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

# # QuTiP example: Calculate the quasi-steadystate of a time-dependent (period) quantum system

# J.R. Johansson and P.D. Nation
# 
# For more information about QuTiP see [http://qutip.org](http://qutip.org)

# Find the steady state of a driven qubit, by finding the eigenstates of the propagator for one driving period

# In[1]:


get_ipython().run_line_magic('matplotlib', 'inline')


# In[2]:


import matplotlib.pyplot as plt


# In[3]:


import numpy as np


# In[4]:


from qutip import *


# In[5]:


def hamiltonian_t(t, args):
    #
    # evaluate the hamiltonian at time t. 
    #
    H0 = args['H0']
    H1 = args['H1']
    w  = args['w']

    return H0 + H1 * np.sin(w * t)


# In[6]:


def sd_qubit_integrate(delta, eps0, A, w, gamma1, gamma2, psi0, tlist):

    # Hamiltonian
    sx = sigmax()
    sz = sigmaz()
    sm = destroy(2)

    H0 = - delta/2.0 * sx - eps0/2.0 * sz
    H1 = - A * sx
        
    H_args = {'H0': H0, 'H1': H1, 'w': w}
    # collapse operators
    c_op_list = []

    n_th = 0.5 # zero temperature

    # relaxation
    rate = gamma1 * (1 + n_th)
    if rate > 0.0:
        c_op_list.append(np.sqrt(rate) * sm)

    # excitation
    rate = gamma1 * n_th
    if rate > 0.0:
        c_op_list.append(np.sqrt(rate) * sm.dag())

    # dephasing 
    rate = gamma2
    if rate > 0.0:
        c_op_list.append(np.sqrt(rate) * sz)


    # evolve and calculate expectation values
    output = mesolve(hamiltonian_t, psi0, tlist, c_op_list, [sm.dag() * sm], H_args)  

    T = 2 * np.pi / w

    U = propagator(hamiltonian_t, T, c_op_list, H_args)

    rho_ss = propagator_steadystate(U)

    return output.expect[0], expect(sm.dag() * sm, rho_ss) * np.ones(shape(tlist))


# In[7]:


delta = 0.3  * 2 * np.pi   # qubit sigma_x coefficient
eps0  = 1.0  * 2 * np.pi   # qubit sigma_z coefficient
A     = 0.05 * 2 * np.pi   # driving amplitude (sigma_x coupled)
w     = 1.0  * 2 * np.pi   # driving frequency

gamma1 = 0.15           # relaxation rate
gamma2 = 0.05           # dephasing  rate

# intial state
psi0 = basis(2,0)
tlist = np.linspace(0,50,500)


# In[8]:


p_ex, p_ex_ss = sd_qubit_integrate(delta, eps0, A, w, gamma1, gamma2, psi0, tlist)


# In[9]:


fig, ax = plt.subplots(figsize=(12,6))

ax.plot(tlist, np.real(p_ex))
ax.plot(tlist, np.real(p_ex_ss))
ax.set_xlabel('Time')
ax.set_ylabel('P_ex')
ax.set_ylim(0,1)
ax.set_title('Excitation probabilty of qubit');


# ## Software version:

# In[10]:


from qutip.ipynbtools import version_table

version_table()