Hybrid graphene - silver nanoantenna to control THz emission from polar quantum systems

Saeid Izadshenas; Piotr Gładysz; Karolina Słowik

doi:10.1364/OE.496435

1. Introduction

The unique optical properties of polar quantum systems have triggered a growing interest in investigating their response and exploiting them as radiation sources [1–12]. Specifically, when resonantly driven, these systems sustain a dipolar response at both the illumination frequency and at a frequency reduced by orders of magnitude. The latter corresponds to the interaction strength with the driving field, known as the Rabi frequency [1]. In the electric dipole approximation, it linearly depends on the amplitude of the driving field [1,6,13]. This observation gave rise to the concept of all-optically tunable, efficient, coherent radiation sources based on polar quantum systems. The ultimate goal is to tune the frequency of the emitted radiation to the THz regime [2,7,13], and to contribute to bridging the THz gap [14–17]. This approach has led to proposing laser-driven polar molecules or quantum dots for THz lasing [2,7,13] or generating squeezed states of radiation [6,8]. Arrays of quantum dots or molecular ensembles have been investigated, where coherent radiation of macroscopic powers has been predicted [10,13].

Another approach to enhancing the emission power involves plasmonic nanoantennas. Recent proposals exploit surface plasmons to modify the light interaction strength with polar molecules, particularly to influence the molecular optical rectification coefficient [18], or to boost the intensity of THz lasing [7]. In the current scheme, the emission frequency scales the driving field amplitude, hence, the tunability of plasmonic devices is required to support the emission efficiently. To address this requirement, we propose a tunable nanoantenna design that brings the radiation generated by polar molecules to the THz range. The proposed hybrid antenna is made of metallic and graphene components. The metallic elements support the molecular excitation with visible light, leading to Rabi oscillations. For electronic transitions in atoms and molecules, these oscillations typically occur at the MHz to GHz rates. Here, the field amplitude enhancement by the silver nanoparticles blue-shifts the Rabi frequency to the THz regime. The tunable graphene components boost the emission within the THz band. The proposed scheme stands against the paradigmatic approach, where plasmonic nanoantennas are exploited as passive elements merely enhancing quantum transition rates. Here, the antenna actively impacts the spectral properties of the emitted light.

Below, we explain the concept in detail and introduce the proposed nanoantenna. We also engineer the antenna to operate with BaF molecules and demonstrate its tunability. Our proposed design shows promise for developing efficient and tunable radiation sources in the THz regime.

2. Concept

Below we describe the mechanisms of low-frequency radiation generation in polar systems and the impact of plasmonic nanoantennas on the emitted radiation intensity and spectral characteristics.

2.1 Radiation emission from polar quantum systems

The electric dipole moment operator of a two-level quantum system takes the form

(1)$$\begin{aligned}\vec{d} = \frac{1}{2}\left(\vec{d}_{ee} - \vec{d}_{gg}\right)\sigma_z +\frac{1}{2}\left(\vec{d}_{ee}+\vec{d}_{gg}\right)I + \vec{d}_{eg}\sigma^+{+} \vec{d}_{ge}\sigma^-,\end{aligned}$$

where $\sigma _z$, $\sigma ^-$ and $\sigma ^+=(\sigma ^-)^\dagger$ are Pauli inversion, lowering and raising operators. The dipole moment has off-diagonal, induced elements $\vec {d}_{ge} = \vec {d}^\star _{eg}$ responsible for transitions between the excited $\left| e \right\rangle$ and ground $\left| g \right\rangle$ upon illumination with electromagnetic field and diagonal, permanent elements $\vec {d}_{ee}, \vec {d}_{gg}$ which are nonzero for polar systems, i.e. systems breaking the spatial inversion symmetry. The latter elements, referred to as "permanent dipole moments" (PDMs), are routinely neglected in light-matter interaction problems; however, they may give rise to interesting physics, including low-frequency radiation emission (Fig. 1). Below, for simplicity, we will assume real values of the dipole moment elements.

Upon electromagnetic illumination with a monochromatic beam and in the electric dipole approximation, a polar two-level system is described by the time-dependent Hamiltonian given below. It consists of terms that we refer to as "non-polar" since they are independent of PDMs and describe interactions of non-polar systems with light, and terms called "polar" since they require nonzero PDMs.

