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# @author: Arne L. Grimsmo
# @email1: arne.grimsmo@gmail.com
# @organization: University of Sherbrooke
"""
This module contains an implementation of the non-Markovian transfer tensor
method (TTM), introduced in [1].
[1] Javier Cerrillo and Jianshu Cao, Phys. Rev. Lett 112, 110401 (2014)
"""
import numpy as np
from qutip import (Options, spre, vector_to_operator, operator_to_vector,
ket2dm, isket)
from qutip.solver import Result
from qutip.expect import expect_rho_vec
[docs]class TTMSolverOptions:
"""Class of options for the Transfer Tensor Method solver.
Attributes
----------
dynmaps : list of :class:`qutip.Qobj`
List of precomputed dynamical maps (superoperators),
or a callback function that returns the
superoperator at a given time.
times : array_like
List of times :math:`t_n` at which to calculate :math:`\\rho(t_n)`
learningtimes : array_like
List of times :math:`t_k` to use as learning times if argument
`dynmaps` is a callback function.
thres : float
Threshold for halting. Halts if :math:`||T_{n}-T_{n-1}||` is below
treshold.
options : :class:`qutip.solver.Options`
Generic solver options.
"""
def __init__(self, dynmaps=None, times=[], learningtimes=[],
thres=0.0, options=None):
if options is None:
options = Options()
self.dynmaps = dynmaps
self.times = times
self.learningtimes = learningtimes
self.thres = thres
self.store_states = options.store_states
[docs]def ttmsolve(dynmaps, rho0, times, e_ops=[], learningtimes=None, tensors=None,
**kwargs):
"""
Solve time-evolution using the Transfer Tensor Method, based on a set of
precomputed dynamical maps.
Parameters
----------
dynmaps : list of :class:`qutip.Qobj`
List of precomputed dynamical maps (superoperators),
or a callback function that returns the
superoperator at a given time.
rho0 : :class:`qutip.Qobj`
Initial density matrix or state vector (ket).
times : array_like
list of times :math:`t_n` at which to compute :math:`\\rho(t_n)`.
Must be uniformily spaced.
e_ops : list of :class:`qutip.Qobj` / callback function
single operator or list of operators for which to evaluate
expectation values.
learningtimes : array_like
list of times :math:`t_k` for which we have knowledge of the dynamical
maps :math:`E(t_k)`.
tensors : array_like
optional list of precomputed tensors :math:`T_k`
kwargs : dictionary
Optional keyword arguments. See
:class:`qutip.nonmarkov.ttm.TTMSolverOptions`.
Returns
-------
output: :class:`qutip.solver.Result`
An instance of the class :class:`qutip.solver.Result`.
"""
opt = TTMSolverOptions(dynmaps=dynmaps, times=times,
learningtimes=learningtimes, **kwargs)
diff = None
if isket(rho0):
rho0 = ket2dm(rho0)
output = Result()
e_sops_data = []
if callable(e_ops):
n_expt_op = 0
expt_callback = True
else:
try:
tmp = e_ops[:]
del tmp
n_expt_op = len(e_ops)
expt_callback = False
if n_expt_op == 0:
# fall back on storing states
opt.store_states = True
for op in e_ops:
e_sops_data.append(spre(op).data)
if op.isherm and rho0.isherm:
output.expect.append(np.zeros(len(times)))
else:
output.expect.append(np.zeros(len(times), dtype=complex))
except TypeError:
raise TypeError("Argument 'e_ops' should be a callable or" +
"list-like.")
if tensors is None:
tensors, diff = _generatetensors(dynmaps, learningtimes, opt=opt)
rho0vec = operator_to_vector(rho0)
K = len(tensors)
states = [rho0vec]
for n in range(1, len(times)):
states.append(None)
for k in range(n):
if n-k < K:
states[-1] += tensors[n-k]*states[k]
for i, r in enumerate(states):
if opt.store_states or expt_callback:
if r.type == 'operator-ket':
states[i] = vector_to_operator(r)
else:
states[i] = r
if expt_callback:
# use callback method
e_ops(times[i], states[i])
for m in range(n_expt_op):
if output.expect[m].dtype == complex:
output.expect[m][i] = expect_rho_vec(e_sops_data[m], r, 0)
else:
output.expect[m][i] = expect_rho_vec(e_sops_data[m], r, 1)
output.solver = "ttmsolve"
output.times = times
output.ttmconvergence = diff
if opt.store_states:
output.states = states
return output
def _generatetensors(dynmaps, learningtimes=None, **kwargs):
"""
Generate the tensors :math:`T_1,\dots,T_K` from the dynamical maps
:math:`E(t_k)`.
A stationary process is assumed, i.e., :math:`T_{n,k} = T_{n-k}`.
Parameters
----------
dynmaps : list of :class:`qutip.Qobj`
List of precomputed dynamical maps (superoperators) at the times
specified in `learningtimes`, or a callback function that returns the
superoperator at a given time.
learningtimes : array_like
list of times :math:`t_k` to use if argument `dynmaps` is a callback
function.
kwargs : dictionary
Optional keyword arguments. See
:class:`qutip.nonmarkov.ttm.TTMSolverOptions`.
Returns
-------
Tlist: list of :class:`qutip.Qobj.`
A list of transfer tensors :math:`T_1,\dots,T_K`
"""
# Determine if dynmaps is callable or list-like
if callable(dynmaps):
if learningtimes is None:
raise TypeError("Argument 'learnintimes' required when 'dynmaps'" +
"is a callback function.")
def dynmapfunc(n): return dynmaps(learningtimes[n])
Kmax = len(learningtimes)
else:
try:
tmp = dynmaps[:]
del tmp
def dynmapfunc(n): return dynmaps[n]
Kmax = len(dynmaps)
except TypeError:
raise TypeError("Argument 'dynmaps' should be a callable or" +
"list-like.")
if "opt" not in kwargs:
opt = TTMSolverOptions(dynmaps=dynmaps, learningtimes=learningtimes,
**kwargs)
else:
opt = kwargs['opt']
Tlist = []
diff = [0.0]
for n in range(Kmax):
T = dynmapfunc(n)
for m in range(1, n):
T -= Tlist[n-m]*dynmapfunc(m)
Tlist.append(T)
if n > 1:
diff.append((Tlist[-1]-Tlist[-2]).norm())
if diff[-1] < opt.thres:
# Below threshold for truncation
print('breaking', (Tlist[-1]-Tlist[-2]).norm(), n)
break
return Tlist, diff