{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Steady-state superradiance, lasing and cooperative resonance fluorescence\n",
    "\n",
    "Notebook author: Nathan Shammah (nathan.shammah at gmail.com)\n",
    "\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} &=&-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 $N^2$.\n",
    "\n",
    "It is then natural to ask the question of whether the system sustains a superradiant light emission steady state also when local losses are included, just like for the transient superfluorescent light emission [4]. We thus study also\n",
    "\\begin{eqnarray}\n",
    "\\dot{\\rho} &=&-i\\lbrack \\omega_{0}J_z,\\rho \\rbrack\n",
    "+\\frac{\\gamma_\\text {CE}}{2}\\mathcal{L}_{J_{-}}[\\rho]\n",
    "+\\sum_{n=1}^{N}\\left(\\frac{\\gamma_\\text{P}}{2}\\mathcal{L}_{J_{+,n}}[\\rho]+\n",
    "\\frac{\\gamma_\\text{E}}{2}\\mathcal{L}_{J_{-,n}}[\\rho]\\right)\n",
    "\\end{eqnarray}\n",
    "and apply the detailed balance condition to the local emission and pumping rates. This study has relevance to the application of superradiance in light-harvesting devices. \n",
    "\n",
    "We will also assess the possibility of coherently driving the system $\\propto J_x$ in the Hamiltonian, which leads to cooperative resonance fluorescence. \n",
    "\n",
    "See Refs. [1-12] for more information. Simulations performed with QuTiP [13] and PIQS [8], imported as $\\texttt{qutip.piqs}$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "from qutip import *\n",
    "from qutip.piqs import *\n",
    "import numpy as np\n",
    "from scipy import constants"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1) Time evolution\n",
    "\n",
    "We study the system of Eq. (1) by using the $\\texttt{qutip.piqs}$ module 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. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### System properties\n",
    "We initialize an object of the $\\texttt{Dicke}$ class using $\\texttt{qutip.piqs}$. The attributes of this object are used to define the rates of the Liouvillian superoperator. The Hamiltonian is constructed using the $\\texttt{jspin}$ functions in the Dicke basis. The collective and local dissipation rates are defined according to a list of keywords:\n",
    "\n",
    "<table>\n",
    "    <tr>\n",
    "<td> Keyword</td>\n",
    "<td> Rate  $\\gamma_j$</td>\n",
    "<td> Lindbladian $\\mathcal{L}[\\rho]$</td>\n",
    "</tr>\n",
    "\n",
    "<tr>\n",
    "<td> $\\texttt{emission}$ </td>\n",
    "<td> $\\gamma_\\text{E}$</td>\n",
    "<td> \\begin{eqnarray}\\mathcal{L}[\\rho]&=&\\sum_n^N \\left(J_{-,n}\\rho J_{+,n} - \\frac{1}{2}J_{+,n}J_{-,n}\\rho - \\frac{1}{2}\\rho J_{+,n}J_{-,n} \\right)\\end{eqnarray}</td>\n",
    "</tr>\n",
    "\n",
    "<tr>\n",
    "<td> $\\texttt{pumping}$ </td>\n",
    "<td> $\\gamma_\\text{P}$</td>\n",
    "<td> \\begin{eqnarray}\\mathcal{L}[\\rho]&=&\\sum_n^N \\left(J_{+,n}\\rho J_{-,n} - \\frac{1}{2}J_{-,n}J_{+,n}\\rho - \\frac{1}{2}\\rho J_{-,n}J_{+,n} \\right)\\end{eqnarray}</td>\n",
    "</tr>\n",
    "\n",
    "<tr>\n",
    "<td> $\\texttt{dephasing}$ </td>\n",
    "<td> $\\gamma_\\text{D}$</td>\n",
    "<td> \\begin{eqnarray}\\mathcal{L}[\\rho]&=&\\sum_n^N \\left(J_{z,n}\\rho J_{z,n} - \\frac{1}{2}J_{z,n}J_{z,n}\\rho - \\frac{1}{2}\\rho J_{z,n}J_{z,n} \\right)\\end{eqnarray}</td>\n",
