def __init__(
self,
H_sys,
bath,
max_depth,
options=None,
progress_bar=None,
):
self.H_sys = self._convert_h_sys(H_sys)
self.options = Options() if options is None else options
self._is_timedep = isinstance(self.H_sys, QobjEvo)
self._H0 = self.H_sys.to_list()[0] if self._is_timedep else self.H_sys
self._is_hamiltonian = self._H0.type == "oper"
self._L0 = liouvillian(self._H0) if self._is_hamiltonian else self._H0
self._sys_shape = (self._H0.shape[0] if self._is_hamiltonian else int(
np.sqrt(self._H0.shape[0])))
self._sup_shape = self._L0.shape[0]
self._sys_dims = (self._H0.dims
if self._is_hamiltonian else self._H0.dims[0])
self.ados = HierarchyADOs(
self._combine_bath_exponents(bath),
max_depth,
)
self._n_ados = len(self.ados.labels)
self._n_exponents = len(self.ados.exponents)
# pre-calculate identity matrix required by _grad_n
self._sId = fast_identity(self._sup_shape)
# pre-calculate superoperators required by _grad_prev and _grad_next:
Qs = [exp.Q for exp in self.ados.exponents]
self._spreQ = [spre(op).data for op in Qs]
self._spostQ = [spost(op).data for op in Qs]
self._s_pre_minus_post_Q = [
self._spreQ[k] - self._spostQ[k] for k in range(self._n_exponents)
]
self._s_pre_plus_post_Q = [
self._spreQ[k] + self._spostQ[k] for k in range(self._n_exponents)
]
self._spreQdag = [spre(op.dag()).data for op in Qs]
self._spostQdag = [spost(op.dag()).data for op in Qs]
self._s_pre_minus_post_Qdag = [
self._spreQdag[k] - self._spostQdag[k]
for k in range(self._n_exponents)
]
self._s_pre_plus_post_Qdag = [
self._spreQdag[k] + self._spostQdag[k]
for k in range(self._n_exponents)
]
if progress_bar is None:
self.progress_bar = BaseProgressBar()
if progress_bar is True:
self.progress_bar = TextProgressBar()
self._configure_solver()
def _check_progress_bar(progress_bar):
"""
Check instance of progress_bar and return the object.
"""
if progress_bar is None:
pbar = BaseProgressBar()
if progress_bar is True:
pbar = TextProgressBar()
return pbar
def serial_map(task, values, task_args=tuple(), task_kwargs={}, **kwargs):
"""
Serial mapping function with the same call signature as parallel_map, for
easy switching between serial and parallel execution. This
is functionally equivalent to:
result = [task(value, *task_args, **task_kwargs) for value in values]
This function work as a drop-in replacement of :func:`qutip.parallel_map`.
Parameters
----------
task: a Python function
The function that is to be called for each value in ``task_vec``.
values: array / list
The list or array of values for which the ``task`` function is to be
evaluated.
task_args: list / dictionary
The optional additional argument to the ``task`` function.
task_kwargs: list / dictionary
The optional additional keyword argument to the ``task`` function.
progress_bar: ProgressBar
Progress bar class instance for showing progress.
Returns
--------
result : list
The result list contains the value of
``task(value, *task_args, **task_kwargs)`` for each
value in ``values``.
"""
try:
progress_bar = kwargs['progress_bar']
if progress_bar is True:
progress_bar = TextProgressBar()
except:
progress_bar = BaseProgressBar()
progress_bar.start(len(values))
results = []
for n, value in enumerate(values):
progress_bar.update(n)
result = task(value, *task_args, **task_kwargs)
results.append(result)
progress_bar.finished()
return results
def __init__(self,
H_sys,
coup_op,
coup_strength,
temperature,
N_cut,
N_exp,
cut_freq,
planck=1.0,
boltzmann=1.0,
renorm=True,
bnd_cut_approx=True,
options=None,
progress_bar=None,
stats=None):
self.reset()
if options is None:
self.options = Options()
else:
self.options = options
self.progress_bar = False
if progress_bar is None:
self.progress_bar = BaseProgressBar()
elif progress_bar == True:
self.progress_bar = TextProgressBar()
# the other attributes will be set in the configure method
self.configure(H_sys,
coup_op,
coup_strength,
temperature,
N_cut,
N_exp,
cut_freq,
planck=planck,
boltzmann=boltzmann,
renorm=renorm,
bnd_cut_approx=bnd_cut_approx,
stats=stats)
def propagator(H, t, c_op_list, args=None, options=None, sparse=False, progress_bar=None):
"""
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 : qobj or list
Hamiltonian as a Qobj instance of a nested list of Qobjs and
coefficients in the list-string or list-function format for
time-dependent Hamiltonians (see description in :func:`qutip.mesolve`).
t : float or array-like
Time or list of times for which to evaluate the propagator.
c_op_list : list
List of qobj collapse operators.
args : list/array/dictionary
Parameters to callback functions for time-dependent Hamiltonians and
collapse operators.
options : :class:`qutip.Options`
with options for the ODE solver.
progress_bar: BaseProgressBar
Optional instance of BaseProgressBar, or a subclass thereof, for
showing the progress of the simulation. By default no progress bar
is used, and if set to True a TextProgressBar will be used.
Returns
-------
a : qobj
Instance representing the propagator :math:`U(t)`.
"""
if progress_bar is None:
progress_bar = BaseProgressBar()
elif progress_bar is True:
progress_bar = TextProgressBar()
if options is None:
options = Options()
options.rhs_reuse = True
rhs_clear()
if isinstance(t, (int, float, np.integer, np.floating)):
tlist = [0, t]
else:
tlist = t
if isinstance(H, (types.FunctionType, types.BuiltinFunctionType, functools.partial)):
H0 = H(0.0, args)
elif isinstance(H, list):
H0 = H[0][0] if isinstance(H[0], list) else H[0]
else:
H0 = H
if len(c_op_list) == 0 and H0.isoper:
# calculate propagator for the wave function
N = H0.shape[0]
dims = H0.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
progress_bar.start(N)
for n in range(0, N):
progress_bar.update(n)
psi0 = basis(N, n)
output = sesolve(H, psi0, tlist, [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = output.states[k].full().T
progress_bar.finished()
# todo: evolving a batch of wave functions:
# psi_0_list = [basis(N, n) for n in range(N)]
# psi_t_list = mesolve(H, psi_0_list, [0, t], [], [], args, options)
# for n in range(0, N):
# u[:,n] = psi_t_list[n][1].full().T
elif len(c_op_list) == 0 and H0.issuper:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
N = H0.shape[0]
dims = H0.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
progress_bar.start(N)
for n in range(0, N):
progress_bar.update(n)
psi0 = basis(N, n)
rho0 = Qobj(vec2mat(psi0.full()))
output = mesolve(H, rho0, tlist, [], [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
progress_bar.finished()
else:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
N = H0.shape[0]
dims = [H0.dims, H0.dims]
u = np.zeros([N * N, N * N, len(tlist)], dtype=complex)
if sparse:
progress_bar.start(N * N)
for n in range(N * N):
progress_bar.update(n)
psi0 = basis(N * N, n)
psi0.dims = [dims[0], 1]
rho0 = vector_to_operator(psi0)
output = mesolve(H, rho0, tlist, c_op_list, [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = operator_to_vector(output.states[k]).full(squeeze=True)
progress_bar.finished()
else:
progress_bar.start(N * N)
for n in range(N * N):
progress_bar.update(n)
psi0 = basis(N * N, n)
rho0 = Qobj(vec2mat(psi0.full()))
output = mesolve(H, rho0, tlist, c_op_list, [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
progress_bar.finished()
if len(tlist) == 2:
return Qobj(u[:, :, 1], dims=dims)
else:
return [Qobj(u[:, :, k], dims=dims) for k in range(len(tlist))]
def mesolve(H,
rho0,
tlist,
c_ops=None,
e_ops=None,
args=None,
options=None,
progress_bar=None,
_safe_mode=True):
"""
Master equation evolution of a density matrix for a given Hamiltonian and
set of collapse operators, or a Liouvillian.
Evolve the state vector or density matrix (`rho0`) using a given
Hamiltonian or Liouvillian (`H`) and an optional set of collapse operators
(`c_ops`), by integrating the set of ordinary differential equations
that define the system. In the absence of collapse operators the system is
evolved according to the unitary evolution of the Hamiltonian.
The output is either the state vector at arbitrary points in time
(`tlist`), or the expectation values of the supplied operators
(`e_ops`). If e_ops is a callback function, it is invoked for each
time in `tlist` with time and the state as arguments, and the function
does not use any return values.
If either `H` or the Qobj elements in `c_ops` are superoperators, they
will be treated as direct contributions to the total system Liouvillian.
This allows the solution of master equations that are not in standard
Lindblad form.
**Time-dependent operators**
For time-dependent problems, `H` and `c_ops` can be a specified in a
nested-list format where each element in the list is a list of length 2,
containing an operator (:class:`qutip.qobj`) at the first element and where
the second element is either a string (*list string format*), a callback
function (*list callback format*) that evaluates to the time-dependent
coefficient for the corresponding operator, or a NumPy array (*list
array format*) which specifies the value of the coefficient to the
corresponding operator for each value of t in `tlist`.
Alternatively, `H` (but not `c_ops`) can be a callback function with the
signature `f(t, args) -> Qobj` (*callback format*), which can return the
Hamiltonian or Liouvillian superoperator at any point in time. If the
equation cannot be put in standard Lindblad form, then this time-dependence
format must be used.
*Examples*
H = [[H0, 'sin(w*t)'], [H1, 'sin(2*w*t)']]
H = [[H0, f0_t], [H1, f1_t]]
where f0_t and f1_t are python functions with signature f_t(t, args).
H = [[H0, np.sin(w*tlist)], [H1, np.sin(2*w*tlist)]]
In the *list string format* and *list callback format*, the string
expression and the callback function must evaluate to a real or complex
number (coefficient for the corresponding operator).
In all cases of time-dependent operators, `args` is a dictionary of
parameters that is used when evaluating operators. It is passed to the
callback functions as their second argument.
**Additional options**
Additional options to mesolve can be set via the `options` argument, which
should be an instance of :class:`qutip.solver.Options`. Many ODE
integration options can be set this way, and the `store_states` and
`store_final_state` options can be used to store states even though
expectation values are requested via the `e_ops` argument.
.. note::
If an element in the list-specification of the Hamiltonian or
the list of collapse operators are in superoperator form it will be
added to the total Liouvillian of the problem without further
transformation. This allows for using mesolve for solving master
equations that are not in standard Lindblad form.
.. note::
On using callback functions: mesolve transforms all :class:`qutip.Qobj`
objects to sparse matrices before handing the problem to the integrator
function. In order for your callback function to work correctly, pass
all :class:`qutip.Qobj` objects that are used in constructing the
Hamiltonian via `args`. mesolve will check for :class:`qutip.Qobj` in
`args` and handle the conversion to sparse matrices. All other
:class:`qutip.Qobj` objects that are not passed via `args` will be
passed on to the integrator in scipy which will raise a NotImplemented
exception.
Parameters
----------
H : :class:`qutip.Qobj`
System Hamiltonian, or a callback function for time-dependent
Hamiltonians, or alternatively a system Liouvillian.
rho0 : :class:`qutip.Qobj`
initial density matrix or state vector (ket).
tlist : *list* / *array*
list of times for :math:`t`.
c_ops : None / list of :class:`qutip.Qobj`
single collapse operator, or list of collapse operators, or a list
of Liouvillian superoperators.
e_ops : None / list / callback function, optional
A list of operators as `Qobj` and/or callable functions (can be mixed)
or a single callable function. For operators, the result's expect will
be computed by :func:`qutip.expect`. For callable functions, they are
called as ``f(t, state)`` and return the expectation value.
A single callback's expectation value can be any type, but a callback
part of a list must return a number as the expectation value.
args : None / *dictionary*
dictionary of parameters for time-dependent Hamiltonians and
collapse operators.
options : None / :class:`qutip.Options`
with options for the solver.
progress_bar : None / BaseProgressBar
Optional instance of BaseProgressBar, or a subclass thereof, for
showing the progress of the simulation.
Returns
-------
result: :class:`qutip.Result`
An instance of the class :class:`qutip.Result`, which contains
either an *array* `result.expect` of expectation values for the times
specified by `tlist`, or an *array* `result.states` of state vectors or
density matrices corresponding to the times in `tlist` [if `e_ops` is
an empty list], or nothing if a callback function was given in place of
operators for which to calculate the expectation values.
