Abstract
Based on cavity quantum electrodynamics (QED), we investigate the lightmatter interaction between surface plasmon polaritons (SPP) in a metal nanoparticle (MNP) and the excitons in semiconductor quantum dots (SQDs) in an SQDMNP coupled system. We propose a quantum transformation method to strongly reveal the exciton energy shift and the modified decay rate of SQD as well as the coupling among SQDs. To obtain these parameters, a simple system composed of an SQD, an MNP, and a weak signal light is designed. Furthermore, we consider a model to demonstrate the coupling of two SQDs mediated by SPP field under two cases. It is shown that two SQDs can be entangled in the presence of MNP. A high concurrence can be achieved, which is the best evidence that the coupling among SQDs induced by SPP field in MNP. This scheme may have the potential applications in alloptical plasmonenhanced nanoscale devices.
1 Introduction
Due to the advances in modern nanoscience, various nanostructures such as metal nanopartities (MNPs), semiconductor quantum dots (SQDs) and nanowires can be constructed for the applications in photonics and optoelectronics [1,2]. Studies of these nanostructures are essential for further development of nanotechnology. MNPs can be excited to produce surface plasmon polaritons (SPP) [3]. The energy transfer effect in a hybrid nanostruction complex composed of MNPs and SQDs has been observed, which implies the lightmatter interaction between SPP field in MNPs and the excitons in SQDs [4,5]. To display the interaction between the exciton and SPP field, the vacuum Rabi splitting has been studied theoretically [6,7] and experimentally [8]. However, in the SQDMNP coupled system a nonlinear Fano effect can be produced by a strong incident light [9]. Various theoretical [10,11] and experimental [1214] reports have shown a decrease of the exciton lifetime of SQD placed in the vicinity of MNP. The decrease is related to the distance between SQD and MNP as a result of the coupling of the exciton and SPP field [15]. Moreover, the exciton energy level of SQD can be shifted because of the influence of SPP field [14]. Recently, the coupling among SQDs mediated by SPP field has received increasing 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 the solidstate of SQDs and integrated circuits of these nanostructures, the complex system is a promising candidate to implement the quantum information processing. However, more details about the coupling among SQDs and the role of SPP field need to be further studied. To illustrate clearly these quantum effects, a full quantum mechanics method to describe the coupled SQDMNP system have to be developed.
In the present article, cavity QED as a quantum optics toolbox provides a full quantum mechanics description of the coupled SQDMNP system. Under the description we develop a novel quantum transformation method that is suitable for the coupling SQDs to SPP field with large decay rate. The quantum transformation is used to treat master equation of the entire system. Under a certain condition, we obtain an effective Hamiltonian in SQDs' subsystem, and show a modified decay rate for each SQD. The effective Hamiltonian demonstrates an exciton energy shift and the coupling among SQDs. A crossdecay rate is induced by SPP field. It not only changes the decay rate of each SQD but also makes decay between every two SQDs. We analyze the exciton energy shift and the crossdecay rate of every SQD and the coupling among SQDs, and find that these parameters are related to the distance between SQD and MNP. An experimental scheme to obtain these parameters is proposed by the observation of the signal light absorption spectrum of SQD in 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 for demonstrating the coupling of two SQDs.
2 Theory
We consider multiple SQDs in the vicinity of an MNP. Each SQD consists of the electronic
ground state 0〉 and the first excited state ex〉. They interact
with SPP field in the MNP. First, we need to quantize SPP field based on the cavity
quantum electrodynamics (QED). Recently, a good deal of study had been devoted to
quantize SPP field in the metal [2024]. SPP field in the MNP can be considered as a multiplemodes field. After the
second quantization of SPP field, the Hamiltonian can be written as
where
with the Liouvillian terms [26,27],
If π_{k} ≪ 1, the secondorder term remains, and
the higherorder terms can be ignored safely. To obtain the reduce density operation
of
the SQDs' subsystem, we take a trace over the SPP field of the both hands of Eq. (3)
by
using Tr_{SPP}[.]. Here, we assume that the multimode plasmon
field can be consider as a thermal reservoir and the reservoir variables are distributed
in the uncorrelated thermal equilibrium mixture of states,
where
The effective Hamiltonian to reveal the exciton energy shift and the coupling among SQDs is given by
where
Γ_{i,j }= κ + 2τ if i =
j, Γ_{ij }= 2τ if i ≠
j, where
Our method to treat the Hamiltonian is similar with SchriefferWolff transformation
[28]. In cavity (circuit) QED system, when the decay rate of cavity mode is very
small as compared to the detuning between the cavity mode frequency and the transition
frequency of qubits so that it can be ignored safely, the effective Hamiltonian can
be
obtained by using SchriefferWolff transformation [18,19]. Under the treatment of SchriefferWolf transformation, one can obtain
3 Coupling an SQD to an MNP
Now, we consider a simple complex system composed of an SQD and an MNP. As illustrated
in inset of Figure 1, an SQD with radius r is placed in
