Based on cavity quantum electrodynamics (QED), we investigate the light-matterinteraction between surface plasmon polaritons (SPP) in a metal nanoparticle (MNP)and the excitons in semiconductor quantum dots (SQDs) in an SQD-MNP coupled system.We propose a quantum transformation method to strongly reveal the exciton energyshift and the modified decay rate of SQD as well as the coupling among SQDs. Toobtain these parameters, a simple system composed of an SQD, an MNP, and a weaksignal light is designed. Furthermore, we consider a model to demonstrate thecoupling of two SQDs mediated by SPP field under two cases. It is shown that two SQDscan be entangled in the presence of MNP. A high concurrence can be achieved, which isthe best evidence that the coupling among SQDs induced by SPP field in MNP. Thisscheme may have the potential applications in all-optical plasmon-enhanced nanoscaledevices.

1 Introduction

Due to the advances in modern nanoscience, various nanostructures such as metalnanopartities (MNPs), semiconductor quantum dots (SQDs) and nanowires can be constructedfor the applications in photonics and optoelectronics [1, 2]. Studies of these nanostructures are essential for further development ofnanotechnology. MNPs can be excited to produce surface plasmon polaritons (SPP) [3]. The energy transfer effect in a hybrid nanostruction complex composed ofMNPs and SQDs has been observed, which implies the light-matter interaction between SPPfield in MNPs and the excitons in SQDs [4, 5]. To display the interaction between the exciton and SPP field, the vacuumRabi splitting has been studied theoretically [6, 7] and experimentally [8]. However, in the SQD-MNP coupled system a nonlinear Fano effect can beproduced by a strong incident light [9]. Various theoretical [10, 11] and experimental [12–14] reports have shown a decrease of the exciton lifetime of SQD placed in thevicinity of MNP. The decrease is related to the distance between SQD and MNP as a resultof the coupling of the exciton and SPP field [15]. Moreover, the exciton energy level of SQD can be shifted because of theinfluence of SPP field [14]. Recently, the coupling among SQDs mediated by SPP field has receivedincreasing attention [16, 17]. The complex system like cavity QED system [18] and circuit QED system [19] may be applied in quantum information. Owing to the advantages of thesolid-state of SQDs and integrated circuits of these nanostructures, the complex systemis a promising candidate to implement the quantum information processing. However, moredetails about the coupling among SQDs and the role of SPP field need to be furtherstudied. To illustrate clearly these quantum effects, a full quantum mechanics method todescribe the coupled SQD-MNP system have to be developed.

In the present article, cavity QED as a quantum optics toolbox provides a full quantummechanics description of the coupled SQD-MNP system. Under the description we develop anovel quantum transformation method that is suitable for the coupling SQDs to SPP fieldwith large decay rate. The quantum transformation is used to treat master equation ofthe entire system. Under a certain condition, we obtain an effective Hamiltonian inSQDs' subsystem, and show a modified decay rate for each SQD. The effective Hamiltoniandemonstrates an exciton energy shift and the coupling among SQDs. A cross-decay rate isinduced by SPP field. It not only changes the decay rate of each SQD but also makesdecay between every two SQDs. We analyze the exciton energy shift and the cross-decayrate of every SQD and the coupling among SQDs, and find that these parameters arerelated to the distance between SQD and MNP. An experimental scheme to obtain theseparameters is proposed by the observation of the signal light absorption spectrum of SQDin a system consisted of an SQD and an MNP. Based on the achievement of thes parameters,we design a simple model that two identical SQDs interact with an Au MNP fordemonstrating the coupling of two SQDs.

