# This file is part of QuTiP: Quantum Toolbox in Python.
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"""
This module provides solvers for the Lindblad master equation and von Neumann
equation.
"""
__all__ = ['mesolve', 'odesolve']
import os
import types
from functools import partial
import numpy as np
import scipy.sparse as sp
import scipy.integrate
import warnings
from qutip.qobj import Qobj, isket, isoper, issuper
from qutip.superoperator import spre, spost, liouvillian, mat2vec, vec2mat
from qutip.expect import expect_rho_vec
from qutip.solver import Options, Result, config, _solver_safety_check
from qutip.cy.spmatfuncs import cy_ode_rhs, cy_ode_rho_func_td
from qutip.cy.spconvert import dense2D_to_fastcsr_fmode
from qutip.cy.codegen import Codegen
from qutip.cy.utilities import _cython_build_cleanup
from qutip.rhs_generate import rhs_generate
from qutip.states import ket2dm
from qutip.rhs_generate import _td_format_check, _td_wrap_array_str
from qutip.interpolate import Cubic_Spline
from qutip.settings import debug
from qutip.sesolve import (_sesolve_list_func_td, _sesolve_list_str_td,
_sesolve_list_td, _sesolve_func_td, _sesolve_const)
from qutip.ui.progressbar import BaseProgressBar, TextProgressBar
if debug:
import inspect
# -----------------------------------------------------------------------------
# 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=[], 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.
"""
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 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 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()
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
# -----------------------------------------------------------------------------
# A time-dependent dissipative master equation on the list-function format
#
def _mesolve_list_func_td(H_list, rho0, tlist, c_list, e_ops, args, opt,
progress_bar):
"""
Internal function for solving the master equation. See mesolve for usage.
"""
if debug:
print(inspect.stack()[0][3])
#
# check initial state
#
if isket(rho0):
rho0 = rho0 * rho0.dag()
#
# construct liouvillian in list-function format
#
L_list = []
if opt.rhs_with_state:
constant_func = lambda x, y, z: 1.0
else:
constant_func = lambda x, y: 1.0
# add all hamitonian terms to the lagrangian list
for h_spec in H_list:
if isinstance(h_spec, Qobj):
h = h_spec
h_coeff = constant_func
elif isinstance(h_spec, list) and isinstance(h_spec[0], Qobj):
h = h_spec[0]
h_coeff = h_spec[1]
else:
raise TypeError("Incorrect specification of time-dependent " +
"Hamiltonian (expected callback function)")
if isoper(h):
L_list.append([(-1j * (spre(h) - spost(h))).data, h_coeff, False])
elif issuper(h):
L_list.append([h.data, h_coeff, False])
else:
raise TypeError("Incorrect specification of time-dependent " +
"Hamiltonian (expected operator or superoperator)")
# add all collapse operators to the liouvillian list
for c_spec in c_list:
if isinstance(c_spec, Qobj):
c = c_spec
c_coeff = constant_func
c_square = False
elif isinstance(c_spec, list) and isinstance(c_spec[0], Qobj):
c = c_spec[0]
c_coeff = c_spec[1]
c_square = True
else:
raise TypeError("Incorrect specification of time-dependent " +
"collapse operators (expected callback function)")
if isoper(c):
L_list.append([liouvillian(None, [c], data_only=True),
c_coeff, c_square])
elif issuper(c):
L_list.append([c.data, c_coeff, c_square])
else:
raise TypeError("Incorrect specification of time-dependent " +
"collapse operators (expected operator or " +
"superoperator)")
#
# setup integrator
#
initial_vector = mat2vec(rho0.full()).ravel('F')
if issuper(rho0):
if opt.rhs_with_state:
r = scipy.integrate.ode(dsuper_list_td_with_state)
else:
r = scipy.integrate.ode(dsuper_list_td)
else:
if opt.rhs_with_state:
r = scipy.integrate.ode(drho_list_td_with_state)
else:
r = scipy.integrate.ode(drho_list_td)
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)
r.set_initial_value(initial_vector, tlist[0])
r.set_f_params(L_list, args)
#
# call generic ODE code
#
return _generic_ode_solve(r, rho0, tlist, e_ops, opt, progress_bar)
