# QuTiP mesolve: demonstate how to use the e_ops argument¶

Copyright (C) 2011 and later, Paul D. Nation & Robert J. Johansson

In [1]:
%pylab inline
Populating the interactive namespace from numpy and matplotlib
In [2]:
from qutip import *

## Example: the Janyes-Cummings model¶

In [3]:
wc = 1.0  * 2 * pi  # cavity frequency
wa = 1.0  * 2 * pi  # atom frequency
g  = 0.05 * 2 * pi  # coupling strength
kappa = 0.005       # cavity dissipation rate
gamma = 0.05        # atom dissipation rate
N = 15              # number of cavity fock states
tlist = linspace(0,25,100)

### Operators, initial state, Hamiltonian and collapse operators¶

In [4]:
a  = tensor(destroy(N), qeye(2))
sm = tensor(qeye(N), destroy(2))
In [5]:
psi0 = tensor(basis(N,0), basis(2,1))
In [6]:
H = wc * a.dag() * a + wa * sm.dag() * sm + g * (a.dag() * sm + a * sm.dag())
In [7]:
c_ops = [sqrt(kappa) * a, sqrt(gamma) * sm]

## Expectation value operators¶

In [8]:
e_ops = [a.dag() * a, sm.dag() * sm]
In [9]:
result = mesolve(H, psi0, tlist, c_ops, e_ops)
In [10]:
len(result.expect)
Out[10]:
2
In [11]:
result.expect[0].shape
Out[11]:
(100,)
In [12]:
plot_expectation_values(result);

## State-vectors¶

In [13]:
e_ops = []
In [14]:
result = mesolve(H, psi0, tlist, c_ops, e_ops)
In [15]:
len(result.states)
Out[15]:
100
In [16]:
result.states[0]
Out[16]:
$\text{Quantum object: dims = [[15, 2], [15, 2]], shape = [30, 30], type = oper, isHerm = True}\\[1em]\begin{pmatrix}0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 1.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \vdots\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\end{pmatrix}$
In [17]:
# loop over states and calculate the atom entropy
entropy_atom_vec = zeros(len(result.states))

for idx, state in enumerate(result.states):
entropy_atom_vec[idx] = entropy_linear(ptrace(state, 0))
In [18]:
fig, ax = plt.subplots(1,1)

ax.plot(tlist, entropy_atom_vec)

ax.set_xlabel('time', fontsize=16)
ax.set_ylabel(r'$E(\rho_{\rm atom})$', fontsize=16);

## Callback function¶

In [19]:
entropy_cavity_vec = zeros(len(tlist))
In [20]:
# calculate the entropy of the cavity in a callback function

def e_ops_func(t, rho):
# rho is only the data, so do Qobj(rho) to obtain a Qobj representation
# of rho
entropy_cavity_vec[e_ops_func.idx] = entropy_linear(ptrace(Qobj(rho), 1))
e_ops_func.idx += 1

e_ops_func.idx = 0
In [21]:
result = mesolve(H, psi0, tlist, c_ops, e_ops_func)
In [22]:
fig, ax = plt.subplots(1,1)

ax.plot(tlist, entropy_atom_vec, label='atom')
ax.plot(tlist, entropy_cavity_vec, label='cavity')

ax.legend()
ax.set_xlabel('time', fontsize=16)
ax.set_ylabel(r'$E$', fontsize=16);

### Software versions¶

In [23]:
from qutip.ipynbtools import version_table
version_table()
Out[23]:
SoftwareVersion
Python3.3.1 (default, Apr 17 2013, 22:30:32) [GCC 4.7.3]
OSposix [linux]
SciPy0.14.0.dev-2a4ba40
QuTiP2.3.0.dev-2283478
IPython2.0.0-dev
Cython0.19
matplotlib1.4.x
Numpy1.7.1
Sat Sep 28 11:41:47 2013 JST