Harnessing complexity: Nonlinear optical phenomena in L-shapes, nanocrescents, and split-ring resonators.

We conduct systematic studies of the optical characteristics of plasmonic nanoparticles that exhibit C2v symmetry. In particular, we analyze three distinct geometric configurations: an L-type shape, a crescent, and a split-ring resonator shaped like the Greek letter π. Optical properties are examined using the finite-difference time-domain method. It is demonstrated that all three shapes exhibit two prominent plasmon modes associated with the two axes of symmetry. This is in addition to a wide range of resonances observed at high frequencies corresponding to quadrupole modes and peaks due to sharp corners. Next, to facilitate nonlinear analysis, we employ a semiclassical hydrodynamic model, where the electron pressure term is explicitly accounted for. This model goes beyond the standard Drude description and enables capturing nonlocal and nonlinear effects. Employing this model enables us to rigorously examine the second-order angular resolved nonlinear optical response of these nanoparticles in each of the three configurations. Two pumping regimes are considered, namely, continuous wave (CW) and pulsed excitations. For CW pumping, we explore the properties of the second harmonic generation (SHG). Polarization and angle-resolved SHG spectra are obtained, revealing strong dependence on the nanoparticle geometry and incident wave polarization. The C2v symmetry is shown to play a key role in determining the polarization states and selection rules of the SHG signal. For pulsed excitations, we discuss the phenomenon of broadband terahertz (THz) generation induced by the difference-frequency generation . It is shown that the THz emission spectra exhibit unique features attributed to the plasmonic resonances and symmetry of the nanoparticles. The polarization of the generated THz waves is also examined, revealing interesting patterns tied to the nanoparticle geometry. To gain deeper insight, we propose an analytical theory that agrees very well with the numerical experiments. The theory shows that the physical origin of the THz radiation is the mixing of various frequency components of the fundamental pulse by the second-order nonlinear susceptibility. An expression for the far-field THz intensity is derived in terms of the incident pulse parameters and the nonlinear response tensor of the nanoparticle. The results presented in this work offer new insights into the linear and nonlinear optical properties of nanoparticles with C2v symmetry. The demonstrated strong SHG response and efficient broadband THz generation hold great promise for applications in nonlinear spectroscopy, nanophotonics, and optoelectronics. The proposed theoretical framework also provides a valuable tool for understanding and predicting the nonlinear behavior of other related nanostructures.

Interplay between gain and loss in arrays of nonlinear plasmonic nanoparticles: toward parametric downconversion and amplification.

With the help of a theoretical model and finite-difference time-domain (FDTD) simulations based on the hydrodynamic-Maxwell model, we examine the effect of difference-frequency generation (DFG) in an array of L-shaped metal nanoparticles (MNPs) characterized by intrinsic plasmonic nonlinearity. The outcomes of the calculations reveal the spectral interplay between gain and loss in the vicinity of the fundamental frequency of the localized surface plasmon resonances. Subsequently, we identify different array thicknesses and pumping regimes facilitating parametric amplification and spontaneous parametric downconversion. Our results suggest that the parametric amplification regime becomes feasible on a scale of hundreds of nanometers and spontaneous parametric downconversion on the scale of tens of nanometers, opening up new exciting opportunities for developing building blocks of photonic metasurfaces.

On the nature of two-photon transitions for a collection of molecules in a Fabry-Perot cavity.

We investigate the effect of a cavity on nonlinear two-photon transitions of a molecular system and we analyze how such an effect depends on the cavity quality factor, the field enhancement, and the possibility of dephasing. We find that the molecular response to strong light fields in a cavity with a variable quality factor can be understood as arising from a balance between (i) the ability of the cavity to enhance the field of an external probe and promote multiphoton transitions more easily and (ii) the fact that the strict selection rules on multiphoton transitions in a cavity support only one resonant frequency within the excitation range. Although our simulations use a classical level description of the radiation field (i.e., we solve Maxwell-Bloch or Maxwell-Liouville equations within the Ehrenfest approximation for the field-molecule interaction), based on experience with this level of approximation in the past studies of plasmonic and polaritonic systems, we believe that our results are valid over a wide range of external probing.

Comparing semiclassical mean-field and 1-exciton approximations in evaluating optical response under strong light-matter coupling conditions.

