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Physical Review A

Weinberg’s nonlinear wave mechanics

K. Wódkiewicz, Marlan O. Scully

Published Nov 1, 1990• 59 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.42.5111

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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.
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.
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.
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 report new results in neoclassical quantum optics by applying the central theme of the Jaynes–Cummings paper of 1963. This theme, which Jaynes and Cummings emphasized in several ways, is the key role that must be assigned to radiative back reaction in quantum radiation dynamics. Its consequences are well known in subjects related to cavity quantum electrodynamics, and its influence on optical pulse evolution theory is equally strong. Here we provide a new solitonic extension of the radiation reaction theme to short-pulse broad-bandwidth propagation theory. We report a single formula encapsulating a three-pulse scenario in which the stable imprint of signal-pulse coherence, placed in an atomic medium at a predictable location by an initial pulse pair, is subsequently transferred by a third pulse to a different location. This is analysed by reference to our new analytic solution of the nonlinearly coupled density matrix equations that incorporate the radiation reaction fields obtained from the Maxwell equations for the pulses in the atomic medium.
When a monochromatic laser resonantly excites a multi-level atom the optical Bloch equations for the reduced atomic density matrix, which describe the evolution of such a system, may exhibit generalized damping terms. We justify the origin of these damping terms within the radiative cascade description of resonance fluorescence in the dressed-atom picture and in the spin-orbit coupling picture. We describe the connection between generalized damping terms and quantum interference in the process of spontaneous emission.
The Jaynes-Cummings paper treats the problem of one-atom interacting with one cavity mode with two alternative theories: quantum electrodynamics and semiclassical theory, and compares the predictions. The semiclassical theory was later extended to treat a single atom in free space interacting with a classical field including radiation reaction. The resulting theory was called Neoclassical theory, and is described by nonlinear optical Bloch equations. These nonlinear equations have shown up in a number of physical systems and were the subject of the famous Jaynes-Franken bet that was a prominent feature of the early Rochester Coherence and Quantum Optics Conferences. We will touch on all of these topics in a historical review.