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Journal of Optics

Quantum description of light propagation in generalized media

Teppo Häyrynen, Jani Oksanen

Published Jan 5, 2016• 3 citations
Publisher

Abstract

Abstract Linear quantum input–output relation based models are widely applied to describe the light propagation in a lossy medium. The details of the interaction and the associated added noise depend on whether the device is configured to operate as an amplifier or an attenuator. Using the traveling wave (TW) approach, we generalize the linear material model to simultaneously account for both the emission and absorption processes and to have point-wise defined noise field statistics and intensity dependent interaction strengths. Thus, our approach describes the quantum input–output relations of linear media with net attenuation, amplification or transparency without pre-selection of the operation point. The TW approach is then applied to investigate materials at thermal equilibrium, inverted materials, the transparency limit where losses are compensated, and the saturating amplifiers. We also apply the approach to investigate media in nonuniform states which can be e.g. consequences of a temperature gradient over the medium or a position dependent inversion of the amplifier. Furthermore, by using the generalized model we investigate devices with intensity dependent interactions and show how an initial thermal field transforms to a field having coherent statistics due to gain saturation.

DOI
10.1088/2040-8978/18/2/025401

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Abstract Linear quantum input–output relation based models are widely applied to describe the light propagation in a lossy medium. The details of the interaction and the associated added noise depend on whether the device is configured to operate as an amplifier or an attenuator. Using the traveling wave (TW) approach, we generalize the linear material model to simultaneously account for both the emission and absorption processes and to have point-wise defined noise field statistics and intensity dependent interaction strengths. Thus, our approach describes the quantum input–output relations of linear media with net attenuation, amplification or transparency without pre-selection of the operation point. The TW approach is then applied to investigate materials at thermal equilibrium, inverted materials, the transparency limit where losses are compensated, and the saturating amplifiers. We also apply the approach to investigate media in nonuniform states which can be e.g. consequences of a temperature gradient over the medium or a position dependent inversion of the amplifier. Furthermore, by using the generalized model we investigate devices with intensity dependent interactions and show how an initial thermal field transforms to a field having coherent statistics due to gain saturation.
We use a continuous-mode quantization scheme to derive relations between the output- and input-field operators for traveling-wave propagation along attenuating and amplifying optical fibers. These relations provide complete information on the temporal and longitudinal spatial developments of the signal field. They are used here to obtain the effects of propagation on the first and second moments of the photocount in direct detection and of the signal field measured in balanced homodyne detection. Some of the results are similar to those obtained for attenuation or amplification of standing waves in cavities, and, for example, the survival of any input squeezing still limits the maximum gain to twofold. There are, however, additional propagation effects for the traveling-wave system. Thus, in direct detection, it is necessary to take account of the changes in gain profile with propagation distance, and in homodyne detection there are fundamental quantum-mechanical restrictions on the minimum field uncertainties that can be achieved in measurements at separated space-time points. These uncertainty properties are derived in detail and illustrated by the example of a squeezed input signal.
Introduction In this chapter we will present the theory of spectral gain in optical amplifiers and the related optical power distribution. The concepts are general, as they apply to every optical amplifier configuration, independent of the specific architecture and implementation technology. We will configure the optical amplifying medium as a traveling-wave device, where the optical field undergoes optical amplification as it propagates from the input to the output sections. Generally speaking, optical amplification is achieved by a net stimulated emission process. Referring to a generic section z of infinitesimal thickness dz , optical gain is determined by the imbalance between net stimulated emission and cavity losses, including extrinsic absorption, reflection and radiated fields. As we know from the theory of light amplification in the three-level atomic system presented in Chapter 5, stimulated absorption of photons is governed by the same transition rate as stimulated emission, and is included in the net stimulated emission process. With this in mind, we note the identity between stimulated emission and the stimulated absorption rates, Γ 12 = Γ 21 , and the net stimulated photon term Γ[ j 2 ( t ) – j 1 ( t )] p ( t ) in the rate equation. In order to develop the traveling-wave model, the photon density becomes a function of the longitudinal position 0 ≤ z ≤ L along the amplifier medium. Accordingly, we will set p ( t , z ) as the relative density of stimulated photons per unit volume (0 ≤ p ( t , z ) ≤ 1) at time t and section z . When photons with appropriate energy proceed along the amplifying medium, they are amplified by means of net stimulated emission. At some section z = z 0 , the steady-state photon density P ( z ) can approach the saturation density P sat ( z ).
A simple model of a traveling-wave parametric amplifier is discussed in quantum-mechanical terms. The amplifier consists of a slab of Raman-active material, illuminated on one face by an intense laser beam that serves as the pump mode; an optical-phonon oscillation serves as the idler mode, and the scattered Stokes light occupies the signal mode. With only a few simplifying assumptions, it is possible to solve the equation of motion for the signal field operator exactly. For large times, both an amplified steady signal and the quantum noise contributed by the material assume simple steady states in an amplifier of finite length. The transient and asymptotic behaviors of the signal and noise intensities are found, together with the corresponding first-order correlation functions. The quasiprobability distribution which specifies the complete density operator for the electric field strength of the amplified signal is also derived.
The properties of a linear optical amplifier or attenuator in which the input light is coupled to a collection of non-saturable atoms are considered. The photon-number factorial moments and probability distribution of the amplifier output are derived for arbitrary input statistics. Relations are obtained between the output and input variances in photon number, phase-angle cosine, and electric-field magnitude. The effects of amplification and attenuation on signal-to-noise ratio are obtained for both direct and homodyne detection. Particular attention is paid to the extents to which the non-classical properties of photon antibunching and squeezing are preserved by amplification and attenuation. It is found that both properties can at best survive only twofold intensity amplification.
We present a three-dimensional quantum theory of light scattering, absorption, and emission in arbitrary planar dielectric systems. The theory is based on a scheme for quantization of the electromagnetic field in dispersive and lossy arbitrary planar dielectrics that takes explicitly into account the finite extent of the medium. We present general quantum-optical input–output relations of the output TE and TM photon operators to the corresponding input photon operators and to the reservoir noise operators. These three-dimensional input–output relations are uniquely determined by knowledge of the classical light modes of the planar dielectric structure.
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.