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Created October 4, 2022 15:56
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Example for using an `info_hook` to access GRAPE pulses
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simple_state_to_state.ipynb
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Example 1: Optimization of a State-to-State Transfer in a Two-Level-System"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$\n",
"\\newcommand{tr}[0]{\\operatorname{tr}}\n",
"\\newcommand{diag}[0]{\\operatorname{diag}}\n",
"\\newcommand{abs}[0]{\\operatorname{abs}}\n",
"\\newcommand{pop}[0]{\\operatorname{pop}}\n",
"\\newcommand{aux}[0]{\\text{aux}}\n",
"\\newcommand{opt}[0]{\\text{opt}}\n",
"\\newcommand{tgt}[0]{\\text{tgt}}\n",
"\\newcommand{init}[0]{\\text{init}}\n",
"\\newcommand{lab}[0]{\\text{lab}}\n",
"\\newcommand{rwa}[0]{\\text{rwa}}\n",
"\\newcommand{bra}[1]{\\langle#1\\vert}\n",
"\\newcommand{ket}[1]{\\vert#1\\rangle}\n",
"\\newcommand{Bra}[1]{\\left\\langle#1\\right\\vert}\n",
"\\newcommand{Ket}[1]{\\left\\vert#1\\right\\rangle}\n",
"\\newcommand{Braket}[2]{\\left\\langle #1\\vphantom{#2}\\mid{#2}\\vphantom{#1}\\right\\rangle}\n",
"\\newcommand{op}[1]{\\hat{#1}}\n",
"\\newcommand{Op}[1]{\\hat{#1}}\n",
"\\newcommand{dd}[0]{\\,\\text{d}}\n",
"\\newcommand{Liouville}[0]{\\mathcal{L}}\n",
"\\newcommand{DynMap}[0]{\\mathcal{E}}\n",
"\\newcommand{identity}[0]{\\mathbf{1}}\n",
"\\newcommand{Norm}[1]{\\lVert#1\\rVert}\n",
"\\newcommand{Abs}[1]{\\left\\vert#1\\right\\vert}\n",
"\\newcommand{avg}[1]{\\langle#1\\rangle}\n",
"\\newcommand{Avg}[1]{\\left\\langle#1\\right\\rangle}\n",
"\\newcommand{AbsSq}[1]{\\left\\vert#1\\right\\vert^2}\n",
"\\newcommand{Re}[0]{\\operatorname{Re}}\n",
"\\newcommand{Im}[0]{\\operatorname{Im}}\n",
"$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This first example illustrates the basic use of the `GRAPE.jl` by solving a simple canonical optimization problem: the transfer of population in a two level system."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"ExecuteTime": {
"end_time": "2022-10-04T11:48:49.259000-04:00",
"start_time": "2022-10-04T15:48:47.997Z"
}
},
"outputs": [],
"source": [
"using DrWatson\n",
"@quickactivate \"GRAPETests\""
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"ExecuteTime": {
"end_time": "2022-10-04T11:48:53.805000-04:00",
"start_time": "2022-10-04T15:48:48.013Z"
}
},
"outputs": [],
"source": [
"using QuantumControl"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Two-level Hamiltonian"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We consider the Hamiltonian $\\op{H}_{0} = - \\frac{\\omega}{2} \\op{\\sigma}_{z}$, representing a simple qubit with energy level splitting $\\omega$ in the basis $\\{\\ket{0},\\ket{1}\\}$. The control field $\\epsilon(t)$ is assumed to couple via the Hamiltonian $\\op{H}_{1}(t) = \\epsilon(t) \\op{\\sigma}_{x}$ to the qubit, i.e., the control field effectively drives transitions between both qubit states.\n",
"\n",
"We we will use"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"ExecuteTime": {
"end_time": "2022-10-04T11:48:53.939000-04:00",
"start_time": "2022-10-04T15:48:48.022Z"
}
},
"outputs": [],
"source": [
"ϵ(t) = 0.2 * QuantumControl.Shapes.flattop(t, T=5, t_rise=0.3, func=:blackman);"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"ExecuteTime": {
"end_time": "2022-10-04T11:48:54.014000-04:00",
"start_time": "2022-10-04T15:48:48.027Z"
}
},
"outputs": [],
"source": [
"\"\"\"Two-level-system Hamiltonian.\"\"\"\n",
"function hamiltonian(Ω=1.0, ϵ=ϵ)\n",
" σ̂_z = ComplexF64[\n",
" 1 0\n",
" 0 -1\n",
" ]\n",
" σ̂_x = ComplexF64[\n",
" 0 1\n",
" 1 0\n",
" ]\n",
" Ĥ₀ = -0.5 * Ω * σ̂_z\n",
" Ĥ₁ = σ̂_x\n",
" return (Ĥ₀, (Ĥ₁, ϵ))\n",
"end;"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"ExecuteTime": {
"end_time": "2022-10-04T11:48:54.103000-04:00",
"start_time": "2022-10-04T15:48:48.029Z"
}
},
"outputs": [],
"source": [
"H = hamiltonian();"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The control field here switches on from zero at $t=0$ to it's maximum amplitude\n",
"0.2 within the time period 0.3 (the switch-on shape is half a [Blackman pulse](https://en.wikipedia.org/wiki/Window_function#Blackman_window)).\n",
"It switches off again in the time period 0.3 before the\n",
"final time $T=5$). We use a time grid with 500 time steps between 0 and $T$:"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"ExecuteTime": {
"end_time": "2022-10-04T11:48:54.202000-04:00",
"start_time": "2022-10-04T15:48:48.032Z"
}
},
"outputs": [],
"source": [
"tlist = collect(range(0, 5, length=500));"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"ExecuteTime": {
"end_time": "2022-10-04T11:48:59.928000-04:00",
"start_time": "2022-10-04T15:48:48.034Z"
}
},
"outputs": [],
"source": [
"using Plots\n",
"Plots.default(\n",
" linewidth = 3,\n",
" size = (550, 300),\n",
" legend = :right,\n",
" foreground_color_legend = nothing,\n",
" background_color_legend = RGBA(1, 1, 1, 0.8)\n",
")"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"ExecuteTime": {
"end_time": "2022-10-04T11:49:00.250000-04:00",
"start_time": "2022-10-04T15:48:48.036Z"
}
},
"outputs": [],
"source": [
"function plot_control(pulse::Vector, tlist)\n",
" plot(tlist, pulse, xlabel=\"time\", ylabel=\"amplitude\", legend=false)\n",
"end\n",
"\n",
"plot_control(ϵ::T, tlist) where {T<:Function} = plot_control([ϵ(t) for t in tlist], tlist);"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"ExecuteTime": {
"end_time": "2022-10-04T11:49:04.866000-04:00",
"start_time": "2022-10-04T15:48:48.041Z"
}
},
"outputs": [
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" \n",
" \"/>\n",
"</svg>\n"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"fig = plot_control(H[2][2], tlist)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Optimization target"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"First, we define a convenience function for the eigenstates."
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"ExecuteTime": {
"end_time": "2022-10-04T11:49:04.868000-04:00",
"start_time": "2022-10-04T15:48:48.049Z"
}
},
"outputs": [],
"source": [
"function ket(label)\n",
" result = Dict(\"0\" => Vector{ComplexF64}([1, 0]), \"1\" => Vector{ComplexF64}([0, 1]))\n",
" return result[string(label)]\n",
"end;"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The physical objective of our optimization is to transform the initial state $\\ket{0}$ into the target state $\\ket{1}$ under the time evolution induced by the Hamiltonian $\\op{H}(t)$."
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"ExecuteTime": {
"end_time": "2022-10-04T11:49:04.998000-04:00",
"start_time": "2022-10-04T15:48:48.052Z"
}
},
"outputs": [],
"source": [
"objectives = [Objective(initial_state=ket(0), generator=H, target_state=ket(1))];"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The full control problem includes this objective, information about the time grid for the dynamics, and the functional to be used (the square modulus of the overlap $\\tau$ with the target state in this case)."
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"ExecuteTime": {
"end_time": "2022-10-04T11:49:05.072000-04:00",
"start_time": "2022-10-04T15:48:48.057Z"
}
},
"outputs": [],
"source": [
"using QuantumControl.Functionals: J_T_sm\n",
"\n",
"problem = ControlProblem(\n",
" objectives=objectives,\n",
" tlist=tlist,\n",
" pulse_options=Dict(),\n",
" iter_stop=500,\n",
" J_T=J_T_sm,\n",
" check_convergence=res -> begin\n",
" ((res.J_T < 1e-3) && (res.converged = true) && (res.message = \"J_T < 10⁻³\"))\n",
" end,\n",
");"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Simulate dynamics under the guess field"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Before running the optimization procedure, we first simulate the dynamics under the guess field $\\epsilon_{0}(t)$. The following solves equation of motion for the defined objective, which contains the initial state $\\ket{\\Psi_{\\init}}$ and the Hamiltonian $\\op{H}(t)$ defining its evolution."
