# Source code for qutip.solver.mesolve

```
"""
This module provides solvers for the Lindblad master equation and von Neumann
equation.
"""
__all__ = ['mesolve', 'MESolver']
import numpy as np
from time import time
from .. import (Qobj, QobjEvo, isket, liouvillian, ket2dm, lindblad_dissipator)
from ..core import stack_columns, unstack_columns
from ..core.data import to
from .solver_base import Solver
from .sesolve import sesolve, SESolver
[docs]def mesolve(H, rho0, tlist, c_ops=None, e_ops=None, args=None, options=None):
"""
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 :class:`QobjEvo` or
object that can be interpreted as :class:`QobjEvo` such as a list of
(Qobj, Coefficient) pairs or a function.
**Additional options**
Additional options to mesolve can be set via the `options` argument. 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::
When no collapse operator are given and the `H` is not a superoperator,
it will defer to :func:`sesolve`.
Parameters
----------
H : :class:`Qobj`, :class:`QobjEvo`, :class:`QobjEvo` compatible format.
Possibly time-dependent system Liouvillian or Hamiltonian as a Qobj or
QobjEvo. List of [:class:`Qobj`, :class:`Coefficient`] or callable that
can be made into :class:`QobjEvo` are also accepted.
rho0 : :class:`Qobj`
initial density matrix or state vector (ket).
tlist : *list* / *array*
list of times for :math:`t`.
c_ops : list of (:class:`QobjEvo`, :class:`QobjEvo` compatible format)
Single collapse operator, or list of collapse operators, or a list
of Liouvillian superoperators. None is equivalent to an empty list.
e_ops : list of :class:`Qobj` / callback function
Single operator or list of operators for which to evaluate
expectation values or callable or list of callable.
Callable signature must be, `f(t: float, state: Qobj)`.
See :func:`expect` for more detail of operator expectation.
args : None / *dictionary*
dictionary of parameters for time-dependent Hamiltonians and
collapse operators.
options : None / dict
Dictionary of options for the solver.
- store_final_state : bool
Whether or not to store the final state of the evolution in the
result class.
- store_states : bool, None
Whether or not to store the state vectors or density matrices.
On `None` the states will be saved if no expectation operators are
given.
- normalize_output : bool
Normalize output state to hide ODE numerical errors.
- progress_bar : str {'text', 'enhanced', 'tqdm', ''}
How to present the solver progress.
'tqdm' uses the python module of the same name and raise an error
if not installed. Empty string or False will disable the bar.
- progress_kwargs : dict
kwargs to pass to the progress_bar. Qutip's bars use `chunk_size`.
- method : str ["adams", "bdf", "lsoda", "dop853", "vern9", etc.]
Which differential equation integration method to use.
- atol, rtol : float
Absolute and relative tolerance of the ODE integrator.
- nsteps :
Maximum number of (internally defined) steps allowed in one ``tlist``
step.
- max_step : float, 0
Maximum lenght of one internal step. When using pulses, it should be
less than half the width of the thinnest pulse.
Other options could be supported depending on the integration method,
see `Integrator <./classes.html#classes-ode>`_.
Returns
-------
result: :class:`qutip.Result`
An instance of the class :class:`qutip.Result`, which contains
a *list of array* `result.expect` of expectation values for the times
specified by `tlist`, and/or a *list* `result.states` of state vectors
or density matrices corresponding to the times in `tlist` [if `e_ops`
is an empty list of `store_states=True` in options].
"""
H = QobjEvo(H, args=args, tlist=tlist)
c_ops = c_ops if c_ops is not None else []
if not isinstance(c_ops, (list, tuple)):
c_ops = [c_ops]
c_ops = [QobjEvo(c_op, args=args, tlist=tlist) for c_op in c_ops]
use_mesolve = len(c_ops) > 0 or (not rho0.isket) or H.issuper
if not use_mesolve:
return sesolve(H, rho0, tlist, e_ops=e_ops, args=args,
options=options)
solver = MESolver(H, c_ops, options=options)
return solver.run(rho0, tlist, e_ops=e_ops)
[docs]class MESolver(SESolver):
"""
Master equation evolution of a density matrix for a given Hamiltonian and
set of collapse operators, or a Liouvillian.
Evolve the 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.
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.
Parameters
----------
H : :class:`Qobj`, :class:`QobjEvo`
Possibly time-dependent system Liouvillian or Hamiltonian as a Qobj or
QobjEvo. List of [:class:`Qobj`, :class:`Coefficient`] or callable that
can be made into :class:`QobjEvo` are also accepted.
c_ops : list of :class:`Qobj`, :class:`QobjEvo`
Single collapse operator, or list of collapse operators, or a list
of Liouvillian superoperators. None is equivalent to an empty list.
options : dict, optional
Options for the solver, see :obj:`MESolver.options` and
`Integrator <./classes.html#classes-ode>`_ for a list of all options.
attributes
----------
stats: dict
Diverse diagnostic statistics of the evolution.
"""
name = "mesolve"
_avail_integrators = {}
solver_options = {
"progress_bar": "text",
"progress_kwargs": {"chunk_size":10},
"store_final_state": False,
"store_states": None,
"normalize_output": True,
'method': 'adams',
}
def __init__(self, H, c_ops=None, *, options=None):
_time_start = time()
if not isinstance(H, (Qobj, QobjEvo)):
raise TypeError("The Hamiltonian must be a Qobj or QobjEvo")
c_ops = c_ops or []
c_ops = [c_ops] if isinstance(c_ops, (Qobj, QobjEvo)) else c_ops
for c_op in c_ops:
if not isinstance(c_op, (Qobj, QobjEvo)):
raise TypeError("All `c_ops` must be a Qobj or QobjEvo")
self._num_collapse = len(c_ops)
rhs = H if H.issuper else liouvillian(H)
rhs += sum(c_op if c_op.issuper else lindblad_dissipator(c_op)
for c_op in c_ops)
Solver.__init__(self, rhs, options=options)
def _initialize_stats(self):
stats = super()._initialize_stats()
stats.update({
"solver": "Master Equation Evolution",
"num_collapse": self._num_collapse,
})
return stats
```