QuTiP development notebook for testing extracting and eliminating states from a quantum object



In [1]:

    
%pylab inline



In [2]:

    
from qutip import *



In [3]:

    
def visualize(result):
    fig, ax = subplots(figsize=(12,6))
    for idx, e in enumerate(result.expect):
        ax.plot(result.times, e, label="%d" % idx)
    ax.legend()

Consider two coupled qubits:



In [4]:

    
H0 = -2*pi*(tensor(sigmaz(), identity(2)) + tensor(identity(2), sigmaz()))
Hint = 2*pi*0.1 * tensor(sigmax(), sigmax())

H = H0 + Hint



In [5]:

    
H









    Out[5]:





\begin{equation}\text{Quantum object: dims = [[2, 2], [2, 2]], shape = [4, 4], type = oper, isHerm = True}\\[1em]\begin{pmatrix}-12.5663706144 & 0.0 & 0.0 & 0.628318530718\\0.0 & 0.0 & 0.628318530718 & 0.0\\0.0 & 0.628318530718 & 0.0 & 0.0\\0.628318530718 & 0.0 & 0.0 & 12.5663706144\\\end{pmatrix}\end{equation}



In [6]:

    
energy_level_diagram([H], figsize=(6,6), show_ylabels=True);

If the highest energy state is not initially occupied, and if we neglect thermal excitations, it will never be occupied, and therefore not really necessary to include in a calculation of for example the dynamics of the three lowest states.



In [7]:

    
psi_gg = tensor(basis(2, 0), basis(2, 0))
psi_eg = tensor(basis(2, 1), basis(2, 0))
psi_ge = tensor(basis(2, 0), basis(2, 1))
psi_ee = tensor(basis(2, 1), basis(2, 1))

c_op1 = tensor(destroy(2), identity(2))
c_op2 = tensor(identity(2), destroy(2))



In [8]:

    
psi0 = psi_eg



In [9]:

    
tlist = linspace(0, 15, 100)
result = mesolve(H, psi0, tlist,
                 [sqrt(0.1) * c_op1, sqrt(0.25) * c_op2],
                 [ket2dm(psi_gg), ket2dm(psi_ge), ket2dm(psi_eg), ket2dm(psi_ee)])



In [10]:

    
visualize(result)

Note that the fourth state is not included in the dynamics at all. So...

Assume we want to create a new Qobj instance using the three lowest states:



In [11]:

    
# We are only interested in these states
keep_states = [0, 1, 2]

# or equivalently, not interested in these states
remove_states = [3]



In [12]:

    
H_sub = H.eliminate_states(remove_states)

H_sub









    Out[12]:





\begin{equation}\text{Quantum object: dims = [[3], [3]], shape = [3, 3], type = oper, isHerm = True}\\[1em]\begin{pmatrix}-12.5663706144 & 0.0 & 0.0\\0.0 & 0.0 & 0.628318530718\\0.0 & 0.628318530718 & 0.0\\\end{pmatrix}\end{equation}



In [13]:

    
H_sub = H.extract_states(keep_states)

H_sub









    Out[13]:





\begin{equation}\text{Quantum object: dims = [[3], [3]], shape = [3, 3], type = oper, isHerm = True}\\[1em]\begin{pmatrix}-12.5663706144 & 0.0 & 0.0\\0.0 & 0.0 & 0.628318530718\\0.0 & 0.628318530718 & 0.0\\\end{pmatrix}\end{equation}



In [14]:

    
energy_level_diagram([H_sub], figsize=(6,6), show_ylabels=True);



In [15]:

    
psi0_sub = psi0.extract_states(keep_states, normalize=True)

psi0_sub









    Out[15]:





\begin{equation}\text{Quantum object: dims = [[3], [1]], shape = [3, 1], type = ket}\\[1em]\begin{pmatrix}0.0\\0.0\\1.0\\\end{pmatrix}\end{equation}



In [16]:

    
c_op1_sub = c_op1.extract_states(keep_states)
c_op2_sub = c_op2.extract_states(keep_states)



In [17]:

    
psi_gg_sub = psi_gg.extract_states(keep_states)
psi_eg_sub = psi_eg.extract_states(keep_states)
psi_ge_sub = psi_ge.extract_states(keep_states)



In [18]:

    
tlist = linspace(0, 15, 100)
result = mesolve(H_sub, psi0_sub, tlist,
                 [sqrt(0.1) * c_op1_sub, sqrt(0.25) * c_op2_sub],
                 [ket2dm(psi_gg_sub), ket2dm(psi_ge_sub), ket2dm(psi_eg_sub)])



In [19]:

    
visualize(result)

Voila! Same results with fewer quantum states in the calculation, and therefore potentially more efficient.

Software versions



In [20]:

    
from qutip.ipynbtools import version_table
version_table()









    Out[20]:




Software Version
Cython 0.19-dev
SciPy 0.13.0.dev-38ad5d2
QuTiP 2.3.0.dev-9e45c8d
Python 2.7.3 (default, Sep 26 2012, 21:51:14) 
[GCC 4.7.2]
IPython 0.13
OS posix [linux2]
Numpy 1.8.0.dev-bd7104c
matplotlib 1.3.x
Wed Mar 13 16:25:43 2013

Software	Version
Cython	0.19-dev
SciPy	0.13.0.dev-38ad5d2
QuTiP	2.3.0.dev-9e45c8d
Python	2.7.3 (default, Sep 26 2012, 21:51:14) [GCC 4.7.2]
IPython	0.13
OS	posix [linux2]
Numpy	1.8.0.dev-bd7104c
matplotlib	1.3.x
Wed Mar 13 16:25:43 2013