# This file is part of QuTiP: Quantum Toolbox in Python.
#
# Copyright (c) 2011 and later, Paul D. Nation and Robert J. Johansson.
# All rights reserved.
#
# Redistribution and use in source and binary forms, with or without
# modification, are permitted provided that the following conditions are
# met:
#
# 1. Redistributions of source code must retain the above copyright notice,
# this list of conditions and the following disclaimer.
#
# 2. Redistributions in binary form must reproduce the above copyright
# notice, this list of conditions and the following disclaimer in the
# documentation and/or other materials provided with the distribution.
#
# 3. Neither the name of the QuTiP: Quantum Toolbox in Python nor the names
# of its contributors may be used to endorse or promote products derived
# from this software without specific prior written permission.
#
# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
# "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
# LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
# PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
# HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
# SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
# LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
# DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
# THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
# (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
# OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
###############################################################################
__all__ = ['brmesolve', 'bloch_redfield_solve', 'bloch_redfield_tensor']
import numpy as np
import scipy.integrate
import scipy.sparse as sp
from qutip.qobj import Qobj, isket
from qutip.superoperator import spre, spost, vec2mat, mat2vec, vec2mat_index
from qutip.expect import expect
from qutip.solver import Options, _solver_safety_check
from qutip.cy.spmatfuncs import cy_ode_rhs
from qutip.cy.spconvert import dense2D_to_fastcsr_fmode
from qutip.solver import Result
from qutip.superoperator import liouvillian
from qutip.cy.spconvert import arr_coo2fast
# -----------------------------------------------------------------------------
# Solve the Bloch-Redfield master equation
#
[docs]def brmesolve(H, psi0, tlist, a_ops, e_ops=[], spectra_cb=[], c_ops=[],
args={}, options=Options(),
_safe_mode=True):
"""
Solve the dynamics for a system using the Bloch-Redfield master equation.
.. note::
This solver does not currently support time-dependent Hamiltonians.
Parameters
----------
H : :class:`qutip.Qobj`
System Hamiltonian.
rho0 / psi0: :class:`qutip.Qobj`
Initial density matrix or state vector (ket).
tlist : *list* / *array*
List of times for :math:`t`.
a_ops : list of :class:`qutip.qobj`
List of system operators that couple to bath degrees of freedom.
e_ops : list of :class:`qutip.qobj` / callback function
List of operators for which to evaluate expectation values.
c_ops : list of :class:`qutip.qobj`
List of system collapse operators.
args : *dictionary*
Placeholder for future implementation, kept for API consistency.
options : :class:`qutip.solver.Options`
Options for the solver.
Returns
-------
result: :class:`qutip.solver.Result`
An instance of the class :class:`qutip.solver.Result`, which contains
either an array of expectation values, for operators given in e_ops,
or a list of states for the times specified by `tlist`.
"""
if _safe_mode:
_solver_safety_check(H, psi0, a_ops, e_ops, args)
if not spectra_cb:
# default to infinite temperature white noise
spectra_cb = [lambda w: 1.0 for _ in a_ops]
R, ekets = bloch_redfield_tensor(H, a_ops, spectra_cb, c_ops)
output = Result()
output.solver = "brmesolve"
output.times = tlist
results = bloch_redfield_solve(R, ekets, psi0, tlist, e_ops, options)
if e_ops:
output.expect = results
else:
output.states = results
return output
# -----------------------------------------------------------------------------
# Evolution of the Bloch-Redfield master equation given the Bloch-Redfield
# tensor.
#
[docs]def bloch_redfield_solve(R, ekets, rho0, tlist, e_ops=[], options=None):
"""
Evolve the ODEs defined by Bloch-Redfield master equation. The
Bloch-Redfield tensor can be calculated by the function
:func:`bloch_redfield_tensor`.
Parameters
----------
R : :class:`qutip.qobj`
Bloch-Redfield tensor.
ekets : array of :class:`qutip.qobj`
Array of kets that make up a basis tranformation for the eigenbasis.
rho0 : :class:`qutip.qobj`
Initial density matrix.
tlist : *list* / *array*
List of times for :math:`t`.
e_ops : list of :class:`qutip.qobj` / callback function
List of operators for which to evaluate expectation values.
options : :class:`qutip.Qdeoptions`
Options for the ODE solver.
Returns
-------
output: :class:`qutip.solver`
An instance of the class :class:`qutip.solver`, which contains either
an *array* of expectation values for the times specified by `tlist`.
"""
if options is None:
options = Options()
if options.tidy:
R.tidyup()
#
# check initial state
#
if isket(rho0):
