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Physica Scripta

Nonlinear Schrödinger-Liouville Equation with Antihermitian Terms

Salomon Sylvain Mizrahi, V. V. Dodonov, D. Otero

Published Jan 1, 1998• 7 citations
Publisher

Abstract

We extend a nonlinear dynamical equation, proposed a few years ago by S. Weinberg to test a possible breakdown in the linearity of Quantum Mechanics, by introducing antihermitian terms (linear and nonlinear) allowing irreversible evolution. Applying the new equation to a two-level atom, we verify that for a particular situation the antihermitian terms lead exactly to the nonlinear Bloch equations obtained in the neoclassical theory of spontaneous emission proposed by Stroud and Jaynes, which do not exhibit pure spontaneous emission. Lifting that particularization we gain additional antihermitian terms in the nonlinear Bloch equation that take care of the pure spontaneous emission problem. In view of this new terms we discuss whether they are meaningful as a "correction" to linear Quantum Mechanics.

DOI
10.1088/0031-8949/57/1/003

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We extend a nonlinear dynamical equation, proposed a few years ago by S. Weinberg to test a possible breakdown in the linearity of Quantum Mechanics, by introducing antihermitian terms (linear and nonlinear) allowing irreversible evolution. Applying the new equation to a two-level atom, we verify that for a particular situation the antihermitian terms lead exactly to the nonlinear Bloch equations obtained in the neoclassical theory of spontaneous emission proposed by Stroud and Jaynes, which do not exhibit pure spontaneous emission. Lifting that particularization we gain additional antihermitian terms in the nonlinear Bloch equation that take care of the pure spontaneous emission problem. In view of this new terms we discuss whether they are meaningful as a "correction" to linear Quantum Mechanics.
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.
From Weinberg's nonlinear wave equation we derive the nonlinear generalization of the optical Bloch equations with radiative damping. We compare these equations with Fermi's nonlinear radiation-reaction theory and with Jaynes's neoclassical theory. The nonlinear constant \ensuremath{\epsilon}, which is small if compared to the binding energy per nucleon, can be comparable to the radiative lifetime of an excited state of an ion stored in an ion trap. We show that in such a case the nonlinear modification of the optical Bloch equations can lead to large-scale effects in coherent transients, in atomic quantum jumps, and in resonance fluorescence and can modify the emission line shape.
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........
We propose a nonlinear quantum master equation, which results in the nonlinear Bloch-like equations, similar to those which are used usually to describe the collective spontaneous emission by a system of two-level atoms or a spin system in a magnetic field. On the one hand, the new equation is the unique simplest homogeneous and nonsingular generalization of the linear master equations for quantum systems in the finite-dimensional Hilbert space. On the other hand, it corresponds to the mean-field approximation, and in this form it can be used for a larger class of systems. Our equations describe both triggered and pure spin emission processes (when the population is completely inverted), and their solutions show the occurrence of secondary emission peaks, as observed in experiments.
We use a self-field approach to quantum electrodynamics (QED) to show how one may obtain spontaneous emission and the Lamb shift in a two-level atom without second quantization of the radiation field. In addition, we compare the self-field formalism to that of the neoclassical theory of electrodynamics advanced by Crisp and Jaynes [Phys. Rev. 179, 1253 (1969)]. We show that the neoclassical model can be obtained from the self-field approach used here, but that the two are not equivalent. In particular, the self-field approach appears to give a more complete description of radiative processes. Finally, we show that the neoclassical theory's prediction of a nonexponential ``chirruped'' decay is most likely a mathematical artifact of the improper application of the superposition principle in a nonlinear model where such a principle does not hold. A correct treatment with self-field QED yields the usual exponential decay dynamics.