## Abstract

We have theoretically investigated a hybrid nanocrystal complex consisted of a metal nanoparticle (MNP) and a semiconductor quantum dot (SQD) embedded in a nanomechanical resonator in the simultaneous presence of a strong control field and a weak probe field. It is shown that the resonance amplification peak of the probe spectrum will enhance dramatically due to the coupling of the plasmon, exciton and nanomechanical resonator. The enhancement increases significantly with decreasing the distance between the metal nanoparticle and a quantum dot, which implies the strong plasmon enhancement effect in this coupled system. The results obtained here may have the potential applications such as tunable Raman lasers and bio-sensors.

©2010 Optical Society of America

## 1. Introduction

Metal-nanoparticle(MNP) plasmonics is an area of considerable current interest in photonics owing to its significant applications in solar cell, biochemical sensing, optical computing and medical field [1–4]. The collective surface charge oscillations on MNP, known as plasmon, have excellent optical properties to strongly confine the optical field within and around MNP, and greatly enhance nonlinear interactions [5–8]. In particular, gold and silver nanoparticles are known to exhibit pronounced resonances in the visible. Recently, the hybrid systems of plasmonics coupled with semiconductor dye molecules, or quantum dots have been intensively studied due to their extraordinary properties such as the enhancement of radiative emission rates, absorption of the exciton light and non-radiative energy transfer [8–10]. K*ü*hn *et al.* [11] investigated the coupling of a single molecule to a single spherical gold nanoparticle acting as a nanoantenna, and measured a strong enhancement in the fluorescence intensity. Artuso *et al.* [12] found the double peaked Fano structure and bistability in the response of strongly coupled quantum dot-metal nanoparticle system. Sadeghi [13] further studied metaresonances which generates Rabi oscillation in quantum dots via plasmons. Lu and Zhu [14, 15] theoretically demonstrated enhanced Kerr nonlinear coefficients and slow light effect in this artificial hybrid nanocrystal system. Moreover, plasmon fields have long been proved to be essential to increase the efficiency of different kinds of physical processes such as Raman scattering [16–18]. Surface plasmon enhanced Raman scattering has already been used to detect Raman signals to those cannot been detected by conventional Raman spectroscopy [19].

On the other hand, kinds of nanomechanics and optomechanics are widely studied and used to measure the extremely small displacement and extremely weak force [20, 21]. Those nanoscale mechanics are also promising to be employed in biology, medicine and chemistry [22–24]. Some of the more exciting transduction effects are focusing on coupling nanomechanics to nanocrystal and mesoscopic devices [25]. In the present article, we will investigate a hybrid nanocrystal complex consisted of a metal nanoparticle (MNP) and a semiconductor quantum dot (SQD) embedded in a nanomechanical resonator in the simultaneous presence of a strong contol field and a weak probe field. Due to the coupling among the plasmon, exciton and nanomechanical resonator mode, the resonance absorption peak and amplification peak of the probe absorption spectrum will be dramatically enhanced. We illustrate that the enhancement can be continuously adjusted by the separation of the metal nanoparticle and quantum dot. Approximate several orders of magnitude of the enhancement in amplification peak can be achieved in this coupled hybrid system. Such an advantage may lead to a potential application in the technique of nanoscale optical devices such as tunable Raman lasers and bio-sensors.

## 2. Theory

We consider a hybrid complex consisted of a MNP and a SQD coupled to a nanomechanical resonator. The hybrid system is subjected by a strong control field and a weak probe field as shown in Fig. 1. In the schematic, the SQD is embedded in the center of the nanomechanical beam. In low temperature, we assume a simple two-level model for the SQD which consists of a ground state ∣0〉 and the first excited state(single exciton)∣*ex*〉 [26, 27]. Usually the two-level states can be characterized by the pseudospin operators *S*
^{±} and *S ^{z}*. A single gold MNP attached to the end of a sharp optical fibre tip is positioned above the SQD. A scanning near-field optical microscopy is used to position the tip and to stabilize its distance [11,28,29]. The gold MNP has the radius

*a*

_{0}and a center-to-center distance

*R*towards the SQD. In the nanomechanical beam whose thickness is smaller than its width, the resonator mode can be described by a single mode phonon with annihilation operator

