__all__ = [
"FloquetBasis",
"floquet_tensor",
"fsesolve",
"fmmesolve",
"FMESolver",
]
import numpy as np
from qutip.core import data as _data
from qutip.core.data import Data
from qutip import Qobj, QobjEvo
from .propagator import Propagator
from .mesolve import MESolver
from .solver_base import Solver
from .integrator import Integrator
from .result import Result
from time import time
from ..ui.progressbar import progress_bars
[docs]class FloquetBasis:
"""
Utility to compute floquet modes and states.
Attributes
----------
U : :class:`.Propagator`
The propagator of the Hamiltonian over one period.
evecs : :class:`.Data`
Matrix where each column is an initial Floquet mode.
e_quasi : np.ndarray[float]
The quasi energies of the Hamiltonian.
"""
def __init__(
self,
H,
T,
args=None,
options=None,
sparse=False,
sort=True,
precompute=None,
):
"""
Parameters
----------
H : :obj:`.Qobj`, :obj:`.QobjEvo`, QobjEvo compatible format.
System Hamiltonian, with period `T`.
T : float
Period of the Hamiltonian.
args : None / *dictionary*
dictionary of parameters for time-dependent Hamiltonians and
collapse operators.
options : dict [None]
Options used by sesolve to compute the floquet modes.
sparse : bool [False]
Whether to use the sparse eigen solver when computing the
quasi-energies.
sort : bool [True]
Whether to sort the quasi-energies.
precompute : list [None]
If provided, a list of time at which to store the propagators
for later use when computing modes and states. Default is
``linspace(0, T, 101)`` corresponding to the default integration
steps used for the floquet tensor computation.
"""
if not T > 0:
raise ValueError("The period need to be a positive number.")
self.T = T
if precompute is not None:
tlist = np.unique(np.atleast_1d(precompute) % self.T)
memoize = len(tlist)
if tlist[0] != 0:
memoize += 1
if tlist[-1] != T:
memoize += 1
else:
# Default computation
tlist = np.linspace(0, T, 101)
memoize = 101
if (
isinstance(H, QobjEvo)
and (H._feedback_functions or H._solver_only_feedback)
):
raise NotImplementedError("FloquetBasis does not support feedback")
self.U = Propagator(H, args=args, options=options, memoize=memoize)
for t in tlist:
# Do the evolution by steps to save the intermediate results.
self.U(t)
U_T = self.U(self.T)
if not sparse and isinstance(U_T.data, _data.CSR):
U_T = U_T.to("Dense")
evals, evecs = _data.eigs(U_T.data)
e_quasi = -np.angle(evals) / T
if sort:
perm = np.argsort(e_quasi)
self.evecs = _data.permute.indices(evecs, col_perm=np.argsort(perm))
self.e_quasi = e_quasi[perm]
else:
self.evecs = evecs
self.e_quasi = e_quasi
def _as_ketlist(self, kets_mat):
"""
Split the Data array in a list of kets.
"""
dims = [self.U(0).dims[0], [1]]
return [
Qobj(ket, dims=dims)
for ket in _data.split_columns(kets_mat)
]
[docs] def mode(self, t, data=False):
"""
Calculate the Floquet modes at time ``t``.
Parameters
----------
t : float
The time for which to evaluate the Floquet mode.
data : bool [False]
Whether to return the states as a single data matrix or a list of
ket states.
Returns
-------
output : list[:obj:`.Qobj`], :class:`.Data`
A list of Floquet states for the time ``t`` or the states as column
in a single matrix.
"""
t = t % self.T
if t == 0.0:
kets_mat = self.evecs
else:
U = self.U(t).data
phases = _data.diag(np.exp(1j * t * self.e_quasi))
kets_mat = U @ self.evecs @ phases
if data:
return kets_mat
else:
return self._as_ketlist(kets_mat)
[docs] def state(self, t, data=False):
"""
Evaluate the floquet states at time t.
Parameters
----------
t : float
The time for which to evaluate the Floquet states.
data : bool [False]
Whether to return the states as a single data matrix or a list of
ket states.
