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Abstract
We investigate the Berry phase in the nanocrystal complex made of a metal nanoparticle and a slowly rotating semiconductor quantum dot under the radiation of a circularly polarized light. The Berry phase in the dynamic system is found to be more effective to manifest the interaction between the plasmon in the metal nanoparticle and the exciton in the quantum dot. The dependences of the Berry phase on the interparticle distance and the relative position are studied in the weak field condition. The methods to observe the Berry phase are also given.
© 2017 Optical Society of America
1. Introduction
Recently much attention has been paid to the nanocrystal complexes, composed of various building blocks such as metal nanoparticles (MNPs) [1–3], semiconductor quantum dots (SQDs) [4, 5], dielectric nanostructures [6], biomolecules [7], colloid quantum dots [8], semiconductor and metal nanowires [9, 10]. One of the most popular system is undoubtedly the MNP-SQD complex, whose optical properties have been investigated intensively, such as but definitely not limited to, the Fano effect [11–17], the Rabi oscillation [18–22], the spin-plasmon coupling [23–25], the optical bistability [14, 26–31], the population dynamics [32–35], the quantum coherence [36], the resonance fluorescence [37–41], the Förster energy transfer [42], the photoluminescence [43], the spontaneous emission [44], and the enhanced difference frequency generation [45]. Later the system has been modified to some more complex composites [16, 17, 31, 46, 47] by including more MNPs or SQDs, which further extends the scope of the exciton-plasmon interactions in this kind of nanocrystals and possible optical applications.
In generally speaking, most of studies relevant to this kind of composites focused on the optical properties and usually in static conditions [48–70]. The strong field enhancement effect is a characteristics of the coupled systems. However, the field enhanced optical processes become outstanding only when the interdistance between the MNP and the two-level system decreases down to order of nanometers [71–73]. Even so, the improvement is still demanding for detection in experiments. For example, the theoretical results showed the typical red-shift of the steady absorption in the MNP-SQD system is about 0.05 meV [74] and could only be improved to about 0.5 meV even if the multipole effect considered [75]. Besides, the concerned optical properties maybe spoiled as the interdistance further decreasing [20,45] due to the rapid increasing dephasing rate. Thus a more effective way to manifest the dipole-dipole interaction is necessary.
In the paper, we will investigate the evolution of the quantum states in the SQD when cycling around the MNP, which is a dynamic process rather than a static one. In contrast to the optical properties, the highlight is put on the phase of quantum states rather than the energy level shift or inter-level transition. Particularly the Berry phase obtained for each energy level after the SQD finishing the cycling will be studied [76–82]. As we expected, the quantum state of SQD does get a nonzero Berry phase under the circularly polarized optical field, similar to the spin of electrons does under the precessed magnetic field [83, 84]. Moreover, the Berry phase in the system has some new characteristics: it not only has a geometric dependence on the path but also changes sensitively with the interparitcle distance when compared with the energy-level shift in the same condition. The results are obtained almost by analytical derivations and the different contributions to the Berry phase are studied in detail. The sensitive and obvious dependence on the configuration of the system means we could observe the dipole-dipole interaction easily via the Berry phase.
The paper is organized as follows. In Section II, the system and its interaction with the external field are introduced. Detailed derivations for the external field, the dipole induced in the SQD and the Berry phase are given orderly in Subsection A, B, C, respectively. In Section III, the numerical results are shown. The contributions from different parts to the Berry phase and its dependences on different parameters are studied. In Section IV, we will discuss how to observe the Berry phase by two different methods. The assumed measurements are demonstrated. Finally the conclusion is given in Section V.
2. The model and Berry phase
The studied system is the nanocrystal complexes composed of an MNP of radius R0 and an SQD of radius Rs (Rs ≪ R0), as shown in Fig. 1. The SQD is assumed to be a two-level system whose polarization is along the z direction, i.e., the interlevel optical transition matrix element μ = μẑ. In contrast to other similar systems, the SQD is not fixed, but designed to rotate around the MNP slowly. With the same origin as the Cartesian coordinates, the spherical coordinates of the SQD is the (Rd, θ, φ), where Rd is the interdistance, θ and φ are polar angle and azimuthal angle of the SQD at the position, respectively. The dynamic path is just the trajectory by changing φ continuously. A circularly polarized light propagating along z direction is employed as the excitation. The external field is considered to be spatially uniform due to the small size of the system. Once the plasmon in the MNP and the exciton in the SQD are excited, the exciton-plasmon interaction takes effect through the modifications of electric fields felt by each other. As the adiabatical condition is satisfied, the system gets the steady state at each moment in the cycle. So let us begin with the exact solution of the steady electric fields.
