# This file is part of QuTiP: Quantum Toolbox in Python.
#
# Copyright (c) 2011 and later, Paul D. Nation and Robert J. Johansson.
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###############################################################################
__all__ = ['essolve', 'ode2es']
import numpy as np
import scipy.linalg as la
import scipy.sparse as sp
from qutip.qobj import Qobj, issuper, isket, isoper
from qutip.eseries import eseries, estidy, esval
from qutip.expect import expect
from qutip.superoperator import liouvillian, mat2vec, vec2mat
from qutip.solver import Result
from qutip.operators import qzero
# -----------------------------------------------------------------------------
# pass on to wavefunction solver or master equation solver depending on whether
# any collapse operators were given.
#
[docs]def essolve(H, rho0, tlist, c_op_list, e_ops):
"""
Evolution of a state vector or density matrix (`rho0`) for a given
Hamiltonian (`H`) and set of collapse operators (`c_op_list`), by
expressing the ODE as an exponential series. The output is either
the state vector at arbitrary points in time (`tlist`), or the
expectation values of the supplied operators (`e_ops`).
Parameters
----------
H : qobj/function_type
System Hamiltonian.
rho0 : :class:`qutip.qobj`
Initial state density matrix.
tlist : list/array
``list`` of times for :math:`t`.
c_op_list : list of :class:`qutip.qobj`
``list`` of :class:`qutip.qobj` collapse operators.
e_ops : list of :class:`qutip.qobj`
``list`` of :class:`qutip.qobj` operators for which to evaluate
expectation values.
Returns
-------
expt_array : array
Expectation values of wavefunctions/density matrices for the
times specified in ``tlist``.
.. note:: This solver does not support time-dependent Hamiltonians.
"""
n_expt_op = len(e_ops)
n_tsteps = len(tlist)
# Calculate the Liouvillian
if (c_op_list is None or len(c_op_list) == 0) and isket(rho0):
L = H
else:
L = liouvillian(H, c_op_list)
es = ode2es(L, rho0)
# evaluate the expectation values
if n_expt_op == 0:
results = [Qobj()] * n_tsteps
else:
results = np.zeros([n_expt_op, n_tsteps], dtype=complex)
for n, e in enumerate(e_ops):
results[n, :] = expect(e, esval(es, tlist))
data = Result()
data.solver = "essolve"
data.times = tlist
data.expect = [np.real(results[n, :]) if e.isherm else results[n, :]
for n, e in enumerate(e_ops)]
return data
# -----------------------------------------------------------------------------
#
#
[docs]def ode2es(L, rho0):
"""Creates an exponential series that describes the time evolution for the
initial density matrix (or state vector) `rho0`, given the Liouvillian
(or Hamiltonian) `L`.
Parameters
----------
L : qobj
Liouvillian of the system.
rho0 : qobj
Initial state vector or density matrix.
Returns
-------
eseries : :class:`qutip.eseries`
``eseries`` represention of the system dynamics.
"""
if issuper(L):
# check initial state
if isket(rho0):
# Got a wave function as initial state: convert to density matrix.
rho0 = rho0 * rho0.dag()
# check if state is below error threshold
if abs(rho0.full().sum()) < 1e-10 + 1e-24:
# enforce zero operator
return eseries(qzero(rho0.dims[0]))
w, v = L.eigenstates()
v = np.hstack([ket.full() for ket in v])
# w[i] = eigenvalue i
# v[:,i] = eigenvector i
rlen = np.prod(rho0.shape)
r0 = mat2vec(rho0.full())
v0 = la.solve(v, r0)
vv = v * sp.spdiags(v0.T, 0, rlen, rlen)
out = None
for i in range(rlen):
qo = Qobj(vec2mat(vv[:, i]), dims=rho0.dims, shape=rho0.shape)
if out:
out += eseries(qo, w[i])
else:
out = eseries(qo, w[i])
elif isoper(L):
if not isket(rho0):
raise TypeError('Second argument must be a ket if first' +
'is a Hamiltonian.')
# check if state is below error threshold
if abs(rho0.full().sum()) < 1e-5 + 1e-20:
# enforce zero operator
dims = rho0.dims
return eseries(Qobj(sp.csr_matrix((dims[0][0], dims[1][0]),
dtype=complex)))
w, v = L.eigenstates()
v = np.hstack([ket.full() for ket in v])
# w[i] = eigenvalue i
# v[:,i] = eigenvector i
rlen = np.prod(rho0.shape)
r0 = rho0.full()
v0 = la.solve(v, r0)
vv = v * sp.spdiags(v0.T, 0, rlen, rlen)
out = None
for i in range(rlen):
qo = Qobj(np.matrix(vv[:, i]).T, dims=rho0.dims, shape=rho0.shape)
if out:
out += eseries(qo, -1.0j * w[i])
else:
out = eseries(qo, -1.0j * w[i])
else:
raise TypeError('First argument must be a Hamiltonian or Liouvillian.')
return estidy(out)