The phase-space approach to time evolution of quantum states in confined systems: The spectral split-operator method

Kołaczek, Damian ; Spisak, Bartłomiej J. ; Wołoszyn, Maciej

International Journal of Applied Mathematics and Computer Science, Tome 29 (2019) no. 3, pp. 439-451.

Résumé

Using the phase space approach, we consider the quantum dynamics of a wave packet in an isolated confined system with three different potential energy profiles. We solve the Moyal equation of motion for the Wigner function with the highly efficient spectral split-operator method. The main aim of this study is to compare the accuracy of the employed algorithm through analysis of the total energy expectation value, in terms of deviation from its exact value. This comparison is performed for the second and fourth order factorizations of the time evolution operator.

Détail

Keywords: Wigner distribution function, Moyal dynamics, spectral split operator method
Mots-clés : funkcja dystrybucji Wignera, dynamika Moyal, metoda podziału operatora

Bibliographie

[1] Baker, G.A. (1958). Formulation of quantum mechanics based on the quasi-probability distribution induced on phase space, Physical Review 109(6): 2198–2206, DOI: 10.1103/PhysRev.109.2198.

[2] Balazs, N.L. and Jennings, B.K. (1984). Wigner’s function and other distribution functions on Mock phase spaces, Physics Reports 104(6): 347–391, DOI: 10.1016/0370-1573(84)90151-0.

[3] Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A. and Sternheimer, D. (1977). Quantum mechanics as a deformation of classical mechanics, Letters in Mathematical Physics 1(6): 521–530, DOI: 10.1007/BF00399745.

[4] Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A. and Sternheimer, D. (1978a). Deformation theory and quantization. I: Deformation of symplectic structures, Annals of Physics 111(1): 61–110, DOI: 10.1016/0003-4916(78)90224-5.

[5] Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A. and Sternheimer, D. (1978b). Deformation theory and quantization. II: Physical applications, Annals of Physics 111(1): 111–151, DOI: 10.1016/0003-4916(78)90225-7.

[6] Benedict, M.G. and Czirják, A. (1999). Wigner functions, squeezing properties, and slow decoherence of a mesoscopic superposition of two-level atoms, Physical Review A 60(5): 4034–4044, DOI: 10.1103/PhysRevA.60.4034.

[7] Berkovitz, L.D. (1974). Optimal Control Theory, Springer-Verlag, New York, NY.

[8] Błaszak, M. and Domański, Z. (2010). Phase space quantum mechanics, Annals of Physics 327(2): 167–211, DOI: 10.1016/j.aop.2011.09.006.

[9] Bondar, D.I., Cabrera, R., Zhdanov, D.V. and Rabitz, H.A. (2013). Wigner phase-space distribution as a wave function, Physical Review A 88(5): 052108–1–052108–6, DOI: 10.1103/PhysRevA.88.052108.

[10] Castellani, L. (2000). Non-commutative geometry and physics: A review of selected recent results, Classical and Quantum Gravity 17(17): 3377–3401, DOI: 10.1088/0264-9381/17/17/301.

[11] Chin, S.A. (1997). Symplectic integrators from composite operator factorizations, Physics Letters A 226(6): 344–348, DOI: 10.1016/S0375-9601(97)00003-0.

[12] Chin, S.A. and Chen, C.R. (2002). Gradient symplectic algorithms for solving the Schr¨odinger equation with time-dependent potentials, The Journal of Chemical Physics 117(4): 1409–1415, DOI: 10.1063/1.1485725.

[13] Ciurla, M., Adamowski, J., Szafran, B. and Bednarek, S. (2002). Modelling of confinement potentials in quantum dots, Physica E: Low-dimensional Systems and Nanostructures 15(4): 261–268, DOI: 10.1016/S1386-9477(02)00572-6.

[14] Curtright, T.L. and Zachos, C.K. (2012). Quantum mechanics in phase space, Asia-Pacific Physics Newsletter 1(1): 37–46, DOI: 10.1142/S2251158X12000069.

[15] Dattoli, G., Giannessi, L., Ottaviani, P.L. and Torre, A. (1995). Split-operator technique and solution of Liouville propagation equations, Physical Review E: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics 51(1): 821–824, DOI: 10.1103/PhysRevE.51.821.

[16] Delius, G.W. and Hüffmann, A. (1996). On quantum Lie algebras and quantum root systems, Journal of Physics A: Mathematical and General 29(8): 1703–1722, DOI: 10.1088/0305-4470/29/8/018.

[17] Feit, M.D., Fleck, J.A. and Steiger, A. (1982). Solution of the Schrödinger equation by a spectral method, Journal of Computational Physics 47(3): 412–433, DOI: 10.1016/0021-9991(82)90091-2.

[18] Gómez, E.A., Thirumuruganandham, S.P. and Santana, A. (2014). Split-operator technique for propagating phase space functions: Exploring chaotic, dissipative and relativistic dynamics, Computer Physics Communications 185(1): 136–143, DOI: 10.1016/j.cpc.2013.08.025.

[19] Hiley, B.J. (2015). On the relationship between the Wigner–Moyal approach and the quantum operator algebra of von Neumann, Journal of Computational Electronics 14(4): 869–878, DOI: 10.1007/s10825-015-0728-7.