(2)$$\begin{aligned}H = \underbrace{\frac{1}{2}\hbar\omega_0\sigma_z - \hbar\Omega\left(\sigma^+{+} \sigma^-\right) \cos{\omega t}}_\text{non-polar terms} -\underbrace{ \frac{1}{2} \vec{E}_0\left(\vec{d}_{ee} - \vec{d}_{gg}\right) \sigma_z \cos{\omega t}}_\text{polar terms}.\end{aligned}$$

Here, $\omega _0$ is a transition frequency between the excited and ground states, $\Omega = \vec {E}_0\cdot \vec {d}_{eg}/\hbar$ is the Rabi frequency, $\omega$ and $\vec {E}_0$ are the illumination frequency and amplitude. We dropped the term proportional to the identity operator $I$ as it trivially shifts the energies of the eigenstates by relatively small values $|\frac {1}{2}\vec {E}_0\cdot \vec {d}_{jj}|\ll \hbar \omega _0$. The electric dipole approximation intrinsically assumes the point-dipole character of the quantum system, and the electric field amplitude is evaluated at the position of the polar system. From now on, we assume all the dipole moments are parallel to the driving field. Hence, we can omit the vector notation in Eqs. (1) and (2).

We now make use of the unitary transformation [1]

(3)$$\begin{aligned}U = \mathrm{e}^{\frac{1}{2}i\left( \omega t - \kappa \sin{\omega t}\right)\sigma_z},\end{aligned}$$

where $\kappa = E_0\left ( d_{ee}-d_{gg}\right )/\hbar \omega$ is a dimensionless parameter connected to the PDMs difference. Note that $\kappa = 0$ for non-polar systems. By using the Jacoby-Auger identity $\exp (i\kappa \sin \omega t) = \sum ^{\infty }_{n=-\infty }J_n(\kappa )\exp (i n\omega t)$ with Bessel functions of the first kind $J_n(\kappa )$, and applying the rotating-wave approximation, we obtain the time-independent Hamiltonian in the interaction picture with corrections induced by the PDMs

(4)$$\begin{aligned}H_{int} ={-}\frac{1}{2} \hbar \delta \sigma_z - \frac{1}{2} \hbar \underbrace{\Omega \frac{2}{\kappa}J_1(\kappa)}_{\substack{\text{effective Rabi} \\ \text{frequency}}}\left( \sigma^+{+}\sigma^{-}\right).\end{aligned}$$

Here, $\delta = \omega - \omega _0$ is the laser detuning. The derived Hamiltonian can be interpreted in an effective picture in which the light-matter coupling strength quantified in terms of effective Rabi frequency is modified by the PDMs. In general, the frequency of population oscillations additionally depends on the detuning and damping in the system. Below, we focus on resonant illumination with $\delta =0$, while the spontaneous emission rate $\gamma$ of the two-level system near a plasmonic nanoantenna can be significant. Following the analytical derivation shown in Ref. [19], the generalised Rabi frequency of a polar system can be shown to take the form:

(5)$$\begin{aligned}\Omega_\text{polar} = \sqrt{\left(\Omega\frac{2}{\kappa}J_1(\kappa)\right)^2 - \left(\frac{\gamma}{4}\right)^2}.\end{aligned}$$

Note that in the non-polar case and for $\gamma =0$, as $\kappa \xrightarrow {} 0$, the Bessel function $\frac {J_1(\kappa )}{\kappa }\rightarrow \frac {1}{2}$ and $\Omega _\text {polar} \xrightarrow {} \Omega$. Despite the general character of the above expression, in our further investigation, $\kappa$ always remains small and the influence of the PDMs on the Rabi frequency is negligible. However, they may significantly influence the emission properties of the system. For $|E(d_{ee}-d_{gg})|\ll \hbar \omega$, the expectation value for the dipole moment of a polar system is given by [1]

(6)$$\begin{aligned}d(t) ={-} \underbrace{d_{eg}\sin(\omega t)\sin(\Omega_\mathrm{polar}t)}_{\mathrm{Mollow\ triplet\ component}} +\underbrace{\frac{1}{2}(d_{ee}-d_{gg})\cos(\Omega_\mathrm{polar} t)}_{\mathrm{low-energy\ component}},\end{aligned}$$

where one can distinguish the components giving rise to the Mollow triplet in the emission spectrum [20] and a low-energy component at the generalised Rabi frequency. In this work, we focus on enhancing the low-frequency contribution with the use of plasmonic nanostructures.

