Lab 2 - Quantum States, SOLUTIONS - Quantum Mechanics with QuTiP

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)]])

Hbra = H.dag()

Hbra*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`.¶

psi.dag()*psi

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

psi_norm.dag() * psi_norm

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

V.dag()*V

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

H.dag()*V

L.dag()*R

P45.dag()*M45

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

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

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

H.dag()*psi

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.”)¶

expect(H*H.dag(),P45)

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

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

HWP(np.radians(45)) * H

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

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

import matplotlib.pyplot as plt

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

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

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

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

plt.plot(phi_list,sin_list)

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

answer = [expect(M45*M45.dag(),psi(phi)) for phi in phi_list]

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

Lab 2 - Quantum States, SOLUTIONS

Example 1) Check that the ∣H⟩|H\rangle∣H⟩ 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⟩|V\rangle∣V⟩ state is normalized¶

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

4) Calculate the horizontal component cHc_HcH​ of the state ψ=15∣H⟩+25∣V⟩\psi = \frac{1}{\sqrt{5}}|H\rangle + \frac{2}{\sqrt{5}}|V\rangleψ=5​1​∣H⟩+5​2​∣V⟩¶

5) Verify Eq. (3.18), P(H∣∣45⟩)=12,P(H||45\rangle)=\frac{1}{2},P(H∣∣45⟩)=21​, (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⟩|H\rangle∣H⟩ to ∣V⟩|V\rangle∣V⟩¶