Observation of long-range dipole-dipole interactions in hyperbolic metamaterials

Abstract

Dipole-dipole interactions (V _dd) between closely spaced atoms and molecules are related to real photon and virtual photon exchange between them and decrease in the near field connected with the characteristic Coulombic dipole field law. The control and modification of this marked scaling with distance have become a long-standing theme in quantum engineering since dipole-dipole interactions govern Van der Waals forces, collective Lamb shifts, atom blockade effects, and Förster resonance energy transfer. We show that metamaterials can fundamentally modify these interactions despite large physical separation between interacting quantum emitters. We demonstrate a two orders of magnitude increase in the near-field resonant dipole-dipole interactions at intermediate field distances (10 times the near field) and observe the distance scaling law consistent with a super-Coulombic interaction theory curtailed only by absorption and finite size effects of the metamaterial constituents. We develop a first-principles numerical approach of many-body dipole-dipole interactions in metamaterials to confirm our theoretical predictions and experimental observations. In marked distinction to existing approaches of engineering radiative interactions, our work paves the way for controlling long-range dipole-dipole interactions using hyperbolic metamaterials and natural hyperbolic two-dimensional materials.

Fig. 1. Comparison of dipole-dipole interactions (V_dd) in metallic waveguides, photonic crystal band-edge structures, and hyperbolic metamaterials.

Here, r is the distance between interacting emitters, v_g is the group velocity of the waveguide mode with wave vector k, ω_cutoff is the cutoff frequency of the metallic waveguide mode or photonic crystal, and ξ is an interaction range. (A) When the transition frequencies of interacting atoms lie above the cutoff, they will have a sinusoidal-type interaction. (B) On the other hand, at the band edge of a photonic crystal, there occur interactions with a divergent strength and range. (C) Hyperbolic media exhibit fundamentally different Coulombic long-range interactions, which diverge for specific angular directions in the low-loss effective medium limit.

Fig. 2. Förster resonance energy transfer (FRET) as a probe for dipole-dipole interactions.

(A) The calculated vacuum fluctuation-induced dipole-dipole interaction potential for two molecules separated by a realistic (dissipative) slab of HMM, SiO₂, and Ag. γ_o is the free-space decay rate. The hyperbolic metamaterial (HMM) provides strong dipole-dipole interactions along the asymptotes of the resonance cone, which show a Coulombic near-field scaling (1/r³) even for distances comparable with the free-space wavelength—orders of magnitude stronger than conventional materials. The HMM dielectric constants are ϵ_x ≈ −4.2 + 0.2i, ϵ_z ≈ 5.4 + 0.01i. (B) FRET is used as a probe for long-range super-Coulombic RDDI. The donor atom’s radiative dipole transition is resonant with the acceptor absorption dipole transition. (C and D) The enhanced RDDI mediated by directional hyperbolic polaritons for a dipole located below a 100-nm slab of SiO₂ and HMM. The HMM allows a single dipole (white arrow, bottom) to interact with many physically separated acceptors (orange arrows, bright regions, top), giving rise to unique super-Coulombic enhancements for thin films of acceptors and donors.

Fig. 3. Evidence of long-range dipole-dipole interactions across metamaterials.

(A) The sample types used to isolate RDDI in various material systems (donors, Alq3, shown green). (B to D) The transmitted continuous-wave excitation PL spectra (CW PL) is shown for the donor and acceptor separated by dielectric, metal, and metamaterial. The spectra are in units of spectrometer charge-coupled-device (CCD) counts. We note that energy transfer is visible in all three material systems; that is, the donor-excited state is causing the acceptor to be excited and subsequently relax and emit a photon. This is concluded by noting an increased intensity of acceptor emission and a quenched donor emission when the emitters are placed in the hybrid geometry (black curve) relative to the donor-only (blue curve) and acceptor-only (red curve) control systems (see the Supplementary Materials for quantitative differential spectra). (E to G) The time-resolved donor fluorescence for donor-only (blue) and hybrid (black) samples are shown for the three material systems. For the donors/acceptors separated by 100 nm of SiO₂ or Ag (E and F), the hybrid decay traces reveal no additional lifetime reduction compared to the donor-only case, indicating no long-range RDDI. When the donor and acceptors are separated by a 100-nm Ag/SiO₂ multilayer metamaterial (G), we observe a marked excited-state lifetime reduction when the acceptor molecules are present, providing evidence of long-range super-Coulombic RDDI. The percentage change in the signal is plotted in the inset. It is seen that the signal is many SDs away from the uncertainty level and the background throughout the time trace. We emphasize that the net time integrated difference in counts is far greater, providing conclusive evidence of the emergence of super-Coulombic interactions.

Fig. 4. Spatial scaling of long-range interactions.

(A) Experimental confirmation of enhanced energy transfer rates due to the long-range dipole-dipole interactions in a hyperbolic metamaterial (green) compared to a silver film (blue) and a SiO₂ film (red). The noise levels are denoted by dashed curves, and the numerically calculated many-body dipole-dipole interaction curves are denoted by the colored bands (no free-fitting parameters). The theoretical predictions include 10% error bands accounting for uncertainty in the independently extracted physical parameters. (B) We now compare the ideal super-Coulombic behavior to the experimental observations. The curves show the numerically simulated spatial dependence of sheet-to-slab (2D sheet of donors and thin slab of acceptors) many-body dipole-dipole interactions demonstrating an enhanced FRET rate of the effective medium model (yellow) with d⁻³ power law dependence. Multilayer lattice structures with unit cell sizes of 40, 20, and 4 nm, respectively, are also shown exhibiting an extended spatial range with enhanced Coulombic interactions beyond the scale of a wavelength. The green stars correspond to the experimentally measured data. It is seen that the ideal EMT (yellow) has excellent agreement with the numerical simulations for 4-nm unit cell sizes. The same numerical simulations show excellent agreement with experimental data points for 40-nm unit cell sizes only limited by nanofabrication of ultrathin layers. The solid gray line shows the ideal limit obtained from Eq. 1 of adsorbed quantum emitters on a hyperbolic medium, whereas the dashed black line presents the analytical scaling law, taking into account the finite distance between the emitter from the metamaterial.