QuTiP example: eseries

J.R. Johansson and P.D. Nation

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



In [1]:

    
from numpy import pi



In [2]:

    
from qutip import *

Example eseries object: $\sigma_x \exp(i\omega t)$



In [3]:

    
omega = 1.0
es1 = eseries(sigmax(), 1j * omega)



In [4]:

    
es1









    Out[4]:





ESERIES object: 1 terms
Hilbert space dimensions: [[2], [2]]
Exponent #0 = 1j
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True
Qobj data =
[[ 0.  1.]
 [ 1.  0.]]

Example eseries object: $\sigma_x \cos(\omega t)$



In [5]:

    
omega = 1.0
es2 = eseries(0.5 * sigmax(), 1j * omega) + eseries(0.5 * sigmax(), -1j * omega)



In [6]:

    
es2









    Out[6]:





ESERIES object: 2 terms
Hilbert space dimensions: [[2], [2]]
Exponent #0 = 1j
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True
Qobj data =
[[ 0.   0.5]
 [ 0.5  0. ]]
Exponent #1 = -1j
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True
Qobj data =
[[ 0.   0.5]
 [ 0.5  0. ]]

Evaluate eseries object at time $t = 0$



In [7]:

    
esval(es2, 0.0)









    Out[7]:





Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}0.0 & 1.0\\1.0 & 0.0\\\end{array}\right)\end{equation*}

Evaluate eseries object at array of times $t = [0, \pi, 2\pi]$



In [8]:

    
tlist = [0.0, 1.0 * pi, 2.0 * pi]
esval(es2, tlist)









    Out[8]:





array([ Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True
Qobj data =
[[ 0.  1.]
 [ 1.  0.]],
       Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True
Qobj data =
[[ 0. -1.]
 [-1.  0.]],
       Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True
Qobj data =
[[ 0.  1.]
 [ 1.  0.]]], dtype=object)

Expectation values of eseries



In [9]:

    
es2









    Out[9]:





ESERIES object: 2 terms
Hilbert space dimensions: [[2], [2]]
Exponent #0 = 1j
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True
Qobj data =
[[ 0.   0.5]
 [ 0.5  0. ]]
Exponent #1 = -1j
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True
Qobj data =
[[ 0.   0.5]
 [ 0.5  0. ]]



In [10]:

    
expect(sigmax(), es2)









    Out[10]:





ESERIES object: 2 terms
Hilbert space dimensions: [[1, 1]]
Exponent #0 = 1j
1.0
Exponent #1 = -1j
1.0

Arithmetics with eseries



In [11]:

    
es1 = eseries(sigmax(), 1j * omega)
es1









    Out[11]:





ESERIES object: 1 terms
Hilbert space dimensions: [[2], [2]]
Exponent #0 = 1j
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True
Qobj data =
[[ 0.  1.]
 [ 1.  0.]]



In [12]:

    
es2 = eseries(sigmax(), -1j * omega)
es2









    Out[12]:





ESERIES object: 1 terms
Hilbert space dimensions: [[2], [2]]
Exponent #0 = -1j
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True
Qobj data =
[[ 0.  1.]
 [ 1.  0.]]



In [13]:

    
es1 + es2









    Out[13]:





ESERIES object: 2 terms
Hilbert space dimensions: [[2], [2]]
Exponent #0 = 1j
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True
Qobj data =
[[ 0.  1.]
 [ 1.  0.]]
Exponent #1 = -1j
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True
Qobj data =
[[ 0.  1.]
 [ 1.  0.]]



In [14]:

    
es1 - es2









    Out[14]:





ESERIES object: 2 terms
Hilbert space dimensions: [[2], [2]]
Exponent #0 = 1j
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True
Qobj data =
[[ 0.  1.]
 [ 1.  0.]]
Exponent #1 = -1j
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True
Qobj data =
[[ 0. -1.]
 [-1.  0.]]



In [15]:

    
es1 * es2









    Out[15]:





ESERIES object: 1 terms
Hilbert space dimensions: [[2], [2]]
Exponent #0 = 0j
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True
Qobj data =
[[ 1.  0.]
 [ 0.  1.]]



In [16]:

    
(es1 + es2) * (es1 - es2)









    Out[16]:





ESERIES object: 2 terms
Hilbert space dimensions: [[2], [2]]
Exponent #0 = 2j
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True
Qobj data =
[[ 1.  0.]
 [ 0.  1.]]
Exponent #1 = -2j
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True
Qobj data =
[[-1.  0.]
 [ 0. -1.]]

Expectation values of eseries



In [17]:

    
es3 = eseries([0.5*sigmaz(), 0.5*sigmaz()], [1j, -1j]) + eseries([-0.5j*sigmax(), 
                                                                  0.5j*sigmax()], [1j, -1j])
es3









    Out[17]:





ESERIES object: 2 terms
Hilbert space dimensions: [[2], [2]]
Exponent #0 = (-0-1j)
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = False
Qobj data =
[[ 0.5+0.j   0.0+0.5j]
 [ 0.0+0.5j -0.5+0.j ]]
Exponent #1 = 1j
Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = False
Qobj data =
[[ 0.5+0.j   0.0-0.5j]
 [ 0.0-0.5j -0.5+0.j ]]



In [18]:

    
es3.value(0.0)









    Out[18]:





Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}1.0 & 0.0\\0.0 & -1.0\\\end{array}\right)\end{equation*}



In [19]:

    
es3.value(pi/2)









    Out[19]:





Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}0.0 & 1.0\\1.0 & 0.0\\\end{array}\right)\end{equation*}



In [20]:

    
rho = fock_dm(2, 1)
es3_expect = expect(rho, es3)

es3_expect









    Out[20]:





ESERIES object: 2 terms
Hilbert space dimensions: [[1, 1]]
Exponent #0 = (-0-1j)
(-0.5+0j)
Exponent #1 = 1j
(-0.5+0j)



In [21]:

    
es3_expect.value([0.0, pi/2])









    Out[21]:





array([ -1.00000000e+00,  -6.12323400e-17])

Versions



In [22]:

    
from qutip.ipynbtools import version_table

version_table()









    Out[22]:




Software Version
QuTiP 4.2.0
Numpy 1.13.1
SciPy 0.19.1
matplotlib 2.0.2
Cython 0.25.2
Number of CPUs 2
BLAS Info INTEL MKL
IPython 6.1.0
Python 3.6.1 |Anaconda custom (x86_64)| (default, May 11 2017, 13:04:09) 
[GCC 4.2.1 Compatible Apple LLVM 6.0 (clang-600.0.57)]
OS posix [darwin]
Wed Jul 19 22:16:28 2017 MDT

Software	Version
QuTiP	4.2.0
Numpy	1.13.1
SciPy	0.19.1
matplotlib	2.0.2
Cython	0.25.2
Number of CPUs	2
BLAS Info	INTEL MKL
IPython	6.1.0
Python	3.6.1 \|Anaconda custom (x86_64)\| (default, May 11 2017, 13:04:09) [GCC 4.2.1 Compatible Apple LLVM 6.0 (clang-600.0.57)]
OS	posix [darwin]
Wed Jul 19 22:16:28 2017 MDT