Steadystate of the Bloch-Redfield Master Equation

In [1]:

    

%matplotlib inline


    

In [2]:

    

import numpy as np
from qutip import *
from IPython.display import display


    

In [3]:

    

def solve(H, psi0, c_ops, a_ops, S_w, e_ops):
    
    result_me = mesolve(H, psi0, times, c_ops, e_ops)
    result_brme = brmesolve(H, psi0, times, a_ops, e_ops, spectra_cb=S_w)
    
    fig, ax = plot_expectation_values([result_me, result_brme])
    display(fig)
    plt.close(fig)
    
    R, ekets = bloch_redfield_tensor(H, a_ops, S_w)
    print("="* 20 + " Bloch-Redfield tensor: ")
    display(R)
    
    
    L = liouvillian(H, c_ops)
    print("="* 20 + " Lindblad liouvilllian: ")
    display(L)
    
    print("="* 20 + " Bloch-Redfield steadystate dm")
    R_rhoss_eb = steadystate(R)
    R_rhoss = R_rhoss_eb.transform(ekets, True)
    display(R_rhoss)
    
    print("="* 20 + " Lindblad steadystate dm")
    L_rhoss = steadystate(L)
    display(L_rhoss)

    print("="* 20 + " Steadystate expectation values")
    print("R_ob: ", [expect(e, R_rhoss) for e in e_ops])
    print("R_eb: ", [expect(e.transform(ekets), R_rhoss_eb) for e in e_ops])
    print("L   : ", [expect(e, L_rhoss) for e in e_ops])

    print("="* 20 + " Dynamics final states")

    print("R: ", [e[-1].real for e in result_brme.expect])
    print("L: ", [e[-1] for e in result_me.expect])


    

In [4]:

    

delta = 0.0 * 2 * pi
epsilon = 0.5 * 2 * pi
gamma = 0.25
times = np.linspace(0, 50, 100)


    

In [5]:

    

H = delta/2 * sigmay() + epsilon/2 * sigmaz()
psi0 = (2 * basis(2, 0) + basis(2, 1)).unit()
c_ops = [sqrt(gamma) * sigmam()]
a_ops = [sigmax()]
S_w = [lambda w : gamma * (w >= 0)]
e_ops = [sigmax(), sigmay(), sigmaz()]


    

In [6]:

    

solve(H, psi0, c_ops, a_ops, S_w, e_ops)


    

    




/usr/lib/python3/dist-packages/numpy/core/numeric.py:460: ComplexWarning: Casting complex values to real discards the imaginary part
  return array(a, dtype, copy=False, order=order)






    













    




==================== Bloch-Redfield tensor: 






    






Quantum object: dims = [[[2], [2]], [[2], [2]]], shape = [4, 4], type = super, isherm = False\begin{equation*}\begin{pmatrix}0.0 & 0.0 & 0.0 & 0.250\\0.0 & (-0.125-3.142j) & 0.0 & 0.0\\0.0 & 0.0 & (-0.125+3.142j) & 0.0\\0.0 & 0.0 & 0.0 & -0.250\\\end{pmatrix}\end{equation*}






    




==================== Lindblad liouvilllian: 






    






Quantum object: dims = [[[2], [2]], [[2], [2]]], shape = [4, 4], type = super, isherm = False\begin{equation*}\begin{pmatrix}-0.250 & 0.0 & 0.0 & 0.0\\0.0 & (-0.125+3.142j) & 0.0 & 0.0\\0.0 & 0.0 & (-0.125-3.142j) & 0.0\\0.250 & 0.0 & 0.0 & 0.0\\\end{pmatrix}\end{equation*}






    




==================== Bloch-Redfield steadystate dm






    






Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isherm = True\begin{equation*}\begin{pmatrix}0.0 & 0.0\\0.0 & 1.0\\\end{pmatrix}\end{equation*}






    




==================== Lindblad steadystate dm






    






Quantum object: dims = [[2], [2]], shape = [2, 2], type = oper, isherm = True\begin{equation*}\begin{pmatrix}0.0 & 0.0\\0.0 & 1.0\\\end{pmatrix}\end{equation*}






