{
 "cells": [
  {
   "attachments": {},
   "cell_type": "markdown",
   "id": "6718709f",
   "metadata": {},
   "source": [
    "# Zeeman basis and Singlet Triplet basis\n",
    "- Author: Vineeth Francis Thalakottoor\n",
    "- Email: vineeth.thalakottoor@ens.psl.eu or vineethfrancis.physics@gmail.com"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "27a57f94",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Define the source path\n",
    "SourcePath = '/media/HD2/Vineeth/PostDoc_Simulations/Github/PyOR_V1/PyOR_Combined/PyOR/Source_Doc'\n",
    "\n",
    "# Add source path\n",
    "import sys\n",
    "sys.path.append(SourcePath)\n",
    "import time\n",
    "%matplotlib ipympl\n",
    "\n",
    "# Import PyOR package\n",
    "from PyOR_QuantumSystem import QuantumSystem as QunS\n",
    "from PyOR_Hamiltonian import Hamiltonian\n",
    "from PyOR_DensityMatrix import DensityMatrix\n",
    "from PyOR_QuantumObject import QunObj\n",
    "from PyOR_HardPulse import HardPulse\n",
    "from PyOR_Basis import Basis\n",
    "from PyOR_Evolution import Evolutions\n",
    "from PyOR_Plotting import Plotting\n",
    "import PyOR_SignalProcessing as Spro\n",
    "from PyOR_Commutators import Commutators\n",
    "from PyOR_QuantumLibrary import QuantumLibrary\n",
    "from PyOR_Relaxation import RelaxationProcess"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "ab0e39a3",
   "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",
   "id": "3a59211f",
   "metadata": {},
   "source": [
    "### Set parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "275d8217",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Larmor Frequency in MHz:  [-400.22802765 -400.22806765]\n"
     ]
    }
   ],
   "source": [
    "# Master Equation\n",
    "QS.PropagationSpace = \"Liouville\"\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 J coupling between Spins \n",
    "QS.JcoupleValue(\"A\",\"B\",5.0)\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",
    "# Define paris of spins coupled by dipolar interaction\n",
    "QS.Dipole_Pairs = [(\"A\",\"B\")]\n",
    "\n",
    "# Relaxation Process\n",
    "QS.Rprocess = \"Auto-correlated Dipolar Homonuclear\"\n",
    "QS.RelaxParDipole_tau = 10.0e-12\n",
    "QS.RelaxParDipole_bIS = [30.0e3]\n",
    "\n",
    "QS.Update()"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "id": "0f7c0f44",
   "metadata": {},
   "source": [
    "### Generate Hamiltonians"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "b27e93bc",
   "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)\n",
    "COMM = Commutators()\n",
    "Hz = Ham.Zeeman_RotFrame()\n",
    "\n",
    "# J coupling Hamiltonian\n",
    "Hj = Ham.Jcoupling()\n",
    "\n",
    "# Generating the commutation superoperator\n",
    "QS.RowColOrder = 'C'\n",
    "QLib = QuantumLibrary(QS)\n",
    "\n",
    "Hz_L = QLib.CommutationSuperoperator(Hz+Hj)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "fb63d498",
   "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": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Hz.Inverse2PI().matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "7792fc79",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1.24999997786618 & 0 & 0 & 0\\\\0 & -1.24999997786618 & 2.49999995573235 & 0\\\\0 & 2.49999995573235 & -1.24999997786618 & 0\\\\0 & 0 & 0 & 1.24999997786618\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1.24999997786618,                 0,                 0,                0],\n",
       "[               0, -1.24999997786618,  2.49999995573235,                0],\n",
       "[               0,  2.49999995573235, -1.24999997786618,                0],\n",
       "[               0,                 0,                 0, 1.24999997786618]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Hj.Inverse2PI().matrix"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "id": "b1475134",
   "metadata": {},
   "source": [
    "## Zeeman Basis Kets"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "5e2241e6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['|1/2,1/2⟩|1/2,1/2⟩',\n",
       " '|1/2,1/2⟩|1/2,-1/2⟩',\n",
       " '|1/2,-1/2⟩|1/2,1/2⟩',\n",
       " '|1/2,-1/2⟩|1/2,-1/2⟩']"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "QS.ZeemanBasis_Ket()"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "id": "eab82b6d",
   "metadata": {},
   "source": [
    "## Zeeman Basis Bras"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "3188bdd9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['⟨1/2,1/2|⟨1/2,1/2|',\n",
       " '⟨1/2,1/2|⟨1/2,-1/2|',\n",
       " '⟨1/2,-1/2|⟨1/2,1/2|',\n",
       " '⟨1/2,-1/2|⟨1/2,-1/2|']"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "QS.ZeemanBasis_Bra()"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "id": "b7a67890",
   "metadata": {},
   "source": [
