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Physical Review Letters

Amplification under the Standard Quantum Noise Limit

Giacomo Mauro D’Ariano, Chiara Macchiavello, Matteo G. A. Paris

Published Dec 12, 1994• 5 citations
Publisher

Abstract

We show that a saturable amplifier can beat the "3 dB" quantum noise figure of linear amplifiers, approaching the ideal "0 dB" Heisenberg limit. Noise suppression is due to nonlinear saturation effects which cut off the high-intensity tail of the photon distribution. A Monte Carlo simulation of an actual semiconductor laser amplifier leads to a minimum noise figure of 0.51 dB, with a still sizable gain of 10.8 dB. The best performance corresponds to an output signal photon number which equals 0.7 times the saturation number.

DOI
10.1103/physrevlett.73.3187

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We show that a saturable amplifier can beat the "3 dB" quantum noise figure of linear amplifiers, approaching the ideal "0 dB" Heisenberg limit. Noise suppression is due to nonlinear saturation effects which cut off the high-intensity tail of the photon distribution. A Monte Carlo simulation of an actual semiconductor laser amplifier leads to a minimum noise figure of 0.51 dB, with a still sizable gain of 10.8 dB. The best performance corresponds to an output signal photon number which equals 0.7 times the saturation number.
Semiclassically, the signal-spontaneous beat noise in an optical amplifier can be reduced by 3 dB via a filtering technique. This creates a paradox since the noise figure's quantum limit of 3 dB could then be reduced to 0 dB, implying noiseless amplification even though noise is still being introduced by the amplifier. We discuss ways in which this paradox might be resolved and present experimental evidence that dispenses (up to the noise floor of our detectors) with one of these. A semiclassical treatment of the effects of a real (rather than ideal) filter is also presented and suggestions are made for improved (multi-mode) quantum models of this noise.
The standard quantum limit to the linewidth of a laser for which the gain medium can be adiabatically eliminated is ${l}_{0}=\ensuremath{\kappa}/2\mathrm{n\ifmmode \bar{}\else \={}\fi{}}.$ Here $\ensuremath{\kappa}$ is the intensity damping rate and $\mathrm{n\ifmmode \bar{}\else \={}\fi{}}$ the mean photon number. This contains equal contributions from the loss and gain processes, so that simple arguments which attribute the linewidth wholly to phase noise from spontaneous gain are wrong. I show that an unstimulated gain process actually introduces no phase noise, so that the ultimate quantum limit to the linewidth comes from the loss alone and is equal to ${l}_{\mathrm{ult}}=\ensuremath{\kappa}/4\mathrm{n\ifmmode \bar{}\else \={}\fi{}}.$ I investigate a number of physical gain mechanisms which attempt to achieve gain without phase noise: a linear atom-field coupling with a finite interaction time, a nonlinear atom-field coupling, and adiabatic photon transfer using a counterintuitive pulse sequence. The first at best reaches the standard limit l0,{l}_{0},l0​, the second reaches 34l0,\frac{3}{4}{l}_{0},43​l0​, and the third reaches the ultimate limit of lult=12l0.{l}_{\mathrm{ult}}=\frac{1}{2}{l}_{0}.lult​=21​l0​.
We present a theoretical analysis and an experimental study of the statistical properties of the noise accompanying an optical pulse propagating in a nonlinear semiconductor optical amplifier. Several degrees of gain saturation corresponding to different levels of optical signal-to-noise ratios (OSNRs) are examined. We employ the Heun numerical procedure to ensure proper convergence to the Stratonovich solution of the multiplicative propagation equation. This algorithm takes also into account the effect of gain saturation due to the amplified spontaneous emission noise. Moreover, the multicanonical Monte Carlo algorithm is used to efficiently calculate the probability density functions (pdfs), including the tails, of the peak of a pulse which emerges at the output of a saturated semiconductor optical amplifier. The results are compared to the corresponding pdfs obtained in a linear amplification system (where the optical noise is additive and Gaussian) having the same gain and under identical OSNR levels. We demonstrate that the pdf of the saturated amplifier is shifted toward lower power levels and is narrower, or equivalently, the mean and variance for the saturated amplifier case are smaller. Also, the difference between the two configurations increases with the degree of gain saturation. The theoretical predictions are confirmed by a series of experiments in which the pdfs at the output of the linear amplification scheme and saturated semiconductor optical amplifier are measured for an optical pulse of /spl sim/ 70-ps duration.
The intensity fluctuations in the output of a saturable optical amplifier are studied as the amplifier length is increased. Characteristics are determined by calculation of the diagonal elements of the density matrix of the optical field. Saturation affecting both upper-state and lower-state atomic populations is considered. Spontaneous emission distributed along the amplifier causes the "photon-number" distribution to approach a Bose-Einstein distribution during linear amplification if the input field is not Bose-Einstein. The onset of saturation reduces the relative fluctuations of the optical field both by introducing nonlinear gain and by reducing the spontaneous emission. Limiting solutions show that strong saturation leads to a major reduction in the fluctuations causing the photon-number distribution to approach a Poisson distribution.
We analyze and compare the characterization of a quantum device in terms of noise, transmitted bit-error-rate (BER) and mutual information, showing how the noise description is meaningful only for Gaussian channels. After reviewing the description of a quantum communication channel, we study the insertion of an amplifier. We focus attention on the case of direct detection, where the linear amplifier has a 3 decibels noise-figure, which is usually considered an unsurpassable limit, referred to as the standard quantum limit (SQL). Both noise and BER could be reduced using an ideal amplifier, which is feasible in principle. However, just a reduction of noise beyond the SQL does not generally correspond to an improvement of the BER or of the mutual information. This is the case of a laser amplifier, where saturation can greatly reduce the noise-figure, although there is no corresponding improvement of the BER. Such mechanism is illustrated on the basis of Monte Carlo simulations.