1. Introduction

Manipulation of the optical properties of semiconductor quantum dots (QDs) as efficient emitters has a broad range of applications in electro-optic devices such as high-efficiency single photon sources [1], quantum information processors [2], light emitting devices [3], photovoltaic cells [4] and biological applications [5]. In many of these applications the interaction between an excited QD and the surrounding environment can be controlled by using different reservoirs such as optical microresonators [6] or photonic crystal nano cavities [7, 8]. The size of these reservoirs has an inherent limitation, as their size have to be at least half the wavelength of the incident field. On the other hand, localized surface plasmon resonances (LSPRs) of metallic nanoparticles (MNPs) can amplify and localize the electromagnetic fields in nanometer length scales, much smaller than the diffraction limit of light. As a result, MNPs have merged as a viable alternative to control electronic and optical properties of QDs at the nanoscale [9]. This allows combining the sub-wavelength feature offered by plasmons with the efficient emission characteristic of QDs for the design of novel optical systems such as plasmonic lasers [10] and fluorescence-based sensors [11].

In the above-mentioned studies, QDs were influenced by the intrinsic resonances of the cavities or LSPRs. The prime feature of such resonances is that they are not influenced by the QDs, except perhaps for second order changes caused by perturbation of the dielectric constant. Recent investigations have shown, however, that when a QD is in the vicinity of a MNP and driven by a coherent light source (a laser field), the combined system supports molecular-like collective resonances, different from LSPRs. Such resonances, called Plasmonic Meta Resonance (PMR) [12], are associated with the coherent exciton-plasmon coupling in the QD-MNP system formed via the quantum coherence generated in the QD by the laser field. When a QD-MNP system is close to its PMR, the near field of the MNP (coherent-plasmonic field) can become significantly stronger than its LSPR, and the optics and dynamics of the QDs become strongly sensitive to the structural parameters of the system, the intensity of the incident laser field, and the dielectric constant of the environment [13, 14].

The main objective of this paper is to study how the nature of the optical responses of QDs, their excitonic states, and energy transfer pathways can be distinctively determined by the dielectric constant of the environment in the presence of quantum coherence. For this, instead of using optical fields of cavities or intrinsic plasmonic fields of MNPs, we study QDs in coherent-plasmonic fields associated with PMR. Our system consists of a QD conjugated to a gold nanorod (AuNR) in the form of to-the-side configuration and is driven by a laser field (See Fig. 1(a)).

 

Fig. 1 (a) Schematic representation of the hybrid system (QD–AuNR) unit. The arrow on top shows the Förster resonance energy transfer (FRET) rate between QD and AuNR. The dipole-dipole coupling is due to an applied external field which induces a polarization on both the AuNR and the QD. The external (E) field is perpendicular to the QD-AuNR axis. (b) Schematic illustration of the two-level QD (left) upper excitonic ($|2〉$) and ground state ($|1〉$). The shaded area shows the energy band of the AuNR (right) where the dark strip shows the broden plasmon resonance energy peak. The transition energy of QD is matched with the resonance energy of Au nanorod.

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Our results suggest that with minuscule changes in the refractive index of the environment, the optical and electronic (excitonic) transition of the QDs undergo unique and more dramatic changes, compared to the cases of the optical cavities or LSPRs. The results also show that such changes lead to unprecedented spectral modifications in the Förster resonance energy transfer (FRET) from the quantum dots to the nanorods. In particular, we demonstrate that when a QD-AuNR system is close to its PMR, we can adjust the refractive index of the environment to (i) control the frequency range where the energy transfer from the QD to the nanorod is inhibited, (ii) manipulate the exciton transition energy shift, causing blue or red shift, and (iii) disengage the QD from the nanorod and laser field. Our results suggest that in the presence of quantum coherence, the way energy is transferred to the metallic nanorods determines the spectral form of the exciton frequency shift. These results open up new possibilities for development of chemical and biological sensors and nanodevices based on the application of quantum coherence in nanoparticle systems.

