{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Load all dependencies" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "%matplotlib notebook\n", "import matplotlib\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "from lmfit import Parameters, Minimizer, fit_report" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Fit functions" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "def lorentz(x, x0, amplitude, gamma):\n", " ''' Lorentzian distribution to model the photon power spectrum.'''\n", " return amplitude / np.pi * 2*gamma / (4*(x - x0)**2 + gamma**2)\n", "\n", "\n", "def lorentz_FPcavity(x, x0, gamma):\n", " ''' Lorentzian distribution to model the transmission of a Fabry-perot cavity'''\n", " return gamma**2 / (4*(x - x0)**2 + gamma**2)\n", "\n", "def absorption(x, x0, OD, gamma):\n", " '''\n", " Optical absorption for an atomic cloud. \n", " Modeled with the on-resonance OD and a frequency dependend lorentzian. \n", " '''\n", " return np.exp(-OD * gamma**2 / (4*(x - x0)**2 + gamma**2))\n", "\n", "\n", "def residuals(params, x_array, data=None, eps=None):\n", " ''' Define residual function for the fit optimizer. '''\n", " \n", " parvals = params.valuesdict()\n", " gamma_photon = parvals['gamma_photon']\n", " gamma_atom = parvals['gamma_atom']\n", " gamma_cavity = parvals['gamma_cavity']\n", " amplitude = parvals['amplitude']\n", " x0 = parvals['x0']\n", " OD = parvals['OD']\n", " \n", " to_omega = 1e6 * 2 * np.pi\n", " freq_array, dfreq = np.linspace(-80, 80, 1601, retstep=True)\n", " domega = dfreq * to_omega\n", "\n", " # create the spectral cavity lorentzian and convolve it with the absorption model\n", " lorentz_cavity = lorentz_FPcavity(freq_array*to_omega, x0, gamma_cavity)\n", " abs_model = lorentz(freq_array*to_omega, x0, amplitude, gamma_photon) * \\\n", " absorption(freq_array*to_omega, x0, OD, gamma_atom) * domega\n", " cavity_tx = np.convolve(lorentz_cavity, abs_model, mode='same')\n", "\n", " model = [cavity_tx[freq_array==x_array[i]].item() for i in range(len(x_array))]\n", "\n", " if data is None:\n", " return model\n", "\n", " if eps is None:\n", " return model - data\n", "\n", " return (model - data) / eps" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Load the data for the spectra and define relevant bandwidths" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "freq, rate, rate_err = np.genfromtxt('./data_spectrum_initial_photon.csv').T[0:3]\n", "freq_co, rate_co, rate_err_co = np.genfromtxt('./data_spectrum_compressed_photon.csv').T[0:3]\n", "\n", "\n", "gamma_photon = 2 * np.pi * 20.623e6 # value from previouse fit to the exponential decay\n", "gamma_cavity = 2 * np.pi * 2.6e6 # measured cavity bandwidth\n", "gamma_disp_cavity = 2 * np.pi * 7.3e6 # measured dispersion cavity bandwidth\n", "gamma_atom = 2 * np.pi * 6.06e6 # atomic linewidth of the rb-87 D-lines" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Fit to initial photon spectrum" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[Fit Statistics]]\n", " # fitting method = leastsq\n", " # function evals = 31\n", " # data points = 46\n", " # variables = 3\n", " chi-square = 235.757331\n", " reduced chi-square = 5.48272863\n", " Akaike info crit = 81.1714346\n", " Bayesian info crit = 86.6573588\n", "[[Variables]]\n", " gamma_photon: 1.295781e+08 (fixed)\n", " gamma_atom: 9174643.59 +/- 10377491.9 (113.11%) (init = 3.80761e+07)\n", " gamma_cavity: 1.633628e+07 (fixed)\n", " x0: 0 (fixed)\n", " amplitude: 21.6595263 +/- 0.80429711 (3.71%) (init = 21)\n", " OD: 1.59372611 +/- 1.92531297 (120.81%) (init = 1)\n", "[[Correlations]] (unreported correlations are < 0.100)\n", " C(gamma_atom, OD) = -0.974\n", " C(gamma_atom, amplitude) = 0.503\n", " C(amplitude, OD) = -0.376\n", "[1.38807957 1.68766224]\n" ] } ], "source": [ "fit_params = Parameters()\n", "fit_params.add('gamma_photon', value = gamma_photon, vary = False)\n", "fit_params.add('gamma_atom', value=gamma_atom, vary = True)\n", "fit_params.add('gamma_cavity', value=gamma_cavity, vary = False)\n", "fit_params.add('x0', value=0, vary=False)\n", "fit_params.add('amplitude', value=21)\n", "fit_params.add('OD', value=1)\n", "\n", "fit = Minimizer(residuals, fit_params, fcn_args=(freq, rate, rate_err))\n", "result = fit.minimize()\n", "\n", "print(fit_report(result))\n", "print(np.array([8721561.13, 10603894.6])/(2*np.pi*1e6))" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support. ' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. 