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Copy file name to clipboardExpand all lines: QuantumToolbox.jl/time_evolution/Dicke.qmd
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---
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title: "The Dicke Model"
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author: Li-Xun Cai
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date: 2025-05-22#last update (keep this comment as a reminder)
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date: last-modified
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---
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Inspirations taken from [this QuTiP tutorial](https://nbviewer.org/urls/qutip.org/qutip-tutorials/tutorials-v5/lectures/Lecture-3A-Dicke-model.ipynb) by J. R. Johansson.
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```{julia}
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using QuantumToolbox
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using CairoMakie
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CairoMakie.activate!(type = "svg")
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```
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```{julia}
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```
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```{julia}
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fig = Figure(size = (800, 300))
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fig = Figure(size=(900, 350))
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axn = Axis(
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fig[1,1],
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xlabel = "interaction strength",
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ylims!(-N0/2, N0/2)
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lines!(axn, gs, real(nvec))
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lines!(axJz, gs, real(Jzvec))
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display(fig);
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fig
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```
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The expectation value of photon number and $\hat{J}_z$ showed a sudden increment around $g_c$.
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As $g$ increases, the cavity ground state's wigner function plot looks more coherent than a thermal state.
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```
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```{julia}
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fig = Figure(size=(800, 400))
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fig = Figure(size=(900, 400))
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ax = Axis(fig[1,1])
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ax.xlabel = "coupling strength"
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ax.ylabel = "mutual entropy"
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end
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Legend(fig[1,2], ax, label = "number of atoms")
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display(fig);
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fig
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```
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We further consult mutual entropy between the cavity and the spins as a measure of their correlation; the result showed that as the number of atoms $N$ increases, the peak of mutual entropy moves closer to $g_c$.
date: 2025-05-22#last update (keep this comment as a reminder)
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date: last-modified
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---
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Inspirations taken from [the QuTiP tutorial](https://nbviewer.org/urls/qutip.org/qutip-tutorials/tutorials-v5/lectures/Lecture-8-Adiabatic-quantum-computing.ipynb) by J. R. Johansson.
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```{julia}
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using QuantumToolbox
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using CairoMakie
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CairoMakie.activate!(type = "svg")
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```
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```{julia}
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```
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```{julia}
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fig = Figure(size=(800, 400))
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fig = Figure(size=(900, 400))
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ax = Axis(fig[1,1], xticks = (0:0.25:1, ["$(t)T" for t in 0:0.25:1]))
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for idx in 1:20 # only check for the lowest 20 eigenvalues
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color = (idx == 1) ? :magenta : (:gray,0.5)
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lines!(ax, range(0,1,length(tlist)), eigs[:,idx], label = string(idx), color = color)
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end
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display(fig)
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fig
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```
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The plot shows that the gap is nonvanishing and thus validates the evolution. So we proceed to check the expectation value dynamics of the final Hamiltonian and the fidelity dynamics to the truthful ground state throughout the evolution.
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```
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```{julia}
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fig = Figure(size=(800, 400))
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fig = Figure(size=(900, 400))
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axs = Axis.([fig[1,1], fig[1,2]])
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axs[1].title = L"\langle H_f \rangle"
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axs[1].xticks = (0:0.25:1, ["$(t)T" for t in 0:0.25:1])
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Legend(fig[1,3], axs[1], L"T")
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display(fig)
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fig
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```
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As the plot showed, the fidelity between the prepared final state and the truthful ground state reaches 1 as the total evolution time $T$ increases, showcasing the requirement of the adiabatic theorem that the change has to be gradual.
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