def correlation_es(H, rho0, tlist, taulist, c_op_list, a_op, b_op):
"""
Internal function for calculating correlation functions using the
exponential series solver. See :func:`correlation` usage.
"""
# contruct the Liouvillian
L = liouvillian(H, c_op_list)
if rho0 is None:
rho0 = steady(L)
C_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex)
solES_t = ode2es(L, rho0)
for t_idx in range(len(tlist)):
rho_t = esval(solES_t, [tlist[t_idx]])
solES_tau = ode2es(L, b_op * rho_t)
C_mat[t_idx, :] = esval(expect(a_op, solES_tau), taulist)
return C_mat
def _correlation_es_2t(H, state0, tlist, taulist, c_ops, a_op, b_op, c_op):
"""
Internal function for calculating the three-operator two-time
correlation function:
<A(t)B(t+tau)C(t)>
using an exponential series solver.
"""
# the solvers only work for positive time differences and the correlators
# require positive tau
if state0 is None:
rho0 = steadystate(H, c_ops)
tlist = [0]
elif isket(state0):
rho0 = ket2dm(state0)
else:
rho0 = state0
if debug:
print(inspect.stack()[0][3])
# contruct the Liouvillian
L = liouvillian(H, c_ops)
corr_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex)
solES_t = ode2es(L, rho0)
# evaluate the correlation function
for t_idx in range(len(tlist)):
rho_t = esval(solES_t, [tlist[t_idx]])
solES_tau = ode2es(L, c_op * rho_t * a_op)
corr_mat[t_idx, :] = esval(expect(b_op, solES_tau), taulist)
return corr_mat
def _correlation_es_2op_1t(H, rho0, tlist, c_ops, a_op, b_op, reverse=False,
args=None, options=Odeoptions()):
"""
Internal function for calculating correlation functions using the
exponential series solver. See :func:`correlation_ss` usage.
"""
if debug:
print(inspect.stack()[0][3])
# contruct the Liouvillian
L = liouvillian(H, c_ops)
# find the steady state
if rho0 is None:
rho0 = steadystate(L)
elif rho0 and isket(rho0):
rho0 = ket2dm(rho0)
# evaluate the correlation function
if reverse:
# <A(t)B(t+tau)>
solC_tau = ode2es(L, rho0 * a_op)
return esval(expect(b_op, solC_tau), tlist)
else:
# default: <A(t+tau)B(t)>
solC_tau = ode2es(L, b_op * rho0)
return esval(expect(a_op, solC_tau), tlist)
def _correlation_es_2op_1t(H,
rho0,
tlist,
c_ops,
a_op,
b_op,
reverse=False,
args=None,
options=Odeoptions()):
"""
Internal function for calculating correlation functions using the
exponential series solver. See :func:`correlation_ss` usage.
"""
if debug:
print(inspect.stack()[0][3])
# contruct the Liouvillian
L = liouvillian(H, c_ops)
# find the steady state
if rho0 is None:
rho0 = steadystate(L)
elif rho0 and isket(rho0):
rho0 = ket2dm(rho0)
# evaluate the correlation function
if reverse:
# <A(t)B(t+tau)>
solC_tau = ode2es(L, rho0 * a_op)
return esval(expect(b_op, solC_tau), tlist)
else:
# default: <A(t+tau)B(t)>
solC_tau = ode2es(L, b_op * rho0)
return esval(expect(a_op, solC_tau), tlist)
def correlation_es(H, rho0, tlist, taulist, c_op_list, a_op, b_op):
"""
Internal function for calculating correlation functions using the
exponential series solver. See :func:`correlation` usage.
"""
# contruct the Liouvillian
L = liouvillian(H, c_op_list)
if rho0 == None:
rho0 = steady(L)
C_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex)
solES_t = ode2es(L, rho0)
for t_idx in range(len(tlist)):
rho_t = esval_op(solES_t, [tlist[t_idx]])
solES_tau = ode2es(L, b_op * rho_t)
C_mat[t_idx, :] = esval(expect(a_op, solES_tau), taulist)
return C_mat
def _correlation_es_2op_2t(H,
rho0,
tlist,
taulist,
c_ops,
a_op,
b_op,
reverse=False,
args=None,
options=Odeoptions()):
"""
Internal function for calculating correlation functions using the
exponential series solver. See :func:`correlation` usage.
