-
Notifications
You must be signed in to change notification settings - Fork 11
Commit
This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository.
- Loading branch information
Documenter.jl
committed
Aug 11, 2024
1 parent
32e8cfb
commit 22a03d5
Showing
25 changed files
with
2,116 additions
and
3 deletions.
There are no files selected for viewing
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -1 +1 @@ | ||
| v0.4.1 | ||
| v0.4.2 |
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -1 +1 @@ | ||
| v0.4.1 | ||
| v0.4.2 |
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1 @@ | ||
| {"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-08-11T22:36:40","documenter_version":"1.5.0"}} |
Large diffs are not rendered by default.
Oops, something went wrong.
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,78 @@ | ||
| <!DOCTYPE html> | ||
| <html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Quantum Harmonic Oscillators · QuantumSymbolics.jl</title><meta name="title" content="Quantum Harmonic Oscillators · QuantumSymbolics.jl"/><meta property="og:title" content="Quantum Harmonic Oscillators · QuantumSymbolics.jl"/><meta property="twitter:title" content="Quantum Harmonic Oscillators · QuantumSymbolics.jl"/><meta name="description" content="Documentation for QuantumSymbolics.jl."/><meta property="og:description" content="Documentation for QuantumSymbolics.jl."/><meta property="twitter:description" content="Documentation for QuantumSymbolics.jl."/><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/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/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="../search_index.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/catppuccin-mocha.css" data-theme-name="catppuccin-mocha"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-macchiato.css" data-theme-name="catppuccin-macchiato"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-frappe.css" data-theme-name="catppuccin-frappe"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-latte.css" data-theme-name="catppuccin-latte"/><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><script src="../assets/init.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="../">QuantumSymbolics.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">QuantumSymbolics.jl</a></li><li><a class="tocitem" href="../introduction/">Getting Started with QuantumSymbolics.jl</a></li><li><a class="tocitem" href="../express/">Express Functionality</a></li><li><a class="tocitem" href="../qubit_basis/">Qubit Basis Choice</a></li><li class="is-active"><a class="tocitem" href>Quantum Harmonic Oscillators</a><ul class="internal"><li><a class="tocitem" href="#States"><span>States</span></a></li><li><a class="tocitem" href="#Operators"><span>Operators</span></a></li><li><a class="tocitem" href="#Numerical-Conversions-to-QuantumOptics.jl"><span>Numerical Conversions to QuantumOptics.jl</span></a></li></ul></li><li><a class="tocitem" href="../API/">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"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Quantum Harmonic Oscillators</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Quantum Harmonic Oscillators</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/QuantumSavory/QuantumSymbolics.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/QuantumSavory/QuantumSymbolics.jl/blob/main/docs/src/QHO.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Quantum-Harmonic-Oscillators"><a class="docs-heading-anchor" href="#Quantum-Harmonic-Oscillators">Quantum Harmonic Oscillators</a><a id="Quantum-Harmonic-Oscillators-1"></a><a class="docs-heading-anchor-permalink" href="#Quantum-Harmonic-Oscillators" title="Permalink"></a></h1><p>In this section, we describe symbolic representations of bosonic systems in QuantumSymbolics, which can be numerically translated to <a href="https://github.com/qojulia/QuantumOptics.jl"><code>QuantumOptics.jl</code></a>.</p><h2 id="States"><a class="docs-heading-anchor" href="#States">States</a><a id="States-1"></a><a class="docs-heading-anchor-permalink" href="#States" title="Permalink"></a></h2><p>A Fock state is a state with well defined number of excitation quanta of a single quantum harmonic oscillator (an eigenstate of the number operator). In the following example, we create a <code>FockState</code> with 3 quanta in an infinite-dimension Fock space:</p><pre><code class="language-julia-repl hljs">julia> f = FockState(3) | ||
