qutip_spectrum_ss.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Bugs in spectrum_es of qutip\n",
    "We discuss here a few bugs/problems with spectrum_es of qutip, and their solution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table><tr><th>Software</th><th>Version</th></tr><tr><td>OS</td><td>posix [darwin]</td></tr><tr><td>Numpy</td><td>1.9.2</td></tr><tr><td>SciPy</td><td>0.15.1</td></tr><tr><td>QuTiP</td><td>3.1.0</td></tr><tr><td>IPython</td><td>3.1.0-dev</td></tr><tr><td>matplotlib</td><td>1.4.3</td></tr><tr><td>Python</td><td>3.4.3 (default, Mar 10 2015, 14:53:35) \n",
       "[GCC 4.2.1 Compatible Apple LLVM 6.0 (clang-600.0.56)]</td></tr><tr><td>Cython</td><td>0.22</td></tr><tr><td colspan='2'>Thu Mar 26 09:57:49 2015 CET</td></tr></table>"
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       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from qutip.ipynbtools import version_table\n",
    "version_table()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_formats = ['retina']\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from qutip import *\n",
    "\n",
    "plt.style.use('ggplot')\n",
    "plt.rc('lines',linewidth=2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Original code\n",
    "This is a copy of `_spectrum_es` from `qutip/correlation.py` (current master as of 25.03.2015, commit `d5ee07073044ab2f9177506eca0f2996160f7028`)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def _spectrum_es(H, wlist, c_ops, a_op, b_op):\n",
    "    \"\"\"\n",
    "    Internal function for calculating the spectrum of the correlation function\n",
    "    :math:`\\left<A(\\\\tau)B(0)\\\\right>`.\n",
    "    \"\"\"\n",
    "    if debug:\n",
    "        print(inspect.stack()[0][3])\n",
    "\n",
    "    # construct the Liouvillian\n",
    "    L = liouvillian(H, c_ops)\n",
    "\n",
    "    # find the steady state density matrix and a_op and b_op expecation values\n",
    "    rho0 = steadystate(L)\n",
    "\n",
    "    a_op_ss = expect(a_op, rho0)\n",
    "    b_op_ss = expect(b_op, rho0)\n",
    "\n",
    "    # eseries solution for (b * rho0)(t)\n",
    "    es = ode2es(L, b_op * rho0)\n",
    "\n",
    "    # correlation\n",
    "    corr_es = expect(a_op, es)\n",
    "\n",
    "    # covariance\n",
    "    cov_es = corr_es - np.real(np.conjugate(a_op_ss) * b_op_ss)\n",
    "\n",
    "    # spectrum\n",
    "    spectrum = esspec(cov_es, wlist)\n",
    "\n",
    "    return spectrum"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Test with driven two-level system\n",
    "\n",
    "Two-level system (transition frequency $\\omega_a$, spontaneous decay rate $\\gamma_a$), driven by a classical laser field (frequency $\\omega_l$, Rabi frequency $\\Omega$), in the frame rotating with the laser frequency."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "σ = destroy(2)\n",
    "ωa = 1.\n",
    "γa = 5e-3\n",
    "ωl = 0.8\n",
    "Ω = 1e-3\n",
    "H = (ωa-ωl)*σ.dag()*σ + Ω*(σ+σ.dag())\n",
    "c_ops = [np.sqrt(γa)*σ]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Do the steps in `_spectrum_es` \"manually\" and print some information. We want to calculate the emission spectrum (Fourier transform of $\\langle\\sigma^\\dagger(\\tau)\\sigma(0)\\rangle$)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Steady state"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<a> = (-0.0049989689626514545+6.248711203314319e-05j)\n",
      "<b> = (-0.0049989689626514545-6.248711203314319e-05j)\n"
     ]
    }
   ],
   "source": [
    "a_op = σ.dag()\n",
    "b_op = σ\n",
    "\n",
    "# construct the Liouvillian\n",
    "L = liouvillian(H, c_ops)\n",
    "# find the steady state density matrix and a_op and b_op expecation values\n",
    "rho0 = steadystate(L)\n",
    "\n",
    "a_op_ss = expect(a_op, rho0)\n",
    "b_op_ss = expect(b_op, rho0)\n",
    "\n",
    "print(\"<a> =\", a_op_ss)\n",
    "print(\"<b> =\", b_op_ss)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### eseries solutions for corr_es and cov_es"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# eseries solution for (b * rho0)(t)\n",
    "es = ode2es(L, b_op * rho0)\n",
    "\n",
    "# correlation\n",
    "corr_es = expect(a_op, es)\n",
    "\n",
    "# covariance\n",
    "cov_es = corr_es - np.real(np.conjugate(a_op_ss) * b_op_ss)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`corr_es` has a zero-frequency component in the rotating frame (at $\\omega_l$ in the lab frame). This corresponds to Rayleigh scattering (or the \"coherent\" part of the emission), and comes from the steady-state component $\\langle\\sigma^\\dagger\\rangle\\langle\\sigma\\rangle$. Note that because of numerics, the frequency is not exactly zero, but $\\approx10^{-17}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  2.00025623e-01   2.00025623e-01   1.44138289e-17]\n",
      "[  6.24759852e-10   6.24759852e-10   2.49935953e-05]\n"
     ]
    }
   ],
   "source": [
    "print(abs(corr_es.rates))\n",
    "print(abs(corr_es.ampl))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`cov_es` in principle should have removed the coherent Rayleigh scattering, to avoid a delta function in the resulting spectrum. However, it mistakenly conjugates `a_op_ss`, and only takes the real part of `conjugate(a_op_ss)*b_op_ss`. As far as I see it, this would be correct if `_spectrum_es` always calculated $\\langle B^\\dagger(\\tau) B(0)\\rangle$ (and only `b_op` was passed as an argument).\n",
    "\n",
    "We see that because of the mistaken conjugation, the amplitudes of the zero-frequency parts are not equal (they should be apart from sign)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  2.00025623e-01   2.00025623e-01   1.44138289e-17   0.00000000e+00]\n",
      "[  6.24759852e-10   6.24759852e-10   2.49935953e-05   2.49857861e-05]\n"
     ]
    }
   ],
   "source": [
    "print(abs(cov_es.rates))\n",