Fig. 1 The nanocrystal complexes composed of an MNP and an SQD, where the corresponding coordinates, the sizes, the angles, and the permittivities are all given explicitly. The system is radiated with a circularly polarized light propagating along z direction. The dynamic path is shown by the dashed line with two arrows.
2.1. The electric field
The incident light is a circularly polarized light propagating along z direction, whose electric field can be written as
At a given position, the electric field felt by the SQD can be split into three parts:
which will be explained in detail as follows. This classification is proposed by us before [75], but note that the center-to-center line here is not always parallel or perpendicular to the polarization of external field so that the electric field felt by the SQD is different.The permittivities of the MNP and the SQD are denoted as εm and εs, respectively, while that of the environment around the nanocrystal complex is εe. The first part E1 is the electric field exerted by the light directly,
where εeff = (εs + 2εe)/3. This part of the electric field would not excite the exciton in the SQD here because μ · E = 0.The second part is the one produced by the polarized MNP. In the paper we will adopt the dipole approximation, which is sound enough to demonstrate the qualitative physics if the interdistance Rd is not so small compared with the radius of the MNP R0. The polarization inside the MNP induced directly by the external field is
where γ1 = [εm(ω) − εe]/[εm(ω) + 2εe]. The polarized MNP will created an additional electric field around and therefore has an influence on the SQD. Since that the first part has no contribution to the exciton in the SQD, this part has the leading contribution. We only need the z component of the electric field in the SQD, which is found to be Note that the positive-frequency part and the negative-frequency one have same weight.The third part comes from the effective electric field produced by the polarization in the MNP induced by the effective dipole of the SQD (εe/εeff) ps (t). Technically, we can decompose the dipole vector into two parts, parallel and perpendicular to the center-to-center line, calculate the induced electric field by each part, and then combine them. By careful calculation, the z component of this part is found as
This part is independent on the azimuthal angle φ as the dipole of the SQD ps (t) has a rotational symmetry. The relationship of debases its contribution compared with that of the second part.To sum up, the interaction item between the SQD and the field is then
2.2. The dipole in the SQD
Now we want to calculate the dipole of the SQD ps (t) analytically. The Hamiltonian of the SQD coupled to the external optical field reads
where ai and are the annihilation and the creation operators of the level i (i=1,2), respectively. The upper and lower levels have the energies of ħω0/2 and −ħω0/2, respectively. As the time dependence of the external field is ∼ e±i(ωt+φ), the induced polarization inside the SQD also has the same time dependence ∼ e±i(ωt+φ). Therefore we can assume the dipole of the SQD has the form where p̃s is time-independent. With the help of rotating wave approximation, the Hamiltonian can be written as where From above we can see that both the second and the third parts of the electric fields have the influence on the felt field of SQD. Unlike the second part, the third part have to be determined self-consistently. Here we only consider the weak-field situation. In this case, the dipole of the SQD is just ps (t) = μ(p + p*), where the interlevel polarization p satisfies Dot above the quantity represents the time derivation in the paper.2.3. The Berry phase
Now we calculate the Berry phase of the excited energy level when the SQD completes a cycle adiabatically. For convenience, we express the Hamiltonian by Pauli operator σ,
The above Hamiltonian is better to be transformed to a frame rotating at the frequency ω to ignore the high frequency part with the help of the following unitary transformation,
where the operator U = exp(iωtσz/2). The transformed Hamiltonian takes the form of where δ = ω0 − ω is the detuning between the transition energy and the excitation.The adiabatical condition means the cycling time period T should satisfy the condition . During the cycle, each quantum state |ψ〉 will pick up a geometric phase as well as the dynamic phase, i.e., eiψD eiγ |ψ〉, where γ is just the well-known Berry phase. Now we give the explicit expression of the Berry phase in the system. The instantaneous eigenvalue of the excited state for Hamiltonian (15) is
and the corresponding normalized eigenstate is where the normalization coefficient is The Berry phase is where Γ is an oriented closed loop in the χ-parameter space. By straight calculation, the Berry phase of the excited state can be expressed as where λ is the instantaneous eigenvalue given above. Note that a linear-polarization light will not bring about Berry phase, because the left and right circularly polarized light produce the opposite Berry phase in the system, which can be easily checked following the above derivations.In the model of spin-1/2 particle under an adiabatically precessing magnetic field, the excited state acquires the Berry phase , where α is the apex angle corresponding to the solid angle. The magnitude of the precessing magnetic field is fixed. In comparison, the Berry phase in our system is caused by the processing of the induced electric field accompanying the SQD. As the magnitude of the electric field has a strong dependence on the interdistance, the Berry phase is not only determined by the solid angle but also by the interdistance.
