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<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Auxiliary Density Operators · Documentation | HierarchicalEOM.jl</title><script data-outdated-warner src="../assets/warner.js"></script><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" 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class="docs-autofit"><a href="../">Documentation | HierarchicalEOM.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><span class="tocitem">Home</span><ul><li><a class="tocitem" href="../">Introduction</a></li><li><a class="tocitem" href="../install/">Installation</a></li><li><a class="tocitem" href="../examples/quick_start/">Quick Start</a></li><li><a class="tocitem" href="../cite/">Cite HierarchicalEOM</a></li></ul></li><li><span class="tocitem">Manual</span><ul><li><a class="tocitem" href="../bosonic_bath/">Bosonic Bath</a></li><li><a class="tocitem" href="../fermionic_bath/">Fermionic Bath</a></li><li class="is-active"><a class="tocitem" href>Auxiliary Density Operators</a><ul class="internal"><li><a class="tocitem" href="#Introduction"><span>Introduction</span></a></li><li><a class="tocitem" href="#Fields"><span>Fields</span></a></li><li><a class="tocitem" href="#Reduced-Density-Operator"><span>Reduced Density Operator</span></a></li><li><a class="tocitem" href="#High-Level-Auxiliary-Density-Operators"><span>High-Level Auxiliary Density Operators</span></a></li><li><a class="tocitem" href="#Expectation-Value"><span>Expectation Value</span></a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2-4" type="checkbox"/><label class="tocitem" for="menuitem-2-4"><span class="docs-label">HEOMLS Matrices</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../heom_matrix/intro/">Introduction</a></li><li><a class="tocitem" href="../heom_matrix/schrodinger_eq/">HEOMLS for Schrödinger Equation</a></li><li><a class="tocitem" href="../heom_matrix/M_Boson/">HEOMLS for Bosonic Bath</a></li><li><a class="tocitem" href="../heom_matrix/M_Fermion/">HEOMLS for Fermionic Bath</a></li><li><a class="tocitem" href="../heom_matrix/M_Boson_Fermion/">HEOMLS for Bosonic and Fermionic Bath</a></li><li><a class="tocitem" href="../heom_matrix/master_eq/">HEOMLS for Master Equation</a></li></ul></li><li><a class="tocitem" href="../hierarchy_dictionary/">Hierarchy Dictionary</a></li><li><a class="tocitem" href="../time_evolution/">Time Evolution</a></li><li><a class="tocitem" href="../stationary_state/">Stationary State</a></li><li><a class="tocitem" href="../spectrum/">Spectrum</a></li><li><input class="collapse-toggle" id="menuitem-2-9" type="checkbox"/><label class="tocitem" for="menuitem-2-9"><span class="docs-label">Examples</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../examples/cavityQED/">Cavity QED system</a></li><li><a class="tocitem" href="../examples/dynamical_decoupling/">Driven Systems and Dynamical Decoupling</a></li><li><a class="tocitem" href="../examples/SIAM/">The single-impurity Anderson model</a></li><li><a class="tocitem" href="../examples/electronic_current/">Electronic Current</a></li></ul></li><li><input class="collapse-toggle" id="menuitem-2-10" type="checkbox"/><label class="tocitem" for="menuitem-2-10"><span class="docs-label">Benchmark Solvers</span><i class="docs-chevron"></i></label><ul class="collapsed"><li><a class="tocitem" href="../examples/benchmark_ODE_solvers/">DifferentialEquations solvers</a></li><li><a class="tocitem" href="../examples/benchmark_LS_solvers/">LinearSolve solvers</a></li></ul></li></ul></li><li><a class="tocitem" href="../libraryAPI/">Library API</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><a class="is-disabled">Manual</a></li><li class="is-active"><a href>Auxiliary Density Operators</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Auxiliary Density Operators</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/NCKU-QFort/HierarchicalEOM.jl/blob/main/docs/src/ADOs.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="doc-ADOs"><a class="docs-heading-anchor" href="#doc-ADOs">Auxiliary Density Operators</a><a id="doc-ADOs-1"></a><a class="docs-heading-anchor-permalink" href="#doc-ADOs" title="Permalink"></a></h1><h2 id="Introduction"><a class="docs-heading-anchor" href="#Introduction">Introduction</a><a id="Introduction-1"></a><a class="docs-heading-anchor-permalink" href="#Introduction" title="Permalink"></a></h2><p>The auxiliary density operators (<a href="../libraryAPI/#HierarchicalEOM.HeomAPI.ADOs"><code>ADOs</code></a>) <span>$\rho_{\textbf{j}\vert\textbf{q}}^{(m,n,p)}(t)$</span> encode environmental effects related to different exponential terms (<a href="../libraryAPI/#HierarchicalEOM.Bath.Exponent"><code>Exponent</code></a>) present in the <a href="../bosonic_bath/#doc-Bosonic-Bath"><code>Bosonic Bath</code></a> and <a href="../fermionic_bath/#doc-Fermionic-Bath"><code>Fermionic Bath</code></a> correlation functions and provide an iterative description of high-order system-baths memory effects.</p><p>In <span>$\rho_{\textbf{j}\vert\textbf{q}}^{(m,n,p)}(t)$</span>, the tuple <span>$(m, n, p)$</span> represents the <span>$m$</span>th-level-bosonic-and-<span>$n$</span>th-level-fermionic ADO with parity <span>$p$</span>, and <span>$\textbf{j}$</span> (<span>$\textbf{q}$</span>) denotes a vector <span>$[j_m,\cdots,j_1]$</span> (<span>$[q_n,\cdots,q_1]$</span>) where each <span>$j$</span> (<span>$q$</span>) represents a specific multi-index ensemble <span>$\{\beta, l\}$</span> (<span>$\{\alpha, h\}$</span>) with</p><ul><li><span>$\beta$</span> : denotes the index of bosonic bath</li><li><span>$\alpha$</span> : denotes the index of fermionic bath</li><li><span>$l$</span> : denotes the index of exponent in the bosonic bath</li><li><span>$h$</span> : denotes the index of exponent in the fermionic bath</li></ul><div class="admonition is-info"><header class="admonition-header">Reduced Density Operator</header><div class="admonition-body"><p>The system reduced density operator refers to <span>$m=n=0$</span>, namely <span>$\rho_{\vert}^{(0,0,p)}(t)$</span>.</p></div></div><p>In <code>HierarchicalEOM.jl</code>, we express all the auxiliary density operators into a single column vector and store it in the object defined as : </p><p><a href="../libraryAPI/#HierarchicalEOM.HeomAPI.ADOs"><code>struct ADOs</code></a>, </p><p>which is usually obtained after solving the time <a href="../time_evolution/#doc-Time-Evolution">evolution</a> or <a href="../stationary_state/#doc-Stationary-State">stationary state</a> by a given <a href="../heom_matrix/intro/#doc-HEOMLS-Matrix">HEOM Liouvillian superoperator Matrix</a>.</p><h2 id="Fields"><a class="docs-heading-anchor" href="#Fields">Fields</a><a id="Fields-1"></a><a class="docs-heading-anchor-permalink" href="#Fields" title="Permalink"></a></h2><p>The fields of the structure <a href="../libraryAPI/#HierarchicalEOM.HeomAPI.ADOs"><code>ADOs</code></a> are as follows:</p><ul><li><code>data</code> : the vectorized auxiliary density operators</li><li><code>dim</code> : the dimension of the system</li><li><code>N</code> : the number of auxiliary density operators</li><li><code>parity</code>: the <a href="../heom_matrix/intro/#doc-Parity">parity</a> label</li></ul><p>One obtain the value of each fields as follows:</p><pre><code class="language-julia hljs"># usually obtained after solving time evolution or stationary state
ados::ADOs

