Nonlinear Atom-Plasmon Interactions Enabled by Nanostructured Graphene

Electrically tunable graphene plasmons are anticipated to enable strong light-matter interactions with resonant quantum emitters. However, plasmon resonances in graphene are typically limited to infrared frequencies, below those of optical excitations in robust quantum light sources and many biologically interesting molecules. Here we propose to utilize near fields generated by the plasmon-assisted nonlinear optical response of nanostructured graphene to resonantly couple with proximal quantum emitters operating in the near infrared. We show that the nonlinear near field produced by a graphene nanodisk can strongly excite and coherently control quantum states in two- and three-level atomic systems when the third harmonic of its plasmon resonance is tuned to a particular electronic transition. In the present scheme, emitter and plasmon resonances are nondegenerate, circumventing strong enhancement of spontaneous emission. We envision potential applications for the proposed nonlinear plasmonic coupling scheme in sensing and temporal quantum control.

Figure 1

Nonlinear atom-plasmon near-field coupling. (a) Illustration of a graphene nanodisk interacting with impinging light at frequency $\omega$ (red beam) by generating a near field at $3\omega$ (blue field) that couples with a point-dipole quantum emitter (QE) located a distance $d$ above the disk center. Direct coupling of the third-harmonic (TH) near field with the QE is quantified by a Rabi frequency $\mathrm{\Omega }$, while $G$ characterizes the QE self-interaction enabled by graphene. (b) Energy-level diagram of the graphene-QE hybrid: Light at frequency $\omega \approx {\omega }_p$ drives plasmon-assisted TH generation in the nanodisk, producing a field at $3\omega$ that couples with a QE electronic transition of similar energy. For a $D=40\mathrm{nm}$ diameter nanodisk with a QE placed $d=20\mathrm{nm}$ above its center and cw light impinging normally to the graphene plane, we plot (c) $\mathrm{\Omega }$ for various light intensities and (d) $G$ as functions of optical frequency, with Fermi energies $E_F$ indicated by the color-coordinated curves. For the same system, the (e) real and (f) imaginary parts of $G$ are plotted over a wider range of emitter resonance energies.

Figure 2

Power absorption via third-harmonic atom-plasmon coupling. (a) We plot the steady-state power absorption in the QE for the system considered in Fig. 1 with a fixed graphene doping $E_F=0.5\mathrm{eV}$ (yielding a plasmon resonance $\hslash {\omega }_p=0.2694\mathrm{eV}$ for a disk of diameter 40 nm), as a function of the illumination frequency when the QE resonance frequency is ${ϵ}_{12}=3{\omega }_p$. In the upper panel we plot $P_{\mathrm{QE}}$ for a fixed driving intensity $I_{\mathrm{ext}}$ and various QE-graphene separations $d$, while in the lower panel we fix $d$ and vary $I_{\mathrm{ext}}$. (b) Graphene doping dependence for the system in (a) at the resonance condition $\hslash \omega =\hslash {\omega }_p=\hslash {ϵ}_{12}/3=0.2694\mathrm{eV}$.

Figure 3

Nonlinear near-field electromagnetically induced transparency. (a) Schematic illustration of the $V$-type QE-graphene nanodisk hybrid system interacting with a strong driving field of frequency $\omega \approx {\omega }_p\approx {ϵ}_{12}/3$ and intensity $I_{\mathrm{ext}}$ along with an auxiliary probe field of frequency ${\omega }_{\text{probe}}$. (b) The nonlinear plasmonic TH near field from the nanodisk couples resonantly with the $|1⟩\leftrightarrow |2⟩$ transition in the QE, while the weak probe field (${\mathrm{\Omega }}_{\text{probe}}=0.01{\mathrm{\Gamma }}_0$ with detuning $\mathrm{\Delta }\equiv {\omega }_{\text{probe}}-{ϵ}_{13}$) couples only with the $|1⟩\leftrightarrow |3⟩$ transition. (c) For the parameters of Fig. 2, taking $E_F=0.5\mathrm{eV}$ and $\hslash \omega =\hslash {\omega }_p=\hslash {ϵ}_{12}/3=0.2694\mathrm{eV}$, we plot the probe field absorption, characterized by the imaginary part of the QE polarizability associated with the $|1⟩\leftrightarrow |3⟩$ transition, as a function of the detuning from resonance. In the upper (lower) panel $I_{\mathrm{ext}}$ ($d$) is fixed and $d$ ($I_{\mathrm{ext}}$) is varied. (d) Same as (c) but fixing $\mathrm{\Delta }=0$ while $E_F$ is varied.

Figure 4

Nonlinear plasmonic quantum control. (a) A schematic of the graphene nanodisk and the relative orientations of the transition dipole moments ${\overrightarrow{\mu }}_{12}$ and ${\overrightarrow{\mu }}_{13}$ of the neighboring QE, along with the incoming electric field polarization. The scheme to the right shows the energy-level structure of the $\mathrm{\Lambda }$-type QE, indicating an equal-magnitude and opposite-sign detuning of the TH central pulse frequency from either transition. (b) For a pulse of peak intensity $I_{\mathrm{ext}}=1\mathrm{TW}/{\mathrm{m}}^2$ and FWHM duration $\sigma =130\mathrm{fs}$, we plot the direct Rabi frequency at a graphene-QE separation distance $d=15\mathrm{nm}$ and graphene Fermi energies $E_F=0.50$, 0.51, and 0.52 eV, with the transition energies ${ϵ}_{12}$ and ${ϵ}_{13}$ indicated by color-coordinated vertical lines. (c) Time evolution of the populations in states $|2⟩$ and $|3⟩$ for the configurations considered in (b). (d) Final metastable state populations (at $t=10\sigma$) plotted as functions of the peak pulse intensity (upper panel) and the Fermi energy (lower panel).