| |3⟩</code></pre><p>Both vacuum (ground) and single-photon states are defined as constants in both unicode and ASCII for convenience:</p><ul><li><code>vac = F₀ = F0</code> <span>$=|0\rangle$</span> in the number state representation,</li><li><code>F₁ = F1</code> <span>$=|1\rangle$</span> in the number state representation.</li></ul><p>To create quantum analogues of a classical harmonic oscillator, or monochromatic electromagnetic waves, we can define a coherent (a.k.a. semi-classical) state <span>$|\alpha\rangle$</span>, where <span>$\alpha$</span> is a complex amplitude, with <code>CoherentState(α::Number)</code>:</p><pre><code class="language-julia-repl hljs">julia> c = CoherentState(im) | ||
| |im⟩</code></pre><div class="admonition is-info"><header class="admonition-header">Naming convention for quantum harmonic oscillator bases</header><div class="admonition-body"><p>The defined basis for arbitrary symbolic bosonic states is a <code>FockBasis</code> object, due to a shared naming interface for Quantum physics packages. For instance, the command <code>basis(CoherentState(im))</code> will output <code>Fock(cutoff=Inf)</code>. This may lead to confusion, as not all bosonic states are Fock states. However, this is simply a naming convention for the basis, and symbolic and numerical results are not affected by it.</p></div></div><h2 id="Operators"><a class="docs-heading-anchor" href="#Operators">Operators</a><a id="Operators-1"></a><a class="docs-heading-anchor-permalink" href="#Operators" title="Permalink"></a></h2><p>Operations on bosonic states are supported, and can be simplified with <code>qsimplify</code> and its rewriter <code>qsimplify_fock</code>. For instance, we can apply the raising (creation) <span>$\hat{a}^{\dagger}$</span> and lowering (annihilation or destroy) <span>$\hat{a}$</span> operators on a Fock state as follows:</p><pre><code class="language-julia-repl hljs">julia> f = FockState(3); | ||
|
|
||
| julia> raise = Create*f | ||
| a†|3⟩ | ||
|
|
||
| julia> qsimplify(raise, rewriter=qsimplify_fock) | ||
| (sqrt(4))|4⟩ | ||
|
|
||
| julia> lower = Destroy*f | ||
| a|3⟩ | ||
|
|
||
| julia> qsimplify(lower, rewriter=qsimplify_fock) | ||
| (sqrt(3))|2⟩</code></pre><p>Or, we can apply the number operator <span>$\hat{n}$</span> to our Fock state:</p><pre><code class="language-julia-repl hljs">julia> f = FockState(3); | ||
|
|
||
| julia> num = N*f | ||
| n|3⟩ | ||
|
|
||
| julia> qsimplify(num, rewriter=qsimplify_fock) | ||
| 3|3⟩</code></pre><p>Constants are defined for number and ladder operators in unicode and ASCII:</p><ul><li><code>N = n̂</code> <span>$=\hat{n}$</span>,</li><li><code>Create = âꜛ</code> <span>$=\hat{a}^{\dagger}$</span>,</li><li><code>Destroy = â</code> <span>$=\hat{a}$</span>.</li></ul><p>Phase-shift <span>$U(\theta)$</span> and displacement <span>$D(\alpha)$</span> operators, defined respectively as <span>$U(\theta) = \exp\left(-i\theta\hat{n}\right) \quad \text{and} \quad D(\alpha) = \exp\left(\alpha\hat{a}^{\dagger} - \alpha\hat{a}\right),$</span> can be defined with usual simplification rules. Consider the following example:</p><pre><code class="language-julia-repl hljs">julia> displace = DisplaceOp(im) | ||
| D(im) | ||
|
|
||
| julia> c = qsimplify(displace*vac, rewriter=qsimplify_fock) | ||
| |im⟩ | ||
|
|
||
| julia> phase = PhaseShiftOp(pi) | ||
| U(π) | ||
|
|
||
| julia> qsimplify(phase*c, rewriter=qsimplify_fock) | ||