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"ExecuteTime": {
"end_time": "2022-10-04T11:49:10.555000-04:00",
"start_time": "2022-10-04T15:48:48.063Z"
}
},
"outputs": [
{
"data": {
"text/plain": [
"2×500 Matrix{Float64}:\n",
" 1.0 1.0 1.0 1.0 … 0.951457 0.951459 0.951459\n",
" 0.0 7.73456e-40 2.03206e-11 2.96638e-10 0.0485427 0.048541 0.048541"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"guess_dynamics = propagate_objective(\n",
" objectives[1],\n",
" problem.tlist;\n",
" storage=true,\n",
" observables=(Ψ -> abs.(Ψ) .^ 2,)\n",
")"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {
"ExecuteTime": {
"end_time": "2022-10-04T11:49:10.557000-04:00",
"start_time": "2022-10-04T15:48:48.065Z"
}
},
"outputs": [],
"source": [
"function plot_population(pop0::Vector, pop1::Vector, tlist)\n",
" fig = plot(tlist, pop0, label=\"0\", xlabel=\"time\", ylabel=\"population\")\n",
" plot!(fig, tlist, pop1; label=\"1\")\n",
"end;"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {
"ExecuteTime": {
"end_time": "2022-10-04T11:49:10.796000-04:00",
"start_time": "2022-10-04T15:48:48.070Z"
}
},
"outputs": [
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" 1378.55,204.347 1382.17,204.406 1385.79,204.459 1389.41,204.504 1393.03,204.541 1396.64,204.571 1400.26,204.593 1403.88,204.608 1407.5,204.616 1411.12,204.615 \n",
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" 1740.38,175.579 1744,174.995 1747.62,174.407 1751.24,173.815 1754.85,173.219 1758.47,172.619 1762.09,172.015 1765.71,171.407 1769.33,170.796 1772.94,170.182 \n",
" 1776.56,169.563 1780.18,168.942 1783.8,168.316 1787.42,167.688 1791.04,167.056 1794.65,166.422 1798.27,165.784 1801.89,165.143 1805.51,164.499 1809.13,163.853 \n",
" 1812.75,163.203 1816.36,162.551 1819.98,161.896 1823.6,161.239 1827.22,160.579 1830.84,159.917 1834.46,159.253 1838.07,158.586 1841.69,157.917 1845.31,157.246 \n",
" 1848.93,156.573 1852.55,155.898 1856.17,155.222 1859.78,154.543 1863.4,153.863 1867.02,153.181 1870.64,152.498 1874.26,151.813 1877.88,151.127 1881.49,150.439 \n",
" 1885.11,149.751 1888.73,149.061 1892.35,148.37 1895.97,147.678 1899.58,146.985 1903.2,146.292 1906.82,145.597 1910.44,144.903 1914.06,144.207 1917.68,143.511 \n",
" 1921.29,142.815 1924.91,142.118 1928.53,141.421 1932.15,140.724 1935.77,140.027 1939.39,139.329 1943,138.632 1946.62,137.935 1950.24,137.238 1953.86,136.542 \n",
" 1957.48,135.846 1961.1,135.15 1964.71,134.455 1968.33,133.761 1971.95,133.067 1975.57,132.375 1979.19,131.683 1982.81,130.992 1986.42,130.302 1990.04,129.613 \n",
" 1993.66,128.926 1997.28,128.246 2000.9,127.58 2004.51,126.931 2008.13,126.307 2011.75,125.711 2015.37,125.147 2018.99,124.619 2022.61,124.129 2026.22,123.678 \n",
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" \n",
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]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"fig = plot_population(guess_dynamics[1, :], guess_dynamics[2, :], tlist)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Optimization with LBFGSB"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In the following we optimize the guess field $\\epsilon_{0}(t)$ such that the intended state-to-state transfer $\\ket{\\Psi_{\\init}} \\rightarrow \\ket{\\Psi_{\\tgt}}$ is solved."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The GRAPE package performs the optimization by calculating the gradient of $J_T$ with respect to the values of the control field at each point in time. This gradient is then fed into a backend solver that calculates an appropriate update based on that gradient."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"By default, this backend is [LBFGSB.jl](https://github.com/Gnimuc/LBFGSB.jl), a wrapper around the true and tested [L-BFGS-B Fortran library](http://users.iems.northwestern.edu/%7Enocedal/lbfgsb.html). L-BFGS-B is a pseudo-Hessian method: it efficiently estimates the second-order Hessian from the gradient information. The search direction determined from that Hessian dramatically improves convergence compared to using the gradient directly as a search direction. The L-BFGS-B method performs its own linesearch to determine how far to go in the search direction."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"It can be quite instructive to see how the improvement in the pseudo-Hessian search direction compares to the gradient, how the linesearch finds an appropriate step width. For this purpose, we have a [GRAPELinesearchAnalysis](https://github.com/JuliaQuantumControl/GRAPELinesearchAnalysis.jl) package that automatically generates plots in every iteration of the optimization showing the linesearch behavior"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {
"ExecuteTime": {
"end_time": "2022-10-04T11:49:10.944000-04:00",
"start_time": "2022-10-04T15:48:48.072Z"
}
},
"outputs": [],
"source": [
"using GRAPELinesearchAnalysis"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We feed this into the optimization as part of the `info_hook`."
]
},
{
"cell_type": "code",
"execution_count": 39,
"metadata": {
"ExecuteTime": {
"end_time": "2022-10-04T11:51:36.682000-04:00",
"start_time": "2022-10-04T15:51:36.679Z"
}
},
"outputs": [
{
"data": {
"text/plain": [
"store_pulses (generic function with 3 methods)"
]
},
"execution_count": 39,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function store_pulses(wrk, iteration, _...)\n",
" L = length(wrk.controls)\n",
" ϵ_opt = reshape(wrk.pulsevals, L, :)\n",
" opt_pulses = [QuantumControl.Controls.discretize_on_midpoints(ϵ_opt[l, :], tlist) for l=1:L]\n",
" return Tuple(opt_pulses)\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {
"ExecuteTime": {
"end_time": "2022-10-04T11:52:05.532000-04:00",
"start_time": "2022-10-04T15:51:37.447Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" iter. J_T |∇J_T| ΔJ_T FG(F) secs\n",
" 0 9.51e-01 5.40e-02 n/a 1(0) 0.0\n",
" 1 4.45e-02 6.57e-02 -9.07e-01 4(0) 0.2\n",
" 2 1.89e-02 4.38e-02 -2.56e-02 3(0) 5.6\n",
" 3 5.75e-03 2.33e-02 -1.31e-02 1(0) 5.7\n",
" 4 1.03e-05 1.04e-03 -5.74e-03 1(0) 11.4\n"
]
}
],
"source": [
"opt_result_LBFGSB = @optimize_or_load(\n",
" datadir(\"TLS\", \"opt_result_LBFGSB.jld2\"),\n",
" problem,\n",
" method = :grape,\n",
" force = true,\n",
" info_hook = chain_infohooks(\n",
" GRAPELinesearchAnalysis.plot_linesearch(datadir(\"TLS\", \"Linesearch\", \"LBFGSB\")),\n",
" QuantumControl.GRAPE.print_table,\n",
" store_pulses\n",
" )\n",
");"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"When going through this tutorial locally, the [generated images for the linesearch](https://github.com/JuliaQuantumControl/GRAPE.jl/tree/data-dump/TLS/Linesearch/LBFGSB) can be found in `docs/TLS/Linesearch/LBFGSB`."
