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Quantum statistical properties of phase-type three-headed Schrodinger cat state

Lin Dun-Qing Zhu Ze-Qun Wang Zu-Jian Xu Xue-Xiang

Quantum statistical properties of phase-type three-headed Schrodinger cat state

Lin Dun-Qing, Zhu Ze-Qun, Wang Zu-Jian, Xu Xue-Xiang
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  • Quantum superposition is a fundamental principle of quantum mechanics, which provides a crucial basis to observe phenomena beyond the predictions of classical physics. For example, a quantum entangled state can exhibit stronger correlation than classically possible one. In quantum state engineering, many new quantum states can be obtained from the superposition of many known states. In recent decades, the superposition of coherent states (CSs) with the same amplitude but two different phases has been a subject of great interest. This superposition state was often called Schrodinger cat state (here, we also name it 2-headed cat state (2HCS)), which becomes an important tool to study a lot of fundamental issues. Surprisingly, some studies have extended the quantum superposition to involving more than two component coherent states. In order to produce the superposition of three photons, people have considered the superposition of coherent states with three different phases (here, we also name it 3-headed cat state (3HCS)). Furthermore, in microwave cavity quantum electrodynamics of bang-bang quantum Zeno dynamics control, people have proposed the superposition of coherent states with four different phases (here, we also name it 4-headed cat state (4HCS)). In this paper, we make a detailed investigation on the quantum statistical properties of a phase-type 3HCS. These properties include photon number distribution, average photon number, sub-Poissionian distribution, squeezing effect, and Wigner function, etc. We derive their analytical expressions and make numerical simulations for these properties. The results are compared with the counterparts of the CS, the 2HCS and the 4HCS. The conclusions are obtained as follows. 1) The CS, the 2HCS, the 3HCS and the 4HCS have k, 2k, 3k and 4k photon number components, respectively (k is an integer); 2) small difference in average photon number among these quantum states in small-amplitude range can be observed, while their average photon numbers become almost equal in large-amplitude range; 3) the CS exhibits Poisson distribution, and the 2HCS, the 3HCS and the 4HCS exhibit super-Poisson distributions in most amplitude ranges, however, sub-Poisson distribution can be seen for the 3HCS and the 4HCS in some specific amplitude ranges; 4) except for the 2HCS that may have the squeezing property, no squeezing properties can be found in the CS, the 3HCS and the 4HCS; 5) negative values can exist in the Wigner functions for the 2HCS, the 3HCS and the 4HCS, while it is not found in the CS. Similar to the 2HCS and 4HCS, the Wigner function of the 3HCS has negative component, which implies its nonclassicality. Different from the 2HCS, the 3HCS exhibits sub-Poisson photon number distribution in a certain amplitude range, it is weaker than that of the 4HCS. At the same time, no squeezing is found in the 3 or 4HCS, which is another difference from the 2HCS.
      Corresponding author: Xu Xue-Xiang, xuxuexiang@jxnu.edu.cn
    • Funds: Project supported by National Natural Science Foundation of China (Grant No. 11665013), Research on Teaching Reform of Jiangxi Higher Education, China (Grant No. JXJG-16-2-2) and the Gaoyuan Plan Project of Jiangxi Normal University, China.
    [1]

    Dirac P A M 1958 The Principles of Quantum Mechanics (4th Ed.) (Oxford: Oxford University Press) pp1-22

    [2]

    Zeng J Y 2007 Quantum Mechanics (4th Ed.) (Beijing: Science Press) pp52-54 [曾谨言 2007 量子力学(第四版) (北京: 科学出版社)] pp52-54

    [3]

    Dell'Anno F, de Siena S, Illuminati F 2006 Phys. Rep. 428 53

    [4]

    Kok P, Lovett B W 2010 Introduction to Optical Quantum Information Processing (Cambridge: Cambridge University Press) pp183-187

    [5]

    Polkinghorne J C 1985 The Quantum World (Princeton: Princeton University Press) p67

    [6]

    John G 2011 In Search of Schrodinger's Cat: Quantum Physics and Reality (Berlin: Random House Publishing Group) pp234

    [7]

