Theory of dipole radiation near a Dirac photonic crystal

J. Perczel and M. D. Lukin

Phys. Rev. A 101, 033822 – Published 16 March 2020

Abstract

We develop an analytic formalism to describe dipole radiation near the Dirac cone of a two-dimensional photonic crystal slab. In contrast to earlier work, we account for all polarization effects and derive a closed-form expression for the dyadic Green's function of the geometry. Using this analytic Green's function, we demonstrate that the dipolar interaction mediated by the slab exhibits winding phases, which are key ingredients for engineering topological systems for quantum emitters. As an example, we study the coherent atomic interactions mediated by the Dirac cone, which were recently shown to be unusually long range with no exponential attenuation. These results pave the way for further, rigorous analysis of emitters interacting in photonic crystals via photonic Dirac cones.

Received 10 September 2019
Accepted 24 February 2020

DOI:https://doi.org/10.1103/PhysRevA.101.033822

Physics Subject Headings (PhySH)

Cavity quantum electrodynamics Nanophotonics Photonic crystals Quantum description of light-matter interaction

Atomic, Molecular & Optical

Authors & Affiliations

J. Perczel ^1,2,* and M. D. Lukin²

¹Physics Department, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
²Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA

^*Corresponding author: jperczel@fas.harvard.edu

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Vol. 101, Iss. 3 — March 2020

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Figure 1
(a) Quantum emitter embedded in a quasi-two-dimensional photonic crystal slab with a cavitylike unit cell. The emitter is assumed to have three degenerate transitions of energy $ω_{A}$ from the ground $| g 〉$ to the excited states $| σ_{\pm} 〉$ and $| π 〉$ . (b) The photonic bands of the slab feature a Dirac-like dispersion cone. The emitter transition frequency $ω_{A}$ is detuned by $δ_{A}$ from the vertex of the Dirac cone $ω_{Dirac}$ .
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Figure 2
Photonic crystal slab unit cell and the corresponding TE-like band structure for four different geometries. The relevant parameters (in units of $a$ ) are the following—the thickness of the slab is $h = 0.5 a$ , the radius of the holes is $r_{s} = 0.0833 a$ , and the dielectric permittivity is $ɛ_{GaP} = 10.5625$ . (a) When six holes are placed $l = 0.2 a$ from the center, the modes in the upper band have a higher field intensity than in the lower band. Color code shows the electric-field strength $| E_{p} {(0) |}^{2} a^{3}$ at the center of the cell for each mode. (b) For $l = 0.333 a$ the modes in the lower band have a higher field intensity than in the upper band. (c) Pushing the holes to the edge of the cell ( $l = 0.4 a$ ) increases the energy gap between the bands at the $M$ point. (d) Increasing the radius of the six holes to $r_{l} = 0.116 a$ makes the Dirac cone energetically separated from all other bands.
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Figure 3
(a) Cross section of the Dirac cone for $ω a / 2 π c = 0.422$ . Numerical results are shown in solid blue; analytic prediction is shown in dashed black. (b) Field intensity $| E_{p} {(0) |}^{2} a^{3}$ in the top band near the Dirac cone. All values fall in a small interval of [0.725,0.765]. (c) Numerically obtained results for the Dirac bands. Color code shows the $x$ component of the field $| E_{p, x} {(0) |}^{2} / {| E_{p} (0) |}^{2}$ . (d) Same as (c), showing the top band from above. (e), (f) Analytic approximation to the band dispersion and the polarization of the modes. Numerical results were obtained for a GaP structure with geometric parameters $h = 0.25 a$ , $r_{s} = 0.0833 a$ , $r_{l} = 0.15 a$ , and $l = 0.4 a$ , for which we obtain the values $ω_{Dirac} a / 2 π c = 0.4172$ and $v_{s} = 0.3 c$ .
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Figure 4
(a) Detuning $ω_{M} - ω_{Dirac}$ as a function of $r_{l}$ and $l$ , the radius and radial displacement of the central holes. Slab height is $h = 1.13 a$ . (b) The slope constant $v_{s}$ as a function of $r_{l}$ and $l$ for $h = 1.13 a$ . (c) The electric-field strength $| E_{0} |^{2}$ as a function of $r_{l}$ and the slab thickness $h$ for $l = 0.4 a$ . (d) The frequency of the Dirac point $ω_{Dirac}$ as a function of $r_{l}$ and $h$ for $l = 0.4 a$ . The inset of (c) provides the color-coded legend for the curves with different hole radii.
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Figure 5
(a) Real and imaginary parts of the Green's function as a function of the detuning, $δ_{A}$ , corresponding to the coherent and dissipative parts of the dipolar interaction, respectively. The dissipative part goes to zero at the Dirac point. (b) Ratio of the coherent and dissipative interactions as a function of the detuning, $δ_{A}$ . The Green's function is evaluated five lattice sites away from the dipole source along the axis aligned with the $a_{+}$ lattice vector. The transition wavelength of the atoms is assumed to be $λ = 738 nm$ and the Green's function parameters are $| E_{0} |^{2} / a^{3} = 0.75$ and $v_{s} = 0.3 c$ .
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Figure 6
(a) Dipole-dipole interaction $(Re [G_{x x} (r)])$ and (b) cooperative decay $(Im [G_{x x} (r)])$ as a function of distance between two $x$ -polarized atoms. The transition wavelength of the atoms is assumed to be $λ = 738 nm$ , the detuning is $δ_{A} = 19.5 GHz$ , and the Green's function parameters used are $| E_{0} |^{2} / a^{3} = 0.75$ and $v_{s} = 0.3 c$ .
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