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Bulletin of the Russian Academy of Sciences Physics

Integrable Models of Quantum Optics

Vladimir I. Yudson, A. A. Makarov

Published Dec 1, 2018• 1 citation
Publisher

Abstract

Abstract A list of exactly solvable many-body models of quantum optics is presented. The problem of the cascade radiation spectrum of an arbitrary multilevel system, the solution to which contains tasks considered earlier as special cases, is highlighted. Examples of spectra for a three-level system, a harmonic oscillator, the Dicke model, and an equidistant system with a constant rate of spontaneous transition are given.

DOI
10.3103/s1062873818120249

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Jun 10, 1996•0 citations
Abstract A list of exactly solvable many-body models of quantum optics is presented. The problem of the cascade radiation spectrum of an arbitrary multilevel system, the solution to which contains tasks considered earlier as special cases, is highlighted. Examples of spectra for a three-level system, a harmonic oscillator, the Dicke model, and an equidistant system with a constant rate of spontaneous transition are given.
A general expression for the spectrum of cascade spontaneous emission from an arbitrary multilevel system is presented. Effects of the quantum interference of photons emitted in different transitions are analyzed. These effects are especially essential when the transition frequencies are close. Several examples are considered: (i) Three-level system; (ii) Harmonic oscillator; (iii) System with equidistant levels and equal rates of the spontaneous decay for all the transitions; (iv) Dicke superradiance model.
A method to build various integrable models for description of coherent excitation of multilevel media by laser pulses is suggested. Distribution functions over the energy levels of quantum systems depending on the time and frequency detuning are obtained. The distributions follow from Schrödinger equation exact solutions and give the complete dynamical description of laser-excited quantum multilevel systems. Interpretation based on the Fourier spectra of the probability amplitudes of a quantum system is presented. The spectra are expressed in terms of orthonormal polynomials and their weight functions. Matrix elements of the dipole transitions between levels are equal to coefficients of the recurrence formula for the orthonormal polynomial system. Some examples are presented. The Kravchuk oscillator family as an integrable model is constructed to describe the coherent excitation dynamics of multilevel resonance media. It is based on the use of the Kravchuk orthogonal polynomials. The Kravchuk oscillator excitation dynamics is described by the binomial distribution of energy level populations and the distribution parameter depends on excitation conditions. Two known basic models in quantum physics -- the harmonic oscillator and two-level system are the special representatives of the Kravchuk oscillator family.
We give an overview of exactly solvable many-body models of quantum optics. Among them is a system of two-level atoms which interact with photons propagating in a one-dimensional (1D) chiral waveguide; exact eigenstates of this system can be explicitly constructed. This approach is used also for a system of closely located atoms in the usual (non-chiral) waveguide or in 3D space. Moreover, it is shown that for an arbitrary atomic system with a cascade spontaneous radiative decay, the fluorescence spectrum can be described by an exact analytic expression which accounts for interference of emitted photons. Open questions related with broken integrability are discussed.
The process of a cascade light emission by an excited multilevel quantum system has been described exactly. A multiphoton wave function of the system at an arbitrary time has been derived, which allows one to find any characteristic of the emitted field. Using this wave function, we have obtained a general expression for the cascade emission spectrum. This expression is of special interest when some transitions have close frequencies; this case has not been treated before. We present several examples of application of the obtained general expression.
The N=3 Dicke model couples three qubits to a single radiation mode via dipole interaction and constitutes the simplest quantum-optical system allowing for Greenberger-Horne-Zeilinger states. In contrast to the case N=1 (the Rabi model), it is non-integrable if the counter-rotating terms are included. The spectrum is determined analytically, employing the singularity structure of an associated differential equation. While quasi-exact eigenstates known from the Rabi model do not exist, a novel type of spectral degeneracy becomes possible which is not associated with a symmetry of the system.
A method based on the use of integral transform and orthogonal polynomials is presented to obtain the analytic solutions of the equations for coherent dynamics of multilevel quantum systems as simple molecular models describing molecule-radiation interactions in the high-power laser field and many other processes. As an example the Jacobi polynomials have been used and exact analytic solution has been obtained for dynamics of the family of multilevel Jacobi systems with both equidistant and nonequidistant energy levels, with various dependencies of dipole transition moments on energy, with both resonant and non-resonant laser excitation. Some special cases of the solution for definite systems from the family are given.