﻿ 具有双模运算符和相位运算符的广义Jaynes-Cummings模型中的几何相位 Geometric Phase in a Generalized Jaynes-Cummings Model with Double Mode Operators and Phase Operators

Vol.06 No.03(2017), Article ID:21887,7 pages
10.12677/CMP.2017.63010

Geometric Phase in a Generalized Jaynes-Cummings Model with Double Mode Operators and Phase Operators

Yuanxin Qiao, Zhaoxian Yu

Department of Physics, Beijing Information Science and Technology University , Beijing

Received: Aug. 15th, 2017; accepted: Aug. 25th, 2017; published: Aug. 31st, 2017

ABSTRACT

By using the Lewis-Riesenfeld invariant theory, the geometric phase in a generalized Jaynes-Cummings model with double mode operators and phase operators has been studied. Compared with the dynamical phase, the geometric phase in a cycle case is independent of the frequency of the double photon field, the coupling coefficient between photons and atoms, and the atom transition frequency. It is apparent that the geometric phase has the pure geometric and topological characteristics, which means that the geometric phase represents the holonomy in the Hermitian linear bundles.

Keywords:Geometric Phase, Lewis-Riesenfeld Invariant Theory, Generalized Jaynes-Cummings Model

1. 引言

2. 模型

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3. 量子系统的几何相因子

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。该系统与时间相关的薛定谔方程为

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4. 结论

Geometric Phase in a Generalized Jaynes-Cummings Model with Double Mode Operators and Phase Operators[J]. 凝聚态物理学进展, 2017, 06(03): 74-80. http://dx.doi.org/10.12677/CMP.2017.63010

1. 1. Gao, X.C., Xu, J.B. and Qian, T.Z. (1991) Geometric Phase and the Generalized Invariant Formulation. Physical Review A, 44, 7016. https://doi.org/10.1103/PhysRevA.44.7016

2. 2. Yu, G., Yu, Z.X. and Zhang, D.X. (1996) Berry Phase of the Finite Deep Column Potential Well with Motive Boundary. Yantai Teachers University Journal, 12, 117.

3. 3. Gao, X.C. et al. (1996) Quantum-Invariant Theory and the Evolution of a Quantum Scalar Field in Robertson-Walker Flat Spacetimes. Physical Review D Particles & Fields, 53, 4374. https://doi.org/10.1103/PhysRevD.53.4374

4. 4. Hou, Y.Z., Yu, Z.X. and Liu, Y.H. (1994) Using Quantum-Positive Transformations to Cause the Scalability of Topological Items to Obtain the Perturbation of Perturbation on the Contribution of Berry Phase factors. Journal of Xinyan Teachers College, 7, 37.

5. 5. Liang, J.Q. and Ding, X. (1991) Broken Gauge Equivalence of Hamiltonians Due to Time-Evolution and the Berry Phase. Physics Letters A, 153, 273-278. https://doi.org/10.1016/0375-9601(91)90942-2

6. 6. Sun, C.P. and Ge, M.L. (1990) Generalizing Born-Oppenheimer Approximations and Observable Effects of an Induced Gauge Field. Physical Review D Part Fields, 41, 1349-1352. https://doi.org/10.1103/PhysRevD.41.1349

7. 7. Berry, M.V. (1984) Quantal Phase Factors Accompanying Adibatic Changes. Proceedings of the Royal Society of Lon-don, 392, 45-57. https://doi.org/10.1098/rspa.1984.0023

8. 8. Sun, C.P. (1993) Quantum Dynamical Model for Wave-Function Re-duction in Classical and Macroscopic Limits. Physical Review A, 48, 898. https://doi.org/10.1103/PhysRevA.48.898

9. 9. Aharonov, Y. and Bohm, D. (1959) Significance of Electromagnetic Potentials in the Quantum Theory. Physical Review, 115, 485-491. https://doi.org/10.1103/PhysRev.115.485

10. 10. Aharonov, Y. and Anandan, J (1987) Phase Change during a Cyclic Quantum Evolution. Physical Review Letters, 58, 1953. https://doi.org/10.1103/PhysRevLett.58.1593

11. 11. Sun, C.P. (1988) Analytic Treatment of High-Order Adiabatic Approx-imations of 2-Neutrino Oscillations in Matter. Physical Review D Particles & Fields, 38, 2908-2910. https://doi.org/10.1103/PhysRevD.38.2908

12. 12. Samuel, J. and Bhandari, R. (1988) General Setting for Berry’s Phase. Physical Review Letters, 60, 2339. https://doi.org/10.1103/PhysRevLett.60.2339

13. 13. Sun, C.P. et al. (1988) High-Order Quantum Adiabatic Approximation and Berry’s Phase Factor. Journal of Physics A General Physics, 21, 1595. https://doi.org/10.1088/0305-4470/21/7/023

14. 14. Yu, Z.X. and Zhang, D.X. (1995) Phase Factor for Quantum Systems with Time-Bound Condition. Journal of Qingdao University, 8, 49.

15. 15. Chen, J.Q., Li, J. and Liang, Q. (2006) Critical Property of the Geo-metric Phase in the Dicke Model. Physical Review A, 74, 150. https://doi.org/10.1103/PhysRevA.74.054101

16. 16. Pati, A.K. (1995) Geometric Aspects of Noncyclic Quantum Evolution. Physical Review A, 52, 2576-2584. https://doi.org/10.1103/PhysRevA.52.2576

17. 17. Fan, H.Y. and Ruan, T.N. (1984) Some New Applications of Co-herent States. China Science, 42, 27.

18. 18. Tsui, D.C., et al. (1982) Two-Dimensional Magnetotransport in the Extreme Quantum Limit. Physical Review Letters, 48, 1559-1562. https://doi.org/10.1103/PhysRevLett.48.1559

19. 19. Semenoff, G.W., et al. (1986) Non-Abelian Adiabatic Phases and the Fractional Quantum Hall Effect, Physical Review Letters, 57, 1195-1198. https://doi.org/10.1103/PhysRevLett.57.1195

20. 20. Chen, C.M., et al. (1991) Quantum Hall Effect and Bery Phase Factor Physics. Acta Physica sinica, 40, 345

21. 21. Fan, H.Y. and Li, L.S. (1996) Supersymmetric Unitary Operator for Some Generalized Jaynes-Cummings Models Communications in Theoretical Physics, 25, 105. https://doi.org/10.1088/0253-6102/25/1/105

22. 22. Uhlmann, A (1986) Parallel Transport and Quantum Holonomy along Density Operators. Reports on Mathematical Physics, 24, 229-240. https://doi.org/10.1016/0034-4877(86)90055-8

23. 23. Wu, Y. and Yang, X. (1997) Jaynes-Cummings Model for a Trapped Ion in Any Position of a Standing Wave. Physical Review Letters, 78, 3086-3088. https://doi.org/10.1103/PhysRevLett.78.3086

24. 24. Sjoqvist, E. et al. (2000) Geometric Phases for Mixed States in In-terferometry. Physical Review Letters, 85, 2845- 2849. https://doi.org/10.1103/PhysRevLett.85.2845

25. 25. Lewis, H.R. and Riesenfeld, W.B. (1969) An Exact Quantum Theory of the Time-Dependent Harmonic Oscillator and of a Charged Particle in a Time-Dependent Electromagnetic Field. Jour-nal of Mathematical Physics, 10, 1458-1473. https://doi.org/10.1063/1.1664991