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arXiv:1210.8132 · Physical Review A

Two-photon scattering of a tightly focused weak light beam from a small atomic ensemble: An optical probe to detect atomic level structures

Dibyendu Roy

Published Jun 13, 2013• 53 citations
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Abstract

We study two-photon scattering of a tightly focused weak light beam from a small atomic ensemble of two-level atoms (2LAs). This is similar to the scattering of photons from an atomic ensemble in a one-dimensional waveguide. The scaling of two-photon nonlinearity at single-photon resonance shows a nonmonotonic behavior with an increasing number of few identical 2LAs. The two-photon nonlinearity decays monotonically with an increasing number of atoms for incident photons detuned from single-photon resonance. Single-photon transport in two 2LAs is similar to that in a single $V$-type three-level atom (3LA). However, two-photon transport in these two systems shows very different line shapes. When single-photon transmission is zero in these systems, two transmitted photons are bunched together in a $V$-type 3LA, while their correlation is zero in two 2LAs. The difference in the two-photon line shape persists for few 2LAs and 3LAs. Therefore, the two-photon scattering of a tightly focused weak light beam can be used as a probe to detect atomic level structures of different atoms with similar transition energies.

DOI
10.1103/physreva.87.063819

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We study two-photon scattering of a tightly focused weak light beam from a small atomic ensemble of two-level atoms (2LAs). This is similar to the scattering of photons from an atomic ensemble in a one-dimensional waveguide. The scaling of two-photon nonlinearity at single-photon resonance shows a nonmonotonic behavior with an increasing number of few identical 2LAs. The two-photon nonlinearity decays monotonically with an increasing number of atoms for incident photons detuned from single-photon resonance. Single-photon transport in two 2LAs is similar to that in a single VVV-type three-level atom (3LA). However, two-photon transport in these two systems shows very different line shapes. When single-photon transmission is zero in these systems, two transmitted photons are bunched together in a VVV-type 3LA, while their correlation is zero in two 2LAs. The difference in the two-photon line shape persists for few 2LAs and 3LAs. Therefore, the two-photon scattering of a tightly focused weak light beam can be used as a probe to detect atomic level structures of different atoms with similar transition energies.
We address correlated few-photon transport in one-dimensional waveguides coupled to a two-level system (TLS), such as an atom or a quantum dot. We derive exactly the single-photon and two-photon current (transmission) for linear and nonlinear (tight-binding sinusoidal) energy-momentum dispersion relations of photons in the waveguides and compare the results for the different dispersions. A large enhancement of the two-photon current for the sinusoidal dispersion has been seen at a certain transition energy of the TLS away from the single-photon resonances.
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.
Realising a platform for efficiently guiding and manipulating light at the singlephoton level lies at the heart of optical quantum information processing. In this thesis we aid in the pursuit of this goal by analysing few-photon transport properties in photonic devices while keeping in mind possible applications and experimental challenges. We especially focus on single and two-photon transport properties in one-dimensional waveguides coupled to (artificial) atoms modelled as two and three-level emitters. We begin by introducing key concepts relevant for this thesis and then explain how to model such waveguide geometries. We show how directional-dependent coupling between a waveguide and a localized quantum system can be modelled from a chain of harmonic oscillators by allowing the system to couple to more than one site. Using the input-output formalism and scattering matrix theory we determine a general form for the single and two-photon S-matrices for a waveguide coupled to a two-level system. We also explain how to engineer a quantum optical Fanoresonance waveguide geometry and calculate how this modifies the S-matrices, the few-photon scattering probabilities, and the scattered output spectra. We show that the nonlinear effects in such a system generates highly correlated directional scattering probabilities. We device a protocol that can generate maximal entanglement between two spectrally distinct solid-state emitters by the use of multi-photon scattering. The results also reveal a rich structure for multi-photon states interacting with non-identical emitters. As a final investigation, we quantify how spectral diffusion in quantum dots affects finite-width single and two-photon scattering. Single-photon scattering probabilities are described by a simple convolution but this is not the case for a two-photon state scattering off a two-level emitter. We show that the effect of spectral wandering depletes the nonlinear scattering properties, but that it affects co and counter-propagating output probabilities differently.
We investigate the single-photon propagation in the one-dimensional waveguide coupled to NNN two-level atoms. For a waveguide coupled to NNN distant atoms, the transparency can be induced by coherent interaction at resonance for an eveneveneven numbernumbernumber of NNN, while for an oddoddodd numbernumbernumber of NNN, the photon can be fully reflected. We then can switch the photon transport by controlling the atomic number. For a waveguide coupled to a small atomic ensemble of NNN atoms in the same position, the transparency disappears, and the photonic transport property displays a jump behavior near the resonance point.
Under usual laboratory conditions the absorption of a beam of light by a dilute two-level medium depends on the intensity and spectrum of the incident light. If the incident photon flux is small compared to the decay rate of the absorber’s excited state, saturation effects can be neglected and only the spectrum of the light is important; thus, weak-field absorption depends only on the firstorder correlation function of the incident light. This result relies, however, on a requirement that the incident beam cross-section be much larger than the absorption cross-section of an atom. Then individual atoms act as low efficiency scatterers and the statistics of the scattering process is governed by the law of large numbers applied to many scattering sites. In contrast, if it is arranged that just one atom significantly absorbs (scatters) a beam of photons, correlation functions of the incident light beyond the first-order must be considered. For a beam that is focused within an absorption cross-section, we might naively predict that every photon will be scattered and none transmitted. In fact, even for a very weak beam there is a finite probability for two photons to arrive at the atom within the excited state lifetime, and in this event at least one of them will be transmitted (statistical saturation). In this paper we calculate the transmitted photon flux for a weak beam of photons focussed strongly onto a single, resonant two-state atom. We study the dependence of the transmitted flux on the statistics of the incident photons. Photon beams derived from sources of coherent, chaotic, squeezed, coherent-antibunched, and broadband-antibunched light are considered. our calculations are based on a recently developed theory of cascaded open systems,1,2 and serve to illustrate the usefulness of this theory.
The one- and two-photon scattering matrix SSS is obtained analytically for a one-dimensional waveguide and a pointlike scatterer with NNN excited levels (generalized V-type\mathsf{\text{V}}\text{-type}V-type atom). We argue that the two-photon scattering matrix contains sufficient information to distinguish between different level structures which are equivalent for single-photon scattering, such as a V\mathsf{\text{V}}V atom with N=2N=2N=2 excited levels and two two-level systems. In particular, we show that the scattering with the V-type\mathsf{\text{V}}\text{-type}V-type atom exhibits a destructive interference effect leading to two-photon coupled-resonator-induced transparency, where the nonlinear part of the two-photon scattering matrix vanishes when each incident photon fulfills a single-photon condition for transparency.