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"""
This module provides solvers for the Lindblad master equation and von Neumann
equation.
"""
__all__ = ['mesolve']
import numpy as np
import scipy.integrate
from qutip.qobj import Qobj, isket, isoper, issuper
from qutip.superoperator import spre, spost, liouvillian, vec2mat, lindblad_dissipator
from qutip.expect import expect_rho_vec
from qutip.solver import Options, Result, solver_safe, SolverSystem
from qutip.cy.spmatfuncs import spmv
from qutip.cy.spconvert import dense2D_to_fastcsr_fmode
from qutip.states import ket2dm
from qutip.sesolve import sesolve
from qutip.ui.progressbar import BaseProgressBar, TextProgressBar
from qutip.qobjevo import QobjEvo
from qutip.cy.openmp.utilities import check_use_openmp
# -----------------------------------------------------------------------------
# pass on to wavefunction solver or master equation solver depending on whether
# any collapse operators were given.
#
[docs]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 of :class:`qutip.Qobj` / callback function single
single operator or list of operators for which to evaluate
expectation values.
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
# -----------------------------------------------------------------------------
# A time-dependent unitary wavefunction equation on the list-function format
#_mesolve_QobjEvo(H, c_ops, tlist, args, options)
def _mesolve_QobjEvo(H, c_ops, tlist, args, opt):
"""
Prepare the system for the solver, H can be an QobjEvo.
"""
H_td = QobjEvo(H, args, tlist=tlist)
if not issuper(H_td.cte):
L_td = liouvillian(H_td)
else:
L_td = H_td
for op in c_ops:
# We want to avoid passing tlist where it isn't necessary, to allow a
# Hamiltonian/Liouvillian which already _has_ time-dependence not equal
# to the mesolve evaluation times to be used in conjunction with
# time-independent c_ops. If we _always_ pass it, it may appear to
# QobjEvo that there is a tlist mismatch, even though it is not used.
if isinstance(op, Qobj):
op_td = QobjEvo(op)
elif isinstance(op, QobjEvo):
op_td = QobjEvo(op, args)
else:
op_td = QobjEvo(op, args, tlist=tlist)
if not issuper(op_td.cte):
op_td = lindblad_dissipator(op_td)
L_td += op_td
if opt.rhs_with_state:
L_td._check_old_with_state()
nthread = opt.openmp_threads if opt.use_openmp else 0
L_td.compile(omp=nthread)
ss = SolverSystem()
ss.H = L_td
ss.makefunc = _qobjevo_set
solver_safe["mesolve"] = ss
return ss
def _test_liouvillian_dimensions(L_dims, rho_dims):
"""
Raise ValueError if the dimensions of the Liouvillian and the density
matrix or superoperator state are incompatible with the master equation.
"""
if L_dims[0] != L_dims[1]:
raise ValueError("Liouvillian had nonsquare dims: " + str(L_dims))
if not ((L_dims[1] == rho_dims) or (L_dims[1] == rho_dims[0])):
raise ValueError("".join([
"incompatible Liouvillian and state dimensions: ",
str(L_dims), " and ", str(rho_dims),
]))
def _qobjevo_set(HS, rho0, args, e_ops, opt):
"""
From the system, get the ode function and args
"""
H_td = HS.H
H_td.solver_set_args(args, rho0, e_ops)
if issuper(rho0):
func = H_td.compiled_qobjevo.ode_mul_mat_f_vec
elif rho0.isket or rho0.isoper:
func = H_td.compiled_qobjevo.mul_vec
else:
# Should be caught earlier in mesolve.
raise ValueError("rho0 must be a ket, density matrix or superoperator")
_test_liouvillian_dimensions(H_td.cte.dims, rho0.dims)
return func, ()
# -----------------------------------------------------------------------------
# Master equation solver for python-function time-dependence.
#
class _LiouvillianFromFunc:
def __init__(self, func, c_ops, rho_dims):
self.f = func
self.c_ops = c_ops
self.rho_dims = rho_dims
def H2L(self, t, rho, args):
Ht = self.f(t, args)
Lt = -1.0j * (spre(Ht) - spost(Ht))
_test_liouvillian_dimensions(Lt.dims, self.rho_dims)
Lt = Lt.data
for op in self.c_ops:
Lt += op(t).data
return Lt
def H2L_with_state(self, t, rho, args):
Ht = self.f(t, rho, args)
Lt = -1.0j * (spre(Ht) - spost(Ht))
_test_liouvillian_dimensions(Lt.dims, self.rho_dims)
Lt = Lt.data
for op in self.c_ops:
Lt += op(t).data
return Lt
def L(self, t, rho, args):
Lt = self.f(t, args)
_test_liouvillian_dimensions(Lt.dims, self.rho_dims)
Lt = Lt.data
for op in self.c_ops:
Lt += op(t).data
return Lt
def L_with_state(self, t, rho, args):
Lt = self.f(t, rho, args)
_test_liouvillian_dimensions(Lt.dims, self.rho_dims)
Lt = Lt.data
for op in self.c_ops:
Lt += op(t).data
return Lt
def _mesolve_func_td(L_func, c_op_list, rho0, tlist, args, opt):
"""
Evolve the density matrix using an ODE solver with time dependent
Hamiltonian.