2.2 Emission enhancement with plasmonic nanoantennas

Plasmons, which are collective oscillations of conduction electrons, can be excited in metallic materials. Near interfaces with dielectrics, they interact strongly with light giving rise to hybrid excitations called surface plasmon polaritons. Their electromagnetic component can be confined down to the nanoscale, which is associated with enormous optical field intensity enhancements. These remarkable properties have many applications, including optical sensing [21–25], modulation of optical signals [26,27], or light-matter interaction enhancement [28,29].

Metallic nanoparticles called plasmonic nanoantennas are routinely exploited in the latter context due to their ability to mediate between propagating radiation and locally confined fields [30]. A quantum system interacts with the electromagnetic field with a strength $\Omega$ proportional to its amplitude. In the confinement volume, the field amplitude and, hence, the interaction strength are enhanced to:

(7)$$\begin{aligned}\Omega_\mathrm{NP} = \Omega \frac{E_\mathrm{NP}}{E_0},\end{aligned}$$

where $\vec {E}_\mathrm {NP}(\vec {r},\omega )=\vec {E}_\mathrm {scat}(\vec {r},\omega )+\vec {E}_0(\omega )$ is the electric field in the vicinity of the nanoantenna, including the illuminating $\vec {E}_0$ and the scattered contributions $\vec {E}_\mathrm {scat}$, and $E_\mathrm {NP}$ is its component along the direction of molecular dipole moments. For polar systems, the field enhancement induces a spectral shift of the emitted radiation according to Eqs. (5), (6) with $\Omega$ replaced by $\Omega _\mathrm {NP}$.

Another consequence of the nanoantenna presence near a molecule is the Purcell effect – an enhancement of the spontaneous emission rate of the system

(8)$$\begin{aligned}\gamma_{NP} = \gamma \frac{P_\text{NP}(\vec{r}_0,\omega_0)}{P_0(\omega_0)}.\end{aligned}$$

For a dipolar transition, the enhancement factor is evaluated as a ratio of $P_\text {NP}$ and $P_0$, representing powers emitted by a dipole oscillating at the transition frequency and positioned near the nanoantenna or in free space, respectively [30,31]. This effect depends on the nanoantenna geometry and material, as well as the respective position of the source $\vec {r}_0$. Hence, contrary to the coupling strength (7), the spontaneous emission rate is independent of the driving field. In order to achieve Rabi oscillations and the resulting low-frequency emission, the intensity of the driving field has to be large enough to compensate for the Purcell-enhanced losses: There is a threshold value of the driving electric field $E_\text {t}$ for which squared Rabi frequency, now expressed as

(9)$$\begin{aligned}\Omega^2_\text{polar, NP} = \Omega^2_\text{NP} - \left(\frac{\gamma_\text{NP}}{4}\right)^2,\end{aligned}$$

becomes positive. We have assumed here the $\kappa \rightarrow 0$ limit, justified in all cases discussed further. Below that value, the system is overdamped and the low-frequency radiation is quenched.

2.3 Concept summary

This work is based on two observations:

1. Polar quantum systems driven at their transition frequency (typically near-infrared to visible), may emit radiation at the Rabi frequency (typically MHz to GHz).
2. Plasmonic nanostructures enhance interactions of light at resonance frequencies with quantum systems positioned in the field-enhancement volume.

We consider a nanoantenna with two operating modes: one in the near-infrared to visible range and one in the THz domain. The mode in the near-infrared to the visible domain tuned to resonance with the polar system’s transition frequency $\omega _0$, is exploited to locally enhance the electric field at this frequency upon plane-wave illumination. In consequence, the Rabi oscillation frequency of the polar system and hence, its emission frequency, are blue-shifted from GHz to THz, according to Eq. (7). The THz antenna mode is tuned to the blue-shifted Rabi frequency, enhancing the emission power of the corresponding molecular dipole. Graphene elements enable a significant degree of tunability for this mode, which allows one to adjust it, e.g., to the intensity of the incoming beam.

Below we characterize the performance of a specific, graphene-based nanoantenna tuned to support a selected example molecule.