2. Model and formalism

In this paper, we assume that the QD-AuNR hybrid is embedded in a background dielectric constant (${\epsilon }_b$) and is driven by a linearly polarized external electric field with an amplitude E₀ and frequency$\omega$($E(t)=\frac{1}{2}E{}_0(t)e^{-i\omega t}$). The laser light polarization is parallel to the AuNR longitudindal axis and perpendicular to the hybrid system axis, as shown in Fig. 1(a). We treat the AuNR classically and represent it as an ellipsoid (prolate spheroid) with a semi-major axis a and semi-minor axis b, where the corresponding aspect ratio is q = a/b [15]. The response of the AuNR to the applied electric field is described by its frequency dependent scalar polarizability. The polarizability of the AuNR in the dipole approximation can be written as a frequency-dependent complex coefficient $\beta \left(\omega \right)=\left[{\epsilon }_m(\omega )-{\epsilon }_b\right]/\left[3{\epsilon }_b+3\kappa \left({\epsilon }_m(\omega )-{\epsilon }_b\right)\right]$ where$\kappa$ is called the depolarization factor of the AuNR, and ${\epsilon }_m(\omega )$ is the dielectric constants of the AuNR. In our calculations, the amplitude of the incident electric field lies along the semi-major axis (a) and depolarization factor is given as

To explicit the effect of exciton-plasmon interaction on the energy-transfer efficiency between the nanoparticles (QD and AuNR), as well as the exciton resonance energy shift, Eq. (4) can be written as:

3. Results and discussion

In our calculations, the aspect ratio of AuNR is taken to be q = 4 where its minor radius is b = 5 nm. For this geometry a maximum field enhancement created near the tips of the rod in the near infrared (NIR) region at $\hslash {\omega }_0$ = 1.6 eV ($\lambda$ = 775 nm) energy. The dielectric constant of AuNR is taken to be ${\epsilon }_m$ = −22.3 + 1.39i which is the bulk dielectric constant for gold as found experimentally at the given energy [20]. For AuNR of such a size, recent calculations have shown that the influence of the particle size on the dielectric constant is negligible. For example, in ref [21], it is shown that in an AuNR the scattered spectrum and the incident laser energy have a peak at the same energy for both the experimental bulk dielectric constant and the Drude model based theoretical dielectric constant including all the corrections regarding the size and the shape of the AuNR. Also in ref [22]. it is shown that the difference between the value of the dielectric constant of the bulk and the nanosized metallic particles becomes only significant for particles smaller than 10 nm. The reference dielectric constant of the background material is ${\epsilon }_b$ = 1.80 (like water). In Fig. 2(a) we show the impact of q on the variation of the maximum field enhancement at the tips of AuNR versus incident laser energy. As shown in this figure by increasing q the maximum field enhancement increases and the peak energy shifts towards NIR region. Figure 2(b) shows the cross section of the electric field enhancement distribution in and around an AuNR for b = 5 nm and q = 4. In this figure, the right legend shows the value of normalized field enhancement. Note that for q less than 1 the maximum field enhancement decreases dramatically.

 

Fig. 2 (a) The maximum filed enhancement on the tips of a prolate spheroid versus the aspect ratio (q) and the incident laser energy, (b) Cross-section of the electric field enhancement distribution in and around a AuNR, for q = 4 when irradiated with a $\hslash {\omega }_0$ = 1.6eV ( = 775 nm) linearly polarized along the ellipsoid’s longitudinal axis. The legend on the right shows the values of the field enhancement.

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The QD can be considered as CdTe/CdS [23] which has an exciton resonance energy ($\hslash {\omega }_0$ = 1.6 eV) in the NIR region. The dielectric constant and dipole moment of the QD are taken as ${\epsilon }_s$ = 6 [24], and $\mu$ = 0.7 e nm [25], respectively. The typical values for the radiative decay rate and the dephasing width parameter are ${\Gamma }_{21}$ = 1.25 ns⁻¹ and ${\gamma }_{21}$ = 2.7 ns⁻¹, respectively [24]. Here, one can consider the effects of the additional nonradiative rate by Ohmic losses in the environment assuming that the QD has a high intrinsic quantum yield [26]. Also the effect of the geometrical properties of the gold nanoparticle on the radiative decay rate can be considered [13]. However, considering these additional terms in Eq. (3) and Eq. (4) has no significant impact on the results presented in this paper.

We set the laser intensity to be I = 64.3 W/cm² to put the system on the verge of the PMR for the desired center-to-center distance between QD and AuNR (R ≈21 nm). Please note that here the quantum dot is treated as a dipole. However, the dimension of the system we used in calculation is reasonable in practice. In the case of CdTe/CdS, to obtain the NIR wavelength emission, the size of the QD is around 10 nm. If this is the case, the radius of the QD and the minor radius of AuNR are 5 nm, the surface to surface distance between the particles would be around 11 nm at which it is reasonable to see the exciton-plasmon effects.

To obtain the optical properties and exciton transition energy shift of the QD as well as the energy transfer rate between the nanoparticles, the coherence terms (${\rho }_{21}$and${\rho }_{12}$) are obtained via solving Eq. (3) and Eq. (4) numerically in steady state.

