{
 "cells": [
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# PyOR Quantum\n",
    "## Author: Vineeth Thalakottoor\n",
    "## Introduction to Dipolar Hamiltonians"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Define the source path\n",
    "SourcePath = '/media/HD2/Vineeth/PostDoc_Simulations/Github/PyOR_V1/PyOR_Combined/Source_Doc'\n",
    "\n",
    "# Add source path\n",
    "import sys\n",
    "sys.path.append(SourcePath)\n",
    "\n",
    "# Import PyOR package\n",
    "from PyOR_QuantumSystem import QuantumSystem as QunS\n",
    "from PyOR_Hamiltonian import Hamiltonian\n",
    "from PyOR_QuantumLibrary import QuantumLibrary\n",
    "import PyOR_SphericalTensors as ST\n",
    "import PyOR_Rotation as Rot\n",
    "QLib = QuantumLibrary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Define the spin system\n",
    "Spin_list = {\"A\" : \"H1\", \"B\" : \"H1\"}\n",
    "QS = QunS(Spin_list,PrintDefault=False)\n",
    "\n",
    "# initialize the system\n",
    "QS.Initialize()"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Set parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Rotating frame frequencies: {'A': -2514706800.0, 'B': -2514706800.0}\n",
      "Offset frequencies: {'A': 10.0, 'B': 50.0}\n",
      "Initial spin temperatures: {'A': 300.0, 'B': 300.0}\n",
      "Final spin temperatures: {'A': 300.0, 'B': 300.0}\n",
      "Radiation damping gain: {'A': 0, 'B': 0}\n",
      "Radiation damping phase: {'A': 0, 'B': 0}\n",
      "\n",
      "Rprocess = Phenomenological\n",
      "RelaxParDipole_tau = 0.0\n",
      "DipolePairs = [('A', 'B')]\n",
      "RelaxParDipole_bIS = []\n"
     ]
    }
   ],
   "source": [
    "# Master Equation\n",
    "QS.PropagationSpace = \"Hilbert\"\n",
    "QS.MasterEquation = \"Redfield\"\n",
    "\n",
    "# B0 Field in Tesla, Static Magnetic field (B0) along Z\n",
    "QS.B0 = 9.4\n",
    "\n",
    "# Offset Frequency in rotating frame (Hz)\n",
    "QS.OFFSET[\"A\"] = 10.0\n",
    "QS.OFFSET[\"B\"] = 50.0\n",
    "\n",
    "# Define paris of spins coupled by dipolar interaction\n",
    "QS.DipoleAngle = [(45,45)]\n",
    "QS.Dipole_Pairs = [(\"A\",\"B\")]\n",
    "QS.Dipole_DipolarAlpabet = [\"All\"]\n",
    "QS.DipolebIS = [25.0e3]\n",
    "\n",
    "# Define initial and final Spin Temperature\n",
    "QS.I_spintemp[\"A\"] = 300.0\n",
    "QS.I_spintemp[\"B\"] = 300.0\n",
    "QS.F_spintemp[\"A\"] = 300.0\n",
    "QS.F_spintemp[\"B\"] = 300.0\n",
    "\n",
    "# Relaxation Process\n",
    "QS.Rprocess = \"Phenomenological\"\n",
    "QS.R1 = 1\n",
    "QS.R2 = 2\n",
    "\n",
    "QS.Update()"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Zeeman Hamiltonians"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Larmor Frequency in MHz:  [-400.22802765 -400.22806765]\n"
     ]
    }
   ],
   "source": [
    "# generate Larmor Frequencies\n",
    "QS.print_Larmor = True\n",
    "Ham = Hamiltonian(QS)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-400228040.565114 & 0 & 0 & 0\\\\0 & 20.3718327157626 & 0 & 0\\\\0 & 0 & -20.3718327157626 & 0\\\\0 & 0 & 0 & 400228040.565114\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[-400228040.565114,                0,                 0,                0],\n",
       "[                0, 20.3718327157626,                 0,                0],\n",
       "[                0,                0, -20.3718327157626,                0],\n",
       "[                0,                0,                 0, 400228040.565114]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Lab Frame Hamiltonian\n",
    "Hz_lab = Ham.Zeeman()\n",
    "Hz_lab.Inverse2PI().matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-30.0000000759161 & 0 & 0 & 0\\\\0 & 20.0000008601147 & 0 & 0\\\\0 & 0 & -20.0000008601147 & 0\\\\0 & 0 & 0 & 30.0000000759161\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[-30.0000000759161,                0,                 0,                0],\n",
       "[                0, 20.0000008601147,                 0,                0],\n",
