Four-wave parametric amplification in semiconductor quantum dot-metallic nanoparticle hybrid molecules

Jian-Bo Li; Meng-Dong He; Li-Qun Chen

doi:10.1364/OE.22.024734

1. Introduction

The optical properties of hybrid molecules composed of metallic nanostructures and semiconductor quantum dots (SQDs) have attracted much interest in recent years [1–8]. Based on the exciton-plasmon interaction, such hybrid molecules exhibit many interesting optical phenomena, such as nonlinear Fano effect [9], exciton-plasmon-photon conversion [10], superradiance [11], gain without inversion [12], intensity-dependent enhancement of saturable absorption [13], slow light effect [14,15], plasmon-enhanced terahertz emission and ac-Stark shifts [16,17], and photoinduced diffraction grating [18]. Taking SQD-metallic nanoparticle (MNP) hybrid nanosystem for example, Artuso et al. found the double peaked Fano structure and bistability when the SQD couples strongly to the MNP [19]. Malyshev et al. showed that optical bistability and hysteresis in a hybrid SQD-MNP nanodimer can be revealed by measuring the optical hysteresis of Rayleigh scattering [20]. Then Malyshev derived the analytical bistability condition in such a system [21]. Singh predicted that the second-harmonic generation (SHG) signals from the QD and MNP in the SQD-MNP hybrid system can be switched on and off by applying an external control field [22]. Based on these optical properties, the SQD-MNP hybrid molecules can be used to develop single-photon transistor [23], nanosensor [24], optical modulator [25], and photodetector [26].

As an attractive nonlinear optical phenomenon, four-wave mixing effect is appealing hosts for the investigation of fundamental physics and novel applications. Resulting from four-wave mixing effect, Kerr nonlinearity of the SQD-MNP hybrid molecule can be enhanced via exciton-plasmon interaction [27]. In our previous work, we revealed that the bistable properties of nonlinear absorption and refraction response of four-wave mixing effect in the same system [28]. Recently, Paspalakis et al. studied the four-wave mixing effect in an SQD-MNP system and found that the four-wave mixing spectrum can display a three- or one-peaked structure by adjusting the distance between the SQD and the MNP [29]. However, the parametric amplification by four-wave mixing in these hybrid systems has not been reported yet.

In this paper, we study four-wave parametric amplification that arises in SQD-MNP hybrid molecules. We find that the efficiency of four-wave parametric amplification is strongly modified by the frequency and intensity of the pump field and the interparticle distance. Additionally, the probe-wave gain induced by four-wave parametric amplification is tunable in a large range. It is possible to utilize such high gain to construct a tunable four-wave parametric oscillator.

2. Model and formalism

We consider a hybrid molecule composed of a SQD of radius r and a spherical MNP of radius R in an environment with dielectric constant ε₀ as shown in Figs. 1(a) and 1(b). The surface-to-surface distance between the two nanoparticles is denoted as d. The hybrid molecule is subjected to two applied fields: a strong pump field with amplitude E₀ and frequency ω₀; a weak probe field with amplitude E₁ and frequency ω₁ (E₀ >> E₁). The SQD is regarded as a two-level system which consists of the ground state |0> and the first excited state |1>.

Fig. 1 (a) Schematic diagram of a hybrid molecule composed of a SQD and a MNP interacting with a strong pump field and a weak probe field. (b) Energy-level diagram of a two-level SQD coupled to a MNP. (c) The phase matching condition for collinear propagation in this nonlinear molecular material.

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The Hamiltonian of the SQD in a rotating frame reads as follows [28]