    




==================== Steadystate expectation values
R_ob:  [0.0, 0.0, -1.0]
R_eb:  [0.0, 0.0, -1.0]
L   :  [0.0, 0.0, -1.0]
==================== Dynamics final states
R:  [0.0015427968283087687, 2.506068854864511e-07, -0.999994037354897]
L:  [0.001542796828343541, 2.5060686974779289e-07, -0.99999403735489634]

In [7]:

    

N = 10

w0 = 1.0 * 2 * pi
g = 0.05 * w0
kappa = 0.15

times = np.linspace(0, 50, 1000)


    

In [8]:

    

a = destroy(N)
H = w0 * a.dag() * a + g * (a + a.dag())
psi0 = ket2dm((basis(N, 4) + basis(N, 2) + basis(N,0)).unit())
a_ops = [a + a.dag()]
e_ops = [a.dag() * a, a + a.dag()]


    

In [9]:

    

c_ops = [sqrt(kappa) * a]
S_w = [lambda w : kappa * (w >= 0)]


    

In [10]:

    

solve(H, psi0, c_ops, a_ops, S_w, e_ops)


    

    













    




==================== Bloch-Redfield tensor: 






    






Quantum object: dims = [[[10], [10]], [[10], [10]]], shape = [100, 100], type = super, isherm = False\begin{equation*}\begin{pmatrix}0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 1.926\times10^{-25}\\0.0 & (-0.075-6.283j) & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & (-0.150-12.566j) & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & (-0.225-18.850j) & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & (-1.165+19.005j) & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & (-1.240+12.722j) & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & (-1.312+6.437j) & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & -1.286\\\end{pmatrix}\end{equation*}






    




==================== Lindblad liouvilllian: 






    






Quantum object: dims = [[[10], [10]], [[10], [10]]], shape = [100, 100], type = super, isherm = False\begin{equation*}\begin{pmatrix}0.0 & -0.314j & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\-0.314j & (-0.075-6.283j) & -0.444j & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & -0.444j & (-0.150-12.566j) & -0.544j & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & -0.544j & (-0.225-18.850j) & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & (-1.125+18.850j) & -0.831j & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & -0.831j & (-1.200+12.566j) & -0.889j & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & -0.889j & (-1.275+6.283j) & -0.942j\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & -0.942j & -1.350\\\end{pmatrix}\end{equation*}






    




==================== Bloch-Redfield steadystate dm






    






Quantum object: dims = [[10], [10]], shape = [10, 10], type = oper, isherm = True\begin{equation*}\begin{pmatrix}0.998 & -0.050 & 0.002 & -5.090\times10^{-05} & \cdots & 5.809\times10^{-10} & -1.098\times10^{-11} & 1.939\times10^{-13} & -3.215\times10^{-15}\\-0.050 & 0.002 & -8.817\times10^{-05} & 2.545\times10^{-06} & \cdots & -2.904\times10^{-11} & 5.488\times10^{-13} & -9.694\times10^{-15} & 1.607\times10^{-16}\\0.002 & -8.817\times10^{-05} & 3.117\times10^{-06} & -8.999\times10^{-08} & \cdots & 1.027\times10^{-12} & -1.940\times10^{-14} & 3.427\times10^{-16} & -5.683\times10^{-18}\\-5.090\times10^{-05} & 2.545\times10^{-06} & -8.999\times10^{-08} & 2.598\times10^{-09} & \cdots & -2.964\times10^{-14} & 5.602\times10^{-16} & -9.894\times10^{-18} & 1.641\times10^{-19}\\\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots\\5.809\times10^{-10} & -2.904\times10^{-11} & 1.027\times10^{-12} & -2.964\times10^{-14} & \cdots & 3.382\times10^{-19} & -6.392\times10^{-21} & 1.129\times10^{-22} & -1.872\times10^{-24}\\-1.098\times10^{-11} & 5.488\times10^{-13} & -1.940\times10^{-14} & 5.602\times10^{-16} & \cdots & -6.392\times10^{-21} & 1.208\times10^{-22} & -2.139\times10^{-24} & 3.578\times10^{-26}\\1.939\times10^{-13} & -9.694\times10^{-15} & 3.427\times10^{-16} & -9.894\times10^{-18} & \cdots & 1.129\times10^{-22} & -2.139\times10^{-24} & 3.922\times10^{-26} & -7.811\times10^{-28}\\-3.215\times10^{-15} & 1.607\times10^{-16} & -5.683\times10^{-18} & 1.641\times10^{-19} & \cdots & -1.872\times10^{-24} & 3.578\times10^{-26} & -7.811\times10^{-28} & 1.006\times10^{-28}\\\end{pmatrix}\end{equation*}