"""
if c_ops is None:
c_ops = []
if isinstance(c_ops, (Qobj, QobjEvo)):
c_ops = [c_ops]
if e_ops is None:
e_ops = []
if isinstance(e_ops, Qobj):
e_ops = [e_ops]
if isinstance(e_ops, dict):
e_ops_dict = e_ops
e_ops = [e for e in e_ops.values()]
else:
e_ops_dict = None
if progress_bar is None:
progress_bar = BaseProgressBar()
if progress_bar is True:
progress_bar = TextProgressBar()
# check if rho0 is a superoperator, in which case e_ops argument should
# be empty, i.e., e_ops = []
# TODO: e_ops for superoperator
if issuper(rho0) and not e_ops == []:
raise TypeError("Must have e_ops = [] when initial condition rho0 is" +
" a superoperator.")
if options is None:
options = Options()
if options.rhs_reuse and not isinstance(H, SolverSystem):
# TODO: deprecate when going to class based solver.
if "mesolve" in solver_safe:
# print(" ")
H = solver_safe["mesolve"]
else:
pass
# raise Exception("Could not find the Hamiltonian to reuse.")
if args is None:
args = {}
check_use_openmp(options)
use_mesolve = ((c_ops and len(c_ops) > 0) or (not isket(rho0))
or (isinstance(H, Qobj) and issuper(H))
or (isinstance(H, QobjEvo) and issuper(H.cte))
or (isinstance(H, list) and isinstance(H[0], Qobj)
and issuper(H[0]))
or (not isinstance(H, (Qobj, QobjEvo)) and callable(H)
and not options.rhs_with_state and issuper(H(0., args)))
or (not isinstance(H, (Qobj, QobjEvo)) and callable(H)
and options.rhs_with_state))
if not use_mesolve:
return sesolve(H,
rho0,
tlist,
e_ops=e_ops,
args=args,
options=options,
progress_bar=progress_bar,
_safe_mode=_safe_mode)
if isket(rho0):
rho0 = ket2dm(rho0)
if (not (rho0.isoper or rho0.issuper)) or (rho0.dims[0] != rho0.dims[1]):
raise ValueError(
"input state must be a pure state vector, square density matrix, "
"or superoperator")
if isinstance(H, SolverSystem):
ss = H
elif isinstance(H, (list, Qobj, QobjEvo)):
ss = _mesolve_QobjEvo(H, c_ops, tlist, args, options)
elif callable(H):
ss = _mesolve_func_td(H, c_ops, rho0, tlist, args, options)
else:
raise Exception("Invalid H type")
func, ode_args = ss.makefunc(ss, rho0, args, e_ops, options)
if _safe_mode:
# This is to test safety of the function before starting the loop.
v = rho0.full().ravel('F')
func(0., v, *ode_args) + v
res = _generic_ode_solve(func,
ode_args,
rho0,
tlist,
e_ops,
options,
progress_bar,
dims=rho0.dims)
res.num_collapse = len(c_ops)
if e_ops_dict:
res.expect = {
e: res.expect[n]
for n, e in enumerate(e_ops_dict.keys())
}
return res
def bloch_redfield_solve(R,
ekets,
rho0,
tlist,
e_ops=[],
options=None,
progress_bar=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()
if progress_bar is None:
progress_bar = BaseProgressBar()
elif progress_bar is True:
progress_bar = TextProgressBar()
#
# 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)
progress_bar.start(n_tsteps)
for t_idx, _ in enumerate(tlist):
progress_bar.update(t_idx)
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])
progress_bar.finished()
return result_list
def propagator(H, t, c_op_list=[], args={}, options=None,
unitary_mode='batch', parallel=False,
progress_bar=None, **kwargs):
"""
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 : qobj or list
Hamiltonian as a Qobj instance of a nested list of Qobjs and
coefficients in the list-string or list-function format for
time-dependent Hamiltonians (see description in :func:`qutip.mesolve`).
t : float or array-like
Time or list of times for which to evaluate the propagator.
c_op_list : list
List of qobj collapse operators.
args : list/array/dictionary
Parameters to callback functions for time-dependent Hamiltonians and
collapse operators.
options : :class:`qutip.Options`
with options for the ODE solver.
unitary_mode = str ('batch', 'single')
Solve all basis vectors simulaneously ('batch') or individually
('single').
parallel : bool {False, True}
Run the propagator in parallel mode. This will override the
unitary_mode settings if set to True.
progress_bar: BaseProgressBar
Optional instance of BaseProgressBar, or a subclass thereof, for
showing the progress of the simulation. By default no progress bar
is used, and if set to True a TextProgressBar will be used.
Returns
-------
a : qobj
Instance representing the propagator :math:`U(t)`.
"""
kw = _default_kwargs()
if 'num_cpus' in kwargs:
num_cpus = kwargs['num_cpus']
else:
num_cpus = kw['num_cpus']
if progress_bar is None:
progress_bar = BaseProgressBar()
elif progress_bar is True:
progress_bar = TextProgressBar()
if options is None:
options = Options()
options.rhs_reuse = True
rhs_clear()
if isinstance(t, (int, float, np.integer, np.floating)):
tlist = [0, t]
else:
tlist = t
td_type = _td_format_check(H, c_op_list, solver='me')
if isinstance(H, (types.FunctionType, types.BuiltinFunctionType,
functools.partial)):
H0 = H(0.0, args)
elif isinstance(H, list):
H0 = H[0][0] if isinstance(H[0], list) else H[0]
else:
H0 = H
if len(c_op_list) == 0 and H0.isoper:
# calculate propagator for the wave function
N = H0.shape[0]
dims = H0.dims
if parallel:
unitary_mode = 'single'
u = np.zeros([N, N, len(tlist)], dtype=complex)
output = parallel_map(_parallel_sesolve,range(N),
task_args=(N,H, tlist,args,options),
progress_bar=progress_bar, num_cpus=num_cpus)
for n in range(N):
for k, t in enumerate(tlist):
u[:, n, k] = output[n].states[k].full().T
else:
if unitary_mode == 'single':
u = np.zeros([N, N, len(tlist)], dtype=complex)
progress_bar.start(N)
for n in range(0, N):
progress_bar.update(n)
psi0 = basis(N, n)
output = sesolve(H, psi0, tlist, [], args, options, _safe_mode=False)
for k, t in enumerate(tlist):
u[:, n, k] = output.states[k].full().T
progress_bar.finished()
elif unitary_mode =='batch':
u = np.zeros(len(tlist), dtype=object)
_rows = np.array([(N+1)*m for m in range(N)])
_cols = np.zeros_like(_rows)
_data = np.ones_like(_rows,dtype=complex)
psi0 = Qobj(sp.coo_matrix((_data,(_rows,_cols))).tocsr())
if td_type[1] > 0 or td_type[2] > 0:
H2 = []
for k in range(len(H)):
if isinstance(H[k], list):
H2.append([tensor(qeye(N), H[k][0]), H[k][1]])
else:
H2.append(tensor(qeye(N), H[k]))
else:
H2 = tensor(qeye(N), H)
output = sesolve(H2, psi0, tlist, [] , args = args, _safe_mode=False,
options=Options(normalize_output=False))
for k, t in enumerate(tlist):
u[k] = sp_reshape(output.states[k].data, (N, N))
unit_row_norm(u[k].data, u[k].indptr, u[k].shape[0])
u[k] = u[k].T.tocsr()
else:
raise Exception('Invalid unitary mode.')
elif len(c_op_list) == 0 and H0.issuper:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
unitary_mode = 'single'
N = H0.shape[0]
sqrt_N = int(np.sqrt(N))
dims = H0.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
if parallel:
output = parallel_map(_parallel_mesolve,range(N * N),
task_args=(sqrt_N,H,tlist,c_op_list,args,options),
progress_bar=progress_bar, num_cpus=num_cpus)
for n in range(N * N):
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output[n].states[k].full()).T
else:
progress_bar.start(N)
for n in range(0, N):
progress_bar.update(n)
col_idx, row_idx = np.unravel_index(n,(sqrt_N,sqrt_N))
rho0 = Qobj(sp.csr_matrix(([1],([row_idx],[col_idx])), shape=(sqrt_N,sqrt_N), dtype=complex))
output = mesolve(H, rho0, tlist, [], [], args, options, _safe_mode=False)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
progress_bar.finished()
else:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
unitary_mode = 'single'
N = H0.shape[0]
dims = [H0.dims, H0.dims]
u = np.zeros([N * N, N * N, len(tlist)], dtype=complex)
if parallel:
output = parallel_map(_parallel_mesolve,range(N * N),
task_args=(N,H,tlist,c_op_list,args,options),
progress_bar=progress_bar, num_cpus=num_cpus)
for n in range(N * N):
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output[n].states[k].full()).T
else:
progress_bar.start(N * N)
for n in range(N * N):
progress_bar.update(n)
col_idx, row_idx = np.unravel_index(n,(N,N))
rho0 = Qobj(sp.csr_matrix(([1],([row_idx],[col_idx])), shape=(N,N), dtype=complex))
output = mesolve(H, rho0, tlist, c_op_list, [], args, options, _safe_mode=False)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
progress_bar.finished()
if len(tlist) == 2:
if unitary_mode == 'batch':
return Qobj(u[-1], dims=dims)
else:
return Qobj(u[:, :, 1], dims=dims)
else:
if unitary_mode == 'batch':
return np.array([Qobj(u[k], dims=dims) for k in range(len(tlist))], dtype=object)
else:
return np.array([Qobj(u[:, :, k], dims=dims) for k in range(len(tlist))], dtype=object)
def bloch_redfield_solve(R, ekets, rho0, tlist, e_ops=[], options=None, progress_bar=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()
if progress_bar is None:
progress_bar = BaseProgressBar()
elif progress_bar is True:
progress_bar = TextProgressBar()
#
# 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)
progress_bar.start(n_tsteps)
for t_idx, _ in enumerate(tlist):
progress_bar.update(t_idx)
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])
progress_bar.finished()
return result_list
def parallel_map(
task,
values,
task_args=None,
task_kwargs=None,
num_cpus=None,
progress_bar=None,
):
"""Map function `task` onto `values`, in parallel.
This function's interface is identical to
:func:`qutip.parallel.parallel_map` as of QuTiP 4.5.0, but has the option
of using :mod:`loky` as a backend (see :func:`set_parallelization`). It
also eliminates internal threads, according to
:obj:`USE_THREADPOOL_LIMITS`.
"""
# TODO: if QuTiP's parallel_map catches up, we can remove this function,
# and put QuTiP's parallel_map into __all__ to maintain krotov's interface.
if task_args is None:
task_args = ()
if task_kwargs is None:
task_kwargs = {}
if num_cpus is None:
num_cpus = multiprocessing.cpu_count()
if progress_bar is None:
progress_bar = BaseProgressBar()
if progress_bar is True:
progress_bar = TextProgressBar()
progress_bar.start(len(values))
nfinished = [0]
def _update_progress_bar(x):
nfinished[0] += 1
progress_bar.update(nfinished[0])
if USE_LOKY:
Executor = LokyReusableExecutor
if USE_THREADPOOL_LIMITS:
Executor = partial(
LokyReusableExecutor,
initializer=_process_threadpool_limits_initializier,
)
else:
Executor = ProcessPoolExecutor
_threadpool_limits = _no_threadpool_limits
if USE_THREADPOOL_LIMITS:
_threadpool_limits = threadpool_limits
with _threadpool_limits(limits=1):
with Executor(max_workers=num_cpus) as executor:
jobs = []
try:
for value in values:
args = (value, ) + tuple(task_args)
job = executor.submit(task, *args, **task_kwargs)
job.add_done_callback(_update_progress_bar)
jobs.append(job)
res = [job.result() for job in jobs]
except KeyboardInterrupt as e:
raise e
progress_bar.finished()
return res
def grape_unitary_adaptive(U,
H0,
H_ops,
R,
times,
eps=None,
u_start=None,
u_limits=None,
interp_kind='linear',
use_interp=False,
alpha=None,
beta=None,
phase_sensitive=False,
overlap_terminate=1.0,
progress_bar=BaseProgressBar()):
"""
Calculate control pulses for the Hamiltonian operators in H_ops so that
the unitary U is realized.
Experimental: Work in progress.
Parameters
----------
U : Qobj
Target unitary evolution operator.
H0 : Qobj
Static Hamiltonian (that cannot be tuned by the control fields).
H_ops: list of Qobj
A list of operators that can be tuned in the Hamiltonian via the
control fields.
R : int
Number of GRAPE iterations.
time : array / list
Array of time coordinates for control pulse evalutation.
u_start : array
Optional array with initial control pulse values.
Returns
-------
Instance of GRAPEResult, which contains the control pulses calculated
with GRAPE, a time-dependent Hamiltonian that is defined by the
control pulses, as well as the resulting propagator.
