# QuTiP example: Wigner function animation for the dynamics of the Jaynes-Cumming model¶

J.R. Johansson and P.D. Nation

For more information about QuTiP see http://qutip.org

In [1]:
%matplotlib inline

In [2]:
import matplotlib.pyplot as plt

In [3]:
import numpy as np

In [4]:
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm
import time

In [5]:
from qutip import *
from qutip.ipynbtools import plot_animation

In [6]:
def jc_integrate(N, wc, wa, g, kappa, gamma, psi0, use_rwa, tlist):

# Hamiltonian
idc = qeye(N)
ida = qeye(2)

a  = tensor(destroy(N), ida)
sm = tensor(idc, destroy(2))

if use_rwa:
# use the rotating wave approxiation
H = wc * a.dag() * a + wa * sm.dag() * sm + g * (a.dag() * sm + a * sm.dag())
else:
H = wc * a.dag() * a + wa * sm.dag() * sm + g * (a.dag() + a) * (sm + sm.dag())

# collapse operators
c_op_list = []

n_th_a = 0.0 # zero temperature

rate = kappa * (1 + n_th_a)
if rate > 0.0:
c_op_list.append(np.sqrt(rate) * a)

rate = kappa * n_th_a
if rate > 0.0:
c_op_list.append(np.sqrt(rate) * a.dag())

rate = gamma
if rate > 0.0:
c_op_list.append(np.sqrt(rate) * sm)

# evolve and calculate return state vectors
result = mesolve(H, psi0, tlist, c_op_list, [])

return result

In [7]:
# parameters
wc = 1.0 * 2 * np.pi   # cavity frequency
wa = 1.0 * 2 * np.pi   # atom frequency
g  = 0.05 * 2 * np.pi  # coupling strength
kappa = 0.05        # cavity dissipation rate
gamma = 0.15        # atom dissipation rate
N = 10              # number of cavity fock states

use_rwa = True

# initial state
psi0 = tensor(basis(N,0), basis(2,1))  # start with an excited atom
#psi0 = tensor(coherent(N,1.5), basis(2,0))  # or a coherent state the in cavity
#psi0 = tensor((coherent(N,2.0)+coherent(N,-2.0)).unit(), basis(2,0))  # or a superposition of coherent states

tlist = np.linspace(0, 30, 150)

In [8]:
result = jc_integrate(N, wc, wa, g, kappa, gamma, psi0, use_rwa, tlist)

In [9]:
xvec = np.linspace(-5.,5.,100)
X,Y = np.meshgrid(xvec, xvec)

In [10]:
def plot_setup(result):

fig = plt.figure(figsize=(12, 6))
ax = Axes3D(fig, azim=-107, elev=49)

return fig, ax

In [11]:
cb = None

def plot_result(result, n, fig=None, axes=None):

global cb

if fig is None or axes is None:
fig, ax = plot_setup(result)

axes.cla()

# trace out the atom
rho_cavity = ptrace(result.states[n], 0)

W = wigner(rho_cavity, xvec, xvec)

surf = axes.plot_surface(X, Y, W, rstride=1, cstride=1, cmap=cm.jet,
alpha=1.0, linewidth=0.05, vmax=0.25, vmin=-0.25)
axes.set_xlim3d(-5, 5)
axes.set_ylim3d(-5, 5)
axes.set_zlim3d(-0.25, 0.25)

if not cb:
cb = plt.colorbar(surf, shrink=0.65, aspect=20)

return fig, axes

In [12]:
plot_animation(plot_setup, plot_result, result)

Out[12]:

## Versions¶

In [13]:
from qutip.ipynbtools import version_table

version_table()

Out[13]:
SoftwareVersion
Cython0.21.1
SciPy0.14.0
QuTiP3.1.0
Python2.7.8 (default, Oct 20 2014, 15:05:19) [GCC 4.9.1]
IPython2.3.0
OSposix [linux2]
Numpy1.9.0
matplotlib1.4.2
Tue Jan 13 16:59:39 2015 JST