       "[                0,                0, -20.0000008601147,                0],\n",
       "[                0,                0,                 0, 30.0000000759161]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Rotating Frame Hamiltonian\n",
    "Hz = Ham.Zeeman_RotFrame()\n",
    "Hz.Inverse2PI().matrix"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Dipolar Hamiltonian with Dipolar alphabets"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}3125.0 & 0 & 0 & 0\\\\0 & -3125.0 & 3125.0 & 0\\\\0 & 3125.0 & -3125.0 & 0\\\\0 & 0 & 0 & 3125.0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[3125.0,       0,       0,      0],\n",
       "[     0, -3125.0,  3125.0,      0],\n",
       "[     0,  3125.0, -3125.0,      0],\n",
       "[     0,       0,       0, 3125.0]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "QS.Dipole_DipolarAlpabet = [\"secular Homonuclear\"]\n",
    "Hd = Ham.DDcoupling()\n",
    "Hd.Inverse2PI().Round(0).matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-3125.0 & 0 & 0 & 0\\\\0 & 3125.0 & 0 & 0\\\\0 & 0 & 3125.0 & 0\\\\0 & 0 & 0 & -3125.0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[-3125.0,      0,      0,       0],\n",
       "[      0, 3125.0,      0,       0],\n",
       "[      0,      0, 3125.0,       0],\n",
       "[      0,      0,      0, -3125.0]])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "QS.Dipole_DipolarAlpabet = [\"secular Hetronuclear\"]\n",
    "Hd = Ham.DDcoupling()\n",
    "Hd.Inverse2PI().Round(0).matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-3125.0 & 0 & 0 & 0\\\\0 & 3125.0 & 0 & 0\\\\0 & 0 & 3125.0 & 0\\\\0 & 0 & 0 & -3125.0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[-3125.0,      0,      0,       0],\n",
       "[      0, 3125.0,      0,       0],\n",
       "[      0,      0, 3125.0,       0],\n",
       "[      0,      0,      0, -3125.0]])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "QS.Dipole_DipolarAlpabet = [\"A\"]\n",
    "Hd = Ham.DDcoupling()\n",
    "Hd.Inverse2PI().Round(0).matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 3125.0 & 0\\\\0 & 3125.0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[0,      0,      0, 0],\n",
       "[0,      0, 3125.0, 0],\n",
       "[0, 3125.0,      0, 0],\n",
       "[0,      0,      0, 0]])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "QS.Dipole_DipolarAlpabet = [\"B\"]\n",
    "Hd = Ham.DDcoupling()\n",
    "Hd.Inverse2PI().Round(0).matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & -6629.0 + 6629.0 i & -6629.0 + 6629.0 i & 0\\\\0 & 0 & 0 & 6629.0 - 6629.0 i\\\\0 & 0 & 0 & 6629.0 - 6629.0 i\\\\0 & 0 & 0 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[0, -6629.0 + 6629.0*I, -6629.0 + 6629.0*I,                 0],\n",
       "[0,                  0,                  0, 6629.0 - 6629.0*I],\n",
       "[0,                  0,                  0, 6629.0 - 6629.0*I],\n",
       "[0,                  0,                  0,                 0]])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "QS.Dipole_DipolarAlpabet = [\"C\"]\n",
    "Hd = Ham.DDcoupling()\n",
    "Hd.Inverse2PI().Round(0).matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\-6629.0 - 6629.0 i & 0 & 0 & 0\\\\-6629.0 - 6629.0 i & 0 & 0 & 0\\\\0 & 6629.0 + 6629.0 i & 6629.0 + 6629.0 i & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[                 0,                 0,                 0, 0],\n",
       "[-6629.0 - 6629.0*I,                 0,                 0, 0],\n",
       "[-6629.0 - 6629.0*I,                 0,                 0, 0],\n",
       "[                 0, 6629.0 + 6629.0*I, 6629.0 + 6629.0*I, 0]])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "QS.Dipole_DipolarAlpabet = [\"D\"]\n",
    "Hd = Ham.DDcoupling()\n",
    "Hd.Inverse2PI().Round(0).matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 9375.0 i\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[0, 0, 0, 9375.0*I],\n",
