\n\n\n\n
Charge Stability Diagrams: The Map to Electron Configuration<\/h3>\n\n\n\n
A critical tool in quantum dot tuning is the charge stability diagram (CSD), a two-dimensional voltage map depicting charge configurations. <\/p>\n\n\n\n These diagrams are governed by the capacitance model, where electrostatic energy determines charge states. Clean and well-defined CSDs enable the extraction of model parameters, such as the size and slopes of Coulomb diamonds, which are essential for understanding and optimizing the system.<\/p>\n\n\n\n As the number of quantum dots increases, the number of CSD cuts grows exponentially, making manual analysis impractical. Current strategies include:<\/p>\n\n\n\n We propose a simpler yet effective alternative: leveraging Fourier transforms to analyze CSDs.<\/p>\n\n\n\n The system can be described using a capacitance matrix that links charges and voltages on the dots and gates, analogous to the relationship \\( Q = CV \\):<\/p>\n\n\n\n $$ Here:<\/p>\n\n\n\n Minimising the electrostatic energy, \\( E \\), determines the charge configuration. This is visualised in a CSD, with gate voltages as axes and charge states represented by color. Parameters such as the size and slope of Coulomb diamonds are directly derived from this model.<\/p>\n\n\n\n Four critical parameters derived from the capacitance model:<\/p>\n\n\n\n Our method focuses on two-dot systems but can be generalized to larger setups. Here\u2019s the step-by-step process:<\/p>\n\n\n\n Charge stability diagrams are obtained experimentally by sweeping gate voltages and measuring responses, such as currents through sensing dots. Alternatively, synthetic data can be generated using the QDarts simulator to explore different noise levels. Below we present 10 CSDs obtained for a different level of noise (x-axis)<\/p>\n\n\n\n We use the Sobel filter, a simple edge-detection tool with 3×3 kernels, to identify horizontal and vertical edges. The resulting gradient magnitudes highlight the edges of Coulomb diamonds.<\/p>\n\n\n\n Applying a two-dimensional Fourier transform (fft2) to the filtered image reveals periodic structures in the CSD. The transform captures prominent \u201cjets” corresponding to the directions perpendicular to the diamond edges.<\/p>\n\n\n\n To identify jets, we:<\/p>\n\n\n\n The period of oscillations along the jets is determined by the distance from the origin to the first peak along each jet. This period is inversely proportional to the size of the Coulomb diamonds.<\/p>\n\n\n\n Using synthetic data generated by the QDarts simulator, we validated our method\u2019s accuracy under varying noise levels. The Fourier-based approach proved robust, efficiently extracting key parameters even in noisy conditions.<\/p>\n\n\n\n Clearly below certain noise level, the relative prediction error in any of the quantites is less than 10%. In general it is likely to be dominated by the “unlucky” CSD.