    "</tr>\n",
    "\n",
    "<tr>\n",
    "<td> $\\texttt{collective}\\_\\texttt{emission}$ </td>\n",
    "<td> $\\gamma_\\text{CE}$</td>\n",
    "<td> \\begin{eqnarray}\\mathcal{L}[\\rho]&=& J_{-}\\rho J_{+} - \\frac{1}{2}J_{+}J_{-}\\rho - \\frac{1}{2}\\rho J_{+}J_{-} \\end{eqnarray}</td>\n",
    "</tr>\n",
    "\n",
    "<tr>\n",
    "<td> $\\texttt{collective}\\_\\texttt{pumping}$ </td>\n",
    "<td> $\\gamma_\\text{CP}$</td>\n",
    "<td> \\begin{eqnarray}\\mathcal{L}[\\rho]&=&  J_{+}\\rho J_{-} - \\frac{1}{2}J_{-}J_{+}\\rho - \\frac{1}{2}\\rho J_{-}J_{+} \\end{eqnarray}</td>\n",
    "</tr>\n",
    "\n",
    "<tr>\n",
    "<td> $\\texttt{collective}\\_\\texttt{dephasing}$ </td>\n",
    "<td> $\\gamma_\\text{CD}$</td>\n",
    "<td> \\begin{eqnarray}\\mathcal{L}[\\rho]&=& J_{z}\\rho J_{z} - \\frac{1}{2}J_{z}^2\\rho - \\frac{1}{2}\\rho J_{z}^2 \\end{eqnarray}</td>\n",
    "</tr>\n",
    "\n",
    "</table>\n",
    "\n",
    "Note that in the table above and in $\\texttt{qutip.piqs}$ functions, the Lindbladian $\\mathcal{L}[\\rho]$ is written with a factor 1/2 with respect to $\\mathcal{L}_{A}[\\rho]$ reported in the LaTeX math equations, in order to have the Lindbladian and full Liouvillian matrix consistently defined by the rates $\\gamma_\\alpha$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = 4\n",
    "system = Dicke(N = N)\n",
    "[jx, jy, jz] = jspin(N)\n",
    "jp = jspin(N, \"+\")\n",
    "jm = jspin(N, \"-\")\n",
    "\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": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.23 ms ± 41.7 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%timeit(system.liouvillian())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Liouvillian and steady state $\\rho_\\text{ss}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "L = system.liouvillian()\n",
    "rhoss = steadystate(L)\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",
    "\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",
    "#### $g^{(2)}(\\tau) = \\frac{\\langle: J^\\dagger(\\tau) a^\\dagger(0) a(\\tau) a(0) :\\rangle}{|\\langle: a^\\dagger(0) a(0) :\\rangle|^2}= \\frac{\\langle: J_{+}(\\tau) J_{+}(0) J_{-}(\\tau) J_{-}(0) :\\rangle}{|\\langle J_{+}(0) J_{-}(0) \\rangle|^2}\\ \\ \\ \\ \\ \\ (2)$.\n",
    "\n",
    "We apply the Quantum Regression Theorem, which allows to calculate two-time correlation functions for two operators $A$ and $B$ from the Liouvillian $\\mathcal{D}$ of the system,\n",
    "\n",
    "$\\langle A(t+\\tau)B(t)\\rangle = \\text{Tr}[Ae^{\\mathcal{D}\\tau}[B\\rho(t)]]=\\text{Tr}[A\\rho_B(\\tau)]$\n",
    "\n",
    "where we introduced $\\rho_B(\\tau)$. In the case of Eq. (2) $A = B = J_{+}J_{-}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "# time evolution\n",
    "nt = 1001\n",
    "# superradiant delay time\n",
    "td = np.log(N)/(N*gCE)\n",
    "tmax = 5 * td\n",
    "t = np.linspace(0, tmax, nt)\n",
    "\n",
    "# we initialize the collective TLS state in the ground state, given in terms of a Dicke state\n",
    "rho0 = dicke(N, N/2, -N/2)\n",
    "# we can initialize the collective TLS state in the steady state\n",
    "rho0 = steadystate(L)\n",
    "rhoss = steadystate(L)\n",
    "# We implement the \"quantum regression theorem\"\n",
    "A = jp*jm\n",
    "rhoB = jm*rhoss*jp\n",
