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Journal of Physics A Mathematical and General

Direct derivation of a macroscopic NLS equation from the quantum theory

Hervé Leblond

Published Mar 30, 2001• 9 citations
Publisher

Abstract

The use of the matrix density formalism, and of a multiscale expansion, allows the derivation of a macroscopic nonlinear evolution equation for a short light pulse (a nonlinear Schrödinger equation), directly from the microscopic quantum mechanics equations. We consider the simple case of a monochromatic plane wave, interacting with independent two-level atoms, to show that such a computation is possible. For their linear part, the results agree with that of the linear dispersion theory, but the obtained nonlinear coefficient differs appreciably from that derived from the computation of the so-called nonlinear susceptibilities, except in one simple particular situation.

DOI
10.1088/0305-4470/34/14/316

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The use of the matrix density formalism, and of a multiscale expansion, allows the derivation of a macroscopic nonlinear evolution equation for a short light pulse (a nonlinear Schrödinger equation), directly from the microscopic quantum mechanics equations. We consider the simple case of a monochromatic plane wave, interacting with independent two-level atoms, to show that such a computation is possible. For their linear part, the results agree with that of the linear dispersion theory, but the obtained nonlinear coefficient differs appreciably from that derived from the computation of the so-called nonlinear susceptibilities, except in one simple particular situation.
Starting with the same form of atomic nonlinearity, Weinberg [Ann. Phys. (NY) 194, 336 (1989)] and W\'odkiewicz and Scully [Phys. Rev. A 42, 5111 (1990)] obtained contradictory results concerning an evolution of the atomic inversion w in a two-level atom in Weinberg's nonlinear quantum mechanics: If the atom is initially in a ground state then either the evolution of w (1) can be linear if one uses a nonlinear generalization of the Jaynes-Cummings Hamiltonian, or (2) is always nonlinear if one uses the nonlinear Bloch equations derived from the nonlinear atomic Hamiltonian function. It is shown that the difference is rooted in inequivalent descriptions of the composite ``atom-plus-field'' system. The linear evolution of w results from a ``faster-than-light communication'' between the atom and the field. If one applies a description without the ``faster-than-light telegraph'' then the calculations based on a suitably modified Jaynes-Cummings Hamiltonian lead to the same dynamics of w as is found in semiclassical calculations based on Bloch equations. It is shown also that a nonlinear quantum mechanics based on a nonlinear Schr\"odinger equation does not possess a natural probability interpretation.
Abstract The aim of our work was to find an unambiguous connection between irreversible macroscopic dynamics and reversible microdynamics that makes it possible to manifest irreversibility on a submacroscopic level without the use of coarse graining arguments. On this level the state of a physical system can be described by a field amplitude Ψ and the time evolution of this system is determined by a field equation for Ψ. For conservative systems, this field equation is formally identical with the linear Schrödinger equation, which can be constructed with the help of the classic Hamiltonian function for the corresponding problem. Regarding irreversible phenomena like damping due to a frictional force, for those dissipative systems no classic Hamiltonian function exists. Therefore the corresponding field equation cannot be obtained in the usual way. Nevertheless, also for dissipative systems it is possible to obtain a field equation in an unambiguous way using only Newton's form of classic mechanics. The result of our method is a nonlinear Schrödinger‐type field equation with a logarithmic nonlinearity. We discuss in more detail the properties of our logarithmic nonlinearity that corresponds to a macroscopic frictional force in a unique way. A figurative interpretation in terms of environment and interaction can be given. From a more formal point of view, the compatibility of our nonlinear operator with principles known from the theory of linear operators is investigated. One of the surprising results is the fact that although our nonlinear Hamiltonian HNL is “Hermitean” in the usual sense, in contrast to the linear theory an operator exp(i · HNL) is not unitary. Furthermore, in our theory the time‐derivative of the mean value of an operator is not only essentially determined by (the mean value of) its commutator with the Hamiltonian. There also occurs an additional term that causes irreversibility of the evolution and is connected with the feature of our theory that (in general) time derivative and construction of the mean value are no longer commuting operations. This fact shows some similarity with coarse graining theories, but in our theory the reason can be traced back unambiguously to an irreversible physical phenomenon.
In Chap. 1 we introduced the Schrodinger equation and gave solutions for a number of simple cases. The general solution was then obtained by a linear superposition of these solutions. In our work, we are particularly concerned with the way in which atomic systems interact with electromagnetic radiation. These systems are still described by wave functions given by superpositions of simple solutions, but the expansion coefficients Cn (probability amplitudes) become functions of time. In this chapter we derive equations of motion for the expansion coefficients using Schrddinger’s equation and introduce the electric-dipole interaction energy. Then, specializing to a two-level system and a monochromatic electromagnetic field, we solve the equations of motion both by time-dependent perturbation theory and by the more exact Rabi method.
We consider the problem of the propagation of short-duration light pulses in a two-level medium in the regime of strong interaction, near the resonance, with no restriction on the strength of the coupling between field and medium. We prove that the local variations of the population difference created by the electric field actually induce a nonlinear effect by coupling the fundamental of the field Fourier component to its harmonics. As a result, the multiscale averaging limit of the Maxwell-Bloch system is a new system of coupled nonlinear Schrodinger equations for the slowly varying envelope of the electric field. This system has one integrable limit when the electric field is circularly polarized: it is the scalar nonlinear Schrodinger equation. Moreover, it possesses three constants of the motion and a Hamiltonian formulation. Then we prove that our system is nonintegrable as it does not pass the Painleve test. Finally, a stability analysis shows that it is modulationally unstable and thus it propagates steady pulses of coherent light.
Many phenomena in the observable macroscopic world obey nonlinear evolution equations while the microscopic world is governed by quantum mechanics, a fundamental theory that is supposedly linear. In order to combine these two worlds in a common formalism, at least one of them must sacrifice one of its dogmas. Linearizing nonlinear dynamics would destroy the fundamental property of this theory, however, it can be shown that quantum mechanics can be reformulated in terms of nonlinear Riccati equations. In a first step, it will be shown that the information about the dynamics of quantum systems with analytical solutions can not only be obtainable from the time-dependent Schrödinger equation but equally-well from a complex Riccati equation. Comparison with supersymmetric quantum mechanics shows that even additional information can be obtained from the nonlinear formulation. Furthermore, the time-independent Schrödinger equation can also be rewritten as a complex Riccati equation for any potential. Extension of the Riccati formulation to include irreversible dissipative effects is straightforward. Via (real and complex) Riccati equations, other fields of physics can also be treated within the same formalism, e.g., statistical thermodynamics, nonlinear dynamical systems like those obeying a logistic equation as well as wave equations in classical optics, Bose- Einstein condensates and cosmological models. Finally, the link to abstract "quantizations" such as the Pythagorean triples and Riccati equations connected with trigonometric and hyperbolic functions will be shown.