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

where Δ = ω₁₀ - ω₀ is the frequency difference between the exciton and the pump field, ω₁₀ is the frequency of exciton. σ_ij = |i><j| (i, j = 0, 1) is the dipole transition operator between |i> and |j> of the SQD, σ_z = (σ₁₁-σ₀₀)/2. µ₁₀ is the interband dipole moment element. Ẽ_SQD is the total optical field felt by the SQD, Ẽ_SQD = E/ε_eff + S_αP_MNP /[ε₀ε_eff (d + r + R)³] with ε_eff = (2ε₀ + ε_s)/(3ε₀) [30,31], E = E₀ + E₁e^-iδt is the total applied field, where δ = ω₁ – ω₀ is the probe-pump detuning [32,33]. ε_s is the dielectric constant of the SQD. The polarization parameter S_α = 2 (−1) when the applied field is parallel (perpendicular) to the major axis of the hybrid molecule. The dipole of the MNP is P_MNP = α_MNPẼ_MNP where α_MNP = ε₀R³(ε_m-ε₀)/(2ε₀ + ε_m) and Ẽ_MNP = E + S_αP_SQD /[ε₀(d + r + R)³] [30,31,34], and ε_m is the dielectric constant of the MNP [35], P_SQD = µ₁₀σ₁₀ is the dipole moment of the SQD [32]. For simplicity we define p = µσ₁₀ and w = 2σ_z. In terms of the Heisenberg equation of motion dO/dt = -i[O, H]/ћ, we obtain the temporal evolutions of the exciton in the SQD as follows

\dot{p} = - (1 / T_{2} + i Δ) p - i μ_{10}^{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 is the exciton lifetime and T₂ is the exciton dephasing time. Ẽ_SQD = A(E₀ + E₁e^-iδt) + Bp with A = 1/ε_eff + S_αα_MNP/[ε₀ε_eff(d + r + R)³] and B = S_α²α_MNP/[ε₀²ε_eff(d + r + R)⁶]. To obtain a steady-state solution of Eqs. (2) and (3), we make the ansatz p(t) = p₀(ω₀) + p₁(ω₁)e^-iδt + p_-1 (2ω₀ - ω₁)eⁱ^δt, w(t) = w₀(ω₀) + w₁(ω₁)e^-iδt + w_-1(2ω₀- ω₁)eⁱ^δt. p_-1 represents a wave-mixing response, which gives rise to generation of a new optical wave with frequency 2ω₀ - ω₁. Upon working to the lowest order in E₁ but to all orders in E₀, one can obtain

χ_{e f f}^{(1)} (ω_{0} + δ) = \frac{N p_{1}}{ε_{0} E_{1}},

χ_{e f f}^{(3)} (ω_{0} - δ) = \frac{N p_{- 1}}{3 ε_{0} E_{0}^{2} E_{1}^{*}},

p_{- 1} (2 ω_{0} - ω_{1} = ω_{0} - δ) = \frac{2 A^{3} μ_{10}^{4} T_{2}^{3} w_{0}}{ℏ^{3}} \cdot \frac{(i - Δ_{p u}) (2 i - δ_{p r}) [i + (Δ_{p u} + B_{p u} w_{0})]}{[i - (Δ_{p u} + B_{p u} w_{0})] D (δ_{p r})} E_{0}^{2} E_{1}^{*},

p_{- 1} (ω_{1}) = \frac{A μ_{10}^{2} T_{2} w_{0}}{ℏ (δ_{p r} - Δ_{p u} - B_{p u} w_{0} + i)} \cdot [1 - 2 A^{2} Ω_{p u}^{2} \cdot \frac{(i - Δ_{p u}) (2 i + δ_{p r}) (δ_{p r} + Δ_{p u} + B_{p u} w_{0} + i)}{D^{*} (δ_{p r})}] E_{1},

where

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

and δ_pr = δT₂, Δ_pu = ΔT₂, B_pu = µ₁₀²BT₂/ћ, Ω_pu = µ₁₀E₀T₂/ћ is the generalized Rabi frequency of the pump field, N is the number density of SQD-MNP hybrid molecules.

Next we use the nonlinear polarization to study the spatial propagation effects in the hybrid system. The polarization amplitudes at frequencies ω₀ ± δ can be written as [32,33]

P_{1} = P (ω_{0} + δ) = ε_{0} χ_{e f f}^{(1)} (ω_{0} + δ) E_{0} + 3 ε_{0} χ_{e f f}^{(3)} (ω_{0} + δ) E_{0}^{2} E_{- 1}^{*},

P_{- 1} = P (ω_{0} - δ) = ε_{0} χ_{e f f}^{(1)} (ω_{0} - δ) E_{- 1} + 3 ε_{0} χ_{e f f}^{(3)} (ω_{0} - δ) E_{0}^{2} E_{1}^{*} .

Here χ_eff⁽¹⁾(ω₀ - δ) and χ_eff⁽³⁾(ω₀ + δ) can be obtained by formally replacing δ by -δ in Eqs. (4) and (5). E_-1 represents the complex amplitude of the new wave with a frequency ω₀ - δ.

The wave equations are given by

\nabla^{2} E_{1} + k_{1}^{2} E_{1} = {(ω_{0} + δ)}^{2} P_{1} / (ε_{0} c^{2}),

\nabla^{2} E_{- 1} + k_{- 1}^{2} E_{- 1} = {(ω_{0} - δ)}^{2} P_{- 1} / (ε_{0} c^{2}) .

Now it is assumed that the amplitude of the pump wave A₀ is constant, and the probe-field amplitudes A _{± 1} are small variables, the wave equations become

d A_{1} / d z = - α_{1} A_{1} + κ_{1} A_{- 1}^{*} E x p [i Δ k z],

d A_{- 1} / d z = - α_{1} A_{- 1} + κ_{- 1} A_{1}^{*} E x p [i Δ k z] .

Here k _{± 1} = n _{± 1}(ω₀ ± δ)/c denote the propagation constants, where n _{± 1} = (1 + Reχ_eff⁽¹⁾(ω₀ ± δ))^1/2 are the corresponding refraction indexes, c is the speed of light, k₀ is the propagation constant of the pump wave at frequency ω₀. According to four-wave mixing effect, the wavevector mismatch along the z direction can be expressed as

Δ \vec{k} = 2 {\vec{k}}_{0} - {\vec{k}}_{1} - {\vec{k}}_{- 1}

, as shown in Fig. 1(c). The nonlinear absorption and coupling coefficients are given by [32]

α_{\pm 1} = - \frac{1}{2} (\frac{ω_{0} \pm δ}{n_{\pm 1} c}) {Im}_{e f f}^{(1)} (ω_{0} \pm δ),

κ_{\pm 1} = - i \frac{3}{2} (\frac{ω_{0} \pm δ}{n_{\pm 1} c}) χ_{e f f}^{(3)} (ω_{0} \pm δ) A_{0}^{2} .

From Eqs. (13)–(16), we obtain the expressions of E₁ and E_-₁

\begin{array}{l} A_{1} (z) = {(η_{+} - η_{-})}^{- 1} {[κ_{1} A_{- 1}^{*} (0) - (η_{-} + α_{1} + i Δ k / 2) A_{1} (0)] e^{η_{+} z} \\ + [- κ_{1} A_{- 1}^{*} (0) + (η_{+} + α_{1} + i Δ k / 2) A_{1} (0)] e^{η_{-} z}} e^{i Δ k z / 2}, \end{array}

\begin{array}{l} A_{- 1}^{*} (z) = {(η_{+} - η_{-})}^{- 1} {[(η_{+} + α_{1} + i Δ k / 2) A_{- 1}^{*} (0) + κ_{- 1}^{*} A_{1} (0)] e^{η_{+} z} \\ - [(η_{-} + α_{1} + i Δ k / 2) A_{- 1}^{*} (0) + κ_{- 1}^{*} A_{1} (0)] e^{η_{-} z}} e^{- i Δ k z / 2} . \end{array}

where A₁(0) and A_-1(0) denote the field amplitudes at the boundary z = 0 of nonlinear molecular material. η _± are given by

η_{\pm} = - \frac{1}{2} (α_{1} + α_{- 1}^{*}) \pm {[(α_{1} + α_{- 1}^{*} + i Δ k) + 4 κ_{1} κ_{- 1}^{*}]}^{1 / 2} .

To clearly reflect the parametric amplification process of four-wave mixing, we define a gain factor for the probe-wave G = |A₁(L)/A₁(0)| in term of Eq. (17), where L denotes the length of nonlinear molecular material.

