Lecture 16: Gallery of Wigner functions¶

Author: J.R. Johansson, robert@riken.jp, http://jrjohansson.github.io

Latest version of this ipython notebook lecture is available at: http://github.com/jrjohansson/qutip-lectures

In [1]:

%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np

In [2]:

from qutip import *

Introduction¶

Parameters¶

In [3]:

N = 20

In [4]:

def plot_wigner_2d_3d(psi):
    #fig, axes = plt.subplots(1, 2, subplot_kw={'projection': '3d'}, figsize=(12, 6))
    fig = plt.figure(figsize=(17, 8))
    
    ax = fig.add_subplot(1, 2, 1)
    plot_wigner(psi, fig=fig, ax=ax, alpha_max=6);

    ax = fig.add_subplot(1, 2, 2, projection='3d')
    plot_wigner(psi, fig=fig, ax=ax, projection='3d', alpha_max=6);
    
    plt.close(fig)
    return fig

Vacuum state: $\left|0\right>$¶

In [5]:

psi = basis(N, 0)
plot_wigner_2d_3d(psi)

Out[5]:

Thermal states¶

In [6]:

psi = thermal_dm(N, 2)
plot_wigner_2d_3d(psi)

Out[6]:

Coherent states: $\left|\alpha\right>$¶

In [7]:

psi = coherent(N, 2.0)
plot_wigner_2d_3d(psi)

Out[7]:

In [8]:

psi = coherent(N, -1.0)
plot_wigner_2d_3d(psi)

Out[8]:

Superposition of coherent states¶

In [9]:

psi = (coherent(N, -2.0) + coherent(N, 2.0)) / np.sqrt(2)
plot_wigner_2d_3d(psi)

Out[9]:

In [10]:

psi = (coherent(N, -2.0) - coherent(N, 2.0)) / np.sqrt(2)
plot_wigner_2d_3d(psi)

Out[10]:

In [11]:

psi = (coherent(N, -2.0) + coherent(N, -2j) + coherent(N, 2j) + coherent(N, 2.0)).unit()
plot_wigner_2d_3d(psi)

Out[11]:

In [12]:

psi = (coherent(N, -2.0) + coherent(N, -1j) + coherent(N, 1j) + coherent(N, 2.0)).unit()
plot_wigner_2d_3d(psi)

Out[12]:

In [13]:

NN = 8

fig, axes = plt.subplots(NN, 1, figsize=(5, 5 * NN), sharex=True, sharey=True) 
for n in range(NN):
    psi = sum([coherent(N, 2*np.exp(2j * np.pi * m / (n + 2))) for m in range(n + 2)]).unit()
    plot_wigner(psi, fig=fig, ax=axes[n])

    #if n < NN - 1:
    #    axes[n].set_ylabel("")

Mixture of coherent states¶

In [14]:

psi = (coherent_dm(N, -2.0) + coherent_dm(N, 2.0)) / np.sqrt(2)
plot_wigner_2d_3d(psi)

Out[14]:

Fock states: $\left|n\right>$¶

In [15]:

from IPython.display import display

In [16]:

for n in range(6):
    psi = basis(N, n)
    display(plot_wigner_2d_3d(psi))

Superposition of Fock states¶

In [17]:

NN = MM = 5

fig, axes = plt.subplots(NN, MM, figsize=(18, 18), sharex=True, sharey=True) 
for n in range(NN):
    for m in range(MM):
        psi = (fock(N, n) + fock(N, m)).unit()
        plot_wigner(psi, fig=fig, ax=axes[n, m])
        #axes[n, m].set_title(r"$(\left|%d\right> + \left|%d\right>)/\sqrt{2}$" % (n, m))
        if n < NN - 1:
            axes[n, m].set_xlabel("")
        if m > 0:
            axes[n, m].set_ylabel("")

Squeezed vacuum states¶

In [18]:

psi = squeeze(N, 0.5) * basis(N, 0)
display(plot_wigner_2d_3d(psi))

psi = squeeze(N, 0.75j) * basis(N, 0)
display(plot_wigner_2d_3d(psi))

psi = squeeze(N, -1) * basis(N, 0)
display(plot_wigner_2d_3d(psi))

Superposition of squeezed vacuum¶

In [19]:

psi = (squeeze(N, 0.75j) * basis(N, 0) - squeeze(N, -0.75j) * basis(N, 0)).unit()
display(plot_wigner_2d_3d(psi))

Mixture of squeezed vacuum¶

In [20]:

psi = (ket2dm(squeeze(N, 0.75j) * basis(N, 0)) + ket2dm(squeeze(N, -0.75j) * basis(N, 0))).unit()
display(plot_wigner_2d_3d(psi))

Displaced squeezed vacuum¶

In [21]:

psi = displace(N, 2) * squeeze(N, 0.75) * basis(N, 0)
display(plot_wigner_2d_3d(psi))

Superposition of two displaced squeezed states¶

In [22]:

psi = (displace(N, -1) * squeeze(N, 0.75) * basis(N, 0) - displace(N, 1) * squeeze(N, -0.75) * basis(N, 0)).unit()
display(plot_wigner_2d_3d(psi))

Versions¶

In [23]:

from qutip.ipynbtools import version_table

version_table()

Out[23]:

Software	Version
QuTiP	3.1.0.dev-f7fffa7
SciPy	0.14.0
Cython	0.20.1
Numpy	1.10.0.dev-27d73bf
OS	posix [darwin]
matplotlib	1.4.0
IPython	2.3.0
Python	3.4.1 (default, Sep 20 2014, 19:44:17) [GCC 4.2.1 Compatible Apple LLVM 5.1 (clang-503.0.40)]
Wed Oct 08 23:59:44 2014 JST