Returns
-------
output : list[:obj:`.Qobj`], :class:`.Data`
A list of Floquet states for the time ``t`` or the states as column
in a single matrix.
"""
if t:
phases = _data.diag(np.exp(-1j * t * self.e_quasi))
states_mat = self.mode(t, True) @ phases
else:
states_mat = self.evecs
if data:
return states_mat
else:
return self._as_ketlist(states_mat)
[docs] def from_floquet_basis(self, floquet_basis, t=0):
"""
Transform a ket or density matrix from the Floquet basis at time ``t``
to the lab basis.
Parameters
----------
floquet_basis : :obj:`.Qobj`, :class:`.Data`
Initial state in the Floquet basis at time ``t``. May be either a
ket or density matrix.
t : float [0]
The time at which to evaluate the Floquet states.
Returns
-------
output : :obj:`.Qobj`, :class:`.Data`
The state in the lab basis. The return type is the same as the type
of the input state.
"""
is_Qobj = isinstance(floquet_basis, Qobj)
if is_Qobj:
dims = floquet_basis.dims
floquet_basis = floquet_basis.data
if dims[0] != self.U(0).dims[1]:
raise ValueError(
"Dimensions of the state do not match the Hamiltonian"
)
state_mat = self.state(t, True)
lab_basis = state_mat @ floquet_basis
if floquet_basis.shape[1] != 1:
lab_basis = lab_basis @ state_mat.adjoint()
if is_Qobj:
return Qobj(lab_basis, dims=dims)
return lab_basis
[docs] def to_floquet_basis(self, lab_basis, t=0):
"""
Transform a ket or density matrix in the lab basis
to the Floquet basis at time ``t``.
Parameters
----------
lab_basis : :obj:`.Qobj`, :class:`.Data`
Initial state in the lab basis.
t : float [0]
The time at which to evaluate the Floquet states.
Returns
-------
output : :obj:`.Qobj`, :class:`.Data`
The state in the Floquet basis. The return type is the same as the
type of the input state.
"""
is_Qobj = isinstance(lab_basis, Qobj)
if is_Qobj:
dims = lab_basis.dims
lab_basis = lab_basis.data
if dims[0] != self.U(0).dims[1]:
raise ValueError(
"Dimensions of the state do not match the Hamiltonian"
)
state_mat = self.state(t, True)
floquet_basis = state_mat.adjoint() @ lab_basis
if lab_basis.shape[1] != 1:
floquet_basis = floquet_basis @ state_mat
if is_Qobj:
return Qobj(floquet_basis, dims=dims)
return floquet_basis
def _floquet_delta_tensor(f_energies, kmax, T):
"""
Floquet-Markov master equation X matrices.
Parameters
----------
f_energies : np.ndarray
The Floquet energies.
kmax : int
The truncation of the number of sidebands (default 5).
T : float
The period of the time-dependence of the Hamiltonian.
Returns
-------
delta : np.ndarray
Floquet delta tensor.
"""
delta = np.subtract.outer(f_energies, f_energies)
return np.add.outer(delta, np.arange(-kmax, kmax + 1) * (2 * np.pi / T))
def _floquet_X_matrices(floquet_basis, c_ops, kmax, ntimes=100):
"""
Floquet-Markov master equation X matrices.
Parameters
----------
floquet_basis : :class:`FloquetBasis`
The system Hamiltonian wrapped in a FloquetBasis object.
c_ops : list of :obj:`.Qobj`
The collapse operators describing the dissipation.
kmax : int
The truncation of the number of sidebands (default 5).
ntimes : int [100]
The number of integration steps (for calculating X) within one period.
