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Physical Review A

Quantum-mechanical formulation of light propagation: A multiple-scattering approach

Wei Guo

Published Aug 31, 2007• 5 citations
Publisher

Abstract

Since in quantum optics light is represented in terms of photons, light propagation through a linear medium is discussed quantum mechanically in this paper by following the multiple-scattering process of one incident photon from the medium. To treat the photon and the medium on the same quantum footing, the medium is assumed to be an ensemble of uniformly distributed identical two-level atoms. It is found that inside the medium the incident photon follows the same propagation rules as a plane wave does in the classical domain, and has a possibility to become entangled with the atoms. It is also found that when interacting with a two-level test atom outside the medium, the output photon appears to be formally in a single mode identical to that of the incident photon.

DOI
10.1103/physreva.76.023834

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Since in quantum optics light is represented in terms of photons, light propagation through a linear medium is discussed quantum mechanically in this paper by following the multiple-scattering process of one incident photon from the medium. To treat the photon and the medium on the same quantum footing, the medium is assumed to be an ensemble of uniformly distributed identical two-level atoms. It is found that inside the medium the incident photon follows the same propagation rules as a plane wave does in the classical domain, and has a possibility to become entangled with the atoms. It is also found that when interacting with a two-level test atom outside the medium, the output photon appears to be formally in a single mode identical to that of the incident photon.
This thesis concerns the theoretical investigation of interference phenomena related to elastic and inelastic scattering of quantized light. The presented work is naturally divided into two parts, the first is concerning elastic scattering while the second explores inelastic scattering. In the first part we use a scattering-matrix formalism combined with results from random-matrix theory to investigate the interference of quantum optical states on a multiple scattering medium. We investigate a single realization of a scattering medium thereby showing that it is possible to create entangled states by interference of squeezed beams. Mixing photon states on the single realization also shows that quantum interference naturally arises by interfering quantum states. We further investigate the ensemble averaged transmission properties of the quantized light and see that the induced quantum interference survives even after disorder averaging. The quantum interference manifests itself through increased photon correlations. Furthermore, the theoretical description of a measurement procedure is presented. In this work we relate the noise power spectrum of the total transmitted or reflected light to the photon correlations after ensemble averaging. This analysis enables us to describe an experimental observation of the quantum nature of light that survives the averaging over disorder. In the second part we investigate inelastic scattering. This we do by first treating the scattering of light on dipoles embedded in an arbitrary dielectric environment. By considering the two different models for dipole interaction known as the minimal-coupling and electric-dipole interaction Hamiltonians, we find exact relations between the electric field and the dipole operators in the Heisenberg picture, while keeping the model of the dipoles arbitrary. Due to the exact treatment of the electric-field operators, we obtain kernels known from classical scattering theory to describe the propagation of the field from the dipoles. Using the found electric field operators we derive the Heisenberg equations of motion for the dipoles while treating them as quantum two-level systems and using the Born–Markov and rotating-wave approximations. Postponing the rotating-wave approximation to the very end of the formal calculations allows us to identify the different physical parameters of the dipole evolution in terms of physical quantities known from optics. Finally, we use our Heisenberg picture formalism to treat a dilute cloud of two-level atoms, by simplifying the equations of motion using a single-scattering approximation for the interaction between the atoms. This enables us to derive expressions for the steady-state population and fluorescence spectrum, where we find cooperative effects in both the elastic and the inelastic spectra.
Abstract The interaction of light with a single two-level emitter is the most fundamental process in quantum optics, and is key to many quantum applications. As a distinctive feature, two photons are never detected simultaneously in the light scattered by the emitter. This is commonly interpreted by saying that a single two-level quantum emitter can only absorb and emit single photons. However, it has been theoretically proposed that the photon anticorrelations can be thought of as arising from quantum interference between two possible two-photon scattering amplitudes, which one refers to as coherent and incoherent. This picture is in stark contrast to the aforementioned one, in that it assumes that the atom has two different mechanisms at its disposal to scatter two photons at the same time. Here we experimentally validate the interference picture by showing that, when spectrally rejecting only the coherent component of the fluorescence light of a single two-level atom, the remaining light consists of photon pairs that have been simultaneously scattered by the atom. Our results offer fundamental insights into the quantum-mechanical interaction between light and matter and open up novel approaches for the generation of highly non-classical light fields enabling, for example, Fourier-limited photon-pair sources that approach the theoretical limit in brightness.
We suggest a quantum description of Rayleigh light scattering on atoms. We show that an entangled state of the excited atom and the incident photon is formed during the scattering. Due to entanglement, a photon is never completely absorbed by the atom. The formation of the scattering spectrum is considered as a relaxation of incident photons to the reservoir of free space modes that are in thermal equilibrium. Additional excitations of the reservoir modes occurring during scattering are treated as scattered light. We show that even if the frequency of incident photons is incommensurate with an atomic transition frequency, the scattered light spectrum has a maximum at the frequency of incident photons. In addition, the linewidth of the scattered light is much smaller than that of the spontaneous emission of a single atom. Therefore, the process can be considered as elastic.
Classical optics describes quite successfully the propagation of laser light, both in free-space and inside a transparent medium. On the other hand, neither classical optics nor the conventional formulation of quantum optics can address adequately the propagation of nonclassical light. Nonclassical states of light are usually described by second-quantization operators acting onto the vacuum state of the radiation field, and the corresponding expectation values of the electric and magnetic fields are often zero. Clearly, the description of propagation of nonclassical light requires a direct quantum mechanical treatment of the spatial progression of the field operators. This chapter outlines the main features of propagative quantum optics, and examines more particularly the treatment of propagation through a linear (refractive) medium, and across a refractive interface. In order to facilitate the discussion of the physical ideas, a plane-wave propagation geometry, which greatly simplifies the mathematical treatment, is used.
The propagation of a strongly polarised electromagnetic wave through a resonance medium with arbitrary angular momenta of the levels is investigated in the adiabatic following approximation. Exact wavefunctions of the two-level atomic system with degenerate levels in the field of a circularly (linearly) polarised wave are constructed and the picture of the energetic level splitting and shift is discussed. Exact formulae for the non-linear refractive index of the medium in the field of the circularly (linearly) polarised wave are found. The propagation of counterpropagating polarised waves-an intense pump wave and a probe light signal-through the resonance medium is studied. The studies are carried out in the scope of a two-level (one-photon resonance) and a three-level (one- and two-photon resonances) medium. Formulae are found for the induced pump angle of rotation of the ellipse (plane) of polarisation for the probing signal that are exact in terms of the intensity of the pump wave. The spectral dependence of the angle of rotation is studied, and it is shown that a series of Stark-shifted resonance poles of atomic absorption and a three-photon scattering process arise.