Source code for qutip.essolve

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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)