| |1.2246467991473532e-16 - 1.0im⟩</code></pre><p>Here, we generated a coherent state <span>$|i\rangle$</span> from the vacuum state <span>$|0\rangle$</span> by applying the displacement operator defined by <code>DisplaceOp</code>. Then, we shifted its phase by <span>$\pi$</span> with the phase shift operator (which is called with <code>PhaseShiftOp</code>) to get the result <span>$|-i\rangle$</span>.</p><p>Summarized below are supported bosonic operators.</p><ul><li>Number operator: <code>NumberOp()</code>,</li><li>Creation operator: <code>CreateOp()</code>,</li><li>Annihilation operator: <code>DestroyOp()</code>,</li><li>Phase-shift operator: <code>PhaseShiftOp(phase::Number)</code>,</li><li>Displacement operator: <code>DisplaceOp(alpha::Number)</code>.</li></ul><h2 id="Numerical-Conversions-to-QuantumOptics.jl"><a class="docs-heading-anchor" href="#Numerical-Conversions-to-QuantumOptics.jl">Numerical Conversions to QuantumOptics.jl</a><a id="Numerical-Conversions-to-QuantumOptics.jl-1"></a><a class="docs-heading-anchor-permalink" href="#Numerical-Conversions-to-QuantumOptics.jl" title="Permalink"></a></h2><p>Bosonic systems can be translated to the ket representation with <code>express</code>. For instance:</p><pre><code class="language-julia-repl hljs">julia> f = FockState(1); | ||
|
|
||
| julia> express(f) | ||
| Ket(dim=3) | ||
| basis: Fock(cutoff=2) | ||
| 0.0 + 0.0im | ||
| 1.0 + 0.0im | ||
| 0.0 + 0.0im | ||
|
|
||
| julia> express(Create) |> dense | ||
| Operator(dim=3x3) | ||
| basis: Fock(cutoff=2) | ||
| 0.0+0.0im 0.0+0.0im 0.0+0.0im | ||
| 1.0+0.0im 0.0+0.0im 0.0+0.0im | ||
| 0.0+0.0im 1.41421+0.0im 0.0+0.0im | ||
|
|
||
| julia> express(Create*f) | ||
| Ket(dim=3) | ||
| basis: Fock(cutoff=2) | ||
| 0.0 + 0.0im | ||
| 0.0 + 0.0im | ||
| 1.4142135623730951 + 0.0im | ||
|
|
||
| julia> express(Destroy*f) | ||
| Ket(dim=3) | ||
| basis: Fock(cutoff=2) | ||
| 1.0 + 0.0im | ||
| 0.0 + 0.0im | ||
| 0.0 + 0.0im</code></pre><div class="admonition is-warning"><header class="admonition-header">Cutoff specifications for numerical representations of quantum harmonic oscillators</header><div class="admonition-body"><p>Symbolic bosonic states and operators are naturally represented in an infinite dimension basis. For numerical conversions of such quantum objects, a finite cutoff of the highest allowed state must be defined. By default, the basis dimension of numerical conversions is set to 3 (so the number representation cutoff is 2), as demonstrated above. To define a different cutoff, one must customize the <code>QuantumOpticsRepr</code> instance, e.g. provide <code>QuantumOpticsRepr(cutoff=n::Int)</code> to <code>express</code>.</p></div></div><p>If we wish to specify a different numerical cutoff, say 4, to the previous examples, then we rewrite them as follows:</p><pre><code class="language-julia-repl hljs">julia> f = FockState(1); | ||
|
|
||
| julia> express(f, QuantumOpticsRepr(cutoff=4)) | ||
| Ket(dim=5) | ||
| basis: Fock(cutoff=4) | ||
| 0.0 + 0.0im | ||
| 1.0 + 0.0im | ||
| 0.0 + 0.0im | ||
| 0.0 + 0.0im | ||
| 0.0 + 0.0im | ||
|
|
||
| julia> express(Create, QuantumOpticsRepr(4)) |> dense | ||
| Operator(dim=5x5) | ||
| basis: Fock(cutoff=4) | ||
| 0.0+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im | ||
| 1.0+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im | ||
| 0.0+0.0im 1.41421+0.0im 0.0+0.0im 0.0+0.0im 0.0+0.0im | ||
| 0.0+0.0im 0.0+0.0im 1.73205+0.0im 0.0+0.0im 0.0+0.0im | ||
| 0.0+0.0im 0.0+0.0im 0.0+0.0im 2.0+0.0im 0.0+0.0im</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../qubit_basis/">« Qubit Basis Choice</a><a class="docs-footer-nextpage" href="../API/">API »</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="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Sunday 11 August 2024 22:36">Sunday 11 August 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> |
Oops, something went wrong.