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arXiv:quant-ph/9904104 · Physical Review Letters

Phase Transitions and the Internal Noise Structure of Nonlinear Schrödinger Equation Solitons

M. J. Werner, Stephen R. Friberg

Published Nov 24, 1997• 39 citations
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Abstract

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.

DOI
10.1103/physrevlett.79.4143

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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 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.
Optical solitons are pulses that propagate in optical fibers without temporal or spectral distortion, owing to balancing of the second-order dispersion of the refractive index by nonlinear self-phase modulation induced by the intensity-dependent refractive index. Classically, this stationary distortion-free propagation only requires the proper choice of pulse width and shape for a given pulse energy. However, quantum fluctuations do not undergo stationary propagation but rather evolve due to dispersion and self-phase modulation. As a soliton pulse propagates, zero-point or vacuum noise associated with the pulse evolves into squeezed noise. Vacuum fluctuations are responsible for the fundamental detector noise floor known as the shot noise limit. Reduction of the noise in the detection of squeezed soliton pulses has achieved a noise level 34% below this limit. In optical fibers, thermal fluctuation of the refractive index limit the noise reduction that can be achieved by adding phase fluctuations in excess of the vacuum noise. At low frequencies thermal noise originates from local structural dynamics of fused silica which produces a 1/f power spectrum and from guided acoustic wage Brillouin scattering which modulates the refractive index at frequencies characteristic of the fiber structure. The bandwidth of these fluctuations is ~2 GHz; and for soliton pulses with spectral bandwidth much >2 GHz, we have shown that this noise can be made insignificant. For very short pulses, however, noise from Raman scattering must be considered. Recent calculations incorporating this noise mechanism are compared with experiment.
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.
The quantum dynamics of coherent optical pulses are studied using a nondiagonal coherent-state generalized PPP representation including photon-phonon interactions. Photon-number squeezing of coherent quantum solitons using spectral filtering is theoretically predicted. It is shown that Raman noise does not significantly reduce photon-number squeezing produced by spectral filtering of 1-ps fundamental coherent quantum solitons in optical fibers. Coherent NsechNsechNsech pulses with N>1N>1N>1 can show a larger reduction in photon-number fluctuations even at room temperature. The reduction in quantum noise for N>1N>1N>1 is not restricted to photon number and an improvement of more than 3 dB is also found for the quadrature-phase squeezed soliton experiments using a fiber Sagnac interferometer at 77 K.
We report the photon-number squeezing of optical solitons. 2.7 ps pulses were launched as solitons down a 1.5 km optical fiber. For energies slightly above that of fundamental solitons, they broadened spectrally due to self-phase modulation caused by ${\ensuremath{\chi}}^{(3)}$ nonlinearities. Filtering away outlying components of the broadened spectra squeezed the soliton's photon-number fluctuations to 2.3 dB (41%) below the shot-noise limit. Accounting for losses, this corresponds to 3.7 dB (57%) photon-number squeezing. A quantum field-theoretic model shows that the outlying spectral components have large energy fluctuations, so that their removal causes squeezing.
We study the quantum-noise properties of spectrally filtered solitons in optical fibers. Perturbation theory, including a quantum description of the continuum, is used to derive a complete analytical expression for the second-order correlator of the amplitude quadrature. This correlator is subsequently used to optimize the frequency response of the filter numerically in order to achieve the minimum photon-number noise. For propagation distances up to three soliton periods, the length at which the best noise reduction occurs, a square filter is found to be approximately optimum. For longer distances, more-complicated filter shapes are predicted for the best noise reduction.