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Chin. Phys. B, 2024, Vol. 33(3): 030306    DOI: 10.1088/1674-1056/acf2fe
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Does the Hartman effect exist in triangular barriers

Qing-Ling Li(李青玲)1, Shuang Zheng(郑爽)1, and Zhi Xiao(肖智)1,2,†
1 Department of Mathematics and Physics, North China Electric Power University, Beijing 102206, China;
2 Hebei Key Laboratory of Physics and Energy Technology, North China Electric Power University, Baoding 071000, Chin
Abstract  We study the phase, Larmor and dwell times of a particle scattered off triangular barriers (TBs). It is interesting that the dependences of dwell, reflective phase and Larmor times on the wave number, barrier width and height for a pair of mirror-symmetric (MS) exact triangular barriers (ETBs) are quite different, as the two ETBs have quite distinct scattering surfaces. In comparison, the dependence of the transmitted phase or Larmor times is exactly the same, since the transmitted amplitudes are the same for a pair of MS TBs. We further study the Hartman effect by defining the phase and Larmor velocities associated with the phase and Larmor times. We find no barrier width saturation effect for the transmitted and reflected times. This is indicated by the fact that all the velocities approach finite constants that are much smaller than the speed of light in vacuum for TBs with positive-slope impact faces. As for ETBs with vertical left edges, the naive velocities seem to also indicate the absence of the Hartman effect. These are quite distinct from rectangular barriers and may shed new light on the clarification of the tunneling time issues.
Keywords:  triangular barrier      Hartman effect  
Received:  21 May 2023      Revised:  02 August 2023      Accepted manuscript online:  23 August 2023
PACS:  03.65.Xp (Tunneling, traversal time, quantum Zeno dynamics)  
Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11974108, 11875127, and 12211530044) and the Fundamental Research Funds for the Central Universities (Grant No. 2020MS052).
Corresponding Authors:  Zhi Xiao     E-mail:  spacecraft@pku.edu.cn

Cite this article: 

Qing-Ling Li(李青玲), Shuang Zheng(郑爽), and Zhi Xiao(肖智) Does the Hartman effect exist in triangular barriers 2024 Chin. Phys. B 33 030306