3. Results

In this section, we introduce the parameters of the BaF molecule we have chosen for demonstration. Next, we present the design of the tunable bimodal nanoantenna and provide a broad range of numerically calculated characteristics of frequency-dependent enhancement. Finally, as proof of principle, we evaluate the resulting field and emission-power enhancements in the real-life case of the BaF molecule. However, we emphasize that the structure could be re-engineered for other polar systems.

3.1 Molecule

For a real-life example of a polar system, we consider a BaF molecule [32]. We approximate it as a two-level system with electronic states: ground $\left| g \right\rangle \sim X^2\Sigma ^+$, and excited $\left| e \right\rangle \sim A^2\Pi _{\frac {1}{2}}$. The following quantities are given in SI units and atomic units a.u. ($\hbar = e = m_e = k_e = 1$). The transition energy between the eigenstates is $\omega _0=2\pi \cdot 348$ THz (1.44 eV or 860 nm; 0.053 a.u.) in the near-infrared regime [33], and the transition dipole moment element is $d_{eg} = d_{ge} = 5.54$ D (2.18 a.u.) [34]. The PDM of the ground state $d_{gg} = 3.17$ D (1.25 a.u.) has been obtained experimentally [35]. The PDM of the excited state has not yet been measured, however, the calculated value yields $d_{ee} = 3.4$ D (1.36 a.u.) [32]. The lifetime of the excited state is $57$ ns [36], and hence, the spontaneous emission rate $\gamma = 110$ MHz (2.66$\cdot 10^{-9}$ a.u.).

3.2 Nanoantenna geometry

Plasmonic nanoantennas are typically made of noble metals whose specific sizes and morphologies allow tailoring the spectral position of plasmonic resonances in the visible and infrared domains [31]. Recently, highly doped graphene has emerged as an alternative plasmonic material, sustaining resonances in the THz regime [37]. Compared to noble metals, graphene plasmons arise through a collective response of a small number of charge carriers, bringing within reach the ability to actively tune the plasmon resonance frequency via electric gating [38].

The combination of graphene and noble-metal elements allows us to design a tunable plasmonic device with two modes: The one operating in the near-infrared domain is fixed at the selected transition frequency of BaF. The other mode appears in the terahertz range, where we expect the blue-shifted Rabi frequency to occur. The corresponding resonance can be adjusted to the Rabi frequency by changing the chemical potential of the graphene elements.

The proposed nanoantenna is composed of a pair of silver nanorods with the length $l$, a circular cross-section of diameter $w=2r$, located on top of two graphene micro-disks of radius $R$ (Fig. 2). The gap between the graphene nanodisks or silver nanorods is denoted as $s$.

Fig. 1. Energy scheme (a) and dynamics (b) of a two-level polar system subject to illumination with a resonant driving field $\vec {E}$. The colour insets in (a) schematically depict charge distributions giving rise to the permanent dipole moments $\vec {d}_{jj}$ in the ground ($j=g$, orange) and excited ($j=e$, cyan) states $|g\rangle$, $|e\rangle$. Upon illumination, the population of the system oscillates between the eigenstates with the Rabi frequency $\Omega$ proportional to the field amplitude $\vec {E}_0$ and the transition dipole element $\vec {d}_{eg}$, as shown in (b) for the population inversion, or the difference between the excited and ground state population. The Rabi oscillation gives rise to an oscillating permanent dipole $\vec {d}_{ee}-\vec {d}_{gg}$, being the source of radiation at Rabi frequency.

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Fig. 2. Schematic of a nanoantenna combining graphene micro-disks and silver nanorods.

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3.3 Nanoantenna performance

We now discuss the performance of the nanoantenna components, i.e. the pair of silver nanorods and the nanodisk dimer. We numerically model the optical response of the nanoantenna using the finite integration method in COMSOL Multiphysics. We employed the user-control mesh method to effectively adjust the mesh size according to the elements present in the simulated structure. Specifically, we increased the mesh size in smaller gaps to ensure accurate representation. By using a free tetrahedral mesh, we employed varying mesh sizes for different elements, with the highest mesh density implemented in the gap region. For the dipole illumination, we utilized a mesh size ranging from 1 nm/3 to 1 nm/15, while for the plane wave illumination, the mesh size was determined as radius/3 to radius/10 between the nanoparticles. To optimize the simulation process, we exploited the symmetries inherent in the structure. By limiting the simulation domain to a quarter of the structure and assigning perfect electric conductor properties to the $yz$ plane, and perfect magnetic conductor properties to the $xz$ plane, we were able to significantly reduce the simulation time and memory requirements. The optical properties of silver are based on the Johnson and Christy measurements [39]. We model graphene conductivity $\sigma (\omega )$ with the Kubo formula through its intra- and interband components [40]

