PyOR Quantum

Author: Vineeth Thalakottoor

Introduction to equlibrium density matrix

[1]:

# Define the source path
SourcePath = '/media/HD2/Vineeth/PostDoc_Simulations/Github/PyOR_V1/PyOR_Combined/Source_Doc'

# Add source path
import sys
sys.path.append(SourcePath)

# Import PyOR package
from PyOR_QuantumSystem import QuantumSystem as QunS
from PyOR_Hamiltonian import Hamiltonian
from PyOR_DensityMatrix import DensityMatrix
from PyOR_QuantumObject import QunObj

[2]:

# Define the spin system
Spin_list = {"A" : "H1", "B" : "H1"}
QS = QunS(Spin_list,PrintDefault=False)

# initialize the system
QS.Initialize()

Set parameters

[3]:

# Master Equation
QS.PropagationSpace = "Hilbert"
QS.MasterEquation = "Redfield"

# B0 Field in Tesla, Static Magnetic field (B0) along Z
QS.B0 = 9.4

# Offset Frequency in rotating frame (Hz)
QS.OFFSET["A"] = 10.0
QS.OFFSET["B"] = 50.0

# Define J coupling between Spins
QS.JcoupleValue("A","B",5.0)

# Define initial and final Spin Temperature
QS.I_spintemp["A"] = 300.0
QS.I_spintemp["B"] = 300.0
QS.F_spintemp["A"] = 300.0
QS.F_spintemp["B"] = 300.0

QS.Update()

Rotating frame frequencies: {'A': -2514706800.0, 'B': -2514706800.0}
Offset frequencies: {'A': 10.0, 'B': 50.0}
Initial spin temperatures: {'A': 300.0, 'B': 300.0}
Final spin temperatures: {'A': 300.0, 'B': 300.0}
Radiation damping gain: {'A': 0, 'B': 0}
Radiation damping phase: {'A': 0, 'B': 0}

Rprocess = No Relaxation
RelaxParDipole_tau = 0.0
DipolePairs = []
RelaxParDipole_bIS = []

Generate Hamiltonians

[4]:

# generate Larmor Frequencies
QS.print_Larmor = True
Ham = Hamiltonian(QS)

Larmor Frequency in MHz:  [-400.22802765 -400.22806765]

[5]:

DM = DensityMatrix(QS,Ham)

Thermal_DensMatrix = True

if Thermal_DensMatrix:
    # High Temperature
    HT_approx = False

    # Initial Density Matrix
    rho_in = DM.EquilibriumDensityMatrix(QS.Ispintemp,HT_approx)

    # Equlibrium Density Matrix
    rhoeq = DM.EquilibriumDensityMatrix(QS.Fspintemp,HT_approx)
else:
    rho_in = QunObj(QS.Az + QS.Bz)
    rhoeq = QunObj(QS.Az + QS.Bz)

Trace of density matrix =  1.0
Trace of density matrix =  1.0

[6]:

# Initial Density Matrix
rho_in.matrix

[6]:

$\displaystyle \left[\begin{matrix}0.250016003847122 & 0 & 0 & 0\\0 & 0.25 & 0 & 0\\0 & 0 & 0.25 & 0\\0 & 0 & 0 & 0.249983996152878\end{matrix}\right]$

[7]:

# Final Density Matrix
rhoeq.matrix

[7]:

$\displaystyle \left[\begin{matrix}0.250016003847122 & 0 & 0 & 0\\0 & 0.25 & 0 & 0\\0 & 0 & 0.25 & 0\\0 & 0 & 0 & 0.249983996152878\end{matrix}\right]$

[8]:

DM = DensityMatrix(QS,Ham)

Thermal_DensMatrix = False

if Thermal_DensMatrix:
    # High Temperature
    HT_approx = False

    # Initial Density Matrix
    rho_in = DM.EqulibriumDensityMatrix(QS.Ispintemp,HT_approx)

    # Equlibrium Density Matrix
    rhoeq = DM.EqulibriumDensityMatrix(QS.Fspintemp,HT_approx)
else:
    rho_in = QS.Az + QS.Bz
    rhoeq = QS.Az + QS.Bz

[9]:

# Initial Density Matrix
rho_in.matrix

[9]:

$\displaystyle \left[\begin{matrix}1.0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & -1.0\end{matrix}\right]$

[10]:

# Final Density Matrix
rhoeq.matrix

[10]:

$\displaystyle \left[\begin{matrix}1.0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & 0\\0 & 0 & 0 & -1.0\end{matrix}\right]$

Individual density matrix

[11]:

QS.Arho.matrix

[11]:

$\displaystyle \left[\begin{matrix}0.500016005441018 & 0\\0 & 0.499983994558982\end{matrix}\right]$

[12]:

QS.Brho.matrix

[12]:

$\displaystyle \left[\begin{matrix}0.500016005442617 & 0\\0 & 0.499983994557383\end{matrix}\right]$

[13]:

rho = QS.Arho.TensorProduct(QS.Brho)
rho.matrix

[13]:

$\displaystyle \left[\begin{matrix}0.250016005697992 & 0 & 0 & 0\\0 & 0.249999999743026 & 0 & 0\\0 & 0 & 0.249999999744626 & 0\\0 & 0 & 0 & 0.249983994814357\end{matrix}\right]$