    "\n",
    "#g2(tau)\n",
    "result1 = mesolve(L, rhoB, t, [], e_ops = [A], options = Options(store_states=False))\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]\n",
    "nphot_ss = expect(jp*jm, rhoss)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Visualization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10fc9c470>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10438c518>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "j2max = (0.5 * N + 1) * (0.5 * N)\n",
    "label_size = 20\n",
    "plt.rc('text', usetex = True)\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/nphot_ss**2, '-')\n",
    "plt.plot(t/td, 1+0*g2t, '--')\n",
    "plt.xlabel(r'$\\tau/t_\\text{D}$', fontsize = label_size)\n",
    "plt.ylabel(r'$g^{(2)}(\\tau)$', fontsize = label_size)\n",
    "plt.title(r'Second-order coherence function', fontsize = label_size)\n",
    "plt.xticks([0,(tmax/2)/td,tmax/td])\n",
    "plt.show()\n",
    "plt.close()\n",
    "\n",
    "\n",
    "fig1 = plt.figure()\n",
    "plt.plot(t/td, jpjmt/j2max, '-')\n",
    "plt.xlabel(r'$t/t_\\text{D}$', 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.ylim([0,1])\n",
    "plt.rc('xtick', labelsize=label_size) \n",
    "plt.rc('ytick', labelsize=label_size)\n",
    "plt.title(r\"Light emission in units of $\\frac{N}{2}(\\frac{N}{2}+1)$\", 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 $\\frac{\\gamma_\\text{P}}{N\\gamma_\\text{CE}}\\simeq 1$ [4,5]. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "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)\n",
    "    jpjm_ss = expect(jp*jm, rho_ss)\n",
    "    jpjmss_max_list.append(jpjm_ss)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Visualization\n",
    "We use the notation $\\gamma_\\uparrow=\\gamma_\\text{P}$ and $\\gamma_\\Downarrow=\\gamma_\\text{CE}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10fe39710>"
      ]
     },
     "metadata": {},
     "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.rc('text', usetex = True)\n",
    "plt.rc('xtick', labelsize=label_size) \n",
    "plt.rc('ytick', labelsize=label_size)\n",
    "plt.xlabel(r'${\\gamma_\\uparrow}/\\left({N\\gamma_\\Downarrow}\\right)$', fontsize = label_size)\n",
    "plt.ylabel(r'$\\langle J_{+}J_{-}\\rangle_\\mathrm{ss}$', fontsize = label_size)\n",
    "fname = 'figures/sssr_maxjpjm{}.pdf'.format(N)\n",
    "plt.title(r'Steady-state light emission', fontsize = label_size)\n",
    "plt.show()\n",
    "\n",
    "plt.close()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Emission spectrum"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Single TLS case: Mollow triplet in the resonance fluorescence spectrum, no cooperative effects"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10ff58518>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "wz = 1\n",
    "om = 5\n",
    "gamm = 2\n",
    "H = wz * sigmaz() + om * sigmax()\n",
    "c_ops = [np.sqrt(gamm)*sigmam()]\n",
    "a_op = sigmap()\n",
    "b_op = sigmam()\n",
    "wmin = - 20\n",
    "wmax = 20\n",
    "nw = 1000\n",
    "wlist = np.linspace(wmin, wmax, nw)\n",
    "liouv_TLS = liouvillian(H,c_ops)\n",
    "sw = spectrum(liouv_TLS, wlist, [], a_op, b_op, solver=\"es\", use_pinv=False)\n",
    "\n",
    "smax = np.max(sw)\n",
    "lw = 2\n",
    "plt.plot(wlist, sw/smax, '-r', linewidth = 2*lw)\n",
