__all__ = ['Propagator', 'propagator', 'propagator_steadystate']
import numbers
import numpy as np
from .. import Qobj, qeye, qeye_like, unstack_columns, QobjEvo, liouvillian
from ..core import data as _data
from .mesolve import mesolve, MESolver
from .sesolve import sesolve, SESolver
from .heom.bofin_solvers import HEOMSolver
from .solver_base import Solver
from .multitraj import MultiTrajSolver
[docs]def propagator(H, t, c_ops=(), args=None, options=None, **kwargs):
r"""
Calculate the propagator U(t) for the density matrix or wave function such
that :math:`\psi(t) = U(t)\psi(0)` or
:math:`\rho_{\mathrm vec}(t) = U(t) \rho_{\mathrm vec}(0)`
where :math:`\rho_{\mathrm vec}` is the vector representation of the
density matrix.
Parameters
----------
H : :obj:`.Qobj`, :obj:`.QobjEvo`, :obj:`.QobjEvo` compatible format
Possibly time-dependent system Liouvillian or Hamiltonian as a Qobj or
QobjEvo. ``list`` of [:obj:`.Qobj`, :obj:`.Coefficient`] or callable
that can be made into :obj:`.QobjEvo` are also accepted.
t : float or array-like
Time or list of times for which to evaluate the propagator.
c_ops : list, optional
List of Qobj or QobjEvo collapse operators.
args : dictionary, optional
Parameters to callback functions for time-dependent Hamiltonians and
collapse operators.
options : dict, optional
Options for the solver.
**kwargs :
Extra parameters to use when creating the
:obj:`.QobjEvo` from a list format ``H``.
Returns
-------
U : :obj:`.Qobj`, list
Instance representing the propagator(s) :math:`U(t)`. Return a single
Qobj when ``t`` is a number or a list when ``t`` is a list.
"""
if isinstance(t, numbers.Real):
tlist = [0, t]
list_output = False
else:
tlist = t
list_output = True
if not isinstance(H, (Qobj, QobjEvo)):
H = QobjEvo(H, args=args, **kwargs)
if c_ops:
H = liouvillian(H, c_ops)
U0 = qeye_like(H)
if H.issuper:
out = mesolve(H, U0, tlist, args=args, options=options).states
else:
out = sesolve(H, U0, tlist, args=args, options=options).states
if list_output:
return out
else:
return out[-1]
[docs]def propagator_steadystate(U):
r"""Find the steady state for successive applications of the propagator
:math:`U`.
Parameters
----------
U : :obj:`.Qobj`
Operator representing the propagator.
Returns
-------
a : :obj:`.Qobj`
Instance representing the steady-state density matrix.
"""
evals, estates = U.eigenstates()
shifted_vals = np.abs(evals - 1.0)
ev_idx = np.argmin(shifted_vals)
rho_data = unstack_columns(estates[ev_idx].data)
rho_data = _data.mul(rho_data, 0.5 / _data.trace(rho_data))
return Qobj(_data.add(rho_data, _data.adjoint(rho_data)),
dims=U.dims[0],
isherm=True,
copy=False)
[docs]class Propagator:
"""
A generator of propagator for a system.
Usage:
U = Propagator(H, c_ops)
psi_t = U(t) @ psi_0
Save some previously computed propagator are stored to speed up subsequent
computation. Changing ``args`` will erase these stored probagator.
Parameters
----------
system : :obj:`.Qobj`, :obj:`.QobjEvo`, :class:`.Solver`
Possibly time-dependent system driving the evolution, either already
packaged in a solver, such as :class:`.SESolver` or :class:`.BRSolver`,
or the Liouvillian or Hamiltonian as a :obj:`.Qobj`,
:obj:`.QobjEvo`. ``list`` of [:obj:`.Qobj`, :obj:`.Coefficient`]
or callable that can be made into :obj:`.QobjEvo` are also accepted.
Solvers that run non-deterministacilly, such as :class:`.MCSolver`, are
not supported.
c_ops : list, optional
List of :obj:`.Qobj` or :obj:`.QobjEvo` collapse operators.
args : dictionary, optional
Parameters to callback functions for time-dependent Hamiltonians and
collapse operators.
options : dict, optional
Options for the solver.
memoize : int, default: 10
Max number of propagator to save.
tol : float, default: 1e-14
Absolute tolerance for the time. If a previous propagator was computed
at a time within tolerance, that propagator will be returned.
