Optical bistability and nonlinearity of coherently coupled exciton-plasmon systems

Jian-Bo Li; Nam-Chol Kim; Mu-Tian Cheng; Li Zhou; Zhong-Hua Hao; Qu-Quan Wang

doi:10.1364/OE.20.001856

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)).

Fig. 1 (a) Schematic diagram of the hybrid system driven by a strong control field with amplitude E_c and frequency ω_c and probed by a weak signal field with amplitude E_s and frequency ω_s (E_c >>E_s). r and R are the radii of SQD and MNP, respectively. d is the surface-to-surface distance between SQD and MNP. ε₀, ε_s and ε_m are the dielectric constants of the background medium, SQD and MNP, respectively. (b) Energy level diagram of the system. ω₁₀ and ω_sp are the frequencies of exciton and surface plasmon, respectively.

Download Full Size | PDF

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

\hat{H} = ℏ Δ {\hat{σ}}_{z} - μ ({\tilde{E}}_{S Q D} {\hat{σ}}_{01} + {\tilde{E}}_{S Q D}^{*} {\hat{σ}}_{10}),

where Δ = ω₁₀ - ω_c is the difference between the exciton frequency ω₁₀ and the control field frequency ω_c.

{\hat{σ}}_{i j}

(i, j = 0,1) is the dipole transition operator between |i〉 and |j〉 of SQD,

{\hat{σ}}_{z} = ({\hat{σ}}_{11} - {\hat{σ}}_{00}) / 2

. µ is the interband dipole moment element. Ẽ_SQD and Ẽ_MNP are the total electric field felt by the SQD and MNP, respectively. Here, Ẽ_SQD = E_c + E_se^-iδt + S_αP_MNP /[ε_eff (d + r + R)³], Ẽ_MNP = E_c + E_se^-iδt + S_αP_SQD /[ε_eff (d + r + R)³] [9,31]. ε_eff = (2ε₀ + ε_s)/3, where ε₀ and ε_s are the dielectric constants of the background medium and SQD, respectively. δ = ω_s – ω_c is the frequency difference of the signal field and the control field. Geometric factor S_α is equal to 2 (−1) when E is parallel to the z (x, y) axis. The z direction corresponds to the axis of the hybrid system. The dipole of the MNP P_MNP = α_MNPẼ_MNP derives from the charge induced on the surface of the MNP, where α_MNP = ε₀R³(ε_m-ε₀)/(2ε₀ + ε_m) [12], and ε_m is the dielectric constant of the Au nanoparticle [32]. The dipole moment of the SQD is P_SQD = µσ₁₀ [33]. Based on the Heisenberg equation of motion, if we set p = µσ₁₀ and w = 2σ_z, and ignore thequantum properties of

{\hat{σ}}_{10}

and

{\hat{σ}}_{z}

[34], then the equations give

\dot{p} = - (1 / T_{2} + i Δ) p - i μ^{2} {\tilde{E}}_{S Q D} w / ℏ \dot{w} = - (w + 1) / T_{1} + 4 Im (p {\tilde{E}}_{S Q D}^{*}) / ℏ,

where T₁ and T₂ are the exciton lifetime and the exciton dephasing time, respectively. Ẽ_SQD = A(E_c + E_se^-iδt) + Bp [31], where A = 1 + S_αα_MNP /[ε_eff (d + r + R)³] and B = S_α²α_MNP /[ε_eff²(d + r + R)⁶]. To obtain a steady-state solution for Eq. (2), we make the ansatz [33,35] p(t) = p₀ + p₁e^-iδt + p_-1e^iδt, w(t) = w₀ + w₁e^-iδt + w_-1e^iδt, where p₀, p₁ and p_-₁ are the dipole moments of the SQD oscillating at frequencies ω_c, ω_s and 2ω_c - ω_s, respectively. Similarly, w₀, w₁ and w_-₁ are the population inversions oscillating at frequencies ω_c, ω_s and 2ω_c - ω_s, respectively. Here, p_-₁ represents a wave-mixing response, because it gives rise to generation of an optical wave with frequency 2ω_c - ω_s. We assume that the other terms are small in the sense that |p₀|>>|p₁|, |p_-1| and |w₀|>>|w₁|, |w_-1|. Upon working to the lowest order in E_s but to all orders in E_c, we can get p_-1, which are related to the effective nonlinear optical susceptibility as follows:

χ_{e f f}^{(3)} = \frac{N p_{- 1}}{3 ε_{0} E_{c}^{2} E_{s}^{*}} = \frac{2 N A^{3} μ^{4} T_{2}^{3} w_{0}}{3 ε_{0} ℏ^{3}} \cdot \frac{(i - Δ_{c}) (2 i - δ_{c}) [i + (Δ_{c} + B_{c} w_{0})]}{[i - (Δ_{c} + B_{c} w_{0})] D (δ_{c})},

where

\begin{matrix} D (δ_{c}) = (δ_{c} - i T_{2} / T_{1}) [1 + {(Δ_{c} + B_{c} w_{0})}^{2}] [{(δ_{c} - i)}^{2} - {(Δ_{c} + B_{c} w_{0})}^{2}] \\ + 4 A^{2} Ω_{c}^{2} [(i - δ_{c}) (1 + Δ_{c}^{2}) - B_{c} w_{0} (i B_{c} w_{0} + Δ_{c} δ_{c})] . \end{matrix}

Here, N is the number density of the hybrid system, Ω_c² = µ²|E_c|²T₂² /ћ² is the generalized intensity of the control field, δ_c = δT₂, Δ_c = ΔT₂ and B_c = µ²BT₂ /ћ². Imχ_eff⁽³⁾ and Reχ_eff⁽³⁾ represent the nonlinear absorption and refraction, respectively. The population inversion of the exciton w₀ is determined by the cubic equation

w_{0} = - 4 A^{2} Ω_{c}^{2} (T_{1} / T_{2}) w_{0} / [1 + {(Δ_{c} + B_{c} w_{0})}^{2}] - 1 .

The Eq. (5) is of the third order in w₀ 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_c². 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 ε₀ = 2.25, ε_s = 6, T₁ = 0.8 ns, T₂ = 0.3 ns and µ = 10⁻²⁸ C∙m [9], and N = 10¹⁴ 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 (ω₁₀ - ω_c)T₂. 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 (ω₁₀ - ω_c)T₂ 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_c corresponding to the stimulated Rayleigh resonance.

Fig. 2 Nonlinear absorption Imχ_eff⁽³⁾ (a) and refraction Reχ_eff⁽³⁾ (b) as a function of the detuning (ω_s - ω_c)T₂ with d = 15 nm and (ω₁₀ - ω_c)T₂ = 0, 1, 2, 5.

Download Full Size | PDF

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 w₀ 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_b₀ ≤ d ≤ d_b₁ (d_b₀ and d_b₁ are the critical values that the optical bistable response exists), population inversion w₀ versus d manifests a standard S-shaped curve, while the induced nonlinear absorption and refraction spectra are exotic coiled curves. When d < d_b₀ or d > d_b₁, the bistability effect disappears. The nonlinear refraction of SQD-MNP system compared with SQD system is obviously enhanced. Speaking clearly, the maximum value of |Reχ_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.

Fig. 3 (a) Population inversion w₀ as a function of interparticle distance between SQD and MNP d for R = 5, 7.5, 10 nm with ω_c = ω_s and (ω₁₀ - ω_c)T₂ = 1. (b) Nonlinear absorption Imχ_eff⁽³⁾ versus d. (c) Imχ_eff⁽³⁾ versus d for R = 7.5 nm in the bistability region of (b). (d) Nonlinear refraction Reχ_eff⁽³⁾ versus d. (e) Reχ_eff⁽³⁾ versus d for R = 7.5 nm in the bistability region of (d).

