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arXiv:math-ph/0302039 · Journal of the Physical Society of Japan

An Approach to Exact Solutions of the Time-dependent Supersymmetric Two-Level Three-Photon Jaynes–Cummings Model

Jian-Qi Shen, Hongyi Zhu, Hong Mao

Published Jun 15, 2002• 15 citations
arXivHTMLPDF

Abstract

By utilizing the property of the supersymmetric structure in the two-level multiphoton Jaynes-Cummings model, an invariant is constructed in terms of the supersymmetric generators by working in the sub-Hilbert-space corresponding to a particular eigenvalue of the conserved supersymmetric generators. We obtain the exact solutions of the time-dependent Schrödinger equation which describes the time-dependent supersymmetric two-level three-photon Jaynes-Cummings model (TLTJCM) by using the invariant-related unitary transformation formulation. The case under the adiabatic approximation is also discussed. Keywords: Supersymmetric Jaynes-Cummings model; exact solutions; invariant theory; geometric phase factor; adiabatic approximation

DOI
10.1143/jpsj.71.1440

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By utilizing the property of the supersymmetric structure in the two-level multiphoton Jaynes-Cummings model, an invariant is constructed in terms of the supersymmetric generators by working in the sub-Hilbert-space corresponding to a particular eigenvalue of the conserved supersymmetric generators. We obtain the exact solutions of the time-dependent Schrödinger equation which describes the time-dependent supersymmetric two-level three-photon Jaynes-Cummings model (TLTJCM) by using the invariant-related unitary transformation formulation. The case under the adiabatic approximation is also discussed. Keywords: Supersymmetric Jaynes-Cummings model; exact solutions; invariant theory; geometric phase factor; adiabatic approximation
The present paper obtains the exact solutions to the time-dependent supersymmetric two-level multiphoton Jaynes-Cummings model and the Chiao-Wu model that describes the propagation of a photon inside an optical fiber. On the basis of the fact that the two-level multiphoton Jaynes-Cummings model possesses a supersymmetric structure, an invariant is constructed in terms of the supersymmetric generators by working in the sub-Hilbert-space corresponding to a particular eigenvalue of the conserved supersymmetric generators (i.e., the time-independent invariant). By constructing the effective Hamiltonian that describes the interaction of the photon with the medium of the optical fiber, it is further verified that the particular solution to the Schrödinger equation is the eigenfunction of the second-quantized momentum operator of photons field. This, therefore, means that the explicit expression (rather than the hidden form that involves the chronological product) for the time-evolution operator of wave function is obtained by means of the invariant theories.
Abstract The present paper obtains the exact solutions to the time‐dependent supersymmetric two‐level multiphoton Jaynes‐Cummings model and the Chiao‐Wu model that describes the propagation of a photon inside an optical fiber. On the basis of the fact that the two‐level multiphoton Jaynes‐Cummings model possesses a supersymmetric structure, an invariant is constructed in terms of the supersymmetric generators by working in the sub‐Hilbert‐space corresponding to a particular eigenvalue of the conserved supersymmetric generators (i.e., the time‐independent invariant). By constructing the effective Hamiltonian that describes the interaction of the photon with the medium of the optical fiber, it is further verified that the particular solution to the Schrödinger equation is the eigenfunction of the second‐quantized momentum operator of photons field. This, therefore, means that the explicit expression (rather than the hidden form that involves the chronological product) for the time‐evolution operator of wave function is obtained by means of the invariant theories.
An approach to exact quantum solutions of the time-dependent two energy level Jaynes–Cummings model with an imaginary photon process is represented in this work. The Lewis–Riesenfeld invariant treatment and the unitary transformation method are used for this purpose. The original Schrödinger equation is reduced to an equivalent solvable one through unitary transformations by using suitable unitary operators. The reduced equation corresponds to a simpler Hamiltonian which is written as a linear combination of the generators of the reduced-dimensional SU(2) algebra. A Hermitian invariant operator is constructed based on the same algebraic formulation and its instantaneous eigenfunctions are obtained. By utilizing such eigenfunctions, the complete quantum wave functions of the system are evaluated. Such wave functions are necessary when we analyze the quantum characteristics of the system.
There exist a number of typical and interesting systems or models which possess three-generator Lie-algebraic structure in atomic physics, quantum optics, nuclear physics and laser physics. The well-known fact that all simple 3-generator algebras are either isomorphic to the algebra sl(2,C)sl(2,C)sl(2,C) or to one of its real forms enables us to treat these time-dependent quantum systems in a unified way. By making use of the Lewis-Riesenfeld invariant theory and the invariant-related unitary transformation formulation, the present paper obtains exact solutions of the time-dependent Schrödinger equations governing various three-generator quantum systems. For some quantum systems whose time-dependent Hamiltonians have no quasialgebraic structures, we show that the exact solutions can also be obtained by working in a sub-Hilbert-space corresponding to a particular eigenvalue of the conserved generator (i.e., the time-independent invariant that commutes with the time-dependent Hamiltonian). The topological property of geometric phase factors in time-dependent systems is briefly discussed.