gpe_fft_v0.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## FFT solver for 1D Gross-Pitaevski equation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We look for the complex function $\\psi(x)$ satisfying the GP equation\n",
    "\n",
    "\n",
    "$-\\frac{1}{2}(i\\partial_x - \\Omega)^2\\psi+g|\\psi|^2\\psi = \\mu \\psi$,\n",
    "\n",
    "with periodic boundary conditions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Import libraries and general definitions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.fftpack import fft, ifft\n",
    "from scipy.integrate import odeint\n",
    "import numpy.linalg as lin\n",
    "%matplotlib inline\n",
    "pi=np.pi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Data block"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "Zmax=2*np.pi;   # Half grid length\n",
    "Npoint=128;   # Number of grid points\n",
    "Nparticle=50; # Number of particles\n",
    "a_s=0.5; # scattering length \n",
    "whoz=1.0; # harmonic oscilator angular frequency\n",
    "#mu=3.27; #Chemical potential\n",
    "#Omega=pi/(2*Zmax); #reference frame velocity"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Derived quantities"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "NormWF=1.0/(2*Zmax); # Wave function (WF) norm\n",
    "gint=2*a_s*Nparticle*NormWF;# nonlinear-term strength\n",
    "Dz=2*Zmax/Npoint;#length step size\n",
    "Dk=pi/Zmax; # momentum step size, maximum wave number is Kmax=Dk*Npoint/2\n",
    "Kmax=Dk*(Npoint//2) # maximum momentum"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Utilities"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def changeFFTposition(f,N,j):  # change the order in vectors from FFT\n",
    "# f(0...N-1) is the vector to order\n",
    "# N is the vector dimension\n",
    "# j is a switch indicating the change direction\n",
    "# physical order is f=[(-(Zmax-Dz):Dz:-Dz) (0:Dz:Zmax) ]\n",
    "# FFT order is f=[(0:Dk:kmax) (-(kmax-kz):kz:-kz)]\n",
    "    f1 = f*1\n",
    "    if (j==1):  # from physical to FFT order\n",
    "        for i in range(0,N//2-1) : \n",
    "            f1[i] = f[N//2-1+i];\n",
    "            f1[N//2+1+i] = f[i];\n",
    "        f1[N//2-1] = f[N-2];        \n",
    "        f1[N//2] = f[N-1];\n",
    "    elif (j==0): # from FFT to physical order\n",
    "        for i in range(0,N//2-1) : \n",
    "            f1[i] = f[N//2+1+i];\n",
    "            f1[N//2+1+i] = f[i+2];\n",
    "        f1[N//2-1] = f[0];        \n",
    "        f1[N//2] = f[1];        \n",
    "    else:\n",
    "        print(\"error in changeFFTposition(f,N,j): j must be 0 or 1...\")\n",
    "    return f1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Grid definitions: physical and momentum space"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "z = np.arange(-Zmax+Dz,Zmax+Dz,Dz) # physical (R-space) grid points in ascending order\n",
    "# zp=[(0:Dz:Zmax) (-(Zmax-Dz):Dz:-Dz)]; \n",
    "zp = changeFFTposition(z,Npoint,1) # (R-space) grid points with FFT order\n",
    "#print(\"grid points (K-order): \"); print(zp)\n",
    "#print(\" R-order: \"); print(z)\n",
    "#\n",
    "# kp=[(0:Dk:Kmax) (-(Kmax-Dk):Dk:-Dk)]; # grid points (K-space), FFT order\n",
    "kp = np.arange(-Kmax+Dk,Kmax+Dk,Dk)\n",
    "kp = changeFFTposition(kp,Npoint,1)\n",
    "#print(\"momentum values: \"); print(kp)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define operators"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "Ekin_K = 0.5*(kp**2); # Kinetic energy in K space\n",
    "# print(\"Ekin: \"); print(Ekin_K)\n",
    "Vpot_R = 0.5*whoz*zp**2; # Potential energy in R space\n",
    "# print(\"Vpot: \"); print(Vpot_R)    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Main functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def Energy(c): # Energy calculation\n",
    "    global gint, Vpot_R, Ekin_K, Npoint\n",
    "    ek = sum(Ekin_K*abs(c)**2) # Kinetic energy in K\n",
    "    psi = ifft(c)*Npoint; # wf FFT to R\n",
    "    ep = sum(Vpot_R*abs(psi)**2)/Npoint; # Potential energy\n",
    "    ei = 0.5*gint*sum(abs(psi)**4)/Npoint; # Interaction energy\n",
    "    em =  ek+ep+ei;\n",
    "    chem_pot = em+ei;\n",
    "    return em, chem_pot, ek, ep, ei\n",
    "#\n",
    "def H1DGP(cr,t,tau): # time evolution operator so that: \\partial_t psi  = -i H1DGP psi\n",
    "    global gint, Vpot_R, Ekin_K, Npoint\n",
    "    #\n",
    "    # cr is the wave function psi in K3 written as a real vector: cr=[psi.real psi.imag]\n",
    "    # t is the time (which is not used if H)\n",
    "    # tau is the imaginary time \n",
    "    #\n",
    "    [c1,c2]=np.split(cr, 2) # turn to complex variables\n",
    "    c=c1+1j*c2\n",
    "    psi = ifft(c)*Npoint; # FFT to R3\n",