"""
if eps is None:
eps = 0.1 * (2 * np.pi) / (times[-1])
eps_vec = np.array([eps / 2, eps, 2 * eps])
eps_log = np.zeros(R)
overlap_log = np.zeros(R)
best_k = 0
_k_overlap = np.array([0.0, 0.0, 0.0])
M = len(times)
J = len(H_ops)
K = len(eps_vec)
Uf = [None for _ in range(K)]
u = np.zeros((R, J, M, K))
if u_limits and len(u_limits) != 2:
raise ValueError("u_limits must be a list with two values")
if u_limits:
warnings.warn("Causion: Using experimental feature u_limits")
if u_limits and u_start:
# make sure that no values in u0 violates the u_limits conditions
u_start = np.array(u_start)
u_start[u_start < u_limits[0]] = u_limits[0]
u_start[u_start > u_limits[1]] = u_limits[1]
if u_start is not None:
for idx, u0 in enumerate(u_start):
for k in range(K):
u[0, idx, :, k] = u0
if beta:
warnings.warn("Causion: Using experimental feature time-penalty")
if phase_sensitive:
_fidelity_function = lambda x: x
else:
_fidelity_function = lambda x: abs(x)**2
best_k = 1
_r = 0
_prev_overlap = 0
progress_bar.start(R)
for r in range(R - 1):
progress_bar.update(r)
_r = r
eps_log[r] = eps_vec[best_k]
logger.debug("eps_vec: {}".format(eps_vec))
_t0 = time.time()
dt = times[1] - times[0]
if use_interp:
ip_funcs = [
interp1d(times,
u[r, j, :, best_k],
kind=interp_kind,
bounds_error=False,
fill_value=u[r, j, -1, best_k]) for j in range(J)
]
def _H_t(t, args=None):
return H0 + sum(
[float(ip_funcs[j](t)) * H_ops[j] for j in range(J)])
U_list = [(-1j * _H_t(times[idx]) * dt).expm()
for idx in range(M - 1)]
else:
def _H_idx(idx):
return H0 + sum(
[u[r, j, idx, best_k] * H_ops[j] for j in range(J)])
U_list = [(-1j * _H_idx(idx) * dt).expm() for idx in range(M - 1)]
logger.debug("Time 1: %fs" % (time.time() - _t0))
_t0 = time.time()
U_f_list = []
U_b_list = []
U_f = 1
U_b = 1
for m in range(M - 1):
U_f = U_list[m] * U_f
U_f_list.append(U_f)
U_b_list.insert(0, U_b)
U_b = U_list[M - 2 - m].dag() * U_b
logger.debug("Time 2: %fs" % (time.time() - _t0))
_t0 = time.time()
for j in range(J):
for m in range(M - 1):
P = U_b_list[m] * U
Q = 1j * dt * H_ops[j] * U_f_list[m]
if phase_sensitive:
du = -cy_overlap(P.data, Q.data)
else:
du = (-2 * cy_overlap(P.data, Q.data) *
cy_overlap(U_f_list[m].data, P.data))
if alpha:
# penalty term for high power control signals u
du += -2 * alpha * u[r, j, m, best_k] * dt
if beta:
# penalty term for late control signals u
du += -2 * beta * k**2 * u[r, j, k] * dt
for k, eps_val in enumerate(eps_vec):
u[r + 1, j, m, k] = u[r, j, m, k] + eps_val * du.real
if u_limits:
if u[r + 1, j, m, k] < u_limits[0]:
u[r + 1, j, m, k] = u_limits[0]
elif u[r + 1, j, m, k] > u_limits[1]:
u[r + 1, j, m, k] = u_limits[1]
u[r + 1, j, -1, :] = u[r + 1, j, -2, :]
logger.debug("Time 3: %fs" % (time.time() - _t0))
_t0 = time.time()
for k, eps_val in enumerate(eps_vec):
def _H_idx(idx):
return H0 + sum(
[u[r + 1, j, idx, k] * H_ops[j] for j in range(J)])
U_list = [(-1j * _H_idx(idx) * dt).expm() for idx in range(M - 1)]
Uf[k] = gate_sequence_product(U_list)
_k_overlap[k] = _fidelity_function(cy_overlap(Uf[k].data,
U.data)).real
best_k = np.argmax(_k_overlap)
logger.debug("k_overlap: ", _k_overlap, best_k)
if _prev_overlap > _k_overlap[best_k]:
logger.debug("Regression, stepping back with smaller eps.")
u[r + 1, :, :, :] = u[r, :, :, :]
eps_vec /= 2
else:
if best_k == 0:
eps_vec /= 2
elif best_k == 2:
eps_vec *= 2
_prev_overlap = _k_overlap[best_k]
overlap_log[r] = _k_overlap[best_k]
if overlap_terminate < 1.0:
if _k_overlap[best_k] > overlap_terminate:
logger.info("Reached target fidelity, terminating.")
break
logger.debug("Time 4: %fs" % (time.time() - _t0))
_t0 = time.time()
if use_interp:
ip_funcs = [
interp1d(times,
u[_r, j, :, best_k],
kind=interp_kind,
bounds_error=False,
fill_value=u[R - 1, j, -1]) for j in range(J)
]
H_td_func = [H0] + [[H_ops[j], lambda t, args, j=j: ip_funcs[j](t)]
for j in range(J)]
else:
H_td_func = [H0] + [[H_ops[j], u[_r, j, :, best_k]] for j in range(J)]
progress_bar.finished()
result = GRAPEResult(u=u[:_r, :, :, best_k], U_f=Uf[best_k], H_t=H_td_func)
result.eps = eps_log
result.overlap = overlap_log
return result
def cy_grape_unitary(U,
H0,
H_ops,
R,
times,
eps=None,
u_start=None,
u_limits=None,
interp_kind='linear',
use_interp=False,
alpha=None,
beta=None,
phase_sensitive=True,
progress_bar=BaseProgressBar()):
"""
Calculate control pulses for the Hamitonian operators in H_ops so that the
unitary U is realized.
Experimental: Work in progress.
Parameters
----------
U : Qobj
Target unitary evolution operator.
H0 : Qobj
Static Hamiltonian (that cannot be tuned by the control fields).
H_ops: list of Qobj
A list of operators that can be tuned in the Hamiltonian via the
control fields.
R : int
Number of GRAPE iterations.
time : array / list
Array of time coordinates for control pulse evalutation.
u_start : array
Optional array with initial control pulse values.
Returns
-------
Instance of GRAPEResult, which contains the control pulses calculated
with GRAPE, a time-dependent Hamiltonian that is defined by the
control pulses, as well as the resulting propagator.
"""
if eps is None:
eps = 0.1 * (2 * np.pi) / (times[-1])
M = len(times)
J = len(H_ops)
u = np.zeros((R, J, M))
H_ops_data = [H_op.data for H_op in H_ops]
if u_limits and len(u_limits) != 2:
raise ValueError("u_limits must be a list with two values")
if u_limits:
warnings.warn("Causion: Using experimental feature u_limits")
if u_limits and u_start:
# make sure that no values in u0 violates the u_limits conditions
u_start = np.array(u_start)
u_start[u_start < u_limits[0]] = u_limits[0]
u_start[u_start > u_limits[1]] = u_limits[1]
if u_limits:
use_u_limits = 1
u_min = u_limits[0]
u_max = u_limits[1]
else:
use_u_limits = 0
u_min = 0.0
u_max = 0.0
if u_start is not None:
for idx, u0 in enumerate(u_start):
u[0, idx, :] = u0
if beta:
warnings.warn("Causion: Using experimental feature time-penalty")
alpha_val = alpha if alpha else 0.0
beta_val = beta if beta else 0.0
progress_bar.start(R)
for r in range(R - 1):
progress_bar.update(r)
dt = times[1] - times[0]
if use_interp:
ip_funcs = [
interp1d(times,
u[r, j, :],
kind=interp_kind,
bounds_error=False,
fill_value=u[r, j, -1]) for j in range(J)
]
def _H_t(t, args=None):
return H0 + sum(
[float(ip_funcs[j](t)) * H_ops[j] for j in range(J)])
U_list = [(-1j * _H_t(times[idx]) * dt).expm().data
for idx in range(M - 1)]
else:
def _H_idx(idx):
return H0 + sum([u[r, j, idx] * H_ops[j] for j in range(J)])
U_list = [(-1j * _H_idx(idx) * dt).expm().data
for idx in range(M - 1)]
U_f_list = []
U_b_list = []
U_f = 1
U_b = sp.eye(*(U.shape))
for n in range(M - 1):
U_f = U_list[n] * U_f
U_f_list.append(U_f)
U_b_list.insert(0, U_b)
U_b = U_list[M - 2 - n].T.conj().tocsr() * U_b
cy_grape_inner(U.data, u, r, J, M, U_b_list, U_f_list, H_ops_data, dt,
eps, alpha_val, beta_val, phase_sensitive, use_u_limits,
u_min, u_max)
if use_interp:
ip_funcs = [
interp1d(times,
u[R - 1, j, :],
kind=interp_kind,
bounds_error=False,
fill_value=u[R - 1, j, -1]) for j in range(J)
]
H_td_func = [H0] + [[H_ops[j], lambda t, args, j=j: ip_funcs[j](t)]
for j in range(J)]
else:
H_td_func = [H0] + [[H_ops[j], u[-1, j, :]] for j in range(J)]
progress_bar.finished()
return GRAPEResult(u=u, U_f=Qobj(U_f_list[-1], dims=U.dims), H_t=H_td_func)
def grape_unitary(U,
H0,
H_ops,
R,
times,
eps=None,
u_start=None,
u_limits=None,
interp_kind='linear',
use_interp=False,
alpha=None,
beta=None,
phase_sensitive=True,
progress_bar=BaseProgressBar()):
"""
Calculate control pulses for the Hamiltonian operators in H_ops so that the
unitary U is realized.
Experimental: Work in progress.
Parameters
----------
U : Qobj
Target unitary evolution operator.
H0 : Qobj
Static Hamiltonian (that cannot be tuned by the control fields).
H_ops: list of Qobj
A list of operators that can be tuned in the Hamiltonian via the
control fields.
R : int
Number of GRAPE iterations.
time : array / list
Array of time coordinates for control pulse evalutation.
u_start : array
Optional array with initial control pulse values.
Returns
-------
Instance of GRAPEResult, which contains the control pulses calculated
with GRAPE, a time-dependent Hamiltonian that is defined by the
control pulses, as well as the resulting propagator.
"""
if eps is None:
eps = 0.1 * (2 * np.pi) / (times[-1])
M = len(times)
J = len(H_ops)
u = np.zeros((R, J, M))
if u_limits and len(u_limits) != 2:
raise ValueError("u_limits must be a list with two values")
if u_limits:
warnings.warn("Caution: Using experimental feature u_limits")
if u_limits and u_start:
# make sure that no values in u0 violates the u_limits conditions
u_start = np.array(u_start)
u_start[u_start < u_limits[0]] = u_limits[0]
u_start[u_start > u_limits[1]] = u_limits[1]
if u_start is not None:
for idx, u0 in enumerate(u_start):
u[0, idx, :] = u0
if beta:
warnings.warn("Causion: Using experimental feature time-penalty")
progress_bar.start(R)
for r in range(R - 1):
progress_bar.update(r)
dt = times[1] - times[0]
if use_interp:
ip_funcs = [
interp1d(times,
u[r, j, :],
kind=interp_kind,
bounds_error=False,
fill_value=u[r, j, -1]) for j in range(J)
]
def _H_t(t, args=None):
return H0 + sum(
[float(ip_funcs[j](t)) * H_ops[j] for j in range(J)])
U_list = [(-1j * _H_t(times[idx]) * dt).expm()
for idx in range(M - 1)]
else:
def _H_idx(idx):
return H0 + sum([u[r, j, idx] * H_ops[j] for j in range(J)])
U_list = [(-1j * _H_idx(idx) * dt).expm() for idx in range(M - 1)]
U_f_list = []
U_b_list = []
U_f = 1
U_b = 1
for n in range(M - 1):
U_f = U_list[n] * U_f
U_f_list.append(U_f)
U_b_list.insert(0, U_b)
U_b = U_list[M - 2 - n].dag() * U_b
for j in range(J):
for m in range(M - 1):
P = U_b_list[m] * U
Q = 1j * dt * H_ops[j] * U_f_list[m]
if phase_sensitive:
du = -_overlap(P, Q)
else:
du = -2 * _overlap(P, Q) * _overlap(U_f_list[m], P)
if alpha:
# penalty term for high power control signals u
du += -2 * alpha * u[r, j, m] * dt
if beta:
# penalty term for late control signals u
du += -2 * beta * m * u[r, j, m] * dt
u[r + 1, j, m] = u[r, j, m] + eps * du.real
if u_limits:
if u[r + 1, j, m] < u_limits[0]:
u[r + 1, j, m] = u_limits[0]
elif u[r + 1, j, m] > u_limits[1]:
u[r + 1, j, m] = u_limits[1]
u[r + 1, j, -1] = u[r + 1, j, -2]
if use_interp:
ip_funcs = [
interp1d(times,
u[R - 1, j, :],
kind=interp_kind,
bounds_error=False,
fill_value=u[R - 1, j, -1]) for j in range(J)
]
H_td_func = [H0] + [[H_ops[j], lambda t, args, j=j: ip_funcs[j](t)]
for j in range(J)]
else:
H_td_func = [H0] + [[H_ops[j], u[-1, j, :]] for j in range(J)]
progress_bar.finished()
# return U_f_list[-1], H_td_func, u
return GRAPEResult(u=u, U_f=U_f_list[-1], H_t=H_td_func)
def propagator(H,
t,
c_op_list=[],
args={},
options=None,
parallel=False,
progress_bar=None,
**kwargs):
"""
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 : qobj or list
Hamiltonian as a Qobj instance of a nested list of Qobjs and
coefficients in the list-string or list-function format for
time-dependent Hamiltonians (see description in :func:`qutip.mesolve`).
t : float or array-like
Time or list of times for which to evaluate the propagator.
c_op_list : list
List of qobj collapse operators.
args : list/array/dictionary
Parameters to callback functions for time-dependent Hamiltonians and
collapse operators.
options : :class:`qutip.Options`
with options for the ODE solver.
parallel : bool {False, True}
Run the propagator in parallel mode.
progress_bar: BaseProgressBar
Optional instance of BaseProgressBar, or a subclass thereof, for
showing the progress of the simulation. By default no progress bar
is used, and if set to True a TextProgressBar will be used.
