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Quantum Optics Journal of the European Optical Society Part B

Quantum fluctuations do not annihilate the optical soliton

Dmitrii Kouznetsov

Published Aug 1, 1992• 4 citations
Publisher

Abstract

The behaviour of optical solitons in non-linear fibres with dispersion is discussed. The possibility of quantum decay of the fundamental soliton is investigated. The evolution of the second-order correlation function is analysed. A theorem on the evolution of the fourth-order correlation function is presented. It provides the proof of the stability of the fundamental soliton.

DOI
10.1088/0954-8998/4/4/003

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The behaviour of optical solitons in non-linear fibres with dispersion is discussed. The possibility of quantum decay of the fundamental soliton is investigated. The evolution of the second-order correlation function is analysed. A theorem on the evolution of the fourth-order correlation function is presented. It provides the proof of the stability of the fundamental soliton.
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.
The Q-function of the quantum soliton in a fibre is derived in a new form suitable for the assessment of the possibilities for experimental observations of specific quantum-soliton effects. The characteristic effects of soliton evolution are associated with a cubic term in the nonlinear phase angle and appear at distances well beyond the attenuation length (long range). At the accessible distances, the nonlinear dynamics of a fundamental soliton follows essentially the single-mode dynamics (short and middle range). However, for the higher-order solitons, the form of the evolution parameter suggests strong deviations of soliton dynamics from the single-mode with a square of soliton number. The use of enhanced-nonlinearity fibres and higher-order solitons might make the experimental studies of the specific quantum-soliton features more viable.
A linearized perturbation quantum theory of second-order soliton propagation is developed. The theory shows that the quantum fluctuations of photon number, phase, momentum, and position at an arbitrary propagation distance are linear combinations of these fluctuations at zero distance. The evolutions of second-order soliton quantum fluctuations are evaluated and compared with the quantum-fluctuation evolutions of a fundamental soliton. Based on this theory, the squeezing effect of a second-order soliton is studied. It is shown that, like a fundamental soliton, a second-order soliton also exhibits squeezing along propagation when a proper combination of the number and phase operators is detected.
Summary form only given. Optical solitons in fibres are attractive macroscopical quantum objects, which are particle-like pulses propagating over large distances without distortion due to the balance between Kerr nonlinearity and group velocity dispersion effects. However, in the frame of the quantum theory, they exhibit a complicated internal quantum structure with nontrivial nonlinear dynamics. In the process of the propagation along a fiber, the nonlinear dynamics of secant hyperbolical shaped optical pulses lead to the formation of classically stable soliton pulses with a rich variety of nonclassical properties, such as quantum noise reduction, spectral quantum noise correlations, and quantum non-demolition interactions. We discuss the experimental characterisation of these quantum properties of femtosecond optical solitons at the telecommunication wavelength of 1.5 /spl mu/m experiencing nonlinear-interactions in silica fibres.
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.
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.