    "label_size = 20\n",
    "plt.rc('text', usetex = True)\n",
    "plt.rc('xtick', labelsize=label_size) \n",
    "plt.rc('ytick', labelsize=label_size)\n",
    "plt.xlabel(r'$\\Delta\\omega$', fontsize = label_size)\n",
    "plt.ylabel(r'$S(\\Delta\\omega)$', fontsize = label_size)\n",
    "plt.title(r'Spectrum', fontsize = label_size)\n",
    "plt.show()\n",
    "plt.close()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10ff526d8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "rhoss_TLS = steadystate(liouv_TLS)\n",
    "B = sigmap() * sigmam()\n",
    "jpjm_ss = expect(B, rhoss_TLS)\n",
    "rhoB = sigmam() * rhoss_TLS * sigmap()\n",
    "#Excited\n",
    "t = np.linspace(0, 2, 1000)\n",
    "options = Options(nsteps = 10000, atol = 1e-6)\n",
    "result3 = mesolve(liouv_TLS, rhoB, t, [], e_ops = B, options = Options(store_states=True))\n",
    "g2t = result3.expect[0]\n",
    "\n",
    "taulist = t#np.linspace(0, 20, 100)\n",
    "nphot_ss = expect(sigmap()*sigmam(), rhoss_TLS)\n",
    "plt.plot(t, np.real(g2t)/nphot_ss**2, '-', linewidth = lw, label = r\"$\\mathrm{Re}[g^2(\\tau)]$\")\n",
    "plt.plot(t, np.imag(g2t)/nphot_ss**2, '-o', linewidth = lw, label = r\"$\\mathrm{Im}[g^2(\\tau)]$\")\n",
    "plt.plot(t, 1 +0*t , '--', linewidth = lw)\n",
    "label_size = 20\n",
    "plt.rc('text', usetex = True)\n",
    "plt.rc('xtick', labelsize=label_size) \n",
    "plt.rc('ytick', labelsize=label_size)\n",
    "plt.xlabel(r'$\\tau$', fontsize = label_size)\n",
    "plt.ylabel(r'$g^{2}(\\tau)$', fontsize = label_size)\n",
    "plt.title(r'Correlation function', fontsize = label_size)\n",
    "plt.legend(fontsize = label_size*0.8)\n",
    "plt.show()\n",
    "plt.close() "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### $N>1$: Steady-state superradiance, cooperative resonance fluorescence\n",
    "\n",
    "Here we study the first-order and second-order correlation functions and the spectrum (using the quantum regression formula) of the coherently driven system of $N\\gg1$ two-level systems that can also emit cooperatively [1]. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "N = 10\n",
      "Hilbert space dim = (36, 36)\n",
      "Number of Dicke states = 36\n",
      "Liouvillian space dim = (1296, 1296)\n",
      "pumping = 5.0\n",
      "collective_emission = 1.0\n",
      "ok\n",
      "ok 2\n",
      "ok 3\n",
      "ok 4\n"
     ]
    }
   ],
   "source": [
    "N = 10\n",
    "[jx, jy, jz] = jspin(N)\n",
    "jp = jspin(N, \"+\")\n",
    "jm = jp.dag()\n",
    "\n",
    "w0 = 20\n",
    "g = 0\n",
    "h0 = w0 * jz + g * jx \n",
    "gCE = 1.0\n",
    "gD = .0\n",
    "gP = N/2 * gCE\n",
    "n_therm = 1\n",
    "\n",
    "gE_list = [0, gP*(1+n_therm)]\n",
    "system = Dicke(N = N)\n",
    "gE = 0\n",
    "\n",
    "system.hamiltonian = h0\n",
    "system.collective_emission = gCE\n",
    "system.emission = gE\n",
    "system.pumping = gP\n",
    "system.dephasing = gD\n",
    "L = system.liouvillian()\n",
    "rhoss = steadystate(L)\n",
    "print(system)\n",
    "td = np.log(N)/(N*gCE)\n",
    "tmax = 100 * td\n",
    "t = np.linspace(0, tmax, 2000)\n",
    "dw = 10\n",
    "wmin = w0 - dw\n",
    "wmax = w0 + dw\n",
    "nw = 100\n",
    "wlist = np.linspace(wmin, wmax, nw)\n",
    "lw = 3\n",
    "#1\n",