ados.data
ados.dim
ados.N
ados.parity</code></pre><div class="admonition is-warning"><header class="admonition-header">Warning</header><div class="admonition-body"><p>We express all the auxiliary density operators in only a single column vector <code>ADOs.data</code>. To obtain each auxiliary density operators in matrix form, please use the following methods and functions.</p></div></div><h2 id="Reduced-Density-Operator"><a class="docs-heading-anchor" href="#Reduced-Density-Operator">Reduced Density Operator</a><a id="Reduced-Density-Operator-1"></a><a class="docs-heading-anchor-permalink" href="#Reduced-Density-Operator" title="Permalink"></a></h2><p>In order to obtain the system reduced density operator in matrix form, just simply call <a href="../libraryAPI/#HierarchicalEOM.HeomAPI.getRho"><code>getRho</code></a></p><pre><code class="language-julia hljs">ados::ADOs
ρ = getRho(ados)</code></pre><h2 id="High-Level-Auxiliary-Density-Operators"><a class="docs-heading-anchor" href="#High-Level-Auxiliary-Density-Operators">High-Level Auxiliary Density Operators</a><a id="High-Level-Auxiliary-Density-Operators-1"></a><a class="docs-heading-anchor-permalink" href="#High-Level-Auxiliary-Density-Operators" title="Permalink"></a></h2><p>Although we express all the auxiliary density operators in the vector <code>ADOs.data</code>, we still make the <a href="../libraryAPI/#HierarchicalEOM.HeomAPI.ADOs"><code>ADOs</code></a> like a list where accessing each element would return a specific auxiliary density operator in matrix type. </p><p>In order to obtain the auxiliary density operator in matrix form with a specific index <code>i</code>, just simply call <a href="../libraryAPI/#HierarchicalEOM.HeomAPI.getADO"><code>getADO</code></a></p><pre><code class="language-julia hljs">ados::ADOs
i::Int