    "# Zeeman Basis states"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "fae740ab",
   "metadata": {},
   "outputs": [],
   "source": [
    "BS = Basis(QS)\n",
    "Zstates, DicZ = BS.Zeeman_Basis()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "29936b3b",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['|1/2,1/2⟩|1/2,1/2⟩', '|1/2,1/2⟩|1/2,-1/2⟩', '|1/2,-1/2⟩|1/2,1/2⟩', '|1/2,-1/2⟩|1/2,-1/2⟩']\n"
     ]
    }
   ],
   "source": [
    "print(DicZ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "ffead04a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1.0\\\\0\\\\0\\\\0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1.0],\n",
       "[  0],\n",
       "[  0],\n",
       "[  0]])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Zstates[0].matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "5706a152",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Ket State = 1.0 |1/2,1/2⟩|1/2,1/2⟩\n"
     ]
    }
   ],
   "source": [
    "BS.KetState_Components(Zstates[0], Zstates, DicZ )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "1c40c47b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0\\\\1.0\\\\0\\\\0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[  0],\n",
       "[1.0],\n",
       "[  0],\n",
       "[  0]])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Zstates[1].matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "c51b7e54",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Ket State = 1.0 |1/2,1/2⟩|1/2,-1/2⟩\n"
     ]
    }
   ],
   "source": [
    "BS.KetState_Components(Zstates[1], Zstates, DicZ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "caba7aa6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0\\\\0\\\\1.0\\\\0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[  0],\n",
       "[  0],\n",
       "[1.0],\n",
       "[  0]])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Zstates[2].matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "f0fc407b",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Ket State = 1.0 |1/2,-1/2⟩|1/2,1/2⟩\n"
     ]
    }
   ],
   "source": [
    "BS.KetState_Components(Zstates[2], Zstates, DicZ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "248f6ff0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0\\\\0\\\\0\\\\1.0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[  0],\n",
       "[  0],\n",
       "[  0],\n",
       "[1.0]])"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Zstates[3].matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "6f7f0d83",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Ket State = 1.0 |1/2,-1/2⟩|1/2,-1/2⟩\n"
     ]
    }
   ],
   "source": [
    "BS.KetState_Components(Zstates[3], Zstates, DicZ)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "id": "1b36e187",
   "metadata": {},
   "source": [
    "## Singlet Triplet Basis states"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "6a5dd7aa",
   "metadata": {},
   "outputs": [],
   "source": [
    "STstates, DicST = BS.SingletTriplet_Basis()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "f4b7f2b1",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['S0 ', 'Tp ', 'T0 ', 'Tm ']\n"
     ]
    }
   ],
   "source": [
    "print(DicST)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "cfd1279c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0\\\\0.707106781186547\\\\-0.707106781186547\\\\0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[                 0],\n",
       "[ 0.707106781186547],\n",
       "[-0.707106781186547],\n",
       "[                 0]])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "STstates[0].matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "f7ed6a9e",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Ket State = 0.70711 |1/2,1/2⟩|1/2,-1/2⟩ + -0.70711 |1/2,-1/2⟩|1/2,1/2⟩\n"
     ]
    }
   ],
   "source": [
    "BS.KetState_Components(STstates[0], Zstates, DicZ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "f1a1aa11",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1.0\\\\0\\\\0\\\\0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1.0],\n",
       "[  0],\n",
       "[  0],\n",
       "[  0]])"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "STstates[1].matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "72ad3da6",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Ket State = 1.0 |1/2,1/2⟩|1/2,1/2⟩\n"
     ]
    }
   ],
   "source": [
    "BS.KetState_Components(STstates[1], Zstates, DicZ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "6cc4ee19",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0\\\\0.707106781186547\\\\0.707106781186547\\\\0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[                0],\n",
       "[0.707106781186547],\n",
       "[0.707106781186547],\n",