3. Numerical results
Now we present the numerical results of the Berry phase based on the obtained formulae. The typical parameters used to do the calculations are as follows: R0 = 10nm, εe = 12ε0 and ε0 is the permittivity of vacuum, the transition matrix element μ = er0 and r0 = 0.6nm, the detuning ħδ = 0.5meV and ħω0 = 1.55eV. The material of the MNP is gold (Au) whose permittivity is (−29 + 2i)ε0 [85] at this frequency. The permittivity of the SQD is 12ε0.
We first analyze the dependence of Berry phase on the polar angle θ. In the weak field case, the quantity Δ ≈ 1. Here the intensity of the external field is set as E0 = 105V/m so that Δ ≈ 1 is checked to be satisfied. In Fig. 2, the Berry phases for a certain interparticle distance such as Rd = 12nm are shown. To see the contributions made by different parts of electric fields, the results when only E2 part is considered only (hollowed symbols) and when both parts E2 + E3 are included (solid symbols) are calculated, respectively. For case of only E2 considered, the Berry phase increases firstly and then decreases as the angle θ increasing, with the extremum locating at θ = π/4. The reason is obvious because of the sin 2θ factor in the expression of electric field E2. Note that the Berry phases corresponding to 2π and 0 are equivalent and thus the extremum is actually a maximal change of the phase. The rule is also applied for the other interparticle distances. However, the electric field E3 has a big influence on the Berry phase especially when the interparticle distance is small. As it shows, when both parts are considered, the Berry phase has a complex relation with the polar angle θ, which is determined not only by the unidirectional influence from the MNP to the SQD, but also by the interaction between them. The obvious dependence of the interaction on the angle gives a new way to observe the strong coupling in the system.
Fig. 2 The dependence of the Berry phase on the polar angle θ with different parts of the electric fields considered. The lines with the hallowed circles is the case when only E2 is considered while the lines with solid ones is that when both E2 and E3 are included. The field strength is set as E0 = 105V/m and the interparticle distance is Rd = 12nm.
The difference brought about by E3 will disappear only when the SQD and the MNP are separated by a big interparticle distance as it takes effect according to the relation of . In this case the contribution of E3 can be neglected.
The dependence of the Berry phase on the interparticle distance can be seen from Fig. 3, where the calculated Berry phases with and without E3 part considered are drawn for several different interparticle distances. For a relatively big distance, the results in two cases are almost same, because the electric field of E3 has a decreasing contribution with the increasing distance. When two nanoparticles are close, the Berry phase for these two cases are totally different. The difference is caused by the non-negligible E3. The obvious sensitivity of the Berry phase on the E3 provides an effective way to study or observe the dipole-dipole interaction in the complex.
Fig. 3 The dependence of the Berry phase on the interparticle distance, with both E2 and E3 considered and with only E3 considered. The polar angle is θ = π/4. Other used parameters are same as that in Fig. 2.
4. Observation of the Berry phase
The Berry phase is usually observed by interference. Here we design two schemes to observe the Berry phase by adding another SQD in the system. All the building blocks in the complex could be imbedded in a dielectric slab, which is radiated by the excitation light propagating along the normal direction. Rotating the slab around the light can realize the slow variation of this dynamic system. The principle is to measure the interference result of two produced lights polarized along z direction by two SQDs with different Berry phases.
4.1. By different positions
In this method, we place one SQD above and the other below the MNP with the same interparticle distance and supplementary polar angle. We only need to rotate the sample around z direction along which a circularly polarized light propagates, as shown in Fig. 4. We assume these two SQDs have the identical energy level. If the incident light is right circularly polarized light for one SQD, the other feels the left circularly polarized light according to the symmetry, therefore the Berry phases produced in these two SQDs are exactly opposite.
Fig. 4 The scheme to observe the Berry phase in the complexes with two identical SQDs locating above and below the MNP, respectively. The sample is radiated by a circularly polarized light propagating along z direction. The right part shows the energy levels of these two SQDs and the strength changing of the detected light.
