3] Fan H Y, Chen J H, Zhang P F 2015 Front. Phys. 10 187
[4] Jia F, Xu S, Deng C Z, Liu C J, Hu L Y 2016 Front. Phys. 11 110302
[5] Fan H Y 1997 Representation and Transformation Theory in Quantum Mechanics (Shanghai: Shanghai Scientific and Technical Publishers) (in Chinese) [范洪义 1997 量子力 学表象与变换论 (上海: 上海科学技术出版社)]
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[18] Ye W, Zhang K Z, Zhang H L, Xu X X, Hu L Y 2018 Laser Phys. Lett. 15 025204
[19] Ouyang Y, Wang S, Zhang L J 2016 J. Opt. Soc. Am. B 33 1373
[20] Joo J, Munro W J, Spiller T P 2011 Phys. Rev. Lett. 107 083601
[21] Hu L Y, Wei C P, Huang J H, Liu C J 2014 Opt. Commun. 323 68
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[23] Paris M G A 2000 Phys. Rev. A 62 033813
[24] Xu X X, Hu L Y, Liao Z Y 2018 J. Opt. Soc. Am. B 35 174
[25] Ralph T C, White A G, Munro W J, Milburn G J 2001 Phys. Rev. A 65 012314
[26] Knill E, Laflamme R, Milburn G J 2001 Nature 409 46
[27] Braunstein S L, van Loock P 2005 Rev. Mod. Phys. 77 513
[28] Hu L Y, Fan H Y 2009 Int. J. Mod. Phys. A 24 2689
[29] Jia F, Xu X X, Liu C J, Huang J H, Hu L Y, Fan H Y 2014 Acta Phys. Sin. 63 220301 (in Chinese) [贾芳, 徐学翔, 刘寸金, 黄接辉, 胡利云, 范洪义 2014 物理学报 63 220301]
[30] Skaar J, Garca Escartn J C, Landro H 2004 Am. J. Phys. 72 1385
[31] Hu L Y, Fan H Y 2009 Europhys. Lett. 85 60001
[32] Odemir S K, Miranowicz A, Koashi M, Imoto N 2001 Phys. Rev. A 64 063818
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[1] Fan H Y, Klauder J R 1994 Phys. Rev. A 49 704
[2] Fan H Y, Chen J H 2015 Front. Phys. 10 1
[3] Fan H Y, Chen J H, Zhang P F 2015 Front. Phys. 10 187
[4] Jia F, Xu S, Deng C Z, Liu C J, Hu L Y 2016 Front. Phys. 11 110302
[5] Fan H Y 1997 Representation and Transformation Theory in Quantum Mechanics (Shanghai: Shanghai Scientific and Technical Publishers) (in Chinese) [范洪义 1997 量子力 学表象与变换论 (上海: 上海科学技术出版社)]
[6] Hu L Y, Fan H Y 2009 Opt. Commun. 282 4379
[7] Jia F, Liu C J, Hu Y Q, Fan H Y 2016 Acta Phys. Sin. 65 220302 (in Chinese) [贾芳, 刘寸金, 胡银泉, 范洪义 2016 物理学报 65 220302]
[8] Fan H Y, Liang X T 2001 Phys. Lett. A 291 61
[9] Zheng K M, Liu S Y, Zhang H L, Liu C J, Hu L Y 2014 Front. Phys. 9 451
[10] Li H R, Li F L, Yang Y 2006 Chin. Phys. B 15 2947
[11] He G Q, Zhu S W, Guo H B, Zeng G H 2008 Chin. Phys. B 17 1263
[12] Zhou N R, Li J F, Yu Z B, Gong L H, Farouk A 2017 Quantum Inform. Proc. 16 UNSP4
[13] Gong L H, Song H C, He C S, Liu Y, Zhou N R 2014 Phys. Scripta 89 035101
[14] Bartley T J, Crowley P J D, Datta A, Nunn J, Zhang L, Walmsley I 2013 Phys. Rev. A 87 022313
[15] DellAnno F, de Siena S, Illuminati F 2006 Phys. Rep. 428 53
[16] Liu J B, Wang J J, Xu Z 2017 Chin. Phys. B 26 014201
[17] Hu L Y, Liao Z Y, Zuabiry M S 2017 Phys. Rev. A 95 012310
[18] Ye W, Zhang K Z, Zhang H L, Xu X X, Hu L Y 2018 Laser Phys. Lett. 15 025204
[19] Ouyang Y, Wang S, Zhang L J 2016 J. Opt. Soc. Am. B 33 1373
[20] Joo J, Munro W J, Spiller T P 2011 Phys. Rev. Lett. 107 083601
[21] Hu L Y, Wei C P, Huang J H, Liu C J 2014 Opt. Commun. 323 68
[22] Koniorczyk M, Kurucz Z, Garis A, Janszky J 2000 Phys. Rev. A 62 013802
[23] Paris M G A 2000 Phys. Rev. A 62 033813
[24] Xu X X, Hu L Y, Liao Z Y 2018 J. Opt. Soc. Am. B 35 174
[25] Ralph T C, White A G, Munro W J, Milburn G J 2001 Phys. Rev. A 65 012314
[26] Knill E, Laflamme R, Milburn G J 2001 Nature 409 46
[27] Braunstein S L, van Loock P 2005 Rev. Mod. Phys. 77 513
[28] Hu L Y, Fan H Y 2009 Int. J. Mod. Phys. A 24 2689
[29] Jia F, Xu X X, Liu C J, Huang J H, Hu L Y, Fan H Y 2014 Acta Phys. Sin. 63 220301 (in Chinese) [贾芳, 徐学翔, 刘寸金, 黄接辉, 胡利云, 范洪义 2014 物理学报 63 220301]
[30] Skaar J, Garca Escartn J C, Landro H 2004 Am. J. Phys. 72 1385
[31] Hu L Y, Fan H Y 2009 Europhys. Lett. 85 60001
[32] Odemir S K, Miranowicz A, Koashi M, Imoto N 2001 Phys. Rev. A 64 063818
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