#
# evaluate drho(t)/dt according to the master equation using the
# [Qobj, function] style time dependence API
#
def drho_list_td(t, rho, L_list, args):
L = L_list[0][0] * L_list[0][1](t, args)
for n in range(1, len(L_list)):
#
# L_args[n][0] = the sparse data for a Qobj in super-operator form
# L_args[n][1] = function callback giving the coefficient
#
if L_list[n][2]:
L = L + L_list[n][0] * (L_list[n][1](t, args)) ** 2
else:
L = L + L_list[n][0] * L_list[n][1](t, args)
return L * rho
def drho_list_td_with_state(t, rho, L_list, args):
L = L_list[0][0] * L_list[0][1](t, rho, args)
for n in range(1, len(L_list)):
#
# L_args[n][0] = the sparse data for a Qobj in super-operator form
# L_args[n][1] = function callback giving the coefficient
#
if L_list[n][2]:
L = L + L_list[n][0] * (L_list[n][1](t, rho, args)) ** 2
else:
L = L + L_list[n][0] * L_list[n][1](t, rho, args)
return L * rho
#
# evaluate dE(t)/dt according to the master equation using the
# [Qobj, function] style time dependence API, where E is a superoperator
#
def dsuper_list_td(t, y, L_list, args):
L = L_list[0][0] * L_list[0][1](t, args)
for n in range(1, len(L_list)):
#
# L_args[n][0] = the sparse data for a Qobj in super-operator form
# L_args[n][1] = function callback giving the coefficient
#
if L_list[n][2]:
L = L + L_list[n][0] * (L_list[n][1](t, args)) ** 2
else:
L = L + L_list[n][0] * L_list[n][1](t, args)
return _ode_super_func(t, y, L)
def dsuper_list_td_with_state(t, y, L_list, args):
L = L_list[0][0] * L_list[0][1](t, y, args)
for n in range(1, len(L_list)):
#
# L_args[n][0] = the sparse data for a Qobj in super-operator form
# L_args[n][1] = function callback giving the coefficient
#
if L_list[n][2]:
L = L + L_list[n][0] * (L_list[n][1](t, y, args)) ** 2
else:
L = L + L_list[n][0] * L_list[n][1](t, y, args)
return _ode_super_func(t, y, L)
# -----------------------------------------------------------------------------
# A time-dependent dissipative master equation on the list-string format for
# cython compilation
#
def _mesolve_list_str_td(H_list, rho0, tlist, c_list, e_ops, args, opt,
progress_bar):
"""
Internal function for solving the master equation. See mesolve for usage.
"""
if debug:
print(inspect.stack()[0][3])
#
# check initial state: must be a density matrix
#
if isket(rho0):
rho0 = rho0 * rho0.dag()
#
# construct liouvillian
#
Lconst = 0
Ldata = []
Linds = []
Lptrs = []
Lcoeff = []
Lobj = []
# loop over all hamiltonian terms, convert to superoperator form and
# add the data of sparse matrix representation to
for h_spec in H_list:
if isinstance(h_spec, Qobj):
h = h_spec
if isoper(h):
Lconst += -1j * (spre(h) - spost(h))
elif issuper(h):
Lconst += h
else:
raise TypeError("Incorrect specification of time-dependent " +
"Hamiltonian (expected operator or " +
"superoperator)")
elif isinstance(h_spec, list):
h = h_spec[0]
h_coeff = h_spec[1]
if isoper(h):
L = -1j * (spre(h) - spost(h))
elif issuper(h):
L = h
else:
raise TypeError("Incorrect specification of time-dependent " +
"Hamiltonian (expected operator or " +
"superoperator)")
Ldata.append(L.data.data)
Linds.append(L.data.indices)
Lptrs.append(L.data.indptr)
if isinstance(h_coeff, Cubic_Spline):
Lobj.append(h_coeff.coeffs)
Lcoeff.append(h_coeff)
else:
raise TypeError("Incorrect specification of time-dependent " +
"Hamiltonian (expected string format)")
# loop over all collapse operators
for c_spec in c_list:
if isinstance(c_spec, Qobj):
c = c_spec
if isoper(c):
cdc = c.dag() * c
Lconst += spre(c) * spost(c.dag()) - 0.5 * spre(cdc) \
- 0.5 * spost(cdc)
elif issuper(c):
Lconst += c
else:
raise TypeError("Incorrect specification of time-dependent " +
"Liouvillian (expected operator or " +
"superoperator)")
elif isinstance(c_spec, list):
c = c_spec[0]
c_coeff = c_spec[1]
if isoper(c):
cdc = c.dag() * c
L = spre(c) * spost(c.dag()) - 0.5 * spre(cdc) \
- 0.5 * spost(cdc)
c_coeff = "(" + c_coeff + ")**2"
elif issuper(c):
L = c
else:
raise TypeError("Incorrect specification of time-dependent " +
"Liouvillian (expected operator or " +
"superoperator)")
Ldata.append(L.data.data)
Linds.append(L.data.indices)
Lptrs.append(L.data.indptr)
Lcoeff.append(c_coeff)
else:
