Basis Change of the System (Spin operators and States)

[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()

Spin Operators

  • After (first) QS.Initialize(), spin operators are always in Zeeman basis

[3]:

QS.Ax.matrix

[3]:
$\displaystyle \left[\begin{matrix}0 & 0 & 0.5 & 0\\0 & 0 & 0 & 0.5\\0.5 & 0 & 0 & 0\\0 & 0.5 & 0 & 0\end{matrix}\right]$

Set parameters

[4]:

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

# Operator Basis
#QS.Basis_SpinOperators_Hilbert = "Singlet Triplet" # or
QS.Basis_SpinOperators_Hilbert = "Zeeman"

# 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]

Generate Hamiltonians

[5]:

# generate Larmor Frequencies

Ham = Hamiltonian(QS)
COMM = Commutators(QS)

# Zeeman Hamiltonian in rotating frame
Hz = Ham.Zeeman_RotFrame()

# J coupling Hamiltonian
Hj = Ham.Jcoupling()

# Generating the commutation superoperator
QLib = QuantumLibrary(QS)

Larmor Frequency in MHz:  [-400.22802765 -400.22806765]

[6]:

Hz.Inverse2PI().matrix

[6]:
$\displaystyle \left[\begin{matrix}-30.0000000759161 & 0 & 0 & 0\\0 & 20.0000008601147 & 0 & 0\\0 & 0 & -20.0000008601147 & 0\\0 & 0 & 0 & 30.0000000759161\end{matrix}\right]$
[7]:

Hj.Inverse2PI().Tolarence(1.0e-5).matrix

[7]:
$\displaystyle \left[\begin{matrix}1.24999997786618 & 0 & 0 & 0\\0 & -1.24999997786618 & 2.49999995573235 & 0\\0 & 2.49999995573235 & -1.24999997786618 & 0\\0 & 0 & 0 & 1.24999997786618\end{matrix}\right]$

Calling Hamiltonian class from QunautmSystem class

  • Alternate way of calling Hamiltonian class from QuantuSystem class

[8]:

Hj1 = QS.Class_hamiltonian.Jcoupling()
Hj1.Inverse2PI().Tolarence(1.0e-5).matrix

[8]:
$\displaystyle \left[\begin{matrix}1.24999997786618 & 0 & 0 & 0\\0 & -1.24999997786618 & 2.49999995573235 & 0\\0 & 2.49999995573235 & -1.24999997786618 & 0\\0 & 0 & 0 & 1.24999997786618\end{matrix}\right]$

Spin operators

  • After QS.Update, all the spin operators will be in the new basis

[9]:

QS.Ax.matrix

[9]:
$\displaystyle \left[\begin{matrix}0 & 0 & 0.5 & 0\\0 & 0 & 0 & 0.5\\0.5 & 0 & 0 & 0\\0 & 0.5 & 0 & 0\end{matrix}\right]$

Basis states

  • Singlet and triplet

  • Zeeman

[10]:

# Singlet Triplet states
ST_B, dic_ST = QS.Class_basis.SingletTriplet_Basis()

# Zeeman states
Z_B, dic_Z = QS.Class_basis.Zeeman_Basis()

[11]:

# Singlet triplet states dictionary
dic_ST

[11]:

['S0 ', 'Tp ', 'T0 ', 'Tm ']

[12]:

# Zeeman states dictionary
dic_Z

[12]:

['|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⟩']

[13]:

ST_B[0].matrix

[13]:
$\displaystyle \left[\begin{matrix}0\\0.707106781186547\\-0.707106781186547\\0\end{matrix}\right]$
[14]:

Z_B[1].matrix

[14]:
$\displaystyle \left[\begin{matrix}0\\1.0\\0\\0\end{matrix}\right]$
[15]:

# Decompose the states in basis states
QS.Class_basis.KetState_Components(Z_B[3],ST_B, dic_ST)

Ket State = 1.0 Tm

[32]:

# Create a stat with magnetic quantum number
X1 = QS.State({"A": -1/2,"B": -1/2})
X1.matrix

[32]:
$\displaystyle \left[\begin{matrix}0\\0\\0\\1.0\end{matrix}\right]$
[33]:

QS.Class_basis.KetState_Components(X1, ST_B, dic_ST)

Ket State = 1.0 Tm

[34]:

QS.Class_basis.KetState_Components(X1, Z_B, dic_Z)

Ket State = 1.0 |1/2,-1/2⟩|1/2,-1/2⟩

[35]:

QS.Class_basis.KetState_Components(ST_B[0], Z_B, dic_Z)

Ket State = 0.70711 |1/2,1/2⟩|1/2,-1/2⟩ + -0.70711 |1/2,-1/2⟩|1/2,1/2⟩

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