Optical force induced by strong exciton-plasmon coupling

Wanjun Li; Wanjun Li; Yang Yu; Yang Yu; Yang Yu; Haochen Yan; Qingguang Zeng; Ting-Hui Xiao; Ting-Hui Xiao

doi:10.1364/OE.443686

1. Introduction

Plasmonic nanoantennas provide a powerful tool to manipulate light-matter interaction due to their capability of realizing subwavelength localization of incident light [1,2]. By virtue of their high electromagnetic field enhancement and small electromagnetic mode volumes [3], plasmonic nanoantennas are ideal for achieving coupling with various quantum emitters, such as atoms, molecules, nanomaterials, and quantum dots [4–9]. This enables a wide range of intriguing applications. In the weak-coupling regime, the light emission properties of the quantum emitters are modified due to the Purcell effect, which has been widely used for surface-enhanced spectroscopy [10–14]. The emitters are usually treated as perturbative probes that hardly influence resonances of plasmonic nanoantennas. In the strong-coupling regime, the emitters are no longer treated as perturbative probes, but are used to interplay with plasmonic nanoantennas to produce new hybrid resonances or states, which has been widely employed to investigate quantum electrodynamic (QED) effects for quantum information applications [15,16].

To construct a coupling system for the above applications, positioning a single quantum emitter in the vicinity of a plasmonic antenna is a fundamental step in which optical forces induced by the high electromagnetic field enhancement of plasmonic nanoantennas start to play a crucial role [17]. Due to tiny sizes of quantum emitters that are much smaller than the wavelength of incident light, the quantum emitters are conventionally treated as perturbative probes whose minimal Rayleigh scattering is negligible [18]. The optical forces experienced by the quantum emitter positioned in the vicinity of a plasmonic antenna are approximately calculated by perturbative gradient forces arising from the inhomogeneous intensity of incident light localized by the plasmonic nanoantenna [19,20]. Recently, the optical forces that include the influence of the quantum emitters themselves on the localized electromagnetic field have also been investigated, such as the self-induced back-action optical trapping that utilizes the optical resonance of a quantum emitter approximated as a dielectric nanoparticle [21]. However, optical forces are rarely explored by taking the influence of electronic properties of the quantum emitter into consideration, especially in the strong-coupling regime.

In this article, we study the optical forces induced by the strong exciton-plasmon coupling between a single quantum dot and a plasmonic nanoantenna, in which vacuum Rabi splitting and anti-crossing phenomena are evident. Both attractive and repulsive optical forces are systematically investigated by tuning the interplay between the excitons of the quantum dot and the surface plasmons of the nanoantenna. Our work elucidates the non-perturbative optical forces induced by strong exciton-plasmon coupling, which opens up a new avenue for optical manipulation and optical assembly of strong-coupling systems for quantum information applications.

2. Results and Discussion

2.1 Strong exciton-plasmon coupling

To study the optical forces induced by the strong exciton-plasmon coupling, we firstly design a strong coupling system composed of a plasmonic nanoantenna and a quantum dot. As shown in Fig. 1(a), the plasmonic nanoantenna is a bowtie structure with a gap size d of 10 nm, which is composed of two mirror-symmetric cones with an inclination angle $\theta $ of 25 degrees, a tip radius r of 40 nm, and a length l of 150 nm. The quantum dot is modeled as a sphere with a radius R of 4 nm. The electronic property of the quantum dot is included in its permittivity model, which is expressed as [22]:

(1)$${\varepsilon _{\textrm{QD}}}(\omega )= {\varepsilon _\textrm{b}} + f\frac{{N[{{f_\textrm{C}}({{E_\textrm{e}}} )- {f_\textrm{v}}({{E_\textrm{h}}} )} ]{e^2}/{m_0}{\varepsilon _0}}}{{({{\omega^2} + i2\omega {\gamma_\textrm{X}} - \omega_0^2} )}},$$

where ${\varepsilon _\textrm{b}}$ is the background permittivity, N is the dipole density of electron-hole pairs, ${f_\textrm{C}}$ (${f_\textrm{v}}$) is the electron (hole) distribution function with electron (hole) confinement energy ${E_\textrm{e}}$ (${E_\textrm{h}}$), ${\gamma _\textrm{X}}$ is the decay rate of oscillation, e is the elementary charge, ${m_0}$ is the electron mass, ${\varepsilon _0}$ is the permittivity of vacuum, $\omega $ is the frequency of excitation light, ${\omega _0}$ is the transition eigenfrequency, and $f\; $ is the oscillator strength. The oscillator strength f is expressed as [22]:

(2)$$f = \frac{2}{{{m_0}\hbar {\omega _0}}}{M_\textrm{b}}^2{\left|{\smallint {f_\textrm{e}}({\boldsymbol r} ){f_\textrm{h}}({\boldsymbol r} ){d^3}{\boldsymbol r}} \right|^2},$$

where $\hbar $ is the reduced Planck constant, ${\boldsymbol r}$ is the radial coordinate vector, ${M_\textrm{b}}$ is the dipole matrix element of the quantum dot and ${f_\textrm{e}}({\boldsymbol r} )$ (${f_\textrm{h}}({\boldsymbol r} )$) is the electron (hole) wave function while the integral means the electron-hole overlap. For simplicity, the term $fN[{{f_\textrm{C}}({{E_\textrm{e}}} )- {f_\textrm{v}}({{E_\textrm{h}}} )} ]{e^2}/{m_0}{\varepsilon _0}$ in Eq. (1) is rewritten as ${\varepsilon _{\textrm{ex}}}\omega _0^2$ in which ${\varepsilon _{\textrm{ex}}}$ is the effective permittivity induced by the above excitonic property of the quantum dot. The Eq. (1) is thus simplified to

(3)$${\varepsilon _{\textrm{QD}}}(\omega )= {\varepsilon _\textrm{b}} + \frac{{{\varepsilon _{\textrm{ex}}}\omega _0^2}}{{({{\omega^2} + i2\omega {\gamma_\textrm{X}} - \omega_0^2} )}}\; ,$$

Fig. 1. Strong exciton-plasmon coupling between a plasmonic nanoantenna and a quantum dot. (a) Schematic diagram of the strong-coupling system composed of a plasmonic nanoantenna and a quantum dot. (b) Scattering spectrum of the plasmonic nanoantenna. The inset shows the electric field magnitude distribution at the wavelength of LSPR. (c) Rabi splitting in the scattering spectrum of the strong-coupling system when the frequency of excitonic resonance in the quantum dot coincides with that of LSPR. The two insets depict the electric field magnitude distributions at the wavelengths of two split peaks that correspond to two hybrid resonances or states. (d) Scattering spectra of the strong-coupling system with a varying detuning of excitonic resonance in the quantum dot. (e) Peak wavelengths of two polariton branches and LSPR. Blue circles, red circles and triangles denote extracted peak wavelengths for the upper and lower polariton branches, and the estimated uncoupled QD exciton emission, respectively. The dotted line denotes the bare LSPR. The calculated peak positions are plotted as the continuous solid curves.

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The strong coupling system is excited by linearly polarized light with its polarization along x axis. To ensure the excitation of localized surface plasmon resonance (LSPR), the scattering spectrum of the plasmonic nanoantenna is calculated and shown in Fig. 1(b). A characteristic scattering peak with a full width at half maximum Δλ of 225 nm is observed at the wavelength ${\lambda _{\textrm{res}}}$ of 790.19 nm, which corresponds to a quality factor Q of about 3.5. With this quality factor Q, the decay rate ${\gamma _\textrm{C}}$ of LSPR is calculated by its definition ${\gamma _\textrm{C}} = {\omega _{\textrm{res}}}/Q = $ 9.6 ${\times} $ 10¹⁴ rad/s, where ${\omega _{\textrm{res}}}$ is the resonant frequency of LSPR in the plasmonic nanoantenna. The inset shows the electric field magnitude distribution at the wavelength of LSPR, indicating a 35-fold electric-field magnitude enhancement within the gap. In comparison, when the plasmonic nanoantenna is strongly coupled with a quantum dot with the background permittivity ${\varepsilon _\textrm{b}}$ of 6.1 [21], the effective permittivity ${\varepsilon _{\textrm{ex}}}$ of 0.6 [23], the decay rate ${\gamma _\textrm{X}}$ of 10¹³ rad/s, and the transition eigenfrequency ${\omega _0}$ of 2.4 ${\times} $ 10¹⁵ rad/s, the Rabi splitting in its scattering spectrum is evident as shown in Fig. 1(c). Two hybrid resonances at the wavelengths of 782.72 nm and 818.42 nm are observed. In the strong-coupling regime, the coupling strength can be evaluated from the Rabi splitting by using the following equation [24–26]:

(4)$${\omega _{1,2}} = \frac{{{\omega _\textrm{X}} + {\omega _\textrm{C}}}}{2} - i\frac{{({{\gamma_\textrm{X}} + {\gamma_\textrm{C}}} )}}{4} \pm {\left[ {{g^2} + \frac{1}{4}{{\left( {{\omega_\textrm{C}} - {\omega_\textrm{X}} - i\frac{{{\gamma_\textrm{X}} - {\gamma_\textrm{C}}}}{2}} \right)}^2}} \right]^{\frac{1}{2}}},$$

where ${\omega _{1,2}}$ is the eigenfrequencies of the two splitted hybrid resonances, ${\omega _\textrm{X}}$ is the frequency of excitonic resonance (${\omega _0}$), ${\omega _\textrm{C}}$ is the resonant frequency (${\omega _{\textrm{res}}}$) of the plasmonic nanoantenna, g is the coupling strength. To achieve strong coupling, it is essential to compare the coupling strength and the decay rates. For the model shown in Fig. 1(c), the value of $g/{\gamma _\textrm{C}}$ is 0.261 which satisfies the strong-coupling criterion of $g/{\gamma _\textrm{C}} > $ 0.25 [24]. The two insets depict the electric field magnitude distributions at the wavelengths of two split peaks, indicating two hybrid modes. To further study the exciton-plasmon interaction, we calculated the scattering spectra of the strong-coupling system by tuning the frequency of excitonic resonance from $\textrm{1}\textrm{.91} \times \textrm{1}{\textrm{0}^{\textrm{15}}}$ rad/s to $\textrm{2}\textrm{.85} \times \textrm{1}{\textrm{0}^{\textrm{15}}}$ rad/s as shown in Fig. 1(d). Two side peaks are evident in the scattering spectrum, which originates from the coherent coupling between the excitonic resonance and the LSPR. The wavelengths of the two side peaks are extracted and shown with the correspondent detuning between the excitonic resonance and the LSPR in Fig. 1(e). The anti-crossing of the two polariton is clearly identified, indicating the system is in the strong-coupling regime.

2.2 Optical forces induced by the strong coupling

In the above strong-coupling system, we study the non-perturbative optical forces on the quantum dot by using a finite-difference time-domain (FDTD) method realized by a homemade code. The minimum mesh size is 0.8 nm to ensure the high accuracy of simulations. The scattering boundary condition combined with perfectly matched layers is used to calculate the scattering spectra of the coupled system while avoid the influence of reflection. The average running time for each simulation is ∼20 hours. The optical force $\vec{F}$ on the quantum dot is calculated by integrating the Maxwell’s stress tensor $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over T} $ over a surface $\vec{S}$ that encloses the quantum dot [27] as:

(5)$$\vec{F} = \mathop{{\int\!\!\!\!\!\int}\mkern-21mu \bigcirc} {\mathrm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}}\ \over T}\ }} \cdot d\vec{S},$$

$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over T} $ is defined as:

(6)$$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over T} = \varepsilon \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} + \mu \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over H} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over H} - \frac{1}{2}({\varepsilon {E^2} + \mu {H^2}} )\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over I} ,$$

where $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over I} $ is the intensity tensor, $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} $ is the electric field, $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over H} $ is the magnetic field, $\varepsilon $, $\mu $ are the permittivity and permeability of the surrounding medium, respectively. To calculate the optical forces on the quantum dot, it is essential to shift the quantum dot away from its equilibrium position in the plasmonic nanoantenna. With the shift of the quantum dot along z axis, the scattering spectra of the coupled system are calculated and shown in Fig. 2(a). With the z-axis shift from 10 nm to 30 nm, the coupling strength decreases from 0.261 to 0.252. The correspondent optical forces on the quantum dot with the optical power of incident light of 1 mW are shown in Fig. 2(b) See (Supplement 1 for more details). Interestingly, a strong optical attractive force (negative) is generated at the wavelength of ∼802 nm while significantly decreases with the z-axis shift from 10 nm to 30 nm. In comparison, a repulsive force (positive) is generated at the wavelength of ∼798 nm and moderately decreases with the z-axis shift. To explore their origins, we compare the optical forces on the quantum dot with and without the strong exciton-plasmon coupling in Fig. 2(c). For the case without the strong exciton-plasmon coupling, the effective permittivity ${\varepsilon _{\textrm{ex}}}$ induced by the excitonic property of the quantum dot is set to be zero, with which the quantum dot is identical to a dielectric particle. It is evident from the comparison that both the strong attractive force and the repulsive force originate from the strong exciton-plasmon coupling. The plasmons enable the high enhancement and localization of incident light to generate the high light-intensity gradient while the excitons provide the essential permittivity environment to generate both attractive and repulsive forces [28]. The strong coupling ensures the high-efficiency enhancement of the optical forces by the high-efficiency and low-loss optical energy transfer between the excitons of the quantum dot and the plasmons of the nanoantenna. This is also verified by the electric field magnitude distributions in the insets of Fig. 2(c), in which the localized electric-field magnitude around the quantum dot with the strong coupling is much higher than that without the strong coupling. To further study the transition process from the attractive force without the strong coupling to the repulsive force with the strong coupling, we calculate the optical forces on the quantum dot with various coupling strengths tailored by the core-shell ratio of the quantum dot as shown in Fig. 2(d). The permittivity of the core/shell is identical to that of the quantum dot with/without the strong exciton-plasmon coupling in Fig. 2(c). With the decrease of core-shell ratio, the optical force is also decreased. When the core-shell ratio decreases to 1/3, the optical force at the wavelength of 798 nm is transited from the repulsive force to the attractive force as the attractive contribution of light-intensity gradient induced by the plasmons is larger than the repulsive contribution of the permittivity environment induced by the excitons. The insets in Fig. 2(d) show their electric field magnitude distributions. Compared with that of the attractive force in the insets of Fig. 2(c), the electric field magnitude distribution of the repulsive force is more localized within the quantum dot, reflecting the dominant contribution of the permittivity environment induced by the excitons. With the decrease of core-shell ratio, the electric-field magnitude within the quantum dot is also decreased, which agrees with the variation of the repulsive force.