    




==================== Lindblad steadystate dm






    






Quantum object: dims = [[10], [10]], shape = [10, 10], type = oper, isherm = True\begin{equation*}\begin{pmatrix}0.998 & (-0.050+5.953\times10^{-04}j) & (0.002-4.208\times10^{-05}j) & (-5.086\times10^{-05}+1.822\times10^{-06}j) & \cdots & (5.791\times10^{-10}-4.155\times10^{-11}j) & -1.093\times10^{-11} & 1.931\times10^{-13} & -3.213\times10^{-15}\\(-0.050-5.953\times10^{-04}j) & 0.002 & (-8.814\times10^{-05}+1.052\times10^{-06}j) & (2.544\times10^{-06}-6.074\times10^{-08}j) & \cdots & (-2.898\times10^{-11}+1.731\times10^{-12}j) & 5.471\times10^{-13} & -9.662\times10^{-15} & 1.608\times10^{-16}\\(0.002+4.208\times10^{-05}j) & (-8.814\times10^{-05}-1.052\times10^{-06}j) & 3.116\times10^{-06} & (-8.995\times10^{-08}+1.074\times10^{-09}j) & \cdots & 1.025\times10^{-12} & -1.936\times10^{-14} & 3.419\times10^{-16} & -5.691\times10^{-18}\\(-5.086\times10^{-05}-1.822\times10^{-06}j) & (2.544\times10^{-06}+6.074\times10^{-08}j) & (-8.995\times10^{-08}-1.074\times10^{-09}j) & 2.597\times10^{-09} & \cdots & -2.960\times10^{-14} & 5.591\times10^{-16} & -9.877\times10^{-18} & 1.644\times10^{-19}\\\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots\\(5.791\times10^{-10}+4.155\times10^{-11}j) & (-2.898\times10^{-11}-1.731\times10^{-12}j) & 1.025\times10^{-12} & -2.960\times10^{-14} & \cdots & 2.950\times10^{-19} & 3.230\times10^{-21} & -6.446\times10^{-22} & 3.730\times10^{-23}\\-1.093\times10^{-11} & 5.471\times10^{-13} & -1.936\times10^{-14} & 5.591\times10^{-16} & \cdots & 3.230\times10^{-21} & -2.375\times10^{-21} & 2.451\times10^{-22} & -1.555\times10^{-23}\\1.931\times10^{-13} & -9.662\times10^{-15} & 3.419\times10^{-16} & -9.877\times10^{-18} & \cdots & -6.446\times10^{-22} & 2.451\times10^{-22} & -2.989\times10^{-23} & 2.228\times10^{-24}\\-3.213\times10^{-15} & 1.608\times10^{-16} & -5.691\times10^{-18} & 1.644\times10^{-19} & \cdots & 3.730\times10^{-23} & -1.555\times10^{-23} & 2.228\times10^{-24} & -1.921\times10^{-25}\\\end{pmatrix}\end{equation*}






    




==================== Steadystate expectation values
R_ob:  [0.0024999999999999628, -0.09999999999999923]
R_eb:  [0.0024999999999999628, -0.09999999999999924]
L   :  [0.0024996438434600815, -0.09998575373839912]
==================== Dynamics final states
R:  [0.0034899627568905876, -0.097648226112854564]
L:  [0.0034896231351718215, -0.097634314487055257]

In [11]:

    

n_th = 1.5
c_ops = [sqrt(kappa * (n_th + 1)) * a, sqrt(kappa * n_th) * a.dag()]
w_th = w0/log(1 + 1/n_th)

def S_w_func(w):
    if w >= 0:
        return (n_th + 1) * kappa
    else:
        return (n_th + 1) * kappa * exp(w / w_th)
    
S_w = [S_w_func]


    

In [12]:

    

solve(H, psi0, c_ops, a_ops, S_w, e_ops)


    

    













    




==================== Bloch-Redfield tensor: 






    






Quantum object: dims = [[[10], [10]], [[10], [10]]], shape = [100, 100], type = super, isherm = False\begin{equation*}\begin{pmatrix}-0.225 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 4.815\times10^{-25}\\0.0 & (-0.525-6.283j) & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & (-0.825-12.566j) & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & (-1.125-18.850j) & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & (-3.700+19.005j) & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & (-3.999+12.722j) & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & (-4.231+6.437j) & 0.0\\4.792\times10^{-27} & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & -3.215\\\end{pmatrix}\end{equation*}






    




==================== Lindblad liouvilllian: 






    






Quantum object: dims = [[[10], [10]], [[10], [10]]], shape = [100, 100], type = super, isherm = False\begin{equation*}\begin{pmatrix}-0.225 & -0.314j & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\-0.314j & (-0.525-6.283j) & -0.444j & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & -0.444j & (-0.825-12.566j) & -0.544j & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & -0.544j & (-1.125-18.850j) & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & (-3.600+18.850j) & -0.831j & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & -0.831j & (-3.900+12.566j) & -0.889j & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & -0.889j & (-4.200+6.283j) & -0.942j\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & -0.942j & -3.375\\\end{pmatrix}\end{equation*}






    




==================== Bloch-Redfield steadystate dm






    






Quantum object: dims = [[10], [10]], shape = [10, 10], type = oper, isherm = True\begin{equation*}\begin{pmatrix}0.402 & -0.008 & 1.137\times10^{-04} & -1.313\times10^{-06} & \cdots & 9.582\times10^{-13} & -7.302\times10^{-15} & -2.331\times10^{-15} & 1.775\times10^{-15}\\-0.008 & 0.241 & -0.007 & 1.182\times10^{-04} & \cdots & -1.726\times10^{-10} & 1.521\times10^{-12} & -9.690\times10^{-15} & 2.836\times10^{-14}\\1.137\times10^{-04} & -0.007 & 0.145 & -0.005 & \cdots & 1.831\times10^{-08} & -1.938\times10^{-10} & 1.827\times10^{-12} & -1.787\times10^{-14}\\-1.313\times10^{-06} & 1.182\times10^{-04} & -0.005 & 0.087 & \cdots & -1.269\times10^{-06} & 1.679\times10^{-08} & -1.899\times10^{-10} & 1.899\times10^{-12}\\\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots\\9.582\times10^{-13} & -1.726\times10^{-10} & 1.831\times10^{-08} & -1.269\times10^{-06} & \cdots & 0.019 & -9.946\times10^{-04} & 2.811\times10^{-05} & -5.619\times10^{-07}\\-7.302\times10^{-15} & 1.521\times10^{-12} & -1.938\times10^{-10} & 1.679\times10^{-08} & \cdots & -9.946\times10^{-04} & 0.011 & -6.382\times10^{-04} & 1.912\times10^{-05}\\-2.331\times10^{-15} & -9.690\times10^{-15} & 1.827\times10^{-12} & -1.899\times10^{-10} & \cdots & 2.811\times10^{-05} & -6.382\times10^{-04} & 0.007 & -4.059\times10^{-04}\\1.775\times10^{-15} & 2.836\times10^{-14} & -1.787\times10^{-14} & 1.899\times10^{-12} & \cdots & -5.619\times10^{-07} & 1.912\times10^{-05} & -4.059\times10^{-04} & 0.004\\\end{pmatrix}\end{equation*}






    




==================== Lindblad steadystate dm






    