(10)$$\begin{aligned}\sigma(\omega)=\sigma_\mathrm{intra}(\omega) + \sigma_\mathrm{inter}(\omega),\end{aligned}$$

with the Drude model describing the intraband contribution accounting for the plasmonic response [40]

(11)$$\begin{aligned}\sigma_\mathrm{intra}(\omega)= \frac{2ie^{2}T}{\pi\hbar(\omega+i\tau^{{-}1})}\ln\left[2\cosh(\frac{\mu}{2T})\right],\end{aligned}$$

and the interband contribution described by [40]

(12)$$\begin{aligned}\sigma_\mathrm{inter}(\omega) =\frac{e^{2}}{4\hbar}\left[G(\omega/2)-\frac{4\omega}{i\pi} \int_{0}^{\infty} \frac{G(\epsilon)-G(\frac{\omega}{2})}{\omega^{2}-4\epsilon^2} \,d\epsilon \right].\end{aligned}$$

Here, $i$ is the imaginary unit, $e$ represents the elementary charge, $T$ is the electron temperature, $\mu$ stands for the chemical potential tunable with gating voltage, $\tau =1$ ps is carrier relaxation time in graphene, and

(13)$$\begin{aligned}G(\epsilon) =\frac{\sinh(\frac{\epsilon}{T})}{\cosh(\frac{\mu}{T})+\cosh(\frac{\epsilon}{T})}.\end{aligned}$$

The dielectric function of graphene is evaluated as

(14)$$\begin{aligned}\epsilon(\omega) = 1+\frac{i\sigma(\omega)}{\epsilon_{0}\omega t_g}\end{aligned}$$

with $t_g=1$ nm being the graphene thickness.

The dimensions of graphene and silver antenna components differ by order of magnitude and the material properties sustain resonances at distinct spectral regimes. As we have verified numerically, by using a 4 nm TiO2 layer as a spacer that is sandwiched between silver NPs and graphene disks, the graphene components have a negligible influence on the plasmonic peak in the visible domain, and, vice versa, the silver elements have no notable impact on the spectral position of the peak in the THz regime. However, silver absorption would suppress the radiated power ratio shown in Figs. 8-10 by about 2 orders of magnitude. The results have been obtained for separately modeled near-infrared to visible and THz problems, with identical, COMSOL-built-in scattering boundary conditions at a $250\ \mu$m-diameter sphere that prevents reflection back from infinite space, with an additional perfectly matched layer.

We use Poynting’s theorem to calculate the power radiated from and absorbed by the nanoantenna [41,42]:

(15)$$\begin{aligned}P_\mathrm{rad}(\omega) \quad=\quad \oint\ \langle\vec{E}_\mathrm{sca}(\vec{r},\omega) \times \vec{H}_\mathrm{sca}(\vec{r},\omega)\rangle\ d\vec{A}, \end{aligned}$$

(16)$$\begin{aligned} P_\mathrm{abs}(\omega) \quad=\quad \int\ \langle\vec{J}_\mathrm{ind}(\vec{r},\omega) \cdot \vec{E}_\mathrm{ind}(\vec{r},\omega)\rangle\ dV, \end{aligned}$$

where similarly to $\vec {E}_\mathrm {sca}(\vec {r},\omega )$, the symbol $\vec {H}_\mathrm {sca}(\vec {r},\omega )$ denotes the scattered part of the magnetic field, and $\vec {J}_\mathrm {ind}(\vec {r},\omega )$ and $\vec {E}_\mathrm {ind}(\vec {r},\omega )$ represent the currents and electric field in the nanoantenna volume. The integrals are evaluated, respectively, at the spherical surface of the simulation volume and inside the volume of the nanoantenna elements.