raise TypeError("Incorrect specification of time-dependent " +
"collapse operators (expected string format)")
# add the constant part of the lagrangian
if Lconst != 0:
Ldata.append(Lconst.data.data)
Linds.append(Lconst.data.indices)
Lptrs.append(Lconst.data.indptr)
Lcoeff.append("1.0")
# the total number of liouvillian terms (hamiltonian terms +
# collapse operators)
n_L_terms = len(Ldata)
#
# setup ode args string: we expand the list Ldata, Linds and Lptrs into
# and explicit list of parameters
#
string_list = []
for k in range(n_L_terms):
string_list.append("Ldata[%d], Linds[%d], Lptrs[%d]" % (k, k, k))
# Add object terms to end of ode args string
for k in range(len(Lobj)):
string_list.append("Lobj[%d]" % k)
for name, value in args.items():
if isinstance(value, np.ndarray):
string_list.append(name)
else:
string_list.append(str(value))
parameter_string = ",".join(string_list)
#
# generate and compile new cython code if necessary
#
if not opt.rhs_reuse or config.tdfunc is None:
if opt.rhs_filename is None:
config.tdname = "rhs" + str(os.getpid()) + str(config.cgen_num)
else:
config.tdname = opt.rhs_filename
cgen = Codegen(h_terms=n_L_terms, h_tdterms=Lcoeff, args=args,
config=config)
cgen.generate(config.tdname + ".pyx")
code = compile('from ' + config.tdname + ' import cy_td_ode_rhs',
'<string>', 'exec')
exec(code, globals())
config.tdfunc = cy_td_ode_rhs
#
# setup integrator
#
initial_vector = mat2vec(rho0.full()).ravel('F')
if issuper(rho0):
r = scipy.integrate.ode(_td_ode_rhs_super)
code = compile('r.set_f_params([' + parameter_string + '])',
'<string>', 'exec')
else:
r = scipy.integrate.ode(config.tdfunc)
code = compile('r.set_f_params(' + parameter_string + ')',
'<string>', 'exec')
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)
r.set_initial_value(initial_vector, tlist[0])
exec(code, locals(), args)
#
# call generic ODE code
#
return _generic_ode_solve(r, rho0, tlist, e_ops, opt, progress_bar)
def _td_ode_rhs_super(t, y, arglist):
N = int(np.sqrt(len(y)))
out = np.zeros(N, dtype=complex)
y2 = np.zeros(len(y), dtype=complex)
for i in range(N):
out = cy_td_ode_rhs(t, y[i*N:(i+1)*N], *arglist)
y2[i*N:(i+1)*N] = out
return y2
# -----------------------------------------------------------------------------
# Master equation solver
#
def _mesolve_const(H, rho0, tlist, c_op_list, e_ops, args, opt,
progress_bar):
"""
Evolve the density matrix using an ODE solver, for constant hamiltonian
and collapse operators.
"""
if debug:
print(inspect.stack()[0][3])
#
# check initial state
#
if isket(rho0):
# if initial state is a ket and no collapse operator where given,
# fall back on the unitary schrodinger equation solver
if len(c_op_list) == 0 and isoper(H):
return _sesolve_const(H, rho0, tlist, e_ops, args, opt,
progress_bar)
# Got a wave function as initial state: convert to density matrix.
rho0 = ket2dm(rho0)
#
# construct liouvillian
#
if opt.tidy:
H = H.tidyup(opt.atol)
L = liouvillian(H, c_op_list)
#
# setup integrator
#
initial_vector = mat2vec(rho0.full()).ravel('F')
if issuper(rho0):
r = scipy.integrate.ode(_ode_super_func)
r.set_f_params(L.data)
else:
r = scipy.integrate.ode(cy_ode_rhs)
r.set_f_params(L.data.data, L.data.indices, L.data.indptr)
# r = scipy.integrate.ode(_ode_rho_test)
# r.set_f_params(L.data)
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)
r.set_initial_value(initial_vector, tlist[0])
#
# call generic ODE code
#
return _generic_ode_solve(r, rho0, tlist, e_ops, opt, progress_bar)
#
# evaluate drho(t)/dt according to the master eqaution
# [no longer used, replaced by cython function]
#
def _ode_rho_func(t, rho, L):
return L * rho
def _ode_rho_test(t, rho, data):
# for performance comparison of cython code
return data*(np.transpose(rho))
#
# Evaluate d E(t)/dt for E a super-operator
#
def _ode_super_func(t, y, data):
ym = vec2mat(y)
return (data*ym).ravel('F')
# -----------------------------------------------------------------------------
# Master equation solver for python-function time-dependence.
#
def _mesolve_func_td(L_func, rho0, tlist, c_op_list, e_ops, args, opt,
progress_bar):
"""
Evolve the density matrix using an ODE solver with time dependent
Hamiltonian.