Quantum object: dims = [[10], [10]], shape = [10, 10], type = oper, isherm = True\begin{equation*}\begin{pmatrix}0.402 & (-0.008+9.596\times10^{-05}j) & (1.137\times10^{-04}-2.714\times10^{-06}j) & (-1.312\times10^{-06}+4.705\times10^{-08}j) & \cdots & 9.423\times10^{-13} & -4.294\times10^{-15} & -7.474\times10^{-17} & 1.605\times10^{-18}\\(-0.008-9.596\times10^{-05}j) & 0.241 & (-0.007+8.146\times10^{-05}j) & (1.181\times10^{-04}-2.818\times10^{-06}j) & \cdots & (-1.757\times10^{-10}+1.454\times10^{-11}j) & 1.853\times10^{-12} & -2.153\times10^{-14} & 4.078\times10^{-17}\\(1.137\times10^{-04}+2.714\times10^{-06}j) & (-0.007-8.146\times10^{-05}j) & 0.145 & (-0.005+5.991\times10^{-05}j) & \cdots & (1.800\times10^{-08}-7.440\times10^{-10}j) & (-1.758\times10^{-10}+2.574\times10^{-11}j) & (1.790\times10^{-12}-1.344\times10^{-12}j) & -4.383\times10^{-14}\\(-1.312\times10^{-06}-4.705\times10^{-08}j) & (1.181\times10^{-04}+2.818\times10^{-06}j) & (-0.005-5.991\times10^{-05}j) & 0.087 & \cdots & (-1.279\times10^{-06}+4.505\times10^{-08}j) & (1.710\times10^{-08}+1.921\times10^{-10}j) & (-1.533\times10^{-10}-4.118\times10^{-11}j) & -6.294\times10^{-13}\\\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots\\9.423\times10^{-13} & (-1.757\times10^{-10}-1.454\times10^{-11}j) & (1.800\times10^{-08}+7.440\times10^{-10}j) & (-1.279\times10^{-06}-4.505\times10^{-08}j) & \cdots & 0.019 & (-0.001+1.302\times10^{-05}j) & (2.875\times10^{-05}+9.020\times10^{-07}j) & (-5.015\times10^{-07}-5.299\times10^{-08}j)\\-4.294\times10^{-15} & 1.853\times10^{-12} & (-1.758\times10^{-10}-2.574\times10^{-11}j) & (1.710\times10^{-08}-1.921\times10^{-10}j) & \cdots & (-0.001-1.302\times10^{-05}j) & 0.011 & (-6.459\times10^{-04}+2.824\times10^{-05}j) & (2.157\times10^{-05}-9.632\times10^{-07}j)\\-7.474\times10^{-17} & -2.153\times10^{-14} & (1.790\times10^{-12}+1.344\times10^{-12}j) & (-1.533\times10^{-10}+4.118\times10^{-11}j) & \cdots & (2.875\times10^{-05}-9.020\times10^{-07}j) & (-6.459\times10^{-04}-2.824\times10^{-05}j) & 0.007 & (-3.735\times10^{-04}+5.327\times10^{-05}j)\\1.605\times10^{-18} & 4.078\times10^{-17} & -4.383\times10^{-14} & -6.294\times10^{-13} & \cdots & (-5.015\times10^{-07}+5.299\times10^{-08}j) & (2.157\times10^{-05}+9.632\times10^{-07}j) & (-3.735\times10^{-04}-5.327\times10^{-05}j) & 0.004\\\end{pmatrix}\end{equation*}






    




==================== Steadystate expectation values
R_ob:  [1.441172521173383, -0.09593514655834628]
R_eb:  [1.4411725211733832, -0.09593514655834628]
L   :  [1.4415837865925498, -0.09581420257265141]
==================== Dynamics final states
R:  [1.4412697964611789, -0.094946706436501091]
L:  [1.4416807381001973, -0.094807209086397559]

In [13]:

    

N = 10
a = tensor(destroy(N), identity(2))
sm = tensor(identity(N), destroy(2))
psi0 = ket2dm(tensor(basis(N, 1), basis(2, 0)))
a_ops = [(a + a.dag())]
e_ops = [a.dag() * a, sm.dag() * sm]


    

In [14]:

    

w0 = 1.0 * 2 * pi
g = 0.05 * 2 * pi
kappa = 0.05
times = np.linspace(0, 150 * 2 * pi / g, 1000)

c_ops = [sqrt(kappa) * a]
S_w = [lambda w : kappa*(w > 0)]
H = w0 * a.dag() * a + w0 * sm.dag() * sm + g * (a + a.dag()) * (sm + sm.dag())


    

In [15]:

    

solve(H, psi0, c_ops, a_ops, S_w, e_ops)


    

    













    




==================== Bloch-Redfield tensor: 






    






Quantum object: dims = [[[10, 2], [10, 2]], [[10, 2], [10, 2]]], shape = [400, 400], type = super, isherm = False\begin{equation*}\begin{pmatrix}0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & (-0.013-5.969j) & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & (-0.012-6.597j) & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & (-0.039-12.122j) & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & (-0.413+11.758j) & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & (-0.441+7.263j) & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & (-0.436+5.382j) & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & -0.457\\\end{pmatrix}\end{equation*}






    




==================== Lindblad liouvilllian: 






    






Quantum object: dims = [[[10, 2], [10, 2]], [[10, 2], [10, 2]]], shape = [400, 400], type = super, isherm = False\begin{equation*}\begin{pmatrix}0.0 & 0.0 & 0.0 & -0.314j & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & -6.283j & -0.314j & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & -0.314j & (-0.025-6.283j) & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\-0.314j & 0.0 & 0.0 & (-0.025-12.566j) & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & (-0.425+12.566j) & 0.0 & 0.0 & -0.942j\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & (-0.425+6.283j) & -0.942j & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & -0.942j & (-0.450+6.283j) & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & -0.942j & 0.0 & 0.0 & -0.450\\\end{pmatrix}\end{equation*}






    




==================== Bloch-Redfield steadystate dm






    






Quantum object: dims = [[10, 2], [10, 2]], shape = [20, 20], type = oper, isherm = True\begin{equation*}\begin{pmatrix}0.999 & 0.0 & 0.0 & -0.025 & \cdots & 5.302\times10^{-14} & 0.0 & 0.0 & -3.974\times10^{-15}\\0.0 & 1.178\times10^{-16} & 1.068\times10^{-16} & 0.0 & \cdots & 0.0 & -2.567\times10^{-27} & 6.629\times10^{-29} & 0.0\\0.0 & 1.068\times10^{-16} & 1.123\times10^{-16} & 0.0 & \cdots & 0.0 & -9.345\times10^{-26} & 3.427\times10^{-27} & 0.0\\-0.025 & 0.0 & 0.0 & 6.254\times10^{-04} & \cdots & -1.326\times10^{-15} & 0.0 & 0.0 & 9.942\times10^{-17}\\\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots\\5.302\times10^{-14} & 0.0 & 0.0 & -1.326\times10^{-15} & \cdots & 7.156\times10^{-19} & 0.0 & 0.0 & -8.045\times10^{-20}\\0.0 & -2.567\times10^{-27} & -9.345\times10^{-26} & 0.0 & \cdots & 0.0 & 1.109\times10^{-18} & 4.541\times10^{-18} & 0.0\\0.0 & 6.629\times10^{-29} & 3.427\times10^{-27} & 0.0 & \cdots & 0.0 & 4.541\times10^{-18} & 8.001\times10^{-19} & 0.0\\-3.974\times10^{-15} & 0.0 & 0.0 & 9.942\times10^{-17} & \cdots & -8.045\times10^{-20} & 0.0 & 0.0 & 8.080\times10^{-21}\\\end{pmatrix}\end{equation*}






    




==================== Lindblad steadystate dm






    






