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arXiv:2410.21202 · Physical Review A

Simple analytical model describing the collective nonlinear response of an ensemble of two-level emitters weakly coupled to a waveguide

Max Schemmer, Martin Cordier, Lucas Pache, Philipp Schneeweiß, Jürgen Volz, Arno Rauschenbeutel

Published Apr 27, 2026• 0 citations
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Abstract

We model and investigate the collective nonlinear optical response of an ensemble of two-level emitters that are weakly coupled to a waveguide featuring a single spatial mode. Our approach generalizes the insight that photon-photon correlations in the light scattered by a single two-level emitter result from two-photon interference to the case of many emitters. Using our model, developed for coherent driving in the weak-saturation regime, we study different configurations for probing the nonlinear response of the ensemble, e.g., through the waveguide or via external illumination, and derive analytical expressions for the second-order quantum coherence function, g ( 2 ) ( τ ) , as well as for the squeezing spectrum of the output light in the waveguide, S θ ( ω ) . For the transmission of resonant guided light, we recover the same predictions as previously made with more involved theoretical models when analyzing experimental results regarding g ( 2 ) ( τ ) [Prasad , ] and S θ ( ω ) [Hinney , ]. We also study the transmission of light that is detuned from the transition of the two-level emitter, a situation that we recently studied experimentally [Cordier , ]. Our model predictions show how the collectively enhanced nonlinear response of weakly coupled emitters can be harnessed to generate nonclassical states of light using ensembles ranging from a few to many emitters.

DOI
10.1103/svb4-98c2

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We model and investigate the collective nonlinear optical response of an ensemble of two-level emitters that are weakly coupled to a waveguide featuring a single spatial mode. Our approach generalizes the insight that photon-photon correlations in the light scattered by a single two-level emitter result from two-photon interference to the case of many emitters. Using our model, developed for coherent driving in the weak-saturation regime, we study different configurations for probing the nonlinear response of the ensemble, e.g., through the waveguide or via external illumination, and derive analytical expressions for the second-order quantum coherence function, g ( 2 ) ( τ ) , as well as for the squeezing spectrum of the output light in the waveguide, S θ ( ω ) . For the transmission of resonant guided light, we recover the same predictions as previously made with more involved theoretical models when analyzing experimental results regarding g ( 2 ) ( τ ) [Prasad , ] and S θ ( ω ) [Hinney , ]. We also study the transmission of light that is detuned from the transition of the two-level emitter, a situation that we recently studied experimentally [Cordier , ]. Our model predictions show how the collectively enhanced nonlinear response of weakly coupled emitters can be harnessed to generate nonclassical states of light using ensembles ranging from a few to many emitters.
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Quantum nonlinearity is an essential ingredient for many quantum technologies, but often the nonlinearity is too weak to be exploited at the few-photon level. However, few photons interacting strongly with single quantum emitters in a waveguide environment can impact a significant nonlinear response, opening up a wide range of photon-photon correlations. Using a waveguide-QED system containing a single atom (treated as a two-level system) chirally coupled to a waveguide, we theoretically investigate two-photon nonlinearities with delay-controlled temporal correlations. We use both matrix product states (MPS) and a frequency-dependent scattering theory approach to analyze the exact population dynamics, as well as the first-order and second-order photon correlation functions in transmission of the system, when pumped by a two-photon Fock-state pulse with a bimodal temporal pulse envelope. The two-photon Fock-state pulses are considered to be either two single photons localized to each peak of the pulse, or both photons delocalized (but correlated) between the two peaks. We consider the regimes of a short, moderate, and (relatively) long distance between the two pulse peaks, comparing the important differences in the temporal correlations with the two types of two-photon pulses. We demonstrate the strikingly different nonlinear features and quantum correlations that occur for uncorrelated and correlated two-photon pairs in experimentally accessible regimes.
Analogous to Coulomb blockade for electrons, photon blockade is a key quantum optical effect in which the presence of one photon prevents the transmission of subsequent ones through a nonlinear medium. Beyond its fundamental interest, photon and multi-photon blockade are actively studied as mechanisms for generating technologically-relevant quantum states of light. Although photon blockade typically requires achieving strong light-matter coupling, increasing the number of atoms fails to enhance antibunching. Here, we analyze the optical transmission properties of a quantum resonator that embeds a two-photon-coupled ensemble of emitters, combining an approximate analytical approach with full quantum numerical simulations. We show that when light and matter are coupled via a two-photon interaction, both single- and multi-photon blockade can benefit from a collective enhancement. We propose different driving schemes in which the second or third-order correlation functions are strongly suppressed with increasing atom number. Differently from established methods, this collective enhancement of non-classical properties occurs with unitary transmission and is ultimately constrained only by decoherence. This demonstrates that collective two-photon couplings are a powerful mechanism for realizing photon blockade even in platforms where individual strong coupling is not achievable.
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