2 Theory

We consider multiple SQDs in the vicinity of an MNP. Each SQD consists of the electronicground state |0〉 and the first excited state |ex〉. They interactwith SPP field in the MNP. First, we need to quantize SPP field based on the cavityquantum electrodynamics (QED). Recently, a good deal of study had been devoted toquantize SPP field in the metal [20–24]. SPP field in the MNP can be considered as a multiple-modes field. After thesecond quantization of SPP field, the Hamiltonian can be written as${H}_{SPP}=\sum _{k}{\omega}_{k}{a}_{k}^{+}{a}_{k}$[20, 21], where ω_{
k
}is the frequency of SPP mode$k,{a}_{k}^{+}\left({a}_{k}\right)$ is the creation (annihilation) operation of SPP modek. Next, we consider the interaction between each SQD and SPP modes. Weassume that the coupling strength between each SQD and SPP field is identical forsimplicity. The interaction Hamiltonian, under the rotating-wave approximation, can bewritten as ${H}_{int}=-\sum _{i,k}\left({g}_{k}{a}_{k}{\sigma}_{+}^{i}+{g}_{k}^{*}{a}_{k}^{+}{\sigma}_{-}^{i}\right)$[22, 25], where g_{
k
}is the coupling strength between eachSQD and SPP mode $k,{\sigma}_{+}^{i}=|ex{\u3009}_{i}\u30080|({\sigma}_{-}^{i}{=|0\u3009}_{i}\u3008ex|)$ is the raising (lowering) operator of the i thSQD. Therefore, the Hamiltonian of the entire system can be written as (ħ = 1)

where ${\sigma}_{z}^{i}=(1/2)\times {(|ex\u3009}_{i}{\u3008ex|-|0\u3009}_{i}\u30080|)$. The full quantum dynamics of the coupled nanosystem canbe derived from the following master equation for the density operation

with the Liouvillian terms [26, 27], ${\varsigma}_{SQD}=\left(\kappa /2\right)\times \sum _{i}\left(2{\sigma}_{-}^{i}{\rho}_{+}^{i}-\rho {\sigma}_{+}^{i}{\sigma}_{-}^{i}-{\sigma}_{+}^{i}{\sigma}_{-}^{i}\rho \right)$ describes the decay of each SQD to Markovian reservoirs,κ is the exciton radiative decay rate in SQDs, ${\varsigma}_{SPP}=\sum _{i}\left({\gamma}_{k}/2\right)\times \left(2{a}_{k}\rho {a}_{k}^{+}-\rho {a}_{k}^{+}{a}_{k}-{a}_{k}^{+}{a}_{k}\rho \right)$ describes the relaxation of SPP mode k with decayrate γ_{
k
}. Next, we take a time-independent unitytransformation e^{
is
}on the density operator, where$s=\sum _{i,k}\left({\pi}_{k}{a}_{k}{\sigma}_{+}^{i}+{\pi}_{k}^{*}{a}_{k}^{+}{\sigma}_{-}^{i}\right),{\pi}_{k}={2}_{gk}/\left({\gamma}_{k}+2i{\delta}_{k}\right),{\delta}_{k}={\omega}_{k}-{\omega}_{ex}$, so that $\stackrel{\u0303}{\rho}={e}^{is}\rho {e}^{-is}$,

If |π_{
k
}| ≪ 1, the second-order term remains, andthe higher-order terms can be ignored safely. To obtain the reduce density operation ofthe SQDs' subsystem, we take a trace over the SPP field of the both hands of Eq. (3) byusing Tr_{
SPP
}[.]. Here, we assume that the multi-mode plasmonfield can be consider as a thermal reservoir and the reservoir variables are distributedin the uncorrelated thermal equilibrium mixture of states, $<{a}_{k}>=<{a}_{k}^{+}>=0,<{a}_{k}^{+}{a}_{l}>={\stackrel{\u0304}{n}}_{k}{\partial}_{kl}$, where the thermal average boson number${\left({\stackrel{\u0304}{n}}_{k}\right)}^{-1}=exp\left[\left({\omega}_{k}\right)/\left({k}_{B}T\right)\right]-1$, k_{
B
}is the Boltzmannconstant, and T is the temperature. Therefore,

where ${\rho}_{SQD}=T{r}_{SPP}\left[\stackrel{\u0303}{\rho}\right]$. After some algebraic calculation, the master equation ofthe reduce density operation of the SQDs' subsystem can be written as

where ${\eta}_{0}=\eta +\sum _{k}{8\left|{g}_{k}\right|}^{2}{\delta}_{k}{\stackrel{\u0304}{n}}_{k}/\left({\gamma}_{k}^{2}+4{\delta}_{k}^{2}\right),{\stackrel{\u0304}{n}}_{k}=<{a}_{k}^{+}{a}_{k}>$, and $\eta =\sum _{k}4|{g}_{k}{|}^{2}{\delta}_{k}/\left(4{\delta}_{k}^{2}+{\gamma}_{k}^{2}\right)$ are the coupling strength among SQDs induced by quantizedSPP modes. We can see that η_{0} represents the exciton energyshift as a result of the coupling SQD to all quantized SPP modes. In the bosonic bathcomposed of all SPP modes, according to the Bose-Einstein distribution function,${\stackrel{\u0304}{n}}_{k}\ll 1$ at low temperature so that ${\eta}_{0}\cong \eta $. The dissipation term is given by

Γ_{
i,j
}= κ + 2τ if i =j, Γ_{
ij
}= 2τ if i ≠j, where $\tau =\sum _{k}2{\left|{g}_{k}\right|}^{2}{\gamma}_{k}/\left(4{\delta}_{k}^{2}+{\gamma}_{k}^{2}\right)$. We note that a cross-decay rate 2τ betweenevery two SQDs appears and the exciton lifetime decreases because of the presence of SPPfield. The cross-decay rate represents the nonradiative decay rate that can bedecomposed into different contributions for each SPP mode, i.e., $2\tau \cong {\Gamma}_{MNP}^{nr}$[22].

Our method to treat the Hamiltonian is similar with Schrieffer-Wolff transformation [28]. In cavity (circuit) QED system, when the decay rate of cavity mode is verysmall as compared to the detuning between the cavity mode frequency and the transitionfrequency of qubits so that it can be ignored safely, the effective Hamiltonian can beobtained by using Schrieffer-Wolff transformation [18, 19]. Under the treatment of Schrieffer-Wolf transformation, one can obtain$\eta =\sum _{k}{\left|{g}_{k}\right|}^{2}/{\delta}_{k},\tau =0$. But it is well-known that the decay of SPP field is toolarge to be ignored in the coupled SQD-MNP system. Taking this fact fully into account,our method is suitable for revealing the exciton energy shift, the modify decay rate andthe coupling strength among SQDs.

3 Coupling an SQD to an MNP

Now, we consider a simple complex system composed of an SQD and an MNP. As illustratedin inset of Figure 1, an SQD with radius r is placed inthe vicinity of an MNP with radius R. The center-to-center distance isd. The modified decay rate of the SQD includes the radiative decay rateκ and the nonradiative decay rate ${\mathrm{\Gamma}}_{MNP}^{nr}$ induced by MNP. Owing to the ohmic losses within the metala significant fraction of absorbed power has be dissipated as heat [3]. We first estimate the parameters η and τ. In thecomplex system, the SQD can induce polarization of MNP ${P}_{MNP}=[\gamma {s}_{\alpha}{R}^{3}{P}_{SQD}]/[{\epsilon}_{eff1}{d}^{3}]$, where $\gamma =[{\epsilon}_{M}(\omega )-{\epsilon}_{0}]/[{\epsilon}_{M}(\omega )+2{\epsilon}_{0}],{\epsilon}_{eff1}=[{\epsilon}_{s}+2{\epsilon}_{0}]/[3{\epsilon}_{0}],{P}_{SQD}=\mu (<{\sigma}_{+}>+<{\sigma}_{-}>)$[9], ε_{0}, ε_{
s
}, andε_{
M
}are the dielectric constants of the backgroundmedium, the SQD and the MNP, respectively, μ is the electric dipole momentof the exciton, s_{
α
}is related to the direction of thecoupling. The SPP field induced by the SQD can be expressed as ${E}_{MNP}=[{s}_{\alpha}{P}_{MNP}]/[4\pi {\epsilon}_{0}{\epsilon}_{eff2}{d}^{3}]$ that is the mean value of the electric field operator${\hat{E}}_{MNP}$, where ${\epsilon}_{eff2}=[{\epsilon}_{M}(\omega )+2{\epsilon}_{0}]/[3{\epsilon}_{0}]$. The operator can be split into two contributions${\hat{E}}_{MNP}^{+}+{\hat{E}}_{MNP}^{-}$ evolving with positive and negative frequencies [29]. Based on the principle of second quantization for SPP field, we have$<\mu {\hat{E}}_{MNP}^{+}>=\sum _{k}{g}_{k}<{a}_{k}>$[26]. The above result is under the dipole approximation when the distance islarge comparing to the radius of the MNP. However, if the distance is comparable to theradius of the MNP, we need to consider the multipole polarization in the MNP, includingdipole, quadrupole, octopole, and so on. So the multipole polarization can be expressedas ${P}_{MNP,tot}={\displaystyle \sum _{n=1}({s}_{n}{\epsilon}_{0}{\gamma}_{n}{R}^{2n+1}{P}_{SQD})}/({\epsilon}_{eff1}{d}^{2n+1})$[30], where s_{
n
}= (n + 1)^{2} for thepolarization parallel to the axis of the complex system, γ_{
n
}= [ε_{
M
}(ω) -ε_{0}]/[ε_{
M
}(ω) +ε_{0}(n + 1)/n]. For simplicity we assumethat the distance is larger than the radius of the MNP so that the dipole approximation(n = 1) is reasonable. In the dissipative system, the expectation value<a_{
k
}> = Tr[ρa_{
k
}] of each SPP mode satisfies the equation,${\partial}_{t}<{a}_{k}>=\left({\delta}_{k}-i{\gamma}_{k}/2\right)<{a}_{k}>-{g}_{k}^{*}<{\sigma}_{-}>$. At steady state, we can obtain

