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Conference on Lasers and Electro-Optics

Quantum solitons of the nonlinear Schrodinger equations and their quantum noise

R. Bullough, N. M. Bogoliubov, JUSSI TIMONEN

Published Aug 29, 2005• 0 citations
Publisher

Abstract

Summary form only given. Quantum solitons appear in the optical communications literature as classical solitons vested with quantum fluctuations. A generic model is the quantum nonlinear Schrodinger model which may describe soliton transmission in an optical fibre. We show how the natural 'quantum soliton' of the system derives from the matrix element of the quantised field operator under a condition of very large particle number. We also comment on its quantum noise and stability.

DOI
10.1109/iqec.2000.908152

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Summary form only given. Quantum solitons appear in the optical communications literature as classical solitons vested with quantum fluctuations. A generic model is the quantum nonlinear Schrodinger model which may describe soliton transmission in an optical fibre. We show how the natural 'quantum soliton' of the system derives from the matrix element of the quantised field operator under a condition of very large particle number. We also comment on its quantum noise and stability.
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.
Abstract Optical solitons classically are stationary solutions of the nonlinear Schrödinger equation. We perform a quantum field theoretic treatment by quantising a linearised fluctuation field around the classical soliton solution which can be seen as providing a background spacetime for the field. The linearised fluctuation modifies the soliton background, which is often neglected, reminiscent of the nondepleted-pump approximation. Going beyond this approximation and by using a number-conserving Bogoliubov approach, we find unstable modes that grow as the soliton propagates. Eventually, these unstable modes induce a considerable (backreaction) effect in the soliton. We calculate the backreaction in the classical field fully analytically in the leading second order. The result is a quadratic local decrease of the soliton photon number in propagation due to the backreaction effect of the unstable mode. Provided the initial pulse is close to the classical soliton solution, the unstable mode contributions always become dominant. We also consider practical scenarios for observing this quantum-induced soliton distortion, in the spectral domain. The backreaction, which we expect to be present in bright and dark, discrete and continuous solitons and other nonlinear pulses plays an important role in future optical analogue gravity experiments, for soliton lasers, and optical communications.
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.
We predict phase transitions in the quantum noise characteristics of systems described by the quantum nonlinear Schr\"odinger equation, showing them to be related to the solitonic field transition at half the fundamental soliton amplitude. These phase transitions are robust with respect to Raman noise and scattering losses. We also describe the rich internal quantum noise structure of the solitonic fields in the vicinity of the phase transition. For optical coherent quantum solitons, this leads to the prediction that eliminating the peak sideband noise due to the electronic nonlinearity of silica fiber by spectral filtering leads to the optimal photon-number noise reduction of a fundamental soliton.
We return to the notion of optical solitons as these were first introduced—namely as exact soliton and multi-soliton solutions of the semi-classical self-induced transparency (SIT) equations. The SIT equations, even in their most general inhomogeneously broadened forms, are infinite dimensional, completely integrable Hamiltonian systems, and thus display much mathematical structure, yet their exact soliton solutions display well the nonlinear (self-induced) transparency features of the physical phenomenon. The mathematical feature of complete integrability is handed down from the SIT equations to the nonlinear Schrödinger equations and from this point of view the nonlinear Schrödinger (NLS) equations equations are studied first of all at classical level and then at quantum level. The ‘quantum soliton’ of the quantum attractive NLS model is defined and its properties investigated in detail, particularly in a comparison with the particular properties ascribed to the ‘quantum soliton’ as described in a recently published article (Abraham 1999 Phys. World (February) 21) with this title; as suggested in that article, the quantum NLS soliton is seen as a qubit for quantum information purposes. The covariant form of the quantum attractive NLS model is the quantum sine-Gordon model and this is briefly studied in the context of quantum information also; new results for large area Josephson junctions are reported in this context. The quantum soliton of the one dimensional attractive NLS model has apparently been observed in 2002 in a Bose condensed metal vapour ( 7 Li, (Khaykovich et al 2002 Science 296 1290)) and a detailed calculation suggests that the condensate undergoes a ‘collapse’ from three to one space dimensions thus forming an exact soliton solution of the one dimensional quantum attractive NLS model.