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {
"ExecuteTime": {
"end_time": "2022-10-04T11:50:59.523000-04:00",
"start_time": "2022-10-04T15:50:59.514Z"
}
},
"outputs": [
{
"data": {
"text/plain": [
"GRAPE Optimization Result\n",
"-------------------------\n",
"- Started at 2022-10-04T11:50:17.517\n",
"- Number of objectives: 1\n",
"- Number of iterations: 4\n",
"- Number of pure func evals: 0\n",
"- Number of func/grad evals: 10\n",
"- Value of functional: 1.02765e-05\n",
"- Reason for termination: J_T < 10⁻³\n",
"- Ended at 2022-10-04T11:50:41.448 (23 seconds, 931 milliseconds)\n"
]
},
"execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"opt_result_LBFGSB"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {
"ExecuteTime": {
"end_time": "2022-10-04T11:51:19.292000-04:00",
"start_time": "2022-10-04T15:51:18.752Z"
}
},
"outputs": [
{
"data": {
"text/plain": [
"5-element Vector{Tuple}:\n",
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]
},
"execution_count": 37,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"opt_result_LBFGSB.records"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can plot the optimized field:"
]
},
{
"cell_type": "code",
"execution_count": 49,
"metadata": {
"ExecuteTime": {
"end_time": "2022-10-04T11:55:21.702000-04:00",
"start_time": "2022-10-04T15:55:21.645Z"
}
},
"outputs": [
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" 1129.47,115.07 1133.08,114.094 1136.68,113.169 1140.28,112.294 1143.89,111.47 1147.49,110.697 1151.09,109.975 1154.7,109.305 1158.3,108.685 1161.9,108.117 \n",
" 1165.51,107.6 1169.11,107.135 1172.71,106.721 1176.32,106.359 1179.92,106.049 1183.53,105.79 1187.13,105.583 1190.73,105.427 1194.34,105.324 1197.94,105.272 \n",
" 1201.54,105.272 1205.15,105.324 1208.75,105.427 1212.35,105.583 1215.96,105.79 1219.56,106.049 1223.16,106.359 1226.77,106.721 1230.37,107.135 1233.97,107.6 \n",
" 1237.58,108.117 1241.18,108.685 1244.78,109.305 1248.39,109.975 1251.99,110.697 1255.6,111.47 1259.2,112.294 1262.8,113.169 1266.41,114.094 1270.01,115.07 \n",
" 1273.61,116.096 1277.22,117.173 1280.82,118.299 1284.42,119.476 1288.03,120.702 1291.63,121.979 1295.23,123.304 1298.84,124.679 1302.44,126.103 1306.04,127.576 \n",
" 1309.65,129.098 1313.25,130.668 1316.85,132.286 1320.46,133.953 1324.06,135.667 1327.66,137.429 1331.27,139.238 1334.87,141.095 1338.48,142.998 1342.08,144.948 \n",
" 1345.68,146.944 1349.29,148.986 1352.89,151.074 1356.49,153.207 1360.1,155.386 1363.7,157.61 1367.3,159.878 1370.91,162.19 1374.51,164.547 1378.11,166.947 \n",
" 1381.72,169.391 1385.32,171.877 1388.92,174.407 1392.53,176.979 1396.13,179.592 1399.73,182.248 1403.34,184.944 1406.94,187.682 1410.55,190.461 1414.15,193.279 \n",
" 1417.75,196.138 1421.36,199.036 1424.96,201.973 1428.56,204.948 1432.17,207.963 1435.77,211.015 1439.37,214.104 1442.98,217.231 1446.58,220.395 1450.18,223.595 \n",
" 1453.79,226.83 1457.39,230.102 1460.99,233.408 1464.6,236.749 1468.2,240.124 1471.8,243.533 1475.41,246.976 1479.01,250.451 1482.61,253.959 1486.22,257.498 \n",
" 1489.82,261.07 1493.43,264.672 1497.03,268.305 1500.63,271.968 1504.24,275.661 1507.84,279.383 1511.44,283.134 1515.05,286.914 1518.65,290.721 1522.25,294.555 \n",
" 1525.86,298.416 1529.46,302.304 1533.06,306.217 1536.67,310.156 1540.27,314.12 1543.87,318.108 1547.48,322.12 1551.08,326.155 1554.68,330.214 1558.29,334.294 \n",
" 1561.89,338.397 1565.5,342.521 1569.1,346.665 1572.7,350.831 1576.31,355.015 1579.91,359.22 1583.51,363.442 1587.12,367.684 1590.72,371.943 1594.32,376.219 \n",
" 1597.93,380.511 1601.53,384.82 1605.13,389.145 1608.74,393.484 1612.34,397.838 1615.94,402.206 1619.55,406.587 1623.15,410.982 1626.75,415.388 1630.36,419.807 \n",
" 1633.96,424.237 1637.56,428.677 1641.17,433.128 1644.77,437.588 1648.38,442.057 1651.98,446.535 1655.58,451.021 1659.19,455.514 1662.79,460.015 1666.39,464.521 \n",
" 1670,469.033 1673.6,473.551 1677.2,478.073 1680.81,482.599 1684.41,487.129 1688.01,491.661 1691.62,496.196 1695.22,500.733 1698.82,505.271 1702.43,509.81 \n",
" 1706.03,514.35 1709.63,518.889 1713.24,523.426 1716.84,527.963 1720.45,532.497 1724.05,537.029 1727.65,541.557 1731.26,546.082 1734.86,550.603 1738.46,555.119 \n",
" 1742.07,559.629 1745.67,564.133 1749.27,568.631 1752.88,573.122 1756.48,577.606 1760.08,582.081 1763.69,586.547 1767.29,591.004 1770.89,595.451 1774.5,599.888 \n",
" 1778.1,604.314 1781.7,608.729 1785.31,613.131 1788.91,617.521 1792.52,621.898 1796.12,626.262 1799.72,630.611 1803.33,634.945 1806.93,639.265 1810.53,643.568 \n",
" 1814.14,647.855 1817.74,652.125 1821.34,656.378 1824.95,660.613 1828.55,664.83 1832.15,669.028 1835.76,673.206 1839.36,677.364 1842.96,681.502 1846.57,685.619 \n",
" 1850.17,689.714 1853.77,693.788 1857.38,697.839 1860.98,701.866 1864.58,705.871 1868.19,709.851 1871.79,713.807 1875.4,717.738 1879,721.644 1882.6,725.523 \n",
" 1886.21,729.376 1889.81,733.203 1893.41,737.002 1897.02,740.773 1900.62,744.516 1904.22,748.23 1907.83,751.915 1911.43,755.57 1915.03,759.195 1918.64,762.79 \n",
" 1922.24,766.354 1925.84,769.886 1929.45,773.387 1933.05,776.855 1936.65,780.291 1940.26,783.694 1943.86,787.063 1947.47,790.398 1951.07,793.699 1954.67,796.965 \n",
" 1958.28,800.196 1961.88,803.392 1965.48,806.552 1969.09,809.676 1972.69,812.763 1976.29,815.813 1979.9,818.826 1983.5,821.802 1987.1,824.739 1990.71,827.689 \n",
" 1994.31,831.09 1997.91,835.401 2001.52,840.668 2005.12,846.833 2008.72,853.815 2012.33,861.517 2015.93,869.824 2019.53,878.612 2023.14,887.749 2026.74,897.099 \n",
" 2030.35,906.528 2033.95,915.908 2037.55,925.12 2041.16,934.054 2044.76,942.62 2048.36,950.739 2051.97,958.353 2055.57,965.423 2059.17,971.925 2062.78,977.853 \n",
" 2066.38,983.216 2069.98,988.035 2073.59,992.341 2077.19,996.17 2080.79,999.565 2084.4,1002.56 2088,1005.21 2091.6,1007.53 2095.21,1009.57 2098.81,1010.52 \n",
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" 300.67,563.349 304.274,563.218 307.877,562.85 311.481,562.244 315.084,561.358 318.688,560.161 322.291,558.615 325.895,556.678 329.498,554.309 333.102,551.471 \n",
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" 372.74,490.013 376.344,483.161 379.947,476.563 383.551,470.356 387.154,464.67 390.758,459.63 394.361,455.35 397.965,451.927 401.568,449.442 405.172,447.952 \n",
" 408.775,447.411 412.378,447.361 415.982,447.361 419.585,447.361 423.189,447.361 426.792,447.361 430.396,447.361 433.999,447.361 437.603,447.361 441.206,447.361 \n",