Therefore, η = Re[G], τ = Im[G], where$G=[\gamma {(\mu {s}_{\alpha})}^{2}{R}^{3}]/[4\pi {\epsilon}_{0}{\epsilon}_{eff1}{\epsilon}_{eff2}{d}^{6}]$. We note that, here, η, τ ~d^{-6}. So, it is reasonable that g_{
k
}~d^{-3}. The verdict is in good agreement with the coupling strengthbetween a two-level system and a single mode of SPP field [24, 27]. In [9], Zhang et al. found that the interaction between an SQD and an MNP leads tothe formation of a hybird exciton with the shifted exciton frequency and the decreasedlifetime in which the SPP field is treated as a classical field rather than a quantizedfield. Here, we make a same conclusion under the quantized SPP field.

An experimental scheme to measure the two parameters is proposed by observation on theabsorption spectrum of SQD in the system. Now, we consider an SQD in the vicinity of anAu MNP excited a weak signal light E_{
s
}with frequencyω_{
s
}. According to master equation∂_{
t
}ρ_{
SQD
}=-i[H',ρ_{
SQD
}] +ς'_{
SQD
}, where ${H}^{\prime}=\left({\omega}_{ex}-\eta \right){\sigma}_{z}-\mu \left({E}_{s}{\sigma}_{+}{e}^{-i{\omega}_{s}t}+{E}_{s}^{*}{\sigma}_{-}{e}^{i{\omega}_{s}t}\right),{\varsigma}_{SQD}=\left({\gamma}_{SQD}^{tot}/2\right)\times \left(2{\sigma}_{-}\rho {\sigma}_{+}-\rho {\sigma}_{+}{\sigma}_{-}-\sigma +{\sigma}_{-}\rho \right),{\gamma}_{SQD}^{tot}=\kappa +2\tau $, we have

where $T=1/{\gamma}_{SQD}^{tot}$. The polarization induced by the signal light can beexpressed as $p={\epsilon}_{0}\chi {E}_{s}{e}^{-i{\omega}_{s}t}$[31], where χ is the total susceptibility to all order. We canobtain the total susceptibility: $\chi =\left({\left|\mu \right|}^{2}/{\epsilon}_{0}\right)\times \left[T-\left({\omega}_{s}+\eta -{\omega}_{ex}\right){T}^{2}\right]/\left[1+{\left({\omega}_{s}+\eta -{\omega}_{ex}\right)}^{2}{T}^{2}+8{\left|\mu {E}_{s}\right|}^{2}{T}^{2}\right]$. It can be expended in powers of the electric fieldχ = χ^{(1)} + 3 χ^{(3)}|E_{
s
}|^{2} +···,where

