## Abstract

We theoretically investigated optical third-order nonlinearity of a coherently coupled exciton-plasmon hybrid system under a strong control field with a weak probe field. The analytic formulas of exciton population and effective third-order optical susceptibility of the hybrid of a metal nanoparticle (MNP) and a semiconductor quantum dot (SQD) were deduced. The bistable exciton population and the induced bistable nonlinear absorption and refraction response were revealed. The bistability region can be tuned by adjusting the size of metal nanoparticle, interparticle distance and intensity of control field. Our results have perspective applications in optical information processing based on resonant coupling of exciton-plasmon.

© Optical Society of America

## 1. Introduction

Interactions between exciton systems and plasmonic structures have attracted considerable attention [1–20]. Because of these interactions, one can enhance emission and fluorescence [3,4], control energy transfer [5], generate single plasmons [6], induce exciton-plasmon-photon conversion [7], modify the spontaneous emission in semiconductor quantum dots (SQDs) [8], etc. Moreover, the exciton coherent dynamics building on the exciton-plasmon interaction have been extensively theoretically studied, including nonlinear Fano resonances [9,10], formation and control of Rabi oscillation [11,12], enhancement of Rabi flopping [11], mid-infrared generation [14] and SQD-induced transparency [15] and so on. Many photonic devices based on exciton-plasmon hybrid structures have already been predicted in theory and demonstrated experimentally, including optical switches and transistors [16,17], nanoscale lasers [18–20], photodetectors [21] and optical modulators [22].

Many investigations of the third-order optical nonlinearity and bistability that arise in the coherently coupled exciton-plasmon system have also been reported. Recently, Wang *et al.* experimentally demonstrated that the effective nonlinear absorption and refraction of CdS-Ag core-shell QDs comparing with CdS QDs were greatly enhanced [23]. Furthermore, Ji’s group observed the intensity-dependent enhancement of saturable absorption in a PbS-Au nanohybrid composite [24]. In theoretical research, Xiong *et al*. reported that third-order nonlinear optical susceptibility of CdTe QDs in the hybrid SQD-MNP system can be effectively modified by local field enhancement and dipole interaction [25]. Enhancement of Kerr nonlinearity in the exciton-plasmon hybrid system has also been theoretically demonstrated [26]. Optical bistability has been exhibited in some strong coupled exciton-plasmon hybrid systems, which could not been manifested in strongly coupled two-level molecules [27]. Artuso *et al.* found a regime of bistability in the hybrid system when SQD strongly coupled to MNP by increasing the sizes of both the SQD and the MNP. Then, they probed the transition to bistability that existed in such system, and saw a discontinuous jump in the response of the system [28,29]. Most recently, optical bistability and hysteresis in a hybrid SQD-MNP nanodimer could be realized by adjusting the incident field intensity and was manifested by the internal degrees of freedom of such system, there properties could be revealed by measuring the intensity of the Rayleigh scattering [30]. However, the nonlinear absorption and refraction response related to the optical bistability arose from the exciton-plasmon interaction have not been reported.

In this paper, we theoretically deduced the analytic formulas for exciton population and effective third-order optical susceptibility of the SQD-MNP hybrid system under a strong control field with a weak probe field. We studied the third-order optical nonlinearity of the system and found the bistable exciton population and the induced bistable nonlinear absorption and refraction response.

## 2. Model and formalism

We consider a system comprising a spherical SQD with radius *r* and a spherical MNP with radius *R*, separated by a surface-to-surface distance *d* (Fig. 1(a)
). The SQD and MNP are described by a density matrix formalism and classical electrodynamics, respectively. The hybrid is subject to a strong control field and a weak signal field (Fig. 1(b)).

The Hamiltonian of the system in the rotating-wave approximation can be written as

*ω*

_{10}-

*ω*is the difference between the exciton frequency

_{c}*ω*

_{10}and the control field frequency

*ω*.${\widehat{\sigma}}_{ij}$(

_{c}*i*,

*j*= 0,1) is the dipole transition operator between |

*i*〉 and |

*j*〉 of SQD, ${\widehat{\sigma}}_{z}=({\widehat{\sigma}}_{11}-{\widehat{\sigma}}_{00})/2$.

*µ*is the interband dipole moment element.

*Ẽ*and

_{SQD}*Ẽ*are the total electric field felt by the SQD and MNP, respectively. Here,

_{MNP}*Ẽ*=

_{SQD}*E*+

_{c}*E*[

_{s}e^{-iδt}+ S_{α}P_{MNP}/*ε*(

_{eff}*d*+

*r*+

*R*)

^{3}],

*Ẽ*=

_{MNP}*E*+

_{c}*E*[

_{s}e^{-iδt}+ S_{α}P_{SQD}/*ε*(

_{eff}*d*+

*r*+

*R*)

^{3}] [9,31].