To investigate optics and electronic (excitonic) states of the QD in the presence of the plasmonic field of the AuNR when it is normalized by quantum coherence, we start by reviewing PMR in the QD-AuNR hybrid system for to-the-side configuration. Figure 3 shows the results for the coherent–plasmonic field enhancement (CPFE) factor defined in Eq. (6) as a function of the background dielectric constant (${\epsilon }_b$). In this figure, for R = 21 nm, we calculate the CPFE for an interval of ${\epsilon }_b$ from 1.7 to 1.9. The solid, dash-dotted and the dotted curve show the CPFE for ${\Delta }_{12}$ = 0, 5 and 10 ns⁻¹, respectively. Initially, depending on the values of ${\Delta }_{12}$, as ${\epsilon }_b$ increases, CPFE is suppressed (dark state) until ${\epsilon }_b$ approaches a certain value. Note that the dark state here is the sign of PMR. After this value CPFE raises abruptly reaching a maximum value (bright state). The rationale behind this is to show how these features in the CPFE profile in the QD-AuNR hybrid system not only supports the existence of PMR but also shows the existence of switching of the system when the dielectric constant of the background material changes slightly. In Fig. 3 we also show this process depends on the value of ${\Delta }_{12}$. For ${\Delta }_{12}$ = 0, 5 and 10 ns⁻¹ the state transition from dark to bright happens for ${\epsilon }_b$ = 1.788 (solid line), 1.785 (dash-dotted line) and 1.765 (dotted line), respectively. In this figure, to show the impact of coherent exciton-plasmon coupling and the quantum coherence on CPFE in QD, we also consider the case where the center-to-center distance between QD and AuNR is large and therefore the effects of quantum coherence have vanished (bare QD). The dashed line in Fig. 3 shows that for this condition (R = 100 nm), CPFE is basically equal to unity, i.e., no significant field enhancement occurs.

 

Fig. 3 The coherent–plasmonic field enhancement (CPFE) versus the background dielectric constant (${\epsilon }_b$) with I = 64.3 W/cm² and R = 21 nm. The solid, dash-dotted and dotted curve shows ${\Delta }_{12}$ = 0, 5 and 10 ns⁻¹, respectively. The dashed line shows the case when the particles are far from each other (R = 100 nm).

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It is worth mentioning, that in the absence of quantum coherence (or very large R), as seen in Fig. 2(b), the center of the AuNR is the nod point of plasmon polaritons. Therefore, at this point electromagnetic field enhancement is very weak. Under this condition, field enhancement at the tips of the AuNR is rather significant. Due to coherent exciton-plasmon coupling in the QD-AuNR system, however, our results show that the near field of the AuNR (coherent-plasmonic field) can become significantly stronger or weaker than bare AuNR, depending on the molecular states of QD-AuNR system. This allows efficient FRET from the QD to the AuNR.

As mentioned, the real (${\eta }_{\mathrm{Re}}$) and imaginary (${\eta }_{\mathrm{Im}}$) part of the QD self-feedback term ($\eta$) in Eq. (5)(are related to the exciton energy shift of QD and the FRET rate between QD and AuNR, respectively. As seen in this equation in dipole-dipole limit the $\eta$ term depends on the geometrical and environmental properties of the hybrid systems. In the dipole-dipole limit, the ratio of b to R should be small since the multipole expansion is a power series in (b/R)². In this paper for b = 5 nm, R = 21 nm, the first-order correction is nearly 12% [27, 28]. To show the impact of R and ${\epsilon }_b$ on the exciton energy shift and the FRET rate (See Eq. (7)), we illustrate in Fig. 4 the exciton transition energy shift (${\eta }_{\mathrm{Re}}{\Delta }_{\rho }$) and FRET rate factor (${\eta }_{\mathrm{Im}}{\Delta }_{\rho }$) versus R for two values of ${\Delta }_{21}$ = 0 and ${\Delta }_{21}$ = 600, (i.e. when the laser is on and off resonance with the exciton transition of the QD) and for different values of ${\epsilon }_b$.

 

Fig. 4 (a) The exciton transition energy shift (${\eta }_{\mathrm{Re}}{\Delta }_{\rho }$) and (b) Förster enhanced broadening factor (${\eta }_{\mathrm{Im}}{\Delta }_{\rho }$) versus center-to-center parameter (R) for ${\Delta }_{21}$ = 0. The solid, dotted and the dashed curve shows ${\epsilon }_b$ = 1.8, 1.825 and 1.85, respectively. The insets show the same quantities for each case where ${\Delta }_{21}$ = 600 ns⁻¹.

