{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Steady-state superradiance\n", "\n", "Author: Nathan Shammah (nathan.shammah@gmail.com)\n", "\n", "We consider a system of $N$ two-level systems (TLSs) with identical frequency $\\omega_{0}$, incoherently pumped at a rate $\\gamma_\\text{P}$ and de-excitating at a collective emission rate $\\gamma_\\text{CE}$,\n", "\n", "\\begin{eqnarray}\n", "\\dot{\\rho} &=& \n", "-i\\lbrack \\omega_{0}J_z,\\rho \\rbrack\n", "+\\frac{\\gamma_\\text {CE}}{2}\\mathcal{L}_{J_{-}}[\\rho]\n", "+\\frac{\\gamma_\\text{P}}{2}\\sum_{n=1}^{N}\\mathcal{L}_{J_{+,n}}[\\rho]\n", "\\end{eqnarray}\n", "This system can sustain superradiant light emission and line narrowing [1-3], whose peak intensity scales proportionally to $N^2$." ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import matplotlib.pyplot as plt\n", "from qutip import *\n", "from qutip.piqs import *" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 1) Time evolution\n", "\n", "We study the system of Eq. (1) (above) by using the *Permutational Invariant Quantum Solver (PIQS)* to build the Liouvillian of the system. Using QuTiP's $\\texttt{mesolve}()$ we can calculate operators expectation values in time as well as higher order correlation functions [4,5]. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### System properties" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "N = 10\n", "system = Dicke(N = N)\n", "[jx, jy, jz] = jspin(N)\n", "jp = jspin(N,\"+\")\n", "jm = jp.dag()\n", "w0 = 1\n", "h0 = w0 * jz\n", "gCE = 1\n", "gP = N * gCE\n", "system.hamiltonian = h0\n", "system.collective_emission = gCE\n", "system.pumping = gP" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Liouvillian and steady state $\\rho_\\text{ss}$" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "L = system.liouvillian()\n", "rhoss = steadystate(L, method=\"direct\")\n", "jpjm_ss = expect(jp*jm, rhoss)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Time integration for $g^{(2)}(\\tau)$ and $\\langle J_{+}J_{-}\\rangle (t)$\n", "We define the $g^{(2)}(\\tau)$ of the system as the two-time correlation function mapping the photonic degrees of freedom onto the TLS collective operators\n", "\n", "\\begin{eqnarray*}\n", "g^{(2)}(\\tau) = \\frac{\\langle: J^\\dagger(\\tau) a^\\dagger(0) a(\\tau) a(0) :\\rangle}{|\\langle: a^\\dagger(0) a(0) :\\rangle|^2}= \\frac{\\langle: J_{+}(\\tau) J_{+}(0) J_{-}(\\tau) J_{-}(0) :\\rangle}{|\\langle J_{+}(0) J_{-}(0) \\rangle|^2}\n", "\\end{eqnarray*}" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "# time evolution parameters\n", "nt = 1000\n", "td = np.log(N)/(N*gCE)\n", "tmax = 5 * td\n", "t = np.linspace(0, tmax, nt)\n", "# initial state\n", "rho0= dicke(N, N/2, -N/2)\n", "# calculate g2(tau)\n", "A = jp*jm\n", "rhoA = jm*rhoss*jp\n", "#g2(tau)\n", "result1 = mesolve(L, rhoA, t, [], e_ops = [A], options = Options(store_states=True))\n", "g2t = result1.expect[0]\n", "#rho(t)\n", "result2 = mesolve(L, rho0, t, [], e_ops = A, options = Options(store_states=True))\n", "rhot = result2.states\n", "jpjmt = result2.expect[0]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Visualization" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": 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4wuY3APSb82y+JiIf4uiz2ASKspGfmYrWc/1Ol0JEZBsrguZhAA8BeADGApcPAGgXkV4RedgMnu0WvE/C8XgEN1f4GDRElNBivkezAA15ONpjwOxD03bCWJBzE4BqEWkDUK+qT1jwngmjuqIAT568jMGxSeRnpjpdDhGR5axo0RSGbzAneB6CETafg7Hy8z4AB9m6mau6ogCqwIu8T0NECcqKoGkXkb+JtMNcNaAgJHhqYDxygEw3rc+HCNDawe4zIkpMKw4ac57NbeZggL+OMBCgJuTYARiTPsmUm5GKa0pzeZ+GiBKWFS0amM+oeQjA/wTQJiLTZvD0Avh5+OFWvGciubmiAC+eH+BKzkSUkCwJGgBQ1QOq6gdwK4APA9gPoEZVHweMAQIiMgOu9jxPdYUPQ+NTOH152OlSiIgsZ8WoszlUtRURusdUdVBE9kbal+xu2VAAAGjp6MeWUq4QQESJxbIWTSxU9aGQB6aRqbIoG0U5aXjuTJ/TpRARWW5Vg4YiExHcVunHb9p7ocr7NESUWBg0LvG6ykJcHBzHhf4xp0shIrIUg8Ylbg8Y816fbe91uBIiImsxaFyiqiQHBVmp+A3v0xBRgmHQuITHY96nOcMWDRElFgaNi7yushDn+8ZwcYD3aYgocTBoXOR1AT8AsFVDRAmFQeMiW8vykJ+ZiqdPM2iIKHEwaFzE4xHcsbkQvz7dw/k0RJQwXBk0IlInIrWrdZ6bvHFzMToHx9HGdc+IKEG4LmjMoDi4Wue5zZuqigAAvzzZ43AlRETWcE3QiEhARJoABADEPJlkuee51Xp/FiqLsvHr0wwaIkoMrgkaVW1X1QZVPbAa57nZm6qK8ExbL65OTTtdChHRirkmaOg1b9xchLHJabR2DDhdChHRijFoXOj1mwrh9Qh+deqy06UQEa1Y0gaNiNSLSLOINF++7K4P9NyMVNxSUYBfnHBXXUREy5G0QWM+erpGVWuKi4udLmeeHVtL8PvOK3iVy9EQUZxL2qBxu9rrSgEAx453OVwJEdHKMGhcalNxDiqLsnH0eLfTpRARrQiDxsVqt5bg2bZeDF+dcroUIqJlY9C42I6tpZiYnsGvTnJQABHFL7cGjR+AL3yjiPhEpE1E6pdyXryq2VCA/MxUHOF9GiKKYylOFxAkIj4A+2AsJeMD0CgiOwEcUdVDIYf6l3le3EnxerDj2hIcO96NiakZpKW49fcCIqLoXBM0qjoAYG8MxxQs9bx49q4byvH4C6/iqbYevPWaEqfLISJaMv6K7HJv2lKE3PQU/OR3nU6XQkS0LAwal0tP8WLntlIcfuUSJqZmnC6HiGjJGDRx4D03lmNofAq/Ps3RZ0QUfxg0ceCNm4uRl5GCH7P7jIjiEIMmDqSlePD2bWU48koXxif5jBoiii8Mmjjxvuq1GLo6hcOvXHK6FCKiJWHQxInbKwux1peJQy0XnC6FiGhJGDRxwuMR3F29Fr8+3YPOQT46gIjiB4Mmjtx9yzqoAo+3vup0KUREMWPQxJENhdm4baMfj7VcgKo6XQ4RUUwYNHGmrmYd2ntG8JszfU6XQkQUEwZNnHnvjWuQn5mK7zzT4XQpREQxYdDEmcw0L3bXrMPhVy7h0uC40+UQES2KQROH7r19A6ZV8b3nzjldChHRohg0cWhDYTbu3FKMf33uHBfaJCLXY9DEqT9/w0ZcHrqKH/32otOlEBEtiEETp+7cUoxry3Lx9SfbMDPDoc5E5F4MmjglIrjvzk041T2MY3/odrocIqKoGDRx7N03lGO9PxNf/cVpTuAkItdi0MSxFK8H9W/ehBfODeCZ9l6nyyEiiohBE+d23bIOZXkZ+MLhE2zVEJErMWjiXEaqFx+rrULruQEcO857NUTkPgyaBLDrlnWoLMrGQ4dPYJoj0IjIZRg0CSDF68Endm7Bia4h/OAFPkKAiNyFQZMg3n1DOW5al4/Gn/0BQ+OTTpdDRDSLQZMgPB7BZ/74evQMX8U/HD3ldDlERLMYNAnkpvU+vP/W9fjW02dx4tKQ0+UQEQFg0CScv337tcjNSMG+x3/HgQFE5AoMmgTjz07Dp9+7Da3nBnDwV+1Ol0NE5M6gEZE6EaldwvHVIlIvIrXB73bW53Z/vH0N3rGtDH//85M42cUuNCJyluuCxgyJg0s4PgCgUVUPqOpRVT0AoMHcnpREBP/nfdcjJyMFH/v+ixifnHa6JCJKYq4JGhEJiEgTgACAviWc2gCgKWxbE4BGq2qLR0U56fi73TfheOcVPPjDl50uh4iSmGuCRlXbVbXBbJEsRR2A1rBtzeb2pPbWa0rw0T/ajEeaL+D7fOwzETnENUGzHCLiQ4QWkKoOmPuTtvss6K9qt+CNm4vwqR++jKfbepwuh4iSUIrTBayQH3gtWCIIAEjqoVdej+Arf1aNuq8/jYZvt+DR+16Pa8vynC6LEoiqYkaBGVVo+HeY32fm/hw8JnhcpJ+N0fmxDdGPdeHypQz4t2IxdJEo25d4fLQzlv76c/ekeATr/VnR3tQy8R40PqcLwDffPX/btj8BbtsDTIwC/7Jr/v7tfwbc/AFgpBd45M/n77/1vwHX3w0MXgAeb5i//w1/AVzzTqDnFPCjv5q//81/A2x6K9D5O+Bn+5AP4Me503h56ApGm4DOuz6L8hvuBM79Bjj2mfnnv+NzQPmNQNsTwC+/MH//e/8BKKoCTvwUePqf5u+/qwnIXwe8/Bjw/D/P37/720B2IfDCvwAvfm/+/g88CqRlAc8dBF75t/n7/+tPjO9PfQk4eXjuvtQM4N7HjD8/uR9of3Lu/qwC4J7vGn8++mng/PNz9+etAe42x6L89AHg0ktz9xduAv7Tl4w///tfAr1tc/eX3QC88/PGnx/bA1y5OHf/+luB2k8bf374XmC0f+7+wFuAt9xv/Pm7dwOT43P3b3k7cMdfGh/u33w3pmcU0zPGB/P0jKJz3TvRvvH9mBwfxu3PfHg2AIIf3C/434Xnfe+Ad6wP95771JyAUCh+lvEu/DLtzcif7MZfD/8dFEaQGPuB78h78YTegnXTF/CgHJj3YfzlqT/BUzM34Do5iwdTv4Nw+yfvQatuQbWcxP2pD8/b/5nJ/4zf60bc4XkJH02Z/9/+k5P/He26Bjs8LdiT8v/m7f/4xP9AJwrxHs8zuDfl6Lz99018DP3IQ533SdR5fzlv/wcn7sc40nGv9wje43123v73T3wKALDH+2Ps8L4wZ9+4puGDk3sBAB/1Po47vK/M2d+vObhv8uMAgPtTvo9qz9wVPDrVj49PfgQA8GDKt3Gdp2PO/vaZMnxyag8A4LMpBxHwXJqz//czG/CZKePz5IupX0G5zL3V3TpThf1T7wcAfC31iyiQYXwi8//i6X075v09rRbvQbNsIlIPoB4AKioqHK7GfukpXlxblovjnVfwqR++jE+W1SDp+xUdpGYwTE9NY2pGja/pGbx6YQBP/aINA2MTuOvSEGRqHFNmmEyr4pcXTuHgzw9jZGIK30udP2bmxxc68N2nWpGBq/hW2sicfQLguYFe/DT1Ekq9w3jf1DQ8YvyW6xHjgLQUL3xZaSjRdGRc9ULEOC94zC3FPvj8a1EyMY2yjgyIiLnfeI/dG9bjDv81KB4WVJzOmn3f4B8+VFWJ3oJtKB6YQuWp7Ln7AXzi+i0YKtiK0suj2HwiZ96v5v9r+3UYya1EaecAAqdz5v39/nfNNoxnlaP8Qic2ts/dDwCff92NmEr3Y13HWazrmL//i2/YjpmUTGxo+wPKX50/iOarb6kGAFSefAElnXODYtqbga++0di/+fizKOyeGxQTaT585fZqKBTXvvwECvo65+xfl+nHl2tuBgBs+91PkTc492GGZTlF+NLNxv4bXyhCztDwnP3F+cX4xxu3AwCqmwuQOTYxZ7/fX4I11xn7b3rOh7QJ4MG3bpv3d7SDuPFhWSLSBqBBVef/SjL3uGoALQAKwrvPREQB7FzsNQCgpqZGm5ubV1Jy3PjDpSv4wMHfwOMRfPODt+L6tflOl5RQpqZn0D10FV1XxtEzPIHLQ1eNr+Fx9AxN4PLw1dltYwsMO0/zeuDLSoUvKxV5GanIyUhBdloKstO9yE5PQU56CrLTU5CdZvwc3JaV5kVmmhfpKV6kp3iQkWp8T0/xIMUb17dkyWVEpEVVa2I5Nt5bNMH7L34As0FjDhII3U+ma8vy8P362/Ff/vk57Pr6M/jH92/H27aVOV1W3Bgcm8Sr/WO4ODCGzsExvDowjosDY7NfXUNXIy79U5CViqKcdBTnpuPmCh+Kc9JRmJOOAjNM8jPTZoPFl5mGjFTPvP50ongV10GjqgMi0o7592r8AAZUlUETQVVpLv7tL+7Anm+3oOG7LfjInZvxsdoqpPI3Xqgq+kcncbZ3BB29IzjbM2p87zW+94/OfQRDqldQnp+JNb4M3L6pEGt9mSjPz0RZfvpssBRmpyMthf+2lLziOmhMRwHUYO5cmmpzO0VRkpuBh+tvx4M/fBn/9MRpPNXWg7/fvR2VRdlOl7YqpqZn0NE3ilNdwzjVNYRT3cM40zOCs70jGBqfmj1OBFiTn4mNRVl45w3l2ODPwnp/Ftb4jHApyk6Hx8OWB9FC3HqPph/AHlU9FLbdB+OeTGNwYqe57VFV3Rly3BEY93hiatEk0z2aSH7024v45A9ewtXJGTS8JYCPvHUzMlK9TpdlidcCZQgnu4ZxqtsIlvbLI5iYnpk9bq0vE4HibFQWZWNDYTY2FmZhQ2E21vszkZ6SGP8WRFaKy3s0ZmDsgzH3xQegUUR2AjgSFjj+0PPM7rO9ItII4Hnz/EZ2m8XuvTetwesCfnz2J8fx5f84jUMtF3DfnZuwu2Z93ATO5PQMOnpHZ1snJ7uGcLp7eF6grCvIRFVJDt6ypRhVpbmoKsnB5pIcZKe75n8FooTjyhbNakv2Fk2oZ9t78YXDJ9Dc0Y/SvHT86W0V2F2zHmt8mU6XBiAYKCM41TWMk13DONk9hNNdw2jvGcbk9GvX8np/JqpKclFVmoOqklxsKc3BpmIGCpFVltKiYdCAQRNOVfFMWy++9mQbfn26BwLg9ZsKsePaUtRuLUVFof0ziYfGJ9HRO4qO3lGc7jYC5VTXEM70jMwLlC0ludhcmoMtJbnYUpqLTSXZyEpjoBDZiUGzRAya6M73jeKR5vP46cuXcLrbmCC2Jj8D2yt8uGmdD4HiHGwozEKFPyvmbraZGUXf6AS6royj+4ox5+TSlXGc6x1FR58xuqtn+LXJZiLA+oIsVJXkoKrUaJ1UlTBQiJzEoFkiBk1szvaM4IkT3Wjp6MeL5wdwoX9szv7MVC/yM425