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Physical Review A

Second-quantized models for optical solitons in nonlinear fibers: Equal-time versus equal-space commutation relations

Andrey B. Matsko, V. V. Kozlov

Published Aug 15, 2000• 22 citations
Publisher

Abstract

The dynamics of a quantum soliton propagating in nonlinear optical fibers is analyzed in the second-quantized picture in a fully canonical way. The quantum nonlinear Schr\"odinger equation with equal-time commutation relations for the field operators is solved using the formalism developed by Haus and Lai [J. Opt. Soc. Am. B $7,$ 386 (1990)]. The canonical approach is compared with the treatment of Haus and Lai based on phenomenological equal-space commutation relations. The two theories are shown to provide similar outcomes of a homodyne measurement. This result justifies the use of equal-space commutation relations in quantum studies of soliton propagation.

DOI
10.1103/physreva.62.033811

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The dynamics of a quantum soliton propagating in nonlinear optical fibers is analyzed in the second-quantized picture in a fully canonical way. The quantum nonlinear Schr"odinger equation with equal-time commutation relations for the field operators is solved using the formalism developed by Haus and Lai [J. Opt. Soc. Am. B 7,7,7, 386 (1990)]. The canonical approach is compared with the treatment of Haus and Lai based on phenomenological equal-space commutation relations. The two theories are shown to provide similar outcomes of a homodyne measurement. This result justifies the use of equal-space commutation relations in quantum studies of soliton propagation.
In the preceding paper [paper I of a two-part study; Lai and Haus, Phys. Rev. A 40, 844 (1989)] we have used the time-dependent Hartree approximation to solve the quantum nonlinear Schr\"odinger equation. In the present paper, the eigenstates of the Hamiltonian are constructed exactly by Bethe's ansatz method and are superimposed to construct exact soliton states. Both fundamental and higher-order soliton states are constructed and their mean fields are calculated. The quantum effects of soliton propagation and soliton collisions are studied in the framework of this construction. It is shown that a soliton experiences dispersion as well as phase spreading. The magnitude of this dispersion is estimated and is shown to be very small when the average photon number of the soliton is much larger than unity. The phase and position shifts due to a collision and the uncertainty of these shifts are also calculated.
An introduction to the quantum theory of nonlinear optics This article is provides an introduction to the quantum theory of optics in nonlinear dielectric media. We begin with a short summary of the classical theory of nonlinear optics, that is nonlinear optics done with classical fields. We then discuss the canonical formalism for fields and its quantization. This is applied to quantizing the electromagnetic field in free space. The definition of a nonclassical state of the electromagnetic field is presented, and several examples are examined. This is followed by a brief introduction to entanglement in the context of field modes. The next task is the quantization of the electromagnetic field in an inhomogeneous, linear dielectric medium. Before going on to field quantization in nonlinear media, we discuss a number of commonly employed phenomenological models for quantum nonlinear optical processes. We then quantize the field in both nondispersive and dispersive nonlinear media. Flaws in the most commonly used methods of accomplishing this task are pointed out and discussed. Once the quantization has been completed, it is used to study a multimode theory of parametric down conversion and the propagation of quantum solitons.
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.
Based on the linearization approximation and the conservation of commutation brackets, a general, self-consistent scheme is developed to quantize nonlinear optical-pulse propagation problems. A general computation procedure is developed to calculate the quantum uncertainties of the inner product between any given function and the (perturbed) field operator. As an illustration, a self-consistent quantum theory of the self-Raman effect in optical fibers is presented. The influence of the self-Raman effect on soliton squeezing is examined.
We present a canonical quantum theory of radiation in nonlinear media taking into account the effects of linear dispersion and absorption in a consistent way. We start from a microscopic model and apply recently developed concepts of the quantization of radiation in dispersive and absorbing linear media and extend the theory in order to include nonlinear optical processes. We derive propagation equations in space and time for the quantized radiation field, with special emphasis on the propagation of quantum-light pulses in Kerr media. In particular, the method enables us to derive systematically the noise sources that arise naturally in the nonlinear terms.
Quantum fluctuations of optical solitons in fibers are discussed on the basis of the exact quantum states in the frame of the quantum nonlinear Schr\"odinger equation. Position and momentum fluctuations as well as quadrature phase fluctuations are calculated, and explicit results are expressed in terms of the intrinsic quantum parameters. Soliton wave-packet spreading and phase-diffusion phenomena are clearly demonstrated, and the corresponding squeezing effects are analyzed. In particular, soliton amplitude squeezing is investigated in an experimental setup of a Mach-Zehnder interferometer containing an optical fiber. Coherent states for quantum solitons are discussed with attention given to their dynamical preservability. Compared with previous linearized approaches, the present study may provide a more fundamental treatment to quantum fluctuations of solitons.