Returns
-------
X : list of dict of :class:`.Data`
A dict of the sidebands ``k`` for the X matrices of each c_ops
"""
T = floquet_basis.T
N = floquet_basis.U(0).shape[0]
omega = (2 * np.pi) / T
tlist = np.linspace(T / ntimes, T, ntimes)
ks = np.arange(-kmax, kmax + 1)
out = {k: [_data.csr.zeros(N, N)] * len(c_ops) for k in ks}
for t in tlist:
mode = floquet_basis.mode(t, data=True)
FFs = [mode.adjoint() @ c_op.data @ mode for c_op in c_ops]
for k, phi in zip(ks, np.exp(-1j * ks * omega * t) / ntimes):
out[k] = [
_data.add(prev, new, phi) for prev, new in zip(out[k], FFs)
]
return [{k: out[k][i] for k in ks} for i in range(len(c_ops))]
def _floquet_gamma_matrices(X, delta, J_cb):
"""
Floquet-Markov master equation gamma matrices.
Parameters
----------
X : list of dict of :class:`.Data`
Floquet X matrices created by :func:`_floquet_X_matrices`.
delta : np.ndarray
Floquet delta tensor created by :func:`_floquet_delta_tensor`.
J_cb : list of callables
A list callback functions that compute the noise power spectrum as
a function of frequency. The list should contain one callable for each
collapse operator `c_op`, in the same order as the elements of `X`.
Each callable should accept a numpy array of frequencies and return a
numpy array of corresponding noise power.
Returns
-------
gammas : dict of :class:`.Data`
A dict mapping the sidebands ``k`` to their gamma matrices.
"""
N = delta.shape[0]
kmax = (delta.shape[2] - 1) // 2
gamma = {k: _data.csr.zeros(N, N) for k in range(-kmax, kmax + 1, 1)}
for X_c_op, sp in zip(X, J_cb):
response = sp(delta) * ((2 + 0j) * np.pi)
response = [
_data.Dense(response[:, :, k], copy=False)
for k in range(2 * kmax + 1)
]
for k in range(-kmax, kmax + 1, 1):
gamma[k] = _data.add(
gamma[k],
_data.multiply(
_data.multiply(X_c_op[k].conj(), X_c_op[k]),
response[k + kmax],
),
)
return gamma
def _floquet_A_matrix(delta, gamma, w_th):
"""
Floquet-Markov master equation rate matrix.
Parameters
----------
delta : np.ndarray
Floquet delta tensor created by :func:`_floquet_delta_tensor`.
gamma : dict of :class:`.Data`
Floquet gamma matrices created by :func:`_floquet_gamma_matrices`.
w_th : float
The temperature in units of frequency.
"""
kmax = (delta.shape[2] - 1) // 2
if w_th > 0.0:
deltap = np.copy(delta)
deltap[deltap == 0.0] = np.inf
thermal = 1.0 / (np.exp(np.abs(deltap) / w_th) - 1.0)
thermal = [_data.Dense(thermal[:, :, k]) for k in range(2 * kmax + 1)]
gamma_kk = _data.add(gamma[0], gamma[0].transpose())
A = _data.add(gamma[0], _data.multiply(thermal[kmax], gamma_kk))
for k in range(1, kmax + 1):
g_kk = _data.add(gamma[k], gamma[-k].transpose())
thermal_kk = _data.multiply(thermal[kmax + k], g_kk)
A = _data.add(A, _data.add(gamma[k], thermal_kk))
thermal_kk = _data.multiply(thermal[kmax - k], g_kk.transpose())
A = _data.add(A, _data.add(gamma[-k], thermal_kk))
else:
# w_th is 0, thermal = 0s
A = gamma[0]
for k in range(1, kmax + 1):
A = _data.add(gamma[k], A)
A = _data.add(gamma[-k], A)
return A
def _floquet_master_equation_tensor(A):
"""
Construct a tensor that represents the master equation in the floquet
basis (with constant Hamiltonian and collapse operators?).
Simplest RWA approximation [Grifoni et al, Phys.Rep. 304 229 (1998)]
Parameters
----------
A : :class:`.Data`
Floquet-Markov master equation rate matrix.
Returns
-------
output : array
The Floquet-Markov master equation tensor `R`.
