Eigenvalue decomposition method for photon statistics of frequency-filtered fields and its application to quantum dot emitters

Kenji Kamide, Satoshi Iwamoto, and Yasuhiko Arakawa

Phys. Rev. A 92, 033833 – Published 18 September 2015

Abstract

A simple calculation method for photon statistics of frequency-filtered fields is proposed. This method, based on eigenvalue decompositions of superoperators, allows us to study effects on the photon statistics of spectral filtering by various types of filters, such as Gaussian and rectangular filters as well as Lorentzian filters, which is not possible by conventional approaches. As an example, this method is applied to a simulation of quantum dot single-photon emitters, where we found that the efficient choice of the filter types to have pure single photons depends on the excitation conditions, i.e., incoherent or coherent (and resonant) excitations.

Received 19 May 2015

DOI:https://doi.org/10.1103/PhysRevA.92.033833

Authors & Affiliations

Kenji Kamide^1,*, Satoshi Iwamoto^1,2, and Yasuhiko Arakawa^1,2

¹Institute for Nano Quantum Information Electronics (NanoQuine), University of Tokyo, Tokyo 153-8505, Japan
²Institute of Industrial Science, University of Tokyo, Tokyo 153-8505, Japan

^*kamide@iis.u-tokyo.ac.jp

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Vol. 92, Iss. 3 — September 2015

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Images

Figure 1
Quantum emitter and detection system. The emission dynamics of the quantum emitter is given by the quantum master equation, $\frac{d}{d t} \hat{ρ} = L \hat{ρ}$ . The effect of the spectral filter is described by the correlation function $f (τ)$ , and the photons passed through the filter enter into the HBT setup [13] for the $g^{(2)}$ correlation measurement.
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Figure 2
Schematics of the filter function: (a) $f (τ)$ in the time domain and (b) $F (ω)$ in the frequency domain.
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Figure 3
A neutral QD model [14, 26, 27]. (a) Among 16 electronic states at the QD ground levels, six neutral states with up to two excitons (G, BX1, BX2, DX1, DX2, and XX) are taken into account. (b) Incoherent pump ( $P$ ) and decay processes ( $γ_{s p}, γ_{S}^{e}, γ_{S}^{h}$ ).
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Figure 4
(a) The emission spectrum of a QD SP emitter (solid) for XX binding energy $χ = 2000 μ eV$ is shown with the normalized filter functions ${| F (ω) |}^{2}$ (dashed) for Lorentzian (black), Gaussian (green), and rectangular (magenta) filters with $λ = 300 μ eV$ . (b, c) $g^{(2)} (0)$ of the QD emission after spectral filtering by the three types of filters, Lorentzian (black), Gaussian (green), and rectangular (magenta). (b) Filter bandwidth ( $λ$ ) dependence for XX binding energy $χ = 2000 μ eV$ . (c) The $χ$ dependence of $g^{(2)} (0)$ at the optimal filter bandwidth $λ_{opt} [\approx 100 μ eV$ for the Lorentzian filter in (b); $g^{(2)} (0)$ at $λ_{opt}$ are also indicated in (b)]. We set $(γ_{s p}, Γ_{ph}) = (0.67, 20) μ eV$ , small pump rate $P$ in the linear regime [14], and the spin-flip time $τ_{S} \equiv γ_{S} = 10$ ns for all figures.
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Figure 5
Illustration of the single-photon emission from coherently driven emitters (the Rabi frequency $Ω_{R}$ and $ω_{0} \equiv ω_{L} = ω_{X}$ , laser frequency $ω_{L}$ , and the emitter transition frequency $ω_{X}$ ). (a) The bare and dressed energy levels with the radiative transitions marked by arrows corresponding to the emission peaks in (b), and (b) the fluorescence spectrum showing Mollow triplets, a central peak (C) at $ω = ω_{0}$ , and upper (U) and lower (L) side peaks. The $g^{(2)} (0)$ of the upper side peak emission (U) spectrally selected by three types of filters with $ω_{F} = ω_{0} + 2 Ω_{R}$ , Lorentzian (black), Gaussian (green), and rectangular (magenta). (c) $g^{(2)} (0)$ as a function of the normalized filter bandwidth $λ / Ω_{R}$ . (d) $g^{(2)} (0)$ minimized in the range $0 < λ < 2 Ω_{R}$ as a function of the normalized Rabi energy $Ω_{R} / γ_{sp}$ (the values inside brackets and arrows indicate the minima in (c); the upper limit of the range is set in order to avoid the detection of the driving laser light in the shaded area, $λ > 2 Ω_{R}$ ). We set the spontaneous emission $γ_{s p} = 0.3 Ω_{R}$ and dephasing rate $Γ_{ph} = 0$ for (b) and (c), and the dephasing rate $Γ_{ph} = 0$ for all figures.
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Figure 6
$g^{(2)} (0)$ as a function of the filter bandwidth $λ / Ω_{R}$ for a rectangular filter with different decay parameters $(2 Ω_{R} / γ_{s p}, Γ_{ph} / Ω_{R})$ : (100, 0) for the solid line, (100, 0.1) for the dashed line, and (6.7, 0) for the dotted line [same as Fig. 5]. For the solid line, the $g^{(2)} (0)$ values at the dip and two peaks are indicated.
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