the vicinity of an MNP with radius R. The centertocenter distance is
d. The modified decay rate of the SQD includes the radiative decay rate
κ and the nonradiative decay rate
Figure 1. The signal light absorption spectrum of SQD. The signal light absorption spectrum of SQD for different distance d. The inset shows a complex system composed of a SQD to a MNP. A SQD with radius r is placed in the vicinity of a MNP with radius R. The centertocenter distance is d.
Therefore, η = Re[G], τ = Im[G], where
An experimental scheme to measure the two parameters is proposed by observation on
the
absorption spectrum of SQD in the system. Now, we consider an SQD in the vicinity
of an
Au MNP excited a weak signal light E_{s }with frequency
ω_{s}. According to master equation
∂_{t}ρ_{SQD }=
i[H',ρ_{SQD}] +
ς'_{SQD}, where
where p = μρ_{ex,0}, w = ρ_{ex,ex } ρ_{0,0}.
The steady state solution can be obtained by setting the lefthand sides of Eqs. (9) and (10) equal to zero. Thus,
where
is the firstorder (linear) susceptibility.
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
Figure 2. The probability and the concurrence in one case. The probability of each state, the concurrence of the two SQDs as a function of time when the initial state of the two SQDs is the state ex, 0〉. The left inset shows a model composed of two SQDs and a MNP. The right inset shows the dissipation channels of the two SQDs.
4 Coupling of two SQDs
We consider a simple model composed of two identical SQDs and an Au MNP for revealing
the coupling between two SQDs induced by SPP field, as shown in left inset of Figure
2. The interaction between the two identical SQDs can be
neglected 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 expression
of two parameters η, τ. If one of the two distances changed, the
expressions of the crossdecay rate and the coupling constant between the two SQDs
need
to be modified. As mentioned above, g_{k }~
d^{3}. The expression of the crossdecay rate and the coupling
strength can be rewritten as Im[G'] and Re[G'], respectively, where
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)ρρ(2〉 〈2 + 4〉 〈4)] + (κ/2) × [2(1〉 〈3 3〉 〈4)ρ(3〉 〈14〉 〈3)(3〉 〈3 + 4〉 〈4)ρ  ρ(3〉 〈3 + 4〉 〈4)]. It shows two dissipated channels. The first term describes dissipation through one cascade channel 4〉 → 2〉 → 1〉 with fast decay rate κ + 4τ. The second term describes dissipation through another cascade channel 4〉 → 3〉 → 1〉 with slow decay rate κ (see inset of Figure 3).
Figure 3. The probability and the concurrence in another case. The probability of each state, the concurrence of the two SQDs as a function of time when the initial state of every SQD is in their excited state. The inset shows the dissipation channels of the two SQDs.
In order to illustrate the coupling of the two SQDs, we analyze the following two
parameters: (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,
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
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); and
P_{2}(t_{0}) =
P_{3}(t_{0}) (see the figure of probability). This
is because two entangled states 2〉 and 3〉 make a product state
ex, 0〉 or 0, ex〉. The absence of the oscillation in
the figure of the concurrence implies that the coupling of the two SQDs cannot play
a
role in the creation of the concurrence. In the two cases, we can generate the entangled
state of the two SQDs because the quantized SPP modes are act as the platform of the
energy transfer between the two SQDs. If the MNP is absent (d → ∞
), the coupling strength η and the crossdecay rate τ of the
two SQDs are equal to zero so that the SQDs cannot be entangled. We can tune the
concurrence of the two SQDs by changing the distance d. In our theoretical
calculations presented above, we do not consider size distribution of the SQD. A
numerical averaging of the obtained results for different spatial dispersions of the
distance will give a perfect prediction of the dispersion effects on the concurrence.
Because of size inhomogeneities of CdSe SQD, we assume that the position distribution
density satisfies the Gaussian distribution
Figure 4. Comparison between the original results and the modified results. The concurrence as a function of time: the original results with a fixed distance d = 16 nm (solid curves), the modified results to reveal the dispersion effects of size distribution of the SQD (dash curves).
In conclusion, we have clearly demonstrated the interaction of SQDs and SPP field in MNP via a novel quantum transformation. The SPP field can induce the exciton energy shift and the decay rate modification of each SQD. The expressions of them is given by analysis. They can be measured by the designed scheme. Moreover, the coupling of two SQDs mediated by SPP field has been revealed strongly under two cases. With respect to the coupling among three or more SQDs, it is very significant for multipartite entanglement. The entanglement due to the lightmatter interaction in the coupled SQDMNP system may be applied in alloptical plasmonenhanced nanoscale devices.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
YH finished the main work of this paper, including deducing the formulas, plotting the figures, and drafting the manuscript. KDZ participated in the discussion and provided some useful suggestion. All authors are involved in revising the manuscript and approved the final version.
Acknowledgements
Part of this study had been supported by the National Natural Science Foundation of China (No. 10774101 and No. 10974133) and the Ministry of Education Program for Training Ph.D.
References

