Chapter 12 - Simple Harmonic Oscillator - Quantum Mechanics with QuTiP

Several examples that make use of the built-in Simple Harmonic Oscillator tools:

from numpy import sqrt,linspace,exp
from qutip import *
import matplotlib.pyplot as plt
%matplotlib inline

N = 40 # size of the Hilbert space.
       # N needs to be large enough that matrix is not truncated
a = destroy(N)
n = a.dag()*a

Define a coherent state with $\alpha = 2:$

psi = coherent(N,1)

psi

Expectation value $\langle n\rangle$ :

psi.dag()*n*psi

(1+0j)

# Convert to a density matrix:
psi_dm = ket2dm(psi)

psi*psi.dag()

# Fock state distribution of the coherent state:
plot_fock_distribution(psi_dm)

(<Figure size 640x480 with 1 Axes>,
 <Axes: xlabel='Fock number', ylabel='Occupation probability'>)

Also look at other coherent states:

# With larger N and α:
plot_fock_distribution(coherent_dm(100,6))

(<Figure size 640x480 with 1 Axes>,
 <Axes: xlabel='Fock number', ylabel='Occupation probability'>)

And some other density matrices, like a fock-state (or n eigenstate):

# With a Fock state (number state):
plot_fock_distribution(fock_dm(20,2))

(<Figure size 640x480 with 1 Axes>,
 <Axes: xlabel='Fock number', ylabel='Occupation probability'>)

fock(20,2)

And a thermal state:

# A thermal state has decaying amplitudes
plot_fock_distribution(thermal_dm(20,2))

(<Figure size 640x480 with 1 Axes>,
 <Axes: xlabel='Fock number', ylabel='Occupation probability'>)

Finally, can visualize the phasor using what is called the Wigner function:¶

xvec = linspace(-10,10,200) # Create an array for the phase space coordinates.
plt.imshow(wigner(coherent(40,5j)+coherent(40,-5j),xvec,xvec), extent=[-10,10,-10,10]) # contour plot of Wigner function