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本文利用代数动力学方法研究了一类Rosen-Zener模型及其多能级推广系统的精确解. 不同于以往将二能级系统薛定谔方程转化为超几何方程的方法, 本文证明这类特殊的Rosen-Zener模型可以通过引入正则变换或规范变换予以解析求解, 并进一步揭示该方法可以推广求解其多能级系统. 在此基础上详细刻画了随时间演化系统各能级间的跃迁几率, 讨论了这类体系所具备的动力学不变量, 以及这类模型存在的对偶系统及其可解性.
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关键词:
- Rosen-Zener模型 /
- 正则变换 /
- 非绝热跃迁 /
- 动力学不变量
Exact solution to the driven quantum system with an explicitly time-dependent Hamiltonian is not only an issue of fundamental importance to quantum mechanics itself, but also a ubiquitous problem in the design for quantum control. In particular, the nonadiabatic transition induced by the time-dependent external field is often involved in order to target the quantum state for the atomic and molecular systems. In this paper we investigate the exact dynamics and the associated nonadiabatic transition in a typical driven model, the Rosen-Zener model and its multi-level extension, by virtue of the algebraic dynamical method. Previously, this kind of driven models, especially of the two-level case, were solved by converting the corresponding Schrödinger equation to a hypergeometric equation. The property of the dynamical transition of the system was then achieved by the asymptotic behavior of the yielded hypergeometric function. A critical drawback related to such methods is that they are very hard to be developed so as to treat the multi-level extension of the driven model. Differing from the above mentioned method, we demonstrate that the particular kind of the Rosen-Zener model introduced here could be solved analytically via a canonical transformation or a gauge transformation approach. In comparison, we show that the present method at least has two aspects of advantages. Firstly, the method enables one to describe the evolution of the wavefunction of the system analytically over any time interval of the pulse duration. Moreover, we show that the method could be exploited to deal with the multi-level extensions of the model. The explicit expression of the dynamical basis states, including the three-level system and the four-level system, is presented and the transition probabilities induced by the nonadiabatic evolution among different levels are then characterized for the model during the time evolution. In addition, our study reveals further that the dual model of the driven system can be constructed. Since the dynamical invariant of a solvable system can always be obtained within the framework of the algebraic dynamical method, the general connection between the dual model and the original one, including the solvability and their dynamical invariants, are established and characterized distinctly.-
Keywords:
- Rosen-Zener model /
- canonical transformation /
- nonadiabaticity-induced transition /
- dynamical invariant
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图 1 (a)二能级系统布居数随时间演化； (b)绝热态的保留几率
$F_+(t)$ 以及非绝热跃迁几率$F_-(t)$ . 系统初态均取$|\uparrow\rangle$ Fig. 1. (a) Time evolution of the population of the two-level system; (b) the survival probability
$F_+(t)$ of the adiabatic state and the transition probability$F_-(t)$ induced by the nonadiabaticity. In both cases the initial state of the system is in$|\uparrow\rangle$ .
