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Resonance transmission of one-dimensional quantum walk with phase defects

Wang Dan-Dan Li Zhi-Jian

Resonance transmission of one-dimensional quantum walk with phase defects

Wang Dan-Dan, Li Zhi-Jian
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  • In this paper, the resonance transmission of discrete time quantum walk is studied when it walks on one-dimensional lattice in which two-phase defects or a piece of phase defects exists. The quasi energy of discrete time quantum walk has a unique dispersion relation with the momentum, from which we first discuss the wave velocity direction versus the values of momentum, and distinguish the incident wave and the reflected wave. The gap between two energy bands depends on the parameters of coincident operator, so the phase defects, which break down the translation invariance of quantum walk on uniform lattices, can be regarded as an analogue of quantum potential. Then we use the condition of energy conversion at the boundary points to obtain the transmission rate and discuss its variation with the incident momentum for different strengths and widths of defects in detail. The multiple resonant peaks are observed due to the enhanced interference effect. Different resonant behaviors are shown when the strength of defect is less or greater than /2, correspondingly the resonances occur in a wide region of incident momentum or the sharp resonant peaks appear at discrete values of momentum. Under the condition of strong defect strength, i.e., approaching to , the qualitative relation between the number of resonant peaks and the widths of defect region is given. The number of resonant peaks is 2(N-1) when the two phase defects are located at N sites symmetric about the origin, while the number is 2N when a piece of phase defects is located at -N to N sites. In the case of a piece of phase defects, we also present the phase diagram in parameter space of (k, ) to show the discrete time of quantum walk propagating or tunneling through the defect region. In terms of this phase diagram, the variations of transmission rate with the incident momentum are reasonably explained. One special phenomenon is that the quantum walk is almost totally reflected in the tunneling case except for =/2 and k being slightly off -/2. Moreover, this behavior seems little affecting the defect strength, just similar to a classical particle. As a result of this research, we hope to deepen the insight of the quantum walk and provide methods to control the spreading of quantum walk through artificial defects.
      Corresponding author: Li Zhi-Jian, zjli@sxu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 10974124, 11274208) and the Shanxi Scholarship Council of China (2015-012).
    [1]

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    [2]

    Zaburdaev V, Denisov S, Klafter J 2015 Rev. Mod. Phys. 87 483

    [3]

    Ambainis A 2003 Int. J. Quantum Inf. 01 507

    [4]

    Shenvi N, Kempe J, Whaley K B 2003 Phys. Rev. A 67 052307

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    Lovett N B, Cooper S, Everitt M, Trevers M, Kendon V 2010 Phys. Rev. A 81 042330

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    Kurzyński P, Wjcik A 2011 Phys. Rev. A 83 062315

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    Schmitz H, Matjeschk R, Schneider Ch, Glueckert J, Enderlein M, Huber T, Schaetz T 2009 Phys. Rev. Lett. 103 090504

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    Du J F, Li H, Xu X D, Shi M J, Wu J H, Zhou X Y, Han R D 2003 Phys. Rev. A 67 042316

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    Karski M, Frster L, Choi J M, Steffen A, Alt W, Meschede D, Widera A 2009 Science 325(5937) 174

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    Bouwmeester D, Marzoli I, Karman G P, Schleich W, Woerdman J P 1999 Phys. Rev. A 61 013410

    [12]

    Xue P, Qin H, Tang B, Zhan X, Bian Z H, Li J 2014 Chin. Phys. B 23 110307

    [13]

    Schreiber A, Gbris A, Rohde P P, Laiho K, tefaňk M, Potoček V, Hamilton C, Jex I, Silberhorn C 2012 Science 336 55

    [14]

    Poulios K, Keil R, Fry D, Meinecke J D A, Matthews J C F, Politi A, Lobino M, Grfe M, Heinrich M, Nolte S, Szameit A, O'Brien J L 2013 Phys. Rev. Lett. 112(14) 143604

    [15]

    Farhi E, Gutmann S 1998 Phys. Rev. A 58 915

    [16]

    Aharonov Y, Davidovich L, Zagury N 1993 Phys. Rev. A 48 1687

    [17]

    Strauch F W 2006 Phys. Rev. A 74 030301(R)

    [18]

    Chandrashekar C M 2013 Sci. Rep. 3 2829

    [19]

    Yin Y, Katsanos D E, Evangelou S N 2008 Phys. Rev. A 77 022302

    [20]

    Trm P, Jex I, Schleich W P 2002 Phys. Rev. A 65 052110

    [21]

    Schreiber A, Cassemiro K N, Potoek V, Gbris A, Jex I, Silberhorn Ch 2011 Phys. Rev. Lett. 106 180403