3.3.1 Optical response of silver components

We first characterize the impact of the silver nanorod dimer on the electric field in the gap region. We assume the plane wave illumination scenario in the visible and infrared range, with the illuminating field polarization along the $x$ axis in Fig. 2. As we are interested in the impact of the nanoantenna on the light-matter interaction strength, we quantify the nanoantenna optical response through the local field enhancement at the high-symmetry point $\vec {r}_0$ in the middle of the gap indicated by the red dot in Fig. 2. Figures 3, 4, 5 show the local field enhancement spectra $\frac {|\vec{E}_\mathrm {scat}(\vec {r}_0,\omega )|}{|\vec{E}_0(\omega )|}$, with the plasmonic resonance positions shifted as the geometry parameters are varied. Unless otherwise stated, the nanorod length is $l=115$ nm, radius $r = 10$ nm, and the gap is set to $s=3$ nm. As expected, we find a redshift with the increasing nanorod length and gap size and a blueshift for the growing radius.

Fig. 3. Electric field enhancement spectra for different nanorod radii: (a) results for selected values of radii, (b) results for a greater number of radii demonstrate the scaling of resonance position with the nanorod radius.

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Fig. 4. Electric field enhancement spectra for different gap sizes between nanorods: (a) results for selected gap sizes, (b) results for a greater number of gap sizes demonstrate the strong influence of the gap size on the field enhancement factor.

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Fig. 5. Electric field enhancement spectra for different nanorod lengths: (a) results for selected values of the lengths, (b) results for a greater number of lengths demonstrate the scaling of the resonance position with the nanorod length.

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These investigations allow us to optimize the silver nanoantenna elements and tailor them to match the near-infrared to visible resonance of the nanoantenna with the transition frequency of the BaF molecule. Figure 6(a) demonstrates that with the optimized geometry, the resonance is indeed achieved at $\omega _0 = 2\pi \cdot 348$ THz with a high electric field enhancement factor of $\frac {|\vec{E}_\mathrm {scat}(\vec {r}_0,\omega )|}{|\vec{E}_0(\omega )|}=531$. Figure 6(b) illustrates the field enhancement distribution around the nanorods, highlighting the best performance in the middle of the gap. Note that, according to Eq. (7), this distribution corresponds to the blue-shifted Rabi frequency distribution as a function of the molecular position. Importantly, the nanorods were designed to have a much larger scattering cross-section than the absorption cross-section by carefully adjusting the shape and size of the nanoparticles to mitigate losses. As illustrated in Fig. 6(c), the cross-section ratio on resonance is found to be approximately 3.78.

Fig. 6. (a) Electric field enhancement spectrum for the optimized nanorod dimer with $l=115$ nm, $s=3$ nm, $r=10$ nm shows a resonant peak at the BaF transition frequency $\omega _0 = 2\pi \cdot 348$ THz. (b) Electric field enhancement factor distribution for plane-wave illumination at the resonant frequency. Note the enhancement is shown in logarithmic scale. (c) Scattering (RCS) and absorption (ACS) cross-sections of the nanorod dimer upon plane-wave illumination.

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Next, we investigate the impact of the Purcell enhancement of the spontaneous emission rate on the Rabi frequency $\Omega _\text {polar}$. For all results shown in Fig. 7, we assume a resonance condition so that the drive and molecular transition frequencies coincide $\omega =\omega _0$. In other words, the frequency was modified for evaluation of both the field enhancement and the gamma rate. The Purcell-enhanced $\gamma _\text {NP}$ rates, evaluated according to Eq. (8) are significant especially around the resonance frequency [Fig. 7(a)]. In Fig. 7(b), we plot the squared Rabi frequency as a function of the electric field $E_0$, and drive frequency $\omega$, calculated using Eq. (9). The blue regions correspond to negative values, i.e. overdamped conditions in which the Rabi oscillations and low-frequency emission are quenched. Increasing the incident field amplitude allows for achieving positive values and tuning the Rabi frequency. Figure 7(c) shows the squared Rabi frequency for three selected values of $E_0$ [dashed lines in Fig. 7(b)], from the highly damped regime where emission is suppressed by the presence of the nanorods (blue dashed line, $E_0=5.5$ kV/cm), through the region of two peaks with a local minimum at resonance (black dashed line $E_0=7.5$ kV/cm), and eventually in the most interesting regime of a single peak at resonance (red dashed line, $E_0=12.0$ kV/cm). The threshold field value can be read out from Fig. 7(d), which concludes the above observations by showing Rabi frequency for the molecular resonance $\omega _0 = 2\pi \cdot 348$ THz as a function of the driving field [solid line in Fig. 7(b)]. The red arrow points at the evaluated threshold $E_t \approx 7.5$ kV/cm for the electric field above which low-frequency radiation can be emitted by the system. For moderate field values, the achieved Rabi frequency depends nonlinearly on the illumination amplitude. Finally, for relatively strong drives, as the damping rate becomes negligible, the linear scaling of Rabi frequency on the illumination amplitude is restored.