    "g2_norm, G2 = coherence_function_g2(L, None, t, [], jm, solver=\"me\", args={},\n",
    "                          options=Options(ntraj=[20, 100]))\n",
    "\n",
    "g1_norm, G1 = coherence_function_g1(L, None, t, [], jm, solver=\"me\", args={},\n",
    "                          options=Options(ntraj=[20, 100]))\n",
    "\n",
    "print(\"ok\")\n",
    "#swss = spectrum(L, wlist, [], jp, jm)\n",
    "wlist, swss = spectrum_correlation_fft(t, g1_norm)\n",
    "\n",
    "print(\"ok 2\")\n",
    "#2\n",
    "g2_ss = correlation_2op_1t(L, rhoss, t, [], jp, jm)\n",
    "#3\n",
    "rho0 = dicke(N, N/2, -N/2)\n",
    "#g2(tau)\n",
    "rhoB = jm*rhoss*jp\n",
    "result1 = mesolve(L, rhoB, t, [], e_ops = [jp*jm], options = Options(store_states=True))\n",
    "print(\"ok 3\")\n",
    "g2t = result1.expect[0]\n",
    "result2 = mesolve(L, rho0, t, [], e_ops = jp*jm, options = Options(store_states=True))\n",
    "print(\"ok 4\")\n",
    "rhot = result2.states\n",
    "jpjmt = result2.expect[0]\n",
    "#g2t nat\n",
    "Bn = jp * jm\n",
    "jpjm_ss = expect(Bn, rhoss)\n",
    "rhoBn = Bn * rhoss\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109b1c9e8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109b174e0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10fbbbef0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#1 Plot - Spectrum in frequency\n",
    "plt.plot(wlist, swss, 'o-', linewidth = lw)\n",
    "label_size = 20\n",
    "plt.rc('text', usetex = True)\n",
    "plt.rc('xtick', labelsize = label_size) \n",
    "plt.rc('ytick', labelsize = label_size)\n",
    "plt.xlabel(r'$\\Delta\\omega$', fontsize = label_size)\n",
    "plt.ylabel(r'$S(\\Delta\\omega)$', fontsize = label_size)\n",
    "plt.title(r'Spectrum', fontsize = label_size)\n",
    "plt.xlim([w0-10,w0+10])\n",
    "\n",
    "plt.show()\n",
    "plt.close()\n",
    "\n",
    "#2 Plot \n",
    "j2max = (0.5 * N + 1) * (0.5 * N)\n",
    "\n",
    "plt.rc('text', usetex = True)\n",
    "\n",
    "fig1 = plt.figure()\n",
    "plt.plot(t/td, np.real(g2t)/jpjm_ss**2, '-')\n",
    "plt.plot(t/td, np.real(g2_norm), '--r')\n",
    "plt.plot(t/td, np.real(g1_norm), '--b')\n",
    "plt.plot(t/td, 1+np.imag(g2_norm), 'ro')\n",
    "plt.plot(t/td, 1+np.imag(g2t), '-k')\n",
    "\n",
    "plt.plot(t/td, 1+0*t, '-.')\n",
    "plt.xlabel(r'$\\tau/t_\\text{D}$', fontsize = label_size)\n",
    "plt.ylabel(r'$g^{(1,2)}(\\tau)$', fontsize = label_size)\n",
    "plt.title(r'First and Second Coherence', fontsize = label_size)\n",
    "plt.xticks([0,(tmax/2)/td,tmax/td])\n",
    "plt.rc('xtick', labelsize=label_size) \n",
    "plt.rc('ytick', labelsize=label_size)\n",
    "plt.show()\n",
    "plt.close()\n",
    "\n",
    "#3 Plot Light emission as a function of time\n",
    "fig1 = plt.figure()\n",
    "plt.plot(t/td, jpjmt/j2max, '-')\n",
    "plt.plot(t/td, 0*t+0.5, '-')\n",
    "plt.xlabel(r'$t/t_\\text{D}$', 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.rc('xtick', labelsize=label_size) \n",
    "plt.rc('ytick', labelsize=label_size)\n",
    "#plt.ylim([0,N])\n",
    "plt.show()\n",
    "plt.close()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "occupation number, n0 =  13091.5295848\n",
      "15.0\n",
      "5.0\n"
     ]
    }
   ],
   "source": [
    "## general formual for detailed balance (not used below, giving here a qualitative estimate)\n",