[1] Balantekin A B and Takigawa N 1998 Rev. Mod. Phys. 77 70
[2] Golo V L and Volkov V L 2003 Rev. Mod. Phys. 131 14
[3] Britnell L, Gorbachev R V, Jalil R, Belle B D, Schedin F, Mishchenko A, Georgiou T, Katsnelson M I, Eaves L, Morozov S V, Peres N M R, Leist J, Geim A K, Novoselov K S and Ponomarenko L A 2012 Science 947 335
[4] Coleman S 1977 Phys. Rev. D 15 2929
[5] Gen U and Sasaki M 2000 Phys. Rev. D 61 103508
[6] Condon E U and Morse P M 1931 Rev. Mod. Phys. 3 43
[7] Maccoll L A 1932 Phys. Rev. 40 621
[8] Spierings D C and Steinberg A M 2021 Phys. Rev. Lett. 127 133001
[9] Ramos R, Spierings D, Racicot I and Steinberg A M 2020 Nature 583 529
[10] Rivlin T, Pollak E and Dumont R S 2021 New J. Phys. 23 063044
[11] Rivlin T, Pollak E and Dumont R S 2021 Phys. Rev. A 103 012225
[12] Jia L, Xing H and Fu L 2022 Phys. Rev. A 105 062804
[13] Suzuki F and Unruh W G 2023 Phys. Rev. A 107 042216
[14] Landsman A S and Keller U 2015 Phys. Rep. 547 1
[15] Sainadh U S, Xu H, Wang X, Atia-Tul-Noor A, Wallace W C, Douguet N, Bray A, Ivanov I, Bartschat K, Kheifets A, Sang R T and Litvinyuk I V 2019 Nature 568 75
[16] Eisenbud L 1948 Formal properties of nuclear collisions (Ph. D. Dissertation) (Princeton)
[17] Wigner E P 1955 Phys. Rev. 98 145
[18] Bohm D 1951 Quantum Theory (New York: Prentice-Hall)
[19] Smith F T 1960 Phys. Rev. 118 349
[20] Baz' A I 1967 Sov. J. Nucl. Phys. 5 161
[21] Baz' A I 1967 Sov. J. Nucl. Phys. 4 182
[22] Rybachenko V F 1967 Sov. J. Nucl. Phys. 5 635
[23] Yamada N 2004 Phys. Rev. Lett. 93 170401
[24] Bauer M 2017 Phys. Rev. A 96 022139
[25] Demir D and Güner T 2017 Ann. Phys. 386 291
[26] Han M, Ge P, Wang J, Guo Z, Fang Y, Ma X, Yu X, Deng Y, Wörner H J, Gong Q and Liu Y 2021 Nat. Photonics 15 765
[27] Pollak E and Miller W H 1984 Phys. Rev. Lett. 53 115
[28] Buttiker M and Landauer R 1982 Phys. Rev. Lett. 49 1739
[29] Aharonov Y, Albert D Z and Vaidman L 1988 Phys. Rev. Lett. 60 1351
[30] Aharonov Y and Vaidman L 1990 Phys. Rev. A 41 11
[31] Steinberg A M 1995 Phys. Rev. Lett. 74 2405
[32] Steinberg A M 1995 Phys. Rev. A 52 32
[33] Sokolovski D 2010 Phys. Rev. A 81 042115
[34] Hartman T E 1962 J. Appl. Phys. 33 3427
[35] Winful H G 2003 Phys. Rev. Lett. 90 023901
[36] Winful H G 2006 Phys. Rev. 436 1
[37] Landauer R and Martin Th 1994 Phys. Rev. 66 217
[38] Xiao Z, Huang H and Lu X X 2015 Int. J. Mod. Phys. B 29 1550052
[39] Xiao Z and Huang H 2016 J. Math. Phys. 57 032102
[40] Gutiérrez de la Cal X, Pons M and Sokolovski D 2022 Sci. Rep. 12 3842
[41] Yin C, Wu Z J, Wang X P, Sun J J and Cao Z Q 2010 Chin. Phys. B 19 117305
[42] Davies P C W 1986 J. Phys. A 19 2114
[43] Davies P C W 2005 Am. J. Phys. 73 73
[44] Xiao Z, Zheng S and Liu J C 2021 J. Phys. B 54 105602
[45] Zheng S and Xiao Z 2021 Adv. Appl. Math. 10 1197
[46] Fowler R L and Nordheim L 1928 Proc. R. Soc. Lond. A 119 173
[47] Nordheim L W 1928 Proc. R. Soc. Lond. A 121 626
[48] Pfeiffer A N, Cirelli C, Smolarski1 M, Dimitrovski D, Abu-samha M, Madsen L B and Keller U 2012 Nat. Phys. 8 76
[49] Jensen K L, Riga J, Lebowitz J L, Seviour R and Shiffler D A 2022 Proc. R. Soc. Lond. A 132 124303
[50] Jensen K L, Shabaev A, Riga J, Shiffler D A, Lebowitz J L and Seviour R 2021 Phys. Rev. A 104 062203
[51] Camus N, Yakaboylu E, Fechner L, Klaiber M, Laux M, Mi Y, Hatsagortsyan K Z, Pfeifer T, Keitel C H and Moshammer R 2017 Phys. Rev. Lett. 119 023201
[52] Though to have a better resolution of time issue, it is also necessary to require the spread of the wave packet in coordinate space to be much smaller than the barrier width, which means that the spread σk in k space cannot be too narrow.
[53] Cohen-Tannoudji C, Diu B and Laloe F 2019 Quantum Mechanics Vol. 1 (Wiley-VCH)
[54] Büttiker M 1983 Phys. Rev. B 27 6178
[55] Winful H G 2003 Phys. Rev. Lett. 91 260401
[56] Winful H G, Ngom M and Litchinitser N M 2004 Phys. Rev. A 70 052112
[57] Falck J P and Hauge E H 1988 Phys. Rev. B 38 3287
[58] Hauge E H and Stovneng J A 1989 Rev. Mod. Phys. 61 917
[59] Sokolovski D and Baskin L M 1987 Phys. Rev. A 36 4604
[60] Sokolovski D, Msezane A Z and Shaginyan V R 2005 Phys. Rev. A 71 064103
[61] Dumont R S, Rivlin T and Pollak E 2000 New J. Phys. 22 093060
[62] Zhou L, Gong ZR, Liu Y X, Sun C P and Nori F 2008 Phys. Rev. Lett. 101 100501
[63] Zhou L, Dong H, Liu Y X, Sun C P and Nori F 2008 Phys. Rev. A 78 063827
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