" 444.81,447.361 448.413,447.361 452.017,447.361 455.62,447.361 459.224,447.361 462.827,447.361 466.431,447.361 470.034,447.361 473.638,447.361 477.241,447.361 \n",
" 480.845,447.361 484.448,447.361 488.052,447.361 491.655,447.361 495.259,447.361 498.862,447.361 502.466,447.361 506.069,447.361 509.673,447.361 513.276,447.361 \n",
" 516.88,447.361 520.483,447.361 524.087,447.361 527.69,447.361 531.294,447.361 534.897,447.361 538.501,447.361 542.104,447.361 545.708,447.361 549.311,447.361 \n",
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" 588.949,447.361 592.553,447.361 596.156,447.361 599.76,447.361 603.363,447.361 606.967,447.361 610.57,447.361 614.174,447.361 617.777,447.361 621.381,447.361 \n",
" 624.984,447.361 628.588,447.361 632.191,447.361 635.795,447.361 639.398,447.361 643.002,447.361 646.605,447.361 650.209,447.361 653.812,447.361 657.416,447.361 \n",
" 661.019,447.361 664.623,447.361 668.226,447.361 671.83,447.361 675.433,447.361 679.037,447.361 682.64,447.361 686.244,447.361 689.847,447.361 693.451,447.361 \n",
" 697.054,447.361 700.658,447.361 704.261,447.361 707.865,447.361 711.468,447.361 715.072,447.361 718.675,447.361 722.279,447.361 725.882,447.361 729.486,447.361 \n",
" 733.089,447.361 736.693,447.361 740.296,447.361 743.9,447.361 747.503,447.361 751.107,447.361 754.71,447.361 758.313,447.361 761.917,447.361 765.52,447.361 \n",
" 769.124,447.361 772.727,447.361 776.331,447.361 779.934,447.361 783.538,447.361 787.141,447.361 790.745,447.361 794.348,447.361 797.952,447.361 801.555,447.361 \n",
" 805.159,447.361 808.762,447.361 812.366,447.361 815.969,447.361 819.573,447.361 823.176,447.361 826.78,447.361 830.383,447.361 833.987,447.361 837.59,447.361 \n",
" 841.194,447.361 844.797,447.361 848.401,447.361 852.004,447.361 855.608,447.361 859.211,447.361 862.815,447.361 866.418,447.361 870.022,447.361 873.625,447.361 \n",
" 877.229,447.361 880.832,447.361 884.436,447.361 888.039,447.361 891.643,447.361 895.246,447.361 898.85,447.361 902.453,447.361 906.057,447.361 909.66,447.361 \n",
" 913.264,447.361 916.867,447.361 920.471,447.361 924.074,447.361 927.678,447.361 931.281,447.361 934.884,447.361 938.488,447.361 942.091,447.361 945.695,447.361 \n",
" 949.298,447.361 952.902,447.361 956.505,447.361 960.109,447.361 963.712,447.361 967.316,447.361 970.919,447.361 974.523,447.361 978.126,447.361 981.73,447.361 \n",
" 985.333,447.361 988.937,447.361 992.54,447.361 996.144,447.361 999.747,447.361 1003.35,447.361 1006.95,447.361 1010.56,447.361 1014.16,447.361 1017.76,447.361 \n",
" 1021.37,447.361 1024.97,447.361 1028.58,447.361 1032.18,447.361 1035.78,447.361 1039.39,447.361 1042.99,447.361 1046.59,447.361 1050.2,447.361 1053.8,447.361 \n",
" 1057.4,447.361 1061.01,447.361 1064.61,447.361 1068.21,447.361 1071.82,447.361 1075.42,447.361 1079.02,447.361 1082.63,447.361 1086.23,447.361 1089.83,447.361 \n",
" 1093.44,447.361 1097.04,447.361 1100.65,447.361 1104.25,447.361 1107.85,447.361 1111.46,447.361 1115.06,447.361 1118.66,447.361 1122.27,447.361 1125.87,447.361 \n",
" 1129.47,447.361 1133.08,447.361 1136.68,447.361 1140.28,447.361 1143.89,447.361 1147.49,447.361 1151.09,447.361 1154.7,447.361 1158.3,447.361 1161.9,447.361 \n",
" 1165.51,447.361 1169.11,447.361 1172.71,447.361 1176.32,447.361 1179.92,447.361 1183.53,447.361 1187.13,447.361 1190.73,447.361 1194.34,447.361 1197.94,447.361 \n",
" 1201.54,447.361 1205.15,447.361 1208.75,447.361 1212.35,447.361 1215.96,447.361 1219.56,447.361 1223.16,447.361 1226.77,447.361 1230.37,447.361 1233.97,447.361 \n",
" 1237.58,447.361 1241.18,447.361 1244.78,447.361 1248.39,447.361 1251.99,447.361 1255.6,447.361 1259.2,447.361 1262.8,447.361 1266.41,447.361 1270.01,447.361 \n",
" 1273.61,447.361 1277.22,447.361 1280.82,447.361 1284.42,447.361 1288.03,447.361 1291.63,447.361 1295.23,447.361 1298.84,447.361 1302.44,447.361 1306.04,447.361 \n",
" 1309.65,447.361 1313.25,447.361 1316.85,447.361 1320.46,447.361 1324.06,447.361 1327.66,447.361 1331.27,447.361 1334.87,447.361 1338.48,447.361 1342.08,447.361 \n",
" 1345.68,447.361 1349.29,447.361 1352.89,447.361 1356.49,447.361 1360.1,447.361 1363.7,447.361 1367.3,447.361 1370.91,447.361 1374.51,447.361 1378.11,447.361 \n",
" 1381.72,447.361 1385.32,447.361 1388.92,447.361 1392.53,447.361 1396.13,447.361 1399.73,447.361 1403.34,447.361 1406.94,447.361 1410.55,447.361 1414.15,447.361 \n",
" 1417.75,447.361 1421.36,447.361 1424.96,447.361 1428.56,447.361 1432.17,447.361 1435.77,447.361 1439.37,447.361 1442.98,447.361 1446.58,447.361 1450.18,447.361 \n",
" 1453.79,447.361 1457.39,447.361 1460.99,447.361 1464.6,447.361 1468.2,447.361 1471.8,447.361 1475.41,447.361 1479.01,447.361 1482.61,447.361 1486.22,447.361 \n",
" 1489.82,447.361 1493.43,447.361 1497.03,447.361 1500.63,447.361 1504.24,447.361 1507.84,447.361 1511.44,447.361 1515.05,447.361 1518.65,447.361 1522.25,447.361 \n",
" 1525.86,447.361 1529.46,447.361 1533.06,447.361 1536.67,447.361 1540.27,447.361 1543.87,447.361 1547.48,447.361 1551.08,447.361 1554.68,447.361 1558.29,447.361 \n",
" 1561.89,447.361 1565.5,447.361 1569.1,447.361 1572.7,447.361 1576.31,447.361 1579.91,447.361 1583.51,447.361 1587.12,447.361 1590.72,447.361 1594.32,447.361 \n",
" 1597.93,447.361 1601.53,447.361 1605.13,447.361 1608.74,447.361 1612.34,447.361 1615.94,447.361 1619.55,447.361 1623.15,447.361 1626.75,447.361 1630.36,447.361 \n",
" 1633.96,447.361 1637.56,447.361 1641.17,447.361 1644.77,447.361 1648.38,447.361 1651.98,447.361 1655.58,447.361 1659.19,447.361 1662.79,447.361 1666.39,447.361 \n",
" 1670,447.361 1673.6,447.361 1677.2,447.361 1680.81,447.361 1684.41,447.361 1688.01,447.361 1691.62,447.361 1695.22,447.361 1698.82,447.361 1702.43,447.361 \n",
" 1706.03,447.361 1709.63,447.361 1713.24,447.361 1716.84,447.361 1720.45,447.361 1724.05,447.361 1727.65,447.361 1731.26,447.361 1734.86,447.361 1738.46,447.361 \n",
" 1742.07,447.361 1745.67,447.361 1749.27,447.361 1752.88,447.361 1756.48,447.361 1760.08,447.361 1763.69,447.361 1767.29,447.361 1770.89,447.361 1774.5,447.361 \n",
" 1778.1,447.361 1781.7,447.361 1785.31,447.361 1788.91,447.361 1792.52,447.361 1796.12,447.361 1799.72,447.361 1803.33,447.361 1806.93,447.361 1810.53,447.361 \n",
" 1814.14,447.361 1817.74,447.361 1821.34,447.361 1824.95,447.361 1828.55,447.361 1832.15,447.361 1835.76,447.361 1839.36,447.361 1842.96,447.361 1846.57,447.361 \n",
" 1850.17,447.361 1853.77,447.361 1857.38,447.361 1860.98,447.361 1864.58,447.361 1868.19,447.361 1871.79,447.361 1875.4,447.361 1879,447.361 1882.6,447.361 \n",
" 1886.21,447.361 1889.81,447.361 1893.41,447.361 1897.02,447.361 1900.62,447.361 1904.22,447.361 1907.83,447.361 1911.43,447.361 1915.03,447.361 1918.64,447.361 \n",