3. Results and discussion

For numerical calculations, we consider a hybrid system composed of CdSe QD-Au NP molecules. For CdSe QD, we take ε_s = 6ε₀, T₁ = 0.8 ns, T₂ = 0.3 ns and µ₁₀ = 10⁻²⁸ C·m, hω₁₀ = 2.5 eV. For Au NP, the broad plasmon frequency is near the exciton resonant frequency and the radius R is taken as 7.5 nm [16]. Other parameters are N = 10²⁰/m³, L = 10μm, S_α = −1. In the following, we assume A₁(0) = A_-1(0) and consider the case of perfect phase matching (Δk = 0).

We are mainly interested in the probe-wave gain achieved by four-wave parametric amplification in the hybrid system. Before investigating the probe-wave gain, we first explore the four-wave mixing response. Figures 2(a) and 2(b) show the variation of four-wave mixing response |p_-1(2ω₀-ω₁)| as a function of the probe-pump detuning δ_pr for d = 15 nm and Ω_pu² = 20. In Fig. 2(a), we divide the region of mixing response spectra into three parts corresponding to three-photon resonance (TP), Rayleigh resonance (RL) and ac-Stark resonance (AC) [28,33]. For Δ_pu ≤ 0, the magnitudes of the peaks attributed to three-photon resonance and ac-Stark resonance decrease as Δ_pu increases. In Fig. 2(b), for Δ_pu > 0, we obtain an opposite result with further increase of Δ_p. Additionally, the magnitude of the peak induced by Rayleigh resonance changes relatively little when Δ_pu ≤ 0 (Δ_pu > 0). These results suggest that the role of three resonance mechanisms will change as Δ_pu varies. Moreover, we find that these mixing response spectra are symmetrical about the axis δ_pr = 0. Such behavior is further revealed in Fig. 2(c). Similar curves have also been observed by Paspalakis et al. in a metal-semiconductor nanosystem [29].

Fig. 2 (a) The mixing response spectrum as a function of probe-pump detuning δ_pr when (a) Δ_pu = −6, −4, −2, 0 (b) Δ_pu = 2.8, 4, 6. The positions of the peaks induced by three-photon resonance (TP), Rayleigh resonance (RL) and ac-Stark resonance (AC) as a function of Δ_pu. Other parameters are d = 15 nm and Ω_pu² = 20.

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To estimate the impact of the frequencies of the pump and probe fields on four-wave parametric amplification, Fig. 3 shows how the probe-wave gain G changes with δ_pr at perfect phase matching Δk = 0. When the system is excited with a constant pumping (Ω_pu² = 20), the peak of G, depicted in Figs. 3(a) and 3(b), reaches its largest value at the position of three-photon resonance, so this gain can be assigned to the three-photon gain. The gain peak moves to higher δ_pr with the increasing of Δ_pu. For Δ_pu ≤ 0, the maximal gain G_max is 137121, 110012, 77750.9, 42667.9 when Δ_pu = −6, 4, 2, 0, respectively. While for Δ_pu > 0, G_max becomes relatively small, and G_max is equal to 19.2, 67.6, 425.1 when Δ_pu = 2.8, 4, 6, respectively. It is obvious that the parametric amplification process becomes rather significant when Δ_pu = −6, and the corresponding G_max is 322 times larger than that in the case of Δ_pu = 6.

Fig. 3 (a) The probe-wave gain G as a function of δ_pr when (a) Δ_pu ≤ 0 and (b) Δ_pu > 0. All other parameters are the same as those in Fig. 2.

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In the following we mainly consider the case of Δ_pu = −6. To better visualize the dependence of mixing response |p_-1(2ω₀-ω₁)| on the interparticle distance d, we plot |p_-1(2ω₀-ω₁)| as a function of d for different probe-pump detuning δ_pr in Fig. 4(a). In our calculations we find that the probe-wave gain G becomes infinite when δ_pr = 0 and Δk = 0. As δ_pr = −18.75, 17.5, |p_-1(2ω₀-ω₁)| reaches a maximum at d = 15.1, 15.4 nm, respectively. When δ_pr = −5, 1 and δ_pr ≈0, |p_-1(2ω₀-ω₁)| increases with the increase of d. The results imply that the hybrid system may undergo an amplification process by four-wave mixing for some specific values of d. In order to clarify it, in Fig. 4(b) we show how the probe-wave gain G changes with d under the same conditions as those in Fig. 4(a). Note that in Fig. 4(b), for δ_pr = −18.75, G almost keeps invariant until d approaches a certain amount, then attains its maximum at d = 15.1 nm (G_max = 137125). As we expected, |p_-1(2ω₀-ω₁)| and G both reach their maximums at the same positions. For δ_pr = −17.5, the gain curve exhibits a sharp peak similar to that in the case of δ_pr = −18.75 and the peak position has a visible shift. While for other four cases, d almost has no effect on the nature of the gain curves which suggests that the parametric amplification process of four-wave mixing is significantly suppressed despite d changes.