Noginov M, Zhu G, Belgrave A, Bakker R, Shalaev V, Narimanov E, Stout S, Herz E, Suteewong T, Wiesner U: Demonstration of a spaserbased nanolaser.
Nature 2009, 460:1110. PubMed Abstract  Publisher Full Text

Schuller JA, Barnard ES, Cai W, Jun YC, White JS, Brongersma ML: Plasmonics for extreme light concentration and manipulation.
Nat Mater 2010, 9:193. PubMed Abstract  Publisher Full Text

Akimov A, Mukherjee A, Yu C, Chang D, Zibrov A, Hemmer P, Park H, Lukin M: Generation of single optical plasmons in metallic nanowires coupled to quantum dots.
Nature 2007, 450:402. PubMed Abstract  Publisher Full Text

Govorov AO, Bryant GW, Zhang W, Skeini T, Lee J, Kotov NA, Slocik JM, Naik RR: Excitonplasmon interaction and hybrid excitons in semiconductormetal nanoparticle assemblies.
Nano lett 2006, 6:984. Publisher Full Text

Hosoki K, Tayagaki T, Yamamoto S, Matsuda K, Kanemitsu Y: Direct and stepwise energy transfer from excitons to plasmons in closepacked metal and semiconductor nanoparticle monolayer films.
Phys Rev Lett 2008, 100:207404. PubMed Abstract  Publisher Full Text

Savasta S, Saija R, Ridolfo A, Di Stefano O, Denti P, Borghese F: Nanopolaritons: vacuum rabi splitting with a single quantum dot in the center of a dimer nanoantenna.
ACS nano 2010, 4:6369. PubMed Abstract  Publisher Full Text

Manjavacas A, Abajo FJG, Nordlander P: Quantum plexcitonics: strongly interacting plasmons and excitons.
Nano lett 2011, 11:2318. PubMed Abstract  Publisher Full Text

Passmore BS, Adams DC, Ribaudo T, Wasserman D, Lyon S, Davids P, Chow WW, Shaner EA: Observation of rabi splitting from surface plasmon coupled conduction state transitions in electrically excited InAs quantum dots.
Nano lett 2011, 11:338. PubMed Abstract  Publisher Full Text

Zhang W, Govorov AO, Bryant GW: Semiconductormetal nanoparticle molecules: hybrid excitons and the nonlinear Fano effect.
Phys Rev Lett 2006, 97:146804. PubMed Abstract  Publisher Full Text

Merten H, Koenderink A, Polman A: Plasmonenhanced luminescence near noblemetal nanospheres: comparison of exact theory and an improved Gersten and Nitzan model.