" 1922.24,447.361 1925.84,447.361 1929.45,447.361 1933.05,447.361 1936.65,447.361 1940.26,447.361 1943.86,447.361 1947.47,447.361 1951.07,447.361 1954.67,447.361 \n",
" 1958.28,447.361 1961.88,447.361 1965.48,447.361 1969.09,447.361 1972.69,447.361 1976.29,447.361 1979.9,447.361 1983.5,447.361 1987.1,447.361 1990.71,447.411 \n",
" 1994.31,447.952 1997.91,449.442 2001.52,451.927 2005.12,455.35 2008.72,459.63 2012.33,464.67 2015.93,470.356 2019.53,476.563 2023.14,483.161 2026.74,490.013 \n",
" 2030.35,496.987 2033.95,503.954 2037.55,510.794 2041.16,517.4 2044.76,523.679 2048.36,529.556 2051.97,534.971 2055.57,539.884 2059.17,544.274 2062.78,548.133 \n",
" 2066.38,551.471 2069.98,554.309 2073.59,556.678 2077.19,558.615 2080.79,560.161 2084.4,561.358 2088,562.244 2091.6,562.85 2095.21,563.218 2098.81,563.349 \n",
" \n",
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" 300.67,959.722 304.274,958.665 307.877,956.424 311.481,953.904 315.084,951.067 318.688,947.879 322.291,944.303 325.895,940.297 329.498,935.82 333.102,930.837 \n",
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" 444.81,742.62 448.413,739.393 452.017,736.139 455.62,732.856 459.224,729.546 462.827,726.21 466.431,722.846 470.034,719.456 473.638,716.04 477.241,712.599 \n",
" 480.845,709.132 484.448,705.641 488.052,702.125 491.655,698.586 495.259,695.023 498.862,691.437 502.466,687.828 506.069,684.197 509.673,680.545 513.276,676.871 \n",
" 516.88,673.176 520.483,669.46 524.087,665.724 527.69,661.969 531.294,658.194 534.897,654.401 538.501,650.589 542.104,646.76 545.708,642.913 549.311,639.049 \n",
" 552.915,635.168 556.518,631.272 560.122,627.359 563.725,623.432 567.329,619.489 570.932,615.533 574.536,611.563 578.139,607.579 581.743,603.582 585.346,599.573 \n",
" 588.949,595.552 592.553,591.52 596.156,587.476 599.76,583.422 603.363,579.358 606.967,575.285 610.57,571.202 614.174,567.11 617.777,563.011 621.381,558.903 \n",
" 624.984,554.789 628.588,550.667 632.191,546.54 635.795,542.406 639.398,538.268 643.002,534.124 646.605,529.977 650.209,525.825 653.812,521.67 657.416,517.512 \n",
" 661.019,513.352 664.623,509.19 668.226,505.026 671.83,500.861 675.433,496.696 679.037,492.531 682.64,488.367 686.244,484.203 689.847,480.041 693.451,475.881 \n",
" 697.054,471.724 700.658,467.569 704.261,463.418 707.865,459.27 711.468,455.127 715.072,450.989 718.675,446.857 722.279,442.73 725.882,438.609 729.486,434.495 \n",
" 733.089,430.389 736.693,426.29 740.296,422.199 743.9,418.117 747.503,414.045 751.107,409.982 754.71,405.929 758.313,401.886 761.917,397.855 765.52,393.835 \n",
" 769.124,389.828 772.727,385.832 776.331,381.85 779.934,377.881 783.538,373.926 787.141,369.986 790.745,366.06 794.348,362.149 797.952,358.254 801.555,354.375 \n",
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" 841.194,312.901 844.797,309.251 848.401,305.622 852.004,302.016 855.608,298.433 859.211,294.872 862.815,291.335 866.418,287.822 870.022,284.334 873.625,280.87 \n",
" 877.229,277.431 880.832,274.018 884.436,270.631 888.039,267.27 891.643,263.936 895.246,260.629 898.85,257.349 902.453,254.098 906.057,250.875 909.66,247.68 \n",
" 913.264,244.514 916.867,241.378 920.471,238.272 924.074,235.195 927.678,232.15 931.281,229.135 934.884,226.151 938.488,223.199 942.091,220.278 945.695,217.39 \n",
" 949.298,214.535 952.902,211.712 956.505,208.922 960.109,206.166 963.712,203.444 967.316,200.756 970.919,198.103 974.523,195.484 978.126,192.901 981.73,190.353 \n",
" 985.333,187.84 988.937,185.364 992.54,182.924 996.144,180.52 999.747,178.154 1003.35,175.824 1006.95,173.532 1010.56,171.277 1014.16,169.061 1017.76,166.882 \n",
" 1021.37,164.742 1024.97,162.641 1028.58,160.579 1032.18,158.555 1035.78,156.572 1039.39,154.627 1042.99,152.723 1046.59,150.859 1050.2,149.035 1053.8,147.252 \n",
" 1057.4,145.509 1061.01,143.807 1064.61,142.147 1068.21,140.528 1071.82,138.95 1075.42,137.414 1079.02,135.919 1082.63,134.467 1086.23,133.057 1089.83,131.69 \n",
" 1093.44,130.365 1097.04,129.082 1100.65,127.843 1104.25,126.646 1107.85,125.493 1111.46,124.383 1115.06,123.316 1118.66,122.293 1122.27,121.313 1125.87,120.377 \n",
" 1129.47,119.485 1133.08,118.637 1136.68,117.833 1140.28,117.073 1143.89,116.358 1147.49,115.686 1151.09,115.06 1154.7,114.477 1158.3,113.939 1161.9,113.446 \n",
" 1165.51,112.997 1169.11,112.593 1172.71,112.234 1176.32,111.92 1179.92,111.65 1183.53,111.426 1187.13,111.246 1190.73,111.111 1194.34,111.021 1197.94,110.976 \n",
" 1201.54,110.976 1205.15,111.021 1208.75,111.111 1212.35,111.246 1215.96,111.426 1219.56,111.65 1223.16,111.92 1226.77,112.234 1230.37,112.593 1233.97,112.997 \n",
" 1237.58,113.446 1241.18,113.939 1244.78,114.477 1248.39,115.06 1251.99,115.686 1255.6,116.358 1259.2,117.073 1262.8,117.833 1266.41,118.637 1270.01,119.485 \n",
" 1273.61,120.377 1277.22,121.313 1280.82,122.293 1284.42,123.316 1288.03,124.383 1291.63,125.493 1295.23,126.646 1298.84,127.843 1302.44,129.082 1306.04,130.365 \n",
" 1309.65,131.69 1313.25,133.057 1316.85,134.467 1320.46,135.919 1324.06,137.414 1327.66,138.95 1331.27,140.528 1334.87,142.147 1338.48,143.807 1342.08,145.509 \n",
" 1345.68,147.252 1349.29,149.035 1352.89,150.859 1356.49,152.723 1360.1,154.627 1363.7,156.572 1367.3,158.555 1370.91,160.579 1374.51,162.641 1378.11,164.742 \n",
" 1381.72,166.882 1385.32,169.061 1388.92,171.277 1392.53,173.532 1396.13,175.824 1399.73,178.154 1403.34,180.52 1406.94,182.924 1410.55,185.364 1414.15,187.84 \n",
" 1417.75,190.353 1421.36,192.901 1424.96,195.484 1428.56,198.103 1432.17,200.756 1435.77,203.444 1439.37,206.166 1442.98,208.922 1446.58,211.712 1450.18,214.535 \n",
" 1453.79,217.39 1457.39,220.278 1460.99,223.199 1464.6,226.151 1468.2,229.135 1471.8,232.15 1475.41,235.195 1479.01,238.272 1482.61,241.378 1486.22,244.514 \n",
" 1489.82,247.68 1493.43,250.875 1497.03,254.098 1500.63,257.349 1504.24,260.629 1507.84,263.936 1511.44,267.27 1515.05,270.631 1518.65,274.018 1522.25,277.431 \n",
" 1525.86,280.87 1529.46,284.334 1533.06,287.822 1536.67,291.335 1540.27,294.872 1543.87,298.433 1547.48,302.016 1551.08,305.622 1554.68,309.251 1558.29,312.901 \n",
" 1561.89,316.573 1565.5,320.266 1569.1,323.979 1572.7,327.713 1576.31,331.466 1579.91,335.239 1583.51,339.03 1587.12,342.84 1590.72,346.668 1594.32,350.513 \n",
" 1597.93,354.375 1601.53,358.254 1605.13,362.149 1608.74,366.06 1612.34,369.986 1615.94,373.926 1619.55,377.881 1623.15,381.85 1626.75,385.832 1630.36,389.828 \n",