The detected light is determined by the polarizations in the SQDs. Based on the above analysis, the polarizations after finishing a cycle obtain the Berry phases γa and γb, respectively, which have the relation γa = −γb. The intensity of polarization can be expressed as P ∝ eiφD eiγi E0, where φD is the dynamical phase and γi is the Berry phase [86]. Then intensity of the detected light is
where the interference factor can be further derived as Based on the theory, Using the parameters θ = π/4 and Rd = 15nm, stated in caption of Fig. 4, the corresponding Berry phase through a circle is about γa = 1.6π. The intensity of the detected light after a cycle according changes the factor cos 2γa, as shown in right part of Fig. 4. By measuring the strength changing, the Berry’s phase in the nanocrystal complexes can be obtained.4.2. By different detunings
As the Berry phase is also determined by the detuning δ, we can use the sign of detuning to produce the opposite Berry phases. Now we add another SQD in the same pathway with respect to the original one, and rotate the whole sample around z axis, as shown in Fig. 5. In this case, the energy level of these two SQDs are different, satisfying (ω1 + ω2)/2 = ω0. Based on the theory, the polarizations of these two SQDs acquire the opposite Berry phases and consequently the intensity of the detected light is determined by
If we rotate the system around z direction by the time period of T = π/δ, then the intensity is just , which changes in the same way as the first method. Similarly, by measuring the strength of the detected light, the Berry phase in the nanocrystal complex can be obtained.Fig. 5 The scheme to observe the Berry phase in the nanocrystal complex including two SQDs with the different detuning. The transition energies of these two SQDs satisfy ω2 + ω1 = 2ω0.
5. Conclusion
We investigate the Berry phase in the rotating MNP-SQD system radiated by a circularly polarized light. Exact solutions to the electric fields, the dipole in the SQDs and the Berry phase of the quantum states are presented analytically. The Berry phase of the quantum states in such dynamic system are found to be more sensitive than the shift of eigen energies and therefore is easy to be detected. We also design two methods to observe the Berry phase by interference and present the theoretical analyses. The properties of the Berry phase are beneficial to study the dipole-dipole interaction in the system.
Funding
National Key Research and Development Program of China (Grant 2016YFA0301300); National Natural Science Foundation of China (NSFC) (Grant 11004015).
References and links
1. G. Bachelier and A. Mlayah, “Surface plasmon mediated Raman scattering in metal nanoparticles,” Phys. Rev. B 69, 205408 (2004). [CrossRef]
2. I. L. Rasskazov, S. V. Karpov, and V. A. Markel, “Surface plasmon polaritons in curved chains of metal nanoparticles,” Phys. Rev. B 90, 075405 (2014). [CrossRef]
3. P. J. Compaijen, V. A. Malyshev, and J. Knoester, “Surface-mediated light transmission in metal nanoparticle chains,” Phys. Rev. B 87, 205437 (2013). [CrossRef]
4. C. Schulhauser, D. Haft, R. J. Warburton, K. Karrai, A. O. Govorov, A. V. Kalameitsev, A. Chaplik, W. Schoenfeld, J. M. Garcia, and P. M. Petroff, “Magneto-optical properties of charged excitons in quantum dots,” Phys. Rev. B 66, 193303 (2002). [CrossRef]
5. V. N. Gladilin, S. N. Klimin, V. M. Fomin, and J. T. Devreese, “Optical properties of polaronic excitons in stacked quantum dots,” Phys. Rev. B 69, 155325 (2004). [CrossRef]
6. Z. Y. Jia, J. N. Li, H. W. Wu, C. Wang, T. Y. Chen, R. W. Peng, and M. Wang, “Dipole coupling and dual Fano resonances in a silicon nanodimer,” J. Appl. Phys. 119, 074302 (2016). [CrossRef]