Returns
-------
a : qobj
Instance representing the propagator :math:`U(t)`.
"""
kw = _default_kwargs()
if 'num_cpus' in kwargs:
num_cpus = kwargs['num_cpus']
else:
num_cpus = kw['num_cpus']
if progress_bar is None:
progress_bar = BaseProgressBar()
elif progress_bar is True:
progress_bar = TextProgressBar()
if options is None:
options = Options()
options.rhs_reuse = True
rhs_clear()
if isinstance(t, (int, float, np.integer, np.floating)):
tlist = [0, t]
else:
tlist = t
td_type = _td_format_check(H, c_op_list, solver='me')[2]
if td_type > 0:
rhs_generate(H, c_op_list, args=args, options=options)
if isinstance(
H,
(types.FunctionType, types.BuiltinFunctionType, functools.partial)):
H0 = H(0.0, args)
elif isinstance(H, list):
H0 = H[0][0] if isinstance(H[0], list) else H[0]
else:
H0 = H
if len(c_op_list) == 0 and H0.isoper:
# calculate propagator for the wave function
N = H0.shape[0]
dims = H0.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
if parallel:
output = parallel_map(_parallel_sesolve,
range(N),
task_args=(N, H, tlist, args, options),
progress_bar=progress_bar,
num_cpus=num_cpus)
for n in range(N):
for k, t in enumerate(tlist):
u[:, n, k] = output[n].states[k].full().T
else:
progress_bar.start(N)
for n in range(0, N):
progress_bar.update(n)
psi0 = basis(N, n)
output = sesolve(H,
psi0,
tlist, [],
args,
options,
_safe_mode=False)
for k, t in enumerate(tlist):
u[:, n, k] = output.states[k].full().T
progress_bar.finished()
# todo: evolving a batch of wave functions:
# psi_0_list = [basis(N, n) for n in range(N)]
# psi_t_list = mesolve(H, psi_0_list, [0, t], [], [], args, options)
# for n in range(0, N):
# u[:,n] = psi_t_list[n][1].full().T
elif len(c_op_list) == 0 and H0.issuper:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
N = H0.shape[0]
sqrt_N = int(np.sqrt(N))
dims = H0.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
if parallel:
output = parallel_map(_parallel_mesolve,
range(N * N),
task_args=(sqrt_N, H, tlist, c_op_list, args,
options),
progress_bar=progress_bar,
num_cpus=num_cpus)
for n in range(N * N):
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output[n].states[k].full()).T
else:
progress_bar.start(N)
for n in range(0, N):
progress_bar.update(n)
col_idx, row_idx = np.unravel_index(n, (sqrt_N, sqrt_N))
rho0 = Qobj(
sp.csr_matrix(([1], ([row_idx], [col_idx])),
shape=(sqrt_N, sqrt_N),
dtype=complex))
output = mesolve(H,
rho0,
tlist, [], [],
args,
options,
_safe_mode=False)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
progress_bar.finished()
else:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
N = H0.shape[0]
dims = [H0.dims, H0.dims]
u = np.zeros([N * N, N * N, len(tlist)], dtype=complex)
if parallel:
output = parallel_map(_parallel_mesolve,
range(N * N),
task_args=(N, H, tlist, c_op_list, args,
options),
progress_bar=progress_bar,
num_cpus=num_cpus)
for n in range(N * N):
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output[n].states[k].full()).T
else:
progress_bar.start(N * N)
for n in range(N * N):
progress_bar.update(n)
col_idx, row_idx = np.unravel_index(n, (N, N))
rho0 = Qobj(
sp.csr_matrix(([1], ([row_idx], [col_idx])),
shape=(N, N),
dtype=complex))
output = mesolve(H,
rho0,
tlist,
c_op_list, [],
args,
options,
_safe_mode=False)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
progress_bar.finished()
if len(tlist) == 2:
return Qobj(u[:, :, 1], dims=dims)
else:
return np.array(
[Qobj(u[:, :, k], dims=dims) for k in range(len(tlist))],
dtype=object)
def mcsolve(H, psi0, tlist, c_ops, e_ops, ntraj=None,
args={}, options=None, progress_bar=True,
map_func=None, map_kwargs=None):
"""Monte Carlo evolution of a state vector :math:`|\psi \\rangle` for a
given Hamiltonian and sets of collapse operators, and possibly, operators
for calculating expectation values. Options for the underlying ODE solver
are given by the Options class.
mcsolve supports time-dependent Hamiltonians and collapse operators using
either Python functions of strings to represent time-dependent
coefficients. Note that, the system Hamiltonian MUST have at least one
constant term.
As an example of a time-dependent problem, consider a Hamiltonian with two
terms ``H0`` and ``H1``, where ``H1`` is time-dependent with coefficient
``sin(w*t)``, and collapse operators ``C0`` and ``C1``, where ``C1`` is
time-dependent with coeffcient ``exp(-a*t)``. Here, w and a are constant
arguments with values ``W`` and ``A``.
Using the Python function time-dependent format requires two Python
functions, one for each collapse coefficient. Therefore, this problem could
be expressed as::
def H1_coeff(t,args):
return sin(args['w']*t)
def C1_coeff(t,args):
return exp(-args['a']*t)
H = [H0, [H1, H1_coeff]]
c_ops = [C0, [C1, C1_coeff]]
args={'a': A, 'w': W}
or in String (Cython) format we could write::
H = [H0, [H1, 'sin(w*t)']]
c_ops = [C0, [C1, 'exp(-a*t)']]
args={'a': A, 'w': W}
Constant terms are preferably placed first in the Hamiltonian and collapse
operator lists.
Parameters
----------
H : :class:`qutip.Qobj`
System Hamiltonian.
psi0 : :class:`qutip.Qobj`
Initial state vector
tlist : array_like
Times at which results are recorded.
ntraj : int
Number of trajectories to run.
c_ops : array_like
single collapse operator or ``list`` or ``array`` of collapse
operators.
e_ops : array_like
single operator or ``list`` or ``array`` of operators for calculating
expectation values.
args : dict
Arguments for time-dependent Hamiltonian and collapse operator terms.
options : Options
Instance of ODE solver options.
progress_bar: BaseProgressBar
Optional instance of BaseProgressBar, or a subclass thereof, for
showing the progress of the simulation. Set to None to disable the
progress bar.
map_func: function
A map function for managing the calls to the single-trajactory solver.
map_kwargs: dictionary
Optional keyword arguments to the map_func function.
Returns
-------
results : :class:`qutip.solver.Result`
Object storing all results from the simulation.
.. note::
It is possible to reuse the random number seeds from a previous run
of the mcsolver by passing the output Result object seeds via the
Options class, i.e. Options(seeds=prev_result.seeds).
"""
if debug:
print(inspect.stack()[0][3])
if options is None:
options = Options()
if ntraj is None:
ntraj = options.ntraj
config.map_func = map_func if map_func is not None else parallel_map
config.map_kwargs = map_kwargs if map_kwargs is not None else {}
if not psi0.isket:
raise Exception("Initial state must be a state vector.")
if isinstance(c_ops, Qobj):
c_ops = [c_ops]
if isinstance(e_ops, Qobj):
e_ops = [e_ops]
if isinstance(e_ops, dict):
e_ops_dict = e_ops
e_ops = [e for e in e_ops.values()]
else:
e_ops_dict = None
config.options = options
if progress_bar:
if progress_bar is True:
config.progress_bar = TextProgressBar()
else:
config.progress_bar = progress_bar
else:
config.progress_bar = BaseProgressBar()
# set num_cpus to the value given in qutip.settings if none in Options
if not config.options.num_cpus:
config.options.num_cpus = qutip.settings.num_cpus
if config.options.num_cpus == 1:
# fallback on serial_map if num_cpu == 1, since there is no
# benefit of starting multiprocessing in this case
config.map_func = serial_map
# set initial value data
if options.tidy:
config.psi0 = psi0.tidyup(options.atol).full().ravel()
else:
config.psi0 = psi0.full().ravel()
config.psi0_dims = psi0.dims
config.psi0_shape = psi0.shape
# set options on ouput states
if config.options.steady_state_average:
config.options.average_states = True
# set general items
config.tlist = tlist
if isinstance(ntraj, (list, np.ndarray)):
config.ntraj = np.sort(ntraj)[-1]
else:
config.ntraj = ntraj
# set norm finding constants
config.norm_tol = options.norm_tol
config.norm_steps = options.norm_steps
# convert array based time-dependence to string format
H, c_ops, args = _td_wrap_array_str(H, c_ops, args, tlist)
# SETUP ODE DATA IF NONE EXISTS OR NOT REUSING
# --------------------------------------------
if not options.rhs_reuse or not config.tdfunc:
# reset config collapse and time-dependence flags to default values
config.soft_reset()
# check for type of time-dependence (if any)
time_type, h_stuff, c_stuff = _td_format_check(H, c_ops, 'mc')
c_terms = len(c_stuff[0]) + len(c_stuff[1]) + len(c_stuff[2])
# set time_type for use in multiprocessing
config.tflag = time_type
# check for collapse operators
if c_terms > 0:
config.cflag = 1
else:
config.cflag = 0
# Configure data
_mc_data_config(H, psi0, h_stuff, c_ops, c_stuff, args, e_ops,
options, config)
# compile and load cython functions if necessary
_mc_func_load(config)
else:
# setup args for new parameters when rhs_reuse=True and tdfunc is given
# string based
if config.tflag in [1, 10, 11]:
if any(args):
config.c_args = []
arg_items = list(args.items())
for k in range(len(arg_items)):
config.c_args.append(arg_items[k][1])
# function based
elif config.tflag in [2, 3, 20, 22]:
config.h_func_args = args
# load monte carlo class
mc = _MC(config)
# Run the simulation
mc.run()
# Remove RHS cython file if necessary
if not options.rhs_reuse and config.tdname:
_cython_build_cleanup(config.tdname)
# AFTER MCSOLVER IS DONE
# ----------------------
# Store results in the Result object
output = Result()
output.solver = 'mcsolve'
output.seeds = config.options.seeds
# state vectors
if (mc.psi_out is not None and config.options.average_states
and config.cflag and ntraj != 1):
output.states = parfor(_mc_dm_avg, mc.psi_out.T)
elif mc.psi_out is not None:
output.states = mc.psi_out
# expectation values
if (mc.expect_out is not None and config.cflag
and config.options.average_expect):
# averaging if multiple trajectories
if isinstance(ntraj, int):
output.expect = [np.mean(np.array([mc.expect_out[nt][op]
for nt in range(ntraj)],
dtype=object),
axis=0)
for op in range(config.e_num)]
elif isinstance(ntraj, (list, np.ndarray)):
output.expect = []
for num in ntraj:
expt_data = np.mean(mc.expect_out[:num], axis=0)
data_list = []
if any([not op.isherm for op in e_ops]):
for k in range(len(e_ops)):
if e_ops[k].isherm:
data_list.append(np.real(expt_data[k]))
else:
data_list.append(expt_data[k])
else:
data_list = [data for data in expt_data]
output.expect.append(data_list)
else:
# no averaging for single trajectory or if average_expect flag
# (Options) is off
if mc.expect_out is not None:
output.expect = mc.expect_out
# simulation parameters
output.times = config.tlist
output.num_expect = config.e_num
output.num_collapse = config.c_num
output.ntraj = config.ntraj
output.col_times = mc.collapse_times_out
output.col_which = mc.which_op_out
if e_ops_dict:
output.expect = {e: output.expect[n]
for n, e in enumerate(e_ops_dict.keys())}
return output
def odesolve(H, rho0, tlist, c_op_list, e_ops, args=None, options=None):
"""
Master equation evolution of a density matrix for a given Hamiltonian.
Evolution of a state vector or density matrix (`rho0`) for a given
Hamiltonian (`H`) and set of collapse operators (`c_op_list`), by
integrating the set of ordinary differential equations that define the
system. The output is either the state vector at arbitrary points in time
(`tlist`), or the expectation values of the supplied operators
(`e_ops`).
For problems with time-dependent Hamiltonians, `H` can be a callback
function that takes two arguments, time and `args`, and returns the
Hamiltonian at that point in time. `args` is a list of parameters that is
passed to the callback function `H` (only used for time-dependent
Hamiltonians).
Parameters
----------
H : :class:`qutip.qobj`
system Hamiltonian, or a callback function for time-dependent
Hamiltonians.
rho0 : :class:`qutip.qobj`
initial density matrix or state vector (ket).
tlist : *list* / *array*
list of times for :math:`t`.
c_op_list : list of :class:`qutip.qobj`
list of collapse operators.
e_ops : list of :class:`qutip.qobj` / callback function
list of operators for which to evaluate expectation values.
args : *dictionary*
dictionary of parameters for time-dependent Hamiltonians and
collapse operators.
options : :class:`qutip.Options`
with options for the ODE solver.
Returns
-------
output :array
Expectation values of wavefunctions/density matrices
for the times specified by `tlist`.
Notes
-----
On using callback function: odesolve transforms all :class:`qutip.qobj`
objects to sparse matrices before handing the problem to the integrator
function. In order for your callback function to work correctly, pass
all :class:`qutip.qobj` objects that are used in constructing the
Hamiltonian via args. odesolve will check for :class:`qutip.qobj` in
`args` and handle the conversion to sparse matrices. All other
:class:`qutip.qobj` objects that are not passed via `args` will be
passed on to the integrator to scipy who will raise an NotImplemented
exception.