*a*and creation operator

*a*

^{+}[25]. Since the flexion induces extensions and compressions in the structure, this longitudinal strain will modify the energy of the electronic states of SQD through the deformation potential coupling, then the SQD will couple to the nanomechanical resonator. Considering the exciton in SQD interacts with a strong control field

*E*with frequency

_{c}*ω*and a weak probe field

_{c}*E*with frequency

_{s}*ω*simultaneously, the total Hamiltonian of the coupled system reads as follows [9, 25, 30]

_{s}where *ω _{ex}* and

*ω*are the frequency of exciton and resonator mode respectively,

_{n}*β*is the coupling strength of the resonator mode and SQD,

*μ*is the interband dipole matrix element and

*E*is the total optical field felt by the SQD. In a rotating frame at the control field frequency

_{SQD}*ω*, the total Hamiltonian is given by

_{c}where Δ = *ω _{ex}* −

*ω*is the frequency difference between the exciton and control field. ${\tilde{E}}_{\mathrm{SQD}}={E}_{c}+{E}_{s}{e}^{-i\delta t}+\frac{{S}_{\alpha}{P}_{\mathrm{MNP}}}{{\epsilon}_{\mathrm{eff}1}{R}^{3}}$ , with ${\epsilon}_{\mathrm{eff}1}=\frac{2{\epsilon}_{0}+{\epsilon}_{s}}{3{\epsilon}_{0}}$ ,

_{c}*ε*

_{0}and

*ε*are the dielectric constants of the background medium and SQD, respectively.

_{s}*δ*=

*ω*−

_{c}*ω*is the detuning of the probe and control field.

_{s}*S*is polar factor for electric field polarization and

_{α}*S*= 2 corresponds that the polar direction is along the

_{α}*z*axis of the hybrid system.

*P*is the dipole which comes from the charge induced by the probe field. For a spherical particle whose radius is much smaller than the wavelength of light, the electric field is uniform across the particle and the electrostatic(Rayleigh) approximation is a good one. Then the

_{MNP}*P*is given by ${P}_{\mathrm{MNP}}=\gamma {a}^{3}[{E}_{c}+{E}_{s}{e}^{-i\delta t}+\frac{{S}_{\alpha}{P}_{\mathrm{SQD}}}{{\epsilon}_{\mathrm{eff}2}{R}^{3}}]$ , where $\gamma =\frac{{\epsilon}_{\mathrm{Au}}\left(\omega \right)-{\epsilon}_{0}}{2{\epsilon}_{0}+{\epsilon}_{\mathrm{Au}}\left(\omega \right)}$ , ${\epsilon}_{\mathrm{eff}2}=\frac{2{\epsilon}_{0}+{\epsilon}_{\mathrm{Au}}\left(\omega \right)}{3{\epsilon}_{0}}$ [9, 30], ${\epsilon}_{\mathrm{Au}}\left(\omega \right)=1-\frac{{\omega}_{p}^{2}}{\omega (\omega +i{\gamma}_{\mathrm{Au}})}$ is the MNP’s dielectric constant,

_{MNP}*ω*and

_{p}*γ*are the bulk metal plasma frequency and the frequency-dependent damping, respectively. The imaginary part of relative permittivity

_{Au}*ε*determines the metallic losses [31, 32]. The dipole moment of the SQD is expressed via the off-diagonal elements of the density matrix:

_{Au}*P*=

_{SQD}*μS*

^{−}[33]. The dipole approximation used here is reasonable when the distance

*R*is large and the exciton-plasmon interaction is relatively weak [30]. Therefore the total optical field felt by the SQD is

*E*=

_{SQD}*A*(

*E*+

_{c}*E*

_{s}e^{−iδt}) +

*μBS*

^{−}, where $A=1+\frac{{\gamma a}^{3}{S}_{\alpha}}{{\epsilon}_{\mathrm{eff}1}{R}^{3}}$ , $B=\frac{\gamma {a}^{3}{S}_{\alpha}^{2}}{{\epsilon}_{\mathrm{eff}1}{\epsilon}_{\mathrm{eff}2}{R}^{6}}$ .

