Source code for qutip.mesolve

This module provides solvers for the Lindblad master equation and von Neumann

__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 import spmv
from 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 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 / 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.solver.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.solver.Result` An instance of the class :class:`qutip.solver.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 = 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 = 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 = 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 = 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 = [] need_qobj_state = opt.store_states if callable(e_ops): n_expt_op = 0 expt_callback = True output.num_expect = 1 need_qobj_state = True 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 need_qobj_state = True else: for op in e_ops: if not isinstance(op, Qobj) and callable(op): output.expect.append(np.zeros(n_tsteps, dtype=complex)) need_qobj_state = True e_ops_data.append(None) continue 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 need_qobj_state: 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): if not isinstance(e_ops[m], Qobj) and callable(e_ops[m]): output.expect[m][t_idx] = e_ops[m](t, rho_t) else: 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