"""
if debug:
print(inspect.stack()[0][3])
#
# check initial state
#
if isket(rho0):
rho0 = ket2dm(rho0)
#
# construct liouvillian
#
new_args = None
if len(c_op_list) > 0:
L_data = liouvillian(None, c_op_list).data
else:
n, m = rho0.shape
if issuper(rho0):
L_data = sp.csr_matrix((n, m), dtype=complex)
else:
L_data = sp.csr_matrix((n ** 2, m ** 2), dtype=complex)
if type(args) is dict:
new_args = {}
for key in args:
if isinstance(args[key], Qobj):
if isoper(args[key]):
new_args[key] = (
-1j * (spre(args[key]) - spost(args[key])))
else:
new_args[key] = args[key]
else:
new_args[key] = args[key]
elif type(args) is list or type(args) is tuple:
new_args = []
for arg in args:
if isinstance(arg, Qobj):
if isoper(arg):
new_args.append((-1j * (spre(arg) - spost(arg))).data)
else:
new_args.append(arg.data)
else:
new_args.append(arg)
if type(args) is tuple:
new_args = tuple(new_args)
else:
if isinstance(args, Qobj):
if isoper(args):
new_args = (-1j * (spre(args) - spost(args)))
else:
new_args = args
else:
new_args = args
#
# setup integrator
#
initial_vector = mat2vec(rho0.full()).ravel('F')
if issuper(rho0):
if not opt.rhs_with_state:
r = scipy.integrate.ode(_ode_super_func_td)
else:
r = scipy.integrate.ode(_ode_super_func_td_with_state)
else:
if not opt.rhs_with_state:
r = scipy.integrate.ode(cy_ode_rho_func_td)
else:
r = scipy.integrate.ode(_ode_rho_func_td_with_state)
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)
r.set_initial_value(initial_vector, tlist[0])
r.set_f_params(L_data, L_func, new_args)
#
# call generic ODE code
#
return _generic_ode_solve(r, rho0, tlist, e_ops, opt, progress_bar)
#
# evaluate drho(t)/dt according to the master equation
#
def _ode_rho_func_td(t, rho, L0, L_func, args):
L = L0 + L_func(t, args)
return L * rho
#
# evaluate drho(t)/dt according to the master equation
#
def _ode_rho_func_td_with_state(t, rho, L0, L_func, args):
L = L0 + L_func(t, rho, args)
return L * rho
#
# evaluate dE(t)/dt according to the master equation, where E is a
# superoperator
#
def _ode_super_func_td(t, y, L0, L_func, args):
L = L0 + L_func(t, args).data
return _ode_super_func(t, y, L)
#
# evaluate dE(t)/dt according to the master equation, where E is a
# superoperator
#
def _ode_super_func_td_with_state(t, y, L0, L_func, args):
L = L0 + L_func(t, y, args).data
return _ode_super_func(t, y, L)
# -----------------------------------------------------------------------------
# Generic ODE solver: shared code among the various ODE solver
# -----------------------------------------------------------------------------
def _generic_ode_solve(r, rho0, tlist, e_ops, opt, progress_bar):
"""
Internal function for solving ME. Solve an ODE which solver parameters
already setup (r). Calculate the required expectation values or invoke
callback function at each time step.
