Nanocavities for Molecular Optomechanics: Their Fundamental Description and Applications

Abstract

Vibrational Raman scattering-a process where light exchanges energy with a molecular vibration through inelastic scattering-is most fundamentally described in a quantum framework where both light and vibration are quantized. When the Raman scatterer is embedded inside a plasmonic nanocavity, as in some sufficiently controlled implementations of surface-enhanced Raman scattering (SERS), the coupled system realizes an optomechanical cavity where coherent and parametrically amplified light-vibration interaction becomes a resource for vibrational state engineering and nanoscale nonlinear optics. The purpose of this Perspective is to clarify the connection between the languages and parameters used in the fields of molecular cavity optomechanics (McOM) versus its conventional, "macroscopic" counterpart and to summarize the main results achieved so far in McOM and the most pressing experimental and theoretical challenges. We aim to make the theoretical framework of molecular cavity optomechanics practically usable for the SERS and nanoplasmonics community at large. While quality factors (Q) and mode volumes (V) essentially describe the performance of a nanocavity in enhancing light-matter interaction, we point to the light-cavity coupling efficiencies (η) and optomechanical cooperativities () as the key parameters for molecular optomechanics. As an illustration of the significance of these quantities, we investigate the feasibility of observing optomechanically induced transparency with a molecular vibration-a measurement that would allow for a direct estimate of the optomechanical cooperativity.

Figure 1

Simplified setting for single-mode molecular cavity optomechanics (McOM). (a) A plasmonic cavity (here sketched as a metallic dimer) supports a plasmonic resonance with radiative and nonradiative decay rates κ_ex and κ₀, respectively. The surface-enhanced local field E(t) couples to the induced Raman dipole μ_R(t) of the vibrating molecule, resulting in the vacuum optomechanical coupling rate g₀. The molecular vibration is damped at a rate γ. (b) Frequency domain schematic showing the plasmonic resonance at frequency ω_p of width κ = κ_ex + κ₀. Under a single-frequency pump at ω_L, the optomechanical coupling gives rise to two Raman sidebands at ω_L ± Ω_ν, where Ω_ν is the molecular vibration frequency. Panel (c) is inspired from ref (26).

Figure 2

Overview of representative plasmonic (i–iv) and dielectric (a–d) optomechanical cavities in terms of cooperativity and radiative coupling efficiency (defined in Figure 3). For plasmonic cavities, we assume that a monolayer of biphenyl–thiol molecules covers the nanoparticle or the substrate and that the incoming laser power is set as 100 μW in a diffraction limited spot. Empty symbols represent the single-photon cooperativity , while full blue symbols represent . The data for dielectric cavities are compiled from published experiments: (a) ref (41), (b) ref (42), (c) ref (43), and (d) ref (44).

Figure 3

Correspondence between macroscopic and molecular cavity optomechanics. (a) Sketch of general cavity optomechanics scenario where all input and output coupling losses can be modeled by beam splitters with transmissions η_in, η_out, respectively; the total decay rate of the cavity is the sum of an external decay rate κ_ex allowing for photon exchange with radiation modes and an internal decay rate κ₀ that accounts for absorption and scattering losses. (b) In McOM, the input coupling efficiency corresponds to the ratio of focal spot size to the extinction cross section of the nanocavity (pictured here as a simple metal nanoparticle coated with molecules). (c) The external coupling efficiency is given by the ratio of scattering to extinction cross section in the single mode limit. (d) The optomechanical coupling occurs in the near field through the Raman polarizability, leading to scattering rates Γ_+,–, respectively, for Stokes (add a quantum of vibration) and anti-Stokes (removes one) processes. The existence of a spectrally overlapping quasi-continuum of dark modes is responsible for additional Raman scattering contributions in the near field that are, however, not detected in the far field (quenching).

Figure 4

(a) Breakdown of decay channels for a dipole emitter optimally coupled to the bright mode of an NPoM cavity. The share 1 – β corresponds to quenching; it is not a modal quantity (but depends on wavelength and position of the emitter). Mode volume is computed at the position of highest dipolar coupling to the mode, and Q factor is evaluated by the spectral line width. Results were confirmed by QNM calculations. (b) Relative contributions of dispersive and dissipative coupling rates, estimated by varying the permittivity in the nanogap and computing its effect on ω_p, κ_ex and κ₀. (c–f) Breakdown of decay channels for the dominant bright mode of several nanostructures depicted as insets. A more complete list of parameters for various geometries is provided in Figure 6, Appendix.

Figure 5

Optomechanically induced transparency: concept and measurement schemes. (a) Frequency representation of all involved fields and resonances with the definitions of detunings Δ and δ. (b) Same parameters represented on an energy level diagram in the single-excitation subspace. The notation |p, ν⟩ designates a state with p plasmons and ν vibrations. (c) Computed map of OMIT signal strength (normalized to the input probe power) vs cooperativity and external coupling efficiency η_rad. (d) Possible OMIT measurement scheme in macroscopic cOM. Adapted with permission from ref (140). Copyright 2020 American Physical Society. PLL, phase-locked loop; FC, filter cavity; PM, phase modulator. (e) Proposed implementation of OMIT measurement in McOM using a dark-field geometry for pump and probe excitation. A spectral filter blocks the pump, and the probe signal is sent to the detector. The pump (pulsed or cw) is intensity-modulated so that the demodulated scattered probe intensity senses the pump-induced change in cavity scattering, which can have thermal (broad plasmonic response) or optomechanical (narrow vibrational response) origins.

Figure 6

Full list of parameters extracted from the simulations of various plasmonic nanostructures. The parameters characterize the dipolar plasmonic mode for the single NP cases, the L01 mode for the NPoM cases, and the antisymmetric L01 mode for the NP-in-groove (NPiG) case. E_cav is here defined as the spatial maximum of the norm of the electric field at a distance d = 1 nm from the nanoparticle’s surface, and E₀ is the norm of the background electric field generated or by a plane-wave excitation or by a point dipole source. For nanogap structures (NPoM and NPiG), E_cav is taken on the midgap plane, which is at a distance of 0.65 nm from the surface considering a single layer of molecules 1.3 nm thick. To calculate A_focus, a focusing objective of numerical aperture NA = 0.9 is considered. The NPs are modeled by a sphere of 80 nm diameter. The NPoMs are composed by a truncated sphere of 80 nm with a facet size of 10 nm and a 300 nm thick metallic film separated from the sphere by a dielectric layer (1.3 nm thickness, refractive index of 1.4). For the NPiG case, the dimensions of the truncated sphere are modified (150 nm diameter without facets), and the groove dimensions considered are as follows: 1.4 μm long with a trapezoidal cross section whose bases are 180 and 80 nm; the height is 150 nm. The corners are rounded by 20 nm. For plane-wave excitation simulations of the nanogap structures, we consider an incident wave at an angle of 60 deg from the gold plane’s normal (p-polarized for the NPoM case, orthogonal to the groove for the NPiG case).

Figure 7

Mie calculations for a spherical silver NP of 80 nm diameter under plane-wave excitation. (a) Spatial distribution of the intensity enhancement factor around the silver NP for a plane-wave excitation tuned to the dipole resonance. (b) Spectra of extinguished, scattered and absorbed power. (c) Multi-Lorentzian fit of the extinguished power with the dipolar and quadrupolar contributions labeled (dip) and (quad), respectively.