The Heisenberg equation of motion *dO*/*dt* = *−i*[*O*,*H*]/*h̅* gives the temporal evolutions of the exciton in the SQD and the nanomechanical resonator. The commutation relation [*S ^{z}*,

*S*

^{±}] = ±

*S*

^{±}, [

*S*

^{†},

*S*

^{−}] = 2

*S*and [

^{z}*a*

^{†},

*a*] = 1 can be used. If we set

*Q*=

*a*

^{†}+

*a*, and ignore the quantum properties of

*S*,

^{z}*S*

^{−}and

*Q*, then the semiclassical equations read as follows

where Ω = *μE _{c}*/

*h̅*is the Rabi frequency of the control field, Γ

_{1}and Γ

_{2}are the exciton relaxation rate and the exciton dephasing rate, respectively.

*γ*is the decay rate of the nanomechanical resonator due to the coupling to a reservoir of “background” mode and other intrinsic processes [22, 25]. In order to solve these equations, we make the ansatz [34]:

_{n}*S*=

^{z}*S*

^{z}_{0}+

*S*

^{z}_{+}

*e*

^{−iδt}+

*S*

^{z}_{−}

*e*,

^{iδt}*S*

^{−}=

*S*

_{0}+

*S*

_{+}

*e*+

^{−iδt}*S*

_{−}

*e*,

^{iδt}*Q*=

^{−}*Q*

_{0}+

*Q*

_{+}

*e*+

^{−iδt}*Q*

_{−}

*e*. Upon substituting these equations to Eqs. (3)−(5), we can obtain the steady state equations and finally

^{iδt}*S*

_{+}, which is related to the linear optical susceptibility as

*χ*

^{(s)}(

*ω*) =

_{s}*μS*

_{+}/

*E*=

_{s}*μ*

^{2}/Γ

_{2}

*h̅χ*(

*ω*), where the dimensionless susceptibility is given by

_{s}where *C* = Δ_{0}−*δ*
_{0}−*ω*
_{n0}
*β*
^{2}
*w*
_{0} + *B*
_{R0}
*w*
_{0}−*i*(1−*B*
_{I0}
*w*
_{0}), *D*=Δ_{0}+*δ*
_{0} − *ω*
_{n0}
*β*
^{2}
*w*
_{0} + *B*
_{R0}
*w*
_{0} + *i*(1 − *B*
_{I0}
*w*
_{0}), *E* = Ω^{2}
_{0}
*ω*
_{n0}
*β*
^{2}
*w*
_{0}
*η*/(Δ_{0} − *ω*
_{n0}
*β*
^{2}
*w*
_{0} + *B*
_{R0}
*w*
_{0} − *i*(1 − *B*
_{I0}
*w*
_{0})), *F* = Ω^{2}
_{0}
*ω*
_{n0}
*β*
^{2}
*w*
_{0}
*η*/(Δ_{0} − *ω*
_{n0}
*β*
^{2}
*w*
_{0} + *B*
_{R0}
*w*
_{0} + *i*(1 − *B*
_{I0}
*w*
_{0})), *G* = [*A*(*C* + *δ*
_{0}) − *i*2*B*
_{I0}
*Aw*
_{0}] [(2*F* + 2Ω^{2}
_{0}
*A*) (*D* − *δ*
_{0}) − 2*B*
^{*}
_{0}Ω^{2}
_{0}
*Aw*
_{0}] − *iD*(Γ_{10} − *iδ*
_{0}) (*C*+*δ*
_{0}) (*D* − *δ*
_{0}), where *η* = *ω*
^{2}
_{n}/(*ω*
^{2}
_{n} − *δ*
^{2} − *iγ _{n}δ*),

*w*

_{0}= 2

*S*

^{z}_{0},

*δ*

_{0}=

*δ*/Γ

_{2}, Ω

_{0}= Ω/Γ

_{2},

*ω*

_{n0}=

*ω*/Γ

_{n}_{2}, Δ

_{0}= Δ/Γ

_{2}, Γ

_{10}= Γ

_{1}/Γ

_{2},

*B*

_{0}=

*μ*

^{2}

*B*/(

*h̅*Γ

_{2}),

*B*

_{R0}=

*Re*(

*B*

_{0}), and

*B*

_{I0}=

*Im*(

*B*

_{0}). Due to the presence of the MNP, the decay rate of the exciton will increase. Zhang

*et al.*[9] have shown that the plasmon-exciton interaction leads to the formation of a hybrid exciton with the shifted exciton frequency and the decreased lifetime, which are determined by

*B*and

_{R}*B*respectively. The population inversion of exciton(

_{I}*w*

_{0}) is determined by the equation

## 3. Numerical results and discussion

In what follows we choose the realistic coupled system of an InAs quantum dot, gold MNP and a GaAs nanomechanical resonator. In this case Γ_{1} = 2Γ_{2} = 0.3*GHz*, *ω _{n}* = 1.2

*GHz*,

*γ*= 1 × 10

_{n}^{−4}

*GHz*,

*β*= 0.06 [25].