IMHhvAJj6ZKRq1MYuTqNkatTGJ6YiriO1Jr8DFQUZmFjoXFDPhhggWIGCpHbMGiWiEGzPAOjE7PzS873jWJgdBIDY5MYHJvE1PSMuU6WcWx2und2NntOegqKctJRmpeOkrwMlOZloDiHc02I4klcjjqj+OPLSsP2rDRsX+/82qZE5F78FZKIiGzFoCEiIlsxaIiIyFYMGiIishWDhoiIbMWgISIiWzFoiIjIVgwaIiKyFVcGACAilwF0LPP0IgA9FpZDFIrXF9lpJdfXBlUtjuVABs0KiUhzrMswEC0Vry+y02pdX+w6IyIiWzFoiIjIVgyalTvgdAGU0Hh9kZ1W5friPRoiIrIVWzRERGQrBg0REdmKDz6LQkSqAdQAaAcQANCuqkftOo+Si4jUwbg+Npnfm1T10CLn+ADUAzgEoA+AH0ADgCO8xihoJdeJbZ9fqsqvsC/zH/hI2LZHAQTsOI9fyfUFoA5AdcjPPgBtAOoXOS8AGE/INr/6AdQ5/ffhl7u+lnud2Pn5xa6zyBoANIVtawLQaNN5lFwCqtoa/EFVB2BcI+HXTiQ7ARQA2KSqBbpIK4iS1nKuE9s+vxg0kdUBaA3b1mxut+M8ShJmt8Y95vdQR839gcVeQ1UHVLXdjvoocSzjOrHt84tBE8b8AAjA6NucZf7WGfWDYLnnUXIxr4eA+UXkCnZ/fnEwwHx+4LV/4AgCMG6UWXUeJRlVLYiwuRZALL+BBsyBBIBxzfWx+4wiWOp1YuvnF4NmvvAXq5YSAAAD2ElEQVQuDbvPIwKM/vHPLXJMHwCEfmCIyKMiAoYNhVjOdWLr5xe7zogcJiL1MH7j3L/QcWafe/iSIRxsQnO48Tph0EQR4WatredRcjL7vhtUdecyX6IdRjcJrztaSEzXiV3XEYNmvmA/pD90Y8h/gGj9lMs9j5JbI4AdsRwoIvdH2By8ecvBBQRg2deJrZ9fDJow5s2wdszvs/RjgZu1yz2PkpeINAHYu8AN2NBjAwAaI4z+CX4w8PqiZV8ndn9+MWgiOwpjGYZQ1eZ2O86jJGPel2kM/R9YRGqjDSM1j2uI8D98LYDWWMKKEt8KrxPbPr8YNJHtBbArbFuDuR2A0aQUkTbzAyPm84hChp36RKTa/KoFsCv4ARHl+uoLDSKzW6MBwJ5VK57iwaLXyWp/fnF4cwSqOiAie0WkEcDzMPo1GyP8luBf5nmUpMz/6R+Nsnux6+uQiNSZQVUIo5tjF68vCrWE62TVPr/44DMiIrIVu86IiMhWDBoiIrIVg4aIiGzFoCEiIlsxaIhczBz2TBTXGDRELmUOT612ug6ileI8GiL3ugecjEkJgC0aIvfyRVoyxFxJoM2JgoiWg0FD5EJmt1m0FQTuAcC1zShusOuMyJ0aMH/dqaBacKFWiiNs0RC5TPAZIAustFsN4MjqVUS0MgwaIpuISJOIaJSvlgVO3Y2wbjPzEQJNIhIMmF3mzxyVRq7HRTWJbGA+1OwIgFYYTzdsRMhy6ws9F8QMk11RBgLcD+AeVb3F8qKJbMJ7NEQWM1sZTaraGvJzW4xP0vTBeKJhtGN3gvdnKM4waIgsFgyYELUwWjaxqAfw8AL7a2G0jmaZD7lqAHA/jFbTAIxnkBQCeF5VD8X43kS2YNcZkc3M+zE7YmzRHFHVnVH2VQNoAVAQ/lpmS6hfVSVseyOAXlXdv+y/ANEKcTAAkY3MAIg48TLCsQEsPD9mznPfg6PTFqKqewHsC320L9FqY9AQ2Ws3Yu82qwPQtMD+WwE0h/xcH+3AMEfN1yZyBIOGyF47YTx/PRb3qOpiN/pbgNlVnWMdFNAHI6SIHMGgIbJXADEEgtm11bzIYZ8DsNNcnibSoINo/Ig97Igsx8EARC5g3rQ/EkOLJtr5EQcDmPv6YQxGiDWYiCzFoCFyARFpWckkzAVGnd0PoNAcFEDkCM6jIXKYOWx5sW6zhc4PzqMJBkvoPJo2Dm0mp7FFQ+Qwc7maJnZtUaLiYAAi5/kZMpTI2KIhIiJbsUVDRES2YtAQEZGtGDRERGQrBg0REdmKQUNERLZi0BARka0YNEREZCsGDRER2er/A2R7BmOyC8vjAAAAAElFTkSuQmCC\n", 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WAJgrImt8thUDcIlIJYCakfTF0PCerT2C1w4246E7FrI5jIiG5fgMfQDVsJKF71yXYrM9IFXd4L9NRB4EsFRV19oeYZprOtuNh178AFfNyWdzGBGFxelmMQBYC2CV37YKsx2A1UwmInUiUh6inEmxCI6A7/5+Lzp7+vDQHQvZHEZEYYmo5iIiE2HNQWkD0KKq7ZEGYJrG1nqbs0y5lQGaxAIu72ImWq4FsBpArohUAahSVf9Z/xSBVw804bdvH8NXll+MosmcLElE4QkruYjIfbBqFyUAFFZHe6vZ7RKRPLPtGQCbVLVhJEGYRBA0GZhJknlB9rlh1XS4GrLNunr78e3nd2NOwXh84WNznQ6HiJJI0ORiaiffBLAYwEYAq1T1dKjCRGQ5gHUm2VSp6h/tDJbi6yfbDqKhuRNP3Xs1xmRnOh0OESWRgMlFRBYDKAewXlXrwy1MVbcC2GrKuE9EilX1P2yJlOLqwMkzePSVOqxYPB3XFHEpfSIamSHJRUTmAHCp6uejKVhVHxORHBFZoarPRVMWxZeq4lvPv4fxo7PwrdvmOx0OESWhIcnF1FTCrq2EYprRmFiSzO/e+RBv1bfg4RULMWnCaKfDIaIkNKKhyCKyKFaBUGLo6O7DQy/uxcLpOVhdwjktRBSZkc5zGTIiyzR9fZ2JJzX8eNtBnGzvxr/efjkyMzinhYgiEzK5iMhBEakRkYdEZFmgY1T1tKp+H8CkYMdQcqhv6sDjO9y4s3gGimcGHPlNRBSW4Wouq2CN/roJ1nIs5SbZPCwiy8xwZQADI8Wiej4LOes7/7MHo7MysfaWS5wOhYiSXMhJlKr6NoC3ve9F5GVYz6ovgzUrXkVkF6wVinea7ezAT0Jb957Etn2N+Nat8zH5gjFOh0NESW6kfS5uVf2+qt6kqhkAlsJ6fv0kWLWb79kdIMVeV28/vvPC+5hbOB5/e81sp8MhohQw0rXFNvq+GW7ZFkoOP3u1HoeaO/HLv78Ko7ISYS1TIkp2I7qTmH4VSiGnznThkW0HUXbZFFx3caHT4RBRihiSXERkjoissKNwEZkoIvfaURbFxv/dcgDdfR5845ZLnQ6FiFLIkOTinaEvIj8VkdmRFmxWUv6Gqj4eeXgUS/tOnMEzNYfxmY/MgquQy+kTkX0CNouZUWLrANwvIi+JyL2+w46DEZFFIvKoiLwEoE5Vv2FzvGSj7764FxNGZ+Gryy92OhQiSjFBO/TNumDrAEBE7gTwuIgUw3qeSxuAFnPoXFjPWnHDGpJcZZITJbBX9jdi+/5GfPu2+cgbP8rpcIgoxYQ1WkxVNwPY7H0vIjkY/CTKkM95ocTS71E89Pu9mJk/Dp/96CynwyGiFBTRY45NMmHtJEk9W3sE+06ewSN3F2N0Fh8CRkT246SGNHO2uw8/eHk/Smbl4ZYFFzodDhGlKCaXNFP1Sh2aznbjW7fNhwhXPSai2Ig4uYjICs5hSS4n27vw2A43/vrKaVjMVY+JKIZG/LAwM9S4BlaHfp6I1Jo5MXyeS4L7z60H0NeveOCmeU6HQkQpbtgOfTO/pQLAGgB1ADao6v0+h3xfRBbDmhOzBMAz5pj2WARMkalv6sDTNUdw99UzMWvSeKfDIaIUFzS5mLktFbDmsFSpakmwY828lvvNefcB+KOINJvzuAR/AvjBy/swKjMDX1pW5HQoRJQGAiYXEXkUQCuACrMcTNhU9TEAj4nIHAAVIrJGVddEHypF6r1jp/HCu8fx5WVFfFYLEcVFwOTi1+wVEZOU1kVbDkWv8n8/QO64bNx3vcvpUIgoTXAocop7/WATdhxowhc/VoSJY7KdDoeI0oStyUVEltlZHkVHVVH50j5MzRnDZV6IKK7srrmU2VweReGlPSfwzpE2fK10HsZkc5kXIoofu5MLp3wniL5+D9a/tA9FkydgRfF0p8MhojRjd3JRm8ujCG3edRTuxg48cNMlyMpk1xoRxRfvOimoq7cfP6w+gEUX5eLmy6c4HQ4RpSEmlxT01JuHcfx0F75+8yVcnJKIHMHkkmLO9fTjkT/V4SOufFxbVOB0OESUptihn2J++ecGNJ3txj/ddInToRBRGguaXERkdgTl1UUcCUXtbHcfHn3FjesuLsDS2flOh0NEaSxUzaVypIWZdcXIIT9/vQEtHT34xzIuqU9Ezgq15P4qEVFYS+hv5RL6ia29qxcbtrux7NLJfBAYETluuOe53ARgNQAVETeAagAvg8km4fxsRz1On+tlrYWIEkKoZrFNqpoPoAjA5wG8DeuBYZsBtIrIAfMEyjviECeF0NbZgydercfNl0/Bguk5TodDRBQyubQAgKq6VXWDqq42yWYu/JKNiDSLyI2xD5cC2bDdjbM9ffgaay1ElCCCJpdgz3RR1foAyeZxANUiwqV346z5bDeefL0Bty2ciksvnOh0OEREAGyY52KSzVoASwGsjz4kGomq7W509fbjH0pZayGixGHbJEpV3QWg1q7yaHinznThF2804FOLpqNo8gSnwyEiGhB1chGRFSKyyLzlqshx9NM/1aG3X/GV5Rc7HQoR0SDDDUUOx3oAeSJyH4DTNpRHYTjV3oWn3jyMO4unY3bBeKfDISIaxI7ksgTWXBi3qj5nQ3kUhqrtbvR5FF+8scjpUIiIhog6uajqaQBc9iWOGs9049dvHsKnFk3HrEmstRBR4uGS+0no8R1u9PR58KVlrLUQUWKyNbmIyL12lkdDNZ/txi/eOITbF03HHPa1EFGCsrvmMtfm8sjP46/Wo6uvn30tRJTQ+LCwJNLa0YNfvN6AT1wxjfNaiCih2Z1cOM8lhn72aj06e/vxZfa1EFGCi2q0mIjcCWvZF4VVa1kuIjDfK4AaDk+2x+nOXjz5egNuXTAV86Zc4HQ4REQhRZVcVHUzrCX4AQAi8j1V/UbUUdEQT7xWj7PdffjyctZaiCjxsVksCZw+14snXqvHxy+/kCsfE1FSsGOGftREpBhACQA3ABes2f7VYZy30hw/13ytUtVNsYzVCT9/vQFnulhrIaLkETS5iMhsVW0YYXktIw1ARFwAKlW1zGfbRhFxq6o7xHkrYSWhTeZ9LoCdIpKvqhtGGkeiOtPVi5+9Wo+yy6bg8ml8yiQRJYdQzWKVEZQXyU29AkCV37aqMD7fZZb5BwCoaps5x7+spPaLNw7h9LlefGUZVz4mouQRKrmsEpGnReQOEQmrod+sMzZSKwHs8ttWa7YHZGopa8xXX9VmvyuCOBLOuZ5+/OzVetx4SSEWzmCthYiSx3Ad+jfBGg3WKiIHROSnI0k2wzHJwQW/5jRTCwmaJMx+l3mlrKdrDqOlo4driBFR0gmVXDapaj6AIgCfB/A2gDUIkGyi+Px84HwyCSBo8lDVPN9mMaMUQFuovppk0dPnwYbtblw1Jx9LZuU7HQ4R0YiESi4tAKCqblXdoKqrTbKZC79kIyLNInJjBJ/v36wVrQoADwfbKSLlIlIrIrWNjY02f7S9nn/7GI6f7uIaYkSUlIImF1W9P8j2+gDJ5nEA1SIyK0ZxDktEygG0qOr6YMeYuEtUtaSwsDCO0Y1Mv0