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Optics Letters

Time dependence of quantum fluctuations in solitons

P. D. Drummond, S. J. Carter, Robert M. Shelby

Published Apr 1, 1989• 44 citations
Publisher

Abstract

We investigate quantum fluctuations in coherent solitons propagating in a dispersive nonlinear waveguide. Optimal squeezing or noise reduction occurs for a local oscillator width near the soliton width in time, i.e., the quantum fluctuations are localized.
DOI
10.1364/ol.14.000373

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We develop a quantum theory of propagation in dispersive nonlinear media from the foundations of a correctly quantized field theory. Quantum fluctuations are handled by coherent-state expansions of localized field states. A stochastic nonlinear Schrödinger equation in the field variables is obtained for media with an intensity-dependent refractive index. This predicts squeezing for a continuous-wave input, over a wide bandwidth with anomalous dispersion, and over a gradually reducing bandwidth with normal dispersion. The equation is easily modified to include thermal-noise sources as well. For solitons, fluctuations are reduced over the soliton bandwidth. This leads to quantum solitons that have quadrature fluctuations less than the level of vacuum fluctuations. The complementary quadrature has a correspondingly increased fluctuation level.
Classical optical solitons propagate in optical fibers without temporal or spectral distortion as a result of a balance between phase shifts due to group velocity dispersion and those due to nonlinear self-phase modulation. This balance is achieved by the proper choice of pulse shape and duration for a given pulse energy. The quantum fluctuations associated with a coherent soliton do not undergo stationary propagation, however, and a number of effects are predicted, including the evolution of vacuum noise associated with the input coherent pulse into squeezed fluctuations and the temporal spreading of the soliton pulse because of dispersion acting on quantum fluctuations of the pulse frequency.
We develop a quantum theory of propagation in dispersive nonlinear media via a quantized field theory and a coherent state expansion. The resulting stochastic nonlinear Schrodinger equation includes both quantum fluctuations and thermal noise due to refractive-index fluctuations in the medium. The theory can describe either cw propagation with dispersion or soliton propagation. In the cw case, enhanced squeezing occurs with anomalous dispersion. This is reduced by thermal noise, just as it is in the normal dispersion case. The noise-induced reduction in squeezing is more severe with normal dispersion owing to the intrinsic lack of phase matching in the normal dispersion regime. With anomalous dispersion, the cancellation of nonlinear by linear dispersion allows squeezing to occur over a wide bandwidth even in the presence of thermal fluctuations. Our theory is also able to treat soliton propagation in the anomalous dispersion case. A substantial degree of pulsed squeezing is obtained, and we point out that an optimal detection scheme requires a pulsed local oscillator. Results for the effects of thermal noise on soliton propagation are presented, in which case improvement is obtained by reducing the soliton time duration.
We treat the scaling of thermal phase-noise effects and quantum fluctuations for solitons in optical fibers. The central result is that the size of the thermal noise relative to quantum noise is reduced in proportion to the soliton duration for solitons whose duration is less than the thermal correlation times, suggesting that squeezed quantum solitons should be realizable in optical fibers. The physical basis for this effect is largely the fact that squeezing, which is nonlinear, occurs more rapidly for shorter pulses.
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 requires only the proper choice of pulse width and shape for a given pulse energy. However, quantum mechanical zero-point fluctuations of the amplitude and phase of the optical pulse limit the precision with which this pulse shape can be produced. These quantum fluctuations do not undergo stationary propagation but, rather, evolve because of dispersion and self-phase modulation, As a soliton pulse propagates, this zero-point or vacuum noise associated with the pulse evolves into squeezed noise owing to self-phase modulation in the fiber.1,2