Lab 2 - Quantum States

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*}

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`.

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

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

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

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.")

Lab 2 - Quantum States

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

Example 2) Converting from ket to bra:

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

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

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

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

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.")

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

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

Lab 2 - Quantum States

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

Example 2) Converting from ket to bra:

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

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

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

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

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.")

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

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

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