fx01fqsGD6RFx/cYHT4RARjVjUkyhNslkLaxmYoDf2UAJ0zI/0fBeACt/hzMnsD+8dR31TB774sSKY5XSIiJKKbTP0Tf9H7QhP8/aNDOpU8Ek24fadVAJYPsLPTkiqike21cFVOB43X36h0+EQEUUk6uQiIitEZJF5O6LlX0xHvhtD+17yEWbHvIhUAVgbYlBAUvnT/ka8f7wdn79hLjIyWGshouRkR81lPYCtIrICQCTzXKphLf3iq9hsD8n0s1T6JiERKU3meS6PbDuI6blj8anF050OhYgoYnYklyUA1sFaiuWxCM5fC2CV37YKsx2A1UwmInUmmXi3eSdZ5opIsXmVAliVrEOR36pvQU1DK8qvdyE70+41RYmI4ifqhSvNrPxIkor3/DYRWSsilQBqYM1tqQyQIAb6ZUyfzMYgRSZlYgGAn2w7iEnjR2F1yUVOh0JEFJWEWBXZDAbwnxDpu78NQJ7f+5TqkHjv2Gm8sr8RX7/5Eowdlel0OEREUQnY9iIiXxeRZdEWLiKLROSBaMtJB1Xb3bhgdBY++1HHpgoREdkmWMP+JgCrzRIvD4vI7HALFJGJJjnVwuo72Rp9mKntSEsnXtx9HH9z9UxMHJPtdDhERFEL2CymqvUA7gcAEbkTwAYRUVgP43ou0DnmuApYw5E3qqr/CDAK4onX6iEAPnftbKdDISKyxbB9Lqq6Gdb6YXMAVIjIegBbYD03pQ1WEloOa+hwhUlMFKbTnb14puYIPrloGqbmjHU6HCIiW4TdoW+SxjoA60RkOYBvwqqlbFDVdTGKL+X96s1D6Ozpx33XJe3UHCKiISIaLaaqW8G+lKh19/XjydcbcP28QsyfassjcoiIEgJn6jnov9/+EI1nulHOWgsRpRgmF4d4PIoNO9y4bOpEXFs0yelwiIhsxeTikD/tP4WDp86i/HoXl9UnopTD5OKQqlfcmJYzBrddMdXpUIiIbMfk4oD3jp3Gm/Ut+Ltr53CBSiJKSbyzOeDJ1xswblQmVi/lApVElJqYXOKs+Ww3fvfOh7izeAZyxnKpFyJKTUwucfZ0zRH09Hnwt9dwgUoiSl1MLnHU2+/BL984hOsuLkDR5AucDoeIKGaYXOLopT0ncKK9C5+7ZrbToRARxRSTSxw9+VoDZk0ahxsvmex0KEREMcXkEie7j55G7aFW3PPR2cjI4KRJIkptTC5x4h1+vKpkhtOhEBHFHJNLHJzu7MUL736ITy2ezidNElFaYHKJg9++fRTdfR78zVUznQ6FiCgumFxiTFXx1FuHceWMHCyYnuN0OEREccHkEmO7Drdi/8mzuIu1FiJKI0wuMfbUm0cwYXQW/vrKaU6HQkQUN0wuMeTtyL990TSMHx3RE6WJiJISk0sMDXTkX80mMSJKL0wuMfRM7VFcMSMHl09jRz4RpRcmlxh5/8N27D3ejlVLOGmSiNIPk0uMPLfrKLIzBZ+4gh35RJR+mFxioK/fg+f/8iGWXToZeeNHOR0OEVHcMbnEwI6DTWg6240VxWwSI6L0xOQSA8/tOobccdlcWp+I0haTi83au3rx8p4T+OSV0zAqi79eIkpPvPvZ7A+7j6O7z8MmMSJKa0wuNnvh3eOYPWkcrpzBuS1ElL6YXGzU0tGD1+uacevCqRDh0yaJKH0xudjopT0n0O9R3LpwqtOhEBE5isnFRi/utprELp820elQiIgcxeRiEzaJERGdx+Rik5fZJEZENIDJxSa/330cs9gkRkQEgMnFFu1dvXijrhm3LGCTGBERwORii+37G9HnUZRdxuVeiIgAJhdbbN17CnnjsrHoojynQyEiSghMLlHq9yi27TuFGy+ZjMwMNokREQFMLlHbdbgVbZ29WDafTWJERF5MLlHauvcUsjIE188rdDoUIqKEweQSpT9+cBJXzcnHxDHZTodCRJQwmFyicKztHPafPItll7JJjIjIF5NLFF470AQAuO5iNokREfliconCqwebUHjBaMybMsHpUIiIEgqTS4Q8HsVrB5vwV0UFnJVPROSHySVC+06eQXNHD64tKnA6FCKihMPkEqFXTX/LtUWTHI6EiCjxMLlE6NWDTZhbOB5Tc8Y6HQoRUcJhcolAX78HNQ0tbBIjIgqCySUC7x9vR2dPP5bOznc6FCKihJTldAAAICLFAEoAuAG4ALhVtTpW50WrpqEVAJhciIiCcDy5iIgLQKWqlvls2ygiblV1232eHWrqW3BR/lhcmDMmlh9DRJS0EqFZrAJAld+2KgCVMTovKqqK2kMtWDqLtRYiomASIbmsBLDLb1ut2R6L86LS0NyJprM9WDqHyYWIKBhHk4uI5MLqK2nx3a6qbWa/y87z7FBTb33k0tl86iQRUTBO11zygfNJIYBgSSLS86JW09CCvHHZmFvI9cSIiIJxOrnkxvM8ESkXkVoRqW1sbIzogydNGI3brpjK9cSIiEJwfLRYPKnqBgAbAKCkpEQjKWPdLZfaGhMRUSpyuuYCYKAPJW7nERFRbDmdXLzzUQYNvfJJGsHmq0R6HhERxYFapZLiAAAFvklEQVSjycV0yLsxtA8lH0BbsMmQkZ5HRETx4XTNBQCqYS3h4qvYbI/FeUREFGOJkFzWAljlt63CbAdgNXeJSJ2IlI/kPCIicobjo8VUtU1E1opIJYAaWHNUKgM0beVHeB4REcWZ48kFAFR1F4Yu5eK7vw3AkCnxw51HRETOSIRmMSIiSjGiGtFcwqQnIo0ADkV4egGAJhvDIfLF64tiKdrra5aqFg53UNoml2iISK2q+o9UI7IFry+KpXhdX2wWIyIi2zG5EBGR7ZhcIrPB6QAopfH6oliKy/XFPhciIrIday5ERGQ7JhciIrIdkwsREdkuIZZ/SRQiUgxrpWU3rLXK3Ko67CrLkZ5H6UVEVsK6Puaar1WqummYc3IBlAPYBKAF1hp7FQC28BojX9FcKzG5h6kqX9agBpf5n+C7bSMAVyzO4yu9XgBWAij2eZ8LoA5A+TDnuQCoz6sVwEqnfx6+Eu8V6bUSq3sYm8XOqwBQ5betCkBljM6j9OJSa6FVAAOLsVZi6LUTSBmshVvnqmqeDlPbobQWybUSk3sYk8t5KzF0heVasz0W51GaMM0Va3wew+1Vbfa7hitDVfmEVQpLBNdKTO5hTC4Y+MfvgtVOOcD8dRn0H3+k51F6MdeDy7yIEkYs72Hs0LfkA+d/oQG4YHV02XUepRlVHfI8IgClAML5K9NlBgMA1jXXwqYxCmKk10rM7mFMLhb/5opYn0cEWG3dDw9zTAsA+N4gRGSjiIAJhvxEcq3E7B7GZjEiB4hIOay/KteHOs60n/uvBcUBIzREol0rTC4+AnS4xvQ8Sk+mHbtCVcsiLMINq/mD1x0NJ6xrJRbXEpOLxdummO+70ecXHqzNMdLzKL1VAlgezoEi8mCAzd7OVw4QoAERXisxu4cxuWCgM8uNoe2P+QjR4RrpeZS+RKQKwNoQHai+x7oAVAYYseO9EfD6IgCRXyuxvIcxuZxXDWv5A1/FZnsszqM0Y/pZKn3/wYpIabDhnua4igD/wEsB7AonQVF6iPJaick9jMnlvLUAVvltqzDbAVhVRRGpMzeJsM8j8hkemisixeZVCmCV94YQ5Ppq8U0+prmiAsB9cQueksWw10o872EcimyoapuIrBWRSgA1sNooKwP8JZAf4XmUpsw/8o1Bdg93fW0SkZUmOU2C1XyxitcX+RvBtRKXexifRElERLZjsxgREdmOyYWIiGzH5EJERLZjciEiItsxuRAlCDM0mSglMLkQJQAzfLTY6TiI7MJ5LkSJYQ04MZJSCGsuRDFiZuHXhXl4bqAlOkZYBlHCYHIhip01AMJZoHIlgs/gD6sMokTDZjGi2ClFeIv/VWDo2k4jLYMoobDmQhQ7xQC2hDrA+9yMEKvWDlsGUSJiciGykVlCv0pEvAlhlXkfbCTYavg1iUVQBlHC4cKVRDFgngq4RlWXDHPcFlgr1wbqzA+rDKJExJoLUWyUYZi+EtMk1haiSWzYMogSFWsuRDEgIgqgTFWDJgdTM3Gr6qZwyzAPg6oA8CCshzm1wXpuxyQANcHKIoo3Jhcim5m+kZ0A8kI9XlZEtqhq2UjLMDWeVlUVv+2VAJpVdX20PwNRtNgsRmS/Qc8t944I82VqIKHmrwxbhj9VXQvgG76PuiVyCpMLkf2WAqj1eV8e4JiVAKqiLCOQalM2kaOYXIhiYycwsNJxoH6XNaH6Y8IsI5AWWImJyFFMLkT2exhAmVnWBaq6y3enabaqDXRiuGWEkA+gZmThEtmPHfpEcWY63reEUXMJdn7ADn2zrxXA8hEkI6KYYHIhijMR2RnNxMgQo8UeBDDJdOwTOYoLVxLFkRliPFyTWKjzvfNcvMnEd55LHYchU6JgzYUojkSkCkAVm60o1bFDnyi+8plYKB2w5kJERLZjzYWIiGzH5EJERLZjciEiItsxuRARke2YXIiIyHZMLkREZDsmFyIist3/Byc7KNWzNSSJAAAAAElFTkSuQmCC\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "j2max = (0.5 * N + 1) * (0.5 * N)\n", "\n", "plt.rc('text', usetex = True)\n", "label_size = 20\n", "plt.rc('xtick', labelsize=label_size) \n", "plt.rc('ytick', labelsize=label_size)\n", "\n", "fig1 = plt.figure()\n", "plt.plot(t/td, g2t/jpjm_ss**2, '-')\n", "plt.plot(t/td, 1+0*g2t, '--')\n", "plt.xlabel(r'$\\tau/t_\\mathrm{D}$', fontsize = label_size)\n", "plt.ylabel(r'$g^{(2)}(\\tau)$', fontsize = label_size)\n", "plt.xticks([0,(tmax/2)/td,tmax/td])\n", "plt.show()\n", "plt.close()\n", "\n", "\n", "fig2 = plt.figure()\n", "plt.plot(t/td, jpjmt/j2max, '-')\n", "plt.xlabel(r'$t/t_\\mathrm{D}$', fontsize = label_size)\n", "plt.ylabel(r'$\\langle J_{+}J_{-}\\rangle (t)$', fontsize = label_size)\n", "plt.xticks([0,(tmax/2)/td,tmax/td])\n", "plt.title(r'Light emission', fontsize = label_size)\n", "plt.show()\n", "plt.close()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 2) Maximum of light emission as a function of $\\frac{\\gamma_\\text{P}}{N\\gamma_\\text{CE}}$\n", "\n", "We perform a study of the scaling of the steady state light emission of the system as a function of the pumping rate, normalized by the number of TLSs and the collective