图 2 (a)初态为
$|1\rangle$ 态时系统演化过程中的非绝热跃迁$F_{m1}(t)$ $(m=0,\pm 1)$ . 其中$|1\rangle \rightarrow |-1\rangle$ 跃迁几率非常小, 在$t=0$ 时$F_{-11}\approx 0.0028$ ; (b)初态为$|0\rangle$ 态时系统演化过程中的非绝热跃迁$F_{m0}(t)$ , 其中$F_{10}(t)=F_{-10}(t)$ Fig. 2. (a) Nonadiabaticity-induced transition of the initial state
$|1\rangle$ during the evolution. The transition probability from$|1\rangle $ to$|-1\rangle$ is very small with$F_{-11}\approx 0.0028$ at$t=0$ ; (b) nonadiabaticity-induced transition of the initial state$|0\rangle$ during the evolution, where$F_{10}(t)=F_{-10}(t)$ . -
[1] Lewis H R 1967 Phys. Rev. Lett. 18 510
Google Scholar
[2] Lewis H R, Riesenfeld W B 1969 J. Math. Phys. 10 1458
Google Scholar
[3] Berry M V 2009 J. Phys. A 42 365303
Google Scholar
[4] Demirplak M, Rice S A 2003 J. Phys. Chem. A 107 9937
Google Scholar
[5] Chen X, Lizuain I, Ruschhaupt A, Guery-Odelin D, Muga JG 2010 Phys. Rev. Lett. 105 123003
Google Scholar
[6] Bason M G, Viteau M, Malossi N, Huillery P, Arimondo E, Ciampini D, Fazio R, Giovannetti V, Mannella R, Morsch O 2012 Nat. Phys. 8 147
Google Scholar
[7] Cen L X, Li X Q, Yan Y J, Zheng H Z, Wang S J 2003 Phys. Rev. Lett. 90 147902
Google Scholar
[8] Zhang J, Shim J H, Niemeyer I, Taniguchi T, Teraji T, Abe H, Onoda S, Yamamoto T, Ohshima T, Isoya J, Suter D 2013 Phys. Rev. Lett. 110 240501
Google Scholar
[9] Du Y X, Liang Z T, Li Y C, Yue X X, Lv Q X, Huang W, Chen X, Yan H, Zhu S L 2016 Nat. Commun. 7 12479
Google Scholar
[10] 张春玲, 刘文武 2018 67 160302
Google Scholar
Zhang C L, Liu W W 2018 Acta Phys. Sin. 67 160302
Google Scholar
[11] Yang G, Li W, Cen L X 2018 Chin. Phys. Lett. 35 013201
Google Scholar
[12] Li W, Cen L X 2018 Ann. Phys. 389 1
[13] Li W, Cen L X 2018 Quantum Inf. Process. 17 97
Google Scholar
[14] Barnes E, Sarma S D 2012 Phys. Rev. Lett. 109 060401
Google Scholar
[15] Barnes E 2013 Phys. Rev. A 88 013818
Google Scholar
[16] Landau L D 1932 Phys. Z. Sowjetunion. 2 46
[17] Zener C 1932 Proc. R. Soc. London, Ser. A 137 696
Google Scholar
[18] Rosen N, Zener C 1932 Phys. Rev. 40 502
Google Scholar
[19] Rabi I I 1936 Phys. Rev. 49 324
Google Scholar
[20] Rabi I I 1937 Phys. Rev. 51 652
Google Scholar
[21] Zenesini A, Lignier H, Tayebirad G, Radogostowicz J, Ciampini D, Mannella R, Wimberger S, Morsch O, Arimondo E 2009 Phys. Rev. Lett. 103 090403
Google Scholar
[22] Wei L F, Johansson J R, Cen L X, Ashhab S, Nori F 2008 Phys. Rev. Lett. 100 113601
Google Scholar
[23] Wang L, Zhou C, Tu T, Jiang H W, Guo G P, Guo G C 2014 Phys. Rev. A 89 022337
Google Scholar
[24] McKay D C, Naik R, Reinhold P, Bishop L S, Schuster D I 2015 Phys. Rev. Lett. 114 080501
Google Scholar
[25] Thomas G F 1983 Phys. Rev. A 27 2744
Google Scholar
[26] Osherov V I, Voronin A I 1994 Phys. Rev. A 49 265
Google Scholar
[27] Simeonov L S, Vitanov N V 2014 Phys. Rev. A 89 043411
Google Scholar
[28] Ye D F, Fu L B, Liu J 2008 Phys. Rev. A 77 013402
Google Scholar
[29] Li S C, Fu L B, Duan W S, Liu J 2008 Phys. Rev. A 78 063621
Google Scholar
[30] Wang S J, Li F L, Weiguny A 1993 Phys. Lett. A 180 189
Google Scholar
[31] Wang S J, Zuo W 1994 Phys. Lett. A 196 13
Google Scholar
[32] Wang X Q, Cen L X 2011 Phys. Lett. A 375 2220
Google Scholar
[33] Allen L, Eberly J H 1975 Optical Resonance and Two-Level Atoms (New York: Dover Press) pp78-109
[34] Vasilev G S, Vitanov N V 2006 Phys. Rev. A 73 023416
Google Scholar
[35] Lehto J M S, Suominen K A 2016 Phys. Scr. 91 013005
Google Scholar
[36] Wigner E P 1959 Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra (New York: Academic Press) p167
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