def func(rho, h0, c_ops, l_ops):
"""
right-hand side of the master equation
"""
rhs = -1j * commutator(h0, rho)
for i in range(len(c_ops)):
c_op = c_ops[i]
l_op = l_ops[i]
rhs -= commutator(c_op, l_op.dot(rho) - rho.dot(dag(l_op)))
return rhs
def make_lambda(ns, h0, S, T, cutfreq, reorg):
tmax = 1000.0
print(
'time range to numericall compute the bath correlation function = [0, {}]'
.format(tmax))
print('Decay of correlation function at {} fs = {} \n'.format(tmax * au2fs, \
corr(tmax, T, cutfreq, reorg)/corr(0., T, cutfreq, reorg)))
print('!!! Please make sure this is much less than 1 !!!')
time = np.linspace(0, tmax, 10000)
dt = time[1] - time[0]
dt2 = dt / 2.0
l = np.zeros((ns, ns), dtype=np.complex128)
#t = time[0]
#phi = hop * t - Delta * np.sin(omegad * t)/omegad
#Lambda += corr(t) * (np.sin(2. * phi) * sigmay + np.cos(2. * phi) * sigmaz) * dt2
#h0 = Hamiltonian()
Sint = S.copy()
for k in range(len(time)):
t = time[k]
Sint += -1j * commutator(S, h0) * (-dt)
l += dt * Sint * corr(t, T, cutfreq, reorg)
# t = time[len(time)-1]
# phi = hop * t + Delta * np.sin(omegad * t)/omegad
# Lambda += corr(t) * (np.sin(2. * phi) * sigmay + np.cos(2. * phi) * sigmaz) * dt2
# Lambda = cy * sigmay + cz * sigmaz
return l
def _heom_dl(H, rho0, c_ops, e_ops, temperature, cutoff, reorganization,\
nado, dt, nt, fname=None, return_result=True):
'''
terminator : ado[:,:,nado] = 0
INPUT:
T: in units of energy, kB * T, temperature of the bath
reorg: reorganization energy
nado : auxiliary density operators, truncation of the hierachy
fname: file name for output
'''
nst = H.shape[0]
ado = np.zeros((nst, nst, nado),
dtype=np.complex128) # auxiliary density operators
ado[:, :, 0] = rho0 # initial density matrix
gamma = cutoff # cutoff frequency of the environment, larger gamma --> more Makovian
T = temperature / au2k
reorg = reorganization
print('Temperature of the environment = {}'.format(T))
print('High-Temperature check gamma/(kT) = {}'.format(gamma / T))
if gamma / T > 0.8:
print('WARNING: High-Temperature Approximation may fail.')
print('Reorganization energy = {}'.format(reorg))
# D(t) = (a + ib) * exp(- gamma * t)
a = np.pi * reorg * T # initial value of the correlation function D(0) = pi * lambda * kB * T
b = 0.0
print('Amplitude of the fluctuations = {}'.format(a))
#sz = np.zeros((nstate, nstate), dtype=np.complex128)
sz = c_ops # collapse opeartor
f = open(fname, 'w')
fmt = '{} ' * 5 + '\n'
# propagation time loop - HEOM
t = 0.0
for k in range(nt):
t += dt # time increments
ado[:,:,0] += -1j * commutator(H, ado[:,:,0]) * dt - \
commutator(sz, ado[:,:,1]) * dt
for n in range(nado - 1):
ado[:,:,n] += -1j * commutator(H, ado[:,:,n]) * dt + \
(- commutator(sz, ado[:,:,n+1]) - n * gamma * ado[:,:,n] + n * \
(a * commutator(sz, ado[:,:,n-1]) + \
1j * b * anticommutator(sz, ado[:,:,n-1]))) * dt
# store the reduced density matrix
f.write(
fmt.format(t, ado[0, 0, 0], ado[0, 1, 0], ado[1, 0, 0], ado[1, 1,
0]))
#sz += -1j * commutator(sz, H) * dt
f.close()
return ado[:, :, 0]