Fig. 4 (a) The mixing response spectrum as a function of the interparticle distance d for different values of the probe-pump detuning δ_pr. (b) The probe-wave gain G as a function of d. Other parameters are Δ_pu = −6 and Ω_pu² = 20.

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Figure 5(a) presents the impact of the pump-field intensity Ω_pu² on the mixing response |p_-1(2ω₀-ω₁)|. For δ_pr = −18.75 and −17.5, |p_-1(2ω₀-ω₁)| displays a rather wide peak; when δ_pr = −5, 1 and δ_pr ≈0, |p_-1(2ω₀-ω₁)| slightly increases monotonically as Ω_pu² increases. These results suggest that for the case of G versus Ω_pu², we can expect to obtain similar gain curves shown in Fig. 4(b). In Fig. 5(b), for δ_pr = −18.75, G_max is equal to 142585 when Ω_pu² = 18.3; for δ_pr = −17.5, G_max reduces to 25% that for δ_pr = −18.75 and the gain peak position moves to larger pump-field intensity. While for δ_pr = −5, 1, 18.75 and δ_pr ≈0, these features of the gain peaks are all gone and replaced with four smooth lines (G ≈1). Therefore, we can obtain high efficient four-wave parametric amplification in the hybrid system by purposefully selecting the values of Ω_pu² and δ_pr.

Fig. 5 (a) The mixing response spectrum as a function of the pump-field intensity Ω_pu² for different detuning δ_pr. (b) The probe-wave gain G as a function of Ω_pu². Other parameters are Δ_pu = −6 and d = 15 nm.

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The results presented in this paper are based on SQD-MNP hybrid molecules. Such system exhibits a large probe-wave gain by four-wave parametric amplification, which can be used to develop four-wave parametric oscillators. We have also discussed the dependence of the mixing response spectrum on the interparticle distance and obtained some results similar with that in [29]. These calculated results could be explained by the molecular-like resonances of Sadeghi [12,16]. As far as we know, in a two-level atomic system, the probe-wave gain can reach a maximum 10⁴ [33]. However, our results show that the maximal probe-wave gain in the SQD-MNP hybrid system is approximately 14 times larger than that in the atomic system. As we expected, it is easily to achieve 1.43 × 10⁵ gain under certain experimental conditions in the realistic CdSe QD-Au NP hybrid molecules. Here Δ = −20 ns⁻¹, δ = −62.5 ns⁻¹, and the pump-field intensity I₀ is equal to 2.06 × 10² W/cm².

4. Conclusion

In summary, four-wave parametric amplification in hybrid molecules composed of semiconductor quantum dots and metallic nanoparticles was theoretically studied. We found that the efficiency of four-wave parametric amplification depends strongly on the frequency, intensity of the pump field and the interparticle distance. In particular, the induced probe-wave gain attributed to three-photon effect is tunable in a large range from 1 to 1.43 × 10⁵. Such high gain allows one to utilize the hybrid system to construct four-wave parametric oscillators. We hope that our calculated results in this work can be experimentally demonstrated in the near future.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11404410 and 11174372), the Hunan Provincial Natural Science Foundation (Grant No. 14JJ3116) and the Foundation of Talent Introduction of Central South University of Forestry and Technology (Grant No. 104-0260).

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Four-wave parametric amplification in semiconductor quantum dot-metallic nanoparticle hybrid molecules

Abstract

1. Introduction

2. Model and formalism

3. Results and discussion

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (19)

Optics Express