Fig. 2. Optical forces induced by strong exciton-plasmon coupling. (a) Scattering spectra of the strong coupling system with various coupling strengths tuned by the position of the quantum dot along z axis. (b) Optical forces on the quantum dot with various coupling strengths tuned by the position of the quantum dot along z axis. (c) Optical forces on the quantum dot with and without the strong exciton-plasmon coupling. The insets show the electric field magnitude distributions at the wavelengths that correspond to the largest attractive optical forces with and without the strong exciton-plasmon coupling. (d) Optical forces on the quantum dot with various coupling strengths tailored by the core-shell ratio of the quantum dot. The insets show the electric field magnitude distributions at the wavelengths that correspond to the largest repulsive optical forces with various coupling strengths tailored by the core-shell ratio of the quantum dot.

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2.3 Optical forces tuned by engineering plasmons and excitons

To explore the controllability and tunability of the optical forces induced by the strong exciton-plasmon coupling, we try to manipulate them by engineering the surface plasmons of the nanoantenna. First, we engineer the surface plasmons by varying the gap size d of the nanoantenna. The scattering spectra of the nanoantenna with various gap sizes are calculated and shown in Fig. 3(a). With the increase of the gap size d, the scattering peak of LSPR has a blueshift while its intensity decreases, which corresponds to the decreased electric-field magnitude in the gap as shown in the insets of Fig. 3(a). This results in the decreased coupling strength g between the excitons and the plasmons with zero detuning, which is reflected by the scattering spectra of the correspondent strong-coupling systems shown in Fig. 3(b). Due to the decreased coupling strength g, the optical force on the quantum dot is also decreased as shown in Fig. 3(c). Second, we further engineer the surface plasmons by varying the tip radius r of the nanoantenna. The scattering spectra of the nanoantenna with various tip radii are shown in Fig. 3(d). With the decrease of the tip radius r, the scattering peak of LSPR has a redshift while its peak width decreases, which corresponds to the increased electric-field magnitude in the gap as shown in the insets of Fig. 3(d). This results in the increased ratio of the coupling strength g to the decay rate of the plasmons ${\gamma _\textrm{C}}$ with zero detuning, which is reflected by the scattering spectra of the correspondent strong-coupling systems shown in Fig. 3(e). Due to the increased ratio $g/{\gamma _\textrm{C}}$, the optical force on the quantum dot is also increased as shown in Fig. 3(f).

Fig. 3. Optical forces tuned by engineering the surface plasmons of the nanoantenna (a) Scattering spectra of the plasmonic nanoantenna with various gap sizes. The insets show the electric field magnitude distributions of LSPR with various gap sizes. (b) Scattering spectra of the strong-coupling system with various gap sizes. (c) Optical forces on the quantum dot with various gap sizes. (d) Scattering spectra of the plasmonic nanoantenna with various tip radii. The insets show the electric field magnitude distributions of LSPR with various tip radii. (e) Scattering spectra of the strong-coupling system with various tip radii. (f) Optical forces on the quantum dot with various tip radii.