Fig. 7. (a) Purcell enhanced emission rate as a function of the molecular frequency $\omega _0$. (b) Squared Rabi frequency [Eq. (9)] as a function of the incident electric field amplitude $E_0$ and frequency $\omega$. For this calculation, both the illumination frequency and the molecular transition frequency are varied but always assumed on resonance $\omega =\omega _0$. Dashed lines for particular values of $E_0$ correspond to the curves plotted in panel (c) and represent several regimes for the Rabi frequency – overdamped (blue), nonlinear (black), and linear (red). (d) Rabi frequency of the system in the vicinity of the nanorods as a function of the incident field $E_0$ for $\omega = 2\pi \cdot 348$ THz corresponding to BaF molecules and marked with the vertical black solid line in (b). The marked threshold value indicates a minimal required field to observe low-frequency radiation for a given geometry.

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The evaluated field enhancement factor of about 500 and characteristics from Fig. 7(d) allow us to estimate the range of Rabi frequencies that can be reached with a moderately strong electric field amplitude of $E_0=8.27$ kV/cm ($1.61\cdot 10^{-6}$ a.u.) by the use of the Eq. (5):

(17)$$\begin{aligned}\Omega_\text{polar} = 2\pi\cdot 5\ \text{THz} = 2\pi \cdot 7.6\cdot10^{{-}4} \text{ a.u.}.\end{aligned}$$

This suggests that emission at Rabi frequencies in the range of $2\pi \cdot 1$ up to $2\pi \cdot 10$ THz can be achieved in this manner. Without the field enhancement, the Rabi frequency for the given above field amplitude would be $\Omega _\text {polar} = 2\pi \cdot 10$ GHz. In this way, the plasmonic field enhancement sustained by the silver nanorods leads to a spectral shift of the emission frequency. This unique effect can be achieved by the integration of nanoantennas with polar molecules.

The threshold amplitude of $E_t\approx 7.5$ kV/cm corresponds to the beam intensity of 750 W/mm$^2$. Estimating the size of the nanorod to be 20 nm $\times$ 100 nm, the fluence applied becomes roughly 1.5 $\mathrm {\mu }$J/s, which is well below the threshold of 20 mJ/s, related to material melting, e.g., in gold nanoantennas [43,44].

With this information at hand, we tune the graphene disk dimer to enhance the THz emission power.

3.3.2 Optical response of graphene disk dimer

We perform the simulations using the dipolar illumination scheme, with the source positioned in the gap between the graphene micro-disks, as indicated by the red dot in Fig. 2 and oriented along the $x$ axis. The radiated power spectra for varied graphene disk radii are illustrated in Fig. 8 and show an impressive enhancement in comparison with the free space values. We find the resonance frequency to approximately scale inversely with the graphene disk radii $\omega _\mathrm {res} \propto \sqrt {\frac {1}{r}}$ (blue dashed line in Fig. 8). This result has been derived analytically in the quasistatic approximation for a single disk in Ref. [45]. For larger disks, we find improved radiated and suppressed absorbed powers [Fig. 9(a,b)]. This implies that the efficiency of the graphene micro-disks is enhanced with larger radii.

Fig. 8. Radiated power ratio with respect to the free space values (a) for selected radii of graphene micro-disks, (b) for more values of radii in a logarithmic scale.The blue dashed line follows the relation $\omega _\mathrm {max} = \frac {2\pi a}{\sqrt {r}}$, where $\omega _\mathrm {max}/2\pi$ is the resonance frequency and $a=6.159 \mathrm {THz}\,\sqrt {\mathrm {nm}}$.