The rigorous quantum mechanical description of the collective interaction of many molecules with the radiation field is usually considered numerically intractable, and approximation schemes must be employed. Standard spectroscopy usually contains some levels of perturbation theory, but under strong coupling conditions, other approximations are used. A common approximation is the 1-exciton model in which processes involving weak excitations are described using a basis comprising the ground state and singly excited states of the molecule cavity-mode system. In another frequently used approximation in numerical investigations, the electromagnetic field is described classically, and the quantum molecular subsystem is treated in the mean-field Hartree approximation with its wavefunction assumed to be a product of single molecules' wavefunctions. The former disregards states that take long time to populate and is, therefore, essentially a short time approximation. The latter is not limited in this way, but by its nature, disregards some intermolecular and molecule-field correlations. In this work, we directly compare results obtained from these approximations when applied to several prototype problems involving the optical response of molecules-in-optical cavities systems. In particular, we show that our recent model investigation [J. Chem. Phys. 157, 114108 (2022)] of the interplay between the electronic strong coupling and molecular nuclear dynamics using the truncated 1-exciton approximation agrees very well with the semiclassical mean-field calculation.

Short-time particle motion in one and two-dimensional lattices with site disorder.

As in the case of a free particle, the initial growth of a broad (relative to lattice spacing) wavepacket placed on an ordered lattice is slow (its time derivative has zero initial slope), and the spread (root mean square displacement) becomes linear in t at a long time. On a disordered lattice, the growth is inhibited for a long time (Anderson localization). We consider site disorder with nearest-neighbor hopping on one- and two-dimensional systems and show via numerical simulations supported by the analytical study that the short time growth of the particle distribution is faster on the disordered lattice than on the ordered one. Such faster spread takes place on time and length scales that may be relevant to the exciton motion in disordered systems.

Dissociation slowdown by collective optical response under strong coupling conditions.

We consider an ensemble of diatomic molecules resonantly coupled to an optical cavity under strong coupling conditions at normal incidence. Photodissociation dynamics is examined via direct numerical integration of the coupled Maxwell-Schrödinger equations with molecular rovibrational degrees of freedom explicitly taken into account. It is shown that the dissociation is significantly affected (slowed down) when the system is driven at its polaritonic frequencies. The observed effect is demonstrated to be of transient nature and has no classical analog. An intuitive explanation of the dissociation slowdown at polaritonic frequencies is proposed.

Degenerate parametric down-conversion facilitated by exciton-plasmon polariton states in a nonlinear plasmonic cavity.

We study the effect of degenerate parametric down-conversion (DPDC) in an ensemble of two-level quantum emitters (QEs) coupled via near-field interactions to a single surface plasmon (SP) mode of a nonlinear plasmonic cavity. For this purpose, we develop a quantum driven-dissipative model capturing non-equilibrium dynamics of the system in which incoherently pumped QEs have transition frequency tuned near the second-harmonic response of the SPs. Considering the strong coupling regime, i.e. the SP-QE interaction rate exceeds system dissipation rates, we find a critical SP-QE coupling attributed to the phase transition between normal and lasing steady states. Examining fluctuations above the system's steady states, we predict new elementary excitations, namely, the exciton-plasmon polaritons formed by the two-SP quanta and single-exciton states of QEs. The contribution of two-SP quanta results in the linear scaling of the SP-QE interaction rate with the number of QEs,o, as opposed to theo-scaling known for the Dicke and Tavis-Cummings models. We further examine how SP-QE interaction scaling affects the polariton dispersions and power spectra in the vicinity of the critical coupling. For this purpose, we compare the calculation results assuming a finite ensemble of QEs and the model thermodynamic limit. The calculated power spectra predict an interplay of coherent photon emission by QEs near the second-harmonic frequency and correlated photon-pair emission at the fundamental frequency by the SPs (i.e. the photonic DPDC effect).

Fano plasmonics goes nonlinear.

We investigate the process of the second harmonic generation by plasmonic nano-antennas that exhibit Fano-like resonances. A rigorous fully vectorial Maxwell-hydrodynamics approach is employed to directly calculate the second order susceptibilities as a function of the pump frequency, considering a periodic array of nanodolmens comprised of three Au nanorods. The results of the numerical simulations demonstrate a noticeable enhancement of the second harmonic efficiency by the antisymmetric mode. Additionally, a simple analytical model based on two coupled nonlinear oscillators is proposed. It is shown that the second order optical response can be significantly enhanced at the frequency of the antisymmetric normal mode, thus supporting our numerical results.

Coupling, lifetimes, and "strong coupling" maps for single molecules at plasmonic interfaces.