*ε*= (2

_{eff}*ε*

_{0}+

*ε*)

_{s}*/*3, where

*ε*

_{0}and

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

_{s}*δ*=

*ω*–

_{s}*ω*is the frequency difference of the signal field and the control field. Geometric factor

_{c}*S*

_{α}is equal to 2 (−1) when

**is parallel to the**

*E**z*(

*x*,

*y*) axis. The z direction corresponds to the axis of the hybrid system. The dipole of the MNP

*P*=

_{MNP}*α*derives from the charge induced on the surface of the MNP, where

_{MNP}Ẽ_{MNP}*α*=

_{MNP}*ε*

_{0}

*R*

^{3}(

*ε*-

_{m}*ε*

_{0})/(2

*ε*

_{0}+

*ε*) [12], and

_{m}*ε*is the dielectric constant of the Au nanoparticle [32]. The dipole moment of the SQD is

_{m}*P*=

_{SQD}*µσ*

_{10}[33]. Based on the Heisenberg equation of motion, if we set

*p*=

*µσ*

_{10}and

*w*= 2

*σ*, and ignore thequantum properties of ${\widehat{\sigma}}_{10}$ and ${\widehat{\sigma}}_{z}$ [34], then the equations give

_{z}*T*

_{1}and

*T*

_{2}are the exciton lifetime and the exciton dephasing time, respectively.

*Ẽ*=

_{SQD}*A*(

*E*) +

_{c}+ E_{s}e^{-iδt}*Bp*[31], where

*A*= 1 +

*S*/[

_{α}α_{MNP}*ε*(

_{eff}*d*+

*r*+

*R*)

^{3}] and

*B*=

*S*

_{α}^{2}

*α*/[

_{MNP}*ε*(

_{eff}^{2}*d*+

*r*+

*R*)

^{6}]. To obtain a steady-state solution for Eq. (2), we make the ansatz [33,35]

*p*(

*t*) =

*p*

_{0}+

*p*

_{1}

*e*+

^{-iδt}*p*

_{-1}

*e*,

^{iδt}*w*(

*t*) =

*w*

_{0}+

*w*

_{1}

*e*+

^{-iδt}*w*

_{-1}

*e*, where

^{iδt}*p*

_{0},

*p*

_{1}and

*p*

_{-}_{1}are the dipole moments of the SQD oscillating at frequencies

*ω*,

_{c}*ω*and 2

_{s}*ω*-

_{c}*ω*, respectively. Similarly,

_{s}*w*

_{0},

*w*

_{1}and

*w*

_{-}_{1}are the population inversions oscillating at frequencies

*ω*,

_{c}*ω*and 2

_{s}*ω*-

_{c}*ω*, respectively. Here,

_{s}*p*

_{-}_{1}represents a wave-mixing response, because it gives rise to generation of an optical wave with frequency 2

*ω*-

_{c}*ω*. We assume that the other terms are small in the sense that |

_{s}*p*

_{0}|>>|

*p*

_{1}|, |

*p*

_{-1}| and |

*w*

_{0}|>>|

*w*

_{1}|, |

*w*

_{-1}|. Upon working to the lowest order in

*E*but to all orders in

_{s}*E*, we can get

_{c}*p*

_{-1}, which are related to the effective nonlinear optical susceptibility as follows:

Here, *N* is the number density of the hybrid system, Ω_{c}^{2} = *µ*^{2}|*E _{c}*|

^{2}

*T*

_{2}

^{2}/

*ћ*

^{2}is the generalized intensity of the control field,

*δ*=

_{c}*δT*

_{2}, Δ

*= Δ*

_{c}*T*

_{2}and

*B*=

_{c}*µ*

^{2}

*BT*

_{2}/

*ћ*

^{2}. Im

*χ*

_{eff}^{(3)}and Re

*χ*

_{eff}^{(3)}represent the nonlinear absorption and refraction, respectively. The population inversion of the exciton

*w*

_{0}is determined by the cubic equation

The Eq. (5) is of the third order in *w*_{0} and therefore may have one or three real solutions, strongly depending on the radius of MNP, surface-to-surface distance *d* and the intensity of the control field Ω_{c}^{2}. The latter case corresponds to the optical bistability arising from the strong coupling between SQD and MNP.

## 3. Results and discussion

To obtain an insight from these complicated expressions, we graphically display them for a wide range of some important parameters. We take the typical values *ε*_{0} = 2.25, *ε _{s}* = 6,

*T*

_{1}= 0.8 ns,

*T*

_{2}= 0.3 ns and

*µ*= 10

^{−28}C∙m [9], and

*N*= 10

^{14}m

^{−3}. The bare exciton frequency is chosen to be 2.5 eV, which is close to the broad plasmon frequency of gold (peak near 2.4 eV with a width of approximately 0.25 eV). As for the Au nanoparticle size region we consider, the plasmon resonance frequency changes little with the size.

