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Quantum walk (QW), the quantum mechanical counterpart of classical random walk, has recently been studied in various fields. The evolution of the discrete time quantum walk can be described as follows: the walker changes its spin state by the coin operator C, then takes one step left or right according to its spin state. For homogeneous quantum walk, the coin operator is independent of time and the standard deviation of the position grows linearly in time. It is quadratically faster than that in the classical random walk. In this work, we numerically study the dynamical behaviors of spreading in a one-dimensional discrete time quasiperiodic quantum walk (DTQQW). The DTQQW is that the coin operator is dependent on time and takes two different coins C() and C() arranged in generalized Fibonacci (GF) sequences. The GF sequences are constructed from A by the recursion relation: AAmBn, BA, for m, n are positive integers. They can be classified into two classes according to the wandering exponent . For 0, they belong to the first class, and for 0, they belong to the second class. For one dimensional system, the behaviors of two classes of GF systems are different either for the electronic spectrum of an electron in quasiperiodic potentials or for the quantum phase transitions of the quasiperiodic spin chains. In this paper, we discuss the cases of two different C operators (C();C()) arranged in GF sequences and find that the spreading behaviors are superdiffusion (the standard deviation of the position ~t; 0:5 1) for the two classes of GF DTQQW. For the second class of GF DTQQW, the exponent values are larger than those of the first class of GF DTQQW in the case of two identical C operators. By exploring the probability distribution in the real space, we find that for the first class of GF DTQQW, the probability distributions are almost the same for different initial states and are similar to the classical Gaussian distribution. For the probability distributions of the second class of GF DTQQW, there are two peaks at the two edges and the height of the two peaks can be different for different initial states. They are similar to the ballistic distribution of the homogeneous quantum walk. Therefore, we conclude that for the first class of GF DTQQW, the spreading behaviors are close to those of the classical random walk ( = 0:5) while for the second class of GF DTQQW, they are close to those of the homogeneous quantum walk ( = 1). This result is quite different from the characteristics of the quantum phase transitions in two classes of GF quasiperiodic quantum spin chains.
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Keywords:
- quantum walk /
- time quasiperiodic /
- superdiffusion
[1] Aharonov Y, Davidovich L, Zagury N 1993 Phys. Rev. A 48 2
[2] Kempe J 2003 Contemp. Phys. 44 307
[3] Childs A M, Farhi V, Gutmann S 2002 Quantum Inf. Process. 1 35
[4] Venegas-Andraca S E 2012 Quantum Inf. Process. 11 1015
[5] Shenvi N, Kempe J, Whaley K B 2003 Phys. Rev. A 67 052307
[6] Oka T, Konno N, Arita R, Aoki H 2005 Phys. Rev. Lett. 94 100602
[7] Chandrashekar C M, Laflamme R 2008 Phys. Rev. A 78 022314
[8] Kitagawa T, Rudner M S, Berg E, Demler E 2010 Phys. Rev. A 82 033429
[9] Zahringer F, Kirchmair G, Gerritsma R, Solano E, Blatt R, Roos C F 2010 Phys. Rev. Lett. 104 100503
[10] Schmitz H, Matjeschk R, Schneider C, Glueckert J, Enderlein M, Huber T, Schaetz T 2009 Phys. Rev. Lett. 103 090504
[11] Dur W, Raussendorf R, Kendon V M, Briegel H J 2002 Phys. Rev. A 66 052319
[12] Ma Z Y, Burnett K, d’Arcy M B, Gardiner S A 2006 Phys. Rev. A 73 013401
[13] Travaglione B C, Milburn G J 2002 Phys. Rev. A 65 032310
[14] 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
[15] Chandrashekar C M 2006 Phys. Rev. A 74 032307
[16] Ryan C A, Laforest M, Boileau J C, Laflamme R 2005 Phys. Rev. A 72 062317
[17] Eckert K, Mompart J, Birkl G, Lewenstein M 2005 Phys. Rev. A 72 012327
[18] Manouchehri K, Wang J B 2008 J. Phys. A 41 065304
[19] Xue P, Sanders C B, Leibfried D 2009 Phys. Rev. Lett. 103 183602
[20] Farhi E, Gutmann S 1998 Phys. Rev. A 58 915
[21] Nayak A, Vishwanath A 2000 arXiv: 0010117v1 [quant-ph]
