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{
"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>"
],
"text/plain": [
"<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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],
"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
}
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