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<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Home · SpectralDensities.jl</title><script data-outdated-warner src="assets/warner.js"></script><link rel="canonical" href="https://quantum-exeter.github.io/SpectralDensities.jl/"/><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.045/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="assets/documenter.js"></script><script src="siteinfo.js"></script><script src="../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href>SpectralDensities.jl</a></span></div><form class="docs-search" action="search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li class="is-active"><a class="tocitem" href>Home</a><ul class="internal"><li><a class="tocitem" href="#Features"><span>Features</span></a></li><li><a class="tocitem" href="#Quick-start"><span>Quick start</span></a></li><li><a class="tocitem" href="#Defining-custom-spectral-densities"><span>Defining custom spectral densities</span></a></li></ul></li><li><a class="tocitem" href="reference/">API Reference</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Home</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Home</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/quantum-exeter/SpectralDensities.jl/blob/main/docs/src/index.md#" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="SpectralDensities.jl"><a class="docs-heading-anchor" href="#SpectralDensities.jl">SpectralDensities.jl</a><a id="SpectralDensities.jl-1"></a><a class="docs-heading-anchor-permalink" href="#SpectralDensities.jl" title="Permalink"></a></h1><p>This package implements commonly used spectral densities for Open Quantum Systems and typical operations on them.</p><h2 id="Features"><a class="docs-heading-anchor" href="#Features">Features</a><a id="Features-1"></a><a class="docs-heading-anchor-permalink" href="#Features" title="Permalink"></a></h2><p>Features of SpectralDensities.jl include:</p><ul><li>Definition of most widely used spectral densities: Ohmic, Sub-Ohmic, Supra-Ohmic, Lorentzian (Underdamped), Debye (Overdamped)</li><li>Flexibility to choose desidred cutoff functions: hard cutoffs, exponential cutoff, gaussian cutoff.</li><li>Calculation of the correlation function of a given spectral density (with specialised methods for the pre-defined types)</li><li>Calculation of reorganisation energy of a given spectral density (with specialised methods for the pre-defined types)</li><li>Calculation of the memory kernel of a given spectral density, both in the time domain, and the imaginary part in the Fourier domain (with specialised methods for some of the pre-defined types)</li><li>Methods to compute the spectral density integrals that appear in the weak coupling (2nd order expansion), both for the dynamics (see e.g. H. Breuer, F. Petruccione, &quot;The Theory of Open Quantum Systems&quot;), and the equilibrium mean-force state (see e.g. J.D. Cresser, J. Anders, Phys. Rev. Lett. 127, 250601 (2021)).</li></ul><h2 id="Quick-start"><a class="docs-heading-anchor" href="#Quick-start">Quick start</a><a id="Quick-start-1"></a><a class="docs-heading-anchor-permalink" href="#Quick-start" title="Permalink"></a></h2><p>The desired spectral density can be constructed by simply passing the desired parameters to their respective constructor. For example, for an Ohmic spectral density <span>$J(\omega) = 3\omega$</span> we have</p><pre><code class="language-Julia hljs">Johmic = OhmicSD(3)</code></pre><p>To add a ctuoff to a base spectral density, one simply passes it to the cutoff constructor. For example, to add an exponential cutoff <span>$e^{-\omega/7}$</span> we can do</p><pre><code class="language-Julia hljs">Jc = ExponentialCutoffSD(Johmic, 7)</code></pre><h2 id="Defining-custom-spectral-densities"><a class="docs-heading-anchor" href="#Defining-custom-spectral-densities">Defining custom spectral densities</a><a id="Defining-custom-spectral-densities-1"></a><a class="docs-heading-anchor-permalink" href="#Defining-custom-spectral-densities" title="Permalink"></a></h2><p>The base <a href="reference/#SpectralDensities.AbstractSD"><code>AbstractSD</code></a> can be easily extended by the user. To do so, one must create a new sub-type for which <em>at least</em> one of the methods <a href="reference/#SpectralDensities.sd-Union{Tuple{T}, Tuple{T, Any}} where T&lt;:AbstractSD"><code>sd</code></a> (the spectral density itself <span>$J(\omega)$</span>) or <a href="reference/#SpectralDensities.sdoverω-Union{Tuple{T}, Tuple{T, Any}} where T&lt;:AbstractSD"><code>sdoverω</code></a> (the spectral density divided by frequency <span>$J(\omega)/\omega$</span>) must be defined. It is strongly recommended to define, if possible, <a href="reference/#SpectralDensities.sdoverω-Union{Tuple{T}, Tuple{T, Any}} where T&lt;:AbstractSD"><code>sdoverω</code></a> since it typically makes it simpler to numerically handle potential singularities.</p><p>This is the only requirement, and all other methods will automatically work on the new subtype. Of course, as is typical of Julia&#39;s super flexible multiple-dispatch system, the user should consider providing custom functions on the new type for the reorganisation energy, etc.</p></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="reference/">API Reference »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Thursday 6 July 2023 16:11">Thursday 6 July 2023</span>. Using Julia version 1.9.