一类多能级Rosen-Zener模型的精确解
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摘要
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.
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 RosenZener 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 Schr?dinger 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.
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 RosenZener 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 Schr?dinger 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.
引文
[1]Lewis H R 1967 Phys. Rev. Lett. 18 510
[2]Lewis H R, Riesenfeld W B 1969 J. Math. Phys. 10 1458
[3]Berry M V 2009 J. Phys. A 42 365303
[4]Demirplak M, Rice S A 2003 J. Phys. Chem. A 107 9937
[5]Chen X, Lizuain I, Ruschhaupt A, Guery-Odelin D, Muga JG2010 Phys. Rev. Lett. 105 123003
[6]Bason M G, Viteau M, Malossi N, Huillery P, Arimondo E,Ciampini D, Fazio R, Giovannetti V, Mannella R, Morsch O2012 Nat. Phys. 8 147
[7]Cen L X, Li X Q, Yan Y J, Zheng H Z, Wang S J 2003 Phys.Rev. Lett. 90 147902
[8]Zhang J, Shim J H, Niemeyer I, Taniguchi T, Teraji T, Abe H, Onoda S, Yamamoto T, Ohshima T, Isoya J, Suter D 2013Phys. Rev. Lett. 110 240501
[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
[10]Zhang C L, Liu W W 2018 Acta Phys. Sin. 67 160302(in Chinese)[张春玲,刘文武2018物理学报67 160302]
[11]Yang G, Li W, Cen L X 2018 Chin. Phys. Lett. 35 013201
[12]Li W, Cen L X 2018 Ann. Phys. 389 1
[13]Li W, Cen L X 2018 Quantum Inf. Process. 17 97
[14]Barnes E, Sarma S D 2012 Phys. Rev. Lett. 109 060401
[15]Barnes E 2013 Phys. Rev. A 88 013818
[16]Landau L D 1932 Phys. Z. Sowjetunion. 2 46
[17]Zener C 1932 Proc. R. Soc. London, Ser. A 137 696
[18]Rosen N, Zener C 1932 Phys. Rev. 40 502
[19]Rabi I I 1936 Phys. Rev. 49 324
[20]Rabi I I 1937 Phys. Rev. 51 652
[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
[22]Wei L F, Johansson J R, Cen L X, Ashhab S, Nori F 2008Phys. Rev. Lett. 100 113601
[23]Wang L, Zhou C, Tu T, Jiang H W, Guo G P, Guo G C 2014Phys. Rev. A 89 022337
[24]Mc Kay D C, Naik R, Reinhold P, Bishop L S, Schuster D I2015 Phys. Rev. Lett. 114 080501
[25]Thomas G F 1983 Phys. Rev. A 27 2744
[26]Osherov V I, Voronin A I 1994 Phys. Rev. A 49 265
[27]Simeonov L S, Vitanov N V 2014 Phys. Rev. A 89 043411
[28]Ye D F, Fu L B, Liu J 2008 Phys. Rev. A 77 013402
[29]Li S C, Fu L B, Duan W S, Liu J 2008 Phys. Rev. A 78063621
[30]Wang S J, Li F L, Weiguny A 1993 Phys. Lett. A 180 189
[31]Wang S J, Zuo W 1994 Phys. Lett. A 196 13
[32]Wang X Q, Cen L X 2011 Phys. Lett. A 375 2220
[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
[35]Lehto J M S, Suominen K A 2016 Phys. Scr. 91 013005
[36]Wigner E P 1959 Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra(New York:Academic Press)p167
[2]Lewis H R, Riesenfeld W B 1969 J. Math. Phys. 10 1458
[3]Berry M V 2009 J. Phys. A 42 365303
[4]Demirplak M, Rice S A 2003 J. Phys. Chem. A 107 9937
[5]Chen X, Lizuain I, Ruschhaupt A, Guery-Odelin D, Muga JG2010 Phys. Rev. Lett. 105 123003
[6]Bason M G, Viteau M, Malossi N, Huillery P, Arimondo E,Ciampini D, Fazio R, Giovannetti V, Mannella R, Morsch O2012 Nat. Phys. 8 147
[7]Cen L X, Li X Q, Yan Y J, Zheng H Z, Wang S J 2003 Phys.Rev. Lett. 90 147902
[8]Zhang J, Shim J H, Niemeyer I, Taniguchi T, Teraji T, Abe H, Onoda S, Yamamoto T, Ohshima T, Isoya J, Suter D 2013Phys. Rev. Lett. 110 240501
[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
[10]Zhang C L, Liu W W 2018 Acta Phys. Sin. 67 160302(in Chinese)[张春玲,刘文武2018物理学报67 160302]
[11]Yang G, Li W, Cen L X 2018 Chin. Phys. Lett. 35 013201
[12]Li W, Cen L X 2018 Ann. Phys. 389 1
[13]Li W, Cen L X 2018 Quantum Inf. Process. 17 97
[14]Barnes E, Sarma S D 2012 Phys. Rev. Lett. 109 060401
[15]Barnes E 2013 Phys. Rev. A 88 013818
[16]Landau L D 1932 Phys. Z. Sowjetunion. 2 46
[17]Zener C 1932 Proc. R. Soc. London, Ser. A 137 696
[18]Rosen N, Zener C 1932 Phys. Rev. 40 502
[19]Rabi I I 1936 Phys. Rev. 49 324
[20]Rabi I I 1937 Phys. Rev. 51 652
[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
[22]Wei L F, Johansson J R, Cen L X, Ashhab S, Nori F 2008Phys. Rev. Lett. 100 113601
[23]Wang L, Zhou C, Tu T, Jiang H W, Guo G P, Guo G C 2014Phys. Rev. A 89 022337
[24]Mc Kay D C, Naik R, Reinhold P, Bishop L S, Schuster D I2015 Phys. Rev. Lett. 114 080501
[25]Thomas G F 1983 Phys. Rev. A 27 2744
[26]Osherov V I, Voronin A I 1994 Phys. Rev. A 49 265
[27]Simeonov L S, Vitanov N V 2014 Phys. Rev. A 89 043411
[28]Ye D F, Fu L B, Liu J 2008 Phys. Rev. A 77 013402
[29]Li S C, Fu L B, Duan W S, Liu J 2008 Phys. Rev. A 78063621
[30]Wang S J, Li F L, Weiguny A 1993 Phys. Lett. A 180 189
[31]Wang S J, Zuo W 1994 Phys. Lett. A 196 13
[32]Wang X Q, Cen L X 2011 Phys. Lett. A 375 2220
[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
[35]Lehto J M S, Suominen K A 2016 Phys. Scr. 91 013005
[36]Wigner E P 1959 Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra(New York:Academic Press)p167