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arXiv:1603.02102 · Progress of Theoretical and Experimental Physics

Nonlinear Schrödinger equation and semiclassical description of the light–atom interaction

Sergey A. Rashkovskiy

Published Nov 10, 2016• 10 citations
arXivHTMLPDF

Abstract

In this paper, we consider the nonlinear Schrödinger equation describing the light-atom interaction while taking into account damping due to spontaneous emission.We demonstrate that the optical Bloch equations with damping due to spontaneous emission can be derived immediately from the nonlinear Schrödinger equation without any phenomenology and quantization of radiation.The steady-state solutions of the nonlinear optical Bloch equations are obtained and compared with those obtained for linear optical equations.In this approximation, the light scattering by an atom is considered........

DOI
10.1093/ptep/ptw177

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In this paper, we consider the nonlinear Schrödinger equation describing the light-atom interaction while taking into account damping due to spontaneous emission.We demonstrate that the optical Bloch equations with damping due to spontaneous emission can be derived immediately from the nonlinear Schrödinger equation without any phenomenology and quantization of radiation.The steady-state solutions of the nonlinear optical Bloch equations are obtained and compared with those obtained for linear optical equations.In this approximation, the light scattering by an atom is considered........
A rigorous description of the light–matter interaction requires the use of quantized Maxwell equations to describe the light field and the Schrödinger equation to describe the material medium. This book does not attempt to deal with quantum optical problems that necessitate a field quantization. Instead we focus on properties that are generally grouped under the umbrella of nonlinear optics . Although many definitions of this expression exist, we use it in the sense that the properties we deal with do not depend in a crucial way on the field quantization. For practical purposes, this means in general that the average photon number is large and that we can neglect spontaneous emission as a dynamical process. Rather, we include the consequence of spontaneous emission, that is, the instability of the atomic levels, in a purely phenomenological way. When this is done, we call the material equations the Bloch equations instead of the Schrödinger equation. References [1] through [4] are the classical textbooks that contain a tentative justification for the transition from the Schrödinger to the Bloch equations. None of these attempts is satisfactory from a fundamental viewpoint because the real difficulty is to incorporate the finite lifetime (or natural linewidth) of the atomic energy levels in the Schrödinger equation. This remains an open problem as of now. The Maxwell–Schrödinger equations We consider atoms interacting with an intense electromagnetic field. By intense we mean a field for which the quantization is not necessary.
Within the self-consistent Maxwell-Pauli theory, a nonlinear Schrödinger equation with a short-range compensating field is derived. Stationary and non-stationary solutions of the obtained nonlinear Schrödinger equation for the hydrogen atom are investigated. It is shown that spontaneous emission and the associated rearrangement of the internal structure of the atom, which is traditionally called a spontaneous transition, have a simple and natural description within the classical field theory without any quantization and additional hypotheses. The solution of the nonlinear Schrödinger equation showed that, depending on the frequency of spontaneous emission, the compensating field behaves differently. At relatively low frequencies of spontaneous emission, there is no radiation (waves) of the short-range compensating field and this field does not carry away energy. In this case, the damping rate of the spontaneous emission coincides with that obtained in quantum electrodynamics (QED). At relatively high frequencies of spontaneous emission, radiation (waves) of the compensating field arises, which, along with electromagnetic radiation, carry away some of the energy. In this case, the damping rate of the spontaneous emission is greater (up 1.5 times) than that predicted by QED.
Interaction of atoms with quantum states of light is a long-standing problem that apart from fundamental physics has potential applications for optical quantum-state storage, quantum communication, and quantum information. A fully quantum mechanical treatment of this problem is usually very complicated mathematically. Here we show, however, that quantum mechanical evolution equations describing single-photon emission (absorption) by atomic ensembles can be written in a form equivalent to the semiclassical Maxwell-Bloch equations. This connection allows us to find exact analytical solutions of the fully quantum mechanical problem. We also found that semiclassical Maxwell-Bloch equations should be written in a form different from those commonly used. Namely, the classical limit of the quantum problem gives a propagation equation with the Laplacian operator on the right-hand side rather than with the second-order time derivative.
Chapter 11 presents the simplest and most intuitive approach to atom-optical interactions: the atom is modelled as a damped harmonic oscillator with spontaneous emission as the damping agent. The semiclassical viewpoint, in which the atom is treated as a quantum object but the fields are classical, is adopted throughout the chapter. Next, the two-level atom is introduced, which sets the stage for establishing light forces at the atomic level: the dipole-gradient force and the radiation pressure force. The Einstein B coefficient is revisited as stimulated dipole emission. Finally, the optical Bloch equations are used to reformulate more generally the appearance of optical forces and to describe the existence of atom Doppler cooling.
This tutorial article provides a detailed mathematical overview of the Optical Bloch equation describing the interaction of the two level atom with the Electromagnetic radi- ation field. The equations have been derived in the weak atom field coupling limit. The notion of the optical Bloch equations has been introduced in a systematic process starting from the interaction of the two level atom with the semiclassical radiation field which cannot completely the idea of the Spontaneous emission process later which was included theoretically by,Weigner-Weisskopf theory of spontaneous emission .In the later part of the article the optical Bloch equations has been derived exactly by treating the radiation field quantum mechanically. The radiation field has been quantized using the free field quantization techniques in the Coloumb Gauge with which the Lindblad Master equation has been derived systematically in the weak coupling limit. The optical Bloch equations has been derived for the resonantly driven two level atom from the Lindblad type master equation and later those equations has been solved in the steady state limit. The article also discuss the analytical aspects of describing the theory of atom matter interaction using full-fledged quantum mechanical approach.