"""
N = A.shape[0]
# R[i+N*i, j+N*j] = A[j, i]
cols = np.arange(N, dtype=np.int32)
rows = np.linspace(1 - 1 / (N + 1), N, N**2 + 1, dtype=np.int32)
data = np.ones(N, dtype=complex)
expand = _data.csr.CSR((data, cols, rows), shape=(N**2, N))
R = expand @ A.transpose() @ expand.transpose()
# S[i+N*j, j+N*i] = -1/2 * sum_k(A[i, k] + A[j, k])
ket_1 = _data.Dense(np.ones(N, dtype=complex))
Asum = A @ ket_1
to_super = _data.add(_data.kron(Asum, ket_1), _data.kron(ket_1, Asum))
S = _data.diag(to_super.to_array().flatten() * -0.5, 0)
return _data.add(R, S)
[docs]def floquet_tensor(H, c_ops, spectra_cb, T=0, w_th=0.0, kmax=5, nT=100):
"""
Construct a tensor that represents the master equation in the floquet
basis.
Simplest RWA approximation [Grifoni et al, Phys.Rep. 304 229 (1998)]
Parameters
----------
H : :obj:`.QobjEvo`, :obj:`.FloquetBasis`
Periodic Hamiltonian a floquet basis system.
T : float, optional
The period of the time-dependence of the hamiltonian. Optional if ``H``
is a ``FloquetBasis`` object.
c_ops : list of :class:`.Qobj`
list of collapse operators.
spectra_cb : list callback functions
List of callback functions that compute the noise power spectrum as
a function of frequency for the collapse operators in `c_ops`.
w_th : float, default: 0.0
The temperature in units of frequency.
kmax : int, default: 5
The truncation of the number of sidebands (default 5).
nT : int, default: 100
The number of integration steps (for calculating X) within one period.
Returns
-------
output : array
The Floquet-Markov master equation tensor `R`.
"""
if isinstance(H, FloquetBasis):
floquet_basis = H
T = H.T
else:
floquet_basis = FloquetBasis(H, T)
energy = floquet_basis.e_quasi
delta = _floquet_delta_tensor(energy, kmax, T)
x = _floquet_X_matrices(floquet_basis, c_ops, kmax, nT)
gamma = _floquet_gamma_matrices(x, delta, spectra_cb)
a = _floquet_A_matrix(delta, gamma, w_th)
r = _floquet_master_equation_tensor(a)
dims = floquet_basis.U(0).dims
return Qobj(r, dims=[dims, dims], superrep="super", copy=False)
[docs]def fsesolve(H, psi0, tlist, e_ops=None, T=0.0, args=None, options=None):
"""
Solve the Schrodinger equation using the Floquet formalism.
Parameters
----------
H : :obj:`.Qobj`, :obj:`.QobjEvo`, :obj:`.QobjEvo` compatible format.
Periodic system Hamiltonian as :obj:`.QobjEvo`. List of
[:obj:`.Qobj`, :obj:`.Coefficient`] or callable that
can be made into :obj:`.QobjEvo` are also accepted.
psi0 : :class:`.Qobj`
Initial state vector (ket). If an operator is provided,
tlist : *list* / *array*
List of times for :math:`t`.
e_ops : list of :class:`.Qobj` / callback function, optional
List of operators for which to evaluate expectation values. If this
list is empty, the state vectors for each time in `tlist` will be
returned instead of expectation values.
T : float, default=tlist[-1]
The period of the time-dependence of the hamiltonian.
args : dictionary, optional
Dictionary with variables required to evaluate H.
options : dict, optional
Options for the results.
- | store_final_state : bool
| Whether or not to store the final state of the evolution in the
result class.
- | store_states : bool, None
| Whether or not to store the state vectors or density matrices.
On `None` the states will be saved if no expectation operators are
given.
- | normalize_output : bool
| Normalize output state to hide ODE numerical errors.
Returns
-------
output : :class:`.Result`
An instance of the class :class:`.Result`, which
contains either an *array* of expectation values or an array of
state vectors, for the times specified by `tlist`.