Quantum object: dims = [[10, 2], [10, 2]], shape = [20, 20], type = oper, isherm = True\begin{equation*}\begin{pmatrix}0.998 & 0.0 & 0.0 & (-0.025+4.974\times10^{-05}j) & \cdots & 5.318\times10^{-14} & 0.0 & 0.0 & -7.975\times10^{-16}\\0.0 & 6.282\times10^{-04} & (-3.128\times10^{-05}+4.971\times10^{-05}j) & 0.0 & \cdots & 0.0 & -2.686\times10^{-16} & 5.694\times10^{-18} & 0.0\\0.0 & (-3.128\times10^{-05}-4.971\times10^{-05}j) & 6.250\times10^{-04} & 0.0 & \cdots & 0.0 & -5.325\times10^{-15} & 9.983\times10^{-17} & 0.0\\(-0.025-4.974\times10^{-05}j) & 0.0 & 0.0 & 6.247\times10^{-04} & \cdots & -1.326\times10^{-15} & 0.0 & 0.0 & 1.984\times10^{-17}\\\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots\\5.318\times10^{-14} & 0.0 & 0.0 & -1.326\times10^{-15} & \cdots & -2.021\times10^{-20} & 0.0 & 0.0 & 1.500\times10^{-21}\\0.0 & -2.686\times10^{-16} & -5.325\times10^{-15} & 0.0 & \cdots & 0.0 & -9.225\times10^{-21} & 7.002\times10^{-22} & 0.0\\0.0 & 5.694\times10^{-18} & 9.983\times10^{-17} & 0.0 & \cdots & 0.0 & 7.002\times10^{-22} & -6.962\times10^{-23} & 0.0\\-7.975\times10^{-16} & 0.0 & 0.0 & 1.984\times10^{-17} & \cdots & 1.500\times10^{-21} & 0.0 & 0.0 & -1.126\times10^{-22}\\\end{pmatrix}\end{equation*}






    




==================== Steadystate expectation values
R_ob:  [0.0006269556893884725, 0.0006253905020129923]
R_eb:  [0.0006269556893884723, 0.0006253905020129925]
L   :  [0.0012531197519054977, 0.0012536697716990645]
==================== Dynamics final states
R:  [0.00062695568938809533, 0.00062539050201271794]
L:  [0.0012531197519057469, 0.0012536697716989808]

In [16]:

    

w0 = 1.0 * 2 * pi
g = 0.75 * 2 * pi
kappa = 0.05
times = np.linspace(0, 150 * 2 * pi / g, 1000)


    

In [17]:

    

c_ops = [sqrt(kappa) * a]
H = w0 * a.dag() * a + w0 * sm.dag() * sm + g * (a + a.dag()) * (sm + sm.dag())


    

In [18]:

    

solve(H, psi0, c_ops, a_ops, S_w, e_ops)


    

    













    




==================== Bloch-Redfield tensor: 






    






Quantum object: dims = [[[10, 2], [10, 2]], [[10, 2], [10, 2]]], shape = [400, 400], type = super, isherm = False\begin{equation*}\begin{pmatrix}0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & (-0.051-1.885j) & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & (-0.052-6.705j) & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & (-0.069-9.264j) & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & (-0.557+16.303j) & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & (-0.681+14.149j) & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & (-0.514+0.513j) & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & -0.948\\\end{pmatrix}\end{equation*}






    




==================== Lindblad liouvilllian: 






    






Quantum object: dims = [[[10, 2], [10, 2]], [[10, 2], [10, 2]]], shape = [400, 400], type = super, isherm = False\begin{equation*}\begin{pmatrix}0.0 & 0.0 & 0.0 & -4.712j & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & -6.283j & -4.712j & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & -4.712j & (-0.025-6.283j) & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\-4.712j & 0.0 & 0.0 & (-0.025-12.566j) & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & (-0.425+12.566j) & 0.0 & 0.0 & -14.137j\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & (-0.425+6.283j) & -14.137j & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & -14.137j & (-0.450+6.283j) & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & -14.137j & 0.0 & 0.0 & -0.450\\\end{pmatrix}\end{equation*}






    




==================== Bloch-Redfield steadystate dm






    






