QuTiP example: Energy-levels of a quantum systems as a function of a single parameter

J.R. Johansson and P.D. Nation

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

In [1]:
%matplotlib inline
import matplotlib.pyplot as plt
In [2]:
import numpy as np
from numpy import pi
In [3]:
from qutip import *

Energy spectrum of three coupled qubits

In [4]:
def compute(w1list, w2, w3, g12, g13):

    # Pre-compute operators for the hamiltonian
    sz1 = tensor(sigmaz(), qeye(2), qeye(2))
    sx1 = tensor(sigmax(), qeye(2), qeye(2))

    sz2 = tensor(qeye(2), sigmaz(), qeye(2))
    sx2 = tensor(qeye(2), sigmax(), qeye(2))

    sz3 = tensor(qeye(2), qeye(2), sigmaz())
    sx3 = tensor(qeye(2), qeye(2), sigmax())
  
    idx = 0
    evals_mat = np.zeros((len(w1list),2*2*2))
    for w1 in w1list:

        # evaluate the Hamiltonian
        H = w1 * sz1 + w2 * sz2 + w3 * sz3 + g12 * sx1 * sx2 + g13 * sx1 * sx3

        # find the energy eigenvalues of the composite system
        evals, ekets = H.eigenstates()

        evals_mat[idx,:] = np.real(evals)

        idx += 1

    return evals_mat
In [5]:
w1  = 1.0 * 2 * pi   # atom 1 frequency: sweep this one
w2  = 0.9 * 2 * pi   # atom 2 frequency
w3  = 1.1 * 2 * pi   # atom 3 frequency
g12 = 0.05 * 2 * pi   # atom1-atom2 coupling strength
g13 = 0.05 * 2 * pi   # atom1-atom3 coupling strength

w1list = np.linspace(0.75, 1.25, 50) * 2 * pi # atom 1 frequency range
In [6]:
evals_mat = compute(w1list, w2, w3, g12, g13)
In [7]:
fig, ax = plt.subplots(figsize=(12,6))

for n in [1,2,3]:
    ax.plot(w1list / (2*pi), (evals_mat[:,n]-evals_mat[:,0]) / (2*pi), 'b')

ax.set_xlabel('Energy splitting of atom 1')
ax.set_ylabel('Eigenenergies')
ax.set_title('Energy spectrum of three coupled qubits');

Versions

In [8]:
from qutip.ipynbtools import version_table

version_table()
Out[8]:
SoftwareVersion
IPython2.3.1
Python3.4.0 (default, Apr 11 2014, 13:05:11) [GCC 4.8.2]
OSposix [linux]
Cython0.21.2
SciPy0.14.1
Numpy1.9.1
matplotlib1.4.2
QuTiP3.1.0
Tue Jan 13 13:11:57 2015 JST