Deprecated in QuTiP 2.0.0. Use :func:`mesolve` instead.
"""
warnings.warn("odesolve is deprecated since 2.0.0. Use mesolve instead.",
DeprecationWarning)
if debug:
print(inspect.stack()[0][3])
if options is None:
options = Options()
if (c_op_list and len(c_op_list) > 0) or not isket(rho0):
if isinstance(H, list):
output = _mesolve_list_td(H, rho0, tlist, c_op_list, e_ops, args,
options, BaseProgressBar())
if isinstance(
H, (types.FunctionType, types.BuiltinFunctionType, partial)):
output = _mesolve_func_td(H, rho0, tlist, c_op_list, e_ops, args,
options, BaseProgressBar())
else:
output = _mesolve_const(H, rho0, tlist, c_op_list, e_ops, args,
options, BaseProgressBar())
else:
if isinstance(H, list):
output = _sesolve_list_td(H, rho0, tlist, e_ops, args, options,
BaseProgressBar())
if isinstance(
H, (types.FunctionType, types.BuiltinFunctionType, partial)):
output = _sesolve_func_td(H, rho0, tlist, e_ops, args, options,
BaseProgressBar())
else:
output = _sesolve_const(H, rho0, tlist, e_ops, args, options,
BaseProgressBar())
if len(e_ops) > 0:
return output.expect
else:
return output.states
def mesolve(H,
rho0,
tlist,
c_ops=[],
e_ops=[],
args={},
options=None,
progress_bar=None,
_safe_mode=True):
"""
Master equation evolution of a density matrix for a given Hamiltonian and
set of collapse operators, or a Liouvillian.
Evolve the state vector or density matrix (`rho0`) using a given
Hamiltonian (`H`) and an [optional] set of collapse operators
(`c_ops`), by integrating the set of ordinary differential equations
that define the system. In the absence of collapse operators the system is
evolved according to the unitary evolution of the Hamiltonian.
The output is either the state vector at arbitrary points in time
(`tlist`), or the expectation values of the supplied operators
(`e_ops`). If e_ops is a callback function, it is invoked for each
time in `tlist` with time and the state as arguments, and the function
does not use any return values.
If either `H` or the Qobj elements in `c_ops` are superoperators, they
will be treated as direct contributions to the total system Liouvillian.
This allows to solve master equations that are not on standard Lindblad
form by passing a custom Liouvillian in place of either the `H` or `c_ops`
elements.
**Time-dependent operators**
For time-dependent problems, `H` and `c_ops` can be callback
functions that takes two arguments, time and `args`, and returns the
Hamiltonian or Liouvillian for the system at that point in time
(*callback format*).
Alternatively, `H` and `c_ops` can be a specified in a nested-list format
where each element in the list is a list of length 2, containing an
operator (:class:`qutip.qobj`) at the first element and where the
second element is either a string (*list string format*), a callback
function (*list callback format*) that evaluates to the time-dependent
coefficient for the corresponding operator, or a NumPy array (*list
array format*) which specifies the value of the coefficient to the
corresponding operator for each value of t in tlist.
*Examples*
H = [[H0, 'sin(w*t)'], [H1, 'sin(2*w*t)']]
H = [[H0, f0_t], [H1, f1_t]]
where f0_t and f1_t are python functions with signature f_t(t, args).
H = [[H0, np.sin(w*tlist)], [H1, np.sin(2*w*tlist)]]
In the *list string format* and *list callback format*, the string
expression and the callback function must evaluate to a real or complex
number (coefficient for the corresponding operator).
In all cases of time-dependent operators, `args` is a dictionary of
parameters that is used when evaluating operators. It is passed to the
callback functions as second argument.
**Additional options**
Additional options to mesolve can be set via the `options` argument, which
should be an instance of :class:`qutip.solver.Options`. Many ODE
integration options can be set this way, and the `store_states` and
`store_final_state` options can be used to store states even though
expectation values are requested via the `e_ops` argument.
.. note::
If an element in the list-specification of the Hamiltonian or
the list of collapse operators are in superoperator form it will be
added to the total Liouvillian of the problem with out further
transformation. This allows for using mesolve for solving master
equations that are not on standard Lindblad form.
.. note::
On using callback function: mesolve transforms all :class:`qutip.qobj`
objects to sparse matrices before handing the problem to the integrator
function. In order for your callback function to work correctly, pass
all :class:`qutip.qobj` objects that are used in constructing the
Hamiltonian via args. mesolve will check for :class:`qutip.qobj` in
`args` and handle the conversion to sparse matrices. All other
:class:`qutip.qobj` objects that are not passed via `args` will be
passed on to the integrator in scipy which will raise an NotImplemented
exception.
Parameters
----------
H : :class:`qutip.Qobj`
System Hamiltonian, or a callback function for time-dependent
Hamiltonians, or alternatively a system Liouvillian.
rho0 : :class:`qutip.Qobj`
initial density matrix or state vector (ket).
tlist : *list* / *array*
list of times for :math:`t`.
c_ops : list of :class:`qutip.Qobj`
single collapse operator, or list of collapse operators, or a list
of Liouvillian superoperators.
e_ops : list of :class:`qutip.Qobj` / callback function single
single operator or list of operators for which to evaluate
expectation values.
args : *dictionary*
dictionary of parameters for time-dependent Hamiltonians and
collapse operators.
options : :class:`qutip.Options`
with options for the solver.
progress_bar : BaseProgressBar
Optional instance of BaseProgressBar, or a subclass thereof, for
showing the progress of the simulation.
Returns
-------
result: :class:`qutip.Result`
An instance of the class :class:`qutip.Result`, which contains
either an *array* `result.expect` of expectation values for the times
specified by `tlist`, or an *array* `result.states` of state vectors or
density matrices corresponding to the times in `tlist` [if `e_ops` is
an empty list], or nothing if a callback function was given in place of
operators for which to calculate the expectation values.
"""
# check whether c_ops or e_ops is is a single operator
# if so convert it to a list containing only that operator
if isinstance(c_ops, Qobj):
c_ops = [c_ops]
if isinstance(e_ops, Qobj):
e_ops = [e_ops]
if isinstance(e_ops, dict):
e_ops_dict = e_ops
e_ops = [e for e in e_ops.values()]
else:
e_ops_dict = None
if _safe_mode:
_solver_safety_check(H, rho0, c_ops, e_ops, args)
if progress_bar is None:
progress_bar = BaseProgressBar()
elif progress_bar is True:
progress_bar = TextProgressBar()
# check if rho0 is a superoperator, in which case e_ops argument should
# be empty, i.e., e_ops = []
if issuper(rho0) and not e_ops == []:
raise TypeError("Must have e_ops = [] when initial condition rho0 is" +
" a superoperator.")
# convert array based time-dependence to string format
H, c_ops, args = _td_wrap_array_str(H, c_ops, args, tlist)
# check for type (if any) of time-dependent inputs
_, n_func, n_str = _td_format_check(H, c_ops)
if options is None:
options = Options()
if (not options.rhs_reuse) or (not config.tdfunc):
# reset config collapse and time-dependence flags to default values
config.reset()
#check if should use OPENMP
check_use_openmp(options)
res = None
#
# dispatch the appropriate solver
#
if ((c_ops and len(c_ops) > 0) or (not isket(rho0))
or (isinstance(H, Qobj) and issuper(H)) or
(isinstance(H, list) and isinstance(H[0], Qobj) and issuper(H[0]))):
#
# we have collapse operators, or rho0 is not a ket,
# or H is a Liouvillian
#
#
# find out if we are dealing with all-constant hamiltonian and
# collapse operators or if we have at least one time-dependent
# operator. Then delegate to appropriate solver...
#
if isinstance(H, Qobj):
# constant hamiltonian
if n_func == 0 and n_str == 0:
# constant collapse operators
res = _mesolve_const(H, rho0, tlist, c_ops, e_ops, args,
options, progress_bar)
elif n_str > 0:
# constant hamiltonian but time-dependent collapse
# operators in list string format
res = _mesolve_list_str_td([H], rho0, tlist, c_ops, e_ops,
args, options, progress_bar)
elif n_func > 0:
# constant hamiltonian but time-dependent collapse
# operators in list function format
res = _mesolve_list_func_td([H], rho0, tlist, c_ops, e_ops,
args, options, progress_bar)
elif isinstance(
H, (types.FunctionType, types.BuiltinFunctionType, partial)):
# function-callback style time-dependence: must have constant
# collapse operators
if n_str > 0: # or n_func > 0:
raise TypeError("Incorrect format: function-format " +
"Hamiltonian cannot be mixed with " +
"time-dependent collapse operators.")
else:
res = _mesolve_func_td(H, rho0, tlist, c_ops, e_ops, args,
options, progress_bar)
elif isinstance(H, list):
# determine if we are dealing with list of [Qobj, string] or
# [Qobj, function] style time-dependencies (for pure python and
# cython, respectively)
if n_func > 0:
res = _mesolve_list_func_td(H, rho0, tlist, c_ops, e_ops, args,
options, progress_bar)
else:
res = _mesolve_list_str_td(H, rho0, tlist, c_ops, e_ops, args,
options, progress_bar)
else:
raise TypeError("Incorrect specification of Hamiltonian " +
"or collapse operators.")
else:
#
# no collapse operators: unitary dynamics
#
if n_func > 0:
res = _sesolve_list_func_td(H, rho0, tlist, e_ops, args, options,
progress_bar)
elif n_str > 0:
res = _sesolve_list_str_td(H, rho0, tlist, e_ops, args, options,
progress_bar)
elif isinstance(
H, (types.FunctionType, types.BuiltinFunctionType, partial)):
res = _sesolve_func_td(H, rho0, tlist, e_ops, args, options,
progress_bar)
else:
res = _sesolve_const(H, rho0, tlist, e_ops, args, options,
progress_bar)
if e_ops_dict:
res.expect = {
e: res.expect[n]
for n, e in enumerate(e_ops_dict.keys())
}
return res
def sesolve(H,
psi0,
tlist,
e_ops=[],
args={},
options=Options(),
progress_bar=BaseProgressBar(),
_safe_mode=True):
"""
Schrodinger equation evolution of a state vector or unitary matrix
for a given Hamiltonian.
Evolve the state vector (`psi0`) using a given
Hamiltonian (`H`), by integrating the set of ordinary differential
equations that define the system. Alternatively evolve a unitary matrix in
solving the Schrodinger operator equation.
The output is either the state vector or unitary matrix at arbitrary points
in time (`tlist`), or the expectation values of the supplied operators
(`e_ops`). If e_ops is a callback function, it is invoked for each
time in `tlist` with time and the state as arguments, and the function
does not use any return values. e_ops cannot be used in conjunction
with solving the Schrodinger operator equation
Parameters
----------
H : :class:`qutip.qobj`, :class:`qutip.qobjevo`, *list*, *callable*
system Hamiltonian as a Qobj, list of Qobj and coefficient, QobjEvo,
or a callback function for time-dependent Hamiltonians.
list format and options can be found in QobjEvo's description.
psi0 : :class:`qutip.qobj`
initial state vector (ket)
or initial unitary operator `psi0 = U`
tlist : *list* / *array*
list of times for :math:`t`.
e_ops : list of :class:`qutip.qobj` / callback function
single operator or list of operators for which to evaluate
expectation values.
For list operator evolution, the overlapse is computed:
tr(e_ops[i].dag()*op(t))
args : *dictionary*
dictionary of parameters for time-dependent Hamiltonians
options : :class:`qutip.Qdeoptions`
with options for the ODE solver.
progress_bar : BaseProgressBar
Optional instance of BaseProgressBar, or a subclass thereof, for
showing the progress of the simulation.
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`, or
an *array* or state vectors corresponding to the
times in `tlist` [if `e_ops` is an empty list], or
nothing if a callback function was given inplace of operators for
which to calculate the expectation values.
