Analysis of the photon indistinguishability in incoherently excited quantum dots

F. Troiani, J. I. Perea, and C. Tejedor

Phys. Rev. B 73, 035316 – Published 17 January 2006

Abstract

The solid-state single-photon sources so far demonstrated require energy relaxation processes, which tend to spoil the coherent nature of the time evolution and with it the photon indistinguishability. We focus our theoretical investigation on semiconductor quantum dots embedded in microcavities. In the slow-dephasing limit, the photon indistinguishability is shown to decrease with the ratio between nonradiative and radiative emission rates, according to a simple analytical relation. The effect of dephasing is then investigated within a wide range of physical parameters, providing clear indications for the device optimization.

Received 14 October 2005

DOI:https://doi.org/10.1103/PhysRevB.73.035316

Authors & Affiliations

F. Troiani^*, J. I. Perea, and C. Tejedor

Departamento de Fisica Teorica de la Materia Condensada, Universidad Autonoma de Madrid, 28049 Madrid, Spain

^*Corresponding author. Email address: filippo.troiani@uam.es

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Vol. 73, Iss. 3 — 15 January 2006

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Figure 1
Schematics of the dot-cavity system’s relevant states: $∣ G, 0 ⟩, ∣ G, 1 ⟩, ∣ X, 0 ⟩, ∣ X^{*}, 0 ⟩$ . The occupation of the remaining states is negligible throughout the system’s time evolution. In all the calculations presented below, we consider the case of a QD resonantly excited from $∣ G ⟩$ to $∣ X^{*} ⟩$ by means of a Gaussian laser pulse $Ω_{L} (t) = Ω_{0} \exp [(t - t_{0})^{2} ∕ 2 σ^{2}]$ , with $σ = 1.1 ps$ and $Ω_{0} = 0.75 meV$ .Reuse & Permissions
Figure 2
(a)–(d) Distinguishability of the photons emitted by the microcavity $(p_{c}^{M C})$ and by the quantum dot $(p_{c}^{Q D})$ as a function of the overall rate emission $R \equiv Γ_{s} + 2 g^{2} ∕ κ$ normalized to $Γ_{r}$ , in the absence of dephasing $(γ = 0)$ . Different symbols stand for $F_{p} = 1$ (squares), $F_{p} = 10$ (circles), and $F_{p} = 100$ (triangles). Solid lines depict the curve $p_{c} = R ∕ (R + Γ_{r})$ . In each plot, $Γ_{s}$ and either $g$ [panels (c), (d)] or $κ$ [(a), (b)] are varied in such a way that $Γ_{s} = R ∕ (F_{p} + 1)$ and $2 g^{2} ∕ κ = R F_{p} ∕ (F_{p} + 1)$ . The fixed parameters are (a), (b) $κ = 10 {ps}^{- 1}$ , $Γ_{r} = 0.2 {ps}^{- 1}$ (black symbols), and 0.05 (white symbols); (c), (d) $Γ_{r} = 0.1 {ps}^{- 1}$ , $g = 0.2 meV$ (black), and 0.05 (white). (e) Empirical relation between the collection efficiency $β$ and $p_{c}$ for different values of $Γ_{r} ∕ Γ_{s}$ .Reuse & Permissions
Figure 3
Degree of purity and coherence of the dot-cavity system as a function of time. The purity (thick curves) is quantified by $f_{\tilde{ρ}} (t + τ) \equiv 1 - Tr [{\tilde{ρ}}^{2} (t + τ)] ∕ Tr [\tilde{ρ} (t + τ)]^{2}$ , calculated within the subspace ${∣ G, 1 ⟩, ∣ X, 0 ⟩, ∣ X^{*}, 0 ⟩}$ . The coherence (narrow curves) is given by $∣ G_{χ}^{(1)} (t, t + τ) ∣^{2} ∕ G_{χ}^{(1)} (t)$ , corresponding to different initial times $t = t_{i}$ and enveloped by $G_{χ}^{(1)} (t + τ)$ (shaded gray); the times $t_{i}$ are such that $G_{χ}^{(1)} (t_{2, 3}) = 3 G_{χ}^{(1)} (t_{1, 4}) = 0.75 \max {G_{χ}^{(1)} (t)}$ . The labels on the vertical axes refer to $f_{\tilde{ρ}}$ , while the other curves are plotted in arbitrary units. $R ∕ Γ_{r}$ is 1.0 in panel (a) and 0.1 in (b)–(d); $g$ is $2.0 meV$ in panels (a), (b) and 0.05 in (c), (d); $Γ_{r} = 0.1 {ps}^{- 1}$ in (a)–(d).Reuse & Permissions
Figure 4
Cavity-photon distinguishability as a function the rate emission $R$ , for different values of the dephasing rate $γ$ (see the legend) and of the Purcell factor $F_{p}$ (squares, circles, and triangles correspond, respectively, to $F_{p} = 1, 10, 100$ ). Other physical parameters: $Γ_{r} = 0.1 {ps}^{- 1}$ , $g = 0.2 meV$ .Reuse & Permissions
Figure 5
Cavity-photon distinguishability as a function of the nonradiative relaxation and of the dephasing rates, $Γ_{r}$ and $γ$ , for different values of the rate emission $R$ . In all cases $Γ_{s} = 10^{- 3} {ps}^{- 1}$ , while $F_{p} = 100, 10, 32$ in panels (a), (b), and (c), respectively. The dot-cavity coupling constant is $g = 0.1 meV$ in (a), (b) and 0.05 in (c). The behavior of $p_{c}^{M C}$ for $R = 0.033$ and $g = 0.1 meV$ (not shown here) is qualitatively that shown in panel (c), though the values are on average higher.Reuse & Permissions

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