Reducing Effective System Dimensionality with Long-Range Collective Dipole-Dipole Interactions

Dimensionality plays a crucial role in long-range dipole-dipole interactions (DDIs). We demonstrate that a resonant nanophotonic structure modifies the apparent dimensionality in an interacting ensemble of emitters, as revealed by population decay dynamics. Our measurements on a dense ensemble of interacting quantum emitters in a resonant nanophotonic structure with long-range DDIs reveal an effective dimensionality reduction to $\overline{d}=2.20(12)$, despite the emitters being distributed in 3D. This contrasts with the homogeneous environment, where the apparent dimension is $\overline{d}=3.00$. Our work presents a promising avenue to manipulate dimensionality in an ensemble of interacting emitters.

Figure 1

Concept of apparent dimensionality of an interacting ensemble of emitters. The apparent dimensionality is related to the noninteger exponent of time in the fluorescence decay dynamics $I(t)/I_0=\exp (-{\gamma }_{D}t)\exp (-\alpha t^{\beta })$. In a homogeneous environment, $\beta =0.5(0.33)$ for the 3D (2D) spatial distribution of emitters. A resonant nanophotonic environment modifies the spatial distribution of the neighboring emitters sensed by each interacting emitter which results in the modification of the temporal decay dynamics. This reduces the apparent dimensionality experienced by the interacting emitters, which is reflected in the noninteger exponent, $\beta <0.5$, though, the emitters are distributed in a 3D volume. The green dipoles represent the donor emitters and the orange represents acceptor dipoles.

Figure 2

(a) Acceptor emitter’s absorption spectrum (blue curve), donor emitter’s emission spectrum (orange curve), extinction spectrum of a resonant plasmonic lattice with lattice constant $\sim 300\mathrm{nm}$ (purple dash curve), and off-resonant plasmonic lattice with lattice constant $\sim 350\mathrm{nm}$ (green dash-dot curve). The extinction spectrum of the resonant plasmonic lattice spectrally overlaps with the emission-absorption spectrum of the donor and acceptor emitters (yellow highlighted region). (b) Calculated dipole-dipole interaction potential $|V_{dd}|$ for the resonant and off-resonant plasmonic lattice. The resonant plasmonic lattice shows a strikingly modified scaling law. (c) Monte Carlo simulations depicting the temporal decay dynamics of donor emitters for $|V_{dd}{|}^2=R_0^6/R^6$ scaling and $R_0\ll L_{\mathrm{sys}}$ with $\beta \sim 0.52$. Inset: values of $\beta$ for randomized spatial distributions of emitters. (d) Monte Carlo simulations showing the temporal decay dynamics of donor emitters for $R_0\sim L_{\mathrm{sys}}$. The reduced dimensionality is evident from the estimated values of $\beta \sim 0.4$. Inset: values of $\beta$ for randomized spatial distributions of emitters.

Figure 3

Measured fluorescence lifetime decay when the interacting emitters are in different electromagnetic environments: (a) glass substrate (i.e., a homogeneous environment), (b) ${\mathrm{TiO}}_2$ dielectric lattice (i.e., an off-resonant inhomogeneous electromagnetic environment), and (c) a plasmonic lattice (i.e., a resonant inhomogeneous electromagnetic environment). The value of $\beta \sim 0.5$ in both inhomogeneous and off-resonant inhomogeneous environments. This is commensurate with a 3D system. In contrast, the faster-than-exponential decay dynamics on a resonant silver (Ag) plasmonic lattice reveals an exponent value of $\sim 0.37$. This is commensurate to an effective lower dimension $\overline{d}\sim 2.20(12)$. The emitters were embedded in $\sim 1\mathrm{\mu }\mathrm{m}$ thick polymer thin films.

Figure 4

Extracted probability density function for the resonance energy transfer rate on various electromagnetic environments: (i) glass, a homogeneous environment (dash-dot red curve), (ii) an inhomogeneous environment, ${\mathrm{TiO}}_2$ nanoparticle lattice having the same lattice constant and dimensions as the resonant plasmonic lattice (dotted yellow curve), (iii) an off-resonant plasmonic lattice (purple curve), and (iv) a resonant plasmonic lattice (blue curve). The PDF of the energy transfer rates on the resonant plasmonic lattice is not only shifted but also broader. Inset: reduced number of events having stringer interaction strength (in the tail).