ρ = getADO(ados, 1) # the first element will always be the reduced density operator
ado = getADO(ados, i) # the i-th auxiliary density operator</code></pre><p>Also, <a href="../libraryAPI/#HierarchicalEOM.HeomAPI.ADOs"><code>ADOs</code></a> supports all the element-wise methods (functions) :</p><p>Supports <code>length(::ADOs)</code> which returns the total number of auxiliary density operators (same as <code>ADOs.N</code>) :</p><pre><code class="language-julia hljs">ados::ADOs
length(ados)</code></pre><p>Supports bracket operation <code>[]</code> which is similar to access the element of a list :</p><pre><code class="language-julia hljs">ados::ADOs

# all the following returned ADO will be in matrix form
ados[1] # returns the first auxiliary density operator (which is always the reduced density operator)
ados[10] # returns the 10-th auxiliary density operator
ados[3:10] # returns a list of auxiliary density operators from index 3 to 10
ados[end] # returns the last auxiliary density operator</code></pre><p>Supports iteration (<code>for</code>-loop) process :</p><pre><code class="language-julia hljs">ados::ADOs

for ado in ados # iteration
ado # each auxiliary density operator in matrix form
end</code></pre><div class="admonition is-info"><header class="admonition-header">Work on high-level auxiliary density operators with Hierarchy Dictionary</header><div class="admonition-body"><p>To find the index of the auxiliary density operator and it&#39;s corresponding bath <a href="../libraryAPI/#HierarchicalEOM.Bath.Exponent"><code>Exponent</code></a>, please refer to <a href="../hierarchy_dictionary/#doc-Hierarchy-Dictionary"><code>Hierarchy Dictionary</code></a> for more details.</p></div></div><h2 id="Expectation-Value"><a class="docs-heading-anchor" href="#Expectation-Value">Expectation Value</a><a id="Expectation-Value-1"></a><a class="docs-heading-anchor-permalink" href="#Expectation-Value" title="Permalink"></a></h2><p>Given an observable <span>$A$</span> and <code>ADOs</code> <span>$\rho^{(m,n,p)}_{\textbf{j} \vert \textbf{q}}$</span>, one can calculate the expectation value by</p><p class="math-container">\[\langle A \rangle = \textrm{Tr}\left[A \rho^{(0,0,p)}_{ \vert }\right],\]</p><p>where, <span>$m=n=0$</span> represents the reduced density operator.</p><p>One can directly calculate the expectation values using the function <a href="../libraryAPI/#HierarchicalEOM.HeomAPI.Expect"><code>Expect</code></a>:</p><pre><code class="language-julia hljs">A::AbstractMatrix # observable

# with a single ADOs
ados::ADOs
E = Expect(A, ados)

# with a list contains many ADOs
ados_list::Vector{ADOs}
Elist = Expect(A, ados_list)</code></pre><p>Here, <code>Elist</code> contains the expectation values corresponding to the <code>ados_list</code>.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../fermionic_bath/">« Fermionic Bath</a><a class="docs-footer-nextpage" href="../heom_matrix/intro/">Introduction »</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="Wednesday 30 August 2023 05:59">Wednesday 30 August 2023</span>. Using Julia version 1.9.3.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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