In what follows, as an example, we consider a CdSe SQD with radius r = 3.75 nm [4] and an Au MNP with radius R = 7.5 nm. We use ε_{0} = 1.8, ε_{
s
}= 7.2 [32] and the electric constant of Au ${\epsilon}_{M}\left(\omega \right)={\epsilon}_{b}-{\omega}_{p}^{2}/\left[\omega \left(\omega +i{\eta}_{p}\right)\right]$ with ϵ_{
b
}= 9.5,ħ_{
ω
}= 2.8eV, ħω_{
p
}= 9eV, ħη_{
p
}= 0.07eV [22, 33]. For the decay rate and dipole moment of the SQD, we take κ =1.25 GHz and μ = er_{0} with r_{0} =0.65 nm. Figure 2 shows the absorption spectrum of the SQD (theimaginary part of linear susceptibility Im[χ^{(1)}]) as a functionof the signal-SQD detuning for d = 30, 21, 18, 16 nm. We note that theabsorption peak is shifting and broadening with the decreasing distance between the SQDand the MNP. The absorption peak shift represents the exciton energy shift, and thebroadened peak implies the increased decay rate of SQD as a result of the presence ofSPP field. So, the exciton energy shift η and the cross-decay rate2τ can be obtained by observation of the absorption spectrum. As shownin Figure 2, the exciton energy shift (full width at half maximum)is about 6.5κ (2.5κ) for a small distance d = 16nm.

4 Coupling of two SQDs

We consider a simple model composed of two identical SQDs and an Au MNP for revealingthe coupling between two SQDs induced by SPP field, as shown in left inset of Figure2. The interaction between the two identical SQDs can beneglected safely in the absence of the MNP if the distance between them is very lager.When the distances between every SQD and the MNP are not equal (d_{1} ≠ d_{2}), we need to make a modification for the expressionof two parameters η, τ. If one of the two distances changed, theexpressions of the cross-decay rate and the coupling constant between the two SQDs needto be modified. As mentioned above, g_{
k
}~d^{-3}. The expression of the cross-decay rate and the couplingstrength can be rewritten as Im[G'] and Re[G'], respectively, where${G}^{\prime}=[\gamma {(\mu {s}_{\alpha})}^{2}{R}^{3}]/[4\pi {\epsilon}_{0}{\epsilon}_{eff1}{\epsilon}_{eff2}{d}_{1}^{3}{d}_{2}^{3}]$. However, here, we assume that d_{1} =d_{2} = d for simplicity. In the SQDs' subsystem, we choosean adequate basic of SQDs' subsystem, i.e., $|1\u3009=|0,0\u3009,|2\u3009=(1/\sqrt{2})\times (|ex,0\u3009+|0,ex\u3009),|3\u3009=(1/\sqrt{2})\times (|ex,0\u3009-|0,ex\u3009),|4\u3009=|ex,ex\u3009$. The four collective states are the eigenstates of the twocoupling SQDs. The master equation of the SQDs' subsystem is given by

where H'' = -(ω_{
ex
}- η)|1〉 〈1| - η |2〉 〈2| + η |3〉〈3| + (ω_{
ex
}- η) |4〉〈4|, ζ_{
SQD
}(ρ) = [(κ +4τ)/2] × [2(|2〉 〈4| + |1〉〈2|)ρ(|4〉 〈2| + |2〉 〈1|)-(|2〉〈2| + |4〉 〈4|)ρ-ρ(|2〉 〈2| +|4〉 〈4|)] + (κ/2) × [2(|1〉 〈3| -|3〉〈4|)ρ(|3〉 〈1|-|4〉 〈3|)-(|3〉〈3| + |4〉 〈4|)ρ - ρ(|3〉 〈3| +|4〉 〈4|)]. It shows two dissipated channels. The first term describesdissipation through one cascade channel |4〉 → |2〉 → |1〉with fast decay rate κ + 4τ. The second term describesdissipation through another cascade channel |4〉 → |3〉 →|1〉 with slow decay rate κ (see inset of Figure 3).

In order to illustrate the coupling of the two SQDs, we analyze the following twoparameters: (1) The probability of the two SQDs being in the state |i〉,P_{
i
}(t) =ρ_{i,i}(t), for i = 1,2, 3, 4. (2) The concurrence for quantifying entanglement of the two SQDs,$C\left(t\right)=\sqrt{{\left[{\rho}_{2,2}\left(t\right)-{\rho}_{3,3}\left(t\right)\right]}^{2}+4\text{Im}\phantom{\rule{1em}{0ex}}{\left[{\rho}_{2,3}\left(t\right)\right]}^{2}}$[17, 34]. Here we use the parameters of the above section, and take d = 16nm.

