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The quantum reflection and interference of Bose-Einstein condensates (BECs) encountering a potential barrier or well is one of the most efficient ways of studying the exotic properties of macroscopic matter waves. As a matter of fact, one can reveal the quantum nature, coherence properties, and many-body effects as well as the potential applications of ultracold atomic gases by virtue of the quantum reflection and interference of BECs. Although there have been extensive investigations regarding the quantum reflection and interference of single-component BECs, so far there have been very few studies regarding those of multi-component BECs. In this work, we investigate the quantum reflections and interferences of spin-dependent BECs in semi-infinite potential wells by using the propagation method and the time-of-flight imaging scheme which is widely used in cold atom experiments. We obtain the exact analytical solutions of the spin-dependent condensate wave functions in the semi-infinite potential wells. It is shown that once the spin-dependent optical lattice is switched off the spin-dependent matter wave packets delocalized in different lattice sites interfere with each other during the free expansion. Consequently, the interference fringes with high contrast are formed. At the same time, the expanded spin-dependent matter waves encounter the hard wall of the semi-infinite potential well, which leads to a quantum reflection. There is a double interference between the reflected wave and the freely expanded incident wave, which is characterized by the significant modulation effect in the interference patterns. Concretely, there exist intense density oscillations in several symmetric and local regions of the interference fringes. Essentially, the double interference is a self-interference of BECs, and it results from the interference between the spin-dependent BEC and the BEC image, where the hard wall severs as a mirror plane. Therefore it is similar to Young's double-slit interference in wave optics, and a standing wave node is formed at the trap wall. In particular, the positions and the intervals of the local density oscillations in the interference patterns are determined by evolution time, laser wavelength and laser intensity, which is verified in the numerical simulations and calculations. In addition, the effects of spin state, transport distance, and relative phase on the interference fringes are analyzed and discussed. The present investigation is helpful in understanding the macroscopic quantum properties of the spin-dependent BECs, and provides a new scheme to test the theoretical model and physical mechanism of the condensate interference in a spin-dependent optical lattice.
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Keywords:
- spin-dependent Bose-Einstein condensates /
- semi-infinite potential well /
- quantum reflection /
- quantum interference
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[1] Andrews M R, Townsend C G, Miesner H J, Durfee D S, Kurn D M, Ketterle W 1997 Science 275 637
[2] Pasquini T A, Shin Y, Sanner C, Saba M, Schirotzek A, Pritchard D E, Ketterle W 2004 Phys. Rev. Lett. 93 223201
[3] Pasquini T A, Saba M, Jo G, Shin Y, Ketterle W, Pritchard D E, Savas T A, Mulders N 2006 Phys. Rev. Lett. 97 093201
[4] Greiner M, Mandel O, Esslinger T, Hänsch T W, Bloch I 2002 Nature 415 39
[5] Hofferberth S, Lesanovsky I, Schumm T, Imambekov A, Gritsev V, Demler E, Schmiedmayer J 2008 Nat. Phys. 4 489
[6] Fang B, Johnson A, Roscilde T, Bouchoule I 2016 Phys. Rev. Lett. 116 050402
[7] Chang R, Bouton Q, Cayla H, Qu C, Aspect A, Westbrook C I, Clement D 2016 Phys. Rev. Lett. 117 235303
[8] Castellanos E, Rivas J I 2015 Phys. Rev. D 91 084019
[9] Wen L H, Wang J S, Feng J, Hu H Q 2008 J. Phys. B 41 135301
[10] Scott R G, Martin A M, Fromhold T M, Sheard F W 2005 Phys. Rev. Lett. 95 073201
[11] Marchant A L, Billam T P, Yu M M H, Rakonjac A, Helm J L, Polo J, Weiss C, Gardiner S A, Cornish S L 2016 Phys. Rev. A 93 021604
[12] Berrada T, van Frank S, Bucker R, Schumm T, Schaff J F, Schmiedmayer J, Julia-Diaz B, Polls A 2016 Phys. Rev. A 93 063620
[13] Fouda M F, Fang R, Ketterson J B, Shahriar M S 2016 Phys. Rev. A 94 063644
[14] Mandel O, Greiner M, Widera A, Rom T, Hänsch T W, Bloch I 2003 Phys. Rev. Lett. 91 010407
[15] Castin Y, Dalibard J 1997 Phys. Rev. A 55 4330
[16] Yang X X, Wu Y 1999 Phys. Lett. A 253 219
[17] Liu W M, Wu B, Niu Q 2000 Phys. Rev. Lett. 84 2294
[18] Xiong H, Liu S, Huang G, Xu Z 2002 J. Phys. B 35 4863
[19] Liu S, Xiong H, Xu Z, Huang G 2003 J. Phys. B 36 2083
[20] Xiong H, Liu S, Zhan M 2006 New J. Phys. 8 245
[21] Bach R, Rzazewski K 2004 Phys. Rev. Lett. 92 200401
[22] Liu S, Xiong H 2007 New J. Phys. 9 412
[23] Hadzibabic Z, Stock S, Battelier B, Bretin V, Dalibard J 2004 Phys. Rev. Lett. 93 180403
[24] Ashhab S 2005 Phys. Rev. A 71 063602
[25] Wen L H, Liu M, Xiong H W, Zhan M S 2005 Eur. Phys. J. D 36 89
[26] Wen L H, Liu M, Kong L B, Chen A X, Zhan M S 2005 Chin. Phys. 14 690
[27] Wen L H, Liu M, Kong L B, Zhan M S 2005 Chin. Phys. Lett. 22 812
[28] Yue X, Liu S, Wu B, Xiong H 2017 Chin. Phys. B 26 050501
[29] Wen L H, Xiong H W, Wu B 2010 Phys. Rev. A 82 053627
[30] Wen L H, Li J H 2014 Phys. Rev. A 90 053621
[31] Feynman R P, Hibbs A R 1965 Quantum Mechanics and Path Integrals (New York:McGraw-Hill Inc.) pp26-74
[32] Akhundova E A, Dodonov V V, Man'ko V I 1985 J. Phys. A 18 467
[33] Pedri P, Pitaevskii L, Stringari S, Fort C, Burger S, Cataliotti F S, Maddaloni P, Minardi F, Inguscio M 2001 Phys. Rev. Lett. 87 220401
[34] Robinett W 2006 Phys. Scr. 73 681
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