Figures 2(a)
and 2(b) show the dynamic evolution of nonlinear absorption and refraction spectra with four different detunings (*ω*_{10} - *ω _{c}*)

*T*

_{2}. In Fig. 2(a), the absorption spectra which are located in regions I, II, III are originated from the three-photon resonance, stimulated Rayleigh resonance and ac-Stark resonance, respectively [35]. In region I, the absorption spectrum attributed to the three-photon effect evolves from a Fano-like lineshape into a negative peak and has a peak redshift as the detuning (

*ω*

_{10}-

*ω*)

_{c}*T*

_{2}increases. In region II, the absorption spectrum assigned to the Rayleigh resonance evolves from a negative peak into a Fano-like lineshape and has an increased magnitude. In region III, the absorption spectrum induced by the ac-Stark effect evolves from the Fano-like lineshape into a positive peak, and the peak first redshifts and then blueshifts. The corresponding refraction spectra are shown in Fig. 2(b). As required by the nonlinear Kramers-Kronig relation, the complementary behavior between the absorptive and refractive features is evident. From resonant excitation to near-resonant excitation, the dynamic evolution of the absorption and refraction spectra implies that the dominant role of three resonance mechanisms changes continuously. In the following discussion, we focus on the nonlinear absorption and refraction response of the hybrid under near-resonant excitation when

*ω*=

_{s}*ω*corresponding to the stimulated Rayleigh resonance.

_{c}Figure 3(a)
shows the bistability of exciton population on the interparticle distance between SQD and MNP *d* for the radius of MNP *R* = 5, 7.5, 10 nm. There are three real roots for *w*_{0} in Eq. (5) for a given *d*, which implies that a regime of bistability emerges. A region of optical bistability is formed by adjusting the value of *d*. In the bistability region of *d _{b}*

_{0}≤

*d*≤

*d*

_{b}_{1}(

*d*

_{b}_{0}and

*d*

_{b}_{1}are the critical values that the optical bistable response exists), population inversion

*w*

_{0}versus

*d*manifests a standard S-shaped curve, while the induced nonlinear absorption and refraction spectra are exotic coiled curves. When

*d*<

*d*

_{b}_{0}or

*d*>

*d*

_{b}_{1}, the bistability effect disappears. The nonlinear refraction of SQD-MNP system compared with SQD system is obviously enhanced. Speaking clearly, the maximum value of |Re

*χ*

_{eff}^{(3)}| of the hybrid system is 3.1, 3.6, 4.3 times that of SQD system, corresponding to

*R*= 5, 7.5, 10 nm, respectively. But the enhancement of nonlinear refraction is negligible. Take

*R*= 7.5 nm for example, Figs. 3(c) and 3(e) show the nonlinear absorption and refraction spectra in the bistability region (12.52 nm ≤

*d*≤ 13.09 nm), respectively. The bistable nonlinear absorption response curve traces out a crossed path ①→②→③→④→⑤→⑥→⑦, and the bistable refraction response curve displays a complementary behavior. When the interaction between SQD and MNP is strong enough by adjusting the interparticle distance

*d*, the interaction can result in bistable optical nonlinear absorption and refraction response.

Figure 4(a)
shows the bistability of exciton population on intensity of control field Ω_{c}^{2} for *R* = 7.5 nm and *d* = 12.8 nm. Population inversion *w*_{0} versus Ω_{c}^{2} manifests a standard S-shaped curve by adjusting the control field intensity, implying a standard bistable behavior. When the control field intensity slowly increases, the system firstly follows the lower (stable) branch and then jumps to the upper (stable) branch at the critical intensity. With sweeping the intensity back, the system remains on the upper branch and then makes a transition to the lower branch when the intensity passes through the other critical value. A hysteresis loop has been completed. The intermediate branch is unstable. Similar optical hysteresis curve has also been observed by Malyshev *et al.* in the metal-semiconductor nanodimer when they measured the Rayleigh scattering intensity as a function of excitation intensity [30]. Both Im*χ _{eff}*

^{(3)}and Re

*χ*

_{eff}^{(3)}behave in a different manner, and display bistable curves (see Figs. 4b–4e). Obviously, the nonlinear absorption and refraction response (Im

*χ*

_{eff}^{(3)}and Re

*χ*

_{eff}^{(3)}) could be used to manifest optical bistability and optical hysteresis. Out of the bistability region, the maximum value of |Re

*χ*

_{eff}^{(3)}| of the hybrid system is 2.8 times that of SQD system.

## 4. Conclusion

In conclusion, we theoretically investigated the third-order optical nonlinearity of a coherently coupled SQD-MNP system in the presence of a strong control field with a weak probe field. We deduced the analytic formulas of exciton population and effective third-order optical susceptibility of the hybrid, and found the bistable exciton population and the induced bistable nonlinear absorption and refraction response. We showed that the bistability region can be tuned by adjusting the size of metal nanoparticle, interparticle distance and intensity of control field. The optical bistability promised possible applications as optical memory cells and optical switches. We hoped that our predictions in this work can be experimentally demonstrated using the optical pump-probe method in the near future.

## Acknowledgments

This work was supported by NSFC (10874134, 61008043 and 11004001), National Program on Key Science Research of China (2011CB922201), and Key Project of Ministry of Education of China (708063).

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