*a*

_{0}= 2.5

*nm*,

*μ*= 40

*D*,

*ε*

_{0}= 1,

*ε*= 6 [9, 30].

_{s}*ω*= 1.37 × 10

_{p}^{3}

*THz*,

*γ*=

_{Au}*ω*/60 [31, 32]. Figure 2 shows the probe absorption spectrum as a function of the detuning Δ

_{p}_{s}(Δ

_{s}=

*ω*−

_{s}*ω*) between probe field and exciton for several separations. At first, we consider the coupled system which only consists of SQD and nanomechanical resonator, it means the MNP is placed far away the SQD. Due to the coupling of exciton and nanomechanical resonator, it makes normal mode splitting in the probe absorption spectrum as shown in the black solid curve in Fig. 2. The result is compatible with the previous work of Li and Zhu [35]. Then we approach the MNP to the SQD, the peak splitting will shrink due to the coupling of MNP and SQD. As the probe field frequency close to the resonance frequency of gold MNP, the probe field will excite the collective surface charge oscillations on MNP. The MNP plays a role as antenna and radiates the exponential spatial distribution field. The enhanced field will promote the absorption of the excitation light by the SQD, and make the SQD much easier to saturation which causes the peak splitting to narrow. The inset shows the relation between splitting and separation. It is obvious that the splitting have a nearly linear dependence on the separation between MNP and SQD. This nearly linear relation can be used to describe the coupling strength of the hybrid system. So this scheme may provide us a method to measure the coupling strength between MNP and SQD.

_{ex}Figure 3(a) shows the absorption spectrum of the probe field as a function of Δ_{s} in the case Δ = 0 and the system without the MNP coupling to the SQD. From this figure, we can see that for each value of *ω _{n}* another two sharp peaks appear at sidebands which exactly locate at

*δ*= ±

*ω*. Part (b) of Fig. 3 shows the absorption peak and amplification peak will be dramatically enhanced as the SQD coupled to the MNP. This is due to the plasmon enhancement effect. The plasmon polarizes the exciton in SQD, and increases the population inversion of exciton. Then absorption of the light by SQD and radiative emission rates will increase. When the exciton decays finally, it will emit more intensive fields than the system without MNP. As it is shown in Fig. 3(b), the enhancement can be continuously adjusted by the separation of the MNP and SQD. When the separation is 16

_{n}*nm*, the enhanced peak will be almost 3 times larger than the system without MNP. From Fig. 3(b), we can also find that the full width at half maximum of the peak will reduce as the separation diminishes.

The gain enhancement and absorption enhancement as a function of separation between SQD and gold MNP are shown in Fig. 4. As the decrease of the separation between MNP and SQD, the gain enhancement will have an exponential increase. Especially, at a separation of 14.8 nm, approximate three orders of magnitude of the enhancement in amplification peak can be achieved in this coupled system. Such an advantage may lead to a potential application in Raman lasers.

## 4. Conclusion

To conclude, we have theoretically investigated a hybrid nanocrystal complex embedded in a nanomechanical resonator. We illustrate that the resonance absorption peak and amplification peak will enhance dramatically owing to the coupling among the plasmon, exciton and nanomechanical resonator mode. We also discuss the relation between the enhancement and the distance of MNP and SQD, and show that the enhancement can be continuously adjusted by the separation of the metal nanoparticle and quantum dot. Approximate several orders of magnitude of the enhancement in amplification peak can be achieved in this coupled system. The results obtained here may have the potential applications of nanoscale optical devices such as tunable Raman lasers and bio-sensors.

## Acknowledgments

The authors gratefully acknowledge support from National Natural Science Foundation of China (No.10774101 and No.10974133) and the National Ministry of Education Program for Training Ph.D.

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