Notes
-----
The :class:`Propagator` is not a :obj:`.QobjEvo` so
it cannot be used for operations with :obj:`.Qobj` or
:obj:`.QobjEvo`. It can be made into a
:obj:`.QobjEvo` with ::
U = QobjEvo(Propagator(H))
"""
def __init__(self, system, *, c_ops=(), args=None, options=None,
memoize=10, tol=1e-14):
if isinstance(system, MultiTrajSolver):
raise TypeError("Non-deterministic solvers cannot be used "
"as a propagator system")
elif isinstance(system, HEOMSolver):
raise NotImplementedError(
"HEOM is not supported by Propagator. "
"Please, tell us on GitHub issues if you need it!"
)
elif isinstance(system, Solver):
self.solver = system
else:
Hevo = QobjEvo(system, args=args)
c_ops = [QobjEvo(op, args=args) for op in c_ops]
if Hevo.issuper or c_ops:
self.solver = MESolver(Hevo, c_ops=c_ops, options=options)
else:
self.solver = SESolver(Hevo, options=options)
self.times = [0]
self.invs = [None]
self.props = [qeye(self.solver.sys_dims)]
self.solver.start(self.props[0], self.times[0])
self.cte = self.solver.rhs.isconstant
H_0 = self.solver.rhs(0)
self.unitary = not H_0.issuper and H_0.isherm
self.args = args
self.memoize = max(3, int(memoize))
self.tol = tol
def _lookup_or_compute(self, t):
"""
Get U(t) from cache or compute it.
"""
idx = np.searchsorted(self.times, t)
if idx < len(self.times) and abs(t-self.times[idx]) <= self.tol:
U = self.props[idx]
elif idx > 0 and abs(t-self.times[idx-1]) <= self.tol:
U = self.props[idx-1]
else:
U = self._compute(t, idx)
self._insert(t, U, idx)
return U
[docs] def __call__(self, t, t_start=0, **args):
"""
Get the propagator from ``t_start`` to ``t``.
Parameters
----------
t : float
Time at which to compute the propagator.
t_start: float [0]
Time at which the propagator start such that:
``psi[t] = U.prop(t, t_start) @ psi[t_start]``
args : dict
Argument to pass to a time dependent Hamiltonian.
Updating ``args`` take effect since ``t=0`` and the new ``args``
will be used in future call.
"""
# We could improve it when the system is constant using U(2t) = U(t)**2
if not self.cte and args and args != self.args:
self.args = args
self.solver._argument(args)
self.times = [0]
self.props = [qeye_like(self.props[0])]
self.solver.start(self.props[0], self.times[0])
if t_start:
if t == t_start:
U = self._lookup_or_compute(0)
if self.cte:
U = self._lookup_or_compute(t - t_start)
else:
Uinv = self._inv(self._lookup_or_compute(t_start))
U = self._lookup_or_compute(t) @ Uinv
else:
U = self._lookup_or_compute(t)
return U
[docs] def inv(self, t, **args):
"""
Get the inverse of the propagator at ``t``, such that
``psi_0 = U.inv(t) @ psi_t``
Parameters
----------
t : float
Time at which to compute the propagator.
args : dict
Argument to pass to a time dependent Hamiltonian.
Updating ``args`` take effect since ``t=0`` and the new ``args``
will be used in future call.
"""
return self._inv(self(t, **args))
def _compute(self, t, idx):
"""
Compute the propagator at ``t``, ``idx`` point to a pair of
(time, propagator) close to the desired time.
"""
t_last = self.solver._integrator.get_state(copy=False)[0]
if self.times[idx-1] <= t_last <= t:
U = self.solver.step(t)
elif idx > 0:
self.solver.start(self.props[idx-1], self.times[idx-1])
U = self.solver.step(t)
else:
# Evolving backward in time is not supported by all integrator.
self.solver.start(qeye_like(self.props[0]), t)
Uinv = self.solver.step(self.times[idx])
U = self._inv(Uinv)
return U
def _inv(self, U):
return U.dag() if self.unitary else U.inv()
def _insert(self, t, U, idx):
"""
Insert a new pair of (time, propagator) to the memorized states.
"""
while len(self.times) >= self.memoize:
rm_idx = self.memoize // 2
if self.times[rm_idx] < t:
idx -= 1
del self.times[rm_idx]
del self.props[rm_idx]
self.times.insert(idx, t)
self.props.insert(idx, U)