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Quantum Electronics and Laser Science Conference

Configuration space methods and quantum optical solitons

John M. Fini, H. A. Haus

Published May 7, 2000• 0 citations
OpenAlex

Abstract

Summary form only given. The term configuration space applies to any approach which treats a quantum system in terms of particle coordinates, as opposed to a second-quantized or field operator approach. Any physical system can be described in terms these two exactly equivalent formulations. Configuration space methods are a natural approach to certain propagation problems, providing simplified calculations and new physical insight. One motivation is the potential to design devices that manipulate wave functions for the collective coordinates of optical pulses. Some examples are discussed.

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Summary form only given. The term configuration space applies to any approach which treats a quantum system in terms of particle coordinates, as opposed to a second-quantized or field operator approach. Any physical system can be described in terms these two exactly equivalent formulations. Configuration space methods are a natural approach to certain propagation problems, providing simplified calculations and new physical insight. One motivation is the potential to design devices that manipulate wave functions for the collective coordinates of optical pulses. Some examples are discussed.
We examine the effects of loss and gain on a quantum soliton using a configuration-space approach. A simple microscopic model of local photon-matter interaction is applied to solitons with arbitrary quantum superpositions of momentum. Such a theory is needed in the analysis and design of systems that manipulate the wave function of soliton center-of-mass coordinates. The formalism is tested by calculating the momentum noise induced by loss and gain, and by comparison with the well-known Gordon-Haus calculation [J. P. Gordon and H. A. Haus, Opt. Lett. 11, 665 (1986)]. The comparison provides physical insight and reproduces the old result as a special case.
Lai and Haus [Phys. Rev. A 40, 854 (1989)] recently introduced the use of a photon configuration-space wave function to analyze optical soliton propagation in a fiber. This is of interest as photon wave functions have long been the source of controversy; the work of Lai and Haus was the first to demonstrate that they could be of use in quantum optics. The systematic reduction of photon configuration-space wave functions directly from QED was recently carried out, which puts the general approach on solid ground. Here we apply the photon configuration-space approach systematically to the dynamics of the soliton center of mass and phase spread, and to squeezing in optical fibers. The predictions of the configuration-space model agree with results from the quantum field theory linearization models, which provide an important benchmark for the new approach. The configuration-space model leads ultimately to a description of the soliton as a quantum object that obeys free particle dynamics in relative position and phase. Phase squeezing appears in the configuration-space model as a consequence of free particle dynamics following initial phase localization due to the use of an initial classical soliton state. \textcopyright{} 1996 The American Physical Society.
The stochastic theory presented by Drummond, Gardiner, and Walls [Phys. Rev. A 24, 914 (1981)] is an interesting approach to problems in quantum optics. In this theory, an exact, quantum evolution is written in terms of classical functions (not operators) driven by explicit, quantum noise. We examine the origin of uncertainty in the formalism through the simple example of a single, nonlinear oscillator. We then test the stochastic theory applied to the problem of soliton propagation. We extend the linearized stochastic model by computing analytically quantum uncertainties in the four basic soliton parameters: photon number, momentum, phase, and position. Agreement with second-quantized and configuration-space soliton theories verifies the stochastic formalism.
This paper deals with certain aspects of the phase-space formulation of quantum mechanics. Noting the ambiguities involved in quantizing the classical phase-space dynamics, a generalized quantum-mechanical phase space is introduced in terms of eigenvalues of two mutually incompatible complete sets of operators. General definitions are given for the phase-space distribution function corresponding to a given abstract state of the system and the phase-space function for an arbitrary operator. Our formulation goes over to the Wigner formulation when the complete sets correspond to position and momentum operators in Cartesian coordinates. Explicit examples are discussed with use of action-angle and polar coordinates for simple one-, two-, and three-dimensional systems. Symmetry changes in phase space are discussed by applying Gel'fand-Levitan transformations and it is shown that, in general, the degeneracies of motion in phase space are not reflected in the energy eigenvalues.
In applied physics and in engineering intuitive understanding is a boost for creativity and almost a necessity for efficient product improvement. The existence of soliton solutions to the quantum equations in the presence of self-interactions allows us to draw an intuitive picture of quantum mechanics. The purpose of this work is to compile a collection of models, some of which are simple, some are obvious, some have already appeared in papers and even in textbooks, and some are new. However this is the first time they are presented (to our knowledge) in a common place attempting to provide an intuitive physical description for one of the basic particles in nature: the electron. The soliton model applied to the electrons can be extended to the electromagnetic field providing an unambiguous description for the photon. In so doing a clear image of quantum mechanics emerges, including quantum optics, QED and field quantization. To our belief this description is of highly pedagogical value. We start with a traditional description of the old quantum mechanics, first quantization and second quantization. We describe an intuitive description of first and second quantization based on the concept of solitons, pioneered by the double solution of de Broglie in 1927. Many aspects of first and second quantization are clarified and visualized intuitively, including the so called vacuum fluctuations.
There are many examples in theoretical physics where a fundamental quantity can be considered a quadratic form ρ=∑iρi=∣Ψ∣2 and the corresponding linear form Ψ=∑iψi is highly relevant for the physical problem under study. This, in particular, is the case of the density and the wave function in quantum mechanics. In the study of N-identical-fermion systems we have the additional feature that Ψ is a function of the 3N configuration space coordinates and ρ is defined in three-dimensional real space. For many-electron systems in the ground state the wave function and the Hamiltonian are to be expressed in terms of the configuration space (CS), a replica of real space for each electron. Here we present a geometric formulation of the CS, of the wave function, of the density, and of the Hamiltonian to compute the electronic structure of the system. Then, using the new geometric notation and the indistinguishability and equivalence of the electrons, we obtain an alternative computational method for the ground state of the system. We present the method and discuss its usefulness and relation to other approaches.