"""
if isinstance(e_ops, Qobj):
e_ops = [e_ops]
elif isinstance(e_ops, dict):
e_ops_dict = e_ops
e_ops = [e for e in e_ops.values()]
else:
e_ops_dict = None
if progress_bar is True:
progress_bar = TextProgressBar()
if not (psi0.isket or psi0.isunitary):
raise TypeError("The unitary solver requires psi0 to be"
" a ket as initial state"
" or a unitary as initial operator.")
if options.rhs_reuse and not isinstance(H, SolverSystem):
# TODO: deprecate when going to class based solver.
if "sesolve" in solver_safe:
# print(" ")
H = solver_safe["sesolve"]
else:
pass
# raise Exception("Could not find the Hamiltonian to reuse.")
#check if should use OPENMP
check_use_openmp(options)
if isinstance(H, SolverSystem):
ss = H
elif isinstance(H, (list, Qobj, QobjEvo)):
ss = _sesolve_QobjEvo(H, tlist, args, options)
elif callable(H):
ss = _sesolve_func_td(H, args, options)
else:
raise Exception("Invalid H type")
func, ode_args = ss.makefunc(ss, psi0, args, options)
if _safe_mode:
v = psi0.full().ravel('F')
func(0., v, *ode_args) + v
res = _generic_ode_solve(func,
ode_args,
psi0,
tlist,
e_ops,
options,
progress_bar,
dims=psi0.dims)
if e_ops_dict:
res.expect = {
e: res.expect[n]
for n, e in enumerate(e_ops_dict.keys())
}
res.SolverSystem = ss
return res
def propagator(H,
t,
c_op_list=[],
args={},
options=None,
unitary_mode='batch',
parallel=False,
progress_bar=None,
_safe_mode=True,
**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 : qobj or list
Hamiltonian as a Qobj instance of a nested list of Qobjs and
coefficients in the list-string or list-function format for
time-dependent Hamiltonians (see description in :func:`qutip.mesolve`).
t : float or array-like
Time or list of times for which to evaluate the propagator.
c_op_list : list
List of qobj collapse operators.
args : list/array/dictionary
Parameters to callback functions for time-dependent Hamiltonians and
collapse operators.
options : :class:`qutip.Options`
with options for the ODE solver.
unitary_mode = str ('batch', 'single')
Solve all basis vectors simulaneously ('batch') or individually
('single').
parallel : bool {False, True}
Run the propagator in parallel mode. This will override the
unitary_mode settings if set to True.
progress_bar: BaseProgressBar
Optional instance of BaseProgressBar, or a subclass thereof, for
showing the progress of the simulation. By default no progress bar
is used, and if set to True a TextProgressBar will be used.
Returns
-------
a : qobj
Instance representing the propagator :math:`U(t)`.
"""
kw = _default_kwargs()
if 'num_cpus' in kwargs:
num_cpus = kwargs['num_cpus']
else:
num_cpus = kw['num_cpus']
if progress_bar is None:
progress_bar = BaseProgressBar()
elif progress_bar is True:
progress_bar = TextProgressBar()
if options is None:
options = Options()
options.rhs_reuse = True
rhs_clear()
if isinstance(t, (int, float, np.integer, np.floating)):
tlist = [0, t]
else:
tlist = t
if _safe_mode:
_solver_safety_check(H, None, c_ops=c_op_list, e_ops=[], args=args)
td_type = _td_format_check(H, c_op_list, solver='me')
if isinstance(
H,
(types.FunctionType, types.BuiltinFunctionType, functools.partial)):
H0 = H(0.0, args)
if unitary_mode == 'batch':
# batch don't work with function Hamiltonian
unitary_mode = 'single'
elif isinstance(H, list):
H0 = H[0][0] if isinstance(H[0], list) else H[0]
else:
H0 = H
if len(c_op_list) == 0 and H0.isoper:
# calculate propagator for the wave function
N = H0.shape[0]
dims = H0.dims
if parallel:
unitary_mode = 'single'
u = np.zeros([N, N, len(tlist)], dtype=complex)
output = parallel_map(_parallel_sesolve,
range(N),
task_args=(N, H, tlist, args, options),
progress_bar=progress_bar,
num_cpus=num_cpus)
for n in range(N):
for k, t in enumerate(tlist):
u[:, n, k] = output[n].states[k].full().T
else:
if unitary_mode == 'single':
output = sesolve(H,
qeye(dims[0]),
tlist, [],
args,
options,
_safe_mode=False)
if len(tlist) == 2:
return output.states[-1]
else:
return output.states
elif unitary_mode == 'batch':
u = np.zeros(len(tlist), dtype=object)
_rows = np.array([(N + 1) * m for m in range(N)])
_cols = np.zeros_like(_rows)
_data = np.ones_like(_rows, dtype=complex)
psi0 = Qobj(sp.coo_matrix((_data, (_rows, _cols))).tocsr())
if td_type[1] > 0 or td_type[2] > 0:
H2 = []
for k in range(len(H)):
if isinstance(H[k], list):
H2.append([tensor(qeye(N), H[k][0]), H[k][1]])
else:
H2.append(tensor(qeye(N), H[k]))
else:
H2 = tensor(qeye(N), H)
options.normalize_output = False
output = sesolve(H2,
psi0,
tlist, [],
args=args,
options=options,
_safe_mode=False)
for k, t in enumerate(tlist):
u[k] = sp_reshape(output.states[k].data, (N, N))
unit_row_norm(u[k].data, u[k].indptr, u[k].shape[0])
u[k] = u[k].T.tocsr()
else:
raise Exception('Invalid unitary mode.')
elif len(c_op_list) == 0 and H0.issuper:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
unitary_mode = 'single'
N = H0.shape[0]
sqrt_N = int(np.sqrt(N))
dims = H0.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
if parallel:
output = parallel_map(_parallel_mesolve,
range(N * N),
task_args=(sqrt_N, H, tlist, c_op_list, args,
options),
progress_bar=progress_bar,
num_cpus=num_cpus)
for n in range(N * N):
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output[n].states[k].full()).T
else:
rho0 = qeye(N, N)
rho0.dims = [[sqrt_N, sqrt_N], [sqrt_N, sqrt_N]]
output = mesolve(H,
psi0,
tlist, [],
args,
options,
_safe_mode=False)
if len(tlist) == 2:
return output.states[-1]
else:
return output.states
else:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
unitary_mode = 'single'
N = H0.shape[0]
dims = [H0.dims, H0.dims]
u = np.zeros([N * N, N * N, len(tlist)], dtype=complex)
if parallel:
output = parallel_map(_parallel_mesolve,
range(N * N),
task_args=(N, H, tlist, c_op_list, args,
options),
progress_bar=progress_bar,
num_cpus=num_cpus)
for n in range(N * N):
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output[n].states[k].full()).T
else:
progress_bar.start(N * N)
for n in range(N * N):
progress_bar.update(n)
col_idx, row_idx = np.unravel_index(n, (N, N))
rho0 = Qobj(
sp.csr_matrix(([1], ([row_idx], [col_idx])),
shape=(N, N),
dtype=complex))
output = mesolve(H,
rho0,
tlist,
c_op_list, [],
args,
options,
_safe_mode=False)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
progress_bar.finished()
if len(tlist) == 2:
if unitary_mode == 'batch':
return Qobj(u[-1], dims=dims)
else:
return Qobj(u[:, :, 1], dims=dims)
else:
if unitary_mode == 'batch':
return np.array([Qobj(u[k], dims=dims) for k in range(len(tlist))],
dtype=object)
else:
return np.array(
[Qobj(u[:, :, k], dims=dims) for k in range(len(tlist))],
dtype=object)
def sesolve(H, psi0, tlist, e_ops=None, args=None, options=None,
progress_bar=None, _safe_mode=True):
"""
Schrödinger equation evolution of a state vector or unitary matrix for a
given Hamiltonian.
Evolve the state vector (``psi0``) using a given Hamiltonian (``H``), by
integrating the set of ordinary differential equations that define the
system. Alternatively evolve a unitary matrix in solving the Schrodinger
operator equation.
The output is either the state vector or unitary matrix at arbitrary points
in time (``tlist``), or the expectation values of the supplied operators
(``e_ops``). If ``e_ops`` is a callback function, it is invoked for each
time in ``tlist`` with time and the state as arguments, and the function
does not use any return values. ``e_ops`` cannot be used in conjunction
with solving the Schrodinger operator equation
Parameters
----------
H : :class:`~Qobj`, :class:`~QobjEvo`, list, or callable
System Hamiltonian as a :obj:`~Qobj , list of :obj:`Qobj` and
coefficient, :obj:`~QObjEvo`, or a callback function for time-dependent
Hamiltonians. List format and options can be found in QobjEvo's
description.
psi0 : :class:`~Qobj`
Initial state vector (ket) or initial unitary operator ``psi0 = U``.
tlist : array_like of float
List of times for :math:`t`.
e_ops : None / list / callback function, optional
A list of operators as `Qobj` and/or callable functions (can be mixed)
or a single callable function. For callable functions, they are called
as ``f(t, state)`` and return the expectation value. A single
callback's expectation value can be any type, but a callback part of a
list must return a number as the expectation value. For operators, the
result's expect will be computed by :func:`qutip.expect` when the state
is a ``ket``. For operator evolution, the overlap is computed by: ::
(e_ops[i].dag() * op(t)).tr()
args : dict, optional
Dictionary of scope parameters for time-dependent Hamiltonians.
options : :obj:`~solver.Options`, optional
Options for the ODE solver.
progress_bar : :obj:`~BaseProgressBar`, optional
Optional instance of :obj:`~BaseProgressBar`, or a subclass thereof,
for showing the progress of the simulation.
Returns
-------
output: :class:`~solver.Result`
An instance of the class :class:`~solver.Result`, which contains either
an array of expectation values for the times specified by ``tlist``, or
an array or state vectors corresponding to the times in ``tlist`` (if
``e_ops`` is an empty list), or nothing if a callback function was
given inplace of operators for which to calculate the expectation
values.
"""
if e_ops is None:
e_ops = []
if isinstance(e_ops, Qobj):
e_ops = [e_ops]
elif isinstance(e_ops, dict):
e_ops_dict = e_ops
e_ops = [e for e in e_ops.values()]
else:
e_ops_dict = None
if progress_bar is None:
progress_bar = BaseProgressBar()
if progress_bar is True:
progress_bar = TextProgressBar()
if not (psi0.isket or psi0.isunitary):
raise TypeError("The unitary solver requires psi0 to be"
" a ket as initial state"
" or a unitary as initial operator.")
if options is None:
options = Options()
if options.rhs_reuse and not isinstance(H, SolverSystem):
# TODO: deprecate when going to class based solver.
if "sesolve" in solver_safe:
H = solver_safe["sesolve"]
if args is None:
args = {}
check_use_openmp(options)
if isinstance(H, SolverSystem):
ss = H
elif isinstance(H, (list, Qobj, QobjEvo)):
ss = _sesolve_QobjEvo(H, tlist, args, options)
elif callable(H):
ss = _sesolve_func_td(H, args, options)
else:
raise Exception("Invalid H type")
func, ode_args = ss.makefunc(ss, psi0, args, e_ops, options)
if _safe_mode:
v = psi0.full().ravel('F')
func(0., v, *ode_args) + v
res = _generic_ode_solve(func, ode_args, psi0, tlist, e_ops, options,
progress_bar, dims=psi0.dims)
if e_ops_dict:
res.expect = {e: res.expect[n]
for n, e in enumerate(e_ops_dict.keys())}
return res
def parallel_map(task, values, task_args=tuple(), task_kwargs={}, **kwargs):
"""
Parallel execution of a mapping of `values` to the function `task`. This
is functionally equivalent to::
result = [task(value, *task_args, **task_kwargs) for value in values]
Parameters
----------
task : a Python function
The function that is to be called for each value in ``task_vec``.
values : array / list
The list or array of values for which the ``task`` function is to be
evaluated.
task_args : list / dictionary
The optional additional argument to the ``task`` function.
task_kwargs : list / dictionary
The optional additional keyword argument to the ``task`` function.
progress_bar : ProgressBar
Progress bar class instance for showing progress.
Returns
--------
result : list
The result list contains the value of
``task(value, *task_args, **task_kwargs)`` for
each value in ``values``.
"""
os.environ['QUTIP_IN_PARALLEL'] = 'TRUE'
kw = _default_kwargs()
if 'num_cpus' in kwargs:
kw['num_cpus'] = kwargs['num_cpus']
try:
progress_bar = kwargs['progress_bar']
if progress_bar is True:
progress_bar = TextProgressBar()
except:
progress_bar = BaseProgressBar()
progress_bar.start(len(values))
nfinished = [0]
def _update_progress_bar(x):
nfinished[0] += 1
progress_bar.update(nfinished[0])
try:
pool = Pool(processes=kw['num_cpus'])
async_res = [pool.apply_async(task, (value,) + task_args, task_kwargs,
_update_progress_bar)
for value in values]
while not all([ar.ready() for ar in async_res]):
for ar in async_res:
ar.wait(timeout=0.1)
pool.terminate()
pool.join()
except KeyboardInterrupt as e:
os.environ['QUTIP_IN_PARALLEL'] = 'FALSE'
pool.terminate()
pool.join()
raise e
progress_bar.finished()
os.environ['QUTIP_IN_PARALLEL'] = 'FALSE'
return [ar.get() for ar in async_res]
def sesolve(H, psi0, tlist, e_ops=[], args={}, options=None,
progress_bar=None,
_safe_mode=True):
"""
Schrodinger equation evolution of a state vector or unitary matrix
for a given Hamiltonian.
Evolve the state vector (`psi0`) using a given
Hamiltonian (`H`), by integrating the set of ordinary differential
equations that define the system. Alternatively evolve a unitary matrix in
solving the Schrodinger operator equation.
The output is either the state vector or unitary matrix at arbitrary points
in time (`tlist`), or the expectation values of the supplied operators
(`e_ops`). If e_ops is a callback function, it is invoked for each
time in `tlist` with time and the state as arguments, and the function
does not use any return values. e_ops cannot be used in conjunction
with solving the Schrodinger operator equation
Parameters
----------
H : :class:`qutip.qobj`
system Hamiltonian, or a callback function for time-dependent
Hamiltonians.
psi0 : :class:`qutip.qobj`
initial state vector (ket)
or initial unitary operator `psi0 = U`
tlist : *list* / *array*
list of times for :math:`t`.
e_ops : list of :class:`qutip.qobj` / callback function single
single operator or list of operators for which to evaluate
expectation values.