7. Y. Zelinskyi, Y. Zhang, and V. May, “Photoinduced dynamics in a molecule metal nanoparticle complex: Mean-field approximation versus exact treatment of the interaction,” J. Chem. Phys. 138, 114704 (2013). [CrossRef]
8. S. M. Sadeghi, R. G. West, and A. Nejat, “Photo-induced suppression of plasmonic emission enhancement of CdSe/ZnS quantum dots,” Nanotechnology 22, 405202 (2011). [CrossRef] [PubMed]
9. J.-Y. Yan, “Strong exciton-plasmon interaction in semiconductor-insulator-metal nanowires,” Phys. Rev. B 86, 075438 (2012). [CrossRef]
10. M.-T. Cheng and Y.-Y. Song, “Fano resonance analysis in a pair of semiconductor quantum dots coupling to a metal nanowire,” Opt. Lett. 37, 978–980 (2012). [CrossRef] [PubMed]
11. G. P. Wiederrecht, G. A. Wurtz, and J. Hranisavljevic, “Coherent coupling of molecular excitons to electronic polarizations of noble metal nanoparticles,” Nano Lett. 4, 2121–2125 (2004). [CrossRef]
12. A. M. Kelley, “A molecular spectroscopic description of optical spectra of J-aggregated dyes on gold nanoparticles,” Nano Lett. 7, 3235–3240 (2007). [CrossRef] [PubMed]
13. W. Zhang and A. O. Govorov, “Quantum theory of the nonlinear Fano effect in hybrid metal-semiconductor nanostructures: The case of strong nonlinearity,” Phys. Rev. B 84, 081405 (2011). [CrossRef]
14. R. Artuso and G. Bryant, “Optical Response of Strongly Coupled Quantum Dot-Metal Nanoparticle Systems: Double Peaked Fano Structure and Bistability,” Nano Lett. 8, 2106–2111 (2008). [CrossRef] [PubMed]
15. S. G. Kosionis, A. F. Terzis, V. Yannopapas, and E. Paspalakis, “Nonlocal Effects in Energy Absorption of Coupled Quantum Dot-Metal Nanoparticle Systems,” J. Phys. Chem. C 116, 23663 (2012). [CrossRef]
16. R. A. Shah, N. F. Scherer, M. Pelton, and S. K. Gray, “Ultrafast reversal of a Fano resonance in a plasmon-exciton system,” Phys. Rev. B 88, 075411 (2013). [CrossRef]
17. X. Wu, S. K. Gray, and M. Pelton, “Quantum-dot-induced transparency in a nanoscale plasmonic resonator,” Opt. Express 18, 23633–23645 (2010). [CrossRef] [PubMed]
18. A. O. Govorov, G. W. Bryant, W. Zhang, T. Skeini, J. Lee, N. A. Kotov, J. M. Slocik, and R. R. Naik, “Exciton-plasmon interaction and hybrid excitons in semiconductor-metal nanoparticle assemblies,” Nano Lett. 6, 984 (2006). [CrossRef]
19. S. M. Sadeghi, “The inhibition of optical excitations and enhancement of Rabi flopping in hybrid quantum dot-metallic nanoparticle systems,” Nanotechnology 20, 225401 (2009). [CrossRef] [PubMed]
20. S. M. Sadeghi and K. D. Patty, “Suppression of quantum decoherence via infrared-driven coherent exciton-plasmon coupling: Undamped field and Rabi oscillations,” Appl. Phys. Lett. 104, 083101 (2014). [CrossRef]
21. M.-T. Cheng, S.-D. Liu, H.-J. Zhou, Z.-H. Hao, and Q.-Q. Wang, “Coherent exciton-plasmon interaction in the hybrid semiconductor quantum dot and metal nanoparticle complex,” Opt. Lett. 32, 2125–2127 (2007). [CrossRef] [PubMed]
22. Y. He, C. Jiang, B. Chen, J.-J. Li, and K.-D. Zhu, “Optical determination of vacuum Rabi splitting in a semiconductor quantum dot induced by a metal nanoparticle,” Opt. Lett. 37, 2943–2945 (2012). [CrossRef] [PubMed]
23. M. A. Antón, F. Carreño, S. Melle, O. G. Calderón, E. Cabrera-Granado, and M. R. Singh, “Optical pumping of a single hole spin in a p-doped quantum dot coupled to a metallic nanoparticle,” Phys. Rev. B 87, 195303 (2013). [CrossRef]
24. A. O. Govorov, “Semiconductor-metal nanoparticle molecules in a magnetic field: Spin-plasmon and exciton-plasmon interactions,” Phys. Rev. B 82, 155322 (2010). [CrossRef]