    "print(abs(cov_es.ampl).astype(float)) # astype(float) gives nicer printing here, not sure why"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "###  Spectrum\n",
    "\n",
    "Furthermore, as `cov_es.tidyup()` is not called in the spectrum routine (although it is on printing, which is a \"hidden\" side-effect), we have a zero-frequency component that gives a divide-by-zero error in the spectrum evaluation when the list of frequencies contains zero."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# define one wlist with a point at zero, and one without\n",
    "wlist1 = np.linspace(-1,1,1001)\n",
    "wlist2 = np.linspace(-1,1,1000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/opt/local/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/site-packages/qutip/eseries.py:305: RuntimeWarning: divide by zero encountered in true_divide\n",
      "  np.dot(self.ampl, 1. / (1.0j * wlist[i] - self.rates)))\n",
      "/opt/local/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/site-packages/qutip/eseries.py:305: RuntimeWarning: invalid value encountered in true_divide\n",
      "  np.dot(self.ampl, 1. / (1.0j * wlist[i] - self.rates)))\n",
      "/opt/local/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/site-packages/qutip/eseries.py:305: RuntimeWarning: invalid value encountered in multiply\n",
      "  np.dot(self.ampl, 1. / (1.0j * wlist[i] - self.rates)))\n"
     ]
    }
   ],
   "source": [
    "# spectrum\n",
    "spectrum1 = esspec(cov_es, wlist1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "spectrum2 = esspec(cov_es, wlist2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we do call `cov_es.tidyup()`, we see that the two rates are indeed identified as being identical, but their amplitudes do not cancel exactly."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  2.00025623e-01   2.00025623e-01   1.44138289e-17]\n",
      "[  6.24759852e-10   6.24759852e-10   7.80927834e-09]\n"
     ]
    }
   ],
   "source": [
    "cov_es.tidyup()\n",
    "print(abs(cov_es.rates))\n",
    "print(abs(cov_es.ampl).astype(float)) # astype(float) gives nicer printing here, not sure why"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we have a component at $|\\omega|\\approx0$ (i.e., no oscillation and no decay), we get a delta-peak in the spectrum. If `cov_es.tidyup()` had chosen the frequency to be exactly $0$, this would again give a divide by zero if `wlist` has a point at zero. As it chose to keep $|\\omega|\\approx10^{-17}$, it becomes an extremely sharp peak instead (negative in this specific case)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-1083581127.93\n"
     ]
    }
   ],
   "source": [
    "# spectrum\n",
    "spectrum1 = esspec(cov_es, wlist1)\n",
    "print(spectrum1[500])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x105fff550>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 279,
       "width": 423
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(wlist1,spectrum1,label='spectrum1')\n",
    "plt.plot(wlist2,spectrum2,label='spectrum2')\n",
    "plt.yscale('log')\n",
    "plt.legend()\n",
    "plt.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Fixes\n",
    "First, we define `cov_es` without np.real and np.conjugate."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "cov_es = corr_es - a_op_ss * b_op_ss"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, the amplitudes of the zero-frequency components are exactly the same apart from the opposite sign (within numerical precision)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  2.00025623e-01   2.00025623e-01   1.44138289e-17   0.00000000e+00]\n",
      "[  6.24759852e-10   6.24759852e-10   2.49935953e-05   2.49935953e-05]\n",
      "(-1.76182853029e-19-7.27918939047e-19j)\n"
     ]
    }
   ],
   "source": [
    "print(abs(cov_es.rates))\n",
    "print(abs(cov_es.ampl).astype(float)) # astype(float) gives nicer printing here, not sure why\n",
    "print(cov_es.ampl[2]+cov_es.ampl[3])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculating the spectrum now still gives the divide-by-zero because `cov_es.tidyup()` is not called and the two terms that cancel are not combined."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/opt/local/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/site-packages/qutip/eseries.py:305: RuntimeWarning: divide by zero encountered in true_divide\n",
      "  np.dot(self.ampl, 1. / (1.0j * wlist[i] - self.rates)))\n",
      "/opt/local/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/site-packages/qutip/eseries.py:305: RuntimeWarning: invalid value encountered in true_divide\n",
      "  np.dot(self.ampl, 1. / (1.0j * wlist[i] - self.rates)))\n",
      "/opt/local/Library/Frameworks/Python.framework/Versions/3.4/lib/python3.4/site-packages/qutip/eseries.py:305: RuntimeWarning: invalid value encountered in cdouble_scalars\n",
      "  np.dot(self.ampl, 1. / (1.0j * wlist[i] - self.rates)))\n"
     ]
    }
   ],
   "source": [
    "spectrum1 = esspec(cov_es, wlist1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "if we do call `cov_es.tidyup()`, the zero-frequency (coherent scattering) part is finally removed."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.00250012+0.20001j -0.00250012-0.20001j]\n"
     ]
    }
   ],
   "source": [
    "cov_es.tidyup()\n",
    "print(cov_es.rates)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "spectrum1 = esspec(cov_es, wlist1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
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       "QmCC\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1062807b8>"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 279,
       "width": 423
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(wlist1,spectrum1,label='spectrum1')\n",
    "plt.yscale('log')\n",
    "plt.legend()\n",
    "plt.tight_layout()"
   ]
  }
 ],
 "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": "3.4.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}