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Physical Review A

Quantum-mechanical stability of solitons and the correspondence principle

Franz X. Kärtner, H. A. Haus

Published Sep 1, 1993• 21 citations
Publisher

Abstract

A Hanbury Brown--Twiss experiment is described that is capable of demonstrating, in principle, the solitary behavior of a quantum soliton in an arbitrary fundamental soliton state. The relevant correlation functions are evaluated and show that the quantum noise does not cause a breakup of the solitons of the quantized nonlinear Schr\"odinger equation. For large photon numbers, the intensity autocorrelation function of the fundamental soliton becomes independent of certain momentum and photon-number distributions, which gives us a correspondence principle for the fundamental soliton. Whereas intensity-correlation measurements based on direct detection show essentially classical behavior for large photon numbers, homodyne-detection experiments can uncover quantum effects such as the generation of Schr\"odinger-cat solitons.

DOI
10.1103/physreva.48.2361

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A Hanbury Brown--Twiss experiment is described that is capable of demonstrating, in principle, the solitary behavior of a quantum soliton in an arbitrary fundamental soliton state. The relevant correlation functions are evaluated and show that the quantum noise does not cause a breakup of the solitons of the quantized nonlinear Schr"odinger equation. For large photon numbers, the intensity autocorrelation function of the fundamental soliton becomes independent of certain momentum and photon-number distributions, which gives us a correspondence principle for the fundamental soliton. Whereas intensity-correlation measurements based on direct detection show essentially classical behavior for large photon numbers, homodyne-detection experiments can uncover quantum effects such as the generation of Schr"odinger-cat solitons.
Solitons of the classical nonlinear Schrödinger equation are stable to noise perturbations. One may ask whether the uncertainties associated with the quantum description of a soliton could cause the soliton to become unstable and to break up. It is not difficult to derive the fact that the first order soliton disperses, so that the probability of detecting the soliton at a specified position approaches zero as the time of propagation approaches infinity. This fact does not prove or disprove breakup. It can be shown by using the formalism of Ref. 1 that the intensity correlation function of a quantum soliton is time independent.2 In this paper we describe a form of a Hanbury, Brown, and Twiss experiment that could demonstrate the stability of the fundamental soliton (in principle), and we develop expressions for the hierarchy of correlation functions measured by such an experimental setup. If the soliton breaks up, the correlation functions should show a strong dependence on the propagation time of the soliton. The calculations demonstrate that all the measurable correlation functions do not depend on the propagation time; therefore the quantum soliton is stable.
Coherence properties of quantum optical solitons in the normalized fundamental soliton states are discussed in terms of the integrated two-point field and intensity autocorrelation functions, which are the relevant quantities in the quantum description of soliton interference experiments and in the Hanbury-Brown and Twiss experiment, respectively. Both the spatial and the temporal correlation properties are considered. The correlation functions are shown to be stationary in general, which demonstrates the solitary behavior of quantum fundamental solitons and indicates their stability in propagation. Explicit evaluation of the correlation functions in some limit cases enables us to make comparisons between the classical and the quantum-mechanical results. Conditions leading to the classical limit are derived and the intrinsic quantum phenomena are identified. Of great importance is the contribution of the momentum superposition, which can make the quantum results differ significantly from the corresponding classical ones.
Optical solitons are known to be classically stable objects which are robust to perturbations. In this work we show that due to quantum-mechanical effects, an optical soliton that is initially in a classical soliton coherent state will shed photons into the continuum and hence decay. The standard formulation of the quantized soliton uses the linearized version of the quantum nonlinear Schr\"odinger equation in the background of the classical soliton, and the quantized soliton remains stable in this approximation. We show that if higher-order interaction terms are taken into account, the soliton is no longer stable, and its photon number decreases quadratically as a function of the number of soliton cycles. We compute the power spectrum for the continuum radiation and find a narrow band that is localized about the initial soliton momentum with a cutoff that is inversely proportional to the initial soliton width.
We investigate the noise in a fundamental soliton state constructed from the exact energy eigenstates of the quantized nonlinear Schr\"odinger equation. It is shown that such a state has statistical properties that are substantially different from those of a coherent state with the same mean field. We can construct, for example, a fundamental soliton state that has initially minimum fluctuations for the four linearized soliton operators (particle number, phase, momentum, and position). We also show that the linearized noise analysis is not valid for nonlinear phase shifts larger than n01/4{\mathit{n}}_{0}^{1/4}n01/4​, where n0{\mathit{n}}_{0}n0​ is the average number of particles in the fundamental soliton. \textcopyright{} 1996 The American Physical Society.
The usual view of a soliton in an optical fiber is that it obeys the classical Maxwell equations, and propagates unchanged down a lossless fiber. Of course, it is well known that an optical amplifier introduces noise in position. It is less well known that even a perfectly lossless fiber, without an amplifier, is unable to support the classical invariant solution. Instead, all solitons experience quantum noise in position and phase, and this noise increases with propagation distance. Experimenters at IBM laboratories have recently observed the first quantum effects in solitons. These experiments focussed on quadrature squeezing, and were able to detect the large increase in phase-quadrature noise that is predicted quantum-mechanically. In this paper we treat the effects of pulse-position noise, and indicate how this noise source can limit information transmission using solitons. While these effects are small in currently proposed communication systems, they place a fundamental limit on future enhancements of soliton communications using pulse-position modulation. Of most scientific interest is the macroscopic nature of the quantum fluctuations, which can alter the location of a huge number of cooperatively positioned photons, and possible implications for QND momentum measurements.