Lab 2 - Quantum States

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

import numpy as np
from qutip import *

These are the polarization states:

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)]])

V

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.

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

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()

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()

QWP(np.pi/4)

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

H.dag()*H

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

np.sin?

Example 2) Converting from ket to bra:¶

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

psi.dag()

psi.dag().dag()

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.”)¶

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¶