Adversarial autoencoder ensemble for fast and probabilistic reconstructions of few-shot photon correlation functions for solid-state quantum emitters

Andrew H. Proppe, Kin Long Kelvin Lee, Cristian L. Cortes, Mari Saif, David B. Berkinsky, Tara Sverko, Weiwei Sun, James Cassidy, Mikhail Zamkov, Taehyung Kim, Eunjoo Jang, Stephen K. Gray, Brett A. McGuire, and Moungi G. Bawendi

Phys. Rev. B 106, 045425 – Published 29 July 2022

Abstract

Second-order photon correlation measurements [ $g^{(2)} (τ)$ functions] are widely used to classify single-photon emission purity in quantum emitters or to measure the multiexciton quantum yield of emitters that can simultaneously host multiple excitations – such as quantum dots – by evaluating the value of $g^{(2)} (τ = 0)$ . Accumulating enough photons to accurately calculate this value is time consuming and could be accelerated by fitting of few-shot photon correlations. Here, we develop an uncertainty-aware, deep adversarial autoencoder ensemble (AAE) that reconstructs noise-free $g^{(2)} (τ)$ functions from noise-dominated, few-shot inputs. The model is trained with simulated $g^{(2)} (τ)$ functions that are facilely generated by Poisson sampling time bins. The AAE reconstructions are performed orders-of-magnitude faster, with reconstruction errors and estimates of $g^{(2)} (τ = 0)$ that are lower in variance and similar in accuracy compared to Maximum likelihood estimation and Levenberg-Marquardt least-squares fitting approaches, for simulated and experimentally measured few-shot $g^{(2)} (τ)$ functions (∼100 two-photon events) of InP/ZnS/ZnSe and CdS/CdSe/CdS quantum dots. The deep-ensemble model comprises eight individual autoencoders, allowing for probabilistic reconstructions of noise-free $g^{(2)} (τ)$ functions, and we show that the predicted variance scales inversely with number of shots, with comparable uncertainties to computationally intensive Markov chain Monte Carlo sampling. This work demonstrates the advantage of machine learning models to perform uncertainty-aware, fast, and accurate reconstructions of simple Poisson-distributed photon correlation functions, allowing for on-the-fly reconstructions and accelerated materials characterization of solid-state quantum emitters.

Received 17 March 2022
Revised 5 July 2022
Accepted 15 July 2022

DOI:https://doi.org/10.1103/PhysRevB.106.045425

Physics Subject Headings (PhySH)

Electronic transitions Light-matter interaction Optical transient phenomena Quantum optics

Semiconductors

Fluorescence spectroscopy Luminescence Optical microscopy Photon counting Single-photon detectors

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & OpticalQuantum Information, Science & Technology

Authors & Affiliations

Andrew H. Proppe^1,*, Kin Long Kelvin Lee ^1,2,*, Cristian L. Cortes², Mari Saif¹, David B. Berkinsky ¹, Tara Sverko ¹, Weiwei Sun ¹, James Cassidy³, Mikhail Zamkov³, Taehyung Kim ⁴, Eunjoo Jang⁴, Stephen K. Gray², Brett A. McGuire ^1,†, and Moungi G. Bawendi^1,‡

¹Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
²Center for Nanoscale Materials, Argonne National Laboratory, Lemont, Illinois 60439, USA
³The Center for Photochemical Sciences and Department of Physics, Bowling Green State University, Bowling Green, Ohio 43403, USA
⁴Samsung Advanced Institute of Technology, Samsung Electronics, Suwon-si, Gyeonggi-do 16678, Republic of Korea

^*These authors contributed equally to this work.
^†Corresponding author: mgb@mit.edu
^‡Corresponding author: brettmc@mit.edu

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Vol. 106, Iss. 4 — 15 July 2022

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Images

Figure 1
(a) A Hanbury Brown-Twiss optical setup, where photons emitted from the sample are passed through a 50:50 beamsplitter towards two single-photon detectors in order to construct $g^{(2)}$ . (b) Illustration of confocal microscope used to probe single QDs. A detailed description of the full optical setup can be found in the Supplemental Material [32]. (c) Biexcitons can recombine radiatively or nonradiatively (Auger) before emission of the remaining exciton, which determines the BXQY [area of the $g^{(2)} (τ = 0)$ peak]. (d) Experimental $g^{(2)} s$ for single CdS/CdSe/CdS (BXQY = 43.4%) and InP/ZnSe/ZnS (BXQY = 6.3%) dots. (e) General architecture of the adversarial autoencoder ensemble (AAE) network used in this work. Only the first two of eight submodels are shown, with the black lines extending down towards the remaining six submodels. The output of each individual submodel is sent to the discriminator and a bagging layer to obtain the mean ( $μ$ ) and variance ( $σ$ ) of the output. (f) Simulated few-shot $g^{(2)}$ input with a $g^{(2)} (τ = 0)$ peak of ∼0.5, true underlying function, and AAE reconstruction. The parameters used to generate this data were randomly sampled from the ranges given in Table S1 [32]. Negative and positive time bins are combined to create functions with fewer numbers of bins to reduce network size.
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Figure 2
(a) Fits to a single simulated, few-shot (∼100 counts) $g^{(2)}$ input from the set of 1000 using maximum likelihood estimation (MLE), MLE with a Poisson likelihood (MLEP), Levenberg-Marquardt (LM), and the adversarial autoencoder ensemble (AAE). (b) Root mean squared error of the function reconstruction (Recon. RMSE) and (c) absolute error of center-to-side peak area ratios ( $R$ Error) vs average number of $g^{(2)}$ counts. Error bars indicate the standard deviation, $σ$ , of the computed errors.
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Figure 3
True vs predicted peak area ratios and $r^{2}$ values for a dataset with ∼100 counts in each histogram, with the solid black line representing ideal correlation (i.e., true equals predicted). The dashed gray box highlights the lack of clustering for the AAE.
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Figure 4
(a) Full experimental $g^{(2)} s$ with >10 000 counts and true distribution fit using MLE with Eq. (1). The $g^{(2)} (0)$ values for each sample are given beside the center peak, and the remaining fitted parameters for each sample can be found in Table S3 [32]. (b) Experimental $g^{(2)}$ input with ∼100 counts and AAE reconstruction, (c) reconstruction RMSE, and (d) $R$ error, with error bars indicating $\pm σ$ , vs average $g^{(2)}$ counts for an InP single dot, an InP aggregate, a CdSe single dot, and a CdSe aggregate.
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Figure 5
(a) Input data, AAE output average ( $μ$ ), and two standard deviations around the average ( $μ \pm 2 σ$ ), (b) outputs from the eight individual models within the ensemble, normalized and offset for clarity, and (c) Hamiltonian Monte Carlo average fit ( $μ$ HMC) and 2000 fits drawn from the chain, plotted in gray with transparency ( $σ$ HMC).
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