[22] Bednarska M, Grudka A, Kurzynski P, Luczak T, Wojcik A 2003 Phys. Lett. A 317 21
[23] Xu X P 2010 Eur. Phys. J. B 77 479
[24] Marquezino F L, Portugal R, Abal G, Donangelo R 2008 Phys. Rev. A 77 042312
[25] Ribeiro P, Milman P, Mosseri R 2004 Phys. Rev. Lett. 93 190503
[26] Zhang R, Xu Y Q, Xue P 2015 Chin. Phys. B 24 010303
[27] Zhang R, Qin H, Tang B, Xue P 2013 Chin. Phys. B 22 110312
[28] Li M, Zhang Y S, Guo G C 2013 Chin. Phys. Lett. 2 020304
[29] Xu X P, Liu F 2008 Phys. Rev. A 77 062318
[30] Zhao J, Hu Y Y, Tong P Q 2015 Chin. Phys. Lett. 32 060501
[31] Shechtman D, Blech I, Gratias D, Cahn J W 1984 Phys. Rev. Lett. 53 1951
[32] Levine D, Steinhardt P J 1984 Phys. Rev. Lett. 53 2477
[33] Merlin R, Bajema K, Clarke R, Juang F Y, Bhattacharya P K 1985 Phys. Rev. Lett. 55 1768
[34] Chakrabarti A, Karmakar S N 1991 Phys. Rev. B 44 896
[35] Kolár M, Ali M K, Nori F 1991 Phys. Rev. B 43 1034
[36] Dulea M, Severin M, Riklund R 1990 Phys. Rev. B 42 3680
[37] Oh G Y, Lee M H 1993 Phys. Rev. B 48 12465
[38] Larcher M, Laptyeva T V, Bodyfelt J D, Dalfovo F, Modugno M, Flach S 2012 New J. Phys. 14 103036
[39] Zhang Z J, Tong P Q, Gong J B, Li B W 2012 Phys. Rev. Lett. 108 070603
[40] Zhang Z J, Li W J, Zhu X, Xiong Y, Tong P Q 2015 Acta Phys. Sin. 64 190501 (in Chinese) [张振俊, 李文娟, 朱璇, 熊烨, 童培庆 2015 64 190501]
[41] Luck J M 1993 J. Stat. Phys. 72 417
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[1] Aharonov Y, Davidovich L, Zagury N 1993 Phys. Rev. A 48 2
[2] Kempe J 2003 Contemp. Phys. 44 307
[3] Childs A M, Farhi V, Gutmann S 2002 Quantum Inf. Process. 1 35
[4] Venegas-Andraca S E 2012 Quantum Inf. Process. 11 1015
[5] Shenvi N, Kempe J, Whaley K B 2003 Phys. Rev. A 67 052307
[6] Oka T, Konno N, Arita R, Aoki H 2005 Phys. Rev. Lett. 94 100602
[7] Chandrashekar C M, Laflamme R 2008 Phys. Rev. A 78 022314
[8] Kitagawa T, Rudner M S, Berg E, Demler E 2010 Phys. Rev. A 82 033429
[9] Zahringer F, Kirchmair G, Gerritsma R, Solano E, Blatt R, Roos C F 2010 Phys. Rev. Lett. 104 100503
[10] Schmitz H, Matjeschk R, Schneider C, Glueckert J, Enderlein M, Huber T, Schaetz T 2009 Phys. Rev. Lett. 103 090504
[11] Dur W, Raussendorf R, Kendon V M, Briegel H J 2002 Phys. Rev. A 66 052319
[12] Ma Z Y, Burnett K, d’Arcy M B, Gardiner S A 2006 Phys. Rev. A 73 013401
[13] Travaglione B C, Milburn G J 2002 Phys. Rev. A 65 032310
[14] 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
[15] Chandrashekar C M 2006 Phys. Rev. A 74 032307
[16] Ryan C A, Laforest M, Boileau J C, Laflamme R 2005 Phys. Rev. A 72 062317
[17] Eckert K, Mompart J, Birkl G, Lewenstein M 2005 Phys. Rev. A 72 012327
[18] Manouchehri K, Wang J B 2008 J. Phys. A 41 065304
[19] Xue P, Sanders C B, Leibfried D 2009 Phys. Rev. Lett. 103 183602
[20] Farhi E, Gutmann S 1998 Phys. Rev. A 58 915
[21] Nayak A, Vishwanath A 2000 arXiv: 0010117v1 [quant-ph]
[22] Bednarska M, Grudka A, Kurzynski P, Luczak T, Wojcik A 2003 Phys. Lett. A 317 21
[23] Xu X P 2010 Eur. Phys. J. B 77 479
[24] Marquezino F L, Portugal R, Abal G, Donangelo R 2008 Phys. Rev. A 77 042312
[25] Ribeiro P, Milman P, Mosseri R 2004 Phys. Rev. Lett. 93 190503
[26] Zhang R, Xu Y Q, Xue P 2015 Chin. Phys. B 24 010303
[27] Zhang R, Qin H, Tang B, Xue P 2013 Chin. Phys. B 22 110312
[28] Li M, Zhang Y S, Guo G C 2013 Chin. Phys. Lett. 2 020304
[29] Xu X P, Liu F 2008 Phys. Rev. A 77 062318
[30] Zhao J, Hu Y Y, Tong P Q 2015 Chin. Phys. Lett. 32 060501
[31] Shechtman D, Blech I, Gratias D, Cahn J W 1984 Phys. Rev. Lett. 53 1951
[32] Levine D, Steinhardt P J 1984 Phys. Rev. Lett. 53 2477
[33] Merlin R, Bajema K, Clarke R, Juang F Y, Bhattacharya P K 1985 Phys. Rev. Lett. 55 1768
[34] Chakrabarti A, Karmakar S N 1991 Phys. Rev. B 44 896
[35] Kolár M, Ali M K, Nori F 1991 Phys. Rev. B 43 1034
[36] Dulea M, Severin M, Riklund R 1990 Phys. Rev. B 42 3680
[37] Oh G Y, Lee M H 1993 Phys. Rev. B 48 12465
[38] Larcher M, Laptyeva T V, Bodyfelt J D, Dalfovo F, Modugno M, Flach S 2012 New J. Phys. 14 103036
[39] Zhang Z J, Tong P Q, Gong J B, Li B W 2012 Phys. Rev. Lett. 108 070603
[40] Zhang Z J, Li W J, Zhu X, Xiong Y, Tong P Q 2015 Acta Phys. Sin. 64 190501 (in Chinese) [张振俊, 李文娟, 朱璇, 熊烨, 童培庆 2015 64 190501]
[41] Luck J M 1993 J. Stat. Phys. 72 417
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