Abstract
The term “quantum carpet” can be observed in many closed quantum systems, where the evolution of a wave function exhibits a carpet-like pattern. Quantum carpet mechanisms are also akin to the classical interference patterns of light. Although the origins of quantum carpets have previously been studied by various researchers, many interesting details are still worth exploring. In this study, we present a unified framework for simultaneously analyzing three different features of quantum carpets: full revival, fractional revival, and diagonal canal. For the fractional revival feature, a complete formula is presented to explain its formation through Gaussian sum theory, in which all essential features, including phases and amplitudes, are captured analytically. We also reveal important relationships between the interference terms of diagonal canals and their geometric interpretations such that a better understanding of the development of diagonal canals can be supported.
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G. Bonneau, J. Faraut, and G. Valent, Am. J. Phys. 69, 322 (2001).
R. Robinett, Phys. Rep. 392, 1 (2004).
O. Fojón, M. Gadella, and L. P. Lara, Comput. Math. Appl. 59, 964 (2010).
M. Waegell, Y. Aharonov, and T. Patti, Entropy 18, 149 (2016), arXiv: 1604.05385.
G. Bastard, Superlattices Microstruct. 1, 265 (1985).
A. S. Polkovnikov, and G. G. Zegrya, Phys. Rev. B 58, 4039 (1998).
L. V. Kotova, V. N. Kats, A. V. Platonov, V. P. Kochereshko, R. André, and L. E. Golub, Phys. Rev. B 97, 125302 (2018), arXiv: 1711.10334.
L. I. Schif, Quantum Mechanics (McGraw-Hill Book Company, New York, 1968).
D. J. Griffiths, Introduction to Quantum Mechanics (Cambridge University Press, Cambridge, 2005).
K. Cai, R. X. Wang, Z. Q. Yin, and G. L. Long, Sci. China-Phys. Mech. Astron. 60, 070311 (2017), arXiv: 1610.09922.
H. Y. Yuan, M. H. Yung, and X. R. Wang, Phys. Rev. B 98, 060407 (2018), arXiv: 1801.09017.
H. Y. Yuan, and M. H. Yung, Phys. Rev. B 97, 060405 (2018), arXiv: 1711.04394.
K. Mills, M. Spanner, and I. Tamblyn, Phys. Rev. A 96, 042113 (2017).
Y. B. Sheng, and L. Zhou, Sci. Bull. 62, 1025 (2017).
X. Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, Phys. Rev. Lett. 105, 123003 (2010), arXiv: 1003.2515.
A. Campo, and M. G. Boshier, Sci. Rep. 2, 648 (2012), arXiv: 1201.6627.
X. K. Song, H. Zhang, Q. Ai, J. Qiu, and F. G. Deng, New J. Phys. 18, 023001 (2016), arXiv: 1509.00097.
X. K. Song, Q. Ai, J. Qiu, and F. G. Deng, Phys. Rev. A 93, 052324 (2016), arXiv: 1602.00050.
S. Iqbal, and F. Saif, J. Russ. Laser Res. 29, 466 (2008).
Y. B. Sheng, and L. Zhou, Sci. Rep. 5, 7815 (2015).
Y. B. Sheng, and L. Zhou, Phys. Rev. A 98, 052343 (2018).
M. Belloni, and R. W. Robinett, Phys. Rep. 540, 25 (2014).
C.-L. Lin, arXiv: 1604.04680.
C.-L. Lin, arXiv: 1705.05517.
A. E. Kaplan, I. Marzoli, W. E. Lamb, and W. P. Schleich, Phys. Rev. A 61, 032101 (2000).
M. Berry, I. Marzoli, and W. Schleich, Phys. World 14, 39 (2001).
M. Nest, Phys. Rev. A 73, 023613 (2006).
K. Hornberger, S. Gerlich, P. Haslinger, S. Nimmrichter, and M. Arndt, Rev. Mod. Phys. 84, 157 (2012), arXiv: 1109.5937.
M. H. Muñoz-Arias, J. Madroñero, and C. A. Parra-Murillo, Phys. Rev. A 93, 043603 (2016), arXiv: 1512.06716.
M. R. Barros, A. Ketterer, O. J. Farías, and S. P. Walborn, Phys. Rev. A 95, 042311 (2017), arXiv: 1702.07391.
I. Yousaf, and S. Iqbal, J. Russ. Laser Res. 37, 328 (2016).
P. Kazemi, S. Chaturvedi, I. Marzoli, R. F. O’Connell, and W. P. Schleich, New J. Phys. 15, 013052 (2013).
T. García, N. A. Cordero, and E. Romera, Phys. Rev. B 89, 075416 (2014).
L. Banchi, E. Compagno, and S. Bose, Phys. Rev. A 91, 052323 (2015), arXiv: 1502.03061.
M. Rohith, and C. Sudheesh, Phys. Rev. A 92, 053828 (2015), arXiv: 1507.03724.
M. Krizanac, D. Altwein, E. Y. Vedmedenko, and R. Wiesendanger, New J. Phys. 18, 033029 (2016), arXiv: 1601.00803.
J. A. Yeazell, M. Mallalieu, and C. R. Stroud, Phys. Rev. Lett. 64, 2007 (1990).
M. Greiner, O. Mandel, T. W. Hänsch, and I. Bloch, Nature 419, 51 (2002).
G. Rempe, H. Walther, and N. Klein, Phys. Rev. Lett. 58, 353 (1987).
G. Della Valle, M. Savoini, M. Ornigotti, P. Laporta, V. Foglietti, M. Finazzi, L. Du`o, and S. Longhi, Phys. Rev. Lett. 102, 180402 (2009).
D. L. Aronstein, and C. R. Stroud, Phys. Rev. A 55, 4526 (1997).
K. R. Naqvi, S. Waldenstrøm, and T. H. Hassan, Eur. J. Phys. 22, 395 (2001).
M. Spanner, E. A. Shapiro, and M. Ivanov, Phys. Rev. Lett. 92, 093001 (2004).
P. A. Bernard, A. Chan, Loranger, C. Tamon, and L. Vinet, Phys. Lett. A 382, 259 (2018), arXiv: 1710.02705.
S. Dooley, and T. P. Spiller, Phys. Rev. A 90, 012320 (2014), arXiv: 1404.4296.
M. Rohith, and C. Sudheesh, J. Phys. B-At. Mol. Opt. Phys. 47, 045504 (2014), arXiv: 1309.0104.
J. M. Lemay, L. Vinet, and A. Zhedanov, J. Phys. A-Math. Theor. 49, 335302 (2016), arXiv: 1509.08965.
M. Christandl, L. Vinet, and A. Zhedanov, Phys. Rev. A 96, 032335 (2017), arXiv: 1607.02639.
P. Stifter, C. Leichtie, W. P. Schleich, and J. Marklof, Z. Naturforschung A 52, 377 (1997).
O. M. Friesch, I. Marzoli, and W. P. Schleich, New J. Phys. 2, 4 (2000).
K. Lin, P. Lu, J. Ma, X. Gong, Q. Song, Q. Ji, W. Zhang, H. Zeng, J. Wu, G. Karras, G. Siour, J. M. Hartmann, O. Faucher, E. Gershnabel, Y. Prior, and I. S. Averbukh, Phys. Rev. X 6, 041056 (2016), arXiv: 1606.08200.
T. M. Apostol, Introduction to Analytic Number Theory (Springer, New York, 1976).
D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg, Phys. Rev. Lett. 100, 170506 (2008), arXiv: 0707.0741.
A. Deitmar, and S. Echterhoff, Principles of Harmonic Analysis (Springer, New York, 2009).
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Xiong, H., Song, X., Yuan, H. et al. Ordered space-time structures: Quantum carpets from Gaussian sum theory. Sci. China Phys. Mech. Astron. 62, 970313 (2019). https://doi.org/10.1007/s11433-018-9339-0
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DOI: https://doi.org/10.1007/s11433-018-9339-0