"""
if isinstance(H, FloquetBasis):
floquet_basis = H
else:
T = T or tlist[-1]
# `fsesolve` is a fallback from `fmmesolve`, for the later, options
# are for the open system evolution.
floquet_basis = FloquetBasis(H, T, args, precompute=tlist)
f_coeff = floquet_basis.to_floquet_basis(psi0)
result_options = {
"store_final_state": False,
"store_states": None,
"normalize_output": True,
}
result_options.update(options or {})
result = Result(e_ops, result_options, solver="fsesolve")
for t in tlist:
state_t = floquet_basis.from_floquet_basis(f_coeff, t)
result.add(t, state_t)
return result
[docs]def fmmesolve(
H,
rho0,
tlist,
c_ops=None,
e_ops=None,
spectra_cb=None,
T=0,
w_th=0.0,
args=None,
options=None,
):
"""
Solve the dynamics for the system using the Floquet-Markov master equation.
Parameters
----------
H : :obj:`.Qobj`, :obj:`.QobjEvo`, :obj:`.QobjEvo` compatible format.
Periodic system Hamiltonian as :obj:`.QobjEvo`. List of
[:obj:`.Qobj`, :obj:`.Coefficient`] or callable that
can be made into :obj:`.QobjEvo` are also accepted.
rho0 / psi0 : :class:`.Qobj`
Initial density matrix or state vector (ket).
tlist : *list* / *array*
List of times for :math:`t`.
c_ops : list of :class:`.Qobj`, optional
List of collapse operators. Time dependent collapse operators are not
supported. Fall back on :func:`fsesolve` if not provided.
e_ops : list of :class:`.Qobj` / callback function, optional
List of operators for which to evaluate expectation values.
The states are reverted to the lab basis before applying the
spectra_cb : list callback functions, default: ``lambda w: (w > 0)``
List of callback functions that compute the noise power spectrum as
a function of frequency for the collapse operators in `c_ops`.
T : float, default=tlist[-1]
The period of the time-dependence of the hamiltonian. The default value
``0`` indicates that the 'tlist' spans a single period of the driving.
w_th : float, default: 0.0
The temperature of the environment in units of frequency.
For example, if the Hamiltonian written in units of 2pi GHz, and the
temperature is given in K, use the following conversion:
temperature = 25e-3 # unit K
h = 6.626e-34
kB = 1.38e-23
args['w_th'] = temperature * (kB / h) * 2 * pi * 1e-9
args : dict, optional
Dictionary of parameters for time-dependent Hamiltonian
options : dict, optional
Dictionary of options for the solver.
- | store_final_state : bool
| Whether or not to store the final state of the evolution in the
result class.
- | store_states : bool, None
| Whether or not to store the state vectors or density matrices.
On `None` the states will be saved if no expectation operators are
given.
- | store_floquet_states : bool
| Whether or not to store the density matrices in the floquet basis
in ``result.floquet_states``.
- | normalize_output : bool
| Normalize output state to hide ODE numerical errors.
- | progress_bar : str {'text', 'enhanced', 'tqdm', ''}
| How to present the solver progress.
'tqdm' uses the python module of the same name and raise an error
if not installed. Empty string or False will disable the bar.
- | progress_kwargs : dict
| kwargs to pass to the progress_bar. Qutip's bars use `chunk_size`.
- | method : str ["adams", "bdf", "lsoda", "dop853", "vern9", etc.]
| Which differential equation integration method to use.
- | atol, rtol : float
| Absolute and relative tolerance of the ODE integrator.
- | nsteps : int
| Maximum number of (internally defined) steps allowed in one
``tlist`` step.
- | max_step : float
| Maximum lenght of one internal step. When using pulses, it should
be less than half the width of the thinnest pulse.
Other options could be supported depending on the integration method,
see `Integrator <./classes.html#classes-ode>`_.
Returns
-------
result: :class:`.Result`
An instance of the class :class:`.Result`, which contains
the expectation values for the times specified by ``tlist``, and/or the
state density matrices corresponding to the times.