# Got a wave function as initial state: convert to density matrix.
rho0 = rho0 * rho0.dag()
#
# prepare output array
#
n_tsteps = len(tlist)
dt = tlist[1] - tlist[0]
result_list = []
#
# transform the initial density matrix and the e_ops opterators to the
# eigenbasis
#
rho_eb = rho0.transform(ekets)
e_eb_ops = [e.transform(ekets) for e in e_ops]
for e_eb in e_eb_ops:
if e_eb.isherm:
result_list.append(np.zeros(n_tsteps, dtype=float))
else:
result_list.append(np.zeros(n_tsteps, dtype=complex))
#
# setup integrator
#
initial_vector = mat2vec(rho_eb.full())
r = scipy.integrate.ode(cy_ode_rhs)
r.set_f_params(R.data.data, R.data.indices, R.data.indptr)
r.set_integrator('zvode', method=options.method, order=options.order,
atol=options.atol, rtol=options.rtol,
nsteps=options.nsteps, first_step=options.first_step,
min_step=options.min_step, max_step=options.max_step)
r.set_initial_value(initial_vector, tlist[0])
#
# start evolution
#
dt = np.diff(tlist)
for t_idx, _ in enumerate(tlist):
if not r.successful():
break
rho_eb.data = dense2D_to_fastcsr_fmode(vec2mat(r.y), rho0.shape[0], rho0.shape[1])
# calculate all the expectation values, or output rho_eb if no
# expectation value operators are given
if e_ops:
rho_eb_tmp = Qobj(rho_eb)
for m, e in enumerate(e_eb_ops):
result_list[m][t_idx] = expect(e, rho_eb_tmp)
else:
result_list.append(rho_eb.transform(ekets, True))
if t_idx < n_tsteps - 1:
r.integrate(r.t + dt[t_idx])
return result_list
# -----------------------------------------------------------------------------
# Functions for calculating the Bloch-Redfield tensor for a time-independent
# system.
#
[docs]def bloch_redfield_tensor(H, a_ops, spectra_cb, c_ops=[], use_secular=True):
"""
Calculate the Bloch-Redfield tensor for a system given a set of operators
and corresponding spectral functions that describes the system's coupling
to its environment.
.. note::
This tensor generation requires a time-independent Hamiltonian.
Parameters
----------
H : :class:`qutip.qobj`
System Hamiltonian.
a_ops : list of :class:`qutip.qobj`
List of system operators that couple to the environment.
spectra_cb : list of callback functions
List of callback functions that evaluate the noise power spectrum
at a given frequency.
c_ops : list of :class:`qutip.qobj`
List of system collapse operators.
use_secular : bool
Flag (True of False) that indicates if the secular approximation should
be used.
Returns
-------
R, kets: :class:`qutip.Qobj`, list of :class:`qutip.Qobj`
R is the Bloch-Redfield tensor and kets is a list eigenstates of the
Hamiltonian.
"""
# Sanity checks for input parameters
if not isinstance(H, Qobj):
raise TypeError("H must be an instance of Qobj")
for a in a_ops:
if not isinstance(a, Qobj) or not a.isherm:
raise TypeError("Operators in a_ops must be Hermitian Qobj.")
# default spectrum
if not spectra_cb:
spectra_cb = [lambda w: 1.0 for _ in a_ops]
if c_ops is None:
c_ops = []
# use the eigenbasis
evals, ekets = H.eigenstates()
N = len(evals)
K = len(a_ops)
#only Lindblad collapse terms
if K==0:
Heb = H.transform(ekets)
L = liouvillian(Heb, c_ops=[c_op.transform(ekets) for c_op in c_ops])
return L, ekets
A = np.array([a_ops[k].transform(ekets).full() for k in range(K)])
Jw = np.zeros((K, N, N), dtype=complex)
# pre-calculate matrix elements and spectral densities
# W[m,n] = real(evals[m] - evals[n])
W = np.real(evals[:,np.newaxis] - evals[np.newaxis,:])
for k in range(K):
# do explicit loops here in case spectra_cb[k] can not deal with array arguments
for n in range(N):
for m in range(N):
Jw[k, n, m] = spectra_cb[k](W[n, m])
dw_min = np.abs(W[W.nonzero()]).min()
# pre-calculate mapping between global index I and system indices a,b
Iabs = np.empty((N*N,3),dtype=int)
for I, Iab in enumerate(Iabs):
# important: use [:] to change array values, instead of creating new variable Iab
Iab[0] = I
Iab[1:] = vec2mat_index(N, I)
# unitary part + dissipation from c_ops (if given):
Heb = H.transform(ekets)
L = liouvillian(Heb, c_ops=[c_op.transform(ekets) for c_op in c_ops])
# dissipative part:
rows = []
cols = []
data = []
for I, a, b in Iabs:
# only check use_secular once per I
if use_secular:
# only loop over those indices J which actually contribute
Jcds = Iabs[np.where(np.abs(W[a, b] - W[Iabs[:,1], Iabs[:,2]]) < dw_min / 10.0)]
else:
Jcds = Iabs
for J, c, d in Jcds:
elem = 0+0j
# summed over k, i.e., each operator coupling the system to the environment
elem += 0.5 * np.sum(A[:, a, c] * A[:, d, b] * (Jw[:, c, a] + Jw[:, d, b]))
if b==d:
# sum_{k,n} A[k, a, n] * A[k, n, c] * Jw[k, c, n])
elem -= 0.5 * np.sum(A[:, a, :] * A[:, :, c] * Jw[:, c, :])
if a==c:
# sum_{k,n} A[k, d, n] * A[k, n, b] * Jw[k, d, n])
elem -= 0.5 * np.sum(A[:, d, :] * A[:, :, b] * Jw[:, d, :])
if elem != 0:
rows.append(I)
cols.append(J)
data.append(elem)
R = arr_coo2fast(np.array(data, dtype=complex),
np.array(rows, dtype=np.int32),
np.array(cols, dtype=np.int32), N**2, N**2)
L.data = L.data + R
return L, ekets