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Physical Review A

Configuration-space quantum-soliton model including loss and gain

John M. Fini, Peter L. Hagelstein, H. A. Haus

Published Sep 1, 1999• 6 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.60.2442

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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.
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.
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.
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.
We introduce two approaches to modeling the quantum dynamics of optical χ(3)χ^{(3)}χ(3) solitons. Taking an open-system viewpoint, we project the underlying quantum field into system (soliton) and residual reservoir components. The reservoir is treated as either (i) a discrete ``Lanczos supermode'' (LSM) expansion which localizes dynamics to a few-supermode basis, or (ii) a non-local environment which can be traced out by deriving a Markovian master equation (ME). Using these methods, we analyze and identify the quantum structure of both the soliton's stability and its hierarchy of perturbations. Through numerical simulations, we confirm both methods effectively capture quantum-induced soliton phase shifts in a concise few-mode (single-mode for ME) basis, and the LSM approach also captures photon loss which arises from non-Markovian dispersive couplings. As neither method is limited to the linearized regime, our approaches provide powerful computational tools to analyze complex non-Gaussian quantum dynamics of solitons where other commonly-used methods fail, providing insight into such non-perturbative regimes. We also investigate radiation that occurs in the presence of higher-order dispersion with ultrashort pulses, deriving a ME that predicts photon loss consistent with classical theory, but find that both classical and ME theory dramatically underestimate the actual amount of dissipation, which we explain in terms of dispersive coupling-induced soliton broadening.
Solitons are localized wave solutions that appear in nonlinear systems when self-focusing effects balance the usual pulse dispersion of common optical media. Their stability and particle-like behavior make them ideal candidates for applications that range from communication to optical computing, but in real world physical systems, dissipative processes makes these otherwise stable solutions unstable, and true solitons are particularly hard to observe in systems featuring non-negligible dissipation. In these cases a special type of localized stable solutions, called dissipative solitons, are still possible to obtain, if in addition to a balance between diffraction and nonlinearity, an equilibrium between gain and loss is also present. In this work we discuss theoretically how a 4-level atomic system and an incoherent pumping process can be an ideal experimental testbed for studying this interesting class of solutions, featuring tunable optical properties and controllable gain/loss dynamics that allow to study both classes of temporal and spatial dissipative optical solitons.