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Physical Review A

Quantum-field superpositions via self-phase modulation of coherent wave packets

T. A. B. Kennedy, P. D. Drummond

Published Aug 1, 1988• 33 citations
Publisher

Abstract

With the use of a quantum theory of optical propagation, a set of nonlinear stochastic partial differential equations may be derived which describes the quantum-statistical properties of traveling waves due to self-phase modulation in a nonlinear medium. We calculate exact moments for the field, which exhibit classical self-phase modulation in the short-interaction limit and periodic quantum evolution through field-superposition states in the long-interaction limit. Reversible and irreversible behaviors of the stochastic description are reviewed. The relation of the present work to the corresponding single-mode nonlinear oscillator is discussed.

DOI
10.1103/physreva.38.1319

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With the use of a quantum theory of optical propagation, a set of nonlinear stochastic partial differential equations may be derived which describes the quantum-statistical properties of traveling waves due to self-phase modulation in a nonlinear medium. We calculate exact moments for the field, which exhibit classical self-phase modulation in the short-interaction limit and periodic quantum evolution through field-superposition states in the long-interaction limit. Reversible and irreversible behaviors of the stochastic description are reviewed. The relation of the present work to the corresponding single-mode nonlinear oscillator is discussed.
We develop a travelling-wave formulation of the theory of quantum optics and apply it to the description of the self-phase modulation of a short coherent pulse of light. The infinite nonlinear phase shift that is produced by the vacuum fluctuations is treated, within our approach, through a renormalization procedure. The renormalized theory successfully describes the nonlinear chirp that the pulse undergoes in the course of its propagation and permits the calculation of the squeezing characteristics of self-phase modulation.
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This paper presents a theory for quantum light propagation in a single-mode fiber which includes the effects of the Kerr nonlinearity, group-velocity dispersion, and linear loss. The theory reproduces the results of classical self-phase modulation, quantum four-wave mixing, and classical solution physics, within their respective regions of validity. It demonstrates the crucial role played by the Kerr-effect material time constant, in limiting the quantum phase shifts caused by the broadband zero-point fluctuations that accompany any quantized input field. Operator moment equations - approximated, numerically, via a terminated cumulant expansion - are used to obtain results for homodyne-measurement noise spectra when dispersion is negligible. More complicated forms of these equations can be used to incorporate dispersion into the noise calculations.
We identify and discuss nonlinear phase noise arising in Kerr self-phase modulation of a coherent light pulse propagating through an attenuating medium with third-order nonlinearity in a dispersion-free setting. This phenomenon, accompanying the standard unitary Kerr transformation of the optical field, is described with high accuracy as Gaussian phase diffusion with parameters given by closed expressions in terms of the system properties. We show that the irreversibility of the nonlinear phase noise ultimately limits the ability to transmit classical information in the phase variable over a lossy single-mode bosonic channel with Kerr-type nonlinearity. Our model can be also used to estimate the amount of squeezing attainable through self- phase modulation in a Kerr medium with distributed attenuation.
Starting with the canonical quantization procedure for the electromagnetic field inside an effective (linear or nonlinear) medium, we present a direct-space formulation of the theory of quantum optics. This approach does not use the conventional modal decomposition of the field, but relies on the electromagnetic momentum operator defined in terms of local electric- and magnetic-field operators. The momentum operator contains all the information on the spatial characteristics of the field, and can describe the translation of a short light pulse in a nonlinear medium, without a modal analysis of the pulse. Propagation is described through an operatorial wave equation that relates the temporal evolution of an electromagnetic pulse to its spatial progression. Through this equation, the direct-space approach to quantum optics can treat traveling-wave nonlinear-optical phenomena and, at the same time, account for their quantum statistics. The theory is applied to squeezed-light generation by the parametric down-conversion of a short laser pulse, as an illustration.