In this paper, we present the analytical solution for the model that describes the interaction between a three-level atom and two systems of N-two level atoms. The effects of the quantum numbers and the coupling parameters between spins on the Pancharatnam phase and the atomic inversion, for some special cases of the initial states, are investigated. The comparison between the two effects shows that the analytic results are well consistent.
References
[1]
Mandel, L. and Wolf, E. (1995) Optical Coherence and Quantum Optics. Cambridge University Press, Cambridge. http://www.cambridge.org/US/academic/subjects/physics/optics-optoelectronics-and-photonics/optical-coherence-and-quantum-optics http://dx.doi.org/10.1017/CBO9781139644105
[2]
Walls, D.F. and Milburn, G.J. (1994) Quantum Optics. Springer, Berlin Heidelberg. http://dx.doi.org/10.1007/978-3-642-79504-6
[3]
Gardiner, C.W. and Zoller, P. (2004) Quantum Noise. Springer, Berlin Heidelberg. http://www.amazon.com/Quantum-Noise-Non-Markovian-Applications-Synergetics/dp/3540223010
[4]
Moya-Cessa, H.M. and Soto-Eguibar, F. (2011) Introduction to Quantum Optics. Rinton Press, New Jersey. http://www.rintonpress.com/books/0611.html
[5]
Scully, M.O. and Zubairy, M.S. (1997) Quantum Optics. Cambridge University Press, Cambridge. http://dx.doi.org/10.1017/CBO9780511813993
[6]
Schleich, W. P. (2001) Quantum Optics in Phase Space. Wiley, Weinheim. http://www.amazon.com/Quantum-Optics-Phase-Wolfgang-Schleich/dp/352729435X
[7]
Kira, M. and Koch, S.W. (2011) Semiconductor Quantum Optics. Cambridge University Press, Cambridge. http://dx.doi.org/10.1017/CBO9781139016926
[8]
Agrawal, G.P. and Mehta, C.L. (1974) Dynamics of Parametric Processes with a Trilinear Hamiltonian. Journal of Physics A: Mathematical and General, 7, 607. http://dx.doi.org/10.1088/0305-4470/7/5/011
[9]
Tucker, J. and Walls, D.F. (1969) Quantum Theory of Parametric Frequency Conversion. Annals of Physics, 52, 1-15. http://dx.doi.org/10.1016/0003-4916(69)90318-2
[10]
Tang, C.L. (1969) Spontaneous Emission in the Frequency Up-Conversion Process in Nonlinear Optics. Physical Review, 182, 367. http://dx.doi.org/10.1103/PhysRev.182.367
[11]
Tucker, J. and Walls, D.F. (1969) Quantum Theory of the Traveling-Wave Frequency Converter. Physical Review, 178, 2036. http://dx.doi.org/10.1103/PhysRev.178.2036
[12]
Walls, D.F. and Barakat, R. (1970) Quantum-Mechanical Amplification and Frequency Conversion with a Trilinear Hamiltonian. Physical Review A, 1, 446. http://dx.doi.org/10.1103/PhysRevA.1.446
[13]
Sebawe Abdalla, M. and Ahmed, M.M.A. (2012) Some Statistical Properties for a Spin-(1/2) Particle Coupled to Two Spirals. Optics Communications, 285, 3578-3586. http://dx.doi.org/10.1016/j.optcom.2012.04.014
[14]
Eleuch, H. and Bennaceur, R. (2003) An Optical Soliton Pair among Absorbing Three-Level Atoms. Journal of Optics A: Pure and Applied Optics, 5, 528. http://dx.doi.org/10.1088/1464-4258/5/5/315
[15]
Setea, E.A., Svidzinsky, A.A., Eleuch, H., Yang, Z., Nevels, R.D. and Scully, M.O. (2010) Correlated Spontaneous Emission on the Danube. Journal of Modern Optics, 57, 1311-1330. http://dx.doi.org/10.1080/09500341003605445
[16]
Abdel-Aty, M. (2002) Influence of Second-Order Correction to Rayleigh Scattering on Pancharatnam Phase in a Three-Level Atom. Modern Physics Letters B, 16, 319. http://dx.doi.org/10.1142/S0217984902003816
[17]
Abdel-Aty, M., Abdel-Khalek, S. and Obada, A.-S.F. (2000) Pancharatnam Phase of Two-Mode Optical Fields with Kerr Nonlinearity. Optical Review, 7, 499-504. http://dx.doi.org/10.1007/s10043-000-0499-6
[18]
Bhandari, R. and Samuel, J. (1988) Observation of Topological Phase by Use of a Laser Interferometer. Physical Review Letters, 60, 1211. http://dx.doi.org/10.1103/PhysRevLett.60.1211
[19]
Wagh, A.G. and Rakhecha, V.C. (1995) On Measuring the Pancharatnam Phase. I. Interferometry. Physics Letters A, 197, 107-111. http://dx.doi.org/10.1016/0375-9601(94)00914-B
[20]
Wagh, A.G. and Rakhecha, V.C. (1995) On Measuring the Pancharatnam Phase. II. SU(2) Polarimetry. Physics Letters A, 197, 112-115.