Must be empty list operator evolution
args : *dictionary*
dictionary of parameters for time-dependent Hamiltonians
options : :class:`qutip.Qdeoptions`
with options for the ODE solver.
progress_bar : BaseProgressBar
Optional instance of BaseProgressBar, or a subclass thereof, for
showing the progress of the simulation.
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`, or
an *array* or state vectors corresponding to the
times in `tlist` [if `e_ops` is an empty list], or
nothing if a callback function was given inplace of operators for
which to calculate the expectation values.
"""
# check initial state: must be a state vector
if _safe_mode:
if not isinstance(psi0, Qobj):
raise TypeError("psi0 must be Qobj")
if psi0.isket:
pass
elif psi0.isunitary:
if not e_ops == []:
raise TypeError("Must have e_ops = [] when initial condition"
" psi0 is a unitary operator.")
else:
raise TypeError("The unitary solver requires psi0 to be"
" a ket as initial state"
" or a unitary as initial operator.")
_solver_safety_check(H, psi0, c_ops=[], e_ops=e_ops, args=args)
if isinstance(e_ops, Qobj):
e_ops = [e_ops]
if isinstance(e_ops, dict):
e_ops_dict = e_ops
e_ops = [e for e in e_ops.values()]
else:
e_ops_dict = None
if progress_bar is None:
progress_bar = BaseProgressBar()
elif progress_bar is True:
progress_bar = TextProgressBar()
# convert array based time-dependence to string format
H, _, args = _td_wrap_array_str(H, [], args, tlist)
# check for type (if any) of time-dependent inputs
n_const, n_func, n_str = _td_format_check(H, [])
if options is None:
options = Options()
if (not options.rhs_reuse) or (not config.tdfunc):
# reset config time-dependence flags to default values
config.reset()
#check if should use OPENMP
check_use_openmp(options)
if n_func > 0:
res = _sesolve_list_func_td(H, psi0, tlist, e_ops, args, options,
progress_bar)
elif n_str > 0:
res = _sesolve_list_str_td(H, psi0, tlist, e_ops, args, options,
progress_bar)
elif isinstance(H, (types.FunctionType,
types.BuiltinFunctionType,
partial)):
res = _sesolve_func_td(H, psi0, tlist, e_ops, args, options,
progress_bar)
elif isinstance(H, Qobj):
res = _sesolve_const(H, psi0, tlist, e_ops, args, options,
progress_bar)
else:
raise TypeError("Invalid Hamiltonian specification")
if e_ops_dict:
res.expect = {e: res.expect[n]
for n, e in enumerate(e_ops_dict.keys())}
return res
def run(self, num_traj=0, psi0=None, tlist=None,
args={}, e_ops=None, options=None,
progress_bar=True,
map_func=parallel_map, map_kwargs={}):
# %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
# 4 situation for run:
# - first run
# - change parameters
# - add trajectories
# (self.add_traj) Not Implemented
# - continue from the last time and states
# (self.continue_runs) Not Implemented
# %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
options = options if options is not None else self.options
if self.ran and tlist[0] == self.t:
# psi0 is ignored since we restart from a
# different states for each trajectories
self.continue_runs(num_traj, tlist, args, e_ops, options,
progress_bar, map_func, map_kwargs)
return
if args and args != self.ss.args:
self.ss.set_args(self.ss, args)
self.reset()
if e_ops and e_ops != self.e_ops:
self.set_e_ops(e_ops)
self.reset()
if psi0 is not None and psi0 != self.psi0:
self.psi0 = psi0
self.reset()
tlist = np.array(tlist)
if tlist is not None and np.all(tlist != self.tlist):
self.tlist = tlist
self.reset()
if self.ran:
if options.store_states and self._psi_out[0].shape[0] == 1:
self.reset()
else:
# if not reset here, add trajectories
self.add_traj(num_traj, progress_bar, map_func, map_kwargs)
return
if not num_traj:
num_traj = options.ntraj
if options.num_cpus == 1 or num_traj == 1:
map_func = serial_map
if len(self.seeds) != num_traj:
self.seed(num_traj, self.seeds)
if not progress_bar:
progress_bar = BaseProgressBar()
elif progress_bar is True:
progress_bar = TextProgressBar()
# set arguments for input to monte carlo
map_kwargs_ = {'progress_bar': progress_bar,
'num_cpus': options.num_cpus}
map_kwargs_.update(map_kwargs)
map_kwargs = map_kwargs_
if self.e_ops is None:
self.set_e_ops()
if self.ss.type == "Diagonal":
results = map_func(self._single_traj_diag, list(range(num_traj)), **map_kwargs)
else:
results = map_func(self._single_traj, list(range(num_traj)), **map_kwargs)
self.t = self.tlist[-1]
self.num_traj = num_traj
self.ran = True
for result in results:
state_out, ss_out, expect, collapse = result
self._psi_out.append(state_out)
self._ss_out.append(ss_out)
self._expect_out.append(expect)
self._collapse.append(collapse)
self._psi_out = np.stack(self._psi_out)
self._ss_out = np.stack(self._ss_out)
def propagator(H, t, c_op_list=[], args={}, options=None,
parallel=False, progress_bar=None, **kwargs):
"""
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 : qobj or list
Hamiltonian as a Qobj instance of a nested list of Qobjs and
coefficients in the list-string or list-function format for
time-dependent Hamiltonians (see description in :func:`qutip.mesolve`).
t : float or array-like
Time or list of times for which to evaluate the propagator.
c_op_list : list
List of qobj collapse operators.
args : list/array/dictionary
Parameters to callback functions for time-dependent Hamiltonians and
collapse operators.
options : :class:`qutip.Options`
with options for the ODE solver.
parallel : bool {False, True}
Run the propagator in parallel mode.
progress_bar: BaseProgressBar
Optional instance of BaseProgressBar, or a subclass thereof, for
showing the progress of the simulation. By default no progress bar
is used, and if set to True a TextProgressBar will be used.
Returns
-------
a : qobj
Instance representing the propagator :math:`U(t)`.
"""
kw = _default_kwargs()
if 'num_cpus' in kwargs:
num_cpus = kwargs['num_cpus']
else:
num_cpus = kw['num_cpus']
if progress_bar is None:
progress_bar = BaseProgressBar()
elif progress_bar is True:
progress_bar = TextProgressBar()
if options is None:
options = Options()
options.rhs_reuse = True
rhs_clear()
if isinstance(t, (int, float, np.integer, np.floating)):
tlist = [0, t]
else:
tlist = t
td_type = _td_format_check(H, c_op_list, solver='me')[2]
if td_type > 0:
rhs_generate(H, c_op_list, args=args, options=options)
if isinstance(H, (types.FunctionType, types.BuiltinFunctionType,
functools.partial)):
H0 = H(0.0, args)
elif isinstance(H, list):
H0 = H[0][0] if isinstance(H[0], list) else H[0]
else:
H0 = H
if len(c_op_list) == 0 and H0.isoper:
# calculate propagator for the wave function
N = H0.shape[0]
dims = H0.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
if parallel:
output = parallel_map(_parallel_sesolve,range(N),
task_args=(N,H,tlist,args,options),
progress_bar=progress_bar, num_cpus=num_cpus)
for n in range(N):
for k, t in enumerate(tlist):
u[:, n, k] = output[n].states[k].full().T
else:
progress_bar.start(N)
for n in range(0, N):
progress_bar.update(n)
psi0 = basis(N, n)
output = sesolve(H, psi0, tlist, [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = output.states[k].full().T
progress_bar.finished()
# todo: evolving a batch of wave functions:
# psi_0_list = [basis(N, n) for n in range(N)]
# psi_t_list = mesolve(H, psi_0_list, [0, t], [], [], args, options)
# for n in range(0, N):
# u[:,n] = psi_t_list[n][1].full().T
elif len(c_op_list) == 0 and H0.issuper:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
N = H0.shape[0]
sqrt_N = int(np.sqrt(N))
dims = H0.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
if parallel:
output = parallel_map(_parallel_mesolve,range(N * N),
task_args=(sqrt_N,H,tlist,c_op_list,args,options),
progress_bar=progress_bar, num_cpus=num_cpus)
for n in range(N * N):
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output[n].states[k].full()).T
else:
progress_bar.start(N)
for n in range(0, N):
progress_bar.update(n)
col_idx, row_idx = np.unravel_index(n,(sqrt_N,sqrt_N))
rho0 = Qobj(sp.csr_matrix(([1],([row_idx],[col_idx])), shape=(sqrt_N,sqrt_N), dtype=complex))
output = mesolve(H, rho0, tlist, [], [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
progress_bar.finished()
else:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
N = H0.shape[0]
dims = [H0.dims, H0.dims]
u = np.zeros([N * N, N * N, len(tlist)], dtype=complex)
if parallel:
output = parallel_map(_parallel_mesolve,range(N * N),
task_args=(N,H,tlist,c_op_list,args,options),
progress_bar=progress_bar, num_cpus=num_cpus)
for n in range(N * N):
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output[n].states[k].full()).T
else:
progress_bar.start(N * N)
for n in range(N * N):
progress_bar.update(n)
col_idx, row_idx = np.unravel_index(n,(N,N))
rho0 = Qobj(sp.csr_matrix(([1],([row_idx],[col_idx])), shape=(N,N), dtype=complex))
output = mesolve(H, rho0, tlist, c_op_list, [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
progress_bar.finished()
if len(tlist) == 2:
return Qobj(u[:, :, 1], dims=dims)
else:
return np.array([Qobj(u[:, :, k], dims=dims) for k in range(len(tlist))], dtype=object)
def brmesolve(H,
psi0,
tlist,
a_ops=[],
e_ops=[],
c_ops=[],
args={},
use_secular=True,
tol=qset.atol,
spectra_cb=None,
options=None,
progress_bar=None,
_safe_mode=True):
"""
Solves for the dynamics of a system using the Bloch-Redfield master equation,
given an input Hamiltonian, Hermitian bath-coupling terms and their associated
spectrum functions, as well as possible Lindblad collapse operators.
For time-independent systems, the Hamiltonian must be given as a Qobj,
whereas the bath-coupling terms (a_ops), must be written as a nested list
of operator - spectrum function pairs, where the frequency is specified by
the `w` variable.
*Example*
a_ops = [[a+a.dag(),lambda w: 0.2*(w>=0)]]
For time-dependent systems, the Hamiltonian, a_ops, and Lindblad collapse
operators (c_ops), can be specified in the QuTiP string-based time-dependent
format. For the a_op spectra, the frequency variable must be `w`, and the
string cannot contain any other variables other than the possibility of having
a time-dependence through the time variable `t`:
*Example*
a_ops = [[a+a.dag(), '0.2*exp(-t)*(w>=0)']]
Parameters
----------
H : Qobj / list
System Hamiltonian given as a Qobj or
nested list in string-based format.
psi0: Qobj
Initial density matrix or state vector (ket).
tlist : array_like
List of times for evaluating evolution
a_ops : list
Nested list of Hermitian system operators that couple to
the bath degrees of freedom, along with their associated
spectra.
e_ops : list
List of operators for which to evaluate expectation values.
c_ops : list
List of system collapse operators, or nested list in
string-based format.
args : dict (not implimented)
Placeholder for future implementation, kept for API consistency.
use_secular : bool {True}
Use secular approximation when evaluating bath-coupling terms.
tol : float {qutip.setttings.atol}
Tolerance used for removing small values after
basis transformation.
spectra_cb : list
DEPRECIATED. Do not use.
options : :class:`qutip.solver.Options`
Options for the solver.
progress_bar : BaseProgressBar
Optional instance of BaseProgressBar, or a subclass thereof, for
showing the progress of the simulation.
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 isinstance(c_ops, Qobj):
c_ops = [c_ops]
if isinstance(e_ops, Qobj):
e_ops = [e_ops]
if isinstance(e_ops, dict):
e_ops_dict = e_ops
e_ops = [e for e in e_ops.values()]
else:
e_ops_dict = None
if not (spectra_cb is None):
warnings.warn("The use of spectra_cb is depreciated.",
DeprecationWarning)
_a_ops = []
for kk, a in enumerate(a_ops):
_a_ops.append([a, spectra_cb[kk]])
a_ops = _a_ops
if _safe_mode:
_solver_safety_check(H, psi0, a_ops + c_ops, e_ops, args)
# check for type (if any) of time-dependent inputs
_, n_func, n_str = _td_format_check(H, a_ops + c_ops)
if progress_bar is None:
progress_bar = BaseProgressBar()
elif progress_bar is True:
progress_bar = TextProgressBar()
if options is None:
options = Options()
if (not options.rhs_reuse) or (not config.tdfunc):
# reset config collapse and time-dependence flags to default values
config.reset()
#check if should use OPENMP
check_use_openmp(options)
if n_str == 0:
R, ekets = bloch_redfield_tensor(H,
a_ops,
spectra_cb=None,
c_ops=c_ops)
output = Result()
output.solver = "brmesolve"
output.times = tlist
results = bloch_redfield_solve(R,
ekets,
psi0,
tlist,
e_ops,
options,
progress_bar=progress_bar)
if e_ops:
output.expect = results
else:
output.states = results
return output
elif n_str != 0 and n_func == 0:
output = _td_brmesolve(H,
psi0,
tlist,
a_ops=a_ops,
e_ops=e_ops,
c_ops=c_ops,
use_secular=use_secular,
tol=tol,
options=options,
progress_bar=progress_bar,
_safe_mode=_safe_mode)
return output
else:
raise Exception('Cannot mix func and str formats.')
def sesolve(H, rho0, tlist, e_ops, args={}, options=None,
progress_bar=BaseProgressBar()):
"""
Schrodinger equation evolution of a state vector for a given Hamiltonian.
