Zeeman product operator basis and PMZ product operator basis

Author: Vineeth Francis Thalakottoor

Email: vineeth.thalakottoor@ens.psl.eu or vineethfrancis.physics@gmail.com 

[1]:

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

# Add source path
import sys
sys.path.append(SourcePath)
import time
%matplotlib ipympl

# 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
from PyOR_HardPulse import HardPulse
from PyOR_Basis import Basis
from PyOR_Evolution import Evolutions
from PyOR_Plotting import Plotting
import PyOR_SignalProcessing as Spro
from PyOR_Commutators import Commutators
from PyOR_QuantumLibrary import QuantumLibrary
from PyOR_Relaxation import RelaxationProcess

[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 = "Liouville"
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

# Define paris of spins coupled by dipolar interaction
QS.Dipole_Pairs = [("A","B")]

# Relaxation Process
QS.Rprocess = "Auto-correlated Dipolar Homonuclear"
QS.RelaxParDipole_tau = 10.0e-12
QS.RelaxParDipole_bIS = [30.0e3]

QS.Update()

Larmor Frequency in MHz:  [-400.22802765 -400.22806765]

Product Operator Basis: Zeeman (Hilbert Space)

[4]:

BS = Basis(QS)

Basis_Zeeman, dic_Zeeman, coh_Zeeman, coh_Zeeman_array = BS.ProductOperators_Zeeman()

[5]:

dic_Zeeman

[5]:

['|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|',
 '|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|',
 '|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|',
 '|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|',
 '|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|',
 '|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|',
 '|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|',
 '|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|']

[6]:

print(coh_Zeeman)

[0.0, 1.0, 1.0, 2.0, -1.0, 0.0, 0.0, 1.0, -1.0, 0.0, 0.0, 1.0, -2.0, -1.0, -1.0, 0.0]

[7]:

Basis_Zeeman[0].matrix

[7]:

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

[8]:

coh_Zeeman_array.matrix

[8]:

$\displaystyle \left[\begin{matrix}0\\1.0\\1.0\\2.0\\-1.0\\0\\0\\1.0\\-1.0\\0\\0\\1.0\\-2.0\\-1.0\\-1.0\\0\end{matrix}\right]$

Product Operator Basis: PMZ (Hilbert Space)

[9]:

sort = 'negative to positive'
Index = False
Normal = True
Basis_PMZ, coh_PMZ, dic_PMZ = BS.ProductOperators_SpinHalf_PMZ(sort,Index,Normal)

[10]:

print(dic_PMZ)

['Im1 Im2 ', 'Im1 Iz2 ', 'Im1 Id2 ', 'Iz1 Im2 ', 'Id1 Im2 ', 'Im1 Ip2 ', 'Iz1 Iz2 ', 'Iz1 Id2 ', 'Id1 Iz2 ', 'Id1 Id2 ', 'Ip1 Im2 ', 'Iz1 Ip2 ', 'Id1 Ip2 ', 'Ip1 Iz2 ', 'Ip1 Id2 ', 'Ip1 Ip2 ']

[11]:

print(coh_PMZ)

[-2, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2]

[12]:

Basis_PMZ[0].matrix

[12]:

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

Call Product Operator with string index

[13]:

OpB_H = BS.String_to_Matrix(dic_PMZ, Basis_PMZ)

['Im1Im2', 'Im1Iz2', 'Im1', 'Iz1Im2', 'Im2', 'Im1Ip2', 'Iz1Iz2', 'Iz1', 'Iz2', '', 'Ip1Im2', 'Iz1Ip2', 'Ip2', 'Ip1Iz2', 'Ip1', 'Ip1Ip2']

[14]:

OpB_H["Im1Im2"].matrix

[14]:

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

[15]:

Compare_with_SpinDynamica = True

if Compare_with_SpinDynamica:
    Basis_PMZ = [OpB_H['Im1Im2'],OpB_H['Im1Iz2'],OpB_H['Iz1Im2'],OpB_H['Im1'],OpB_H['Im2'],OpB_H['Im1Ip2'],OpB_H['Ip1Im2'],OpB_H['Iz1Iz2'],OpB_H['Iz1'],OpB_H['Iz2'],OpB_H[''],OpB_H['Ip1Iz2'],OpB_H['Iz1Ip2'],OpB_H['Ip1'],OpB_H['Ip2'],OpB_H['Ip1Ip2'] ]
    dic_PMZ = ['Im1Im2','Im1Iz2','Iz1Im2','Im1','Im2','Im1Ip2','Ip1Im2','Iz1Iz2','Iz1','Iz2','ID','Ip1Iz2','Iz1Ip2','Ip1','Ip2','Ip1Ip2']

Convert Product Operators into Liouville Space (Zeeman Basis)

[16]:

if QS.PropagationSpace == "Liouville":
    Basis_Zeeman_L = Basis_Zeeman
else:
    Basis_Zeeman_L = BS.ProductOperators_ConvertToLiouville(Basis_Zeeman)

[17]:

Basis_Zeeman_L[0].matrix

[17]:

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

Convert Product Operators into Liouville Space (PMZ Basis)

[18]:

Basis_PMZ_L = BS.ProductOperators_ConvertToLiouville(Basis_PMZ)

[19]:

Basis_PMZ_L[0].matrix

[19]:

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

Transformation Between Zeeman product operator basis and PMZ product operator basis

[20]:

U_Z_PMZ = BS.BasisChange_TransformationMatrix(Basis_Zeeman_L,Basis_PMZ_L)

[21]:

U_Z_PMZ.matrix

[21]:

$\displaystyle \left[\begin{array}{cccccccccccccccc}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & 0.5 & 0.5 & 0.5 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -0.707106781186547 & 0 & -0.707106781186547 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -0.707106781186547 & 0 & -0.707106781186547 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1.0\\0 & 0 & 0.707106781186547 & 0 & 0.707106781186547 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & -0.5 & 0.5 & -0.5 & 0.5 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & -1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.707106781186547 & 0 & -0.707106781186547 & 0 & 0\\0 & 0.707106781186547 & 0 & 0.707106781186547 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & -1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & -0.5 & -0.5 & 0.5 & 0.5 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.707106781186547 & 0 & -0.707106781186547 & 0\\1.0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & -0.707106781186547 & 0 & 0.707106781186547 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & -0.707106781186547 & 0 & 0.707106781186547 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.5 & -0.5 & -0.5 & 0.5 & 0 & 0 & 0 & 0 & 0\end{array}\right]$

Relaxation Matrix

[22]:

RPro = RelaxationProcess(QS)
R_L = RPro.Relaxation()

Larmor Frequency in MHz:  [-400.22802765 -400.22806765]

[23]:

plot = Plotting(QS)
plot.PlotFigureSize = (10,10)
plot.PlotFontSize = 10
plot.MatrixPlot(R_L.data.real,dic_Zeeman,dic_Zeeman)

Converting Relaxation Superoperator in Liouville Space from Zeeman to PMZ basis

[24]:

R_L_PMZ = BS.BasisChange_Operator(R_L,U_Z_PMZ)

[25]:

plot.MatrixPlot(R_L_PMZ.data.real,dic_PMZ,dic_PMZ)

Zeeman product operator basis and PMZ product operator basis

Author: Vineeth Francis Thalakottoor

Email: vineeth.thalakottoor@ens.psl.eu or vineethfrancis.physics@gmail.com

Set parameters

Product Operator Basis: Zeeman (Hilbert Space)

Product Operator Basis: PMZ (Hilbert Space)

Call Product Operator with string index

Convert Product Operators into Liouville Space (Zeeman Basis)

Convert Product Operators into Liouville Space (PMZ Basis)

Transformation Between Zeeman product operator basis and PMZ product operator basis

Relaxation Matrix

Converting Relaxation Superoperator in Liouville Space from Zeeman to PMZ basis

Email: vineeth.thalakottoor@ens.psl.eu or vineethfrancis.physics@gmail.com 