|
14 | 14 | ops = [m, m', c, c'] # Operators for meanfield evolution |
15 | 15 |
|
16 | 16 | eqs_RWA = meanfield(ops, H_RWA_sym, [m, c]; rates=[γm, γk], order=1) |
17 | | -eqs_completed_RWA = complete(eqs_RWA) # Meanfield equations using QuantumCumulants.jl</code></pre><p>\begin{align} \frac{d}{dt} \langle m\rangle &= -1 i {\Omega}d -1 i \langle c\rangle ^{2} Vk -0.5 {\gamma}m \langle m\rangle -1 i \Delta \langle m\rangle \ |
18 | | -\frac{d}{dt} \langle m^\dagger\rangle &= 1 i {\Omega}d -0.5 {\gamma}m \langle m^\dagger\rangle + 1 i Vk \langle c^\dagger\rangle ^{2} + 1 i \Delta \langle m^\dagger\rangle \ |
19 | | -\frac{d}{dt} \langle c\rangle &= \frac{-1}{2} i \Delta \langle c\rangle -0.5 \langle c\rangle {\gamma}k -2 i \langle m\rangle Vk \langle c^\dagger\rangle \ |
20 | | -\frac{d}{dt} \langle c^\dagger\rangle &= -0.5 {\gamma}k \langle c^\dagger\rangle + 2 i \langle c\rangle Vk \langle m^\dagger\rangle + \frac{1}{2} i \Delta \langle c^\dagger\rangle \end{align}</p><p>We can use this meanfield equations to construct a <code>HarmonicEquation</code> object in HarmonicSteadyState.jl. In the construction, additional information is computed, such as the Jacobian of the equations, which is used to determine the stability if the the steady states.</p><pre><code class="language-julia hljs">harmonic_eq = HarmonicEquation(eqs_completed_RWA, param)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">A set of 4 harmonic equations |
| 17 | +eqs_completed_RWA = complete(eqs_RWA) # Meanfield equations using QuantumCumulants.jl</code></pre><p>\begin{align} \frac{d}{dt} \langle m\rangle &= -1 i \Delta \langle m\rangle -1 i {\Omega}d -1 i Vk \langle c\rangle ^{2} -0.5 {\gamma}m \langle m\rangle \ |
| 18 | +\frac{d}{dt} \langle m^\dagger\rangle &= 1 i \Delta \langle m^\dagger\rangle + 1 i {\Omega}d + 1 i Vk \langle c^\dagger\rangle ^{2} -0.5 {\gamma}m \langle m^\dagger\rangle \ |
| 19 | +\frac{d}{dt} \langle c\rangle &= \frac{-1}{2} i \Delta \langle c\rangle -2 i \langle m\rangle Vk \langle c^\dagger\rangle -0.5 {\gamma}k \langle c\rangle \ |
| 20 | +\frac{d}{dt} \langle c^\dagger\rangle &= 2 i \langle m^\dagger\rangle Vk \langle c\rangle + \frac{1}{2} i \Delta \langle c^\dagger\rangle -0.5 {\gamma}k \langle c^\dagger\rangle \end{align}</p><p>We can use this meanfield equations to construct a <code>HarmonicEquation</code> object in HarmonicSteadyState.jl. In the construction, additional information is computed, such as the Jacobian of the equations, which is used to determine the stability if the the steady states.</p><pre><code class="language-julia hljs">harmonic_eq = HarmonicEquation(eqs_completed_RWA, param)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">A set of 4 harmonic equations |
21 | 21 | Variables: mᵣ(t), mᵢ(t), cᵣ(t), cᵢ(t) |
22 | 22 | Parameters: Δ, Vk, Ωd, γm, γk |
23 | 23 |
|
|
40 | 40 |
|
41 | 41 | plot(plot(result; y="1/sqrt(2)*(mᵣ+ mᵢ)"), plot(result; y="1/sqrt(2)*(cᵣ + cᵢ)")) |
42 | 42 |
|
43 | | -# Linear response and S21</code></pre><img src="d0481f30.svg" alt="Example block output"/><p>To find the response of the driven system to a second, weak probe, we use the method described <a href="https://quantumengineeredsystems.github.io/HarmonicBalance.jl/stable/background/stability_response#linresp_background">here</a>. Here, we calculate the response in the same rotating frame as the Hamiltonian. The linear response is related to the scattering parameter <span>$S_{21}$</span> by <span>$S_{21}(\omega)=1-\sqrt{\kappa_{ext}} \chi(\omega),$</span> where <span>$\kappa_{ext}$</span> is the coupling of the system to the measurement apparatus.</p><p>The result below shows the characteristic splitting of the magnon resonance above the power threshold, which matches the experiment.</p><pre><code class="language-julia hljs">Ω_range = range(-0.1, 0.1, 500) |
| 43 | +# Linear response and S21</code></pre><img src="1ad6991b.svg" alt="Example block output"/><p>To find the response of the driven system to a second, weak probe, we use the method described <a href="https://quantumengineeredsystems.github.io/HarmonicBalance.jl/stable/background/stability_response#linresp_background">here</a>. Here, we calculate the response in the same rotating frame as the Hamiltonian. The linear response is related to the scattering parameter <span>$S_{21}$</span> by <span>$S_{21}(\omega)=1-\sqrt{\kappa_{ext}} \chi(\omega),$</span> where <span>$\kappa_{ext}$</span> is the coupling of the system to the measurement apparatus.</p><p>The result below shows the characteristic splitting of the magnon resonance above the power threshold, which matches the experiment.</p><pre><code class="language-julia hljs">Ω_range = range(-0.1, 0.1, 500) |
44 | 44 | χ3 = get_susceptibility(result, 1, Ω_range, 3); |
45 | 45 | χ1 = get_susceptibility(result, 1, Ω_range, 1); |
46 | 46 | κ_ext = 0.05 |
|
72 | 72 | Ω_range, drive_range, vcat(S21_log_1', S21_log_3'); c=:matter, cbar_title="S21 (dB)" |
73 | 73 | ) |
74 | 74 | ylabel!("Ω_d") |
75 | | -xlabel!("Probe detuning")</code></pre><img src="c4ab457d.svg" alt="Example block output"/><hr/><p><em>This page was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em></p></article><nav class="docs-footer"><a class="docs-footer-prevpage" 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 1.14.1 on <span class="colophon-date" title="Wednesday 23 July 2025 14:21">Wednesday 23 July 2025</span>. Using Julia version 1.10.10.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> |
| 75 | +xlabel!("Probe detuning")</code></pre><img src="e43a42d4.svg" alt="Example block output"/><hr/><p><em>This page was generated using <a href="https://github.com/fredrikekre/Literate.jl">Literate.jl</a>.</em></p></article><nav class="docs-footer"><a class="docs-footer-prevpage" 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 1.14.1 on <span class="colophon-date" title="Tuesday 12 August 2025 07:13">Tuesday 12 August 2025</span>. Using Julia version 1.10.10.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> |
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