Strong coupling among semiconductor quantum dots induced by a metal nanoparticle
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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.
KeywordsMaster Equation Surface Plasmon Polaritons Surface Plasmon Polaritons Mode Nonradiative Decay Rate Exciton Lifetime
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) . 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 . However, in the SQD-MNP coupled system a nonlinear Fano effect can beproduced by a strong incident light . Various theoretical [10, 11] and experimental [12, 13, 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 . Moreover, the exciton energy level of SQD can be shifted because of theinfluence of SPP field . Recently, the coupling among SQDs mediated by SPP field has receivedincreasing attention [16, 17]. The complex system like cavity QED system  and circuit QED system  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.
Γ i,j = κ + 2τ if i =j, Γ ij = 2τ if i ≠j, where . 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., .
Our method to treat the Hamiltonian is similar with Schrieffer-Wolff transformation . 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. 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
Therefore, η = Re[G], τ = Im[G], where. 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 , 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.
where p = μρex,0, w =ρex,ex-ρ0,0.
is the first-order (linear) susceptibility.
4 Coupling of two SQDs
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) =ρi,i(t), for i = 1,2, 3, 4. (2) The concurrence for quantifying entanglement of the two SQDs,[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. 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 P2(t) andP3(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.
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.
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.
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