#!/usr/bin/python
# -*- coding: utf8 -*-

import matplotlib.pyplot as plt
import matplotlib as mpl
import numpy as np
from math import *

code_website = 'http://commons.wikimedia.org/wiki/User:Geek3/mplwp'
try:
    import mplwp
except ImportError, er:
    print 'ImportError:', er
    print 'You need to download mplwp.py from', code_website
    exit(1)

name = 'mplwp_ramsey_fringes_thermal.svg'
fig = mplwp.fig_standard(mpl)

xlim = -5,5; fig.gca().set_xlim(xlim)
ylim = -0.05,0.85; fig.gca().set_ylim(ylim)
mplwp.mark_axeszero(fig.gca())

# free evolution time vs. interaction pulse time ratio
T_over_tau = 8

def flip(delta, v_rel = 1.0):
    # return spin flip probability of atom beam in Ramsey device
    # delta is the detuning in terms of the Rabi frequency Omega0
    # v_rel is the relative particle speed compared the pi/2-pulse case
    if v_rel <= 0.0: return 0.0
    Omega_rel = sqrt(1.0 + delta**2)
    phase_pulse = 0.25 * pi * Omega_rel / v_rel
    phase_free = 0.25 * pi * delta * T_over_tau / v_rel
    Pflip = (sin(phase_pulse) *
        (Omega_rel * cos(phase_pulse) * cos(phase_free)
         - delta * sin(phase_pulse) * sin(phase_free)) 
        * 2. / Omega_rel**2)**2
    return Pflip
    
from scipy.integrate import quad
from scipy.stats import maxwell

# integrate Maxwell-Boltzmann distribution percentiles from 0 to 1
def v(x):
    return maxwell.ppf(x, scale=0.968/sqrt(2))

def integrand(x, delta):
    if x <= 0. or x >= 1.0: return 0.0
    return flip(delta, v(x))

def flip_thermal(delta):
    # return average spin flip probability of thermal ensemble
    print delta
    return quad(integrand, 0, 1, (delta,), epsabs=1e-6, limit=200)[0]

x = np.linspace(xlim[0], xlim[1], 5001)
y = [flip_thermal(xx) for xx in x]

plt.plot(x, y, label='$P_{flip}(\Delta\omega/\Omega_0)$')
plt.legend(loc='upper right').get_frame().set_alpha(0.9)
    
plt.savefig(name)
mplwp.postprocess(name)