       "[                0]])"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "STstates[2].matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "6284c08d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Ket State = 0.70711 |1/2,1/2⟩|1/2,-1/2⟩ + 0.70711 |1/2,-1/2⟩|1/2,1/2⟩\n"
     ]
    }
   ],
   "source": [
    "BS.KetState_Components(STstates[2], Zstates, DicZ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "f07ada0e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0\\\\0\\\\0\\\\1.0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[  0],\n",
       "[  0],\n",
       "[  0],\n",
       "[1.0]])"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "STstates[3].matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "c2046787",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Ket State = 1.0 |1/2,-1/2⟩|1/2,-1/2⟩\n"
     ]
    }
   ],
   "source": [
    "BS.KetState_Components(STstates[3], Zstates, DicZ)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "id": "6a966173",
   "metadata": {},
   "source": [
    "## Transformation Between Zeeman state and Singlet-Triplet state"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "a097dc2a",
   "metadata": {},
   "outputs": [],
   "source": [
    "U_Z_ST = BS.BasisChange_TransformationMatrix(Zstates,STstates)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "8e7cef1e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 1.0 & 0 & 0\\\\0.707106781186547 & 0 & 0.707106781186547 & 0\\\\-0.707106781186547 & 0 & 0.707106781186547 & 0\\\\0 & 0 & 0 & 1.0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[                 0, 1.0,                 0,   0],\n",
       "[ 0.707106781186547,   0, 0.707106781186547,   0],\n",
       "[-0.707106781186547,   0, 0.707106781186547,   0],\n",
       "[                 0,   0,                 0, 1.0]])"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U_Z_ST.matrix"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "id": "94cc92fc",
   "metadata": {},
   "source": [
    "## Transformation of Zeeman and J coupling Hamiltonian into Singlet-Triplet basis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "1613d288",
   "metadata": {},
   "outputs": [],
   "source": [
    "Hj_ST = BS.BasisChange_Operator(Hj,U_Z_ST)\n",
    "Hz_ST = BS.BasisChange_Operator(Hz,U_Z_ST)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "6d3c648a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-3.74999993359853 & 0 & 4.99168987337528 \\cdot 10^{-17} & 0\\\\0 & 1.24999997786618 & 0 & 0\\\\6.78409585029909 \\cdot 10^{-17} & 0 & 1.24999997786618 & 0\\\\0 & 0 & 0 & 1.24999997786618\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[   -3.74999993359853,                0, 4.99168987337528e-17,                0],\n",
       "[                   0, 1.24999997786618,                    0,                0],\n",
       "[6.78409585029909e-17,                0,     1.24999997786618,                0],\n",
       "[                   0,                0,                    0, 1.24999997786618]])"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Hj_ST.Inverse2PI().matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "1720fc59",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-3.74999993359853 & 0 & 4.99168987337528 \\cdot 10^{-17} & 0\\\\0 & 1.24999997786618 & 0 & 0\\\\6.78409585029909 \\cdot 10^{-17} & 0 & 1.24999997786618 & 0\\\\0 & 0 & 0 & 1.24999997786618\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[   -3.74999993359853,                0, 4.99168987337528e-17,                0],\n",
       "[                   0, 1.24999997786618,                    0,                0],\n",
       "[6.78409585029909e-17,                0,     1.24999997786618,                0],\n",
       "[                   0,                0,                    0, 1.24999997786618]])"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Alternate way\n",
    "Hj.BasisChange(U_Z_ST).Inverse2PI().matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "d154b96a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-3.74999993359853 & 0 & 4.99168987337528 \\cdot 10^{-17} & 0\\\\0 & 1.24999997786618 & 0 & 0\\\\6.78409585029909 \\cdot 10^{-17} & 0 & 1.24999997786618 & 0\\\\0 & 0 & 0 & 1.24999997786618\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[   -3.74999993359853,                0, 4.99168987337528e-17,                0],\n",
       "[                   0, 1.24999997786618,                    0,                0],\n",
       "[6.78409585029909e-17,                0,     1.24999997786618,                0],\n",
       "[                   0,                0,                    0, 1.24999997786618]])"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Another alternate way\n",
    "Hj.Basis(\"singlettriplet\").Inverse2PI().matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "546b8e7e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}2.8102595185518 \\cdot 10^{-16} & 0 & 20.0000008601147 & 0\\\\0 & -30.0000000759161 & 0 & 0\\\\20.0000008601147 & 0 & 2.8102595185518 \\cdot 10^{-16} & 0\\\\0 & 0 & 0 & 30.0000000759161\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[2.8102595185518e-16,                 0,    20.0000008601147,                0],\n",
