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American Journal of Physics

Quantum mechanical description of linear optics

Johannes Skaar, Juan Carlos García-Escartín, Harald Landrø

Published Sep 22, 2004• 48 citations
Publisher

Abstract

Linear optical networks with any number of modes are described quantum mechanically. We reformulate the conventional formal description based on input–output operator relations and obtain an intuitive description based on the possible paths taken by the photons. The effect of a linear optical network on localized photons is treated within the same formalism. The potential and limitations of linear optics for application in quantum information processing are briefly discussed.

DOI
10.1119/1.1775241

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Linear optical networks with any number of modes are described quantum mechanically. We reformulate the conventional formal description based on input–output operator relations and obtain an intuitive description based on the possible paths taken by the photons. The effect of a linear optical network on localized photons is treated within the same formalism. The potential and limitations of linear optics for application in quantum information processing are briefly discussed.
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Linear optical networks play a paramount role in current implementations of quantum information protocols using photons. A paradigmatic example of their potential is given by boson sampling experiments, which recently have reached the quantum advantage regime [1], [2]. Such linear devices are essentiallyNNN-mode interferometers, whose operation is entirely captured by anN×NN\times NN×Nunitary matrixUUUwith complex entriesuji=∣uji∣eθjiu_{j i}=\left\vert u_{j i}\right\vert e^{\theta_{j i}}uji​=∣uji​∣eθji​. This matrix characterizes, in a classical framework, the transformation of the phasorsεj(in)\varepsilon_{j}^{(in)}εj(in)​of the input field across the different modes into the phasorsεj(out)\varepsilon_{j}^{(out)}εj(out)​of the output fields according to the linear relationεi(out)=∑iujiεi(in)\varepsilon_i^{({out })}=\sum_i u_{j i} \varepsilon_i^{({in })}εi(out)​=∑i​uji​εi(in)​).
We present a quantum optical analysis of waveguides directionally coupled to ring resonators, an architecture realizable using silicon nanophotonics. The innate scalability of the silicon platform allows for the possibility of “on-chip” quantum computation and information processing. In this paper, we briefly review a comprehensive method for analyzing the quantum mechanical output of such a network for an arbitrary input state of the quantized, traveling electromagnetic field in the continuous wave (cw) limit. Specifically, we briefly review a recent theoretical result identifying a particular device topology that yields, via Passive Quantum Optical Feedback (PQOF), dramatic and unexpected enhancements of the Hong-Ou-Mandel Effect, an effect central to the operation of many quantum information processing systems. Next, we extend the analysis to our proposal for a scalable, on-chip realization of the Nonlinear Sign (NS) shifter essential for implementation of the Knill-Laflamme-Milburn (KLM) protocol for Linear Optical Quantum Computing (LOQC). Finally, we discuss generalizations to arbitrary networks of directionally coupled ring resonators along with possible applications is the areas of quantum metrology and sensitive photon detection.
We have developed a quantum-mechanical formalism which permits the treatment of light propagation within the conceptual framework of quantum optics. The formalism rests on the calculation of the momentum operator for the radiation field, and yields directly a description for the spatial progression of the electromagnetic waves. In this paper we give a quantum-mechanical treatment for refraction and reflection by applying our formalism to propagation through a linear dielectric. The fidelity with which this formalism reproduces all results known from classical optics demonstrates its validity.