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
Hbra*V
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.
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)
H.dag()*H
To show more information on an object, use the question mark after the function or object:
np.sin?
psi = Qobj([[1+1j],[2-1j]])
psi
psi.dag()
psi.dag().dag()
the .dag()
python method computes the "daggar" or the complex transpose.
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
V.dag()*V
H.dag()*V
L.dag()*R
P45.dag()*M45
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
(H.dag()*P45).norm()**2
0.4999999999999999
np.conjugate(H.dag()*P45) * (H.dag()*P45)
array([[0.5+0.j]])
HWP(np.radians(45)) * H
HWP(np.pi/4) * H == V
True
import matplotlib.pyplot as plt
plt.plot([1,2,3,2,3,4])
[<matplotlib.lines.Line2D at 0x1a1fb031d0>]
phi_list = np.linspace(0,8*np.pi,num=100)
plt.plot(phi,np.sin(phi))
[<matplotlib.lines.Line2D at 0x1a20429c10>]
sin_list = [np.sin(p) for p in phi] # notes
add text
plt.plot(phi,sin_list)
[<matplotlib.lines.Line2D at 0x1a205661d0>]
def psi(phi):
return 1/np.sqrt(2)*(H + np.exp(1j*phi)*V)
answer = [(M45.dag()*psi(phi)).norm()**2 for phi in phi_list]
plt.plot(phi_list/np.pi,answer,"-o")
plt.xlabel("$\phi$")
plt.ylabel("P(-45)")
Text(0, 0.5, 'P(-45)')