    "    HRpsi = (Vpot_R+gint*(abs(psi)**2))*psi; # evolution Epot+Eint in R3\n",
    "    HKpsi = fft(HRpsi)/Npoint; # + FFT to K3\n",
    "    if (tau!=0.0): \n",
    "        c = -tau *( Ekin_K*c + HKpsi )\n",
    "    else:\n",
    "        c = -1j *( Ekin_K*c + HKpsi )\n",
    "    return np.concatenate([c.real,c.imag])\n",
    "#\n",
    "def gaussian(x,n,x0,w): # gaussian wf in  K3\n",
    "    fx = np.pi**0.25*np.exp(-0.5*((x-x0)/w)**2); # gausian in R3\n",
    "    return fft(fx)/n;  # FFT to K3\n",
    "#\n",
    "def normaliza(c): # normalization to 1\n",
    "    norm = lin.norm(c)\n",
    "    if ((norm-1.0)>1.0e-4): # check norm\n",
    "        print(\"normalization from: \",norm)\n",
    "    return c/norm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "Choose initial wafe function and evolve in time"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Energies:          Emed    mu    Ekin    Epot    Eint\n",
      "         initial = 10.4736 20.4471 0.25 0.25 9.97356\n",
      "         final   = 5.39137 8.92526 0.0494713 1.808 3.53389\n",
      "Energy change at last step  = -3.06871e-09\n"
     ]
    }
   ],
   "source": [
    "# initial wf: Gaussian centered at x=0 and width=1\n",
    "c0=normaliza(gaussian(zp,Npoint,0,1)); # wf at t=0\n",
    "# evolve in time: parameters\n",
    "t0=0.0 \n",
    "evolution_time=25.0\n",
    "Nstep = 5000; # number of intermediate steps with outputs\n",
    "tevol=np.empty([Nstep+1]) # time vector\n",
    "energy_cicle=np.empty([Nstep+1,5]) # put the energies in a matrix\n",
    "energy_cicle[0,:] = Energy(c0) # Energies at t=0\n",
    "print(\"Energies:          Emed    mu    Ekin    Epot    Eint\")\n",
    "print(\"         initial = %g %g %g %g %g\"%(Energy(c0)))\n",
    "# print(\"$\\psi(t=0)$: \"); print(ct) \n",
    "c0r=np.concatenate([c0.real,c0.imag]) # prepare real vector for odeint\n",
    "tevol[0]=t0\n",
    "Dt = evolution_time/(1.0*Nstep)\n",
    "Dtint=[0, Dt]\n",
    "tau = 0.1 # imaginary time\n",
    "for i in range(1, Nstep+1): # time evolution cicle\n",
    "    tevol[i] = tevol[i-1] + Dt\n",
    "    ctr = odeint(H1DGP,c0r,Dtint,args=(tau,)); # calculate ctr(t)\n",
    "    [c1r,c2r]=np.split(ctr[1,:], 2)\n",
    "    c = normaliza(c1r+1j*c2r); # check norm in the wf\n",
    "    energy_cicle[i,:] = Energy(c)\n",
    "    c0r=np.concatenate([c.real,c.imag])\n",
    "print(\"         final   = %g %g %g %g %g\"%(Energy(c))) # check energies\n",
    "print(\"Energy change at last step  = %g\"%(energy_cicle[Nstep,0]-energy_cicle[Nstep-1,0]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot convergence during the evolution in the average energy per particle"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fc85dc7fd30>]"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAY0AAAEjCAYAAADOsV1PAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nmZOGmZmVzEnDzMxK9v8BK+V9Ov0cbuEAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fc85ffd7d68>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.title('Convergence',fontsize=15)\n",
    "plt.xlabel('time ($t \\, \\\\omega_{ho}$)',fontsize=15)\n",
    "plt.ylabel('Energy per particle ($E/\\\\hbar \\,\\\\omega_{ho}$)',fontsize=15)\n",
    "#plt.axis([-Zmax,Zmax,0, 8])\n",
    "plt.xticks(np.arange(0, tevol[Nstep]+1,tevol[Nstep]/5))\n",
    "plt.locator_params('y',nbins=3)\n",
    "plt.plot(tevol, energy_cicle[:,0], 'r-')\n",
    "#plt.plot(z, psi, 'r.')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the final density (or wave function)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7fc85d6b4e80>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXwAAAEnCAYAAACqrvj+AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nLyKSEUr4IiIZ8f8Bu+jlS5QXvuoAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fc85d6b4940>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cc = ifft(c)*Npoint*NormWF**0.5 # FFT from K3 to R3 and include the wf norm\n",
    "psi = changeFFTposition(cc,Npoint,0) # psi is the final wave function\n",
    "# plot features\n",
    "plt.title('Final state at $t \\,\\\\omega_{ho}=%g$'%(tevol[Nstep]),fontsize=15)\n",
    "plt.xlabel('$x/a_{ho}$',fontsize=15)\n",
    "#plt.ylabel('$\\\\psi\\,(x)$',fontsize=15)\n",
    "#plt.axis([-Zmax,Zmax,0, 8])\n",
    "plt.xticks(np.arange(-Zmax, Zmax+1,Zmax/2))\n",
    "plt.locator_params('y',nbins=3)\n",
    "#plt.plot(z, psi.real, 'r.',label='real$(\\psi)$')\n",
    "#plt.plot(z, psi.imag, 'b--',label='imag$(\\psi)$')\n",
    "plt.plot(z, abs(psi)**2, 'b--',label='$|\\psi|^2$') # plot density\n",
    "plt.legend(fontsize=15)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "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": "3.5.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}