       "[0, 0, 0,        0],\n",
       "[0, 0, 0,        0],\n",
       "[0, 0, 0,        0]])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "QS.Dipole_DipolarAlpabet = [\"E\"]\n",
    "Hd = Ham.DDcoupling()\n",
    "Hd.Inverse2PI().Round(0).matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\- 9375.0 i & 0 & 0 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[        0, 0, 0, 0],\n",
       "[        0, 0, 0, 0],\n",
       "[        0, 0, 0, 0],\n",
       "[-9375.0*I, 0, 0, 0]])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "QS.Dipole_DipolarAlpabet = [\"F\"]\n",
    "Hd = Ham.DDcoupling()\n",
    "Hd.Inverse2PI().Round(0).matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-3125.0 & -6629.0 + 6629.0 i & -6629.0 + 6629.0 i & 9375.0 i\\\\-6629.0 - 6629.0 i & 3125.0 & 3125.0 & 6629.0 - 6629.0 i\\\\-6629.0 - 6629.0 i & 3125.0 & 3125.0 & 6629.0 - 6629.0 i\\\\- 9375.0 i & 6629.0 + 6629.0 i & 6629.0 + 6629.0 i & -3125.0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[           -3125.0, -6629.0 + 6629.0*I, -6629.0 + 6629.0*I,          9375.0*I],\n",
       "[-6629.0 - 6629.0*I,             3125.0,             3125.0, 6629.0 - 6629.0*I],\n",
       "[-6629.0 - 6629.0*I,             3125.0,             3125.0, 6629.0 - 6629.0*I],\n",
       "[         -9375.0*I,  6629.0 + 6629.0*I,  6629.0 + 6629.0*I,           -3125.0]])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "QS.Dipole_DipolarAlpabet = [\"All\"]\n",
    "\n",
    "theta = 45\n",
    "phi = 45\n",
    "QS.DipoleAngle = [(theta,phi)]\n",
    "\n",
    "Hd = Ham.DDcoupling()\n",
    "Hd.Inverse2PI().Round(0).matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Hd.Hermitian()"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Dipole tensor PAF"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}25000.0 & 0 & 0\\\\0 & 25000.0 & 0\\\\0 & 0 & -50000.0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[25000.0,       0,        0],\n",
       "[      0, 25000.0,        0],\n",
       "[      0,       0, -50000.0]])"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "IT_PAF = Ham.InteractionTensor_PAF_Dipole(25.0e3)\n",
    "IT_PAF.Inverse2PI().matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'Isotropic': 0.0, 'Anisotropy': -50000.0, 'Asymmetry': -0.0}"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "PAF_Decom = Ham.InteractionTensor_PAF_Decomposition(IT_PAF)\n",
    "PAF_Decom"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Dipole tensor LAB"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}6250.0 & -18750.0 & -26517.0\\\\-18750.0 & 6250.0 & -26517.0\\\\-26517.0 & -26517.0 & -12500.0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[  6250.0, -18750.0, -26517.0],\n",
       "[-18750.0,   6250.0, -26517.0],\n",
       "[-26517.0, -26517.0, -12500.0]])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "IT_LAB = Ham.InteractionTensor_LAB(IT_PAF,phi,theta,0)\n",
    "IT_LAB.Inverse2PI().Round(0).matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[0, 0, 0],\n",
       "[0, 0, 0],\n",
       "[0, 0, 0]])"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LAB_Decom = Ham.InteractionTensor_LAB_Decomposition(IT_LAB)\n",
    "LAB_Decom['Isotropic'].matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}6250.0 & -18750.0 & -26517.0\\\\-18750.0 & 6250.0 & -26517.0\\\\-26517.0 & -26517.0 & -12500.0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[  6250.0, -18750.0, -26517.0],\n",
       "[-18750.0,   6250.0, -26517.0],\n",
       "[-26517.0, -26517.0, -12500.0]])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LAB_Decom[\"Symmetric\"].Inverse2PI().Round(0).matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[0, 0, 0],\n",
       "[0, 0, 0],\n",