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Moreover, we try to manipulate the optical forces induced by the strong exciton-plasmon coupling by engineering the excitons of the quantum dot. As shown in Eq. (3), the permittivity of the quantum dot is mainly determined by four parameters, namely, the background permittivity ${\varepsilon _\textrm{b}}$, the effective permittivity ${\varepsilon _{\textrm{ex}}}$, the decay rate ${\gamma _\textrm{X}}$, and the transition eigenfrequency ${\omega _0}$. All of them are controllable and tunable by the excitonic property. First, the background permittivity ${\varepsilon _\textrm{b}}$ is determined by optical transitions far away from the excitonic transition eigenfrequency ${\omega _0}$. By engineering the optical transitions far away from the excitonic transition of the quantum dot, the background permittivity ${\varepsilon _\textrm{b}}$ is tunable. The permittivity of the quantum dot with various ${\varepsilon _\textrm{b}}$ values are shown in Fig. 4(a), with which the correspondent optical forces are calculated and shown in Fig. 4(b). With the decrease of the background permittivity ${\varepsilon _\textrm{b}}$, the peaks of the optical forces have a slight blueshift while the magnitude of attractive force increases. Second, the effective permittivity ${\varepsilon _{\textrm{ex}}}$ is determined by the electron (hole) energy distribution function and the electron-hole overlap as we discussed above. By engineering these excitonic parameters, the effective permittivity ${\varepsilon _{\textrm{ex}}}$ is tunable. The permittivity of the quantum dot with various ${\varepsilon _{\textrm{ex}}}$ values are shown in Fig. 4(c), with which the correspondent optical forces are calculated and shown in Fig. 4(d). With the decrease of the effective permittivity ${\varepsilon _{\textrm{ex}}}$, the peaks of the optical forces have a slight redshift while the magnitudes of the optical forces decrease. Third, the decay rate ${\gamma _\textrm{X}}$ is determined by the loss of excitonic resonance. By engineering the loss of excitonic resonance of the quantum dot, the decay rate ${\gamma _\textrm{X}}$ is tunable. The permittivity of the quantum dot with various ${\gamma _\textrm{X}}$ values are shown in Fig. 4(e), with which the correspondent optical forces are calculated and shown in Fig. 4(f). With the increase of the decay rate ${\gamma _\textrm{X}}$, the peaks of the optical forces have no shifts while the magnitudes of optical forces decrease. This is because the increased decay rate ${\gamma _\textrm{X}}$ reduces the electric-field magnitude in the gap, which results in the decrease of the coupling strength. At last, by engineering the transition eigenfrequency ${\omega _0}$, the peak wavelengths of optical forces are also tunable.

Fig. 4. Optical forces tuned by engineering the excitons of the quantum dot (a) Permittivity of the quantum dot with various background permittivity values. (b) Optical forces on the quantum dot with various background permittivity values. (c) Permittivity of the quantum dot with various effective permittivity values. (d) Optical forces on the quantum dot with various oscillator strengths. (e) Permittivity of the quantum dot with various decay rates. (f) Optical forces on the quantum dot with various decay rates.

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3. Conclusion

In conclusion, we study the optical forces induced by the strong exciton-plasmon coupling between a single quantum dot and a plasmonic nanoantenna, in which vacuum Rabi splitting and anti-crossing phenomena are evident. Interestingly, both large attractive and repulsive optical forces are capable of being generated with the assist of strong exciton-plasmon coupling. In addition, the generated optical forces are fully controllable and tunable by engineering the excitons of the quantum dot and the surface plasmons of the nanoantenna. Our work elucidates the non-perturbative optical forces induced by strong exciton-plasmon coupling, which is useful and extendable to some other coupling systems [29,30]. Our work opens up a new avenue for optical manipulation and optical assembly of strong-coupling systems for quantum information applications.

Funding

National Natural Science Foundation of China (11804254); Japan Society for the Promotion of Science (JP20K14785); Murata Science Foundation; The Science and Technology Projects of Jiangmen ((2017) 149, (2017) 307, (2018) 352); Innovative Leading Talents of Jiangmen ((2019) 7); Cooperative education platform of Guangdong Province ((2016) 31); Key Laboratory of Optoelectronic materials and Applications in Guangdong Higher Education (2017KSYS011).

Disclosures

The authors declare no conflicts of interest.

Data availability

The 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.

Supplemental document

See Supplement 1 for supporting content.

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Optical force induced by strong exciton-plasmon coupling

Abstract

1. Introduction

2. Results and Discussion

2.1 Strong exciton-plasmon coupling

2.2 Optical forces induced by the strong coupling

2.3 Optical forces tuned by engineering plasmons and excitons

3. Conclusion

Funding

Disclosures

Data availability

Supplemental document

References

Supplementary Material (1)

Data availability

Cited By

Figures (4)

Equations (6)

Optics Express