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Physical Review Letters

Analytical solution to the quantum field theory of self-phase modulation with a finite response time

Luc Boivin, Franz X. Kärtner, H. A. Haus

Published Jul 11, 1994• 86 citations
Publisher

Abstract

The quantum theory of a field in 1+1 dimensions coupled to localized oscillators is developed. The solution to the Heisenberg equation for the field is given in closed form. It is shown that the nonlinearity of the medium is inevitably accompanied by phase noise of the field. This noise explains the preferential growth of the Stokes wave for short propagation distances.

DOI
10.1103/physrevlett.73.240

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The quantum theory of a field in 1+1 dimensions coupled to localized oscillators is developed. The solution to the Heisenberg equation for the field is given in closed form. It is shown that the nonlinearity of the medium is inevitably accompanied by phase noise of the field. This noise explains the preferential growth of the Stokes wave for short propagation distances.
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.
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.
Quantum theory of self-phase and cross-phase modulation of ultrashort light pulses in the Kerr medium is developed, taking into account the response time of an electronic nonlinearity. The corresponding algebra of time-dependent Bose operators is elaborated. It is established that the spectral region of the pulse where the quadrature fluctuation level is lower than the shot-noise one depends on the value of the nonlinear phase shift, the intensity of another pulse and the relaxation time of the nonlinearity. It is shown that the frequency of the pulse spectrum at which the suppression of fluctuations is maximum can be controlled by adjusting the other pulse intensity.
The systematic theory of the formation of the short light pulses in the squeezed state during the propagation in a medium with inertial Kerr nonlinearity is developed. The algebra of time-dependent Bose-operators is elaborated and the normal-ordering theorem for them is formulated. It is established that the spectral region where the quadrature fluctuations are weaker than the shot-noise, depends on both the relaxation time of the nonlinearity and the magnitude of the nonlinear phase shift. It is also shown that the frequency at which suppression of the fluctuation is greatest can be controlled by adjusting the phase of the initial coherent light pulse. The spectral correlation function of photons is introduced and photon antibunching is found.