" 1633.96,393.835 1637.56,397.855 1641.17,401.886 1644.77,405.929 1648.38,409.982 1651.98,414.045 1655.58,418.117 1659.19,422.199 1662.79,426.29 1666.39,430.389 \n",
" 1670,434.495 1673.6,438.609 1677.2,442.73 1680.81,446.857 1684.41,450.989 1688.01,455.127 1691.62,459.27 1695.22,463.418 1698.82,467.569 1702.43,471.724 \n",
" 1706.03,475.881 1709.63,480.041 1713.24,484.203 1716.84,488.367 1720.45,492.531 1724.05,496.696 1727.65,500.861 1731.26,505.026 1734.86,509.19 1738.46,513.352 \n",
" 1742.07,517.512 1745.67,521.67 1749.27,525.825 1752.88,529.977 1756.48,534.124 1760.08,538.268 1763.69,542.406 1767.29,546.54 1770.89,550.667 1774.5,554.789 \n",
" 1778.1,558.903 1781.7,563.011 1785.31,567.11 1788.91,571.202 1792.52,575.285 1796.12,579.358 1799.72,583.422 1803.33,587.476 1806.93,591.52 1810.53,595.552 \n",
" 1814.14,599.573 1817.74,603.582 1821.34,607.579 1824.95,611.563 1828.55,615.533 1832.15,619.489 1835.76,623.432 1839.36,627.359 1842.96,631.272 1846.57,635.168 \n",
" 1850.17,639.049 1853.77,642.913 1857.38,646.76 1860.98,650.589 1864.58,654.401 1868.19,658.194 1871.79,661.969 1875.4,665.724 1879,669.46 1882.6,673.176 \n",
" 1886.21,676.871 1889.81,680.545 1893.41,684.197 1897.02,687.828 1900.62,691.437 1904.22,695.023 1907.83,698.586 1911.43,702.125 1915.03,705.641 1918.64,709.132 \n",
" 1922.24,712.599 1925.84,716.04 1929.45,719.456 1933.05,722.846 1936.65,726.21 1940.26,729.546 1943.86,732.856 1947.47,736.139 1951.07,739.393 1954.67,742.62 \n",
" 1958.28,745.817 1961.88,748.986 1965.48,752.125 1969.09,755.235 1972.69,758.314 1976.29,761.363 1979.9,764.382 1983.5,767.369 1987.1,770.324 1990.71,773.299 \n",
" 1994.31,776.731 1997.91,781.08 2001.52,786.391 2005.12,792.607 2008.72,799.646 2012.33,807.412 2015.93,815.789 2019.53,824.654 2023.14,833.874 2026.74,843.313 \n",
" 2030.35,852.839 2033.95,862.322 2037.55,871.642 2041.16,880.692 2044.76,889.379 2048.36,897.626 2051.97,905.375 2055.57,912.585 2059.17,919.234 2062.78,925.315 \n",
" 2066.38,930.837 2069.98,935.82 2073.59,940.297 2077.19,944.303 2080.79,947.879 2084.4,951.067 2088,953.904 2091.6,956.424 2095.21,958.665 2098.81,959.722 \n",
" \n",
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" 300.67,1013.95 304.274,1012.83 307.877,1010.47 311.481,1007.82 315.084,1004.84 318.688,1001.52 322.291,997.796 325.895,993.639 329.498,989.007 333.102,983.862 \n",
" 336.705,978.173 340.309,971.92 343.912,965.093 347.516,957.7 351.119,949.762 354.723,941.321 358.326,932.435 361.93,923.18 365.533,913.65 369.137,903.952 \n",
" 372.74,894.206 376.344,884.54 379.947,875.09 383.551,865.989 387.154,857.37 390.758,849.358 394.361,842.066 397.965,835.594 401.568,830.02 405.172,825.405 \n",
" 408.775,821.701 412.378,818.449 415.982,815.212 419.585,811.938 423.189,808.629 426.792,805.283 430.396,801.903 433.999,798.487 437.603,795.037 441.206,791.554 \n",
" 444.81,788.036 448.413,784.485 452.017,780.902 455.62,777.286 459.224,773.638 462.827,769.959 466.431,766.248 470.034,762.507 473.638,758.736 477.241,754.935 \n",
" 480.845,751.105 484.448,747.245 488.052,743.358 491.655,739.442 495.259,735.499 498.862,731.529 502.466,727.532 506.069,723.509 509.673,719.46 513.276,715.387 \n",
" 516.88,711.288 520.483,707.165 524.087,703.019 527.69,698.849 531.294,694.657 534.897,690.442 538.501,686.206 542.104,681.948 545.708,677.67 549.311,673.371 \n",
" 552.915,669.053 556.518,664.715 560.122,660.358 563.725,655.983 567.329,651.591 570.932,647.181 574.536,642.755 578.139,638.312 581.743,633.854 585.346,629.38 \n",
" 588.949,624.892 592.553,620.39 596.156,615.874 599.76,611.346 603.363,606.804 606.967,602.251 610.57,597.687 614.174,593.111 617.777,588.525 621.381,583.929 \n",
" 624.984,579.324 628.588,574.71 632.191,570.088 635.795,565.459 639.398,560.822 643.002,556.178 646.605,551.529 650.209,546.874 653.812,542.214 657.416,537.549 \n",
" 661.019,532.881 664.623,528.209 668.226,523.535 671.83,518.858 675.433,514.18 679.037,509.5 682.64,504.82 686.244,500.14 689.847,495.461 693.451,490.782 \n",
" 697.054,486.105 700.658,481.431 704.261,476.759 707.865,472.09 711.468,467.425 715.072,462.764 718.675,458.108 722.279,453.458 725.882,448.813 729.486,444.176 \n",
" 733.089,439.545 736.693,434.922 740.296,430.307 743.9,425.7 747.503,421.103 751.107,416.516 754.71,411.939 758.313,407.373 761.917,402.818 765.52,398.275 \n",
" 769.124,393.745 772.727,389.227 776.331,384.724 779.934,380.234 783.538,375.759 787.141,371.299 790.745,366.854 794.348,362.426 797.952,358.014 801.555,353.62 \n",
" 805.159,349.243 808.762,344.885 812.366,340.545 815.969,336.225 819.573,331.925 823.176,327.645 826.78,323.385 830.383,319.147 833.987,314.931 837.59,310.737 \n",
" 841.194,306.566 844.797,302.418 848.401,298.295 852.004,294.195 855.608,290.12 859.211,286.071 862.815,282.047 866.418,278.05 870.022,274.079 873.625,270.136 \n",
" 877.229,266.22 880.832,262.332 884.436,258.473 888.039,254.644 891.643,250.843 895.246,247.073 898.85,243.333 902.453,239.624 906.057,235.947 909.66,232.301 \n",
" 913.264,228.688 916.867,225.107 920.471,221.56 924.074,218.046 927.678,214.566 931.281,211.12 934.884,207.71 938.488,204.334 942.091,200.995 945.695,197.691 \n",
" 949.298,194.424 952.902,191.194 956.505,188.001 960.109,184.845 963.712,181.728 967.316,178.649 970.919,175.61 974.523,172.609 978.126,169.648 981.73,166.727 \n",
" 985.333,163.846 988.937,161.006 992.54,158.207 996.144,155.449 999.747,152.733 1003.35,150.059 1006.95,147.428 1010.56,144.839 1014.16,142.293 1017.76,139.791 \n",
" 1021.37,137.332 1024.97,134.918 1028.58,132.547 1032.18,130.221 1035.78,127.941 1039.39,125.705 1042.99,123.515 1046.59,121.37 1050.2,119.272 1053.8,117.219 \n",
" 1057.4,115.214 1061.01,113.255 1064.61,111.343 1068.21,109.478 1071.82,107.661 1075.42,105.892 1079.02,104.171 1082.63,102.498 1086.23,100.873 1089.83,99.297 \n",
" 1093.44,97.7698 1097.04,96.2915 1100.65,94.8625 1104.25,93.4829 1107.85,92.1529 1111.46,90.8726 1115.06,89.6422 1118.66,88.4619 1122.27,87.3318 1125.87,86.2521 \n",
" 1129.47,85.2228 1133.08,84.2442 1136.68,83.3164 1140.28,82.4394 1143.89,81.6135 1147.49,80.8386 1151.09,80.115 1154.7,79.4426 1158.3,78.8216 1161.9,78.2521 \n",
" 1165.51,77.7341 1169.11,77.2678 1172.71,76.8531 1176.32,76.4901 1179.92,76.1789 1183.53,75.9195 1187.13,75.7119 1190.73,75.5562 1194.34,75.4524 1197.94,75.4005 \n",
" 1201.54,75.4005 1205.15,75.4524 1208.75,75.5562 1212.35,75.7119 1215.96,75.9195 1219.56,76.1789 1223.16,76.4901 1226.77,76.8531 1230.37,77.2678 1233.97,77.7341 \n",
" 1237.58,78.2521 1241.18,78.8216 1244.78,79.4426 1248.39,80.115 1251.99,80.8386 1255.6,81.6135 1259.2,82.4394 1262.8,83.3164 1266.41,84.2442 1270.01,85.2228 \n",
