Evolution of quantum states via Weyl expansion in dissipative channel

doi:10.1088/1674-1056/28/8/084206

Abstract Based on the Weyl expansion representation of Wigner operator and its invariant property under similar transformation, we derived the relationship between input state and output state after a unitary transformation including Wigner function and density operator. It is shown that they can be related by a transformation matrix corresponding to the unitary evolution. In addition, for any density operator going through a dissipative channel, the evolution formula of the Wigner function is also derived. As applications, we considered further the two-mode squeezed vacuum as inputs, and obtained the resulted Wigner function and density operator within normal ordering form. Our method is clear and concise, and can be easily extended to deal with other problems involved in quantum metrology, steering, and quantum information with continuous variable.

Keywords: Weyl expansion Wigner operator similar transformation two-mode squeezed state integration within ordered product (IWOP) technique

Received: 25 March 2019
Revised: 27 April 2019
Accepted manuscript online:

PACS:	42.50.-p	(Quantum optics)
	03.65.-w	(Quantum mechanics)

Fund: Project supported by the National Natural Science Foundation of China (Grant No. 11664017), the Outstanding Young Talent Program of Jiangxi Province, China (Grant No. 20171BCB23034), the Degree and Postgraduate Education Teaching Reform Project of Jiangxi Province, China (Grant No. JXYJG-2013-027), and the Science Fund of the Education Department of Jiangxi Province, China (Grant No. GJJ170184).

Corresponding Authors: Li-Yun Hu, Zhi-Ming Rao
E-mail: hlyun@jxnu.edu.cn;raozm24@jxnu.edu.cn

Cite this article:

Li-Yun Hu(胡利云), Zhi-Ming Rao(饶志明), Qing-Qiang Kuang(况庆强) Evolution of quantum states via Weyl expansion in dissipative channel 2019 Chin. Phys. B 28 084206

[1]	Wigner E P 1932 Phys. Rev. 40 749
[2]	Scully M O, Zubairy M S 1997 Quantum Optics (New York: Cambridge University Press)
[3]	Schleich W P 2001 Quantum Optics in Phase Space (Berlin: Wiley-VCH)
[4]	Kenfack A and Zyczkowski K 2004 J. Opt. B: Quantum Semiclass. Opt. 6 396
[5]	Moya-Cessa H 2006 Phys. Rep. 432 1
[6]	Fan H Y and Hu L Y 2008 Opt. Commun. 281 5571
[7]	Fan H Y and Hu L Y 2008 Mod. Phys. Lett. B 22 2435
[8]	Hofmann H F, Ide T and Kobayashi T 2000 Phys. Rev. A 62 062304
[9]	Hu L Y, Wu J N, Liao Z Y and Zuabiry M S 2016 J. Phys. B: At. Mol. Opt. Phys. 49 175504
[10]	Hu L Y, Liao Z Y and Zuabiry M S 2017 Phys. Rev. A 95 012310
[11]	Vidal G and Werner R F 2002 Phys. Rev. A 65 032314
[12]	Fan H Y and Zaidi H R 1987 Phys. Lett. A 124 303
[13]	Fan H Y 2003 J. Opt. B: Quantum Semiclass. Opt. 5 R147
[14]	Wusche A 1999 J. Opt. B: Quantum Semiclass. Opt. 1 R11
[15]	Fan H Y, Lu H L and Fan Y 2006 Ann. Phys. 321 480
[16]	Mehta C L 1967 Phys. Rev. Lett. 18 752
[17]	Fan H Y and Wang J S 2005 Mod. Phys. Lett. A 20 1525
[18]	Wusche A 2017 Advances in Pure Mathematics 7 533
[19]	Hu L Y, Chen F, Wang Z S and Fan H Y 2011 Chin. Phys. B 20 074204
[20]	Vaidman L 1994 Phys. Rev. A 49 1473
[21]	Madsen L S, Usenko V C, Lassen M, Filip R and Andersen U L 2012 Nat. Commun. 3 1083
[22]	He G Q, Zhu S W, Guo H B and Zeng G H 2008 Chin. Phys. B 17 1263
[23]	Anisimov P M, Raterman G M, Chiruvelli A, Plick W N, Huver S D, Lee H and Dowling J P 2010 Phys. Rev. Lett. 104 103602
[24]	Jiang N Q, Fan H Y, Xi S L, Tang L Y and Yuan X Z 2011 Chin. Phys. B 20 120302
[25]	Zhou N R, Hu L Y and Fan H Y 2011 Chin. Phys. B 20 120301
[26]	Xu L and Tan Q S 2018 Chin. Phys. B 27 014203
[27]	Zhao N, Zhu C H and Quan D X 2015 Chin. Phys. Lett. 32 080306
[28]	Jia F, Zhang K Z, Hu Y Q, Zhang H L, Hu L Y and Fan H Y 2018 Acta Phys. Sin. 67 150301 (in Chinese)
[29]	Xiang S H, Zhu X X and Song K F 2018 Chin. Phys. B 27 100305

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[7]	New transformation of Wigner operator in phase space quantum mechanics for the two-mode entangled case Fan Hong-Yi (范洪义), Yuan Hong-Chun (袁洪春). Chin. Phys. B, 2010, 19(7): 070301.
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[10]	New two-fold integration transformation for the Wigner operator in phase space quantum mechanics and its relation to operator ordering Fan Hong-Yi(范洪义). Chin. Phys. B, 2010, 19(4): 040305.
[11]	New equation for deriving pure state density operators by Weyl correspondence and Wigner operator Xu Ye-Jun(许业军), Fan Hong-Yi(范洪义), and Liu Qiu-Yu(刘秋宇). Chin. Phys. B, 2010, 19(2): 020303.
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[14]	Four-mode continuous-variables entangled state: generation from beam splitter, a parametric down-conversion and a polarizer Kuang Mai-Hua(匡买花), Ma Shan-Jun(马善钧), Liu Dong-Mei(刘冬梅), and Wang Shu-Jing(王淑静). Chin. Phys. B, 2009, 18(3): 1065-1071.
[15]	The q-analogues of two-mode squeezed states constructed by virtue of the IWOP technique Meng Xiang-Guo(孟祥国), Wang Ji-Suo(王继锁), and Li Hong-Qi(李洪奇). Chin. Phys. B, 2008, 17(8): 2973-2978.

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