Vandenbem C, Brayer D, FroufePirez L, Carminati R: Controlling the quantum yield of a dipole emitter with coupled plasmonic modes.

Okamoto K, Niki I, Shvartser A, Narukawa Y, Mukai T, Scherer A: Surfaceplasmonenhanced light emitters based on InGaN quantum wells.
Nat Mater 2004, 3:601. PubMed Abstract  Publisher Full Text

Fedutik Y, Temnov V, Schps O, Woggon U, Artemyev M: Excitonplasmonphoton conversion in plasmonic nanostructures.
Phys Rev Lett 2007, 99:136802. PubMed Abstract  Publisher Full Text

Vasa P, Pomraenke R, Schwieger S, Mazur YI, Kunets V, Srinivasan P, Johnson E, Kihm J, Kim D, Runge E: Coherent excitonsurfaceplasmonpolariton interaction in hybrid metalsemiconductor nanostructures.
Phys Rev Lett 2008, 101:116801. PubMed Abstract  Publisher Full Text

Chen C, Wang C, Wei C, Chen Y: Tunable emission based on the composite of Au nanoparticles and CdSe quantum dots deposited on elastomeric film.
Appl phys lett 2009, 94:071906. Publisher Full Text

Lin ZR, Guo GP, Tu T, Li HO, Zou CL, Chen JX, Lu YH, Ren XF, Guo GC: Quantum bus of metal nanoring with surface plasmon polaritons.

GonzalezTudela A, MartinCano D, Moreno E, MartinMoreno L, Tejedor C, GarciaVidal FJ: Entanglement of two qubits mediated by onedimensional plasmonic waveguides.
Phys Rev Lett 2011, 106:020501. PubMed Abstract  Publisher Full Text

Zheng SB, Guo GC: Efficient scheme for twoatom entanglement and quantum information processing in cavity QED.
Phys Rev Lett 2000, 85:2392. PubMed Abstract  Publisher Full Text

Majer J, Chow JM, Gambetta JM, Koch J, Johnson BR, Schreier JA, Frunzio L, Schuster DI, Houck AA, Wallraff A, Blais AM, Devoret H, Girvin SM, Schoelkopf RJ: Coupling superconducting qubits via a cavity bus.
Nature 2007, 449:443. PubMed Abstract  Publisher Full Text

Bergman DJ, Stockman MI: Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems.

Trügler A, Hohenester U: Strong coupling between a metallic nanoparticle and a single molecule.

Hohenester U, Trügler A: Interaction of single molecules with metallic nanoparticles.

Sugakov V, Vertsimakha G: Localized exciton states with giant oscillator strength in quantum well in vicinity of metallic nanoparticle.

Waks E, Sridharan D: Cavity QED treatment of interactions between a metal nanoparticle and a dipole emitter.

Andersen ML, Stobbe S, Sørensen AS, Lodahl P: Strongly modified plasmonmatter interaction with mesoscopic quantum emitters.
Nat Phys 2011, 7:215. Publisher Full Text

Loudon R: The Quantum Theory of Light. Oxford: Oxford University Press; 2000.

Ridolfo A, Di Stefano O, Fina N, Saija R, Savasta S: Quantum Plasmonics with quantum dotmetal nanoparticle molecules: influence of the fano effect on photon statistics.
Phys Rev Lett 2010, 105:263601. PubMed Abstract  Publisher Full Text

Salomaa M: SchriefferWolff transformation for the Anderson Hamiltonian in a superconductor.
Phys Rev B 1988, 37:9312. Publisher Full Text

Yan J, Zhang W, Duan S, Zhao XG, Govorov AO: Optical properties of coupled metalsemiconductor and metalmolecule nanocrystal complexes: role of multipole effects.

Lu Z, Zhu K: Slow light in an artificial hybrid nanocrystal complex.
J Phys B: At Mol Opt Phy 2009, 42:015502. Publisher Full Text

De Abajo FJG: Optical excitations in electron microscopy.
Rev Mod Phys 2010, 82:209. Publisher Full Text

Wootters WK: Entanglement of formation of an arbitrary state of two qubits.
Phys Rev Lett 1998, 80:2245. Publisher Full Text