" 1273.61,86.2521 1277.22,87.3318 1280.82,88.4619 1284.42,89.6422 1288.03,90.8726 1291.63,92.1529 1295.23,93.4829 1298.84,94.8625 1302.44,96.2915 1306.04,97.7698 \n",
" 1309.65,99.297 1313.25,100.873 1316.85,102.498 1320.46,104.171 1324.06,105.892 1327.66,107.661 1331.27,109.478 1334.87,111.343 1338.48,113.255 1342.08,115.214 \n",
" 1345.68,117.219 1349.29,119.272 1352.89,121.37 1356.49,123.515 1360.1,125.705 1363.7,127.941 1367.3,130.221 1370.91,132.547 1374.51,134.918 1378.11,137.332 \n",
" 1381.72,139.791 1385.32,142.293 1388.92,144.839 1392.53,147.428 1396.13,150.059 1399.73,152.733 1403.34,155.449 1406.94,158.207 1410.55,161.006 1414.15,163.846 \n",
" 1417.75,166.727 1421.36,169.648 1424.96,172.609 1428.56,175.61 1432.17,178.649 1435.77,181.728 1439.37,184.845 1442.98,188.001 1446.58,191.194 1450.18,194.424 \n",
" 1453.79,197.691 1457.39,200.995 1460.99,204.334 1464.6,207.71 1468.2,211.12 1471.8,214.566 1475.41,218.046 1479.01,221.56 1482.61,225.107 1486.22,228.688 \n",
" 1489.82,232.301 1493.43,235.947 1497.03,239.624 1500.63,243.333 1504.24,247.073 1507.84,250.843 1511.44,254.644 1515.05,258.473 1518.65,262.332 1522.25,266.22 \n",
" 1525.86,270.136 1529.46,274.079 1533.06,278.05 1536.67,282.047 1540.27,286.071 1543.87,290.12 1547.48,294.195 1551.08,298.295 1554.68,302.418 1558.29,306.566 \n",
" 1561.89,310.737 1565.5,314.931 1569.1,319.147 1572.7,323.385 1576.31,327.645 1579.91,331.925 1583.51,336.225 1587.12,340.545 1590.72,344.885 1594.32,349.243 \n",
" 1597.93,353.62 1601.53,358.014 1605.13,362.426 1608.74,366.854 1612.34,371.299 1615.94,375.759 1619.55,380.234 1623.15,384.724 1626.75,389.227 1630.36,393.745 \n",
" 1633.96,398.275 1637.56,402.818 1641.17,407.373 1644.77,411.939 1648.38,416.516 1651.98,421.103 1655.58,425.7 1659.19,430.307 1662.79,434.922 1666.39,439.545 \n",
" 1670,444.176 1673.6,448.813 1677.2,453.458 1680.81,458.108 1684.41,462.764 1688.01,467.425 1691.62,472.09 1695.22,476.759 1698.82,481.431 1702.43,486.105 \n",
" 1706.03,490.782 1709.63,495.461 1713.24,500.14 1716.84,504.82 1720.45,509.5 1724.05,514.18 1727.65,518.858 1731.26,523.535 1734.86,528.209 1738.46,532.881 \n",
" 1742.07,537.549 1745.67,542.214 1749.27,546.874 1752.88,551.529 1756.48,556.178 1760.08,560.822 1763.69,565.459 1767.29,570.088 1770.89,574.71 1774.5,579.324 \n",
" 1778.1,583.929 1781.7,588.525 1785.31,593.111 1788.91,597.687 1792.52,602.251 1796.12,606.804 1799.72,611.346 1803.33,615.874 1806.93,620.39 1810.53,624.892 \n",
" 1814.14,629.38 1817.74,633.854 1821.34,638.312 1824.95,642.755 1828.55,647.181 1832.15,651.591 1835.76,655.983 1839.36,660.358 1842.96,664.715 1846.57,669.053 \n",
" 1850.17,673.371 1853.77,677.67 1857.38,681.948 1860.98,686.206 1864.58,690.442 1868.19,694.657 1871.79,698.849 1875.4,703.019 1879,707.165 1882.6,711.288 \n",
" 1886.21,715.387 1889.81,719.46 1893.41,723.509 1897.02,727.532 1900.62,731.529 1904.22,735.499 1907.83,739.442 1911.43,743.358 1915.03,747.245 1918.64,751.105 \n",
" 1922.24,754.935 1925.84,758.736 1929.45,762.507 1933.05,766.248 1936.65,769.959 1940.26,773.638 1943.86,777.286 1947.47,780.902 1951.07,784.485 1954.67,788.036 \n",
" 1958.28,791.554 1961.88,795.037 1965.48,798.487 1969.09,801.903 1972.69,805.283 1976.29,808.629 1979.9,811.938 1983.5,815.212 1987.1,818.449 1990.71,821.701 \n",
" 1994.31,825.405 1997.91,830.02 2001.52,835.594 2005.12,842.066 2008.72,849.358 2012.33,857.37 2015.93,865.989 2019.53,875.09 2023.14,884.54 2026.74,894.206 \n",
" 2030.35,903.952 2033.95,913.65 2037.55,923.18 2041.16,932.435 2044.76,941.321 2048.36,949.762 2051.97,957.7 2055.57,965.093 2059.17,971.92 2062.78,978.173 \n",
" 2066.38,983.862 2069.98,989.007 2073.59,993.639 2077.19,997.796 2080.79,1001.52 2084.4,1004.84 2088,1007.82 2091.6,1010.47 2095.21,1012.83 2098.81,1013.95 \n",
" \n",
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" 300.67,1008.57 304.274,1007.71 307.877,1005.85 311.481,1003.71 315.084,1001.25 318.688,998.43 322.291,995.217 325.895,991.569 329.498,987.444 333.102,982.806 \n",
" 336.705,977.624 340.309,971.876 343.912,965.554 347.516,958.664 351.119,951.229 354.723,943.288 358.326,934.901 361.93,926.144 365.533,917.11 369.137,907.906 \n",
" 372.74,898.652 376.344,889.477 379.947,880.515 383.551,871.901 387.154,863.766 390.758,856.237 394.361,849.426 397.965,843.431 401.568,838.334 405.172,834.192 \n",
" 408.775,830.959 412.378,828.176 415.982,825.405 419.585,822.595 423.189,819.747 426.792,816.86 430.396,813.935 433.999,810.973 437.603,807.973 441.206,804.937 \n",
" 444.81,801.864 448.413,798.755 452.017,795.61 455.62,792.43 459.224,789.214 462.827,785.964 466.431,782.68 470.034,779.362 473.638,776.011 477.241,772.627 \n",
" 480.845,769.21 484.448,765.761 488.052,762.28 491.655,758.767 495.259,755.224 498.862,751.651 502.466,748.047 506.069,744.414 509.673,740.752 513.276,737.061 \n",
" 516.88,733.343 520.483,729.596 524.087,725.822 527.69,722.021 531.294,718.195 534.897,714.342 538.501,710.464 542.104,706.561 545.708,702.634 549.311,698.683 \n",
" 552.915,694.709 556.518,690.712 560.122,686.692 563.725,682.651 567.329,678.588 570.932,674.505 574.536,670.402 578.139,666.278 581.743,662.136 585.346,657.974 \n",
" 588.949,653.795 592.553,649.598 596.156,645.384 599.76,641.153 603.363,636.907 606.967,632.644 610.57,628.367 614.174,624.076 617.777,619.771 621.381,615.452 \n",
" 624.984,611.12 628.588,606.777 632.191,602.422 635.795,598.055 639.398,593.678 643.002,589.291 646.605,584.895 650.209,580.49 653.812,576.077 657.416,571.655 \n",
" 661.019,567.227 664.623,562.792 668.226,558.351 671.83,553.904 675.433,549.452 679.037,544.996 682.64,540.537 686.244,536.074 689.847,531.608 693.451,527.14 \n",
" 697.054,522.671 700.658,518.201 704.261,513.73 707.865,509.259 711.468,504.789 715.072,500.321 718.675,495.854 722.279,491.39 725.882,486.928 729.486,482.47 \n",
" 733.089,478.017 736.693,473.568 740.296,469.124 743.9,464.686 747.503,460.254 751.107,455.829 754.71,451.412 758.313,447.003 761.917,442.602 765.52,438.211 \n",
" 769.124,433.829 772.727,429.457 776.331,425.097 779.934,420.747 783.538,416.41 787.141,412.085 790.745,407.773 794.348,403.475 797.952,399.19 801.555,394.921 \n",
" 805.159,390.666 808.762,386.428 812.366,382.205 815.969,378 819.573,373.811 823.176,369.641 826.78,365.489 830.383,361.356 833.987,357.243 837.59,353.149 \n",
" 841.194,349.076 844.797,345.025 848.401,340.994 852.004,336.986 855.608,333 859.211,329.038 862.815,325.099 866.418,321.184 870.022,317.294 873.625,313.429 \n",
