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

# # Development notebook: Tests for QuTiP's stochastic Schrödinger equation solver
# 
# Copyright (C) 2011 and later, Paul D. Nation & Robert J. Johansson
# 
# In this notebook we test the qutip stochastic Schrödinger equation solver (ssesolve) with a few examples. The same examples are solved using the stochastic master equation solver in the notebook development-smesolve-tests.
# 
# <style>
# .rendered_html {
#     font-family: Liberation Serif;
# }
# 
# .rendered_html h1 {
#     font-family: Liberation Sans;
#     margin: 0 0;
# }
# 
# .rendered_html h2 {
#     font-family: Liberation Sans;
#     margin: 0 0;
# }
# </style>

# In[1]:


get_ipython().run_line_magic('matplotlib', 'inline')
import matplotlib.pyplot as plt


# In[2]:


from qutip import *
import numpy as np


# ## Photo-count detection

# ### Theory
# 
# Stochastic Schrödinger equation for photocurrent detection, in Milburn's formulation, is
# 
# $$
# d|\psi(t)\rangle = -iH|\psi(t)\rangle dt + \left(\frac{a}{\sqrt{\langle a^\dagger a \rangle_\psi(t)}} - 1\right) |\psi(t)\rangle dN(t) - \frac{1}{2}\gamma\left(\langle a^\dagger a \rangle_\psi(t) - a^\dagger a \right) |\psi(t)\rangle dt
# $$
# 
# and $dN(t)$ is a Poisson distributed increment with $E[dN(t)] = \gamma \langle a^\dagger a\rangle (t)$.

# ### Formulation in QuTiP
# 
# In QuTiP we write the stochastic Schrödinger equation on the form:
# 
# $$
# d|\psi(t)\rangle = -iH|\psi(t)\rangle dt + D_{1}[c] |\psi(t)\rangle dt  + D_{2}[c] |\psi(t)\rangle dW
# $$
# 
# where $c = \sqrt{\gamma} a$ is the collapse operator including the rate of the process as a coefficient in the operator. We can identify
# 
# $$
# D_{1}[c] |\psi(t)\rangle = - \frac{1}{2}\left(\langle c^\dagger c \rangle_\psi(t) - c^\dagger c \right)
# $$
# 
# $$
# D_{2}[c] |\psi(t)\rangle = \frac{c}{\sqrt{\langle c^\dagger c \rangle_\psi(t)}} - 1
# $$
# 
# $$dW = dN(t)$$

# ### Reference solution: deterministic master equation

# In[3]:


N = 10
w0 = 0.5 * 2 * np.pi
times = np.linspace(0, 15, 150)
dt = times[1] - times[0]
gamma = 0.25
A = 2.5
ntraj = 50
nsubsteps = 100


# In[4]:


a = destroy(N)
x = a + a.dag()


# In[5]:


H = w0 * a.dag() * a


# In[6]:


psi0 = fock(N, 5)


# In[7]:


sc_ops = [np.sqrt(gamma) * a]


# In[8]:


e_ops = [a.dag() * a, a + a.dag(), (-1j)*(a - a.dag())]


# In[9]:


result_ref = mesolve(H, psi0, times, sc_ops, e_ops)


# In[10]:


plot_expectation_values(result_ref);


# ### Solve using stochastic master equation

# $\displaystyle D_{1}[a, |\psi\rangle] = 
# \frac{1}{2} ( \langle \psi|a^\dagger a |\psi\rangle  - a^\dagger a )|\psi\rangle
# $
# 

# $\displaystyle D_{2}[a, |\psi\rangle] = 
# \frac{a|\psi\rangle}{\sqrt{\langle \psi| a^\dagger a|\psi\rangle}} - |\psi\rangle
# $

# ## Using QuTiP built-in photo-current detection functions for $D_1$ and $D_2$

# In[11]:


result = photocurrent_sesolve(H, psi0, times, sc_ops, e_ops,
                  ntraj=ntraj*5, nsubsteps=nsubsteps, store_measurement=True, normalize=0)


# In[12]:


plot_expectation_values([result, result_ref]);