[21]
Zhao, Y.-G. and Li, B.-Z. (1997) Quantum Phases of Pancharatnam Type for a General Spin in a Time-Dependent Magnetic Field. Chinese Physics Letters, 14, 801. http://dx.doi.org/10.1088/0256-307X/14/11/001
[22]
Lawande, Q.V., Lawande, S.V. and Joshi, A. (1999) Pancharatnam Phase for a System of a Two-Level Atom Interacting with a Quantized Field in a Cavity. Physics Letters A, 251, 164-168. http://dx.doi.org/10.1016/S0375-9601(98)00882-2
[23]
Simon, B. (1983) Holonomy, the Quantum Adiabatic theoreM and Berry’s Phase. Physical Review Letters, 51, 2167. http://dx.doi.org/10.1103/PhysRevLett.51.2167
[24]
Facchi, P., Mariano, A. and Pascazio, S. (1999) Wigner Function and Coherence Properties of Cold and Thermal Neutrons. Acta Physica Slovaca, 49, 677-682.
[25]
Jones, J.A., Vedral, V., Ekert, A. and Castagnoli, G. (2000) Geometric Quantum Computation Using Nuclear Magnetic Resonance. Nature, 403, 869-871. http://dx.doi.org/10.1038/35002528
[26]
Pancharatnam (1956) Generalized Theory of Interference and Its Applications. S.Proc. Ind. Acad. Sci, 44, 247.
[27]
Francisco De Zela, F. (2012) The Pancharatnam-Berry Phase: Theoretical and Experimental Aspects. In: Pahlavani, M.R., Ed., Theoretical Concepts of Quantum Mechanics, Intech, 289. http://dx.doi.org/10.5772/34882
[28]
Abdel-Khalek, S., El-Saman, Y.S. and Abdel-Aty, M. (2010) Geometric Phase of a Moving Three-Level Atom. Optics Communications, 283, 1826-1831. http://dx.doi.org/10.1016/j.optcom.2009.12.065
[29]
van Dijk, T., Schouten, H.F., Ubachs, W. and Visser, T.D. (2010) The Pancharatnam-Berry Phase for Non-Cyclic Polarization Changes. Optics Express, 18, 10796-10804. http://dx.doi.org/10.1364/OE.18.010796
[30]
Bhandari, R. (1989) Berry’s Phase and the Pancharatnam Angle-Some Recent Observations. Bulletin of the Calcutta Mathematical Society, 81, 496.
[31]
Hariharan, P. and Roy, M. (1992) A Geometric-Phase Interferometer. Journal of Modern Optics, 39, 1811-1815. http://dx.doi.org/10.1080/09500349214551881
[32]
Masashi, B. (1993) Decomposition Formulas for Su(1, 1) and Su(2) Lie Algebras and Their Applications in Quantum Optics. Journal of the Optical Society of America B, 10, 1347-1359. http://dx.doi.org/10.1364/JOSAB.10.001347
[33]
Sebawe Abdalla, M., Ahmed, M.M.A., Khalil, E.M. and Obada A.-S.F. (2014) Dynamics of an Adiabatically Effective Two-Level Atom Interacting with a Star-Like System. Progress of Theoretical and Experimental Physics, 2014, 073A02. http://dx.doi.org/10.1093/ptep/ptu091