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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Theory of few-photon quantum scattering in nanophotonic structures

Shanshan Xu, Shanhui Fan

Published Feb 20, 2017• 0 citations
Publisher

Abstract

We present a generalization of the input-output formalism that provides an analytic and systematic computation tool for few-photon scattering matrix in waveguide quantum electrodynamics (QED) systems. We also discuss the theoretical constraints on such scattering matrix from the cluster decomposition principle in quantum field theory.
DOI
10.1117/12.2261178

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We provide a systematic treatment of NNN-photon transport in a waveguide coupled to a local system using the input-output formalism. The main result of the paper is a general connection between the NNN-photon S matrix and the Green's functions of the local system. We also show that the computation can be significantly simplified by exploiting the connectedness structure of both the S matrix and the Green's function, and by computing the Green's function using an effective Hamiltonian that involves only the degrees of freedom of the local system. We illustrate our formalism by computing NNN-photon transport through a cavity containing a medium with Kerr nonlinearity, with NNN up to 3.
We show that the form of the cluster decomposition principle, commonly used in quantum field theory, needs to be significantly generalized. As an illustration, we consider the general structure of two-photon SSS matrix for a waveguide coupled to a local quantum system that supports multiple ground states. The presence of the multiple ground states results in a noncommutative aspect of the system with respect to the exchange of the orders of photons. Consequently, the two-photon SSS matrix significantly differs from the standard form in quantum field theory.
In this work we discuss the emergence of approximate causality in a general setup from waveguide QED—i.e. a one-dimensional propagating field interacting with a scatterer. We prove that this emergent causality translates into a structure for the N -photon scattering matrix. Our work builds on the derivation of a Lieb–Robinson-type bound for continuous models and for all coupling strengths, as well as on several intermediate results, of which we highlight: (i) the asymptotic independence of space-like separated wave packets, (ii) the proper definition of input and output scattering states, and (iii) the characterization of the ground state and correlations in the model. We illustrate our formal results by analyzing the two-photon scattering from a quantum impurity in the ultrastrong coupling regime, verifying the cluster decomposition and ground-state nature. Besides, we generalize the cluster decomposition if inelastic or Raman scattering occurs, finding the structure of the -matrix in momentum space for linear dispersion relations. In this case, we compute the decay of the fluorescence (photon–photon correlations) caused by this S -matrix.
Engineering strong photon–photon interactions at the quantum level have been crucial in various areas of research, notably in quantum information processing and quantum simulation. It is often done by coupling matter strongly to light. A promising way to achieve this is via waveguide quantum electrodynamics (QED). Motivated by these advancements, we study few-photon transport in waveguide QED setups. First, we present a diagrammatic technique to systematically study multiphoton scattering based on the scattering formalism and Green’s function approach. We demonstrate our proposal through physically relevant examples involving scattering of few-photon states from two-level emitters as well as from arrays of correlated Kerr nonlinear resonators described by the Bose–Hubbard model. In the second part, we apply the diagrammatic technique that was developed to perform a comprehensive study on a Bose–Hubbard lattice with a quasi-periodic potential. This model exhibits many-body localisation. We compute the two-photon transmission probability and show that it carries signatures of the underlying localisation transition with close agreement to the participation ratio of the eigenstates. The systematic scattering approach provided in this paper provides a foundation for future works at the interface between quantum optics and condensed matter.
We consider a general class of integrated quantum systems where photon-photon interaction occurs in a quantum device that is localized in space. Using techniques that are closely related to cluster decomposition principles in quantum field theory, we provide a general constraint on the analytic properties of a two-photon S matrix in this class of systems. We also show that the photon-photon interaction in these systems inevitably leads to frequency mixing and entanglement and that frequencies of the single photons cannot be preserved in these systems.