# In[13]:


for m in result.measurement:
    plt.step(times, dt * m.real)


# # Homodyne detection

# In[14]:


ntraj = 50
nsubsteps = 200


# In[15]:


H = w0 * a.dag() * a + A * (a + a.dag())


# In[16]:


result_ref = mesolve(H, psi0, times, sc_ops, e_ops)


# ## Theory

# Stochastic master equation for homodyne can be written as (see Gardiner and Zoller, Quantum Noise)
# 
# $$
# d|\psi(t)\rangle = -iH|\psi(t)\rangle dt + 
# \frac{1}{2}\gamma \left(
# \langle a + a^\dagger \rangle_\psi a  - a^\dagger a   - \frac{1}{4}\langle a + a^\dagger \rangle_\psi^2  \right)|\psi(t)\rangle dt  
# + 
# \sqrt{\gamma}\left(
# a - \frac{1}{2} \langle a + a^\dagger \rangle_\psi  
# \right)|\psi(t)\rangle dW
# $$
# 
# where $dW(t)$ is a normal distributed increment with $E[dW(t)] = \sqrt{dt}$.
# 
# In QuTiP format we have:
# 
# $$
# d|\psi(t)\rangle = -iH|\psi(t)\rangle dt + D_{1}[c, |\psi(t)\rangle] dt  + D_{2}[c, |\psi(t)\rangle] dW
# $$
# 
# where $c = \sqrt{\gamma} a$, so we can identify

# $$
# D_{1}[c, |\psi\rangle] = 
# \frac{1}{2}\left(
# \langle c + c^\dagger \rangle_\psi c   - c^\dagger c  - \frac{1}{4}\langle c + c^\dagger \rangle_\psi^2 
# \right) |\psi(t)\rangle 
# $$

# In[17]:


op = sc_ops[0]
opp = (op + op.dag()).data
opn = (op.dag()*op).data
op = op.data
Hd = H.data * -1j
def d1_psi_func(t, psi):
    e_x = cy.cy_expect_psi(opp, psi, 0)
    return  cy.spmv(Hd, psi) + 0.5 * (e_x * cy.spmv(op, psi) - cy.spmv(opn, psi) - 0.25 * e_x ** 2 * psi)


# $$
# D_{2}[c,|\psi\rangle] = 
# \left(c  - \frac{1}{2} \langle c + c^\dagger \rangle_\psi \right)|\psi(t)\rangle  
# $$

# In[18]:


def d2_psi_func(t, psi):
    out = np.zeros((1,len(psi)), dtype=complex)
    e_x = cy.cy_expect_psi(opp, psi, 0)
    out[0,:] = cy.spmv(op,psi)
    out -= 0.5 * e_x * psi
    return out


# In[19]:


result = general_stochastic(psi0, times, d1=d1_psi_func, d2=d2_psi_func, 
                            e_ops=e_ops, ntraj=ntraj, 
                            m_ops=[a + a.dag()], dW_factors=[1/np.sqrt(gamma)],
                            nsubsteps=nsubsteps, store_measurement=True)


# In[20]:


plot_expectation_values([result, result_ref]);


# In[21]:


for m in result.measurement:
    plt.plot(times, m[:, 0].real, 'b', alpha=0.1)

plt.plot(times,  np.array(result.measurement).mean(axis=0)[:,0].real, 'r', lw=2)
    
plt.plot(times, result_ref.expect[1], 'k', lw=2)
plt.ylim(-15, 15);


# ### Solve problem again, this time with a specified noise (from previous run)

# In[22]:


result = general_stochastic(psi0, times, d1=d1_psi_func, d2=d2_psi_func, 
                            e_ops=e_ops, ntraj=ntraj, noise=result.noise,
                            m_ops=[a + a.dag()], dW_factors=[1/np.sqrt(gamma)],
                            nsubsteps=nsubsteps, store_measurement=True)


# In[23]:


plot_expectation_values([result, result_ref]);