       "[                  0, -30.0000000759161,                   0,                0],\n",
       "[   20.0000008601147,                 0, 2.8102595185518e-16,                0],\n",
       "[                  0,                 0,                   0, 30.0000000759161]])"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Hz_ST.Inverse2PI().matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "6e2d91e9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}2.8102595185518 \\cdot 10^{-16} & 0 & 20.0000008601147 & 0\\\\0 & -30.0000000759161 & 0 & 0\\\\20.0000008601147 & 0 & 2.8102595185518 \\cdot 10^{-16} & 0\\\\0 & 0 & 0 & 30.0000000759161\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[2.8102595185518e-16,                 0,    20.0000008601147,                0],\n",
       "[                  0, -30.0000000759161,                   0,                0],\n",
       "[   20.0000008601147,                 0, 2.8102595185518e-16,                0],\n",
       "[                  0,                 0,                   0, 30.0000000759161]])"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Alternate way\n",
    "Hz.BasisChange(U_Z_ST).Inverse2PI().matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "ddbc86d1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}2.8102595185518 \\cdot 10^{-16} & 0 & 20.0000008601147 & 0\\\\0 & -30.0000000759161 & 0 & 0\\\\20.0000008601147 & 0 & 2.8102595185518 \\cdot 10^{-16} & 0\\\\0 & 0 & 0 & 30.0000000759161\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[2.8102595185518e-16,                 0,    20.0000008601147,                0],\n",
       "[                  0, -30.0000000759161,                   0,                0],\n",
       "[   20.0000008601147,                 0, 2.8102595185518e-16,                0],\n",
       "[                  0,                 0,                   0, 30.0000000759161]])"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Another alternate way\n",
    "Hz.Basis(\"singlettriplet\").Inverse2PI().matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ea4e5d4a",
   "metadata": {},
   "source": [
    "### Eigen values and Eigen Vectors of Hj"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "c178fe57",
   "metadata": {},
   "outputs": [],
   "source": [
    "QLib = QuantumLibrary()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "176e7ff7",
   "metadata": {},
   "outputs": [],
   "source": [
    "eigenvalues, eigenvectors = QLib.Eigen_Split(Hj)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "795478b6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1.24999997786618 & -3.74999993359853 & 1.24999997786618 & 1.24999997786618\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([[1.24999997786618, -3.74999993359853, 1.24999997786618, 1.24999997786618]])"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eigenvalues.Inverse2PI().matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "ff4c9ac9",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Ket State = 0.70711 |1/2,1/2⟩|1/2,-1/2⟩ + 0.70711 |1/2,-1/2⟩|1/2,1/2⟩\n"
     ]
    }
   ],
   "source": [
    "BS.KetState_Components(eigenvectors[0], Zstates, DicZ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "99cec59d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Ket State = 0.70711 |1/2,1/2⟩|1/2,-1/2⟩ + -0.70711 |1/2,-1/2⟩|1/2,1/2⟩\n"
     ]
    }
   ],
   "source": [
    "BS.KetState_Components(eigenvectors[1], Zstates, DicZ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "e04b0b00",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Ket State = 1.0 |1/2,1/2⟩|1/2,1/2⟩\n"
     ]
    }
   ],
   "source": [
    "BS.KetState_Components(eigenvectors[2], Zstates, DicZ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "602954e4",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Ket State = 1.0 |1/2,-1/2⟩|1/2,-1/2⟩\n"
     ]
    }
   ],
   "source": [
    "BS.KetState_Components(eigenvectors[3], Zstates, DicZ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "28176f3e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 1.0 & 0\\\\0.707106781186548 & 0.707106781186548 & 0 & 0\\\\0.707106781186547 & -0.707106781186547 & 0 & 0\\\\0 & 0 & 0 & 1.0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[                0,                  0, 1.0,   0],\n",
       "[0.707106781186548,  0.707106781186548,   0,   0],\n",
       "[0.707106781186547, -0.707106781186547,   0,   0],\n",
       "[                0,                  0,   0, 1.0]])"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BS.BasisChange_TransformationMatrix(Zstates,eigenvectors).matrix"
   ]
  }
 ],
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