Download Full Size | PDF

Figure 4(a) shows the bistability of exciton population on intensity of control field Ω_c² for R = 7.5 nm and d = 12.8 nm. Population inversion w₀ versus Ω_c² 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 Imχ_eff⁽³⁾ and Reχ_eff⁽³⁾ behave in a different manner, and display bistable curves (see Figs. 4b–4e). Obviously, the nonlinear absorption and refraction response (Imχ_eff⁽³⁾ and Reχ_eff⁽³⁾) could be used to manifest optical bistability and optical hysteresis. Out of the bistability region, the maximum value of |Reχ_eff⁽³⁾| of the hybrid system is 2.8 times that of SQD system.

Fig. 4 (a) Population inversion w₀ as a function of control field intensity Ω_c² for R = 7.5 nm and d = 12.8 nm with ω_c = ω_s and (ω₁₀ - ω_c)T₂ = 1. (b) Nonlinear absorption Imχ_eff⁽³⁾ versus Ω_c². (c) Imχ_eff⁽³⁾ versus Ω_c² in the bistability region of (b). (d) Nonlinear refraction Reχ_eff⁽³⁾ versus Ω_c². (e) Reχ_eff⁽³⁾ versus Ω_c² in the bistability region of (d).

Download Full Size | PDF

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).

References and links

1. D. E. Chang, A. S. Sørensen, P. R. Hemmer, and M. D. Lukin, “Quantum optics with surface plasmons,” Phys. Rev. Lett. 97(5), 053002 (2006). [CrossRef] [PubMed]

2. N. T. Fofang, N. K. Grady, Z. Y. Fan, A. O. Govorov, and N. J. Halas, “Plexciton dynamics: exciton-plasmon coupling in a J-Aggregate-Au nanoshell complex provides a mechanism for nonlinearity,” Nano Lett. 11(4), 1556–1560 (2011). [CrossRef] [PubMed]

3. A. O. Govorov and I. Carmeli, “Hybrid structures composed of photosynthetic system and metal nanoparticles: plasmon enhancement effect,” Nano Lett. 7(3), 620–625 (2007). [CrossRef] [PubMed]

4. S. Kühn, U. Håkanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97(1), 017402 (2006). [CrossRef] [PubMed]

5. 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(5), 984–994 (2006). [CrossRef]

6. A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature 450(7168), 402–406 (2007). [CrossRef] [PubMed]

7. Y. Fedutik, V. V. Temnov, O. Schöps, U. Woggon, and M. V. Artemyev, “Exciton-plasmon-photon conversion in plasmonic nanostructures,” Phys. Rev. Lett. 99(13), 136802 (2007). [CrossRef] [PubMed]

8. G. Y. Chen, Y. N. Chen, and D. S. Chuu, “Spontaneous emission of quantum dot excitons into surface plasmons in a nanowire,” Opt. Lett. 33(19), 2212–2214 (2008). [CrossRef] [PubMed]

9. W. Zhang, A. O. Govorov, and G. W. Bryant, “Semiconductor-metal nanoparticle molecules: hybrid excitons and the nonlinear fano effect,” Phys. Rev. Lett. 97(14), 146804 (2006). [CrossRef] [PubMed]

10. 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 (R)(2011).

11. S. M. Sadeghi, “Plasmonic metaresonances: molecular resonances in quantum dot-metallic nanoparticle conjugates,” Phys. Rev. B 79(23), 233309 (2009). [CrossRef]

12. 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(15), 2125–2127 (2007). [CrossRef] [PubMed]

13. S. M. Sadeghi, “The inhibition of optical excitations and enhancement of Rabi flopping in hybrid quantum dot-metallic nanoparticle systems,” Nanotechnology 20(22), 225401 (2009). [CrossRef] [PubMed]

14. J. Y. Yan, W. Zhang, S. Duan, and X. G. Zhao, “Plasmon-enhanced midinfrared generation from difference frequency in semiconductor quantum dots,” J. Appl. Phys. 103(10), 104314 (2008). [CrossRef]

15. 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(23), 235406 (2011). [CrossRef]