# In[24]:


for m in result.measurement:
    plt.plot(times, m[:, 0].real, 'b', alpha=0.1)

plt.plot(times,  np.array(result.measurement).mean(axis=0)[:,0].real, 'r', lw=2)
    
plt.plot(times, result_ref.expect[1], 'k', lw=2)
plt.ylim(-15, 15);


# ## Using QuTiP built-in homodyne detection functions for $D_1$ and $D_2$

# In[25]:


result = ssesolve(H, psi0, times, sc_ops, e_ops, ntraj=ntraj, nsubsteps=nsubsteps,
                  method='homodyne', store_measurement=True, dW_factors=[1])


# In[26]:


plot_expectation_values([result, result_ref]);


# In[27]:


for m in result.measurement:
    plt.plot(times, m[:, 0].real, 'b', alpha=0.1)

plt.plot(times,  np.array(result.measurement).mean(axis=0)[:,0].real/np.sqrt(gamma), 'r', lw=2)
    
plt.plot(times, result_ref.expect[1], 'k', lw=2)
plt.ylim(-15, 15);


# In[28]:


plt.plot(times,  np.array(result.measurement).mean(axis=0)[:,0].real/np.sqrt(gamma), 'r', lw=2)
plt.plot(times, result_ref.expect[1], 'k', lw=2)
plt.plot(times, result.expect[1], 'b', lw=2)


# #### Solve problem again, this time with a specified noise (from previous run)

# In[29]:


result = ssesolve(H, psi0, times, sc_ops, e_ops, ntraj=ntraj, nsubsteps=nsubsteps,
                  method='homodyne', store_measurement=True, noise=result.noise)


# In[30]:


plot_expectation_values([result, result_ref]);


# In[31]:


for m in result.measurement:
    plt.plot(times, m[:, 0].real, 'b', alpha=0.1)

plt.plot(times,  np.array(result.measurement).mean(axis=0)[:,0].real/np.sqrt(gamma), 'r', lw=2)
    
plt.plot(times, result_ref.expect[1], 'k', lw=2)
plt.ylim(-15, 15);


# # Heterodyne detection

# Stochastic Schrödinger equation for heterodyne detection can be written as
# 
# $$
# d|\psi(t)\rangle = -iH|\psi(t)\rangle dt 
# -\frac{1}{2}\gamma\left(a^\dagger a - \langle a^\dagger \rangle a +  \frac{1}{2}\langle a \rangle\langle a^\dagger \rangle)\right)|\psi(t)\rangle dt
# + 
# \sqrt{\gamma/2}\left(
# a - \frac{1}{2} \langle a + a^\dagger \rangle_\psi  
# \right)|\psi(t)\rangle dW_1
# -i
# \sqrt{\gamma/2}\left(
# a - \frac{1}{2} \langle a - a^\dagger \rangle_\psi  
# \right)|\psi(t)\rangle dW_2
# $$
# 
# where $dW_i(t)$ is a normal distributed increment with $E[dW_i(t)] = \sqrt{dt}$.
# 
# In QuTiP format we have:
# 
# $$
# d|\psi(t)\rangle = -iH|\psi(t)\rangle dt + D_{1}[c, |\psi(t)\rangle] dt  + \sum_n D_{2}^{(n)}[c, |\psi(t)\rangle] dW_n
# $$
# 
# where $c = \sqrt{\gamma} a$, so we can identify

# $$
# D_1[c, |\psi(t)\rangle] =
# -\frac{1}{2}\left(c^\dagger c - \langle c^\dagger \rangle c +  \frac{1}{2}\langle c \rangle\langle c^\dagger \rangle \right) |\psi(t)\rangle
# $$

# In[32]:


op = sc_ops[0]
opd = (op.dag()).data
opp = (op + op.dag()).data
opm = (op + op.dag()).data
opn = (op.dag()*op).data
op = op.data
Hd = H.data * -1j
def d1_psi_func(t, psi):
    e_xd = cy.cy_expect_psi(opd, psi, 0)
    e_x = cy.cy_expect_psi(op, psi, 0)
    return  cy.spmv(Hd, psi) - 0.5 * (cy.spmv(opn, psi) - e_xd * cy.spmv(op, psi) + 0.5 * e_x * e_xd * psi)