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As seen in Fig. 4(a) for ${\Delta }_{21}$ = 0 by decreasing R, ${\eta }_{\mathrm{Re}}{\Delta }_{\rho }$ is zero until R approaches a certain value and then depends on the value of ${\epsilon }_b$, the sign and the value of ${\eta }_{\mathrm{Re}}{\Delta }_{\rho }$ changes. Here, for ${\epsilon }_b$ = 1.8, 1.825 and 1.85, ${\eta }_{\mathrm{Re}}{\Delta }_{\rho }$ is positive (solid curve), zero (dash-dotted curve) and negative (dotted curve), respectively. In this calculation we show how a slight variation of ${\epsilon }_b$ can cause a switching in the form of a red and blue shift in the QD exciton energy. This bistable trend also is seen in the FRET rate. As shown in Fig. 4(b) for ${\Delta }_{21}$ = 0 by decreasing R, ${\eta }_{\mathrm{Im}}{\Delta }_{\rho }$vanishes until R approaches a certain amount after that the energy exchange between the nanoparticles starts dramatically. Note that here the ${\epsilon }_b$ variations only change the critical values of R slightly and have no effect on the nature of the curves. In insets of Fig. 4(a) and Fig. 4(b), we show the off resonance laser situation where ${\Delta }_{21}$ = 600. As seen in these insets the variation of ${\eta }_{\mathrm{Re}}{\Delta }_{\rho }$ and ${\eta }_{\mathrm{Im}}{\Delta }_{\rho }$ is monotonic and there is no switching in their profile.

To see further how ${\eta }_{\mathrm{Re}}{\Delta }_{\rho }$and ${\eta }_{\mathrm{Im}}{\Delta }_{\rho }$ are related to the laser field detuning parameter and controlled by the environment, in Fig. 5(a) and Fig. 5(b), we show the variation of these quantities versus ${\Delta }_{21}$ for different values of${\epsilon }_b$. In the case of ${\eta }_{\mathrm{Re}}{\Delta }_{\rho }$ as shown in Fig. 5(a) for R = 21 nm there is no exciton energy shift at resonance (${\Delta }_{12}=0$) and the system is on dark state and we have PMR in the system. However, by sweeping the laser frequency around resonance, the exciton energy shifts and the shifting direction depends on the value of ${\epsilon }_b$. In Fig. 5(a) the curves solid to dotted show ${\eta }_{\mathrm{Re}}{\Delta }_{\rho }$ for ${\epsilon }_b$ = 1.80, 1.81, 1.815, 1.825, 1.84, 1.845 and 1.85, respectively. As seen in this figure the top three curves show the red shift for${\Delta }_{12}\ne 0$.

 

Fig. 5 (a) The exciton transition energy shift (${\eta }_{\mathrm{Re}}{\Delta }_{\rho }$) and (b) Förster enhanced broadening factor (${\eta }_{\mathrm{Im}}{\Delta }_{\rho }$) versus the laser detuning parameter (${\Delta }_{21}$).

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For dash-dotted curve where ${\epsilon }_b$ approaches 1.825, ${\eta }_{\mathrm{Re}}{\Delta }_{\rho }$ flattens to almost zero. This means that in this particular value of ${\epsilon }_b$≈1.825 there is no exciton energy shift and it has symmetry around ${\Delta }_{12}=0$. By further increasing ${\epsilon }_b$, the ${\eta }_{\mathrm{Re}}{\Delta }_{\rho }$ sign changes to negative and blue shift happens. For ${\eta }_{\mathrm{Im}}{\Delta }_{\rho }$ as seen in Fig. 5(b) at${\Delta }_{12}=0$, the FRET rate totally vanishes. However, by sweeping the laser frequency around the QD resonance energy this energy transfer rate increases dramatically and eventually flattens. An interesting result here is that by a slight change in ${\epsilon }_b$ the FRET rate can be narrowed or widened. These results show how the FRET rate between nanoparticles can be controlled by a slight modification of the system environment. In this figure the solid, dash-dotted and dotted curves show ${\epsilon }_b$ = 1.80, 1.825 and 1.85, respectively. Please note that the curves in Fig. 5(a) and Fig. 5(b) are in correlation. For example, for${\epsilon }_b$≈1.825 the exciton energy shift and FRET rate have symmetry around ${\Delta }_{12}=0$. Please note that the impact of such a small variation of the background dielectric constant on the QD exciton energy and AuNR plasmon resonance is negligible in our results. Also as shown in Fig. 1(b), compared to exciton energy of the QD, the energy spectrum of the AuNR at the LSPR is broad enough so that the exciton-plasmon coupling stays valid with such a small modification.