" 877.229,309.59 880.832,305.777 884.436,301.99 888.039,298.231 891.643,294.499 895.246,290.795 898.85,287.12 902.453,283.474 906.057,279.858 909.66,276.271 \n",
" 913.264,272.715 916.867,269.19 920.471,265.696 924.074,262.235 927.678,258.805 931.281,255.409 934.884,252.045 938.488,248.716 942.091,245.42 945.695,242.159 \n",
" 949.298,238.933 952.902,235.742 956.505,232.588 960.109,229.469 963.712,226.387 967.316,223.343 970.919,220.336 974.523,217.366 978.126,214.436 981.73,211.543 \n",
" 985.333,208.69 988.937,205.877 992.54,203.103 996.144,200.37 999.747,197.677 1003.35,195.026 1006.95,192.415 1010.56,189.847 1014.16,187.32 1017.76,184.836 \n",
" 1021.37,182.395 1024.97,179.996 1028.58,177.642 1032.18,175.331 1035.78,173.064 1039.39,170.841 1042.99,168.664 1046.59,166.531 1050.2,164.443 1053.8,162.402 \n",
" 1057.4,160.406 1061.01,158.456 1064.61,156.553 1068.21,154.696 1071.82,152.887 1075.42,151.124 1079.02,149.41 1082.63,147.743 1086.23,146.124 1089.83,144.553 \n",
" 1093.44,143.03 1097.04,141.556 1100.65,140.131 1104.25,138.756 1107.85,137.429 1111.46,136.152 1115.06,134.924 1118.66,133.746 1122.27,132.619 1125.87,131.541 \n",
" 1129.47,130.514 1133.08,129.537 1136.68,128.61 1140.28,127.735 1143.89,126.91 1147.49,126.136 1151.09,125.414 1154.7,124.742 1158.3,124.122 1161.9,123.553 \n",
" 1165.51,123.036 1169.11,122.57 1172.71,122.155 1176.32,121.793 1179.92,121.482 1183.53,121.223 1187.13,121.015 1190.73,120.86 1194.34,120.756 1197.94,120.704 \n",
" 1201.54,120.704 1205.15,120.756 1208.75,120.86 1212.35,121.015 1215.96,121.223 1219.56,121.482 1223.16,121.793 1226.77,122.155 1230.37,122.57 1233.97,123.036 \n",
" 1237.58,123.553 1241.18,124.122 1244.78,124.742 1248.39,125.414 1251.99,126.136 1255.6,126.91 1259.2,127.735 1262.8,128.61 1266.41,129.537 1270.01,130.514 \n",
" 1273.61,131.541 1277.22,132.619 1280.82,133.746 1284.42,134.924 1288.03,136.152 1291.63,137.429 1295.23,138.756 1298.84,140.131 1302.44,141.556 1306.04,143.03 \n",
" 1309.65,144.553 1313.25,146.124 1316.85,147.743 1320.46,149.41 1324.06,151.124 1327.66,152.887 1331.27,154.696 1334.87,156.553 1338.48,158.456 1342.08,160.406 \n",
" 1345.68,162.402 1349.29,164.443 1352.89,166.531 1356.49,168.664 1360.1,170.841 1363.7,173.064 1367.3,175.331 1370.91,177.642 1374.51,179.996 1378.11,182.395 \n",
" 1381.72,184.836 1385.32,187.32 1388.92,189.847 1392.53,192.415 1396.13,195.026 1399.73,197.677 1403.34,200.37 1406.94,203.103 1410.55,205.877 1414.15,208.69 \n",
" 1417.75,211.543 1421.36,214.436 1424.96,217.366 1428.56,220.336 1432.17,223.343 1435.77,226.387 1439.37,229.469 1442.98,232.588 1446.58,235.742 1450.18,238.933 \n",
" 1453.79,242.159 1457.39,245.42 1460.99,248.716 1464.6,252.045 1468.2,255.409 1471.8,258.805 1475.41,262.235 1479.01,265.696 1482.61,269.19 1486.22,272.715 \n",
" 1489.82,276.271 1493.43,279.858 1497.03,283.474 1500.63,287.12 1504.24,290.795 1507.84,294.499 1511.44,298.231 1515.05,301.99 1518.65,305.777 1522.25,309.59 \n",
" 1525.86,313.429 1529.46,317.294 1533.06,321.184 1536.67,325.099 1540.27,329.038 1543.87,333 1547.48,336.986 1551.08,340.994 1554.68,345.025 1558.29,349.076 \n",
" 1561.89,353.149 1565.5,357.243 1569.1,361.356 1572.7,365.489 1576.31,369.641 1579.91,373.811 1583.51,378 1587.12,382.205 1590.72,386.428 1594.32,390.666 \n",
" 1597.93,394.921 1601.53,399.19 1605.13,403.475 1608.74,407.773 1612.34,412.085 1615.94,416.41 1619.55,420.747 1623.15,425.097 1626.75,429.457 1630.36,433.829 \n",
" 1633.96,438.211 1637.56,442.602 1641.17,447.003 1644.77,451.412 1648.38,455.829 1651.98,460.254 1655.58,464.686 1659.19,469.124 1662.79,473.568 1666.39,478.017 \n",
" 1670,482.47 1673.6,486.928 1677.2,491.39 1680.81,495.854 1684.41,500.321 1688.01,504.789 1691.62,509.259 1695.22,513.73 1698.82,518.201 1702.43,522.671 \n",
" 1706.03,527.14 1709.63,531.608 1713.24,536.074 1716.84,540.537 1720.45,544.996 1724.05,549.452 1727.65,553.904 1731.26,558.351 1734.86,562.792 1738.46,567.227 \n",
" 1742.07,571.655 1745.67,576.077 1749.27,580.49 1752.88,584.895 1756.48,589.291 1760.08,593.678 1763.69,598.055 1767.29,602.422 1770.89,606.777 1774.5,611.12 \n",
" 1778.1,615.452 1781.7,619.771 1785.31,624.076 1788.91,628.367 1792.52,632.644 1796.12,636.907 1799.72,641.153 1803.33,645.384 1806.93,649.598 1810.53,653.795 \n",
" 1814.14,657.974 1817.74,662.136 1821.34,666.278 1824.95,670.402 1828.55,674.505 1832.15,678.588 1835.76,682.651 1839.36,686.692 1842.96,690.712 1846.57,694.709 \n",
" 1850.17,698.683 1853.77,702.634 1857.38,706.561 1860.98,710.464 1864.58,714.342 1868.19,718.195 1871.79,722.021 1875.4,725.822 1879,729.596 1882.6,733.343 \n",
" 1886.21,737.061 1889.81,740.752 1893.41,744.414 1897.02,748.047 1900.62,751.651 1904.22,755.224 1907.83,758.767 1911.43,762.28 1915.03,765.761 1918.64,769.21 \n",
" 1922.24,772.627 1925.84,776.011 1929.45,779.362 1933.05,782.68 1936.65,785.964 1940.26,789.214 1943.86,792.43 1947.47,795.61 1951.07,798.755 1954.67,801.864 \n",
" 1958.28,804.937 1961.88,807.973 1965.48,810.973 1969.09,813.935 1972.69,816.86 1976.29,819.747 1979.9,822.595 1983.5,825.405 1987.1,828.176 1990.71,830.959 \n",
" 1994.31,834.192 1997.91,838.334 2001.52,843.431 2005.12,849.426 2008.72,856.237 2012.33,863.766 2015.93,871.901 2019.53,880.515 2023.14,889.477 2026.74,898.652 \n",
" 2030.35,907.906 2033.95,917.11 2037.55,926.144 2041.16,934.901 2044.76,943.288 2048.36,951.229 2051.97,958.664 2055.57,965.554 2059.17,971.876 2062.78,977.624 \n",
" 2066.38,982.806 2069.98,987.444 2073.59,991.569 2077.19,995.217 2080.79,998.43 2084.4,1001.25 2088,1003.71 2091.6,1005.85 2095.21,1007.71 2098.81,1008.57 \n",
" \n",
" \"/>\n",
"</svg>\n"
]
},
"execution_count": 49,
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],
"source": [
"fig = plot_control(opt_result_LBFGSB.optimized_controls[1], tlist)\n",
"plot!(fig, tlist, QuantumControl.Controls.discretize(opt_result_LBFGSB.records[1][1], tlist); ls=:dash)\n",
"plot!(fig, tlist, QuantumControl.Controls.discretize(opt_result_LBFGSB.records[2][1], tlist); ls=:dash)\n",
"plot!(fig, tlist, QuantumControl.Controls.discretize(opt_result_LBFGSB.records[3][1], tlist); ls=:dash)\n",
"plot!(fig, tlist, QuantumControl.Controls.discretize(opt_result_LBFGSB.records[4][1], tlist); ls=:dash)"
]
},
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@goerz
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goerz commented Oct 4, 2022

Reference: https://discourse.julialang.org/t/grapes-method-in-quantum-control-grape-jl/88208/6

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