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Physical Review A

Quantum coherence of optical solitons in fibers

Demin Yao

Published Aug 1, 1995• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.52.1574

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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.
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.
This article presents a review of coherence properties of electromagnetic fields and their measurements, with special emphasis on the optical region of the spectrum. Analyses based on both the classical and quantum theories are described. After a brief historical introduction, the elementary concepts which are frequently employed in the discussion of interference phenomena are summarized. The measure of second-order coherence is then introduced in connection with the analysis of a simple interference experiment and some of the more important second-order coherence effects are studied. Their uses in stellar interferometry and interference spectroscopy are described. Analysis of partial polarization from the standpoint of correlation theory is also outlined. The general statistical description of the field is discussed in some detail. The recently discovered universal "diagonal" representation of the density operator for free fields is also considered and it is shown how, with the help of the associated generalized phase-space distribution function, the quantum-mechanical correlation functions may be expressed in the same form as the classical ones. The sections which follow deal with the statistical properties of thermal and nonthermal light, and with the temporal and spatial coherence of blackbody radiation. Later sections, dealing with fourth- and higher-order coherence effects include a discussion of the photoelectric detection process. Among the fourth-order effects described in detail are bunching phenomena, the Hanbury Brown-Twiss effect and its application to astronomy. The article concludes with a discussion of various transient superposition effects, such as light beats and interference fringes produced by independent light beams.
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.
Introduction In this chapter we will put the concept of coherence on a more mathematical basis. This will involve the formal definition of a number of functions that are used to describe the coherence properties of optical fields. These include the analytic signal, various correlation functions, and the degree of coherence for describing both temporal and spatial coherence. We shall see that the degree of temporal coherence is quantitatively related to the lineshape function and that the degree of spatial coherence between two points is determined by the size, intensity distribution, and location of an illuminating source. We will use the wave equation, and special solutions to the wave equation called Green's functions, to show how spatial coherence varies from point to point. The chapter will conclude with a discussion of how the intensity fluctuations of a source depend on its coherence properties, and we will examine a specific scheme, namely the Hanbury Brown–Twiss experiment, by means of which this relationship is studied. This discussion will involve a discussion of “photon statistics,” namely the time variation of the “detection” of photons from a source. In a quantum-mechanical context, square-law detectors respond to these quantized excitations of the optical field, which we call photons. In classical coherence theory it is advantageous to represent the real electromagnetic field by a complex quantity, both for its mathematical simplicity and because it serves to emphasize that coherence theory deals with phenomena that are sensitive to the “envelope” or average intensity of the field.
We study quantum and classical Hanbury Brown-Twiss correlations in waveguide lattices. We develop a theory for the propagation of photon pairs in the lattice, predicting the emergence of nontrivial quantum interferences unique to lattice systems. Experimentally, we observe the classical counterpart of these interferences using intensity-correlation measurements. We discuss the correspondence between the classical and quantum correlations, and consider path-entangled input states which do not have a classical analogue. Our results demonstrate that waveguide lattices can be used as a robust and highly controllable tool for manipulating quantum states, and offer new ways of studying the quantum properties of light.