       "[0, 0, 0]])"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LAB_Decom[\"Antisymmetric\"].Inverse2PI().Round(0).matrix"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Dipolar Hamiltonian in Lab frame (Cartesian, Euler rotation)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-3125.0 & -6629.0 + 6629.0 i & -6629.0 + 6629.0 i & 9375.0 i\\\\-6629.0 - 6629.0 i & 3125.0 & 3125.0 & 6629.0 - 6629.0 i\\\\-6629.0 - 6629.0 i & 3125.0 & 3125.0 & 6629.0 - 6629.0 i\\\\- 9375.0 i & 6629.0 + 6629.0 i & 6629.0 + 6629.0 i & -3125.0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[           -3125.0, -6629.0 + 6629.0*I, -6629.0 + 6629.0*I,          9375.0*I],\n",
       "[-6629.0 - 6629.0*I,             3125.0,             3125.0, 6629.0 - 6629.0*I],\n",
       "[-6629.0 - 6629.0*I,             3125.0,             3125.0, 6629.0 - 6629.0*I],\n",
       "[         -9375.0*I,  6629.0 + 6629.0*I,  6629.0 + 6629.0*I,           -3125.0]])"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = \"A\"\n",
    "B = \"B\"\n",
    "Hdd_LAB_Euler = Ham.Interaction_Hamiltonian_Catesian_Wigner(A,IT_PAF,B,phi,theta,0)\n",
    "Hdd_LAB_Euler.Inverse2PI().Round(0).matrix"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Dipolar Hamiltonian in Lab frame (Cartesian, Wigner rotation)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-3125.0 & -6629.0 + 6629.0 i & -6629.0 + 6629.0 i & 9375.0 i\\\\-6629.0 - 6629.0 i & 3125.0 & 3125.0 & 6629.0 - 6629.0 i\\\\-6629.0 - 6629.0 i & 3125.0 & 3125.0 & 6629.0 - 6629.0 i\\\\- 9375.0 i & 6629.0 + 6629.0 i & 6629.0 + 6629.0 i & -3125.0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[           -3125.0, -6629.0 + 6629.0*I, -6629.0 + 6629.0*I,          9375.0*I],\n",
       "[-6629.0 - 6629.0*I,             3125.0,             3125.0, 6629.0 - 6629.0*I],\n",
       "[-6629.0 - 6629.0*I,             3125.0,             3125.0, 6629.0 - 6629.0*I],\n",
       "[         -9375.0*I,  6629.0 + 6629.0*I,  6629.0 + 6629.0*I,           -3125.0]])"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Hdd_LAB_Wig = Ham.Interaction_Hamiltonian_Catesian_Wigner(A,IT_PAF,B,phi,theta,0)\n",
    "Hdd_LAB_Wig.Inverse2PI().Round(0).matrix"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Dipolar Hamiltonian in Lab frame (Spherical Tensors, Wigner Rotation)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-3125.0 & -6629.0 + 6629.0 i & -6629.0 + 6629.0 i & 9375.0 i\\\\-6629.0 - 6629.0 i & 3125.0 & 3125.0 & 6629.0 - 6629.0 i\\\\-6629.0 - 6629.0 i & 3125.0 & 3125.0 & 6629.0 - 6629.0 i\\\\- 9375.0 i & 6629.0 + 6629.0 i & 6629.0 + 6629.0 i & -3125.0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[           -3125.0, -6629.0 + 6629.0*I, -6629.0 + 6629.0*I,          9375.0*I],\n",
       "[-6629.0 - 6629.0*I,             3125.0,             3125.0, 6629.0 - 6629.0*I],\n",
       "[-6629.0 - 6629.0*I,             3125.0,             3125.0, 6629.0 - 6629.0*I],\n",
       "[         -9375.0*I,  6629.0 + 6629.0*I,  6629.0 + 6629.0*I,           -3125.0]])"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Hdd_LAB_ST = Ham.Interaction_Hamiltonian_SphericalTensor(A,IT_PAF,B,\"spin-spin\",\"all\", phi,theta,0)\n",
    "Hdd_LAB_ST.Inverse2PI().Round(0).matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-3125.0 & 0 & 0 & 0\\\\0 & 3125.0 & 0 & 0\\\\0 & 0 & 3125.0 & 0\\\\0 & 0 & 0 & -3125.0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[-3125.0,      0,      0,       0],\n",
       "[      0, 3125.0,      0,       0],\n",
       "[      0,      0, 3125.0,       0],\n",
       "[      0,      0,      0, -3125.0]])"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Hdd_LAB_ST = Ham.Interaction_Hamiltonian_SphericalTensor(A,IT_PAF,B,\"spin-spin\",\"secular\", phi,theta,0)\n",
    "Hdd_LAB_ST.Inverse2PI().Round(0).matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-3125.0 & 0 & 0 & 0\\\\0 & 3125.0 & 3125.0 & 0\\\\0 & 3125.0 & 3125.0 & 0\\\\0 & 0 & 0 & -3125.0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[-3125.0,      0,      0,       0],\n",
       "[      0, 3125.0, 3125.0,       0],\n",
       "[      0, 3125.0, 3125.0,       0],\n",
       "[      0,      0,      0, -3125.0]])"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Hdd_LAB_ST = Ham.Interaction_Hamiltonian_SphericalTensor(A,IT_PAF,B,\"spin-spin\",\"secular + pseudosecular\", phi,theta,0)\n",
