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

J.R. Johansson and P.D. Nation

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