Light-Matter Interactions Beyond the Dipole Approximation in Extended Systems Without Multipole Expansion

Abstract

We present a general theoretical framework to capture light-matter interactions beyond the electric-dipole approximation (EDA), applicable to extended nano- and microscale materials interacting with spatially structured electric fields without truncation at finite multipolar order. The approach is based on the Power-Zienau-Woolley (PZW) Hamiltonian for light-matter interactions and a representation of the material's Hamiltonian in a basis of maximally localized Wannier functions (MLWFs), obtainable from first-principles calculations. We utilize this approach to clarify the limitations of the ubiquitous dipole approximation. We consider electric fields with both uniform and non-uniform intensities and a range of ratios of system size to the wavelength of light. Through this analysis, we identify the conditions under which the EDA breaks down, leading to significant errors in the light-induced dynamics. Contrary to conventional belief, we find that the EDA is remarkably robust for uniformly illuminated 1-D or 2-D materials when light propagates perpendicular to the material. For 3-D materials or non-perpendicular illumination of lower-dimensional materials, conventional wisdom holds and the EDA begins to break down when the wavelength becomes comparable to the system size. Furthermore, the EDA fails when the material is illuminated partially or non-uniformly. For slowly varying field intensities this failure can be corrected by finite-order multipolar corrections. However, for fields that vary substantially, correcting via multipolar terms becomes computationally impractical. In contrast, our approach captures beyond-dipole light-matter interactions at the computational cost of a standard dipole calculation. This efficiency enables accurate first-principles simulations of spatially structured light-matter dynamics in nanoscale devices, quantum materials, and interfaces.

Figures (12)

• Figure 1: Schematic illustrations of the three light-matter interaction scenarios considered in this work. The 1-D chain is (a) illuminated by a Gaussian (in space and time) beam that propagates perpendicular to the chain with a spot size $s$ smaller than the chain length $L=N\gamma a$; (b) tilted at an angle $\theta$ relative to the $y$ direction of propagation of a Gaussian (in time) pulse, polarized along the $x$ axis; (c) placed in a $100$ nm gap of a bow-tie metal-dielectric-metal junction and illuminated by a Gaussian (in time) pulse that propagates perpendicular to the chain.
• Figure 2: Resonant excitation of the 1-$\mu$m-long 1-D chain illuminated with a Gaussian beam [Eq. \ref{['gaussian']}] with a spot size $s$ of 0.8 $\mu$m as in the geometry of Fig. \ref{['fig:diagrams']}(a). (a) Temporal profile of the laser field $E(t)$. (b) The average energy and (c) polarization of the chain with $N=400$ unit cells comparing the true dynamics generated by the multipolar PZW (blue) Hamiltonian to the dipole approximated Hamiltonian (red).
• Figure 3: (a) Net energy absorbed after photoexcitation $\Delta \langle \hat{H} \rangle$ and (b) the polarization fidelity [Eq. \ref{['Fidelity']}] for varying chain length $L$ with respect to a fixed spot size $s$ as in the geometry of Fig. \ref{['fig:diagrams']}(a). Multipolar result is shown in blue, dipole in red, octupole in black and 32-pole in green. Simulation parameters are the same as in Figure \ref{['fig:AvgE']}, except $N$ is allowed to vary between 80 and 700 unit cells.
• Figure 4: Comparison of orbital populations after resonant excitation between the PZW multipolar (blue) and its finite-order approximations as in the geometry of Fig. \ref{['fig:diagrams']}(a). (a) Orbital populations for a chain with $N=400$. (b) Population difference between the full multipolar Hamiltonian and the finite-order corrections for states around the HOMO-LUMO gap as a function of chain length. The color coding and simulation parameters are the same as in Fig. \ref{['fig:Ncomparison']}.
• Figure 5: The average energy (a), polarization (b) and population (c) of the chain with $N=400$ unit cells comparing the dynamics generated by the multipolar PZW (blue) Hamiltonian to the dynamics obtained with dipole approximation (red). Simulation parameters are the same as in Figure \ref{['fig:AvgE']}, except $s=10$$\mu$m.