    "Hdd_LAB_ST.Inverse2PI().Round(0).matrix"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Cartesian tensor to Spherical tensors and back"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'rank0': (-0+0j),\n",
       " 'rank1': [(-0+0j), -0j, (-0+0j)],\n",
       " 'rank2': [0j, (-0+0j), (-384764.94904855924+0j), 0j, 0j]}"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "carTOst = ST.MatrixToSphericalTensors(IT_PAF )\n",
    "carTOst"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "T0 = carTOst[\"rank0\"]\n",
    "T11, T10, T1m1 = carTOst[\"rank1\"]\n",
    "T22, T21, T20, T2m1, T2m2 = carTOst[\"rank2\"]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}25000.0 & 0 & 0\\\\0 & 25000.0 & 0\\\\0 & 0 & -50000.0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[25000.0,       0,        0],\n",
       "[      0, 25000.0,        0],\n",
       "[      0,       0, -50000.0]])"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "spTOcar = ST.SphericalTensorsToMatrix(carTOst)\n",
    "spTOcar.Inverse2PI().matrix"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Wigner d matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0.854 & -0.5 & 0.146\\\\0.5 & 0.707 & -0.5\\\\0.146 & 0.5 & 0.854\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[0.854,  -0.5, 0.146],\n",
       "[  0.5, 0.707,  -0.5],\n",
       "[0.146,   0.5, 0.854]])"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Rot.Wigner_d_Matrix(rank=1,beta=45).Round(3).matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0.729 & -0.604 & 0.306 & -0.104 & 0.021\\\\0.604 & 0.354 & -0.612 & 0.354 & -0.104\\\\0.306 & 0.612 & 0.25 & -0.612 & 0.306\\\\0.104 & 0.354 & 0.612 & 0.354 & -0.604\\\\0.021 & 0.104 & 0.306 & 0.604 & 0.729\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[0.729, -0.604,  0.306, -0.104,  0.021],\n",
       "[0.604,  0.354, -0.612,  0.354, -0.104],\n",
       "[0.306,  0.612,   0.25, -0.612,  0.306],\n",
       "[0.104,  0.354,  0.612,  0.354, -0.604],\n",
       "[0.021,  0.104,  0.306,  0.604,  0.729]])"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Rot.Wigner_d_Matrix(rank=2,beta=45).Round(3).matrix"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Wigner D matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- 0.854 i & -0.354 + 0.354 i & 0.146\\\\0.354 - 0.354 i & 0.707 & -0.354 - 0.354 i\\\\0.146 & 0.354 + 0.354 i & 0.854 i\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[       -0.854*I, -0.354 + 0.354*I,            0.146],\n",
       "[0.354 - 0.354*I,            0.707, -0.354 - 0.354*I],\n",
       "[          0.146,  0.354 + 0.354*I,          0.854*I]])"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Rot.Wigner_D_Matrix(rank=1,alpha=45,beta=45,gamma=45).Round(3).matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-0.729 & 0.427 + 0.427 i & - 0.306 i & -0.073 + 0.073 i & 0.021\\\\-0.427 - 0.427 i & - 0.354 i & -0.433 + 0.433 i & 0.354 & -0.073 - 0.073 i\\\\- 0.306 i & 0.433 - 0.433 i & 0.25 & -0.433 - 0.433 i & 0.306 i\\\\0.073 - 0.073 i & 0.354 & 0.433 + 0.433 i & 0.354 i & 0.427 - 0.427 i\\\\0.021 & 0.073 + 0.073 i & 0.306 i & -0.427 + 0.427 i & -0.729\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[          -0.729, 0.427 + 0.427*I,         -0.306*I, -0.073 + 0.073*I,            0.021],\n",
       "[-0.427 - 0.427*I,        -0.354*I, -0.433 + 0.433*I,            0.354, -0.073 - 0.073*I],\n",
       "[        -0.306*I, 0.433 - 0.433*I,             0.25, -0.433 - 0.433*I,          0.306*I],\n",
       "[ 0.073 - 0.073*I,           0.354,  0.433 + 0.433*I,          0.354*I,  0.427 - 0.427*I],\n",
       "[           0.021, 0.073 + 0.073*I,          0.306*I, -0.427 + 0.427*I,           -0.729]])"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Rot.Wigner_D_Matrix(rank=2,alpha=45,beta=45,gamma=45).Round(3).matrix"
   ]
  }
 ],
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