<\/p>\n\n\n\n The Fourier transform-based method offers a simplified, yet effective way to extract parameters underlying charge stability diagrams in the autonomous way. <\/p>\n\n\n\n The code is available at: https:\/\/github.com\/jan-a-krzywda\/ftt-charge-stability-diagram<\/a> TODO<\/p>\n\n\n\n The pursuit of practical quantum computing relies on scalable architectures, and transistor-based quantum dots offer a promising path forward. By trapping single electrons in these semiconductor devices, we hope to process information using their internal degree of freedom – spin. However, achieving this requires precise control and tuning Tuning Quantum Dots: A Complex Prelude to […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-170","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/qplai.liacs.nl\/index.php\/wp-json\/wp\/v2\/posts\/170","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/qplai.liacs.nl\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/qplai.liacs.nl\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/qplai.liacs.nl\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/qplai.liacs.nl\/index.php\/wp-json\/wp\/v2\/comments?post=170"}],"version-history":[{"count":18,"href":"https:\/\/qplai.liacs.nl\/index.php\/wp-json\/wp\/v2\/posts\/170\/revisions"}],"predecessor-version":[{"id":207,"href":"https:\/\/qplai.liacs.nl\/index.php\/wp-json\/wp\/v2\/posts\/170\/revisions\/207"}],"wp:attachment":[{"href":"https:\/\/qplai.liacs.nl\/index.php\/wp-json\/wp\/v2\/media?parent=170"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/qplai.liacs.nl\/index.php\/wp-json\/wp\/v2\/categories?post=170"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/qplai.liacs.nl\/index.php\/wp-json\/wp\/v2\/tags?post=170"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"\"](\"https:\/\/qplai.liacs.nl\/wp-content\/uploads\/image-30-1024x442.png\")
![\"\"](\"https:\/\/qplai.liacs.nl\/wp-content\/uploads\/image-28.png\")
\n\n\n\nOvercoming Challenges with Scalable Strategies<\/h3>\n\n\n\n
\n
\n\n\n\nCapacitance Model: The Physics Behind CSDs<\/h3>\n\n\n\n
\\begin{pmatrix}
Q_D \\\\
Q_G
\\end{pmatrix} =
\\begin{pmatrix}
C_{DD} & C_{DG} \\\\
C_{DG}^T & C_{GG}
\\end{pmatrix}
\\begin{pmatrix}
U_D \\\\
V_G
\\end{pmatrix}
$$<\/p>\n\n\n\n\n
\n\n\n\nKey Parameters for Simplified Analysis<\/h3>\n\n\n\n
\n
\\[
\\Delta V_i = \\frac{|e| C_{DD}^{-1}[i,i]}{(C_{DD}^{-1}C_{DG})[i,i]}
\\]<\/li>\n\n\n\n
\\[
\\theta_1 = \\text{arctan}\\left(\\frac{\\alpha[1,1]}{\\alpha[1,2]}\\right),\\quad \\theta_2 = \\text{arctan}\\left(\\frac{\\alpha[2,2]}{\\alpha[2,1]}\\right) \\quad \\alpha = C_{DD}^{-1}C_{DG}
\\]<\/li>\n<\/ol>\n\n\n\n![\"\"](\"https:\/\/qplai.liacs.nl\/wp-content\/uploads\/image-25.png\")
<\/figure>\n\n\n\n
\n\n\n\nOur Procedure: Fourier Transform Analysis of CSDs<\/h3>\n\n\n\n
1. Data Acquisition<\/h4>\n\n\n\n
![\"\"](\"https:\/\/qplai.liacs.nl\/wp-content\/uploads\/image-12.png\")
<\/figure>\n\n\n\n2. Edge Extraction<\/h4>\n\n\n\n
![\"\"](\"https:\/\/qplai.liacs.nl\/wp-content\/uploads\/image-14-1024x560.png\")
<\/figure>\n\n\n\n3. Fourier Transform<\/h4>\n\n\n\n
![\"\"](\"https:\/\/qplai.liacs.nl\/wp-content\/uploads\/image-17.png\")
<\/figure>\n\n\n\n4. Jet Identification<\/h4>\n\n\n\n
\n
![\"\"](\"https:\/\/qplai.liacs.nl\/wp-content\/uploads\/image-15-1024x345.png\")
<\/figure>\n\n\n\n5. Period Extraction<\/h4>\n\n\n\n
![\"\"](\"https:\/\/qplai.liacs.nl\/wp-content\/uploads\/image-16.png\")
<\/figure>\n\n\n\n
\n\n\n\nResults: Analyzing Synthetic Data<\/h3>\n\n\n\n
![\"\"](\"https:\/\/qplai.liacs.nl\/wp-content\/uploads\/image-24.png\")
<\/figure>\n\n\n\n![\"\"](\"https:\/\/qplai.liacs.nl\/wp-content\/uploads\/image-23.png\")
<\/figure>\n\n\n\n
\n\n\n\nConclusion<\/h3>\n\n\n\n
\n\n\n\nCode availability<\/h3>\n\n\n\n
Data: data_csd_fft.zip<\/a><\/p>\n\n\n\n
\n\n\n\nOutlook and limitations<\/h3>\n\n\n\n
\n","protected":false},"excerpt":{"rendered":"