"""
#
# prepare output array
#
n_tsteps = len(tlist)
e_sops_data = []
output = Result()
output.solver = "mesolve"
output.times = tlist
if opt.store_states:
output.states = []
if isinstance(e_ops, types.FunctionType):
n_expt_op = 0
expt_callback = True
elif isinstance(e_ops, list):
n_expt_op = len(e_ops)
expt_callback = False
if n_expt_op == 0:
# fall back on storing states
output.states = []
opt.store_states = True
else:
output.expect = []
output.num_expect = n_expt_op
for op in e_ops:
e_sops_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")
#
# start evolution
#
progress_bar.start(n_tsteps)
rho = Qobj(rho0)
dt = np.diff(tlist)
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:
rho.data = dense2D_to_fastcsr_fmode(vec2mat(r.y), rho.shape[0], rho.shape[1])
if opt.store_states:
output.states.append(Qobj(rho, isherm=True))
if expt_callback:
# use callback method
e_ops(t, rho)
for m in range(n_expt_op):
if output.expect[m].dtype == complex:
output.expect[m][t_idx] = expect_rho_vec(e_sops_data[m],
r.y, 0)
else:
output.expect[m][t_idx] = expect_rho_vec(e_sops_data[m],
r.y, 1)
if t_idx < n_tsteps - 1:
r.integrate(r.t + dt[t_idx])
progress_bar.finished()
if (not opt.rhs_reuse) and (config.tdname is not None):
_cython_build_cleanup(config.tdname)
if opt.store_final_state:
rho.data = dense2D_to_fastcsr_fmode(vec2mat(r.y), rho.shape[0], rho.shape[1])
output.final_state = Qobj(rho, dims=rho0.dims, isherm=True)
return output
# -----------------------------------------------------------------------------
# Old style API below.
# -----------------------------------------------------------------------------
# -----------------------------------------------------------------------------
# Master equation solver: deprecated in 2.0.0. No support for time-dependent
# collapse operators. Only used by the deprecated odesolve function.
#
def _mesolve_list_td(H_func, rho0, tlist, c_op_list, e_ops, args, opt,
progress_bar):
"""
Evolve the density matrix using an ODE solver with time dependent
Hamiltonian.
"""
if debug:
print(inspect.stack()[0][3])
#
# check initial state
#
if isket(rho0):
# if initial state is a ket and no collapse operator where given,
# fall back on the unitary schrodinger equation solver
if len(c_op_list) == 0:
return _sesolve_list_td(H_func, rho0, tlist, e_ops, args, opt,
progress_bar)
# Got a wave function as initial state: convert to density matrix.
rho0 = ket2dm(rho0)
#
# construct liouvillian
#
if len(H_func) != 2:
raise TypeError('Time-dependent Hamiltonian list must have two terms.')
if not isinstance(H_func[0], (list, np.ndarray)) or len(H_func[0]) <= 1:
raise TypeError('Time-dependent Hamiltonians must be a list ' +
'with two or more terms')
if (not isinstance(H_func[1], (list, np.ndarray))) or \
(len(H_func[1]) != (len(H_func[0]) - 1)):
raise TypeError('Time-dependent coefficients must be list with ' +
'length N-1 where N is the number of ' +
'Hamiltonian terms.')
if opt.rhs_reuse and config.tdfunc is None:
rhs_generate(H_func, args)
lenh = len(H_func[0])
if opt.tidy:
H_func[0] = [(H_func[0][k]).tidyup() for k in range(lenh)]
L_func = [[liouvillian(H_func[0][0], c_op_list)], H_func[1]]
for m in range(1, lenh):
L_func[0].append(liouvillian(H_func[0][m], []))
# create data arrays for time-dependent RHS function
Ldata = [L_func[0][k].data.data for k in range(lenh)]
Linds = [L_func[0][k].data.indices for k in range(lenh)]
Lptrs = [L_func[0][k].data.indptr for k in range(lenh)]
# setup ode args string
string = ""
for k in range(lenh):
string += ("Ldata[%d], Linds[%d], Lptrs[%d]," % (k, k, k))
if args:
td_consts = args.items()
for elem in td_consts:
string += str(elem[1])
if elem != td_consts[-1]:
string += (",")
# run code generator
if not opt.rhs_reuse or config.tdfunc is None:
if opt.rhs_filename is None:
config.tdname = "rhs" + str(os.getpid()) + str(config.cgen_num)
else:
config.tdname = opt.rhs_filename
cgen = Codegen(h_terms=n_L_terms, h_tdterms=Lcoeff, args=args,
config=config)
cgen.generate(config.tdname + ".pyx")
code = compile('from ' + config.tdname + ' import cy_td_ode_rhs',
'<string>', 'exec')
exec(code, globals())
config.tdfunc = cy_td_ode_rhs
#
# setup integrator
#
initial_vector = mat2vec(rho0.full()).ravel()
r = scipy.integrate.ode(config.tdfunc)
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)
r.set_initial_value(initial_vector, tlist[0])
code = compile('r.set_f_params(' + string + ')', '<string>', 'exec')
exec(code)
#
# call generic ODE code
#
return _generic_ode_solve(r, rho0, tlist, e_ops, opt, progress_bar)
# -----------------------------------------------------------------------------
# pass on to wavefunction solver or master equation solver depending on whether
# any collapse operators were given.
#
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