25. J. T. Zhang, Y. Tang, K. Lee, and M. Ouyang, “Tailoring light-matter-spin interactions in colloidal hetero-nanostructures,” Nature (London) 466, 91 (2010). [CrossRef]
26. J.-B. Li, N.-C. Kim, M.-T. Cheng, L. Zhou, Z.-H. Hao, and Q.-Q. Wang, “Optical bistability and nonlinearity of coherently coupled exciton-plasmon systems,” Opt. Express 20, 1856–1861 (2012). [CrossRef] [PubMed]
27. R. D. Artuso, G. W. Bryant, A. Garcia-Etxarri, and J. Aizpurua, “Using local fields to tailor hybrid quantum-dot/metal nanoparticle systems,” Phys. Rev. B 83, 235406 (2011). [CrossRef]
28. J.-B. Li, S. Liang, S. Xiao, M.-D. He, N.-C. Kim, L.-Q. Chen, G.-H. Wu, Y.-X. Peng, X.-Y. Luo, and Z.-P. Guo, “Four-wave mixing signal enhancement and optical bistability of a hybrid metal nanoparticle-quantum dot molecule in a nanomechanical resonator,” Opt. Express 24, 2360–2369 (2016). [CrossRef] [PubMed]
29. A. V. Malyshev and V. A. Malyshev, “Optical bistability and hysteresis of a hybrid metal-semiconductor nanodimer,” Phys. Rev. B 84, 035314 (2011). [CrossRef]
30. B. S. Nugroho, A. A. Iskandar, V. A. Malyshev, and J. Knoester, “Bistable optical response of a nanoparticle heterodimer: Mechanism, phase diagram, and switching time,” J. Chem. Phys. 139, 014303 (2013). [CrossRef] [PubMed]
31. B. S. Nugroho, V. A. Malyshev, and J. Knoester, “Tailoring optical response of a hybrid comprising a quantum dimer emitter strongly coupled to a metallic nanoparticle,” Phys. Rev. B 92, 165432 (2015). [CrossRef]
32. M. A. Antón, F. Carreño, S. Melle, O. G. Calderón, E. Cabrera-Granado, J. Cox, and M. R. Singh, “Plasmonic effects in excitonic population transfer in a driven semiconductor-metal nanoparticle hybrid system,” Phys. Rev. B 86, 155305 (2012). [CrossRef]
33. E. Paspalakis, S. Evangelou, and A. F. Terzis, “Control of excitonic population inversion in a coupled semiconductor quantum dot-metal nanoparticle system,” Phys. Rev. B 87, 235302 (2013). [CrossRef]
34. S. M. Sadeghi, “Plasmonic metaresonances: Molecular resonances in quantum dot-metallic nanoparticle conjugates,” Phys. Rev. B 79, 233309 (2009). [CrossRef]
35. W.-X. Yang, A.-X. Chen, Z. Huang, and R.-K. Lee, “Ultrafast optical switching in quantum dot-metallic nanoparticle hybrid systems,” Optics Express 23, 13032 (2015). [CrossRef] [PubMed]
36. S. M. Sadeghi, W. J. Wing, and R. R. Gutha, “Undamped ultrafast pulsation of plasmonic fields via coherent exciton-plasmon coupling,” Nanotechnology 26, 085202 (2015). [CrossRef] [PubMed]
37. R.-C. Ge, C. Van Vlack, P. Yao, Jeff. F. Young, and S. Hughes, “Accessing quantum nanoplasmonics in a hybrid quantum dot-metal nanosystem: Mollow triplet of a quantum dot near a metal nanoparticle,” Phys. Rev. B 87, 205425 (2013). [CrossRef]
38. J. Hakami, L. Wang, and M. S. Zubairy, “Spectral properties of a strongly coupled quantum-dot-metal-nanoparticle system,” Phys. Rev. A 89, 053835 (2014). [CrossRef]
39. F. Carreño, M. A. Antón, V. Yannopapas, and E. Paspalakis, “Resonance fluorescence spectrum of a Λ-type quantum emitter close to a metallic nanoparticle,” Phys. Rev. A 94, 013834 (2016). [CrossRef]
40. F. Carreño, M. A. Antón, and F. Arrieta-Yáñez, “Resonance fluorescence spectrum of a p-doped quantum dot coupled to a metallic nanoparticle,” Phys. Rev. B 88, 195303 (2013). [CrossRef]
41. Y. Gu, L. Huang, O. J. F. Martin, and Q. Gong, “Resonance fluorescence of single molecules assisted by a plasmonic structure,” Phys. Rev. B 81, 193103 (2010). [CrossRef]
42. S. M. Sadeghi and R. G. West, “Coherent control of Forster energy transfer in nanoparticle molecules: energy nanogates and plasmonic heat pulses,” J. Phys.: Condens. Matter 23, 425302 (2011).