The interaction between excited states of a molecule and excited states of a metal nanostructure (e.g., plasmons) leads to hybrid states with modified optical properties. When plasmon resonance is swept through molecular transition frequency, an avoided crossing may be observed, which is often regarded as a signature of strong coupling between plasmons and molecules. Such strong coupling is expected to be realized when 2|⟨U⟩|/â„Γ > 1, where ⟨U⟩ and Γ are the molecule-plasmon coupling and the spectral width of the optical transition, respectively. Because both ⟨U⟩ and Γ strongly increase with decreasing distance between a molecule and a plasmonic structure, it is not obvious that this condition can be satisfied for any molecule-metal surface distance. In this work, we investigate the behavior of ⟨U⟩ and Γ for several geometries. Surprisingly, we find that if the only contributions to Γ are lifetime broadenings associated with the radiative and nonradiative relaxation of a single molecular vibronic transition, including effects on molecular radiative and nonradiative lifetimes induced by the metal, the criterion 2|⟨U⟩|/â„Γ > 1 is easily satisfied by many configurations irrespective of the metal-molecule distance. This implies that the Rabi splitting can be observed in such structures if other sources of broadening are suppressed. Additionally, when the molecule-metal surface distance is varied keeping all other molecular and metal parameters constant, this behavior is mitigated due to the spectral shift associated with the same molecule-plasmon interaction, making the observation of Rabi splitting more challenging.

Strong coupling between an inverse bowtie Nano-Antenna and a J-aggregate.

We demonstrate strong coupling between a single or few J-aggregates and an inverse bowtie plasmonic structure, when the J-aggregate is located at a specific axial distance from the metallic surface. Three hybrid modes are clearly observed, witnessing a strong interaction, with a Rabi splitting of up to 290 meV, the precise value of which significantly depends on the orientation of the J-aggregate with respect to the symmetry axis of the plasmonic structure. We repeated our experiments with a set of triangular hole arrays, showing consistent formation of three or more hybrid modes, in good agreement with numerical simulations.

Second harmonic generation by strongly coupled exciton-plasmons: The role of polaritonic states in nonlinear dynamics.

We investigate second harmonic generation (SHG) from hexagonal periodic arrays of triangular nano-holes of aluminum using a self-consistent methodology based on the hydrodynamics-Maxwell-Bloch approach. It is shown that angular polarization patterns of the far-field second harmonic response abide to threefold symmetry constraints on tensors. When a molecular layer is added to the system and its parameters are adjusted to achieve the strong coupling regime between a localized plasmon mode and molecular excitons, Rabi splitting is observed from the occurrence of both single- and two-photon transition peaks within the SHG power spectrum. It is argued that the splitting observed for both transitions results from direct two-photon transitions between lower and upper polaritonic states of the strongly coupled system. This interpretation can be accounted by a tailored three-level quantum model, with results in agreement with the unbiased numerical approach. Our results suggest that the hybrid states formed in strongly coupled systems directly contribute to the nonlinear dynamics. This opens new directions in designing THz sources and nonlinear frequency converters.

Second-harmonic generation in nonlinear plasmonic lattices enhanced by quantum emitter gain medium.

We report on a theoretical study of second-harmonic generation (SHG) in plasmonic nanostructures interacting with two-level quantum emitters (QEs) under incoherent energy pump. We generalize the driven-dissipative Tavis-Cummings model by introducing the anharmonic surface plasmon-polariton (SPP) mode coupled to QEs and examine physical properties of corresponding SPP-QE polariton states. Our calculations of the SHG efficiency for strong QE-SPP coupling demonstrate orders of magnitude enhancement facilitated by the polariton gain. We further discuss time-domain numerical simulations of SHG in a square lattice comprising Ag nanopillars coupled to QEs utilizing a fully vectorial nonperturbative nonlinear hydrodynamic model for conduction electrons coupled to Maxwell-Bloch equations for QEs. The simulations support the idea of gain enhanced SHG and show orders of magnitude increase in the SHG efficiency as the QEs are tuned in resonance with the lattice plasmon mode and brought above the population inversion threshold by incoherent pumping. By varying pump frequency and tuning QEs to a localized plasmon mode, we demonstrate further enhancement of the SHG efficiency facilitated by strong local electric fields. The incident light polarization dependence of the SHG is examined and related to the symmetries of participating plasmon modes.

Second Harmonic Generation from a Single Plasmonic Nanorod Strongly Coupled to a WSe2 Monolayer.