"""
c_ops = []
for op in c_op_list:
td = QobjEvo(op, args, tlist=tlist, copy=False)
c_ops.append(td if td.cte.issuper else lindblad_dissipator(td))
c_ops_ = [sum(c_ops)] if c_op_list else []
L_api = _LiouvillianFromFunc(L_func, c_ops_, rho0.dims)
if opt.rhs_with_state:
obj = L_func(0., rho0.full().ravel("F"), args)
L_func = L_api.L_with_state if issuper(obj) else L_api.H2L_with_state
else:
obj = L_func(0., args)
L_func = L_api.L if issuper(obj) else L_api.H2L
ss = SolverSystem()
ss.L = L_func
ss.makefunc = _Lfunc_set
solver_safe["mesolve"] = ss
return ss
def _Lfunc_set(HS, rho0, args, e_ops, opt):
"""
From the system, get the ode function and args
"""
L_func = HS.L
if issuper(rho0):
func = _ode_super_func_td
else:
func = _ode_rho_func_td
return func, (L_func, args)
def _ode_rho_func_td(t, y, L_func, args):
L = L_func(t, y, args)
return spmv(L, y)
def _ode_super_func_td(t, y, L_func, args):
L = L_func(t, y, args)
ym = vec2mat(y)
return (L*ym).ravel('F')
# -----------------------------------------------------------------------------
# Generic ODE solver: shared code among the various ODE solver
# -----------------------------------------------------------------------------
def _generic_ode_solve(func, ode_args, rho0, tlist, e_ops, opt,
progress_bar, dims=None):
"""
Internal function for solving ME.
Calculate the required expectation values or invoke
callback function at each time step.
"""
# %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
# This function is made similar to sesolve's one for futur merging in a
# solver class
# %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
# prepare output array
n_tsteps = len(tlist)
output = Result()
output.solver = "mesolve"
output.times = tlist
size = rho0.shape[0]
initial_vector = rho0.full().ravel('F')
r = scipy.integrate.ode(func)
r.set_integrator('zvode', method=opt.method, order=opt.order,
atol=opt.atol, rtol=opt.rtol, nsteps=opt.nsteps,
first_step=opt.first_step, min_step=opt.min_step,
max_step=opt.max_step)
if ode_args:
r.set_f_params(*ode_args)
r.set_initial_value(initial_vector, tlist[0])
e_ops_data = []
output.expect = []
if callable(e_ops):
n_expt_op = 0
expt_callback = True
output.num_expect = 1
elif isinstance(e_ops, list):
n_expt_op = len(e_ops)
expt_callback = False
output.num_expect = n_expt_op
if n_expt_op == 0:
# fall back on storing states
opt.store_states = True
else:
for op in e_ops:
if op.dims != rho0.dims:
raise TypeError(f"e_ops dims ({op.dims}) are not "
f"compatible with the state's "
f"({rho0.dims})")
e_ops_data.append(spre(op).data)
if op.isherm and rho0.isherm:
output.expect.append(np.zeros(n_tsteps))
else:
output.expect.append(np.zeros(n_tsteps, dtype=complex))
else:
raise TypeError("Expectation parameter must be a list or a function")
if opt.store_states:
output.states = []
def get_curr_state_data(r):
return vec2mat(r.y)
#
# start evolution
#
dt = np.diff(tlist)
cdata = None
progress_bar.start(n_tsteps)
for t_idx, t in enumerate(tlist):
progress_bar.update(t_idx)
if not r.successful():
raise Exception("ODE integration error: Try to increase "
"the allowed number of substeps by increasing "
"the nsteps parameter in the Options class.")
if opt.store_states or expt_callback:
cdata = get_curr_state_data(r)
fdata = dense2D_to_fastcsr_fmode(cdata, size, size)
# Try to guess if there is a fast path for rho_t
if issuper(rho0) or not rho0.isherm:
rho_t = Qobj(fdata, dims=dims)
else:
rho_t = Qobj(fdata, dims=dims, fast="mc-dm")
if opt.store_states:
output.states.append(rho_t)
if expt_callback:
# use callback method
output.expect.append(e_ops(t, rho_t))
for m in range(n_expt_op):
output.expect[m][t_idx] = expect_rho_vec(e_ops_data[m], r.y,
e_ops[m].isherm
and rho0.isherm)
if t_idx < n_tsteps - 1:
r.integrate(r.t + dt[t_idx])
progress_bar.finished()
if opt.store_final_state:
cdata = get_curr_state_data(r)
output.final_state = Qobj(cdata, dims=dims,
isherm=rho0.isherm or None)
return output