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Annual Meeting Optical Society of America

Classical and quantum noise transformations generated by a Kerr nonlinearity

Lance G. Joneckis, Jeffrey H. Shapiro

Published Jan 1, 1991• 0 citations
Publisher

Abstract

The Ken nonlinearity is the source of important phenomena in both classical and quantum optics. Previously we had shown that a material dependent time constant plays a pivotal role in the quantum noise transformations produced by the Kerr nonlinearity.1 In that work, we presented the classical and quantum noise correlations for propagation of an initially phase-insensitive beam through a lossless, dispersionless, Kerr medium. In the present paper, we extend the earlier work to an arbitrary Gaussian input state. Central to both developments is the inclusion of a coarse grained time constant, τ q , which models the non-instantaneous nature of the Kerr nonlinearity. Output spectra based on Gaussian as well as other types of input spectra are presented. Also predicted is a quantum recurrence length, equal to an interaction length which yields a 2π nonlinear phase shift for an energy density of the 1 photon/cτ q m. The possibility of experimental verification of this theory is also discussed.

DOI
10.1364/oam.1991.mo5

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The Ken nonlinearity is the source of important phenomena in both classical and quantum optics. Previously we had shown that a material dependent time constant plays a pivotal role in the quantum noise transformations produced by the Kerr nonlinearity.1 In that work, we presented the classical and quantum noise correlations for propagation of an initially phase-insensitive beam through a lossless, dispersionless, Kerr medium. In the present paper, we extend the earlier work to an arbitrary Gaussian input state. Central to both developments is the inclusion of a coarse grained time constant, τ q , which models the non-instantaneous nature of the Kerr nonlinearity. Output spectra based on Gaussian as well as other types of input spectra are presented. Also predicted is a quantum recurrence length, equal to an interaction length which yields a 2π nonlinear phase shift for an energy density of the 1 photon/cτ q m. The possibility of experimental verification of this theory is also discussed.
Intense light beams propagating in a lossless, dispersionless, single-mode optical fiber are subject to the Kerr effect, i.e., to the intensity-dependent refractive index of the fiber’s fused-silica core. Classically, Kerr-effect-induced self-phase modulation (SPM) can be used for spectral broadening of a picosecond pulse for grating-pair pulse compression down to femtosecond duration. Quantum mechanically, Kerr-effect-induced four-wave mixing (FWM) has been used to produce squeezed-state light. We present a quantum propagation theory for a lossless, dispersionless fiber with the Kerr nonlinearity. The theory includes classical SPM and quantum FWM within their regions of validity. It introduces a material time constant for the Kerr interaction, limiting the quantum phase shifts caused by the broadband zero-point fluctuations that accompany any input field, to develop a coarse-grained time multitemporal mode field analysis. Explicit expressions are obtained for the first and the second output-field moments when the fiber’s input field is in an arbitrary Gaussian state. These results are used to obtain homodyne-detection noise spectra, which are employed, in turn, to seek experimentally accessible manifestations of the Kerr time constant.
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.
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.
We consider two copropagating fields in a nonlinear Kerr medium, each with a particular phase and intensity. The Kerr medium possesses an intensity-dependent refractive index and the phase shift of each field thus depends on the intensities of the fields. Classically it is possible to induce an arbitrary phase shift of one field (the signal field) by either increasing the intensity of the other field (the control field) or by increasing the interaction legnth. We show that if the intensity of the control field is low, the phase shift on the signal is limited by the discrete nature of the photon-number distribution in the control field and cannot be increased simply by increasing the interaction length. In general the maximum phase shift of the signal field is \ensuremath{\varphi} if the control field possesses \ensuremath{\varphi} photons. This limit arises as a consequence of quantum recurrence effects.
The weak nonlinear Kerr interaction between single photons and intense laser fields has been recently proposed as a basis for distributed optics-based solutions to few-qubit applications in quantum communication and computation. Here, we analyse the above Kerr interaction by employing a continuous-time multi-mode model for the input/output fields to/from the nonlinear medium. In contrast to previous single-mode treatments of this problem, our analysis takes into account the full temporal content of the free-field input beams as well as the non-instantaneous response of the medium. The main implication of this model, in which the cross-Kerr phase shift on one input is proportional to the photon flux of the other input, is the existence of phase noise terms at the output. We show that these phase noise terms will preclude satisfactory performance of the parity gate proposed by Munro et al (2005 New J. Phys. 7 137).