Research ArticleAPPLIED PHYSICS

Nonradiating and radiating modes excited by quantum emitters in open epsilon-near-zero cavities

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Science Advances  21 Oct 2016:
Vol. 2, no. 10, e1600987
DOI: 10.1126/sciadv.1600987
  • Fig. 1 Spatially electrostatic fields in unbounded ENZ media.

    (A) Geometry and sketch of a QE located at the center of a vacuum spherical bubble of radius r0, embedded in an unbounded ENZ medium, ε(ω0) ≈ 0. (B) Simulated electric and magnetic field magnitude distributions (r0 = 0.25 μm, λ0 = λp = 10.31 μm). The numerical simulations ratify the excitations of fields with a spatially electrostatic distribution and zero magnetic field in the ENZ region. (C) Analytically and numerically computed effective dipole moment enhancement factor at resonance (r00 = 0.715) as a function of losses (imaginary part of the permittivity of the ENZ region).

  • Fig. 2 Nonradiating modes in open ENZ cavities.

    (A) Sketch of a bubble-insulated QE (shown as red arrow) embedded within an open ENZ cavity of arbitrary shape (shown as gray background) with several vacuum bubbles (shown in green). Simulation results for the (B) electric and (C) magnetic field magnitude distributions inside and outside the cavity, when the QE is at the center of its vacuum bubble. The electric field is trapped within the open cavity, whereas the magnetic field is confined to the vacuum bubble.

  • Fig. 3 Switching between nonradiating and radiating modes.

    (A) Geometry and sketch of an arbitrarily shaped open ENZ cavity, with a vacuum spherical bubble (r0 = 5.27 μm), containing a QE attached to a membrane, so that its position may be displaced along the x axis because of an external stimulus, for example, the vibrational modes excited by an external optical/acoustic wave. (B) Simulated (at λ0 = λp = 10.31 μm) electric field magnitude distributions for displacements: Δx = 0 (nonradiating mode) and Δx = 3.5 μm (radiating mode) in the XZ-plane cut. Excitation rate Γexc (normalized to the free-space excitation rate) (C) and quantum yield ηrad (D), as a function of emitter displacement, for different amounts of loss in the ENZ medium (ε″) and assuming an intrinsic quantum yield ηint = 0.5.

  • Fig. 4 Coupling/decoupling QEs in open ENZ cavities.

    (A) Geometry and sketch of an open ENZ cavity of arbitrary shape with two Si spherical bubbles (r0 = 1.505 μm, εSi = 11.7), separated by a distance of d = 6 μm, and containing QEs. The cavity is assumed to be pierced by a Si rod of cross-section 0.25 μm × 0.25 μm, whose position may be externally controlled by MEMSs. (B) Simulated (at λ0 = λp = 10.31 μm) electric field magnitude distribution excited by the QE in the left bubble for displacements: Δx = 0 (decoupled) and Δx = 1 μm (coupled) in the XZ-plane cut. (C) Individual decay rate, Γ11, decay rate related to coupling, Γ21, and cooperative Lamb shift, Δω21, normalized to the free-space individual decay rate, Γ0, as a function of frequency in the coupled (Δx = 1 μm, first row) and decoupled (Δx = 0, second row) states. The ENZ host has been modeled with a dispersive Drude model ε(ω) = 1 − ωp2/ω(ω + iωc), where ωp = 2π × 29.08 × 1012 rad/s, and for three different amounts of loss, characterized by ωc = 0.001ωp (first column), ωc = 0.01ωp (second column), and ωc = 0.1ωp (third column).

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/2/10/e1600987/DC1

    Supplementary Note 1. Derivation of Eqs. 1 to 3.

    Supplementary Note 2. Quasi-static solution to the problem.

    Supplementary Note 3. Derivation of Eq. 4.

    Supplementary Note 4. Magnetic dipole resonances.

    Supplementary Note 5. QEs shifted from the origin of the coordinates.

    Supplementary Note 6. Equivalent circuit model.

    fig. S1. Sketch and dimensions of the system studied in Fig. 2.

    fig. S2. Sketch and dimensions of the system studied in Fig. 3.

    fig. S3. Optimal bubble radius for resonant magnetic dipole excitation.

    fig. S4. Electric and magnetic dipole excitations as a function of emitter displacement.

    fig. S5. Sketch and dimensions of the system studied in Fig. 4.

    fig. S6. Equivalent circuit model.

    fig. S7. Coupling parameters in a cubic cavity.

  • Supplementary Materials

    This PDF file includes:

    • Supplementary Note 1. Derivation of Eqs. 1 to 3.
    • Supplementary Note 2. Quasi-static solution to the problem.
    • Supplementary Note 3. Derivation of Eq. 4.
    • Supplementary Note 4. Magnetic dipole resonances.
    • Supplementary Note 5. QEs shifted from the origin of the coordinates.
    • Supplementary Note 6. Equivalent circuit model.
    • fig. S1. Sketch and dimensions of the system studied in Fig. 2.
    • fig. S2. Sketch and dimensions of the system studied in Fig. 3.
    • fig. S3. Optimal bubble radius for resonant magnetic dipole excitation.
    • fig. S4. Electric and magnetic dipole excitations as a function of emitter displacement.
    • fig. S5. Sketch and dimensions of the system studied in Fig. 4.
    • fig. S6. Equivalent circuit model.
    • fig. S7. Coupling parameters in a cubic cavity.

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