PyOR Quantum

Author: Vineeth Thalakottoor

Introduction to Quadrupole Hamiltonians

[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 numpy as np
# Import PyOR package
from PyOR_QuantumSystem import QuantumSystem as QunS
from PyOR_Hamiltonian import Hamiltonian
from PyOR_QuantumLibrary import QuantumLibrary
import PyOR_SphericalTensors as ST
import PyOR_Rotation as Rot
QLib = QuantumLibrary()

[2]:

# Define the spin system
Spin_list = {"A" : "H2"}
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"] = 0.0

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


# Relaxation Process
QS.Rprocess = "Phenomenological"
QS.R1 = 1
QS.R2 = 2

QS.Update()

Rotating frame frequencies: {'A': -386011000.0}
Offset frequencies: {'A': 0.0}
Initial spin temperatures: {'A': 300.0}
Final spin temperatures: {'A': 300.0}
Radiation damping gain: {'A': 0}
Radiation damping phase: {'A': 0}

Rprocess = Phenomenological
RelaxParDipole_tau = 0.0
DipolePairs = []
RelaxParDipole_bIS = []

Zeeman Hamiltonians

[4]:

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

Larmor Frequency in MHz:  [-61.43555874]

[5]:

# Lab Frame Hamiltonian
Hz_lab = Ham.Zeeman()
Hz_lab.Inverse2PI().matrix

[5]:

$\displaystyle \left[\begin{matrix}-61435560.0110852 & 0 & 0\\0 & 0 & 0\\0 & 0 & 61435560.0110852\end{matrix}\right]$

[6]:

# Rotating Frame Hamiltonian
Hz = Ham.Zeeman_RotFrame()
Hz.Inverse2PI().matrix

[6]:

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

Quadrupole tensor PAF

[7]:

QuadrupoleCoupling = 100 # Hz
eta = 0.2

A = "A"

IT_PAF = Ham.InteractionTensor_PAF_Quadrupole(A,QuadrupoleCoupling,eta)
IT_PAF.Inverse2PI().matrix

[7]:

$\displaystyle \left[\begin{matrix}-30.0 & 0 & 0\\0 & -20.0 & 0\\0 & 0 & 50.0\end{matrix}\right]$

[8]:

PAF_Decom = Ham.InteractionTensor_PAF_Decomposition(IT_PAF)
PAF_Decom

[8]:

{'Isotropic': 0.0, 'Anisotropy': 50.0, 'Asymmetry': 0.19999999999999998}

Quadrupole tensor LAB

[9]:

phi,theta = 20.0,40.0

IT_LAB = Ham.InteractionTensor_LAB(IT_PAF,phi,theta,0)
IT_LAB.Round(3).matrix

[9]:

$\displaystyle \left[\begin{matrix}2.245 & 46.555 & 232.583\\46.555 & -108.719 & 84.653\\232.583 & 84.653 & 106.474\end{matrix}\right]$

[10]:

LAB_Decom = Ham.InteractionTensor_LAB_Decomposition(IT_LAB)
LAB_Decom['Isotropic'].matrix

[10]:

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

[11]:

LAB_Decom["Symmetric"].Inverse2PI().Round(3).matrix

[11]:

$\displaystyle \left[\begin{matrix}0.357 & 7.409 & 37.017\\7.409 & -17.303 & 13.473\\37.017 & 13.473 & 16.946\end{matrix}\right]$

[12]:

LAB_Decom["Antisymmetric"].Inverse2PI().Round(3).matrix

[12]:

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

Quadrupole Hamiltonian in Lab frame (Secular)

[13]:

Hdd_LAB_Truncates = Ham.Interaction_Hamiltonian_LAB_Quadrupole_Secular(A, QuadrupoleCoupling, eta,theta,phi)
Hdd_LAB_Truncates.Inverse2PI().Round(3).matrix

[13]:

$\displaystyle \left[\begin{matrix}8.715 & 0 & 0\\0 & -17.429 & 0\\0 & 0 & 8.715\end{matrix}\right]$

Quadrupole Hamiltonian in Lab frame (Cartesian, Euler rotation)

[14]:

Hdd_LAB_Euler = Ham.Interaction_Hamiltonian_Catesian_Euler(A,IT_PAF,A,0,theta,phi)
Hdd_LAB_Euler.Inverse2PI().Round(3).matrix

[14]:

$\displaystyle \left[\begin{matrix}8.715 & 27.447 - 1.461 i & 12.455 + 2.462 i\\27.447 + 1.461 i & -17.429 & -27.447 + 1.461 i\\12.455 - 2.462 i & -27.447 - 1.461 i & 8.715\end{matrix}\right]$

Quadrupole Hamiltonian in Lab frame (Cartesian, Wigner rotation)

[15]:

Hdd_LAB_Wig = Ham.Interaction_Hamiltonian_Catesian_Wigner(A,IT_PAF,A,0,theta,phi)
Hdd_LAB_Wig.Inverse2PI().Round(3).matrix

[15]:

$\displaystyle \left[\begin{matrix}8.715 & 27.447 - 1.461 i & 12.455 + 2.462 i\\27.447 + 1.461 i & -17.429 & -27.447 + 1.461 i\\12.455 - 2.462 i & -27.447 - 1.461 i & 8.715\end{matrix}\right]$

Quadrupole Hamiltonian in Lab frame (Spherical Tensors, Wigner Rotation)

[16]:

Hdd_LAB_ST = Ham.Interaction_Hamiltonian_SphericalTensor(A,IT_PAF,A,"spin-spin","all", 0,theta,phi)
Hdd_LAB_ST.Inverse2PI().Round(3).matrix

[16]:

$\displaystyle \left[\begin{matrix}8.715 & 27.447 + 1.461 i & 12.455 - 2.462 i\\27.447 - 1.461 i & -17.429 & -27.447 - 1.461 i\\12.455 + 2.462 i & -27.447 + 1.461 i & 8.715\end{matrix}\right]$

[17]:

Hdd_LAB_ST = Ham.Interaction_Hamiltonian_SphericalTensor(A,IT_PAF,A,"spin-spin","secular", 0,theta,phi)
Hdd_LAB_ST.Inverse2PI().Round(3).matrix

[17]:

$\displaystyle \left[\begin{matrix}8.715 & 0 & 0\\0 & -17.429 & 0\\0 & 0 & 8.715\end{matrix}\right]$

[18]:

Hdd_LAB_ST = Ham.Interaction_Hamiltonian_SphericalTensor(A,IT_PAF,A,"spin-spin","secular + pseudosecular", 0,theta,phi)
Hdd_LAB_ST.Inverse2PI().Round(3).matrix

[18]:

$\displaystyle \left[\begin{matrix}8.715 & 0 & 0\\0 & -17.429 & 0\\0 & 0 & 8.715\end{matrix}\right]$