    "# Bose-Einstein distribution determines the occupation number\n",
    "frequency_hertz = 10**(13)\n",
    "temperature_kelvin = 10**(6)\n",
    "x = (frequency_hertz / temperature_kelvin) * (constants.hbar / constants.Boltzmann)\n",
    "n0 = 1/(np.exp(x) -1)\n",
    "print(\"occupation number, n0 = \",n0)\n",
    "\n",
    "# set collective pumping and collective emission rates (coupled ensembles) \n",
    "g0 = 10\n",
    "n0 = gP / g0 \n",
    "gE = g0 * (1 + n0)\n",
    "print(gE)\n",
    "print(gP)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Steady-state superradiance under incoherent local pumping and collective and local losses\n",
    "\n",
    "Below we study steady state superradiant light emission and lasing under incoherent local pumping. We introduce local losses to study the possibility of superradiant effects under thermal excitation (in the high-temperature limit)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 10.  20.  30.  40.]\n"
     ]
    }
   ],
   "source": [
    "# Do two nested cycles to calculate the dependency of the maximum superradiance peak on the pump rate\n",
    "\n",
    "# compare the case of no loss (noloss) to the thermal case with detailed balance\n",
    "#ntls list\n",
    "ntls_list = np.linspace(10,40,4)\n",
    "print(ntls_list)\n",
    "#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",
    "gP_gCE_N_list_log= 10**(gP_list)\n",
    "max_nn_list = []\n",
    "argmax_nn_list = []\n",
    "normalized_intensity_list = []\n",
    "max_nn_list_noloss = []\n",
    "argmax_nn_list_noloss = []\n",
    "normalized_intensity_list_noloss = []\n",
    "lw_i = 0.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "10\n",
      "20\n",
      "30\n",
      "40\n"
     ]
    }
   ],
   "source": [
    "# Do two nested cycles to calculate the dependency of the maximum superradiance peak on the pump rate\n",
    "for nn in ntls_list:\n",
    "    lw_i = lw_i + 0.5\n",
    "    jpjm_max_list = []\n",
    "    jpjm_max_arg_list = []\n",
    "    jpjmss_max_list = []\n",
    "    jpjm_max_list_noloss = []\n",
    "    jpjm_max_arg_list_noloss = []\n",
    "    jpjmss_max_list_noloss = []    \n",
    "    N = int(nn)\n",
    "    print(N)\n",
    "    ntls = N\n",
    "    jmax = (0.5 * N)\n",
    "    j2max = (0.5 * N + 1) * (0.5 * N)\n",
    "    intensity_max = float(N)*gCE/2*(float(N)*gCE/2+1)\n",
    "    [jx, jy, jz] = jspin(N)\n",
    "    jp = jspin(N, \"+\")\n",
    "    jm = jp.dag()\n",
    "    jpjm = jp*jm\n",
    "    #time\n",
    "    for i in gP_gCE_N_list_log:\n",
    "        gP_gCE_N = i*gP0\n",
    "        gP = gP_gCE_N*(gCE*N)\n",
    "        gE = gP # fix the high temperature limit\n",
    "        td0 = np.log(N)/(N*gCE)\n",
    "        h = jz\n",
    "        system = Dicke(hamiltonian = h, N = N, pumping = gP, \n",
    "                       collective_emission = gCE, emission = gE)\n",
    "        system_noloss = Dicke(hamiltonian = h, N = N, pumping = gP, \n",
    "                              collective_emission = gCE, emission = 0.0*gE)\n",
    "        liouv = system.liouvillian()\n",
    "        liouv_noloss = system_noloss.liouvillian()\n",
    "        #steadystate with loss\n",
    "        rho_ss = steadystate(liouv)\n",
    "        jpjm_ss = expect(jpjm, rho_ss)\n",
