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Beam splitter (BS) is a basic linear element in quantum optics, which plays an important role in preparation of entangled states and quantum measurement. On the basis of the transformation relation between operators at input ports and output ports, we derive the natural representations in different representations. Using the natural expression rather than SU(2) Lie algebra relation, as well as the technique of integration within ordered product (IWOP) of operator, we can conveniently and concisely derive the normally ordering form and exponential expression of BS operator. Many forms of decompositions for BS operator can also be directly obtained by its natural representation and the orthogonality of coordinate states. Furthermore, the entangled state representation and corresponding Schmidt decomposition can be conveniently obtained.
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Keywords:
- beam splitter operator /
- entangled state representation /
- IWOP technique /
- Schmidt decomposition
[1] Louisell W H 1990 Quantum Statistical Properties of Radiation (New York: Wiley)
[2] Dirac P A M 1958 The Prnciple of Quantum Mechanics (4th Ed.) (Oxford: Oxford University Press)
[3] Fan H Y, Ye X 1995 Phys. Rev. A 51 3343
[4] Fan H Y 2002 Phys. Lett. A 294 253
[5] Fan H Y, Liang X T 2001 Phys. Lett. A 291 61
[6] Fan H Y, Hu L Y 2012 Front. Phys 7 261
[7] Liang B L, Wang J S, Meng X G, Yang Q Y 2013 Chin. Phys. B 22 016804
[8] Yuan H C, Li H M, Xu X F 2013 Chin. Phys. B 22 060301
[9] Yu H J, Zhong G B, Ma J G, Ren G 2013 Acta Phys. Sin. 62 134205 (in Chinese) [余海军, 钟国宝, 马建国, 任刚 2013 62 134205]
[10] Wang S J, Ma S J 2011 Acta Phys. Sin. 60 030302 (in Chinese) [王淑静, 马善钧 2011 60 030302]
[11] Aharonov Y 1966 Ann. Phys. (NY) 39 498
[12] van Loock P, Braunstein S L 2000 Phys. Rev. Lett. 84 3482
[13] Zheng K M, Liu S Y, Zhang H L, Liu C J, Hu L Y 2014 Front. Phys. 9 451
[14] Hu L Y, Fan H Y 2006 J. Phys. A: Math. Gen. 39 14133
[15] Hu L Y, Fan H Y 2009 Int. J. Mod. Phys. A 24 2689
[16] Fan H Y 1997 Representation and Transformation Theory in Quantum Mechanics (Shanghai: Shanghai Scientific and Technical Publisher) (in Chinese) p27 [范洪义 1997 量子力学表象与变换论——狄拉克符号法进展 (上海: 上海科技出版社) p27]
[17] Leonhardt U 1993 Phys. Rev. A 48 3265
[18] Wang S M, Zhao D M 2000 Matrix Optics (Beijing: China Higher Education Press)
[19] Hu L Y, Fan H Y 2010 Int. J. Mod. Phys. B 24 1271
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[1] Louisell W H 1990 Quantum Statistical Properties of Radiation (New York: Wiley)
[2] Dirac P A M 1958 The Prnciple of Quantum Mechanics (4th Ed.) (Oxford: Oxford University Press)
[3] Fan H Y, Ye X 1995 Phys. Rev. A 51 3343
[4] Fan H Y 2002 Phys. Lett. A 294 253
[5] Fan H Y, Liang X T 2001 Phys. Lett. A 291 61
[6] Fan H Y, Hu L Y 2012 Front. Phys 7 261
[7] Liang B L, Wang J S, Meng X G, Yang Q Y 2013 Chin. Phys. B 22 016804
[8] Yuan H C, Li H M, Xu X F 2013 Chin. Phys. B 22 060301
[9] Yu H J, Zhong G B, Ma J G, Ren G 2013 Acta Phys. Sin. 62 134205 (in Chinese) [余海军, 钟国宝, 马建国, 任刚 2013 62 134205]
[10] Wang S J, Ma S J 2011 Acta Phys. Sin. 60 030302 (in Chinese) [王淑静, 马善钧 2011 60 030302]
[11] Aharonov Y 1966 Ann. Phys. (NY) 39 498
[12] van Loock P, Braunstein S L 2000 Phys. Rev. Lett. 84 3482
[13] Zheng K M, Liu S Y, Zhang H L, Liu C J, Hu L Y 2014 Front. Phys. 9 451
[14] Hu L Y, Fan H Y 2006 J. Phys. A: Math. Gen. 39 14133
[15] Hu L Y, Fan H Y 2009 Int. J. Mod. Phys. A 24 2689
[16] Fan H Y 1997 Representation and Transformation Theory in Quantum Mechanics (Shanghai: Shanghai Scientific and Technical Publisher) (in Chinese) p27 [范洪义 1997 量子力学表象与变换论——狄拉克符号法进展 (上海: 上海科技出版社) p27]
[17] Leonhardt U 1993 Phys. Rev. A 48 3265
[18] Wang S M, Zhao D M 2000 Matrix Optics (Beijing: China Higher Education Press)
[19] Hu L Y, Fan H Y 2010 Int. J. Mod. Phys. B 24 1271
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