"""
if debug:
print(inspect.stack()[0][3])
# contruct the Liouvillian
L = liouvillian(H, c_ops)
if rho0 is None:
rho0 = steadystate(L)
elif rho0 and isket(rho0):
rho0 = ket2dm(rho0)
C_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex)
solES_t = ode2es(L, rho0)
# evaluate the correlation function
if reverse:
# <A(t)B(t+tau)>
for t_idx in range(len(tlist)):
rho_t = esval(solES_t, [tlist[t_idx]])
solES_tau = ode2es(L, rho_t * a_op)
C_mat[t_idx, :] = esval(expect(b_op, solES_tau), taulist)
else:
# default: <A(t+tau)B(t)>
for t_idx in range(len(tlist)):
rho_t = esval(solES_t, [tlist[t_idx]])
solES_tau = ode2es(L, b_op * rho_t)
C_mat[t_idx, :] = esval(expect(a_op, solES_tau), taulist)
return C_mat
def essolve(H, rho0, tlist, c_op_list, expt_op_list):
"""
Evolution of a state vector or density matrix (`rho0`) for a given
Hamiltonian (`H`) and set of collapse operators (`c_op_list`), by
expressing the ODE as an exponential series. The output is either
the state vector at arbitrary points in time (`tlist`), or the
expectation values of the supplied operators (`expt_op_list`).
Parameters
----------
H : qobj/function_type
System Hamiltonian.
rho0 : :class:`qutip.qobj`
Initial state density matrix.
tlist : list/array
``list`` of times for :math:`t`.
c_op_list : list of :class:`qutip.qobj`
``list`` of :class:`qutip.qobj` collapse operators.
expt_op_list : list of :class:`qutip.qobj`
``list`` of :class:`qutip.qobj` operators for which to evaluate
expectation values.
Returns
-------
expt_array : array
Expectation values of wavefunctions/density matrices for the
times specified in ``tlist``.
.. note:: This solver does not support time-dependent Hamiltonians.
"""
n_expt_op = len(expt_op_list)
n_tsteps = len(tlist)
# Calculate the Liouvillian
if c_op_list is None or len(c_op_list) == 0:
L = H
else:
L = liouvillian(H, c_op_list)
es = ode2es(L, rho0)
# evaluate the expectation values
if n_expt_op == 0:
result_list = [Qobj()] * n_tsteps
else:
result_list = np.zeros([n_expt_op, n_tsteps], dtype=complex)
for n in range(0, n_expt_op):
result_list[n, :] = esval(expect(expt_op_list[n], es), tlist)
return result_list
def essolve(H, rho0, tlist, c_op_list, expt_op_list):
"""
Evolution of a state vector or density matrix (`rho0`) for a given
Hamiltonian (`H`) and set of collapse operators (`c_op_list`), by
expressing the ODE as an exponential series. The output is either
the state vector at arbitrary points in time (`tlist`), or the
expectation values of the supplied operators (`expt_op_list`).
Parameters
----------
H : qobj/function_type
System Hamiltonian.
rho0 : :class:`qutip.qobj`
Initial state density matrix.
tlist : list/array
``list`` of times for :math:`t`.
c_op_list : list of :class:`qutip.qobj`
``list`` of :class:`qutip.qobj` collapse operators.
expt_op_list : list of :class:`qutip.qobj`
``list`` of :class:`qutip.qobj` operators for which to evaluate
expectation values.
Returns
-------
expt_array : array
Expectation values of wavefunctions/density matrices for the
times specified in ``tlist``.
.. note:: This solver does not support time-dependent Hamiltonians.
"""
n_expt_op = len(expt_op_list)
n_tsteps = len(tlist)
# Calculate the Liouvillian
if c_op_list is None or len(c_op_list) == 0:
L = H
else:
L = liouvillian(H, c_op_list)
es = ode2es(L, rho0)
# evaluate the expectation values
if n_expt_op == 0:
result_list = [Qobj()] * n_tsteps
else:
result_list = np.zeros([n_expt_op, n_tsteps], dtype=complex)
for n in range(0, n_expt_op):
result_list[n, :] = esval(expect(expt_op_list[n], es), tlist)
return result_list
def _correlation_es_2op_2t(H, rho0, tlist, taulist, c_ops, a_op, b_op,
reverse=False, args=None, options=Odeoptions()):
"""
Internal function for calculating correlation functions using the
exponential series solver. See :func:`correlation` usage.