    "        jpjmss_max_list.append(jpjm_ss)\n",
    "        #steadystate no loss\n",
    "        rho_ss_noloss = steadystate(liouv_noloss)\n",
    "        jpjm_ss_noloss = expect(jpjm, rho_ss_noloss)\n",
    "        jpjmss_max_list_noloss.append(jpjm_ss_noloss)\n",
    "        #print(\"N=\", N,\" gP/(N*gCE)=\", gP/(N*gCE) )\n",
    "    # emitted light with loss\n",
    "    normalized_intensity = np.array(jpjmss_max_list)/intensity_max\n",
    "    normalized_intensity_list.append(normalized_intensity) \n",
    "    argmax_nn = np.argmax(normalized_intensity)\n",
    "    max_nn = np.max(normalized_intensity)\n",
    "    max_nn_list.append(max_nn)\n",
    "    argmax_nn_list.append(gP_list_log[argmax_nn])\n",
    "    # emitted light no loss\n",
    "    normalized_intensity_noloss = np.array(jpjmss_max_list_noloss)/intensity_max\n",
    "    normalized_intensity_list_noloss.append(normalized_intensity_noloss) \n",
    "    argmax_nn_noloss = np.argmax(normalized_intensity_noloss)\n",
    "    max_nn_noloss = np.max(normalized_intensity_noloss)\n",
    "    max_nn_list_noloss.append(max_nn_noloss)\n",
    "    argmax_nn_list_noloss.append(gP_list_log[argmax_nn_noloss])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10fc6af98>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "label_size = 20\n",
    "label_size2 = 20\n",
    "label_size3 = 20\n",
    "\n",
    "plt.rc('text', usetex = True)\n",
    "plt.rc('xtick', labelsize=label_size) \n",
    "plt.rc('ytick', labelsize=label_size)\n",
    "\n",
    "fig1 = plt.figure(1)\n",
    "lw_i = 0.9\n",
    "for i in range(0,len(normalized_intensity_list)):\n",
    "    \n",
    "    #cycle for all N values\n",
    "    \n",
    "    lw_i = lw_i + 0.4*i\n",
    "    plt.semilogx(gP_gCE_N_list_log, normalized_intensity_list_noloss[i], 'k-',\n",
    "                 lw = lw_i)\n",
    "    plt.semilogx(gP_gCE_N_list_log, normalized_intensity_list[i], 'r--', lw = lw_i)\n",
    "    \n",
    "    # have legend with mid-sized line\n",
    "    \n",
    "    if i == len(normalized_intensity_list) -2:\n",
    "        \n",
    "        plt.semilogx(gP_gCE_N_list_log, normalized_intensity_list_noloss[i], 'k-',\n",
    "                 lw = lw_i, label=r\"$\\gamma_\\downarrow=0$\")\n",
    "        plt.semilogx(gP_gCE_N_list_log, normalized_intensity_list[i], 'r--', lw = lw_i,\n",
    "                 label=r\"$\\gamma_\\downarrow=\\gamma_\\uparrow$\")\n",
    "\n",
    "plt.xlabel(r'${\\gamma_\\uparrow}/\\left({N\\gamma_\\Downarrow}\\right)$', fontsize = label_size3)\n",
    "plt.ylabel(r'$\\langle J_{+}J_{-}\\rangle_\\mathrm{ss}/I_\\mathrm{max}$', fontsize = label_size3)\n",
    "plt.ylim([0,0.7])\n",
    "plt.legend(fontsize = label_size2*0.8)\n",
    "plt.title(r'Steady-state light emission', fontsize = label_size2)\n",
    "plt.show()\n",
    "plt.close()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The plot above shows that there is a critical pumping rate when the system is pumped incoherently and dissipates only collectively (black solid curves), a result obtained in Ref. [4], D. Meiser and M.J. Holland, *Steady-state superradiance with alkaline-earth-metal atoms*, *Phys. Rev. A* **81**, 033847 (2010), and here reproduced using $\\texttt{qutip.piqs}$. \n",
    "\n",
    "The plot also show the scaling with $N$ (thinner to thicker curves as $N$ grows from 10 to 40) [8]. \n",