In the absence of AuNR, when the laser field is detuned around the QD transition, it can lead to blue or red AC Stark shift, depending on the detuning of the laser frequency from this transition. As the laser intensity increases the amount of the AC shift increases, while the spectra can be broadened to some extent (power broadening). The results shown in Fig. 5, however, were obtained with a given amount of the laser intensity, while we changed the refractive index of the environment. The results presented in Fig. 5(a) show unique shift caused by PMR when detuning is small. For larger detuning we see blue or red shift, depending on the value of the dielectric constant, while the frequency and intensity of the laser was kept unchanged. While AC shift played a limited role in our case, the transition shift caused by plasmonic effects is quite important. As a result, for a given ${\Delta }_{12}$ we can have red or blue shift or we can choose a dielectric constant such that any detuning of the laser field does not lead to any shift (dotted-dashed line in Fig. 5(a)). Another remarkable phenomenon seen in our calculations is the possibility to modify the nature of the QD’s optical properties by driving it with coherent laser in the vicinity of a AuNR. It is known that the real and imaginary part of the coherence terms (Re(${\rho }_{21}$) and Im(${\rho }_{21}$)) respectively, show the laser light dispersion (refractive index) and absorption by QD [17]. For a bare QD, these quantities take the usual form of the real and imaginary part of a complex Lorentzian function. However, in a hybrid system, when the QD is placed in the vicinity of AuNR, the nature of these quantities changes dramatically. In Fig. 6(a), the Re(${\rho }_{21}$) spectrum has one minimum and two maxima around ${\Delta }_{12}=0$. These curves show that the QD can become dispersive or non-dispersive by detuning the laser frequency. Also, slight ${\epsilon }_b$ modifications do not change the nature of the curves. By increasing and decreasing${\epsilon }_b$, the Re(${\rho }_{21}$) spectrum width broadens and narrows, respectively. It is also worth mentioning, that if the value of ${\epsilon }_b$ decreases, the dip in Re(${\rho }_{21}$) spectrum vanishes completely.

 

Fig. 6 (a) The Re(${\rho }_{21}$) and (b) Im(${\rho }_{21}$) versus the laser detuning parameter (${\Delta }_{21}$) The solid, dotted and the dashed curve shows ${\epsilon }_b$ = 1.8, 1.825 and 1.85, respectively. The insets show the same quantities for each case where for R = 100 nm, where the nano-particles are far from each other.

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In Fig. 6(b), however, the sign of the Im(${\rho }_{21}$) spectrum changes at ${\Delta }_{12}=0$ where the negative values in this spectrum show a gain without inversion in the hybrid system [29]. Here, by increasing ${\epsilon }_b$ the extrema change causing more absorption and less gain. The insets in Fig. 6(a) and Fig. 6(b), also show the impact of exciton-plasmon interaction on Re(${\rho }_{21}$) and Im(${\rho }_{21}$) profile as a function of ${\Delta }_{12}$ where the particles are far from each other (R = 100 nm). As seen in these insets for a large value of R, Re(${\rho }_{21}$) and Im(${\rho }_{21}$) have the regular behavior as expected. At the end, in terms of the fabrication the single QD-AuNR systems can be formed, for example, by attaching an AuNR to an immobilized QD via a rigid polypeptide [30, 31]. They can also be fabricated using e-beam lithography to fabricate AuNRs on a substrate and then conjugate them with colloidal QDs [32]. The results presented in this paper may even be implemented using epitaxially grown QD–AuNR systems. Regarding this, one can use recent investigations of the fabrication of MNPs inside epitaxial structures, or the formation of epitaxially grown QDs very close to the surface [33]. Such techniques provide us with QD-AuNR hybrids with well-defined orientations, allowing us to have specific distances between QDs and AuNRs and well-defined polarization states.

4. Conclusion

In conclusion, in this report we show that the QD-AuNR hybrid system can support molecular-like collective resonances. The existence of such resonances is due to quantum coherence in QD which can be controlled by geometrical and environmental variation in the system. We study how the variation of center-to-center distance and the environmental dielectric constant causes dramatic spectral changes in the energy transfer rate between the particles and optical properties such as red and blue shift in exciton energy, dispersion, absorption and total electric field experienced by QD). These are very interesting results and can be useful for developing nanoscale plasmonic devices, particularly for chemical and biological sensing applications.