Research ArticlePHYSICS

Unconventional quantum optics in topological waveguide QED

See allHide authors and affiliations

Science Advances  26 Jul 2019:
Vol. 5, no. 7, eaaw0297
DOI: 10.1126/sciadv.aaw0297
  • Fig. 1 System schematic.

    (A) Schematic picture of the present setup: Ne two-level QEs interact with the photonic analog of the SSH model. This model is characterized by having alternating hopping amplitudes J(1 ± δ), where J defines their strength, while δ, the so-called dimerization parameter, controls the asymmetry between them. The interaction with photons (in transparent red) induces nontrivial dynamics between the emitters. (B) Bath’s energy bands for a system with a dimerization parameter ∣δ ∣ = 0.2. The main spectral regions of interest for this manuscript are the middle bandgap (green) and the two bands (blue).

  • Fig. 2 Single-QE dynamics.

    Probability to find the emitter in the excited state, ∣Ce(t)∣2, for different values of ∣δ∣. The other parameters are Δ = 0 (middle of the bandgap) and g = 0.4J. As the bandgap closes, i.e., δ → 0, the decay becomes stronger. Dashed lines mark the value of ∣Ce(t → ∞ )∣2 = [1 + g2/(4J2 ∣ δ ∣)]−2.

  • Fig. 3 BS properties.

    (A) BS wave function for a QE placed at j = 0 that couples to the A sublattice; δ = 0.2 and g = 0.4J. Probability amplitudes Cj,a are shown in blue, while the Cj,b are shown in orange. The QE frequency is set to Δ = 2.2J (top row), Δ = 0 (middle row), and Δ = −2.2J (bottom row). The first column corresponds to the model without disorder, the second corresponds to the model with disorder in the couplings between cavities, and the third corresponds to the model with disorder in the cavities’ resonant frequencies. In both cases with disorder, the disorder strength is set to w = 0.5J. For each case, the value of the BS’s energy is shown at the bottom of the plots. (B) Inverse BS localization length for the two different models of disorder as a function of the disorder strength. Parameters: g = 0.4J and δ = 0.5. The dots correspond to the average value computed with a total of 104 instances of disorder, and the error bars mark the value of 1 SD above and below the average value (the blue curves are slightly displaced to the right for better visibility). Two cases are shown, which correspond to Δ ≃ 2.06J (triangles, outer bandgaps) and Δ = 0 (circles, inner bandgap). (C) Absolute value of the dipolar coupling for Δ = 0 and g = 0.4J; Markov, solid line; exact, dots. The insets show the shape of the BSs in the topological and the trivial phases. The situation for the BA configuration is the same, reversing the role of δ.

  • Fig. 4 Spin models: Phase diagram and correlations.

    (A) Ground state average polarization obtained by exact diagonalization for a chain with Ne = 20 emitters with frequency tuned to Δ = 0 as a function of the chemical potential μ and the decay length of the interactions ξ. The different phases discussed in the text, a valence-bond solid (VBS) and a double Néel ordered phase (DN), are shown schematically below, on the left and right, respectively. Interactions of different sign are marked with links of different color. For the VBS, we show two possible configurations corresponding to δ < 0 (top) and δ > 0 (bottom). In the topologically nontrivial phase (δ < 0), two spins are left uncoupled with the rest of the chain. (B) Correlations Cν(r)=σν9σν9+rσν9σν9+r, ν = x, y, z [Cx(r) = Cy(r)] for the same system as in (A) for different interaction lengths, fixing μ = 0 (left column). Correlations for different chemical potentials fixing ξ = 5; darker colors correspond to lower chemical potentials (right column). Note that we have defined a single index r that combines the unit cell position and the sublattice index. The yellow dashed line marks the value of 1/2 expected when the interactions are of infinite range.

  • Fig. 5 Single-photon scattering.

    (A) Pictorial representation of the scattering process: An incident photon impinges into a scatterer, part of which is reflected (transmitted) with probability amplitude r (t). Bottom row: Relevant level structure for the single-photon scattering for both scatterers considered: one and two QEs. ∣gg〉 ≡ ∣ g1g2 denotes the common ground state, while S,A=(e1g2±g1e2)/2 denotes the symmetric (antisymmetric) excited state combination of the two QEs. (B) Transmission probability for a single emitter coupled to both A and B cavities of the same unit cell (left) and two emitters in the AB configuration separated a total of x12 = 2 unit cells (right). The parameters in the single emitter case are g = 0.4J, δ = ±0.5, and Δ = 1.5J. The dashed line corresponds to the case where the emitter couples to a single sublattice (α = 0,1) (does not depend on the sign of δ). When the emitter couples to both sublattices (α = 0.3), the perfect reflection resonance experiences a shift that is different for δ > 0 (purple line) or δ < 0 (blue line). The parameters for the two-emitter case are g = 0.1J, δ = ±0.5, and Δ ≃ 1.65J, for which the two QEs are in a subradiant configuration if δ > 0.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/7/eaaw0297/DC1

    Section S1. Integration of the dynamics

    Section S2. Subexponential decay

    Section S3. Two QE dynamics in the non-Markovian regime

    Section S4. Existence conditions of two QE BSs

    Section S5. Finite-bath dynamics

    Section S6. Middle BSs in 1D baths

    Fig. S1. Schematics showing the contour of integration.

    Fig. S2. Non-Markovian dynamics.

    Fig. S3. Decaying part of the dynamics of a single emitter with parameters Δ = −2J, ∣δ∣ = 0.5, and g = 0.2J.

    Fig. S4. Disappearance of the two-QEs BSs in the trivial and topological cases.

    Fig. S5. Finite-size effects.

    Table S1. Topological properties of several 1D baths, and their corresponding BS features A to C (see text for discussion) when an emitter couples to them.

    References (61, 62)

  • Supplementary Materials

    This PDF file includes:

    • Section S1. Integration of the dynamics
    • Section S2. Subexponential decay
    • Section S3. Two QE dynamics in the non-Markovian regime
    • Section S4. Existence conditions of two QE BSs
    • Section S5. Finite-bath dynamics
    • Section S6. Middle BSs in 1D baths
    • Fig. S1. Schematics showing the contour of integration.
    • Fig. S2. Non-Markovian dynamics.
    • Fig. S3. Decaying part of the dynamics of a single emitter with parameters Δ = −2J, ∣δ∣ = 0.5, and g = 0.2J.
    • Fig. S4. Disappearance of the two-QEs BSs in the trivial and topological cases.
    • Fig. S5. Finite-size effects.
    • Table S1. Topological properties of several 1D baths, and their corresponding BS features A to C (see text for discussion) when an emitter couples to them.
    • References (61, 62)

    Download PDF

    Files in this Data Supplement:

Navigate This Article