    "\n",
    "The red dashed curves show that the critical pumping is not observed is if the system is allowed to dissipate also incoherently and locally, at a rate set by detailed balance. This condition is set in the most favourable case of high temperature, for which the effect of local losses should be less prominent with respect to the collectively enhanced losses. This condition is particularly relevant for superradiantly-enhanced quantum heat engines and toy models of quantum-enhanced bio-inspired light harvesting mechanisms [8]. "
   ]
  },
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    "#### References\n",
    "\n",
    "[1] R. Bonifacio and L.A. Lugiato, Cooperative effects and bistability for resonance fluorescence, *Opt. Comm.* **19**, 172 (1976)\n",
    "\n",
    "[2] F. Haake, M. I. Kolobov, C. Fabre, E. Giacobino, and S. Reynaud,  Superradiant laser, *Phys. Rev. Lett.* **71**, 995 (1993)\n",
    "\n",
    "[3] F. Haake, M. I. Kolobov, C. Seeger, C. Fabre, E. Giacobino, and S. Reynaud, *Phys. Rev. A* **54**, 1625 (1996)\n",
    "\n",
    "[4] D. Meiser and M.J. Holland, Steady-state superradiance with alkaline-earth-metal atoms, *Phys. Rev. A* **81**, 033847 (2010)\n",
    "\n",
    "[5] D. Meiser and M.J. Holland, Intensity fluctuations in steady-state superradiance, *Phys. Rev. A* **81**, 063827 (2010)\n",
    "\n",
    "[6] J.G. Bohnet *et al.*, A steady-state superradiant laser with less than one intracavity photon, *Nature* **484**, 78 (2012)\n",
    "\n",
    "[7]  N. Shammah, N. Lambert, F. Nori, and S. De Liberato,  Superradiance with local phase-breaking effects, *Phys. Rev. A* **96**, 023863 (2017)\n",
    "\n",
    "[8] N. Shammah, S. Ahmed, N. Lambert, S. De Liberato, and F. Nori, \n",
    "Open quantum systems with local and collective incoherent processes: Efficient numerical simulation using permutational invariance https://arxiv.org/abs/1805.05129\n",
    "\n",
    "[9] M. A. Norcia, R. J. Lewis-Swan, J. R. Cline, B. Zhu, A. M. Rey, and J. K. Thompson, Cavity Mediated Collective Spin Exchange Interactions in a Strontium Superradiant Laser, arXiv:1711.03673 (2017)\n",
    "\n",
    "[10] D. A. Tieri, M. Xu, D. Meiser, J. Cooper, and M. J. Holland, Theory of the crossover from lasing to steady state superradiance, arXiv:1702.04830 (2017)\n",
    "\n",
    "[11] P. Kirton and J. Keeling, Superradiant and lasing states in driven-dissipative Dicke models, *New J. Phys.* **20**, 015009 (2018).\n",
    "\n",
    "[12] P. Kirton, M. M. Roses, J. Keeling, and E. G. Dalla Torre, Introduction to the Dicke model: from equilibrium to nonequilibrium, and vice versa, arXiv:1805.09828 (2018)\n",
    "\n",
    "[13] J.R. Johansson, P.D. Nation, and F. Nori, *Comp. Phys. Comm.* **183**, 1760 (2012) http://qutip.org\n"
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      "\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:      4.3.0\n",
      "Numpy Version:      1.13.3\n",
      "Scipy Version:      1.0.0\n",
      "Cython Version:     0.27.3\n",
      "Matplotlib Version: 2.1.1\n",
      "Python Version:     3.6.3\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/Downloads/Applications/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"
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