Quantum object: dims = [[10, 2], [10, 2]], shape = [20, 20], type = oper, isherm = True\begin{equation*}\begin{pmatrix}0.795 & 0.0 & 0.0 & -0.345 & \cdots & 1.696\times10^{-04} & 0.0 & 0.0 & -3.695\times10^{-05}\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\-0.345 & 0.0 & 0.0 & 0.150 & \cdots & -7.370\times10^{-05} & 0.0 & 0.0 & 1.606\times10^{-05}\\\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots\\1.696\times10^{-04} & 0.0 & 0.0 & -7.370\times10^{-05} & \cdots & 3.617\times10^{-08} & 0.0 & 0.0 & -7.882\times10^{-09}\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0\\-3.695\times10^{-05} & 0.0 & 0.0 & 1.606\times10^{-05} & \cdots & -7.882\times10^{-09} & 0.0 & 0.0 & 1.717\times10^{-09}\\\end{pmatrix}\end{equation*}






    




==================== Lindblad steadystate dm






    






Quantum object: dims = [[10, 2], [10, 2]], shape = [20, 20], type = oper, isherm = True\begin{equation*}\begin{pmatrix}0.595 & 0.0 & 0.0 & (-0.256+6.823\times10^{-04}j) & \cdots & (1.136\times10^{-04}-3.328\times10^{-06}j) & 0.0 & 0.0 & (-2.435\times10^{-05}+8.046\times10^{-07}j)\\0.0 & 0.073 & (-0.092+6.046\times10^{-04}j) & 0.0 & \cdots & 0.0 & (4.927\times10^{-05}-5.798\times10^{-07}j) & (-1.237\times10^{-05}+1.515\times10^{-07}j) & 0.0\\0.0 & (-0.092-6.046\times10^{-04}j) & 0.129 & 0.0 & \cdots & 0.0 & (-6.825\times10^{-05}+2.628\times10^{-06}j) & (1.592\times10^{-05}-7.139\times10^{-07}j) & 0.0\\(-0.256-6.823\times10^{-04}j) & 0.0 & 0.0 & 0.114 & \cdots & (-7.023\times10^{-05}+1.572\times10^{-06}j) & 0.0 & 0.0 & (1.610\times10^{-05}-4.199\times10^{-07}j)\\\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots\\(1.136\times10^{-04}+3.328\times10^{-06}j) & 0.0 & 0.0 & (-7.023\times10^{-05}-1.572\times10^{-06}j) & \cdots & 3.769\times10^{-06} & 0.0 & 0.0 & (-1.360\times10^{-06}+9.785\times10^{-09}j)\\0.0 & (4.927\times10^{-05}+5.798\times10^{-07}j) & (-6.825\times10^{-05}-2.628\times10^{-06}j) & 0.0 & \cdots & 0.0 & 3.466\times10^{-06} & (-1.444\times10^{-06}+1.217\times10^{-08}j) & 0.0\\0.0 & (-1.237\times10^{-05}-1.515\times10^{-07}j) & (1.592\times10^{-05}+7.139\times10^{-07}j) & 0.0 & \cdots & 0.0 & (-1.444\times10^{-06}-1.217\times10^{-08}j) & 7.649\times10^{-07} & 0.0\\(-2.435\times10^{-05}-8.046\times10^{-07}j) & 0.0 & 0.0 & (1.610\times10^{-05}+4.199\times10^{-07}j) & \cdots & (-1.360\times10^{-06}-9.785\times10^{-09}j) & 0.0 & 0.0 & 6.148\times10^{-07}\\\end{pmatrix}\end{equation*}






    




==================== Steadystate expectation values
R_ob:  [0.2676453077208649, 0.15602055141290413]
R_eb:  [0.2676453077208649, 0.15602055141290413]
L   :  [0.44655609815402086, 0.22225842348795105]
==================== Dynamics final states
R:  [0.2676472156697185, 0.15602106817515557]
L:  [0.44662603050034932, 0.22228388401493518]

In [19]:

    

from qutip.ipynbtools import version_table

version_table()


    

    
Out[19]:




Software
Version
Cython
0.20.1post0
SciPy
0.13.3
QuTiP
3.0.0.dev-c71b506
OS
posix [linux]
IPython
2.0.0
Numpy
1.8.1
matplotlib
1.3.1
Python
3.4.1 (default, Jun  9 2014, 17:34:49) 
[GCC 4.8.3]
Wed Jun 25 16:56:28 2014 JST