"""
if c_ops is None and rho0.isket:
return fsesolve(
H,
rho0,
tlist,
e_ops=e_ops,
T=T,
args=args,
options=options,
)
if isinstance(H, FloquetBasis):
floquet_basis = H
else:
T = T or tlist[-1]
t_precompute = np.concatenate([tlist, np.linspace(0, T, 101)])
# `fsesolve` is a fallback from `fmmesolve`, for the later, options
# are for the open system evolution.
floquet_basis = FloquetBasis(H, T, args, precompute=t_precompute)
if not w_th and args:
w_th = args.get("w_th", 0.0)
if isinstance(c_ops, Qobj):
c_ops = [c_ops]
if spectra_cb is None:
spectra_cb = [lambda w: (w > 0)]
elif callable(spectra_cb):
spectra_cb = [spectra_cb]
if len(spectra_cb) == 1:
spectra_cb = spectra_cb * len(c_ops)
a_ops = list(zip(c_ops, spectra_cb))
solver = FMESolver(floquet_basis, a_ops, w_th=w_th, options=options)
return solver.run(rho0, tlist, e_ops=e_ops)
class FloquetResult(Result):
def _post_init(self, floquet_basis):
self.floquet_basis = floquet_basis
if self.options["store_floquet_states"]:
self.floquet_states = []
else:
self.floquet_states = None
super()._post_init()
def add(self, t, state):
if self.options["store_floquet_states"]:
self.floquet_states.append(state)
state = self.floquet_basis.from_floquet_basis(state, t)
super().add(t, state)
[docs]class FMESolver(MESolver):
"""
Solver for the Floquet-Markov master equation.
.. note ::
Operators (``c_ops`` and ``e_ops``) are in the laboratory basis.
Parameters
----------
floquet_basis : :class:`.FloquetBasis`
The system Hamiltonian wrapped in a FloquetBasis object. Choosing a
different integrator for the ``floquet_basis`` than for the evolution
of the floquet state can improve the performance.
a_ops : list of tuple(:class:`.Qobj`, callable)
List of collapse operators and the corresponding function for the noise
power spectrum. The collapse operator must be a :obj:`.Qobj` and
cannot be time dependent. The spectrum function must take and return
an numpy array.
w_th : float
The temperature of the environment in units of Hamiltonian frequency.
kmax : int [5]
The truncation of the number of sidebands..
nT : int [20*kmax]
The number of integration steps (for calculating X) within one period.
options : dict, optional
Options for the solver, see :obj:`FMESolver.options` and
`Integrator <./classes.html#classes-ode>`_ for a list of all options.
"""
name = "fmmesolve"
_avail_integrators = {}
_resultclass = FloquetResult
solver_options = {
"progress_bar": "text",
"progress_kwargs": {"chunk_size": 10},
"store_final_state": False,
"store_states": None,
"normalize_output": True,
"method": "adams",
"store_floquet_states": False,
}
def __init__(
self, floquet_basis, a_ops, w_th=0.0, *, kmax=5, nT=None, options=None
):
self.options = options
if isinstance(floquet_basis, FloquetBasis):
self.floquet_basis = floquet_basis
else:
raise TypeError("The ``floquet_basis`` must be a FloquetBasis")
nT = nT or max(100, 20 * kmax)
self._num_collapse = len(a_ops)
c_ops, spectra_cb = zip(*a_ops)
if not all(
isinstance(c_op, Qobj) and callable(spectrum)
for c_op, spectrum in a_ops
):
raise TypeError("a_ops must be tuple of (Qobj, callable)")
self.rhs = QobjEvo(
floquet_tensor(
self.floquet_basis,
c_ops,
spectra_cb,
w_th=w_th,
kmax=kmax,
nT=nT,
)
)
self._integrator = self._get_integrator()
self._state_metadata = {}
self.stats = self._initialize_stats()
def _initialize_stats(self):
stats = Solver._initialize_stats(self)
stats.update(
{
"solver": "Floquet-Markov master equation",
"num_collapse": self._num_collapse,
}
)
return stats
def _argument(self, args):
if args:
raise ValueError("FMESolver cannot update arguments")
[docs] def start(self, state0, t0, *, floquet=False):
"""
Set the initial state and time for a step evolution.
