Introduction to 1D pulse acquisition (Hamiltonian Eigen State)

Author: Vineeth 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
import numpy as np
%matplotlib ipympl

# Import PyOR package
from PyOR_QuantumSystem import QuantumSystem as QunS
from PyOR_HardPulse import HardPulse
from PyOR_Evolution import Evolutions
from PyOR_Plotting import Plotting
import PyOR_SignalProcessing as Spro
from PyOR_Basis import Basis
from PyOR_DensityMatrix import DensityMatrix

[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"

# Operator Basis
QS.Basis_SpinOperators_Hilbert = "Zeeman" # Try "Singlet Triplet" and see what happen to the all spin operators and Hamiltoninas

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

QS.Update()

Larmor Frequency in MHz:  [-400.22801765 -400.22801765]

Spin operators

[4]:

QS.Az

[4]:

<PyOR_QuantumObject.QunObj at 0x7f6c48481be0>

[5]:

QS.Az.matrix

[5]:

$\displaystyle \left[\begin{matrix}0.5 & 0 & 0 & 0\\0 & 0.5 & 0 & 0\\0 & 0 & -0.5 & 0\\0 & 0 & 0 & -0.5\end{matrix}\right]$

[6]:

QS.Az.data

[6]:

array([[ 0.5+0.j,  0. +0.j,  0. +0.j,  0. +0.j],
       [ 0. +0.j,  0.5+0.j,  0. +0.j,  0. +0.j],
       [ 0. +0.j,  0. +0.j, -0.5+0.j, -0. +0.j],
       [ 0. +0.j,  0. +0.j, -0. +0.j, -0.5+0.j]])

[7]:

QS.Az_sub

[7]:

<PyOR_QuantumObject.QunObj at 0x7f6c0b1c16a0>

[8]:

QS.Az_sub.matrix

[8]:

$\displaystyle \left[\begin{matrix}0.5 & 0\\0 & -0.5\end{matrix}\right]$

Generate Hamiltonians

[9]:

# 1. Zeeman Hamiltonian

Hz = 2.0 * np.pi * 10 * QS.Az + 2.0 * np.pi * 50 * QS.Bz
Hz.Inverse2PI().matrix

[9]:

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

[10]:

# 2. J coupling Hamiltonian

Hj = 2.0 * np.pi * 5 * (QS.Ax * QS.Bx + QS.Ay * QS.By + QS.Az * QS.Bz)
Hj.Inverse2PI().matrix

[10]:

$\displaystyle \left[\begin{matrix}1.25 & 0 & 0 & 0\\0 & -1.25 & 2.5 & 0\\0 & 2.5 & -1.25 & 0\\0 & 0 & 0 & 1.25\end{matrix}\right]$

Change basis to Hamiltonian eigen states

[11]:

eigenvectors, Dic = QS.Class_basis.BasisChange_HamiltonianEigenStates(Hj)

Ket State = 0.70711 |1/2,1/2⟩|1/2,-1/2⟩ + 0.70711 |1/2,-1/2⟩|1/2,1/2⟩
Ket State = 0.70711 |1/2,1/2⟩|1/2,-1/2⟩ + -0.70711 |1/2,-1/2⟩|1/2,1/2⟩
Ket State = 1.0 |1/2,1/2⟩|1/2,1/2⟩
Ket State = 1.0 |1/2,-1/2⟩|1/2,-1/2⟩
Larmor Frequency in MHz:  [-400.22801765 -400.22801765]

[12]:

Dic

[12]:

['0.70711 |1/2,1/2⟩|1/2,-1/2⟩ + 0.70711 |1/2,-1/2⟩|1/2,1/2⟩',
 '0.70711 |1/2,1/2⟩|1/2,-1/2⟩ + -0.70711 |1/2,-1/2⟩|1/2,1/2⟩',
 '1.0 |1/2,1/2⟩|1/2,1/2⟩',
 '1.0 |1/2,-1/2⟩|1/2,-1/2⟩']

[13]:

eigenvectors[0].matrix

[13]:

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

[14]:

QS.Basis_SpinOperators_Hilbert

[14]:

'Zeeman to Hamiltonian eigen states'

[15]:

QS.Basis_SpinOperators_TransformationMatrix.matrix

[15]:

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

[16]:

QS.Az.matrix

[16]:

$\displaystyle \left[\begin{matrix}-1.11855713399442 \cdot 10^{-17} & 0.5 & 0 & 0\\0.5 & -1.11855713399442 \cdot 10^{-17} & 0 & 0\\0 & 0 & 0.5 & 0\\0 & 0 & 0 & -0.5\end{matrix}\right]$

Generate Hamiltonians in new basis state

[17]:

Hz = 2.0 * np.pi * 10 * QS.Az + 2.0 * np.pi * 50 * QS.Bz
Hz.Inverse2PI().Tolarence(1.0e-5).matrix

[17]:

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

[18]:

Hj = 2.0 * np.pi * 5 * (QS.Ax * QS.Bx + QS.Ay * QS.By + QS.Az * QS.Bz)
Hj.Inverse2PI().Tolarence(1.0e-5).matrix

[18]:

$\displaystyle \left[\begin{matrix}1.24999991442865 & 0 & 0 & 0\\0 & -3.74999991442865 & 0 & 0\\0 & 0 & 1.25 & 0\\0 & 0 & 0 & 1.25\end{matrix}\right]$

[19]:

(Hz + Hj).Tolarence(1.0e-5).matrix

[19]:

$\displaystyle \left[\begin{matrix}7.85398109631381 & -125.663706143592 & 0 & 0\\-125.663706143592 & -23.5619443642628 & 0 & 0\\0 & 0 & 196.349540849362 & 0\\0 & 0 & 0 & -180.641577581413\end{matrix}\right]$

Product Operator Basis (PMZ / Shift Z basis)

[20]:

BS = Basis(QS)

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

Product Operator Basis Zeeman

[21]:

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

Initialize density matrix

[22]:

DM = DensityMatrix(QS)

# Initial Density Matrix
rho_in = QS.Az + QS.Bz
rho_in.matrix

[22]:

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

[23]:

DM.DensityMatrix_Components(rho_in,Basis_PMZ,dic_PMZ)

Density Matrix = 1.0 Iz1 Id2  + 1.0 Id1 Iz2

[24]:

DM.DensityMatrix_Components(rho_in,Basis_Zeeman, dic_Zeeman)

Density Matrix = 1.0 |1/2,1/2⟩|1/2,1/2⟩⟨1/2,1/2|⟨1/2,1/2| + -1.0 |1/2,-1/2⟩|1/2,-1/2⟩⟨1/2,-1/2|⟨1/2,-1/2|

[25]:

# Final Density Matrix
rhoeq = QS.Az + QS.Bz
rhoeq.matrix

[25]: