Abstract
Quantum recurrence theorem holds for quantum systems with discrete energy eigenvalues and fails to hold in general for systems with continuous energy. We show that during quantum walk process dominated by interference of amplitude corresponding to different paths fail to satisfy the complete quantum recurrence theorem. Due to the revival of the fractional wave packet, a fractional recurrence characterized using quantum Pólya number can be seen.
[1] S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943) http://dx.doi.org/10.1103/RevModPhys.15.110.1103/RevModPhys.15.1Search in Google Scholar
[2] Luis Barreira, Poincaré recurrence: old and new, IVth International Congress on Mathematical Physics (World Scientific, 2006) 415 10.1142/9789812704016_0039Search in Google Scholar
[3] U. Krengel, Ergodic Theorems (Walter de Gruyter, Berlin, New York, 1985) 10.1515/9783110844641Search in Google Scholar
[4] P. Bocchieri, A. Loinger, Phys. Rev. 107, 337 (1957) http://dx.doi.org/10.1103/PhysRev.107.33710.1103/PhysRev.107.337Search in Google Scholar
[5] L. S. Schulman, Phys. Rev. A 18, 2379 (1978) http://dx.doi.org/10.1103/PhysRevA.18.237910.1103/PhysRevA.18.2379Search in Google Scholar
[6] R. W. Robinett, Phys. Rep. 392, 1 (2004) http://dx.doi.org/10.1016/j.physrep.2003.11.00210.1016/j.physrep.2003.11.002Search in Google Scholar
[7] M. V. Berry, J. Phys. A 29, 6617 (1996) http://dx.doi.org/10.1088/0305-4470/29/20/01610.1088/0305-4470/29/20/016Search in Google Scholar
[8] F. Grobmann, J. M. Rost, W. P. Schleich, J. Phys. A 30, L277 (1997) http://dx.doi.org/10.1088/0305-4470/30/9/00410.1088/0305-4470/30/9/004Search in Google Scholar
[9] M. Berry, I. Marzoli, W. Schleich, Phys. World 14, 39 (2001) 10.1088/2058-7058/14/6/30Search in Google Scholar
[10] R. Iwanow, D. A. May-Arrioja, D. N. Christodoulides, G. I. Stegeman, Phys. Rev. Lett. 95, 053902 (2005) http://dx.doi.org/10.1103/PhysRevLett.95.05390210.1103/PhysRevLett.95.053902Search in Google Scholar PubMed
[11] A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, J. Watrous, Proceeding of the 33rd ACM Symposium on Theory of Computing (ACM Press, New York, 2001) 60 Search in Google Scholar
[12] M. Stefanak, I. Jex, T. Kiss, Phys. Rev. Lett. 100, 020501 (2008) http://dx.doi.org/10.1103/PhysRevLett.100.02050110.1103/PhysRevLett.100.020501Search in Google Scholar PubMed
[13] M. Stefanak, T. Kiss, I. Jex, New J. Phys. 11, 043027 (2009) http://dx.doi.org/10.1088/1367-2630/11/4/04302710.1088/1367-2630/11/4/043027Search in Google Scholar
[14] Norio Konno, arXiv:0908.2213v3 Search in Google Scholar
[15] O. Mülken, A. Blumen, Phys. Rev. E 71, 036128 (2005) http://dx.doi.org/10.1103/PhysRevE.71.03612810.1103/PhysRevE.71.036128Search in Google Scholar PubMed
[16] O. Mülken, A. Blumen, Phys. Rev. E 73, 066117 (2006) http://dx.doi.org/10.1103/PhysRevE.73.06611710.1103/PhysRevE.73.066117Search in Google Scholar PubMed
[17] G. V. Riazanov, Sov. Phys. JETP-USSR 6, 1107 (1958) Search in Google Scholar
[18] R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965) Search in Google Scholar
[19] Y. Aharonov, L. Davidovich, N. Zagury, Phys. Rev. A 48, 1687 (1993) http://dx.doi.org/10.1103/PhysRevA.48.168710.1103/PhysRevA.48.1687Search in Google Scholar
[20] A. M. Childs et al., Proceedings of the 35th ACM Symposium on Theory of Computing (ACM Press, New York, 2003) 59 Search in Google Scholar
