Lab 2 - Quantum States, SOLUTIONS

Useful for working examples and problems with photon quantum states. You may notice some similarity to the Jones Calculus ;-)



In [10]:

    
import numpy as np
from qutip import *

These are the polarization states:



In [11]:

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



In [12]:

    
V









    Out[12]:





Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\begin{equation*}\left(\begin{array}{*{11}c}0.0\\1.0\\\end{array}\right)\end{equation*}



In [17]:

    
Hbra*V









    Out[17]:





Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\begin{equation*}\left(\begin{array}{*{11}c}0.0\\\end{array}\right)\end{equation*}



In [16]:

    
Hbra = H.dag()

Devices:

HWP - Half-wave plate axis at $\theta$ to the horizontal

LP - Linear polarizer, axis at $\theta$

QWP - Quarter-wave plate, axis at $\theta$

Note, these are functions so you need to call them with a specific value of theta.



In [18]:

    
def HWP(theta):
    return Qobj([[np.cos(2*theta),np.sin(2*theta)],[np.sin(2*theta),-np.cos(2*theta)]]).tidyup()



In [19]:

    
def LP(theta):
    return Qobj([[np.cos(theta)**2,np.cos(theta)*np.sin(theta)],[np.sin(theta)*np.cos(theta),np.sin(theta)**2]]).tidyup()



In [20]:

    
def QWP(theta):
    return Qobj([[np.cos(theta)**2 + 1j*np.sin(theta)**2,
                 (1-1j)*np.sin(theta)*np.cos(theta)],
                [(1-1j)*np.sin(theta)*np.cos(theta),
                 np.sin(theta)**2 + 1j*np.cos(theta)**2]]).tidyup()



In [22]:

    
QWP(np.pi/4)









    Out[22]:





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

Example 1) Check that the $|H\rangle$ state is normalized



In [23]:

    
H.dag()*H









    Out[23]:





Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\begin{equation*}\left(\begin{array}{*{11}c}1.0\\\end{array}\right)\end{equation*}

To show more information on an object, use the question mark after the function or object:



In [25]:

    
np.sin?

Example 2) Converting from ket to bra:



In [26]:

    
psi = Qobj([[1+1j],[2-1j]])
psi









    Out[26]:





Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\begin{equation*}\left(\begin{array}{*{11}c}(1.0+1.0j)\\(2.0-1.0j)\\\end{array}\right)\end{equation*}



In [27]:

    
psi.dag()









    Out[27]:





Quantum object: dims = [[1], [2]], shape = (1, 2), type = bra\begin{equation*}\left(\begin{array}{*{11}c}(1.0-1.0j) & (2.0+1.0j)\\\end{array}\right)\end{equation*}



In [17]:

    
psi.dag().dag()









    Out[17]:





Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\begin{equation*}\left(\begin{array}{*{11}c}(1.0+1.0j)\\(2.0-1.0j)\\\end{array}\right)\end{equation*}

the .dag() python method computes the "daggar" or the complex transpose.

1) Is `psi` normalized? If not, find the normalization constant and confirm that constant normalizes `psi`.



In [28]:

    
psi.dag()*psi









    Out[28]:





Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\begin{equation*}\left(\begin{array}{*{11}c}7.0\\\end{array}\right)\end{equation*}



In [29]:

    
psi_norm = psi*np.sqrt(1/7)



In [30]:

    
psi_norm.dag() * psi_norm









    Out[30]:





Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\begin{equation*}\left(\begin{array}{*{11}c}1.000\\\end{array}\right)\end{equation*}

2) Verify that the $|V\rangle$ state is normalized



In [31]:

    
V.dag()*V









    Out[31]:





Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\begin{equation*}\left(\begin{array}{*{11}c}1.0\\\end{array}\right)\end{equation*}

3) Verify that the $|H\rangle$ and $|V\rangle$ states are orthogonal. Repeat for the other pairs of states.



In [32]:

    
H.dag()*V









    Out[32]:





Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\begin{equation*}\left(\begin{array}{*{11}c}0.0\\\end{array}\right)\end{equation*}



In [33]:

    
L.dag()*R









    Out[33]:





Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\begin{equation*}\left(\begin{array}{*{11}c}0.0\\\end{array}\right)\end{equation*}



In [34]:

    
P45.dag()*M45









    Out[34]:





Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\begin{equation*}\left(\begin{array}{*{11}c}0.0\\\end{array}\right)\end{equation*}

4) Calculate the horizontal component $c_H$ of the state $\psi = \frac{1}{\sqrt{5}}|H\rangle + \frac{2}{\sqrt{5}}|V\rangle$



In [35]:

    
psi = 1/np.sqrt(5)*H + 2/np.sqrt(5)*V
psi









    Out[35]:





Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\begin{equation*}\left(\begin{array}{*{11}c}0.447\\0.894\\\end{array}\right)\end{equation*}



In [55]:

    
psi2 = Qobj([[1/np.sqrt(5)],[2/np.sqrt(5)]])
psi2









    Out[55]:





Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\begin{equation*}\left(\begin{array}{*{11}c}0.447\\0.894\\\end{array}\right)\end{equation*}



In [57]:

    
H.dag()*psi









    Out[57]:





Quantum object: dims = [[1], [1]], shape = (1, 1), type = bra\begin{equation*}\left(\begin{array}{*{11}c}0.447\\\end{array}\right)\end{equation*}

5) Verify Eq. (3.18), $P(H||45\rangle)=\frac{1}{2},$ (which states "The probability that a photon prepared in the +45 state will leave a PA_HV in the Horizontal state is one half.")



In [38]:

    
(H.dag()*P45).norm()**2









    Out[38]:





0.4999999999999999



In [49]:

    
np.conjugate(H.dag()*P45) * (H.dag()*P45)









    Out[49]:





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

6) Demonstrate that a half-wave plate at 45-degrees converts $|H\rangle$ to $|V\rangle$



In [58]:

    
HWP(np.radians(45)) * H









    Out[58]:





Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket\begin{equation*}\left(\begin{array}{*{11}c}0.0\\1.0\\\end{array}\right)\end{equation*}



In [60]:

    
HWP(np.pi/4) * H == V









    Out[60]:





True

7) Re-create Figure 3.9 by plotting the probability P(+45) vs phase φ



In [61]:

    
import matplotlib.pyplot as plt



In [62]:

    
plt.plot([1,2,3,2,3,4])









    Out[62]:





[<matplotlib.lines.Line2D at 0x1a1fb031d0>]



In [89]:

    
phi_list = np.linspace(0,8*np.pi,num=100)



In [85]:

    
plt.plot(phi,np.sin(phi))









    Out[85]:





[<matplotlib.lines.Line2D at 0x1a20429c10>]



In [87]:

    
sin_list = [np.sin(p) for p in phi] # notes

add text



In [88]:

    
plt.plot(phi,sin_list)









    Out[88]:





[<matplotlib.lines.Line2D at 0x1a205661d0>]



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In [101]:

    
def psi(phi):
    return 1/np.sqrt(2)*(H + np.exp(1j*phi)*V)



In [102]:

    
answer = [(M45.dag()*psi(phi)).norm()**2 for phi in phi_list]



In [107]:

    
plt.plot(phi_list/np.pi,answer,"-o")
plt.xlabel("$\phi$")
plt.ylabel("P(-45)")









    Out[107]:





Text(0, 0.5, 'P(-45)')



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