Emergence of an Epsilon-Near-Zero Medium from Microscopic Atomic Principles

TL;DR

The paper investigates how an epsilon-near-zero (ENZ) regime arises from collective light scattering in a discrete atomic lattice, challenging continuous-medium descriptions that neglect position-dependent dipole-dipole interactions. Using essentially exact microscopic simulations of a $25$-layer cubic lattice with lattice spacing $a=0.66\lambda$, normal incidence, and isotropic $J=0\to J'=1$ dipoles, the authors solve the coupled-dipole equations to extract forward/backward propagating components with a medium wavenumber $k'$ and phase refractive index $n_p = k'/k$. They observe a collective resonance that yields an enormous increase in the effective wavelength inside the medium, with $n_p \to 0$ and $\lambda_{ ext{eff}}=2\pi/k' \approx 33\lambda$, persisting across transmission bands and robust to atomic position fluctuations, while near-field evanescent components are present yet its slowly varying envelope matches a plane-wave description. This work establishes ENZ behavior from first principles, elucidating how macroscopic electromagnetism emerges from atomic-scale interactions and highlighting limitations of standard continuous-medium theories; it also suggests potential applications in spectroscopy, quantum emitter control, and Doppler-free sensing in ENZ environments.

Abstract

We demonstrate that an effective near-zero refractive index can emerge from collective light scattering in a discrete atomic lattice, using essentially exact microscopic simulations. In a 25-layer array, cooperative response leads to over a thirtyfold increase in the effective optical wavelength within the medium, almost eliminating optical phase accumulation, with potential applications in spectroscopy and optical manipulation of quantum emitters. Crucially, the near-zero refractive index arises from first-principles microscopic theory, rather than being imposed through continuous phenomenological effective-medium model - providing conceptually important insight into the emergence of macroscopic electromagnetism from atomic-scale interactions.

Figures (6)

• Figure 1: Light propagation in an effective ENZ atomic medium. (a) Schematic of light transmitted through a cubic atom array, infinitely extended in the $y$ and $z$ directions. The incident beam with wavevector ${\bf k}=k\hat{\bf e}_x$ excites the atomic polarization, Eq. \ref{['eq:planepol']}, consisting of forwards and backwards propagating waves with wavevectors $\pm k'\hat{\bf e}_x$. (b) Magnitude $|\mathcal{P}_{j}|$ [in units of $\mathcal{D}\epsilon_{0}/(\hbar\gamma)$] and (c) phase $\mathrm{arg}(\mathcal{P}_{j})$ of atomic polarization layer amplitudes as a function of layer position $x_{j}$ (marked by vertical dashed lines). The detunings are $\Delta/\gamma = -0.025,0.103, 0.115, 0.119,$ corresponding to the black dotted, maroon dash-dotted, red dashed, and bright red solid lines, respectively, and the faint lines in (c) the corresponding backward wave components. From these we obtain medium wavenumbers $k'/k = 0.25, 0.1, 0.06, 0.03$.
• Figure 2: (a) Effective wavenumber $k'$ of atomic polarization within and (b) power transmission $T$ through the medium as a function of laser detuning $\Delta$. In (a), $k'$ is purely real for $\Delta\alt 0.12\gamma$ and imaginary for $\Delta\agt 0.12\gamma$.
• Figure 3: Microscopic light field on atomic scale. (a) Magnitude and (b) phase of the total microscopic light field (dotted-black line) along the incident field polarisation as the line $(y,z) = (a/2,a/2)$ is continuously traversed in between the atoms, $-5a \leq x \leq 30a$, for the detuning $\Delta/\gamma = 0.119$ giving polarization amplitudes presented in Fig. \ref{['fig:fig1']}(b). Averaged amplitudes $\bar{\mathrm{E}}^{+}$ are obtained by convolution of the microscopic field with a Gaussian of the root-mean-square widths $\eta = 0.25a$ (dashed maroon line), and $\eta = a$ (solid bright red line), converging to the field amplitude Eq. \ref{['eq:planelight']}.
• Figure 4: Collective line shifts $\delta^{(j)}$ of bands $j$ for an infinite lattice, presented for high-symmetry lines passing through the center of the 3D Brillouin zone (inset), where long-wavelength excitations are present. The bold line denotes the collective resonance band, with quasimomenta $q_{y} = q_{z} = 0$, whose polarization lies in the $yz$ plane. The dashed line denotes the edge of this band at $q_{x} = 0$, where $\delta \simeq -0.12\gamma,$ and the red markers correspond to the wavenumbers of resonant excitations extracted in Figs. \ref{['fig:fig1']}(b-c).
• Figure 5: (Real) phase parameter calculated using the wavenumber $k'$ (thick line) from the exact simulations [Fig.\ref{['fig:fig2']}(a)] and a fit based on the reflection and transmission amplitudes of the exact simulations (thin line). The region $\left|\cos\left(k'h\right)\right| \leq 1$ is shaded and the vertical red dashed line denotes $\Delta \simeq 0.12\gamma$, where $k' \simeq 0$ [Fig. \ref{['fig:fig2']}(b)].