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Optics Express

Effects of transmission on Gaussian optical states

Colin J. McKinstrie, Kevin Marshall, Christian Weedbrook

Published Apr 17, 2015• 1 citation
Publisher

Abstract

The noise properties of phase-insensitive and phase-sensitive optical transmission links are described in detail, for Gaussian input signals. Formulas are derived for the quadrature covariance matrices of the output signals, which allow one to quantify the noise figures of the links and the fidelities of transmission. Another formula is derived, which relates the density operator of an output signal, in the number-state representation, to its covariance matrix. This density matrix allows one to quantify the decrease in coherence and changes in photon-number probabilities associated with transmission. Based on the aforementioned performance metrics, links with distributed phase-sensitive amplification perform significantly better than other links.

DOI
10.1364/oe.23.010856

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The noise properties of phase-insensitive and phase-sensitive optical transmission links are described in detail, for Gaussian input signals. Formulas are derived for the quadrature covariance matrices of the output signals, which allow one to quantify the noise figures of the links and the fidelities of transmission. Another formula is derived, which relates the density operator of an output signal, in the number-state representation, to its covariance matrix. This density matrix allows one to quantify the decrease in coherence and changes in photon-number probabilities associated with transmission. Based on the aforementioned performance metrics, links with distributed phase-sensitive amplification perform significantly better than other links.
In this paper the quantum noise properties of phase-insensitive and phase-sensitive parametric processes are studied. Formulas for the field-quadrature and photon-number means and variances are derived, for processes that involve arbitrary numbers of modes. These quantities determine the signal-to-noise ratios associated with the direct and homo-dyne detection of optical signals. The consequences of the aforementioned formulas are described for frequency conversion, amplification, monitoring, and transmission through sequences of attenuators and amplifiers.
We examine the effects of linear amplification and attenuation on the quantum-mechanical phase properties of light for fields with mean photon numbers of at least the order of 10. The phase probability density is found to satisfy a diffusion equation for both phase-insensitive and phase-sensitive amplifiers and attenuators. The solution is a convolution of the initial phase probability density with an infinite series of expanding Gaussians which clearly illustrates the diffusion of the phase. In particular, we find that for phase-insensitive amplification the phase of the field undergoes time-dependent uniform diffusion. In the limit of large amplification the diffusion ceases and the phase variance of the amplified light is given by the input phase variance plus an extra term which is equal to the phase variance of a coherent state of the same intensity as the initial field. We show that the reduced phase variance of phase-optimized states (relative to coherent states of the same intensity) is lost for power gains greater than the photon-cloning value of 2. In contrast, phase-sensitive amplifiers give rise to time-dependent nonuniform phase diffusion. The amount of phase diffusion depends on the relative phase angle between the light and the amplifier. If the peak of a relatively narrow phase probability density is near a minimum in the phase diffusion coefficient, then the phase noise added by the amplifier will be less than that found for a phase-insensitive amplifier. Further squeezing of the amplifier reduces the added phase noise proportionally. We find that it is possible, using phase-sensitive amplifiers, to amplify phase-optimized states by power gains considerably larger than 2 and still retain a reduced phase variance.
Optical communication links are usually made with erbium-doped fiber amplifiers, which amplify the signal waves in a phase-insensitive (PI) manner. They can also be made with parametric fiber amplifiers, in which the signal waves interact with idler waves. If information is transmitted using only the signals, parametric amplifiers are PI and their noise figures are comparable to those of erbium amplifiers. However, transmitting correlated information in the signals and idlers, or copying the signals prior to transmission, allows parametric amplifiers to be phase-sensitive (PS), which lowers their noise figures. The information capacities of two-mode PS links exceed those of the corresponding PI links by 2 b/s-Hz.
Adopting a new classical approach, the additive noise power generated in optical amplifiers is calculated in terms of power spectral density. The classical formalism used combines a coruscular approach to a phase-amplitude description of the optical field. The noise contributions of the input field fluctuations, including zero-point fluctuations, and of the electron momentum fluctuations at optical frequency linked to the amplifier itself, are clearly identified. The excess of noise associated to coupling or built-in losses is determinated. The well-known result of the Amplified Spontaneous Emission (ASE) is obtained for the laser amplifiers. This description is applied to linear phase-insensitive amplifiers and to inhomogeneous, nonlinear phase-insensitive Raman amplifier, pointing out the effects of gain compression and gain distribution. This new approach makes possible the treatment of the squeezed-state of light and the quadrature reduced noise amplifications.
The use of distributed amplifiers may have some potential advantages for the transmission of quantum information through optical fibers. In addition to the quantum noise introduced by the amplifiers, entanglement between atoms in the amplifying media and the optical field corresponds to which-path information that can further reduce the coherence. Here we analyze the effects of decoherence in a phase-insensitive distributed amplifier by using perturbation theory to calculate the state of the entire system including the atomic media. For an initial coherent state, tracing over the atomic states allows the reduced density matrix of the field to be expressed as a mixture of squeezed states with a reduced spread in photon number and an increased phase uncertainty. The amplifier noise and decoherence can be interpreted as being due to entanglement with the environment rather than the amplification of vacuum fluctuation noise.