[20] Hillery, M., O’Connell, R.F., Scully, M.O. and Wigner, E.P. (1984). Distribution functions in physics: Fundamentals, Physics Reports 106(3): 121–167, DOI: 10.1016/0370-1573(84)90160-1.

[21] Isar, A. and Scheid, W. (2004). Deformation of quantum oscillator and of its interaction with environment, Physica A: Statistical Mechanics and Its Applications 335(1–2): 79–93, DOI: 10.1016/j.physa.2003.12.017.

[22] Kaczor, U., Klimas, B., Szydłowski, D., Wołoszyn, M. and Spisak, B. (2016). Phase-space description of the coherent state dynamics in a small one-dimensional system, Open Physics 14(1): 354–359, DOI: 10.1515/phys-2016-0036.

[23] Kenfack, A. (2016). Comment on Nonclassicality indicator for the real phase-space distribution functions, Physical Review A 93(3): 036101-1–036101-2, DOI: 10.1103/PhysRevA.93.036101.

[24] Kenfack, A. and Życzkowski, K. (2004). Negativity of the Wigner function as an indicator of non-classicality, Journal of Optics B Quantum and Semiclassical Optics 6(10): 396–404, DOI: 10.1088/1464-4266/6/10/003.

[25] Khademi, S., Sadeghi, P. and Nasiri, S. (2016). Reply to Comment on Nonclassicality indicator for the real phase-space distribution functions, Physical Review A 93(3): 036102-1–036102-2, DOI: 10.1103/PhysRevA.93.036102.

[26] Kołaczek, D., Spisak, B.J. and Wołoszyn, M. (2018). Phase-space approach to time evolution of quantum states in confined systems. The spectral split-operator method, in P. Kulczycki et al. (Eds), Contemporary Computational Science, AGH-UST Press, Cracow, p. 5.

[27] Kołaczek, D., Spisak, B.J. and Wołoszyn, M. (2020). Phase-space approach to time evolution of quantum states in confined systems. The spectral split-operator method, in P. Kulczycki et al. (Eds), Information Technology, Systems Research, and Computational Physics, Springer, Cham, pp. 307–320.

[28] Kubo, R. (1964). Wigner representation of quantum operators and its applications to electrons in a magnetic field, Journal of the Physical Society of Japan 19(11): 2127–2139, DOI: 10.1143/JPSJ.19.2127.

[29] Lechner, G. (2011). Deformations of quantum field theories and integrable models, Communications in Mathematical Physics 312(1): 265–302, DOI: 10.1007/s00220-011-1390-y.

[30] Lee, H.-W. (1995). Theory and application of the quantum phase-space distribution functions, Physics Reports 259(3): 147–211, DOI: 10.1016/0370-1573(95)00007-4.

[31] Leung, B. and Prodan, E. (2013). A non-commutative formula for the isotropic magneto-electric response, Journal of Physics A: Mathematical and Theoretical 46(15): 085205-1–085205-14, DOI: 10.1088/1751-8113/46/8/085205.

[32] Luenberger, D.G. (1979). Introduction to Dynamic Systems, Theory, Models, and Applications, John Wiley Sons, Inc., New York, NY.

[33] Ozorio de Almeida, A.M. (1998). The Weyl representation in classical and quantum mechanics, Physics Reports 295(6): 265–342, DOI: 10.1016/S0370-1573(97)00070-7.

[34] Polderman, J.W. and Willems, J.C. (1998). Introduction to Mathematical Systems Theory. A Behavioral Approach, Springer-Verlag, New York, NY.

[35] Pool, J.C.T. (1966). Mathematical aspects of the Weyl correspondence, Journal of Mathematical Physics 7(1): 66–76, DOI: 10.1063/1.1704817.

[36] Sadeghi, P., Khademi, S. and Nasiri, S. (2010). Nonclassicality indicator for the real phase-space distribution functions, Physical Review A 83(1): 012102-1–012102-8, DOI: 10.1103/PhysRevA.82.012102.

[37] Sontag, E.D. (1990). Mathematical Control Theory. Deterministic Finite Dimensional Systems, Springer-Verlag, New York, NY.

[38] Tatarskiĭ, V.I. (1983). The Wigner representation of quantum mechanics, Soviet Physics Uspekhi 26(4): 311–327, DOI: 10.1070/PU1983v026n04ABEH004345.

[39] Ter Haar, D. (1961). Theory and applications of the density matrix, Reports on Progress in Physics 24(1): 304–362, DOI: 10.1088/0034-4885/24/1/307.

[40] Torres-Vega, G. and Frederick, J.H. (1982). Numerical method for the propagation of quantum-mechanical wave functions in phase space, Physical Review Letters 67(19): 2601–2604, DOI: 10.1103/PhysRevLett.67.2601.

[41] Walker, J.A. (1980). Dynamical Systems and Evolution Equations. Theory and Applications, Plenum Press, New York, NY.

[42] Wigner, E. (1932). On the quantum correction for thermodynamic equilibrium, Physical Review 40(5): 749–759, DOI: 10.1103/PhysRev.40.749.

[43] Xue, Y. and Prodan, E. (2012). The noncommutative Kubo formula: Applications to transport in disordered topological insulators with and without magnetic fields, Physical Review. B: Condensed Matter 86(15): 155445-1–155445-17, DOI: 10.1103/PhysRevB.86.155445.