43. T. Pons, I. Medintz, K. Sapsford, S. Higashiya, A. Grimes, D. English, and H. Mattoussi, “On the quenching of semiconductor quantum dot photoluminescence by proximal gold nanoparticles,” Nano Lett. 7, 3157 (2007). [CrossRef] [PubMed]
44. C. A. Marocico and J. Knoester, “Effect of surface-plasmon polaritons on spontaneous emission and intermolecular energy-transfer rates in multilayered geometries,” Phys. Rev. A 84, 053824 (2011). [CrossRef]
45. J.-Y. Yan, W. Zhang, S.-Q. Duan, and X.-G. Zhao, “Plasmon-enhanced midinfrared generation from difference frequency in semiconductor quantum dots,” J. Appl. Phys. 103, 104314 (2008). [CrossRef]
46. R. D. Artuso and G. W. Bryant, “Quantum dot-quantum dot interactions mediated by a metal nanoparticle: Towards a fully quantum model,” Phys. Rev. B 87, 125423 (2013). [CrossRef]
47. L. Hayati, C. Lane, B. Barbiellini, A. Bansil, and H. Mosallaei, “Self-consistent scheme for optical response of large hybrid networks of semiconductor quantum dots and plasmonic metal nanoparticles,” Phys. Rev. B 93, 245411 (2016). [CrossRef]
48. A. Ridolfo, O. Di Stefano, N. Fina, R. Saija, and S. Savasta, “Quantum plasmonics with quantum dot-metal nanoparticle molecules: influence of the Fano effect on photon statistics,” Phys. Rev. Lett. 105, 263601 (2010). [CrossRef]
49. K. Kyhm, K.-C. Je, and R. A. Taylor, “Amplified all-optical polarization phase modulator assisted by a local surface plasmon in Au-hybrid CdSe quantum dots,” Opt. Express 20, 19735–19743 (2012). [CrossRef] [PubMed]
50. R. J. McMillan, L. Stella, and M. Grüning, “Projected equations of motion approach to hybrid quantum/classical dynamics in dielectric-metal composites,” Phys. Rev. B 94, 125312 (2016). [CrossRef]
51. R. D. Artuso and G. W. Bryant, “Strongly coupled quantum dot-metal nanoparticle systems: Exciton-induced transparency, discontinuous response, and suppression as driven quantum oscillator effects,” Phys. Rev. B 82, 195419 (2010). [CrossRef]
52. D. Ratchford, F. Shafiei, S. Kim, S. K. Gray, and X. Li, “Manipulating coupling between a single semiconductor quantum dot and single gold nanoparticle,” Nano Lett. 11, 1049–1054 (2011). [CrossRef] [PubMed]
53. W. X. Yang, X. T. Xie, A. X. Chen, Z. Huang, and R. K. Lee, “Coherent control of high-order-harmonic generation via tunable plasmonic bichromatic near fields in a metal nanoparticle,” Phys. Rev. A 93, 053806 (2016). [CrossRef]
54. S. M. Sadeghi, W. J. Wing, and R. R. Gutha, “Control of plasmon fields via irreversible ultrafast dynamics caused by interaction of infrared laser pulses with quantum-dot-metallic-nanoparticle molecules,” Phys. Rev. A 92, 023808 (2015). [CrossRef]
55. A. Hatef, D. G. Schindel, and M. R. Singh, “Dipole-dipole interaction in a quantum dot and metallic nanorod hybrid system,” Appl. Phys. Lett. 99, 181106 (2011). [CrossRef]
56. M.-T. Cheng, S.-D. Liu, and Q.-Q. Wang, “Modulating emission polarization of semiconductor quantum dots through surface plasmon of metal nanorod,” Appl. Phys. Lett. 92, 162107 (2008). [CrossRef]
57. S. G. Kosionis, A. F. Terzis, S. M. Sadeghi, and E. Paspalakis, “Optical response of a quantum dot-metal nanoparticle hybrid interacting with a weak probe field,” J. Phys.: Condens. Matter 25, 045304 (2012).
58. X. N. Liu, D. Z. Yao, H. M. Zhou, F. Chen, and G. G. Xiong, “Third-order nonlinear optical response in quantum dot-metal nanoparticle hybrid structures,” Appl. Phys. B 113, 603 (2013). [CrossRef]
59. A. Hatef, S. M. Sadeghi, and M. R. Singh, “Plasmonic electromagnetically induced transparency in metallic nanoparticle-quantum dot hybrid systems,” Nanotechnology 23, 065701 (2012). [CrossRef] [PubMed]
60. Y. He and K.-D. Zhu, “Strong coupling among semiconductor quantum dots induced by a metal nanoparticle,” Nano. Res. Lett. 7, 95 (2012). [CrossRef]
61. S. M. Sadeghi, “Gain without inversion in hybrid quantum dot-metallic nanoparticle systems,” Nanotechnology 21, 455401 (2010). [CrossRef] [PubMed]
62. M. E. Tasgin, “Metal nanoparticle plasmons operating within a quantum lifetime,” Nanoscale 5, 8616 (2013). [CrossRef] [PubMed]