    [22]

    Chou C I, Ho C L 2014 Chin. Phys. B 23 110302

    [23]

    Zhang R, Qin H, Tang B, Xue P 2013 Chin. Phys. B 22 110312

    [24]

    Li Z J, Izaac J A, Wang J B 2013 Phys. Rev. A 87 012314

    [25]

    Mohseni M, Rebentrost P, Lloyd S, Aspuru-Guzik A 2008 J. Chem. Phys. 129 174106

    [26]

    Marais A, Sinayskiy I, Kay A, Pentruccione F, Ekert A 2013 New J. Phys. 15 013038

    [27]

    Anderson P W 1958 Phys. Rev. 109 1492

    [28]

    Ribeiro P, Milman P, Mosseri R 2004 Phys. Rev. Lett. 93 190503

    [29]

    Keating J P, Linden N, Matthews J C F, Winter A 2007 Phys. Rev. A 76 012315

    [30]

    Joye A, Merkli M 2010 J. Stat. Phys. 140 1025

    [31]

    Ahlbrecht A, Alberti A, Meschede D, Scholz V B, Werner A H, Werner R F 2012 New J. Phys. 14 073050

    [32]

    Kitagawa T, Rudner M S, Berg E, Demler E 2010 Phys. Rev. A 82 033429

    [33]

    Rakovszky T, Asboth J K 2015 Phys. Rev. A 92 052311

    [34]

    Asbth J K, Obuse H 2013 Phys. Rev. B 88 121406(R)

    [35]

    Wjcik A, Łuczak T, Kurzyński P, Grudka A, Gdala T, Bednarska-Bzdęga M 2012 Phys. Rev. A 85 012329

    [36]

    Izaac J A, Wang J B, Li Z J 2013 Phys. Rev. A 88 042334

    [37]

    Zhang R, Xue P, Twamley J 2014 Phys. Rev. A 89 042317

    [38]

    Li Z J, Wang J B 2015 Sci. Rep. 5 13585

    [39]

    Lam H T, Szeto K Y 2015 Phys. Rev. A 92 012323

    [40]

    Li Z J, Wang J B 2015 J. Phys. A: Math. Theor. 48 355301

  • [1]

    Kempe J 2003 Contemp. Phys. 44 307

    [2]

    Zaburdaev V, Denisov S, Klafter J 2015 Rev. Mod. Phys. 87 483

    [3]

    Ambainis A 2003 Int. J. Quantum Inf. 01 507

    [4]

    Shenvi N, Kempe J, Whaley K B 2003 Phys. Rev. A 67 052307

    [5]

    Lovett N B, Cooper S, Everitt M, Trevers M, Kendon V 2010 Phys. Rev. A 81 042330

    [6]

    Kurzyński P, Wjcik A 2011 Phys. Rev. A 83 062315

    [7]

    Plenio M B, Huelga S F 2008 New J. Phys. 10 113019

    [8]

    Schmitz H, Matjeschk R, Schneider Ch, Glueckert J, Enderlein M, Huber T, Schaetz T 2009 Phys. Rev. Lett. 103 090504

    [9]

    Du J F, Li H, Xu X D, Shi M J, Wu J H, Zhou X Y, Han R D 2003 Phys. Rev. A 67 042316

    [10]

    Karski M, Frster L, Choi J M, Steffen A, Alt W, Meschede D, Widera A 2009 Science 325(5937) 174

    [11]

    Bouwmeester D, Marzoli I, Karman G P, Schleich W, Woerdman J P 1999 Phys. Rev. A 61 013410

    [12]

    Xue P, Qin H, Tang B, Zhan X, Bian Z H, Li J 2014 Chin. Phys. B 23 110307

    [13]

    Schreiber A, Gbris A, Rohde P P, Laiho K, tefaňk M, Potoček V, Hamilton C, Jex I, Silberhorn C 2012 Science 336 55

    [14]

    Poulios K, Keil R, Fry D, Meinecke J D A, Matthews J C F, Politi A, Lobino M, Grfe M, Heinrich M, Nolte S, Szameit A, O'Brien J L 2013 Phys. Rev. Lett. 112(14) 143604

    [15]

    Farhi E, Gutmann S 1998 Phys. Rev. A 58 915

    [16]

    Aharonov Y, Davidovich L, Zagury N 1993 Phys. Rev. A 48 1687

    [17]

    Strauch F W 2006 Phys. Rev. A 74 030301(R)

    [18]

    Chandrashekar C M 2013 Sci. Rep. 3 2829

    [19]

    Yin Y, Katsanos D E, Evangelou S N 2008 Phys. Rev. A 77 022302

    [20]

