Quantum dynamics of propagating photons with strong interactions: a generalized input–output formalism

Abstract

There has been rapid development of systems that yield strong interactions between freely propagating photons in one-dimension via controlled coupling to quantum emitters. This raises interesting possibilities such as quantum information processing with photons or quantum many-body states of light, but treating such systems generally remains a difficult task theoretically. Here, we describe a novel technique in which the dynamics and correlations of a few photons can be exactly calculated, based upon knowledge of the initial photonic state and the solution of the reduced effective dynamics of the quantum emitters alone. We show that this generalized 'input-output' formalism allows for a straightforward numerical implementation regardless of system details, such as emitter positions, external driving, and level structure. As a specific example, we apply our technique to show how atomic systems with infinite-range interactions and under conditions of electromagnetically induced transparency enable the selective transmission of correlated multi-photon states.from other emitters appears to be a difficult task. An exception is the weak excitation limit, in which atoms can be treated as linear scatterers and the powerful transfer matrix method of linear optics can be employed [18,19].The full quantum case has been solved exactly in a limited number of situations in which nonlinear systems are coupled to 1D waveguides [20][21][22][23][24][25][26][27][28][29][30]. The formalism employed in [21] is particularly elegant, because it establishes an input-output relation to determine the nonlinear scattering from a two-level atom. Here, we show that this technique can be efficiently generalized to many atoms, chiral or bi-directional waveguides, and arbitrary atomic configurations, providing a powerful tool to investigate nonlinear optical dynamics in all systems of interest.This paper is organized in the following way: first, we present a generalized input-output formalism to treat few-photon propagation in waveguides coupled to many atoms. We show that the infinite degrees of freedom associated with the photonic modes can be effectively integrated out, yielding an open, interacting 'spin' model that involves only the internal degrees of freedom of the atoms. This open system can be solved using a number of conventional, quantum optical techniques. Then, we show that the solution of the spin problem can be used to re-construct the optical fields. In particular, we provide a prescription to map spin correlations to S-matrix elements, which contain full information about the photon dynamics, and give explicit closed-form expressions for the one-and two-photon cases. Importantly, in analogy with the cavity QED case, our technique enables analytical solutions under some scenarios, but in general allows for simple numerical implementation under a wide variety of circumstances of interest, such as different level structures, external driving, atomic positions, atomic motion, etc. Finally, to illustrate the ease of usage, we apply ...

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