Tunneling-Induced Decay of Wave Packets Localized Beyond an Asymmetric Potential Barrier

Main Article Content

Mikhail L. Strekalov

Abstract

The time behavior of a wave packet, transmitted through a potential barrier, is investigated for all times assuming it to be initially located outside the asymmetric barrier. The survival probability of the initial state, which characterizes the decay of the unstable quantum system in three time domains, is discussed. The presence of initial, exponential, and long-time regimes is a general feature of the decay process. We analytically found the three crucial points of time that separate these regimes. These characteristic times are set by simple formulae, including momentum distribution. It is then shown that the radius of convergence of the cumulant expansion for the survival probability is always small. A long-time behavior of survival probability is described by the asymptotic series whose coefficients decrease with increasing particle energy but at different rates. To be specific, we have examined the exactly solvable model of tunneling through a finite range triangular barrier.

Keywords:
Time-dependent tunneling, survival probability, wave packet, cumulant expansion, triangular potential barrier

Article Details

How to Cite
Strekalov, M. L. (2020). Tunneling-Induced Decay of Wave Packets Localized Beyond an Asymmetric Potential Barrier. Physical Science International Journal, 24(4), 32-47. https://doi.org/10.9734/psij/2020/v24i430187
Section
Original Research Article

References

García-Calderón G, Romo R. Interference in the time domain of a decaying particle with itself as the physical mechanism for the exponential-nonexponential transition in quantum decay. Phys. Rev. A. 2019;100:032121.

Available:https://doi.org/10.1103/PhysRevA.100.032121

Crespi A, Pepe FV, Facchi P, Sciarrino F, Mataloni P, Nakazato H, Pascazio S, Osellame R. Experimental investigation of quantum decay at short, intermediate and long times via integrated photonics. Phys. Rev. Lett. 2019;122:130401.

Available:https://doi.org/10.1103/PhysRevLett.122.130401

Fonda L, Ghirardi GC, Rimini A. Decay theory of unstable quantum systems. Rep. Prog. Phys. 1978;41:587-631.

Available:https://doi.org/10.1088/0034-4885/41/4/003

Gorin T, Prosen T, Seligman TH, Žnidarič M. Dynamics of Loschmidt echoes and fidelity decay. Phys. Rep. 2006;435:33-156. Available:https://doi.org/10.1016/j.physrep.2006.09.003

Gogolin C, Eisert J. Equilibration, thermalisation and the emergence of statistical mechanics in closed quantum systems. Rep. Prog. Phys. 2016;79: 056001.

Available:https://doi.org/10.1088/0034-4885/79/5/056001

Sudarshan ECG, Chiu CB, Bhamathi G. Unstable systems in generalized quantum theory. Adv. Chem. Phys. 1997;99:121-210. Available:https://doi.org/10.1002/9780470141588.ch2

Krauss LM, Dent J. Late time behavior of false vacuum decay: Possible implications for cosmology and metastable inflating states. Phys. Rev. Lett. 2008;100:171301.

Available:https://doi.org/10.1103/PhysRevLett.100.171301

Khalfin LA. Contribution to the decay theory of a quasi-stationary state. Sov. Phys. JETP. 1958;6:1053-1063.

Chiu CB, Sudarshan ECG, Misra B. Time evolution of unstable quantum states and a resolution of Zeno's paradox. Phys. Rev. D. 1977;16:520-529.

Available:https://doi.org/10.1103/PhysRevD.16.520

Anastopoulos C. Decays of unstable quantum systems. Int. J. Theor. Phys. 2019;58:890-930.

Available:https://doi.org/10.1007/s10773-018-3984-z

Facchi P, Pascazio S. Quantum Zeno dynamics: Mathematical and physical aspects. J. Phys. A. 2008;41:493001.

Available:https://doi.org/10.1088/1751-8113/41/49/493001

Boyanovsky D. Quantum decay in renormalizable field theories: Quasiparticle formation, Zeno and anti-Zeno effects. Ann. Phys. 2019;405:176-201.

Available:https://doi.org/10.1016/j.aop.2019.03.012

Miyamoto M. The various power decays of the survival probability at long times for a free quantum particle. J. Phys. A. 2002;35: 7159.

Available:https://doi.org/10.1088/0305-4470/35/33/312

Unnikrishnan K. On the asymptotic decay law for wave packets in free space. Am. J. Phys. 1997;65:526.

Available:https://doi.org/10.1119/1.18583

Unnikrishnan K. An exhaustive analysis of the asymptotic time dependence of wave packets in one dimension. Am. J. Phys. 1998;66:632.

Available:https://doi.org/10.1119/1.18918

Damborenea JA, Egusquiza IL, Muga JG. Asymptotic behavior of the probability density in one dimension. Am. J. Phys. 2002;70:738.

Available:https://doi.org/10.1119/1.1473643

Muga JG, Delgado V, Snider RF. Dwell time and asymptotic behavior of the probability density. Phys. Rev. B. 1995;52: 16381-16384.

Available:https://doi.org/10.1103/PhysRevB.52.16381

Cordero S, Garsía-Calderón G. Transient effects and reconstruction of the energy spectra in the time evolution of transmitted Gaussian wave packets. J. Phys. A. 2010;43:185301.

Available:https://doi.org/10.1088/1751-8113/43/18/185301

Miyamoto M. Initial wave packets and the various power-law decreases of scattered wave packets at long times. Phys. Rev. A. 2004;69:042704.

Available:https://doi.org/10.1103/PhysRevA.69.042704

Barton G. Levinson's theorem in one dimension: Heuristics. J. Phys. A. 1985;18: 479-494.

Available:https://doi.org/10.1088/0305-4470/18/3/023

Sassoli de Bianchi M. Levinson’s theorem, zero-energy resonances, and time delay in one dimensional scattering systems. J. Math. Phys. 1994;35:2719-2733.

Available:https//doi:10.1063/1.530481

Josza R. Fidelity for mixed quantum states. J. Mod. Opt. 1994;41:2315-2323.

Available:https://doi.org/10.1080/09500349414552171

Rodríguez A, Tsallis C. A generalization of the cumulant expansion. Application to a scale-invariant probabilistic model. J. Math. Phys. 2010;51:073301.

Available:https://doi.org/10.1063/1.3448944

Rota GC, Shen J. On the combinatorics of cumulants. J. Combin. Theory A. 2000;91:283-304.

Available:https://doi.org/10.1006/jcta.1999.3017

Esposito S. Multibarrier tunneling. Phys. Rev. E. 2003;67:016609.

Available:https://doi.org/10.1103/PhysRevE.67.016609

Kiriushcheva N, Kuzmin S. Scattering of a Gaussian wave packet by a reflectionless potential. Am. J. Phys. 1998;66:867-872.

Available:https://doi.org/10.1119/1.18985

Strekalov ML. Quantum tunneling in an exactly solvable double-barrier potential: Barrier transmission near the zero energy. J. Math. Chem. 2018;56:890-903.

Available:https://doi.org/10.1007/s10910-017-0838-5

Rothe C, Hintschich SI, Monkman AP. Violation of the exponential-decay law at long times. Phys. Rev. Lett. 2006;96: 163601. Available:https://doi.org/10.1103/PhysRevLett.96.163601

Vallée O, Soares M. Airy functions and applications to physics. Imperial College Press, London; 2004.