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Optics and Spectroscopy

Quantum optics in waveguides with resonant atoms: Multi-photon scattering

Vladimir I. Yudson, P. Reineker

Published Mar 1, 2010• 3 citations
Publisher

Abstract

Quantum scattering of photons inside a one-dimensional waveguide caused by a number of closely located resonant two-level atoms is studied using the theory of integrable quantum systems. The multi-particle wave function of scattered photons is represented as a sum of terms of different degrees of “entanglement.” For two-photon scattering we discuss explicitly differences in photon correlations for the single two-level atom case and the case of several atoms.

DOI
10.1134/s0030400x10030082

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Quantum scattering of photons inside a one-dimensional waveguide caused by a number of closely located resonant two-level atoms is studied using the theory of integrable quantum systems. The multi-particle wave function of scattered photons is represented as a sum of terms of different degrees of “entanglement.” For two-photon scattering we discuss explicitly differences in photon correlations for the single two-level atom case and the case of several atoms.
We study multiphoton scattering inside a one-dimensional waveguide caused by a number of resonant two-level atoms located in a region small compared to the photon wavelength. Applying powerful methods of the theory of integrable quantum systems, we analyze photon correlations caused by scattering. In particular, for the important case of two-photon scattering, we describe explicitly the dependence of the scattering parameters on the number of two-level atoms. Also, we outline the method of describing scattering of an arbitrary number of photons and present explicit expressions for the scattering of three photons in the effective chiral one-dimensional model. The photon wave function is represented as a sum of terms of a different degree of ``entanglement.'' We discuss qualitative distinctions in photon correlations for the single two-level atom case and the case of several atoms.
We consider the coherent propagation of nnn photons in a one-dimensional coupled-resonator waveguide for n=2,3,4….n=2,3,4\dots{}.n=2,3,4…. The scattering by a three-level atom, which resides in one of the resonators of the waveguide and gives rise to only two-photon transitions, results in a perfect quantum switch that allows either total reflection or total transmission. This is to be contrasted to the case of a single photon inside a one-dimensional resonant waveguide scattered by a two-level system with single-photon transitions where only total reflection can be accomplished; viz. the system behaves only as a perfect mirror but not as an ideal, transparent medium.
We give an overview of exactly solvable many-body models of quantum optics. Among them is a system of two-level atoms which interact with photons propagating in a one-dimensional (1D) chiral waveguide; exact eigenstates of this system can be explicitly constructed. This approach is used also for a system of closely located atoms in the usual (non-chiral) waveguide or in 3D space. Moreover, it is shown that for an arbitrary atomic system with a cascade spontaneous radiative decay, the fluorescence spectrum can be described by an exact analytic expression which accounts for interference of emitted photons. Open questions related with broken integrability are discussed.
We present a full multimode treatment of the interaction of the quantized radiation field with a single two-level atom in a one-dimensional waveguide configuration. Starting with an incident pulse consisting of an arbitrary (finite) number of photons in a general initial state, we derive the equations of motion and a closed-form expression for the shape of the pulse after its interaction with the atom. We then specialize our results to the two-photon case where a number of analytical results can be derived, for both unidirectional and bidirectional systems. We study the effects of different pulse shapes, the manifestations of the entangled, so-called bound state of the two photons, single- and two-photon detection probabilities, and provide simple approximate results for the strong-coupling (or adiabatic, or long-pulse) regime. We also discuss the requirements for a true unidirectional setup, and the application of such a setup to photon sorting proposed by Witthaut, Lukin, and S\o{}rensen [Europhys. Lett. 97, 50007 (2012)].
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.
Realising a platform for efficiently guiding and manipulating light at the singlephoton level lies at the heart of optical quantum information processing. In this thesis we aid in the pursuit of this goal by analysing few-photon transport properties in photonic devices while keeping in mind possible applications and experimental challenges. We especially focus on single and two-photon transport properties in one-dimensional waveguides coupled to (artificial) atoms modelled as two and three-level emitters. We begin by introducing key concepts relevant for this thesis and then explain how to model such waveguide geometries. We show how directional-dependent coupling between a waveguide and a localized quantum system can be modelled from a chain of harmonic oscillators by allowing the system to couple to more than one site. Using the input-output formalism and scattering matrix theory we determine a general form for the single and two-photon S-matrices for a waveguide coupled to a two-level system. We also explain how to engineer a quantum optical Fanoresonance waveguide geometry and calculate how this modifies the S-matrices, the few-photon scattering probabilities, and the scattered output spectra. We show that the nonlinear effects in such a system generates highly correlated directional scattering probabilities. We device a protocol that can generate maximal entanglement between two spectrally distinct solid-state emitters by the use of multi-photon scattering. The results also reveal a rich structure for multi-photon states interacting with non-identical emitters. As a final investigation, we quantify how spectral diffusion in quantum dots affects finite-width single and two-photon scattering. Single-photon scattering probabilities are described by a simple convolution but this is not the case for a two-photon state scattering off a two-level emitter. We show that the effect of spectral wandering depletes the nonlinear scattering properties, but that it affects co and counter-propagating output probabilities differently.