QuTiP example: Dynamics of an atom-cavity system using three different solvers

In [1]:

    

%matplotlib inline
import matplotlib.pyplot as plt


    

In [2]:

    

import numpy as np


    

In [3]:

    

from qutip import *


    

In [4]:

    

kappa = 2
gamma = 0.2
g = 1 
wc = 0 
w0 = 0 
wl = 0
N = 4
E = 0.5
tlist = np.linspace(0, 10, 200)


    

In [5]:

    

def solve(E,kappa,gamma,g,wc,w0,wl,N,tlist):

    ida    = qeye(N)
    idatom = qeye(2)

    # Define cavity field and atomic operators
    a  = tensor(destroy(N),idatom)
    sm = tensor(ida,sigmam())

    # Hamiltonian
    H = (w0-wl)*sm.dag()*sm + (wc-wl)*a.dag()*a + 1j*g*(a.dag()*sm - sm.dag()*a) \
        + E*(a.dag()+a)

    #collapse operators
    C1 = np.sqrt(2*kappa)*a
    C2 = np.sqrt(gamma)*sm
    C1dC1 = C1.dag()*C1
    C2dC2 = C2.dag()*C2

    #intial state
    psi0 = tensor(basis(N,0),basis(2,1))
    rho0 = psi0.dag() * psi0;

    # evolve and calculate expectation values
    output = mesolve(H, psi0, tlist, [C1, C2], [C1dC1, C2dC2, a])  

    return output.expect[0], output.expect[1], output.expect[2]


    

In [6]:

    

count1, count2, infield = solve(E,kappa,gamma,g,wc,w0,wl,N,tlist)


    

In [7]:

    

fig, ax = plt.subplots(figsize=(12,6))
ax.plot(tlist, np.real(count1))
ax.plot(tlist, np.real(count2))
ax.set_xlabel('Time')
ax.set_ylabel('Transmitted Intensity and Spontaneous Emission');


    

In [8]:

    

def solve(E,kappa,gamma,g,wc,w0,wl,N,tlist):

    # Define cavity field and atomic operators
    a  = tensor(destroy(N),qeye(2))
    sm = tensor(qeye(N),sigmam())

    # Hamiltonian
    H = (w0-wl)*sm.dag()*sm + (wc-wl)*a.dag()*a + 1j*g*(a.dag()*sm - sm.dag()*a) \
        + E*(a.dag()+a)

    #collapse operators
    C1 = np.sqrt(2*kappa)*a
    C2 = np.sqrt(gamma)*sm
    C1dC1 = C1.dag() * C1
    C2dC2 = C2.dag() * C2

    #intial state
    psi0 = tensor(basis(N,0),basis(2,1))
    rho0 = ket2dm(psi0)

    # Calculate the Liouvillian
    L = liouvillian(H, [C1, C2])

    # Calculate solution as an exponential series
    rhoES = ode2es(L,rho0);
    
    # Calculate expectation values
    count1  = esval(expect(C1dC1,rhoES),tlist);
    count2  = esval(expect(C2dC2,rhoES),tlist);
    infield = esval(expect(a,rhoES),tlist);

    # alternative
    expt_list = essolve(H, psi0, tlist, [C1, C2], [C1dC1, C2dC2, a]).expect

    return count1, count2, infield, expt_list[0], expt_list[1], expt_list[2]


    

In [9]:

    

count1, count2, infield, count1_2, count2_2, infield_2 = solve(E, kappa, gamma, g, wc, w0, wl, N, tlist)


    

In [10]:

    

fig, ax = plt.subplots(figsize=(12,6))
ax.plot(tlist, np.real(count1), tlist, np.real(count1_2), '.')
ax.plot(tlist, np.real(count2), tlist, np.real(count2_2), '.')
ax.set_xlabel('Time')
ax.set_ylabel('Transmitted Intensity and Spontaneous Emission');


    

In [11]:

    

ntraj = 500 #number of Monte-Carlo trajectories
# Hamiltonian
ida = qeye(N)
idatom = qeye(2)
a = tensor(destroy(N),idatom)
sm = tensor(ida,sigmam())
H = (w0-wl)*sm.dag()*sm + (wc-wl)*a.dag()*a + 1j*g*(a.dag()*sm-sm.dag()*a) + E*(a.dag()+a)
# collapse operators
C1 = np.sqrt(2*kappa) * a
C2 = np.sqrt(gamma) * sm
C1dC1 = C1.dag() * C1
C2dC2 = C2.dag() * C2
# intial state
psi0=tensor(basis(N,0),basis(2,1))


    

In [12]:

    

avg = mcsolve(H, psi0, tlist, [C1,C2],[C1dC1,C2dC2], ntraj)


    

    




10.0%. Run time:   0.56s. Est. time left: 00:00:00:05
20.0%. Run time:   0.87s. Est. time left: 00:00:00:03
30.0%. Run time:   1.17s. Est. time left: 00:00:00:02
40.0%. Run time:   1.45s. Est. time left: 00:00:00:02
50.0%. Run time:   1.78s. Est. time left: 00:00:00:01
60.0%. Run time:   2.10s. Est. time left: 00:00:00:01
70.0%. Run time:   2.41s. Est. time left: 00:00:00:01
80.0%. Run time:   2.79s. Est. time left: 00:00:00:00
90.0%. Run time:   3.13s. Est. time left: 00:00:00:00
100.0%. Run time:   3.43s. Est. time left: 00:00:00:00
Total run time:   3.53s

In [13]:

    

fig, ax = plt.subplots(figsize=(12, 6))
ax.plot(tlist, avg.expect[0],
        tlist, avg.expect[1],'--')
ax.set_xlabel('Time')
ax.set_ylabel('Photocount rates')
ax.legend(('Cavity ouput', 'Spontaneous emission'));


    

In [14]:

    

from qutip.ipynbtools import version_table

version_table()


    

    
Out[14]:




Software
Version
Cython
0.21.2
OS
posix [linux]
Python
3.4.0 (default, Apr 11 2014, 13:05:11) 
[GCC 4.8.2]
QuTiP
3.1.0
SciPy
0.14.1
IPython
2.3.1
Numpy
1.9.1
matplotlib
1.4.2
Fri Jan 09 11:18:37 2015 JST