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Home · SpectralDensities.jl</title><script data-outdated-warner src="assets/warner.js"></script><link rel="canonical" href="https://quantum-exeter.github.io/SpectralDensities.jl/"/><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.045/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="assets/documenter.js"></script><script src="siteinfo.js"></script><script src="../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href>SpectralDensities.jl</a></span></div><form class="docs-search" action="search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li class="is-active"><a class="tocitem" href>Home</a><ul class="internal"><li><a class="tocitem" href="#Features"><span>Features</span></a></li><li><a class="tocitem" href="#Quick-start"><span>Quick start</span></a></li><li><a class="tocitem" href="#Defining-custom-spectral-densities"><span>Defining custom spectral densities</span></a></li></ul></li><li><a class="tocitem" href="reference/">API Reference</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Home</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Home</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/quantum-exeter/SpectralDensities.jl/blob/main/docs/src/index.md#" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="SpectralDensities.jl"><a class="docs-heading-anchor" href="#SpectralDensities.jl">SpectralDensities.jl</a><a id="SpectralDensities.jl-1"></a><a class="docs-heading-anchor-permalink" href="#SpectralDensities.jl" title="Permalink"></a></h1><p>This package implements commonly used spectral densities for Open Quantum Systems and typical operations on them.</p><h2 id="Features"><a class="docs-heading-anchor" href="#Features">Features</a><a id="Features-1"></a><a class="docs-heading-anchor-permalink" href="#Features" title="Permalink"></a></h2><p>Features of SpectralDensities.jl include:</p><ul><li>Definition of most widely used spectral densities: Ohmic, Sub-Ohmic, Supra-Ohmic, Lorentzian (Underdamped), Debye (Overdamped)</li><li>Flexibility to choose desidred cutoff functions: hard cutoffs, exponential cutoff, gaussian cutoff.</li><li>Calculation of the correlation function of a given spectral density (with specialised methods for the pre-defined types)</li><li>Calculation of reorganisation energy of a given spectral density (with specialised methods for the pre-defined types)</li><li>Calculation of the memory kernel of a given spectral density, both in the time domain, and the imaginary part in the Fourier domain (with specialised methods for some of the pre-defined types)</li><li>Methods to compute the spectral density integrals that appear in the weak coupling (2nd order expansion), both for the dynamics (see e.g. H. Breuer, F. Petruccione, &quot;The Theory of Open Quantum Systems&quot;), and the equilibrium mean-force state (see e.g. J.D. Cresser, J. Anders, Phys. Rev. Lett. 127, 250601 (2021)).</li></ul><h2 id="Quick-start"><a class="docs-heading-anchor" href="#Quick-start">Quick start</a><a id="Quick-start-1"></a><a class="docs-heading-anchor-permalink" href="#Quick-start" title="Permalink"></a></h2><p>The desired spectral density can be constructed by simply passing the desired parameters to their respective constructor. For example, for an Ohmic spectral density <span>$J(\omega) = 3\omega$</span> we have</p><pre><code class="language-Julia hljs">Johmic = OhmicSD(3)</code></pre><p>To add a ctuoff to a base spectral density, one simply passes it to the cutoff constructor. For example, to add an exponential cutoff <span>$e^{-\omega/7}$</span> we can do</p><pre><code class="language-Julia hljs">Jc = ExponentialCutoffSD(Johmic, 7)</code></pre><h2 id="Defining-custom-spectral-densities"><a class="docs-heading-anchor" href="#Defining-custom-spectral-densities">Defining custom spectral densities</a><a id="Defining-custom-spectral-densities-1"></a><a class="docs-heading-anchor-permalink" href="#Defining-custom-spectral-densities" title="Permalink"></a></h2><p>The base <a href="reference/#SpectralDensities.AbstractSD"><code>AbstractSD</code></a> can be easily extended by the user. To do so, one must create a new sub-type for which <em>at least</em> one of the methods <a href="reference/#SpectralDensities.sd-Union{Tuple{T}, Tuple{T, Any}} where T&lt;:AbstractSD"><code>sd</code></a> (the spectral density itself <span>$J(\omega)$</span>) or <a href="reference/#SpectralDensities.sdoverω-Union{Tuple{T}, Tuple{T, Any}} where T&lt;:AbstractSD"><code>sdoverω</code></a> (the spectral density divided by frequency <span>$J(\omega)/\omega$</span>) must be defined. It is strongly recommended to define, if possible, <a href="reference/#SpectralDensities.sdoverω-Union{Tuple{T}, Tuple{T, Any}} where T&lt;:AbstractSD"><code>sdoverω</code></a> since it typically makes it simpler to numerically handle potential singularities.</p><p>This is the only requirement, and all other methods will automatically work on the new subtype. Of course, as is typical of Julia&#39;s super flexible multiple-dispatch system, the user should consider providing custom functions on the new type for the reorganisation energy, etc.</p></article><nav class="docs-footer"><a class="docs-footer-nextpage" href="reference/">API Reference »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Friday 7 July 2023 11:15">Friday 7 July 2023</span>. Using Julia version 1.9.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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