16. D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3(11), 807–812 (2007). [CrossRef]

17. S. M. Sadeghi, “Tunable nanoswitches based on nanoparticle meta-molecules,” Nanotechnology 21(35), 355501 (2010). [CrossRef] [PubMed]

18. D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90(2), 027402 (2003). [CrossRef] [PubMed]

19. M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460(7259), 1110–1112 (2009). [CrossRef] [PubMed]

20. R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009). [CrossRef] [PubMed]

21. J. S. White, G. Veronis, Z. F. Yu, E. S. Barnard, A. Chandran, S. H. Fan, and M. L. Brongersma, “Extraordinary optical absorption through subwavelength slits,” Opt. Lett. 34(5), 686–688 (2009). [CrossRef] [PubMed]

22. D. Pacifici, H. J. Lezec, and H. A. Atwater, “All-optical modulation by plasmonic excitation of CdSe quantum dots,” Nat. Photonics 1(7), 402–406 (2007). [CrossRef]

23. H. M. Gong, X. H. Wang, Y. M. Du, and Q. Q. Wang, “Optical nonlinear absorption and refraction of CdS and CdS-Ag core-shell quantum dots,” J. Chem. Phys. 125(2), 024707 (2006). [CrossRef] [PubMed]

24. H. I. Elim, W. Ji, J. Yang, and J. Y. Lee, “Intensity-dependent enhancement of saturable absorption in PbS–Au₄ nanohybrid composites: evidence for resonant energy transfer by Auger recombination,” Appl. Phys. Lett. 92(25), 251106 (2008). [CrossRef]

25. X. Zhang and G. G. Xiong, “Metal nanoparticle-induced variation of nonlinear optical susceptibility of a CdTe semiconductor quantum dot,” Physica E 41(7), 1258–1262 (2009). [CrossRef]

26. Z. E. Lu and K. D. Zhu, “Enhancing Kerr nonlinearity of a strong coupled exciton-plasmon in hybrid nanocrystal molecules,” J. Phys. B 41(18), 185503 (2008). [CrossRef]

27. V. A. Malyshev, H. Glaeske, and K.-H. Feller, “Absence of bistable behavior in the optical response of a dimer,” Phys. Rev. A 58(2), 1496–1500 (1998). [CrossRef]

28. R. D. Artuso and G. W. Bryant, “Optical response of strongly coupled quantum dot-metal nanoparticle systems: Double peaked Fano structure and bistability,” Nano Lett. 8(7), 2106–2111 (2008). [CrossRef] [PubMed]

29. 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(19), 195419 (2010). [CrossRef]

30. A. V. Malyshev and V. A. Malyshev, “Optical bistability and hysteresis of a hybrid metal-semiconductor nanodimer,” Phys. Rev. B 84(3), 035314 (2011). [CrossRef]

31. H. Wang and K. D. Zhu, “Coherent optical spectroscopy of a hybrid nanocrystal complex embedded in a nanomechanical resonator,” Opt. Express 18(15), 16175–16182 (2010). [CrossRef] [PubMed]

32. E. D. Palik, Handbook of Optical Constants of Solids (Academic, Orlando, Fla., 1985).

33. R. W. Boyd, Nonlinear Optics (Academic, New York, 2008).

34. B. I. Greene, J. F. Mueller, J. Orenstein, D. H. Rapkine, S. Schmitt-Rink, and M. Thakur, “Phonon-mediated optical nonlinearity in polydiacetylene,” Phys. Rev. Lett. 61(3), 325–328 (1988). [CrossRef] [PubMed]

35. R. W. Boyd, M. G. Raymer, P. Narum, and D. J. Harter, “Four-wave parametric interactions in a strongly driven two-level system,” Phys. Rev. A 24(1), 411–423 (1981). [CrossRef]

Optical bistability and nonlinearity of coherently coupled exciton-plasmon systems

Abstract

1. Introduction

2. Model and formalism

3. Results and discussion

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (5)

Optics Express