Vibrationally Induced Resonances in Lasing

Abstract

Optical circuits and light sources, such as lasers, undergo continuous miniaturization. In its extreme, nanolasers might be comprised of only a few molecules confined in plasmonic nanoresonators. Few-emitter lasers promise low energy requirements and fast responses in a footprint that can be inserted into any device or biological tissue. Utilizing the recently developed stacked hierarchy approach, informed from first principles, we demonstrate the impact of vibrational structure on lasing, using the example of few-molecule lasing in plasmonic cavities. Explicitly accounting for the entire vibrational manifold unveils resonances in the laser intensity that depend on the Stokes shift, drive strength, and the number of emitters. Our work identifies the limits of the omnipresent "incoherent drive"-approximation and paves the way for the understanding of nanolasers at the molecular scale.

Figures (3)

• Figure 1: Overview of the few emitter lasing system:(a) Sketch of the plasmonic nanocavity, comprised of a gold nanoparticle above a gold surface. (b) We restrict our model of the electronic states to the first bright transition, leading to a two level system. However, we take the full spectrum of vibrational modes into account, which we approximate as a large set of harmonic oscillators (purple). Each oscillator in turn couples to a continuum of vibrational modes that we model with an ohmic spectral density (blue). All the vibrations can be combined into a single effective bath with the spectral density $J_{eff}(\omega)$ shown in (d) (green solid line). The corresponding bath correlation function shown in the inset is then fitted with exponentials (black dashed line). Graphic (c) shows a sketch of the lasing mechanism, where the states $\ket{2},\,\ket{3}$ are distinguished from the states $\ket{1}, \ket{4}$ by a different vibrational state. Purple parabolas in the background indicate the shifted harmonic potential surfaces.
• Figure 2: Vibrational impact on few-emitter lasing: We compare the qualitative steady state behavior for increasing the coherent drive in the full model \ref{['eq:H_fewEmitterLasing']},\ref{['eq:fewEmitterLasingLocalDiss']} against an effective incoherent drive that neglects the vibrational structure \ref{['eq:incoherentLasing']}. Plots (a) and (b) show the occupation of the excited electronic state, $p_e$, and the cavity mode, $a^{\dagger}a$, respectively, for a plasmonic cavity containing ten molecules. Both plots show maxima arising from resonances with the vibrational spectrum that are missed by an effective treatment with an incoherent drive. Those maxima become more pronounced for increasing $N$ as shown in plot (c).
• Figure 3: Vibrationally induced resonances: Steady state cavity occupation for $N=5$ (purple) and effective vibrational spectrum are shown on a shared x-axis, with $2E_d = \omega$. Maxima in $\langle a^\dagger a\rangle$ line up with peaks in the spectral density. Solid, lighter lines correspond the the data in Fig. \ref{['fig:fewEmitterLasing']}(c) and Fig. \ref{['fig:sketchMoleculeNanocavity']}(d) respectively. Intentionally omitting (shifting) a peak in $J_{eff}(\omega)$ also removes (shifts) the corresponding occupation maxima, as evidenced by the darker dashed (dotted) lines. Insets show dominant vibrational modes at the peak positions.