[21] N. Shenvi, J. Kempe, K. B. Whaley, Phys. Rev. A 67, 052307 (2003) http://dx.doi.org/10.1103/PhysRevA.67.05230710.1103/PhysRevA.67.052307Search in Google Scholar
[22] A. M. Childs, J. Goldstone, Phys. Rev. A 70, 022314 (2004) http://dx.doi.org/10.1103/PhysRevA.70.02231410.1103/PhysRevA.70.022314Search in Google Scholar
[23] A. Ambainis, J. Kempe, A. Rivosh, Proceedings of ACM-SIAM Symp. on Discrete Algorithms (SODA) (AMC Press, New York, 2005) 1099 Search in Google Scholar
[24] C. M. Chandrashekar, R. Laflamme, Phys. Rev. A 78, 022314 (2008) http://dx.doi.org/10.1103/PhysRevA.78.02231410.1103/PhysRevA.78.022314Search in Google Scholar
[25] T. Oka, N. Konno, R. Arita, H. Aoki, Phys. Rev. Lett. 94, 100602 (2005) http://dx.doi.org/10.1103/PhysRevLett.94.10060210.1103/PhysRevLett.94.100602Search in Google Scholar PubMed
[26] G. S. Engel et al., Nature 446, 782 (2007) http://dx.doi.org/10.1038/nature0567810.1038/nature05678Search in Google Scholar PubMed
[27] M. Mohseni, P. Rebentrost, S. Lloyd, A. Aspuru-Guzik, J. Chem. Phys. 129, 174106 (2008) http://dx.doi.org/10.1063/1.300233510.1063/1.3002335Search in Google Scholar PubMed
[28] P. Révész. Random walk in Random and non-rendom Environments (World Scientific, Singapore, 1990) 10.1142/1107Search in Google Scholar
[29] G. Pólya, Math. Ann. 84, 149 (1921) http://dx.doi.org/10.1007/BF0145870110.1007/BF01458701Search in Google Scholar
[30] D. A. Meyer, J. Stat. Phys. 85, 551 (1996) http://dx.doi.org/10.1007/BF0219935610.1007/BF02199356Search in Google Scholar
[31] A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, J. Watrous, Proceeding of the 33rd ACM Symposium on Theory of Computing (ACM Press, New York, 2001) 60 Search in Google Scholar
[32] E. Farhi, S. Gutmann, Phys. Rev. A 58, 915 (1998) http://dx.doi.org/10.1103/PhysRevA.58.91510.1103/PhysRevA.58.915Search in Google Scholar
[33] F. Strauch, Phys. Rev. A 74, 030310 (2006) http://dx.doi.org/10.1103/PhysRevA.74.03030110.1103/PhysRevA.74.030301Search in Google Scholar
[34] C. M. Chandrashekar, Phys. Rev. A 78, 052309 (2008) http://dx.doi.org/10.1103/PhysRevA.78.05230910.1103/PhysRevA.78.052309Search in Google Scholar
[35] C. M. Chandrashekar, R. Srikanth, R. Laflamme, Phys. Rev. A 77 032326 (2008) http://dx.doi.org/10.1103/PhysRevA.77.03232610.1103/PhysRevA.77.032326Search in Google Scholar
[36] M. Stefanak, I. Jex, T. Kiss, Phys. Rev. A 78, 032306 (2008) http://dx.doi.org/10.1103/PhysRevA.78.03230610.1103/PhysRevA.78.032306Search in Google Scholar
[37] D. Aharonov, A. Ambainis, J. Kempe, U. Vazirani, Proceeding of the 33rd ACM Symposium on Theory of Computing (ACM Press, New York, 2001) 50 Search in Google Scholar
[38] B. Tregenna, W. Flanagan, R. Maile, V. Kendon, New J. Phys. 5, 83 (2003) http://dx.doi.org/10.1088/1367-2630/5/1/38310.1088/1367-2630/5/1/383Search in Google Scholar
[39] N. Inui, Y. Konishi, N. Konno, Phys. Rev. A 69, 052323 (2004) http://dx.doi.org/10.1103/PhysRevA.69.05232310.1103/PhysRevA.69.052323Search in Google Scholar
[40] N. Inui, N. Konno, E. Segawa, Phys. Rev. E 72, 056112 (2005) http://dx.doi.org/10.1103/PhysRevE.72.05611210.1103/PhysRevE.72.056112Search in Google Scholar PubMed
© 2010 Versita Warsaw
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.