Monolayer transition metal dichalcogenides, coupled to metal plasmonic nanocavities, have recently emerged as new platforms for strong light-matter interactions. These systems are expected to have nonlinear-optical properties that will enable them to be used as entangled photon sources, compact wave-mixing devices, and other elements for classical and quantum photonic technologies. Here, we report the first experimental investigation of the nonlinear properties of these strongly coupled systems, by observing second harmonic generation from a WSe2 monolayer strongly coupled to a single gold nanorod. The pump-frequency dependence of the second-harmonic signal displays a pronounced splitting that can be explained by a coupled-oscillator model with second-order nonlinearities. Rigorous numerical simulations utilizing a nonperturbative nonlinear hydrodynamic model of conduction electrons support this interpretation and reproduce experimental results. Our study thus lays the groundwork for understanding the nonlinear properties of strongly coupled nanoscale systems.

Plasmon enhanced second harmonic generation by periodic arrays of triangular nanoholes coupled to quantum emitters.

Optical properties of periodic arrays of nanoholes of a triangular shape with experimentally realizable parameters are examined in both linear and nonlinear regimes. By utilizing a fully vectorial three-dimensional approach based on the nonlinear hydrodynamic Drude model describing metal coupled to Maxwell's equations and Bloch equations for molecular emitters, we analyze linear transmission, reflection, and nonlinear power spectra. Rigorous numerical calculations demonstrating second and third harmonic generation by the triangular hole arrays are performed. It is shown that both the Coulomb interaction of conduction electrons and the convective term contribute on equal footing to the nonlinear response of metal. It is demonstrated that the energy conversion efficiency in the second harmonic process is the highest when the system is pumped at the localized surface plasmon resonance. When molecular emitters are placed on a surface of the hole array line shapes, the second harmonic signal exhibits three peaks corresponding to second harmonics of the localized surface plasmon mode and upper and lower polaritonic states.

Energy Transfer and Interference by Collective Electromagnetic Coupling.

The physics of collective optical response of molecular assemblies, pioneered by Dicke in 1954, has long been at the center of theoretical and experimental scrutiny. The influence of the environment on such phenomena is also of great interest due to various important applications in, e.g., energy conversion devices. In this Letter, we demonstrate both experimentally and theoretically the spatial modulations of the collective decay rates of molecules placed in proximity to a metal interface. We show in a very simple framework how the cooperative optical response can be analyzed in terms of intermolecular correlations causing interference between the response of different molecules and the polarization induced on a nearby metallic boundary and predict similar collective interference phenomena in excitation energy transfer between molecular aggregates.

Modeling optical coupling of plasmons and inhomogeneously broadened emitters.

Optically coupling quantum emitters to nanoparticles provides the foundation for many plasmonic applications. Including quantum mechanical effects within the calculations can be crucial for designing new devices, but classical approximations are sometimes sufficient. Comprehending how the classical and quantum mechanical descriptions of quantum emitters alter their calculated optical response will lead to a better understanding of how to design devices. Here, we describe how the semiclassical Maxwell-Liouville method can be used to calculate the optical response from inhomogeneously broadened states. After describing the Maxwell-Liouville algorithm, we use the method to study the photon echoes from quantum dots and compare the results against analytical models. We then modify the quantum dot's state distribution to match a PbS 850 nm quantum dot's absorption spectra to see how the complete quasi-band structure affects their coupling to gold nanoislands. Finally, we compare the results with previously published work to demonstrate where the complete quantum dot description is necessary.

Ehrenfest+R dynamics. I. A mixed quantum-classical electrodynamics simulation of spontaneous emission.

The dynamics of an electronic system interacting with an electromagnetic field is investigated within mixed quantum-classical theory. Beyond the classical path approximation (where we ignore all feedback from the electronic system on the photon field), we consider all electron-photon interactions explicitly according to Ehrenfest (i.e., mean-field) dynamics and a set of coupled Maxwell-Liouville equations. Because Ehrenfest dynamics cannot capture certain quantum features of the photon field correctly, we propose a new Ehrenfest+R method that can recover (by construction) spontaneous emission while also distinguishing between electromagnetic fluctuations and coherent emission.

Ehrenfest+R dynamics. II. A semiclassical QED framework for Raman scattering.

In Paper I [Chen et al., J. Chem. Phys. 150, 044102 (2019)], we introduced Ehrenfest+R dynamics for a two-level system and showed how spontaneous emission can be heuristically included such that, after averaging over an ensemble of Ehrenfest+R trajectories, one can recover both coherent and incoherent electromagnetic fields. In the present paper, we now show that Ehrenfest+R dynamics can also correctly describe Raman scattering, whose features are completely absent from standard Ehrenfest dynamics. Ehrenfest+R dynamics appear to be quantitatively accurate both for resonant and off-resonant Raman signals, as compared with Kramers-Heisenberg-Dirac theory.