emission rate. The results show an optimal point for $\\frac{\\gamma_\\text{P}}{N\\gamma_\\text{CE}}\\simeq 1$. " ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": [ "# Cycle on Coefficients\n", "gCE = 1\n", "gP0 = 1\n", "gP_min_exp = -20\n", "gP_max_exp = 20\n", "gP_stepsize = 0.5\n", "gP_list = np.arange(gP_min_exp, gP_max_exp+1, gP_stepsize)*0.1\n", "gP_list_log = 10**(gP_list)\n", "\n", "jpjmss_max_list = []\n", "for i in gP_list_log:\n", " gP = i*gP0\n", " system = Dicke(hamiltonian = jz, N = N, pumping = gP, collective_emission = gCE)\n", " liouv = system.liouvillian()\n", " #steadystate\n", " rho_ss = steadystate(liouv,method=\"direct\")\n", " jpjm_ss = expect(jp*jm, rho_ss)\n", " jpjmss_max_list.append(jpjm_ss)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Visualization" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "intensity_max = float(N)*gCE/2*(float(N)*gCE/2+1)\n", "normalized_intensity = np.array(jpjmss_max_list)/intensity_max\n", "plt.semilogx(gP_list_log/(gCE*N), normalized_intensity, '-')\n", "label_size = 20\n", "plt.xlabel(r'${\\gamma_\\mathrm{P}}/\\left({N\\gamma_\\mathrm{CE}}\\right)$', fontsize = label_size)\n", "plt.ylabel(r'$\\langle J_{+}J_{-}\\rangle_\\mathrm{ss}$', fontsize = label_size)\n", "plt.title(r'Steady-state light emission', fontsize = label_size)\n", "plt.show()\n", "plt.close()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### References\n", "\n", "[1] D. Meiser and M.J. Holland, *Phys. Rev. A* **81**, 033847 (2010)\n", "\n", "[2] D. Meiser and M.J. Holland, *Phys. Rev. A* **81**, 063827 (2010)\n", "\n", "[3] J.G. Bohnet *et al.* *Nature* **484**, 78 (2012)\n", "\n", "[4] J.R. Johansson, P.D. Nation, and F. Nori, *Comp. Phys. Comm.* **183**, 1760 (2012) http://qutip.org" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "scrolled": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "QuTiP: Quantum Toolbox in Python\n", "Copyright (c) 2011 and later.\n", "A. J. Pitchford, P. D. Nation, R. J. Johansson, A. Grimsmo, and C. Granade\n", "\n", "QuTiP Version: 4.3.1\n", "Numpy Version: 1.14.2\n", "Scipy Version: 1.1.0\n", "Cython Version: 0.28.5\n", "Matplotlib Version: 2.2.3\n", "Python Version: 3.6.7\n", "Number of CPUs: 2\n", "BLAS Info: INTEL MKL\n", "OPENMP Installed: False\n", "INTEL MKL Ext: True\n", "Platform Info: Darwin (x86_64)\n", "Installation path: /Users/nathanshammah/miniconda3/lib/python3.6/site-packages/qutip\n", "==============================================================================\n", "Please cite QuTiP in your publication.\n", "==============================================================================\n", "For your convenience a bibtex file can be easily generated using `qutip.cite()`\n" ] } ], "source": [ "qutip.about()" ] } ], "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": "3.6.7" } }, "nbformat": 4, "nbformat_minor": 2 }