"""
if debug:
print(inspect.stack()[0][3])
# contruct the Liouvillian
L = liouvillian(H, c_ops)
if rho0 is None:
rho0 = steadystate(L)
elif rho0 and isket(rho0):
rho0 = ket2dm(rho0)
C_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex)
solES_t = ode2es(L, rho0)
# evaluate the correlation function
if reverse:
# <A(t)B(t+tau)>
for t_idx in range(len(tlist)):
rho_t = esval(solES_t, [tlist[t_idx]])
solES_tau = ode2es(L, rho_t * a_op)
C_mat[t_idx, :] = esval(expect(b_op, solES_tau), taulist)
else:
# default: <A(t+tau)B(t)>
for t_idx in range(len(tlist)):
rho_t = esval(solES_t, [tlist[t_idx]])
solES_tau = ode2es(L, b_op * rho_t)
C_mat[t_idx, :] = esval(expect(a_op, solES_tau), taulist)
return C_mat
def correlation_ss_es(H, tlist, c_op_list, a_op, b_op, rho0=None):
"""
Internal function for calculating correlation functions using the
exponential series solver. See :func:`correlation_ss` usage.
"""
# contruct the Liouvillian
L = liouvillian(H, c_op_list)
# find the steady state
if rho0 is None:
rho0 = steady(L)
# evaluate the correlation function
solC_tau = ode2es(L, b_op * rho0)
return esval(expect(a_op, solC_tau), tlist)
def correlation_ss_es(H, tlist, c_op_list, a_op, b_op, rho0=None):
"""
Internal function for calculating correlation functions using the
exponential series solver. See :func:`correlation_ss` usage.
"""
# contruct the Liouvillian
L = liouvillian(H, c_op_list)
# find the steady state
if rho0 == None:
rho0 = steady(L)
# evaluate the correlation function
solC_tau = ode2es(L, b_op * rho0)
return esval(expect(a_op, solC_tau), tlist)
def essolve(H, rho0, tlist, c_op_list, e_ops):
"""
Evolution of a state vector or density matrix (`rho0`) for a given
Hamiltonian (`H`) and set of collapse operators (`c_op_list`), by
expressing the ODE as an exponential series. The output is either
the state vector at arbitrary points in time (`tlist`), or the
expectation values of the supplied operators (`e_ops`).
.. deprecated:: 4.6.0
:obj:`~essolev` will be removed in QuTiP 5. Please use :obj:`~sesolve`
or :obj:`~mesolve` for general-purpose integration of the
Schroedinger/Lindblad master equation. This will likely be faster than
:obj:`~essolve` for you.
Parameters
----------
H : qobj/function_type
System Hamiltonian.
rho0 : :class:`qutip.qobj`
Initial state density matrix.
tlist : list/array
``list`` of times for :math:`t`.
c_op_list : list of :class:`qutip.qobj`
``list`` of :class:`qutip.qobj` collapse operators.
e_ops : list of :class:`qutip.qobj`
``list`` of :class:`qutip.qobj` operators for which to evaluate
expectation values.
Returns
-------
expt_array : array
Expectation values of wavefunctions/density matrices for the
times specified in ``tlist``.
.. note:: This solver does not support time-dependent Hamiltonians.
"""
n_expt_op = len(e_ops)
n_tsteps = len(tlist)
# Calculate the Liouvillian
if (c_op_list is None or len(c_op_list) == 0) and isket(rho0):
L = H
else:
L = liouvillian(H, c_op_list)
es = ode2es(L, rho0)
# evaluate the expectation values
if n_expt_op == 0:
results = [Qobj()] * n_tsteps
else:
results = np.zeros([n_expt_op, n_tsteps], dtype=complex)
for n, e in enumerate(e_ops):
results[n, :] = expect(e, esval(es, tlist))
data = Result()
data.solver = "essolve"
data.times = tlist
data.expect = [
np.real(results[n, :]) if e.isherm else results[n, :]
for n, e in enumerate(e_ops)
]
return data