63. Z. Lu and K.-D. Zhu, “Slow light in an artificial hybrid nanocrystal complex,” J. Phys. B 42, 015502 (2008). [CrossRef]
64. J.-B. Li, M.-D. He, and L.-Q. Chen, “Four-wave parametric amplification in semiconductor quantum dot-metallic nanoparticle hybrid molecules,” Opt. Express 22, 24734–24741 (2014). [CrossRef] [PubMed]
65. Y. He, J. J. Li, and K. D. Zhu, “A tunable optical response of a hybrid semiconductor quantum dot-metal nanoparticle complex in the presence of optical excitations,” J. Opt. Soc. Am. B 29, 997 (2012). [CrossRef]
66. Z. H. Xiao, L. Zheng, and H. Lin, “Photoinduced diffraction grating in hybrid artificial molecule,” Opt. Express 20, 1219–1229 (2012). [CrossRef] [PubMed]
67. S. M. Sadeghi, “Coherent control of metallic nanoparticles near fields: Nanopulse controllers and functional nanoamplifiers,” Phys. Rev. B 82, 035413 (2010). [CrossRef]
68. F. Carreño, M. A. Antón, S. Melle, O. G. Calderón, E. Cabrera-Granado, J. Cox, M. R. Singh, and A. Egatz-Gómez, “Plasmon-enhanced terahertz emission in self-assembled quantum dots by femtosecond pulses,” J. Appl. Phys. 115, 064304 (2014). [CrossRef]
69. H. Wang and K. D. Zhu, “Coherent optical spectroscopy of a hybrid nanocrystal complex embedded in a nanomechanical resonator,” Opt. Express 18, 16175–16182 (2010). [CrossRef] [PubMed]
70. H. J. Chen and K. D. Zhu, “Surface plasmon enhanced sensitive detection for possible signature of majorana fermions via a hybrid semiconductor quantum Dot-Metal nanoparticle system,” Sci. Rep. 5, 13518 (2015). [CrossRef] [PubMed]
71. E. Paspalakis, S. Evangelou, S. G. Kosionis, and A. F. Terzis, “Strongly modified four-wave mixing in a coupled semiconductor quantum dot-metal nanoparticle system,” J. Appl. Phys. 115, 083106 (2014). [CrossRef]
72. Z. Lu and K. Zhu, “Enhancing Kerr nonlinearity of a strongly coupled exciton-plasmon in hybrid nanocrystal molecules,” J. Phys. B 41, 185503 (2008). [CrossRef]
73. X. Feng, Y. Chen, and D. Hou, “Optical nonlinearity enhanced by metal nanoparticle in CdTe quantum dots,” Phys. B: Condens. Matter 406, 1702 (2011). [CrossRef]
74. W. Zhang, A. O. Govorov, and G. W. Bryant, “Semiconductor-metal nanoparticle molecules: Hybrid excitons and the nonlinear Fano effect,” Phys. Rev. Lett. 97, 146804 (2006). [CrossRef] [PubMed]
75. J.-Y. Yan, W. Zhang, S. Duan, X.-G. Zhao, and A. O. Govorov, “Optical properties of coupled metal-semiconductor and metal-molecule nanocrystal complexes: Role of multipole effects,” Phys. Rev. B 77, 165301 (2008). [CrossRef]
76. S. P. Tewari, “Berry’s phase in a two-level atom,” Phys. Rev. A 39, 6082 (1989). [CrossRef]
77. A. C. Aguiar Pinto, M. Moutinho, and M. T. Thomaz, “Berry’s phase in the two-level model,” Braz. J. Phys. 39, 326 (2009). [CrossRef]
78. M. T. Thomaz, A. C. Aguiar Pinto, and M. Moutinho, “Phases of the electronic two-level model under rotating wave approximation,” Phys. Scr. 86, 025001 (2012). [CrossRef]
79. F. Yang and R.-B. Liu, “Berry phases of quantum trajectories of optically excited electron-hole pairs in semiconductors under strong terahertz fields,” New J. Phys. 15, 115005 (2013). [CrossRef]
80. P. J. Leek, J. M. Fink, A. Blais, R. Bianchetti, M. Goppl, J. M. Gambetta, D. I. Schuster, L. Frunzio, R. J. Schoelkopf, and A. Wallraff, “Observation of Berry’s phase in a solid-state qubit,” Science 318, 1889 (2007). [CrossRef] [PubMed]
81. X. X. Yi, L. C. Wang, and T. Y. Zheng, “Berry phase in a composite system,” Phys. Rev. Lett. 92, 150406 (2004). [CrossRef] [PubMed]
82. P. V. Mironova, M. A. Efremov, and W. P. Schleich, “Berry phase in atom optics,” Phys. Rev. A 87, 013627 (2013). [CrossRef]
83. M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. London, Ser. A 392, 45 (1984). [CrossRef]
84. S. Joshi and S. R. Jain, “Geometric phase for neutrino propagation in magnetic field,” Phys. Lett. B 754, 135 (2016). [CrossRef]
85. E. D. Palik, Handbook of Optical Constant of Solids (Academic, 1985).
86. M. Koch, J. Feldmann, G. von Plessen, E. O. Göbel, P. Thomas, and K. Köhler, “Quantum beats versus polarization interference: An experimental distinction,” Phys. Rev. Lett. 69, 3631 (1992). [CrossRef] [PubMed]