If the initial state of the two SQDs is prepared in a product state |ex,0〉, only two dissipation channels |2〉 → |1〉 and |3〉→ |1〉 should been considered (see right inset of Figure 2). To obtain the probability of each state, Eq. (14) can be rewritten as${\partial}_{t}{\rho}_{i,j}\left(t\right)=-i\sum _{k}\left({{H}^{\u2033}}_{i,k}{\rho}_{k,j}-{{H}^{\u2033}}_{k,j}{\rho}_{i,k}\right)+<i\left|{\varsigma}_{SQD}\right|j>$. According to the initial state density matrixρ(0) = (|2〉 + |3〉)(〈2| + 〈3|)/2, we can obtainthe the probability of each state and the concurrence. As shown in Figure 2, with the decrease of P_{2}(t) andP_{3}(t), the probability of the two SQDs in the state|1〉 increases. At about t = 0.08 ns, the concurrence ofthe two SQDsreaches the maximal value. In the figure of the concurrence, a weak oscillation ispresented as a result of the coupling of the two SQDs.

Another case is that the initial state is in another product state |ex,ex〉 (ρ(0) = |4〉 〈4|). Figure 3 shows the probability of each state, the concurrence as a function of time.It shows that the two SQDs can be entangled. Only at about t_{0} =0.275 ns the concurrence is equal to zero (see the figure of the concurrence); andP_{2}(t_{0}) =P_{3}(t_{0}) (see the figure of probability). Thisis because two entangled states |2〉 and |3〉 make a product state|ex, 0〉 or |0, ex〉. The absence of the oscillation inthe figure of the concurrence implies that the coupling of the two SQDs cannot play arole in the creation of the concurrence. In the two cases, we can generate the entangledstate of the two SQDs because the quantized SPP modes are act as the platform of theenergy transfer between the two SQDs. If the MNP is absent (d → ∞), the coupling strength η and the cross-decay rate τ of thetwo SQDs are equal to zero so that the SQDs cannot be entangled. We can tune theconcurrence of the two SQDs by changing the distance d. In our theoreticalcalculations presented above, we do not consider size distribution of the SQD. Anumerical averaging of the obtained results for different spatial dispersions of thedistance will give a perfect prediction of the dispersion effects on the concurrence.Because of size inhomogeneities of CdSe SQD, we assume that the position distributiondensity satisfies the Gaussian distribution $\rho \left(r\right)=exp\left[-{r}^{2}/\left(2{\sigma}^{2}\right)\right]/\left(\sqrt{2\pi \sigma}\right)$, with the the half-width of Gaussian distributionσ = 16Å. Figure 4 shows acomparison between the original results and the modified results considering thedispersion effects on the concurrence under the two cases. We can see the differencebetween the two results. The difference becomes slighter with decreasingσ. When the half-width σ is much smaller than the radius ofthe SQD, there is good agreement between the two results. Moreover, a stationary statewith a high concurrence can be achieved by continuous pumping [17].

In conclusion, we have clearly demonstrated the interaction of SQDs and SPP field in MNPvia a novel quantum transformation. The SPP field can induce the exciton energy shiftand the decay rate modification of each SQD. The expressions of them is given byanalysis. They can be measured by the designed scheme. Moreover, the coupling of twoSQDs mediated by SPP field has been revealed strongly under two cases. With respect tothe coupling among three or more SQDs, it is very significant for multipartiteentanglement. The entanglement due to the light-matter interaction in the coupledSQD-MNP system may be applied in all-optical plasmon-enhanced nanoscale devices.

Declarations

Acknowledgements

Part of this study had been supported by the National Natural Science Foundation ofChina (No. 10774101 and No. 10974133) and the Ministry of Education Program forTraining Ph.D.

Authors’ Affiliations

(1)

Key Laboratory of Artificial Structures and Quantum Control (Ministry ofEducation), Department of Physics, Shanghai Jiao Tong University

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