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arXiv:1502.06049 · Physical Review A

Input-output formalism for few-photon transport: A systematic treatment beyond two photons

Shanshan Xu, Shanhui Fan

Published Apr 30, 2015• 107 citations
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Abstract

We provide a systematic treatment of $N$-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 $N$-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 $N$-photon transport through a cavity containing a medium with Kerr nonlinearity, with $N$ up to 3.

DOI
10.1103/physreva.91.043845

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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 how to use the input-output formalism to compute the propagator for a low-dimensional quantum system coupled to an optical field under the rotating wave and Markovian approximation. The propagator for this coupled system is expressed in terms of the Green's functions of the low-dimensional system, and it is shown that these Green's functions can be computed entirely from the evolution of the low-dimensional system under an effective non-Hermitian Hamiltonian. Our formalism generalizes the previous works that have focused on time-independent Hamiltonians to systems with time-dependent Hamiltonians (e.g., coherently driven systems), making it a suitable computational tool for the analysis of a number of experimentally interesting quantum systems. We illustrate our formalism by applying it to analyze photon emission and scattering from the driven and undriven two-level system and the three-level lambda system.
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 present a method to systematically study multiphoton transmission in one-dimensional systems comprised of correlated quantum emitters coupled to input and output waveguides. Within the Green's function approach of the scattering matrix (S(S(S matrix), we develop a diagrammatic technique to analytically obtain the system's scattering amplitudes while at the same time visualize all the possible absorption and emission processes. Our method helps to reduce the significant effort in finding the general response of a many-body bosonic system, particularly the nonlinear response embedded in the Green's functions. We demonstrate our proposal through physically relevant examples involving scattering of multiphoton states from two-level emitters as well as from arrays of correlated Kerr nonlinear resonators in the Bose-Hubbard model.