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');\n", " var button = $('');\n", " button.click(function (evt) { fig.handle_close(fig, {}); } );\n", " button.mouseover('Stop Interaction', toolbar_mouse_event);\n", " buttongrp.append(button);\n", " var titlebar = this.root.find($('.ui-dialog-titlebar'));\n", " titlebar.prepend(buttongrp);\n", "}\n", "\n", "mpl.figure.prototype._root_extra_style = function(el){\n", " var fig = this\n", " el.on(\"remove\", function(){\n", "\tfig.close_ws(fig, {});\n", " });\n", "}\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(el){\n", " // this is important to make the div 'focusable\n", " el.attr('tabindex', 0)\n", " // reach out to IPython and tell the keyboard manager to turn it's self\n", " // off when our div gets focus\n", "\n", " // location in version 3\n", " if (IPython.notebook.keyboard_manager) {\n", " IPython.notebook.keyboard_manager.register_events(el);\n", " }\n", " else {\n", " // location in version 2\n", " IPython.keyboard_manager.register_events(el);\n", " }\n", "\n", "}\n", "\n", "mpl.figure.prototype._key_event_extra = function(event, name) {\n", " var manager = IPython.notebook.keyboard_manager;\n", " if (!manager)\n", " manager = IPython.keyboard_manager;\n", "\n", " // Check for shift+enter\n", " if (event.shiftKey && event.which == 13) {\n", " this.canvas_div.blur();\n", " // select the cell after this one\n", " var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n", " IPython.notebook.select(index + 1);\n", " }\n", "}\n", "\n", "mpl.figure.prototype.handle_save = function(fig, msg) {\n", " fig.ondownload(fig, null);\n", "}\n", "\n", "\n", "mpl.find_output_cell = function(html_output) {\n", " // Return the cell and output element which can be found *uniquely* in the notebook.\n", " // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n", " // IPython event is triggered only after the cells have been serialised, which for\n", " // our purposes (turning an active figure into a static one), is too late.\n", " var cells = IPython.notebook.get_cells();\n", " var ncells = cells.length;\n", " for (var i=0; i= 3 moved mimebundle to data attribute of output\n", " data = data.data;\n", " }\n", " if (data['text/html'] == html_output) {\n", " return [cell, data, j];\n", " }\n", " }\n", " }\n", " }\n", "}\n", "\n", "// Register the function which deals with the matplotlib target/channel.\n", "// The kernel may be null if the page has been refreshed.\n", "if (IPython.notebook.kernel != null) {\n", " IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n", "}\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.figure()\n", "ax = plt.subplot(111)\n", "\n", "# plot data of initial photon and corresponing errors\n", "plt.errorbar(freq, y=rate, yerr=rate_err,\n", " fmt='b.',\n", " alpha=1,\n", " label='Data, initial photon')\n", "\n", "# plot initial photon power spectrum convolved with the spectral cavity transmission\n", "plt.plot(freq_fft_array/1e6, cavity_tx_init_photon, \n", " 'b', label='Theory, with convolution')\n", "\n", "# plot data of compressed photon and corresponding errors\n", "plt.errorbar(freq_co, y=rate_co, yerr=rate_err_co,\n", " fmt='r.', alpha=1, label='Data, compressed photon')\n", "\n", "# plot compressed photon power spectrum convolved with the spectral cavity transmission\n", "plt.plot(freq_fft_array/1e6, cavity_tx_cmprsd_photon, 'r', label='Theory, compressed with convolution')\n", "\n", "plt.plot(freq_array, residuals(result.params, x_array=freq_array), 'b--', label='Fit')\n", "\n", "plt.xlim(-60, 60)\n", "ax.legend()\n", "\n", "\n", "np.savetxt('theory_initial_photon_power_spectrum_convolved_with_spec_cavity.csv',np.c_[freq_fft_array/1e6, cavity_tx_init_photon], header='frequency(MHz) coincidence_rate(per second)')\n", "np.savetxt('theory_compressed_photon_power_spectrum_convolved_with_spec_cavity.csv',np.c_[freq_fft_array/1e6, cavity_tx_cmprsd_photon], header='frequency(MHz) coincidence_rate(per second)')\n", "np.savetxt('fit_initial_photon_power_spectrum_with_OD_convolved_with_spec_cavity.csv',np.c_[freq_array, residuals(result.params, x_array=freq_array)], header='frequency(MHz) coincidence_rate(per second)')" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.2" }, "toc": { "base_numbering": 1, "nav_menu": {}, "number_sections": true, "sideBar": true, "skip_h1_title": false, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": false, "toc_position": {}, "toc_section_display": true, "toc_window_display": true } }, "nbformat": 4, "nbformat_minor": 2 }