    Trm P, Jex I, Schleich W P 2002 Phys. Rev. A 65 052110

    [21]

    Schreiber A, Cassemiro K N, Potoek V, Gbris A, Jex I, Silberhorn Ch 2011 Phys. Rev. Lett. 106 180403

    [22]

    Chou C I, Ho C L 2014 Chin. Phys. B 23 110302

    [23]

    Zhang R, Qin H, Tang B, Xue P 2013 Chin. Phys. B 22 110312

    [24]

    Li Z J, Izaac J A, Wang J B 2013 Phys. Rev. A 87 012314

    [25]

    Mohseni M, Rebentrost P, Lloyd S, Aspuru-Guzik A 2008 J. Chem. Phys. 129 174106

    [26]

    Marais A, Sinayskiy I, Kay A, Pentruccione F, Ekert A 2013 New J. Phys. 15 013038

    [27]

    Anderson P W 1958 Phys. Rev. 109 1492

    [28]

    Ribeiro P, Milman P, Mosseri R 2004 Phys. Rev. Lett. 93 190503

    [29]

    Keating J P, Linden N, Matthews J C F, Winter A 2007 Phys. Rev. A 76 012315

    [30]

    Joye A, Merkli M 2010 J. Stat. Phys. 140 1025

    [31]

    Ahlbrecht A, Alberti A, Meschede D, Scholz V B, Werner A H, Werner R F 2012 New J. Phys. 14 073050

    [32]

    Kitagawa T, Rudner M S, Berg E, Demler E 2010 Phys. Rev. A 82 033429

    [33]

    Rakovszky T, Asboth J K 2015 Phys. Rev. A 92 052311

    [34]

    Asbth J K, Obuse H 2013 Phys. Rev. B 88 121406(R)

    [35]

    Wjcik A, Łuczak T, Kurzyński P, Grudka A, Gdala T, Bednarska-Bzdęga M 2012 Phys. Rev. A 85 012329

    [36]

    Izaac J A, Wang J B, Li Z J 2013 Phys. Rev. A 88 042334

    [37]

    Zhang R, Xue P, Twamley J 2014 Phys. Rev. A 89 042317

    [38]

    Li Z J, Wang J B 2015 Sci. Rep. 5 13585

    [39]

    Lam H T, Szeto K Y 2015 Phys. Rev. A 92 012323

    [40]

    Li Z J, Wang J B 2015 J. Phys. A: Math. Theor. 48 355301

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Publishing process
  • Received Date:  08 November 2015
  • Accepted Date:  27 December 2015
  • Published Online:  20 March 2016

Resonance transmission of one-dimensional quantum walk with phase defects

    Corresponding author: Li Zhi-Jian, zjli@sxu.edu.cn
  • 1. Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China
Fund Project:  Project supported by the National Natural Science Foundation of China (Grant Nos. 10974124, 11274208) and the Shanxi Scholarship Council of China (2015-012).

Abstract: In this paper, the resonance transmission of discrete time quantum walk is studied when it walks on one-dimensional lattice in which two-phase defects or a piece of phase defects exists. The quasi energy of discrete time quantum walk has a unique dispersion relation with the momentum, from which we first discuss the wave velocity direction versus the values of momentum, and distinguish the incident wave and the reflected wave. The gap between two energy bands depends on the parameters of coincident operator, so the phase defects, which break down the translation invariance of quantum walk on uniform lattices, can be regarded as an analogue of quantum potential. Then we use the condition of energy conversion at the boundary points to obtain the transmission rate and discuss its variation with the incident momentum for different strengths and widths of defects in detail. The multiple resonant peaks are observed due to the enhanced interference effect. Different resonant behaviors are shown when the strength of defect is less or greater than /2, correspondingly the resonances occur in a wide region of incident momentum or the sharp resonant peaks appear at discrete values of momentum. Under the condition of strong defect strength, i.e., approaching to , the qualitative relation between the number of resonant peaks and the widths of defect region is given. The number of resonant peaks is 2(N-1) when the two phase defects are located at N sites symmetric about the origin, while the number is 2N when a piece of phase defects is located at -N to N sites. In the case of a piece of phase defects, we also present the phase diagram in parameter space of (k, ) to show the discrete time of quantum walk propagating or tunneling through the defect region. In terms of this phase diagram, the variations of transmission rate with the incident momentum are reasonably explained. One special phenomenon is that the quantum walk is almost totally reflected in the tunneling case except for =/2 and k being slightly off -/2. Moreover, this behavior seems little affecting the defect strength, just similar to a classical particle. As a result of this research, we hope to deepen the insight of the quantum walk and provide methods to control the spreading of quantum walk through artificial defects.

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