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Fig. 9. Radiated (a) and absorbed (b) power spectra for varying radii of graphene micro-disks.

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According to Eq. (5), the Rabi frequency of the emitted radiation can be modulated with the intensity of the laser field illuminating the molecule. Figure 6(b) reveals the resulting Rabi frequency’s sensitivity to the position of the molecule with respect to the antenna. One of the most significant advantages of graphene is its tunability, which enables adjustment of the resonance frequency of the antenna and its fine-tuning to match the achieved Rabi frequency by changing the chemical potential. As we vary the chemical potential in the range of 0.2 eV to 1.2 eV, we observe a blueshift of the resonance from 3 THz to 11 THz and a decrease in the radiated power ratio with respect to the free space (Fig. 10). In the quasistatic approximation, the predicted scaling of the plasmonic resonance is $\omega \propto \sqrt {\mu _{c}}$ [45] and applies here to a very good approximation, as demonstrated by the blue dashed line in Fig. 10.

Fig. 10. Radiated power spectra (a) for selected values of the chemical potential of graphene micro-disks, (b) for more values of the potential. The blue dashed line follows the relation $\omega _\mathrm {max} = 2\pi a\,\sqrt {\mathrm {\mu _{c}}}$, where $\omega _\mathrm {max}/2\pi$ is the resonance frequency and $a=7.498\ \frac {\mathrm {THz}}{\sqrt {\mathrm {eV}}}$.

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We have thus confirmed that the graphene-based nanoantenna component can be designed to achieve a plasmonic resonance in the desired frequency range of several THz, promising emission power enhancement by up to 8 orders of magnitude. The spectral position of the resonance can be fine-tuned with the chemical potential to adjust to the Rabi frequency variations.

3.4 Discussion

As shown in the Concept section, upon resonant near-infrared illumination, the polar molecule of BaF becomes a source of radiation at the Rabi frequency. The nanoantenna characterized above was designed to influence the emission properties of the molecule. Its impact is two-fold:

1. The nanoantenna shifts the emission frequency to the THz range. This occurs as the Rabi frequency increases proportionally to the enhancement of the electric field at the driving, near-infrared frequency. In our example, the field in the middle of the gap is enhanced by more than two orders of magnitude, shifting the Rabi frequency from the GHz to the THz range. The field enhancement is achieved with the silver components of the antenna.
2. The nanoantenna increases the power of the signal. This is achieved through an enhanced emission power at the THz frequency using the graphene components of the antenna. Note that the emission frequency depends on the applied illumination amplitude of the near-infrared field. The tunability of the graphene disks with the chemical potential enables matching the THz resonance to that amplitude. On the other hand, the tunability can also be exploited to counteract the effect of spatial field fluctuations in the close vicinity of the nanoantenna.

4. Conclusion

We have proposed a hybrid noble-metal – graphene nanoantenna with a bimodal optical response to tune radiation from polar quantum systems into the THz range and improve their emission power. The silver components sustain resonance in the near-infrared to the visible regime, which is used to improve the light-matter coupling strength quantified in terms of Rabi frequency. The enhanced Rabi frequency, at which the polar system emits, is thus blue-shifted to the THz regime. The graphene components are exploited to support the emission from the polar system in the THz range. Graphene tunability plays a crucial role for the proposed antenna since it allows us to adjust its operational frequencies to counteract imperfections in molecule positioning, to adjust to temporal changes in laser intensity, or to modulate the signal on purpose.

We have thoroughly investigated the antenna’s optical response as a function of its geometry parameters to eventually engineer a nanostructure adjusted to the specific target molecule of BaF. Similar scenarios could be realized with alternative polar systems, including molecules, quantum dots, or solid-state defects.

Funding

National Science Centre, Poland (2018/31/D/ST3/01487).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Hybrid graphene - silver nanoantenna to control THz emission from polar quantum systems

Abstract

1. Introduction

2. Concept

2.1 Radiation emission from polar quantum systems

2.2 Emission enhancement with plasmonic nanoantennas

2.3 Concept summary

3. Results

3.1 Molecule

3.2 Nanoantenna geometry

3.3 Nanoantenna performance

3.3.1 Optical response of silver components

3.3.2 Optical response of graphene disk dimer

3.4 Discussion

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (10)

Equations (17)

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