Quantum states

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



In [1]:

    
from numpy import sqrt
from qutip import *

These are the polarization states:



In [2]:

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

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 [3]:

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



In [4]:

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



In [5]:

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

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



In [6]:

    
H.dag()*H









    Out[6]:





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



In [7]:

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









    Out[7]:





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 [8]:

    
psi.dag()









    Out[8]:





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 [9]:

    
psi.dag().dag()









    Out[9]:





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*}

1) verify that the $|V\rangle$ state is normalized

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

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

4) Verify Eq. (3.18)



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