#!/usr/bin/env python
# coding: utf-8

# # Lab 2 - Quantum States
# Useful for working examples and problems with photon quantum states. You may notice some similarity to the Jones Calculus ;-)

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import numpy as np
from qutip import *


# These are the polarization states:

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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)]])


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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.

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


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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()


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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()


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QWP(np.pi/4)


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

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H.dag()*H


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

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get_ipython().run_line_magic('pinfo', 'np.sin')


# ### Example 2) Converting from ket to bra:

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


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psi.dag()


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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