``options`` for the evolutions are read at this step.
Parameters
----------
state0 : :obj:`.Qobj`
Initial state of the evolution.
t0 : double
Initial time of the evolution.
floquet : bool, optional {False}
Whether the initial state is in the floquet basis or laboratory
basis.
"""
if not floquet:
state0 = self.floquet_basis.to_floquet_basis(state0, t0)
super().start(state0, t0)
[docs] def step(self, t, *, args=None, copy=True, floquet=False):
"""
Evolve the state to ``t`` and return the state as a :obj:`.Qobj`.
Parameters
----------
t : double
Time to evolve to, must be higher than the last call.
copy : bool, optional {True}
Whether to return a copy of the data or the data in the ODE solver.
floquet : bool, optional {False}
Whether to return the state in the floquet basis or laboratory
basis.
args : dict, optional {None}
Not supported
Notes
-----
The state must be initialized first by calling ``start`` or
``run``. If ``run`` is called, ``step`` will continue from the last
time and state obtained.
"""
if args:
raise ValueError("FMESolver cannot update arguments")
state = super().step(t)
if not floquet:
state = self.floquet_basis.from_floquet_basis(state, t)
elif copy:
state = state.copy()
return state
[docs] def run(self, state0, tlist, *, floquet=False, args=None, e_ops=None):
"""
Calculate the evolution of the quantum system.
For a ``state0`` at time ``tlist[0]`` do the evolution as directed by
``rhs`` and for each time in ``tlist`` store the state and/or
expectation values in a :class:`.Result`. The evolution method and
stored results are determined by ``options``.
Parameters
----------
state0 : :obj:`.Qobj`
Initial state of the evolution.
tlist : list of double
Time for which to save the results (state and/or expect) of the
evolution. The first element of the list is the initial time of the
evolution. Each times of the list must be increasing, but does not
need to be uniformy distributed.
floquet : bool, optional {False}
Whether the initial state in the floquet basis or laboratory basis.
args : dict, optional {None}
Not supported
e_ops : list {None}
List of Qobj, QobjEvo or callable to compute the expectation
values. Function[s] must have the signature
f(t : float, state : Qobj) -> expect.
Returns
-------
results : :class:`FloquetResult`
Results of the evolution. States and/or expect will be saved. You
can control the saved data in the options.
"""
if args:
raise ValueError("FMESolver cannot update arguments")
if not floquet:
state0 = self.floquet_basis.to_floquet_basis(state0, tlist[0])
_time_start = time()
_data0 = self._prepare_state(state0)
self._integrator.set_state(tlist[0], _data0)
stats = self._initialize_stats()
results = self._resultclass(
e_ops,
self.options,
solver=self.name,
stats=stats,
floquet_basis=self.floquet_basis,
)
results.add(tlist[0], self._restore_state(_data0, copy=False))
stats["preparation time"] += time() - _time_start
progress_bar = progress_bars[self.options["progress_bar"]](
len(tlist) - 1, **self.options["progress_kwargs"]
)
for t, state in self._integrator.run(tlist):
progress_bar.update()
results.add(t, self._restore_state(state, copy=False))
progress_bar.finished()
stats["run time"] = progress_bar.total_time()
# TODO: It would be nice if integrator could give evolution statistics
# stats.update(_integrator.stats)
return results
[docs] @classmethod
def ExpectFeedback(cls):
"""
Expect of the state of the evolution to be used in a time-dependent
operator.
Not not implemented for FMESolver
"""
raise NotImplementedError(
"Feedback not implemented for floquet solver."
)
[docs] @classmethod
def StateFeedback(cls):
"""
State of the evolution to be used in a time-dependent operator.
Not not implemented for FMESolver
"""
raise NotImplementedError(
"Feedback not implemented for floquet solver."
)