    Glauber R J 1963 Phys. Rev. 131 2766

    [8]

    Gerry C C, Knight P 2005 Introductory Quantum Optics (Cambridge: Cambridge University Press) pp174-181

    [9]

    Yukawa M, Miyata K, Mizuta T, Yonezawa H, Marek P, Filip R, Furusawa A 2013 Opt. Express 21 5529

    [10]

    Vlastakis B, Kirchmair G, Leghtas Z, Nigg S E, Frunzio L, Girvin S M, Mirrahimi M, Devoret M H, Schoelkopf R J 2013 Science 342 607

    [11]

    Raimond J M, Facchi P, Peaudecerf B, Pascazio S, Sayrin C, Dotsenko I, Gleyzes S, Brune M, Haroche S 2012 Phys. Rev. A 86 032120

    [12]

    Lee S Y, Lee C W, Nha H, Kaszlikowski D 2015 J. Opt. Soc. Am. B 32 1186

    [13]

    Mandel L 1979 Opt. Lett. 4 205

    [14]

    Walls D F, Milburn G J 1994 Quantum Optics (Berlin: Springer-Verlag) pp81-82

    [15]

    Wigner E P 1932 Phys. Rev. 40 749

    [16]

    Xu X X, Yuan H C, Hu L Y 2010 Acta Phys. Sin. 59 4661

    [17]

    Xu X X, Yuan H C 2016 Phys. Lett. A 380 2342

    [18]

    Lutterbach L, Davidovich L 1997 Phys. Rev. Lett. 78 2547

    [19]

    Kenfack A, Zyczkowski K 2004 J. Opt. B: Quantum Semi-Class. Opt. 6 396

    [20]

    Gerry C C, Mimih J 2010 Contemp. Phys. 51 497

    [21]

    Leghtas Z, Kirchmair G, Vlastakis B, Schoelkopf R J, Devorett M H, Mirrahimi M 2013 Phys. Rev. Lett. 111 120501

    [22]

    Ralph T C, Gilchrist A, Milburn G J, Munro W J, Glancy S 2003 Phys. Rev. A 68 042319

  • [1]

    Dirac P A M 1958 The Principles of Quantum Mechanics (4th Ed.) (Oxford: Oxford University Press) pp1-22

    [2]

    Zeng J Y 2007 Quantum Mechanics (4th Ed.) (Beijing: Science Press) pp52-54 [曾谨言 2007 量子力学(第四版) (北京: 科学出版社)] pp52-54

    [3]

    Dell'Anno F, de Siena S, Illuminati F 2006 Phys. Rep. 428 53

    [4]

    Kok P, Lovett B W 2010 Introduction to Optical Quantum Information Processing (Cambridge: Cambridge University Press) pp183-187

    [5]

    Polkinghorne J C 1985 The Quantum World (Princeton: Princeton University Press) p67

    [6]

    John G 2011 In Search of Schrodinger's Cat: Quantum Physics and Reality (Berlin: Random House Publishing Group) pp234

    [7]

    Glauber R J 1963 Phys. Rev. 131 2766

    [8]

    Gerry C C, Knight P 2005 Introductory Quantum Optics (Cambridge: Cambridge University Press) pp174-181

    [9]

    Yukawa M, Miyata K, Mizuta T, Yonezawa H, Marek P, Filip R, Furusawa A 2013 Opt. Express 21 5529

    [10]

    Vlastakis B, Kirchmair G, Leghtas Z, Nigg S E, Frunzio L, Girvin S M, Mirrahimi M, Devoret M H, Schoelkopf R J 2013 Science 342 607

    [11]

    Raimond J M, Facchi P, Peaudecerf B, Pascazio S, Sayrin C, Dotsenko I, Gleyzes S, Brune M, Haroche S 2012 Phys. Rev. A 86 032120

    [12]

    Lee S Y, Lee C W, Nha H, Kaszlikowski D 2015 J. Opt. Soc. Am. B 32 1186

    [13]

    Mandel L 1979 Opt. Lett. 4 205

    [14]

    Walls D F, Milburn G J 1994 Quantum Optics (Berlin: Springer-Verlag) pp81-82

    [15]

