Exciton-polariton mediated interaction between two nitrogen-vacancy color centers in diamond using two-dimensional transition metal dichalcogenides

J. C. G. Henriques, B. Amorim, and N. M. R. Peres

Phys. Rev. B 103, 085407 – Published 4 February 2021

Abstract

In this paper, starting from a quantum master equation, we discuss the interaction between two negatively charged nitrogen-vacancy color centers in diamond via exciton-polaritons propagating in a two-dimensional transition metal dichalcogenide layer in close proximity to a diamond crystal. We focus on the optical 1.945 eV transition and model the nitrogen-vacancy color centers as two-level (artificial) atoms. We find that the interaction parameters and the energy-level renormalization constants are extremely sensitive to the distance of the nitrogen-vacancy centers to the transition-metal dichalcogenide layer. Analytical expressions are obtained for the spectrum of the exciton-polaritons and for the damping constants entering the Lindblad equation. The conditions for occurrence of exciton mediated superradiance are discussed.

Received 25 September 2020
Revised 26 January 2021
Accepted 28 January 2021

DOI:https://doi.org/10.1103/PhysRevB.103.085407

Physics Subject Headings (PhySH)

Excitons NV centers

2-dimensional systems

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

J. C. G. Henriques ¹, B. Amorim ¹, and N. M. R. Peres ^1,2

¹Department and Centre of Physics, and QuantaLab, University of Minho, Campus of Gualtar, 4710-057 Braga, Portugal
²International Iberian Nanotechnology Laboratory (INL), Av. Mestre José Veiga, 4715-330 Braga, Portugal

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Vol. 103, Iss. 8 — 15 February 2021

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Images

Figure 1
System considered in this paper: Two NV centers each carrying an electric dipole moment $μ_{j}$ (arrows), associated with the optical transition of 1.945 eV, are embedded in a diamond slab with a dielectric constant $ε_{1}$ . The slab is on top of a single layer of a TMD, and the whole system is on top of a dielectric with dielectric function $ε_{2}$ (the value of $ε_{2}$ allows the tuning of the exciton frequency). Each NV center is at a distance $z_{i}$ from the TMD layer and is separated from one another by a distance $ρ$ . On the right panel, we depicted the crystal structure of diamond carrying a NV center: nitrogen (orange) and vacancy (pink). In the bottom panel, the energy diagram of a NV center is depicted together with the different mechanisms leading to the energy-level splitting (adapted from Ref. [9]). The transition we will be considering is the optical transition of 1.945 eV.
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Figure 2
Left: Real and imaginary parts of the susceptibility according to Eq. (10). We observe a resonance at the exciton energy $ω_{ex} = 1.94 eV$ . The real part of the susceptibility is negative in a small interval from approximately 1.940 eV up to 1.965 eV (shaded region). Only in this spectral region does the monolayer TMD support exciton-polaritons. Right: Exciton-polariton dispersion relation $ω_{q}$ using the approximate expression given in Eq. (13) and the susceptibility presented in the left panel. The spectrum is only finite in the energy window where $Re χ$ is negative. The polariton's momentum is much bigger than that of a photon for a given energy, which agrees with the argument used prior to Eq. (13). The parameters of Table 1 were used.
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Figure 3
Interaction parameters, energy shifts, and damping constants for different configurations of the system. In panel (a), we depict $γ_{11}$ as a function of $θ$ , the angle the dipole makes with the $z$ axis, for three different NV-center TMD separations. In panel (b), we depict $γ_{12}$ as a function of $ρ / λ_{p}$ , where $ρ$ is the in-plane distance between the NV centers, and $λ_{p}$ is the polariton wavelength for an energy $ℏ ω_{0}$ ; in the present case we have $λ_{p} \approx 37$ nm. We consider that both dipoles lie in the x-z plane, with an angle with the $z$ axis given by $θ = 0, π / 4$ , and $π / 2$ as depicted in the panel. In panel (c), we show $γ_{12}$ as a function of $ρ / λ_{p}$ , for the configurations shown in the panel: with one dipole along the $z$ axis while the other lies in the $x - y$ plane (aligned along $x, y$ , and $- x$ axes). Panels (d)–(f) show $Δ_{i}$ and $g_{12}$ for the same dipole orientations as in panels (a)–(c). In all plots, the NV centers were assumed to be at a distance $z = 2$ nm from the TMD. The parameters of Table 1 were used.
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Figure 4
Superradiance and subradiance in the two-coupled NV-center systems. Three orientations of the two electric dipole moments, relatively $z$ axis, are considered with both dipoles 2 nm away from the TMD. The parameters are the same as in Fig. 3.
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