# $$D_{2}^{(1)}[c, |\psi(t)\rangle] = \sqrt{1/2} (c - \langle c + c^\dagger \rangle / 2) \psi$$
# 
# $$D_{2}^{(1)}[c, |\psi(t)\rangle] = -i\sqrt{1/2} (c - \langle c - c^\dagger \rangle /2) \psi$$

# In[33]:


sqrt2 = np.sqrt(0.5)
def d2_psi_func(t, psi):
    out = np.zeros((2,len(psi)), dtype=complex)
    e_p = cy.cy_expect_psi(opp, psi, 0)
    e_m = cy.cy_expect_psi(opm, psi, 0)
    out[0,:] = (cy.spmv(op,psi) - e_p * 0.5 * psi)*sqrt2
    out[1,:] = (cy.spmv(op,psi) - e_m * 0.5 * psi)*sqrt2*-1j
    return out


# In[34]:


result = general_stochastic(psi0, times, d1=d1_psi_func, d2=d2_psi_func, 
                            e_ops=e_ops, ntraj=ntraj, len_d2=2,
                            m_ops=[(a + a.dag()), (-1j)*(a - a.dag())], dW_factors=[2/np.sqrt(gamma), 2/np.sqrt(gamma)],
                            nsubsteps=nsubsteps, store_measurement=True)


# In[35]:


plot_expectation_values([result, result_ref]);


# In[36]:


#fig, ax = subplots()

for m in result.measurement:
    plt.plot(times, m[:, 0].real, 'r', alpha=0.025)
    plt.plot(times, m[:, 1].real, 'b', alpha=0.025)
    
plt.plot(times, result_ref.expect[1], 'k', lw=2);
plt.plot(times, result_ref.expect[2], 'k', lw=2);
plt.ylim(-10, 10)

plt.plot(times, np.array(result.measurement).mean(axis=0)[:,0].real, 'r', lw=2);
plt.plot(times, np.array(result.measurement).mean(axis=0)[:,1].real, 'b', lw=2);


# ## Using QuTiP built-in heterodyne detection functions for $D_1$ and $D_2$

# In[37]:


result = ssesolve(H, psi0, times, sc_ops, e_ops, ntraj=ntraj, nsubsteps=nsubsteps,
                  method='heterodyne', store_measurement=True)


# In[38]:


plot_expectation_values([result, result_ref]);


# In[39]:


for m in result.measurement:
    plt.plot(times, m[:, 0, 0].real, 'r', alpha=0.025)
    plt.plot(times, m[:, 0, 1].real, 'b', alpha=0.025)
    
plt.plot(times, result_ref.expect[1], 'k', lw=2)
plt.plot(times, result_ref.expect[2], 'k', lw=2)

plt.plot(times, np.array(result.measurement).mean(axis=0)[:,0,0].real/np.sqrt(gamma), 'r', lw=2)
plt.plot(times, np.array(result.measurement).mean(axis=0)[:,0,1].real/np.sqrt(gamma), 'b', lw=2)


# ### Solve problem again, this time with a specified noise (from previous run)

# In[40]:


result = ssesolve(H, psi0, times, sc_ops, e_ops, ntraj=ntraj, nsubsteps=nsubsteps,
                  method='heterodyne', store_measurement=True, noise=result.noise)


# In[41]:


plot_expectation_values([result, result_ref]);


# In[42]:


for m in result.measurement:
    plt.plot(times, m[:, 0, 0].real, 'r', alpha=0.025)
    plt.plot(times, m[:, 0, 1].real, 'b', alpha=0.025)
    
plt.plot(times, result_ref.expect[1], 'k', lw=2);
plt.plot(times, result_ref.expect[2], 'k', lw=2);

plt.plot(times, np.array(result.measurement).mean(axis=0)[:,0,0].real/np.sqrt(gamma), 'r', lw=2);
plt.plot(times, np.array(result.measurement).mean(axis=0)[:,0,1].real/np.sqrt(gamma), 'b', lw=2);


# # Software version

# In[43]:


from qutip.ipynbtools import version_table

version_table()