Lab 3: Operators

An overview of operator properties



In [1]:

    
import matplotlib.pyplot as plt
from numpy import sqrt,cos,sin,pi,arange
from qutip import *



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H = Qobj([[1],[0]])
V = Qobj([[0],[1]])
P45 = Qobj([[1/sqrt(2)],[1/sqrt(2)]])
M45 = Qobj([[1/sqrt(2)],[-1/sqrt(2)]])
R = Qobj([[1/sqrt(2)],[-1j/sqrt(2)]])
L = Qobj([[1/sqrt(2)],[1j/sqrt(2)]])

Example 1: the outer product and the projection operator

We already have the $|H\rangle$ state represented as a vector in the HV basis, so the $\hat{P}_H$ operator is the outer product $|H\rangle\langle H|$:



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Ph = H*H.dag()
Ph









    Out[3]:





Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}1.0 & 0.0\\0.0 & 0.0\\\end{array}\right)\end{equation*}

Same with the $\hat{P}_V$ operator:



In [4]:

    
Pv = V*V.dag()
Pv









    Out[4]:





Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}0.0 & 0.0\\0.0 & 1.0\\\end{array}\right)\end{equation*}

1) Verify Eq. 4.38 for the HV basis states. Repeat for the 45, and ± basis



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2) Represent the $\hat{R}_p(\theta)$ operator in the HV basis and verify your representation by operating on $|H\rangle$ and $|V\rangle$ states.



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def Rp(theta):
    return Qobj([[cos(theta),-sin(theta)],[sin(theta),cos(theta)]]).tidyup()



In [6]:

    
Rp(pi/2)









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Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isherm = False\begin{equation*}\left(\begin{array}{*{11}c}0.0 & -1.0\\1.0 & 0.0\\\end{array}\right)\end{equation*}



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V == Rp(pi/2)*H









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True

3) Using your $\hat{R}_p(\theta)$ operator, verify the operator properties described in Sections 4.1 and 4.2. Specifically, verify Eqns. 4.5, 4.6, 4.7, 4.16, 4.18, 4.22, and 4.27



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Example: similarity transform

The following defines a function that creates a similarity transform matrix. It takes the two old basis vectors and the two new basis vectors.



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def sim_transform(o_basis1, o_basis2, n_basis1, n_basis2):
    a = n_basis1.dag()*o_basis1
    b = n_basis1.dag()*o_basis2
    c = n_basis2.dag()*o_basis1
    d = n_basis2.dag()*o_basis2
    return Qobj([[a.data[0,0],b.data[0,0]],[c.data[0,0],d.data[0,0]]])



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Shv45 = sim_transform(H,V,P45,M45)  # as found in Example 4.A.1, Eq. 4.A.10.
Shv45









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Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}0.707 & 0.707\\0.707 & -0.707\\\end{array}\right)\end{equation*}



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Shv45 * L  # compare to Eq.









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Quantum object: dims = [[2], [1]], shape = [2, 1], type = ket\begin{equation*}\left(\begin{array}{*{11}c}(0.500+0.500j)\\(0.500-0.500j)\\\end{array}\right)\end{equation*}

4) Represent $\hat{P}_H$ in the ±45 basis.

Check your answer against Eqns. 4.A.17 and 4.72



In [11]:

    
Shv45*Ph*Shv45.dag()









    Out[11]:





Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}0.500 & 0.500\\0.500 & 0.500\\\end{array}\right)\end{equation*}



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