Evolve the state vector or density matrix (`rho0`) using a given
Hamiltonian (`H`), by integrating the set of ordinary differential
equations that define the system.
The output is either the state vector at arbitrary points in time
(`tlist`), or the expectation values of the supplied operators
(`e_ops`). If e_ops is a callback function, it is invoked for each
time in `tlist` with time and the state as arguments, and the function
does not use any return values.
Parameters
----------
H : :class:`qutip.qobj`
system Hamiltonian, or a callback function for time-dependent
Hamiltonians.
rho0 : :class:`qutip.qobj`
initial density matrix or state vector (ket).
tlist : *list* / *array*
list of times for :math:`t`.
e_ops : list of :class:`qutip.qobj` / callback function single
single operator or list of operators for which to evaluate
expectation values.
args : *dictionary*
dictionary of parameters for time-dependent Hamiltonians and
collapse operators.
options : :class:`qutip.Qdeoptions`
with options for the ODE solver.
Returns
-------
output: :class:`qutip.odedata`
An instance of the class :class:`qutip.odedata`, which contains either
an *array* of expectation values for the times specified by `tlist`, or
an *array* or state vectors or density matrices corresponding to the
times in `tlist` [if `e_ops` is an empty list], or
nothing if a callback function was given inplace of operators for
which to calculate the expectation values.
"""
if isinstance(e_ops, Qobj):
e_ops = [e_ops]
if isinstance(e_ops, dict):
e_ops_dict = e_ops
e_ops = [e for e in e_ops.values()]
else:
e_ops_dict = None
# check for type (if any) of time-dependent inputs
n_const, n_func, n_str = _ode_checks(H, [])
if options is None:
options = Odeoptions()
if (not options.rhs_reuse) or (not odeconfig.tdfunc):
# reset odeconfig time-dependence flags to default values
odeconfig.reset()
if n_func > 0:
res = _sesolve_list_func_td(H, rho0, tlist, e_ops, args, options,
progress_bar)
elif n_str > 0:
res = _sesolve_list_str_td(H, rho0, tlist, e_ops, args, options,
progress_bar)
elif isinstance(H, (types.FunctionType,
types.BuiltinFunctionType,
partial)):
res = _sesolve_func_td(H, rho0, tlist, e_ops, args, options,
progress_bar)
else:
res = _sesolve_const(H, rho0, tlist, e_ops, args, options,
progress_bar)
if e_ops_dict:
res.expect = {e: res.expect[n] for n, e in enumerate(e_ops_dict.keys())}
return res
def propagator(H,
t,
c_op_list,
args=None,
options=None,
sparse=False,
progress_bar=None):
"""
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 : qobj or list
Hamiltonian as a Qobj instance of a nested list of Qobjs and
coefficients in the list-string or list-function format for
time-dependent Hamiltonians (see description in :func:`qutip.mesolve`).
t : float or array-like
Time or list of times for which to evaluate the propagator.
c_op_list : list
List of qobj collapse operators.
args : list/array/dictionary
Parameters to callback functions for time-dependent Hamiltonians and
collapse operators.
options : :class:`qutip.Options`
with options for the ODE solver.
progress_bar: BaseProgressBar
Optional instance of BaseProgressBar, or a subclass thereof, for
showing the progress of the simulation. By default no progress bar
is used, and if set to True a TextProgressBar will be used.
Returns
-------
a : qobj
Instance representing the propagator :math:`U(t)`.
"""
if progress_bar is None:
progress_bar = BaseProgressBar()
elif progress_bar is True:
progress_bar = TextProgressBar()
if options is None:
options = Options()
options.rhs_reuse = True
rhs_clear()
if isinstance(t, (int, float, np.integer, np.floating)):
tlist = [0, t]
else:
tlist = t
if isinstance(
H,
(types.FunctionType, types.BuiltinFunctionType, functools.partial)):
H0 = H(0.0, args)
elif isinstance(H, list):
H0 = H[0][0] if isinstance(H[0], list) else H[0]
else:
H0 = H
if len(c_op_list) == 0 and H0.isoper:
# calculate propagator for the wave function
N = H0.shape[0]
dims = H0.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
progress_bar.start(N)
for n in range(0, N):
progress_bar.update(n)
psi0 = basis(N, n)
output = sesolve(H, psi0, tlist, [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = output.states[k].full().T
progress_bar.finished()
# todo: evolving a batch of wave functions:
# psi_0_list = [basis(N, n) for n in range(N)]
# psi_t_list = mesolve(H, psi_0_list, [0, t], [], [], args, options)
# for n in range(0, N):
# u[:,n] = psi_t_list[n][1].full().T
elif len(c_op_list) == 0 and H0.issuper:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
N = H0.shape[0]
dims = H0.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
progress_bar.start(N)
for n in range(0, N):
progress_bar.update(n)
psi0 = basis(N, n)
rho0 = Qobj(vec2mat(psi0.full()))
output = mesolve(H, rho0, tlist, [], [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
progress_bar.finished()
else:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
N = H0.shape[0]
dims = [H0.dims, H0.dims]
u = np.zeros([N * N, N * N, len(tlist)], dtype=complex)
if sparse:
progress_bar.start(N * N)
for n in range(N * N):
progress_bar.update(n)
psi0 = basis(N * N, n)
psi0.dims = [dims[0], 1]
rho0 = vector_to_operator(psi0)
output = mesolve(H, rho0, tlist, c_op_list, [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = operator_to_vector(
output.states[k]).full(squeeze=True)
progress_bar.finished()
else:
progress_bar.start(N * N)
for n in range(N * N):
progress_bar.update(n)
psi0 = basis(N * N, n)
rho0 = Qobj(vec2mat(psi0.full()))
output = mesolve(H, rho0, tlist, c_op_list, [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
progress_bar.finished()
if len(tlist) == 2:
return Qobj(u[:, :, 1], dims=dims)
else:
return [Qobj(u[:, :, k], dims=dims) for k in range(len(tlist))]
def mesolve(H,
rho0,
tlist,
c_ops,
e_ops,
args={},
options=None,
progress_bar=BaseProgressBar()):
"""
Master equation evolution of a density matrix for a given Hamiltonian.
Evolve the state vector or density matrix (`rho0`) using a given
Hamiltonian (`H`) and an [optional] set of collapse operators
(`c_op_list`), by integrating the set of ordinary differential equations
that define the system. In the absense of collase operators the system is
evolved according to the unitary evolution of the Hamiltonian.
The output is either the state vector at arbitrary points in time
(`tlist`), or the expectation values of the supplied operators
(`e_ops`). If e_ops is a callback function, it is invoked for each
time in `tlist` with time and the state as arguments, and the function
does not use any return values.
**Time-dependent operators**
For problems with time-dependent problems `H` and `c_ops` can be callback
functions that takes two arguments, time and `args`, and returns the
Hamiltonian or Liuovillian for the system at that point in time
(*callback format*).
Alternatively, `H` and `c_ops` can be a specified in a nested-list format
where each element in the list is a list of length 2, containing an
operator (:class:`qutip.qobj`) at the first element and where the
second element is either a string (*list string format*) or a callback
function (*list callback format*) that evaluates to the time-dependent
coefficient for the corresponding operator.
*Examples*
H = [[H0, 'sin(w*t)'], [H1, 'sin(2*w*t)']]
H = [[H0, f0_t], [H1, f1_t]]
where f0_t and f1_t are python functions with signature f_t(t, args).
In the *list string format* and *list callback format*, the string
expression and the callback function must evaluate to a real or complex
number (coefficient for the corresponding operator).
In all cases of time-dependent operators, `args` is a dictionary of
parameters that is used when evaluating operators. It is passed to the
callback functions as second argument
.. note::
If an element in the list-specification of the Hamiltonian or
the list of collapse operators are in super-operator for it will be
added to the total Liouvillian of the problem with out further
transformation. This allows for using mesolve for solving master
equations that are not on standard Lindblad form.
.. note::
On using callback function: mesolve transforms all :class:`qutip.qobj`
objects to sparse matrices before handing the problem to the integrator
function. In order for your callback function to work correctly, pass
all :class:`qutip.qobj` objects that are used in constructing the
Hamiltonian via args. odesolve will check for :class:`qutip.qobj` in
`args` and handle the conversion to sparse matrices. All other
:class:`qutip.qobj` objects that are not passed via `args` will be
passed on to the integrator to scipy who will raise an NotImplemented
exception.
Parameters
----------
H : :class:`qutip.qobj`
system Hamiltonian, or a callback function for time-dependent
Hamiltonians.
rho0 : :class:`qutip.qobj`
initial density matrix or state vector (ket).
tlist : *list* / *array*
list of times for :math:`t`.
c_ops : list of :class:`qutip.qobj`
single collapse operator, or list of collapse operators.
e_ops : list of :class:`qutip.qobj` / callback function single
single operator or list of operators for which to evaluate
expectation values.
args : *dictionary*
dictionary of parameters for time-dependent Hamiltonians and
collapse operators.
options : :class:`qutip.Odeoptions`
with options for the ODE solver.
Returns
-------
output: :class:`qutip.odedata`
An instance of the class :class:`qutip.odedata`, which contains either
an *array* of expectation values for the times specified by `tlist`, or
an *array* or state vectors or density matrices corresponding to the
times in `tlist` [if `e_ops` is an empty list], or
nothing if a callback function was given inplace of operators for
which to calculate the expectation values.
"""
# check whether c_ops or e_ops is is a single operator
# if so convert it to a list containing only that operator
if isinstance(c_ops, Qobj):
c_ops = [c_ops]
if isinstance(e_ops, Qobj):
e_ops = [e_ops]
if isinstance(e_ops, dict):
e_ops_dict = e_ops
e_ops = [e for e in e_ops.values()]
else:
e_ops_dict = None
# check for type (if any) of time-dependent inputs
n_const, n_func, n_str = _ode_checks(H, c_ops)
if options is None:
options = Odeoptions()
if (not options.rhs_reuse) or (not odeconfig.tdfunc):
# reset odeconfig collapse and time-dependence flags to default values
odeconfig.reset()
res = None
#
# dispatch the appropriate solver
#
if ((c_ops and len(c_ops) > 0) or (not isket(rho0))
or (isinstance(H, Qobj) and issuper(H)) or
(isinstance(H, list) and isinstance(H[0], Qobj) and issuper(H[0]))):
#
# we have collapse operators
#
#
# find out if we are dealing with all-constant hamiltonian and
# collapse operators or if we have at least one time-dependent
# operator. Then delegate to appropriate solver...
#
if isinstance(H, Qobj):
# constant hamiltonian
if n_func == 0 and n_str == 0:
# constant collapse operators
res = _mesolve_const(H, rho0, tlist, c_ops, e_ops, args,
options, progress_bar)
elif n_str > 0:
# constant hamiltonian but time-dependent collapse
# operators in list string format
res = _mesolve_list_str_td([H], rho0, tlist, c_ops, e_ops,
args, options, progress_bar)
elif n_func > 0:
# constant hamiltonian but time-dependent collapse
# operators in list function format
res = _mesolve_list_func_td([H], rho0, tlist, c_ops, e_ops,
args, options, progress_bar)
elif isinstance(
H, (types.FunctionType, types.BuiltinFunctionType, partial)):
# old style time-dependence: must have constant collapse operators
if n_str > 0: # or n_func > 0:
raise TypeError("Incorrect format: function-format " +
"Hamiltonian cannot be mixed with " +
"time-dependent collapse operators.")
else:
res = _mesolve_func_td(H, rho0, tlist, c_ops, e_ops, args,
options, progress_bar)
elif isinstance(H, list):
# determine if we are dealing with list of [Qobj, string] or
# [Qobj, function] style time-dependencies (for pure python and
# cython, respectively)
if n_func > 0:
res = _mesolve_list_func_td(H, rho0, tlist, c_ops, e_ops, args,
options, progress_bar)
else:
res = _mesolve_list_str_td(H, rho0, tlist, c_ops, e_ops, args,
options, progress_bar)
else:
raise TypeError("Incorrect specification of Hamiltonian " +
"or collapse operators.")
else:
#
# no collapse operators: unitary dynamics
#
if n_func > 0:
res = _sesolve_list_func_td(H, rho0, tlist, e_ops, args, options,
progress_bar)
elif n_str > 0:
res = _sesolve_list_str_td(H, rho0, tlist, e_ops, args, options,
progress_bar)
elif isinstance(
H, (types.FunctionType, types.BuiltinFunctionType, partial)):
res = _sesolve_func_td(H, rho0, tlist, e_ops, args, options,
progress_bar)
else:
res = _sesolve_const(H, rho0, tlist, e_ops, args, options,
progress_bar)
if e_ops_dict:
res.expect = {
e: res.expect[n]
for n, e in enumerate(e_ops_dict.keys())
}
return res