    Wigner E P 1932 Phys. Rev. 40 749

    [16]

    Xu X X, Yuan H C, Hu L Y 2010 Acta Phys. Sin. 59 4661

    [17]

    Xu X X, Yuan H C 2016 Phys. Lett. A 380 2342

    [18]

    Lutterbach L, Davidovich L 1997 Phys. Rev. Lett. 78 2547

    [19]

    Kenfack A, Zyczkowski K 2004 J. Opt. B: Quantum Semi-Class. Opt. 6 396

    [20]

    Gerry C C, Mimih J 2010 Contemp. Phys. 51 497

    [21]

    Leghtas Z, Kirchmair G, Vlastakis B, Schoelkopf R J, Devorett M H, Mirrahimi M 2013 Phys. Rev. Lett. 111 120501

    [22]

    Ralph T C, Gilchrist A, Milburn G J, Munro W J, Glancy S 2003 Phys. Rev. A 68 042319

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  • Received Date:  25 December 2016
  • Accepted Date:  08 March 2017
  • Published Online:  05 May 2017

Quantum statistical properties of phase-type three-headed Schrodinger cat state

    Corresponding author: Xu Xue-Xiang, xuxuexiang@jxnu.edu.cn
  • 1. College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang 330022, China;
  • 2. Center for Quantum Science and Technology, Jiangxi Normal University, Nanchang 330022, China
Fund Project:  Project supported by National Natural Science Foundation of China (Grant No. 11665013), Research on Teaching Reform of Jiangxi Higher Education, China (Grant No. JXJG-16-2-2) and the Gaoyuan Plan Project of Jiangxi Normal University, China.

Abstract: Quantum superposition is a fundamental principle of quantum mechanics, which provides a crucial basis to observe phenomena beyond the predictions of classical physics. For example, a quantum entangled state can exhibit stronger correlation than classically possible one. In quantum state engineering, many new quantum states can be obtained from the superposition of many known states. In recent decades, the superposition of coherent states (CSs) with the same amplitude but two different phases has been a subject of great interest. This superposition state was often called Schrodinger cat state (here, we also name it 2-headed cat state (2HCS)), which becomes an important tool to study a lot of fundamental issues. Surprisingly, some studies have extended the quantum superposition to involving more than two component coherent states. In order to produce the superposition of three photons, people have considered the superposition of coherent states with three different phases (here, we also name it 3-headed cat state (3HCS)). Furthermore, in microwave cavity quantum electrodynamics of bang-bang quantum Zeno dynamics control, people have proposed the superposition of coherent states with four different phases (here, we also name it 4-headed cat state (4HCS)). In this paper, we make a detailed investigation on the quantum statistical properties of a phase-type 3HCS. These properties include photon number distribution, average photon number, sub-Poissionian distribution, squeezing effect, and Wigner function, etc. We derive their analytical expressions and make numerical simulations for these properties. The results are compared with the counterparts of the CS, the 2HCS and the 4HCS. The conclusions are obtained as follows. 1) The CS, the 2HCS, the 3HCS and the 4HCS have k, 2k, 3k and 4k photon number components, respectively (k is an integer); 2) small difference in average photon number among these quantum states in small-amplitude range can be observed, while their average photon numbers become almost equal in large-amplitude range; 3) the CS exhibits Poisson distribution, and the 2HCS, the 3HCS and the 4HCS exhibit super-Poisson distributions in most amplitude ranges, however, sub-Poisson distribution can be seen for the 3HCS and the 4HCS in some specific amplitude ranges; 4) except for the 2HCS that may have the squeezing property, no squeezing properties can be found in the CS, the 3HCS and the 4HCS; 5) negative values can exist in the Wigner functions for the 2HCS, the 3HCS and the 4HCS, while it is not found in the CS. Similar to the 2HCS and 4HCS, the Wigner function of the 3HCS has negative component, which implies its nonclassicality. Different from the 2HCS, the 3HCS exhibits sub-Poisson photon number distribution in a certain amplitude range, it is weaker than that of the 4HCS. At the same time, no squeezing is found in the 3 or 4HCS, which is another difference from the 2HCS.

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