def _correlation_me_4op_2t(H, rho0, tlist, taulist, c_ops,
a_op, b_op, c_op, d_op, reverse=False,
args=None, options=Odeoptions()):
"""
Calculate the four-operator two-time correlation function on the form
<A(t)B(t+tau)C(t+tau)D(t)>.
See, Gardiner, Quantum Noise, Section 5.2.1
"""
if debug:
print(inspect.stack()[0][3])
if rho0 is None:
rho0 = steadystate(H, c_ops)
elif rho0 and isket(rho0):
rho0 = ket2dm(rho0)
C_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex)
rho_t = mesolve(
H, rho0, tlist, c_ops, [], args=args, options=options).states
for t_idx, rho in enumerate(rho_t):
C_mat[t_idx, :] = mesolve(H, d_op * rho * a_op, taulist,
c_ops, [b_op * c_op],
args=args, options=options).expect[0]
return C_mat
def _correlation_me_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 master
equation solver. See :func:`correlation` for usage.
"""
if debug:
print(inspect.stack()[0][3])
if rho0 is None:
rho0 = steadystate(H, c_ops)
elif rho0 and isket(rho0):
rho0 = ket2dm(rho0)
C_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex)
rho_t_list = mesolve(
H, rho0, tlist, c_ops, [], args=args, options=options).states
if reverse:
# <A(t)B(t+tau)>
for t_idx, rho_t in enumerate(rho_t_list):
C_mat[t_idx, :] = mesolve(H, rho_t * a_op, taulist,
c_ops, [b_op], args=args,
options=options).expect[0]
else:
# <A(t+tau)B(t)>
for t_idx, rho_t in enumerate(rho_t_list):
C_mat[t_idx, :] = mesolve(H, b_op * rho_t, taulist,
c_ops, [a_op], args=args,
options=options).expect[0]
return C_mat
def _correlation_me_2t(H,
state0,
tlist,
taulist,
c_ops,
a_op,
b_op,
c_op,
args=None,
options=Options()):
"""
Internal function for calculating the three-operator two-time
correlation function:
<A(t)B(t+tau)C(t)>
using a master equation 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])
rho_t = mesolve(H, rho0, tlist, c_ops, [], args=args,
options=options).states
corr_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex)
H_shifted, _args = _transform_H_t_shift(H, args)
for t_idx, rho in enumerate(rho_t):
if not isinstance(H, Qobj):
_args["_t0"] = tlist[t_idx]
corr_mat[t_idx, :] = mesolve(H_shifted,
c_op * rho * a_op,
taulist,
c_ops, [b_op],
args=_args,
options=options).expect[0]
if t_idx == 1:
options.rhs_reuse = True
return corr_mat
def _correlation_me_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 master
equation solver. See :func:`correlation` for usage.
"""
if debug:
print(inspect.stack()[0][3])
if rho0 is None:
rho0 = steadystate(H, c_ops)
elif rho0 and isket(rho0):
rho0 = ket2dm(rho0)
C_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex)
rho_t_list = mesolve(H, rho0, tlist, c_ops, [], args=args,
options=options).states
if reverse:
# <A(t)B(t+tau)>
for t_idx, rho_t in enumerate(rho_t_list):
C_mat[t_idx, :] = mesolve(H,
rho_t * a_op,
taulist,
c_ops, [b_op],
args=args,
options=options).expect[0]
else:
# <A(t+tau)B(t)>
for t_idx, rho_t in enumerate(rho_t_list):
C_mat[t_idx, :] = mesolve(H,
b_op * rho_t,
taulist,
c_ops, [a_op],
args=args,
options=options).expect[0]
return C_mat
def _parallel_mesolve(n, N, H, tlist, c_op_list, args, options):
col_idx, row_idx = np.unravel_index(n, (N, N))
rho0 = Qobj(sp.csr_matrix(([1], ([row_idx], [col_idx])),
shape=(N, N), dtype=complex))
output = mesolve(H, rho0, tlist, c_op_list, [], args, options,
_safe_mode=False)
return output
def _correlation_me_4op_1t(H,
rho0,
tlist,
c_ops,
a_op,
b_op,
c_op,
d_op,
args=None,
options=Odeoptions()):
"""
Calculate the four-operator two-time correlation function on the form
<A(0)B(tau)C(tau)D(0)>.
See, Gardiner, Quantum Noise, Section 5.2.1
"""
if debug:
print(inspect.stack()[0][3])
if rho0 is None:
rho0 = steadystate(H, c_ops)
elif rho0 and isket(rho0):
rho0 = ket2dm(rho0)
return mesolve(H,
d_op * rho0 * a_op,
tlist,
c_ops, [b_op * c_op],
args=args,
options=options).expect[0]
def _parallel_mesolve(n, N, H, tlist, c_op_list, args, options):
col_idx, row_idx = np.unravel_index(n, (N, N))
rho0 = Qobj(sp.csr_matrix(([1], ([row_idx], [col_idx])),
shape=(N,N), dtype=complex))
output = mesolve(H, rho0, tlist, c_op_list, [], args, options,
_safe_mode=False)
return output
def _correlation_me_2t(H, state0, tlist, taulist, c_ops, a_op, b_op, c_op,
args={}, options=Options()):
"""
Internal function for calculating the three-operator two-time
correlation function:
<A(t)B(t+tau)C(t)>
using a master equation 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])
rho_t = mesolve(H, rho0, tlist, c_ops, [],
args=args, options=options).states
corr_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex)
H_shifted, c_ops_shifted, _args = _transform_L_t_shift(H, c_ops, args)
if config.tdname:
_cython_build_cleanup(config.tdname)
rhs_clear()
for t_idx, rho in enumerate(rho_t):
if not isinstance(H, Qobj):
_args["_t0"] = tlist[t_idx]
corr_mat[t_idx, :] = mesolve(
H_shifted, c_op * rho * a_op, taulist, c_ops_shifted,
[b_op], args=_args, options=options
).expect[0]
if t_idx == 1:
options.rhs_reuse = True
if config.tdname:
_cython_build_cleanup(config.tdname)
rhs_clear()
return corr_mat
def correlation_ode(H, rho0, tlist, taulist, c_op_list, a_op, b_op):
"""
Internal function for calculating correlation functions using the master
equation solver. See :func:`correlation` usage.
"""
if rho0 == None:
rho0 = steadystate(H, co_op_list)
C_mat = np.zeros([np.size(tlist),np.size(taulist)],dtype=complex)
rho_t = mesolve(H, rho0, tlist, c_op_list, []).states
for t_idx in range(len(tlist)):
C_mat[t_idx,:] = mesolve(H, b_op * rho_t[t_idx], taulist, c_op_list, [a_op]).expect[0]
return C_mat
def coherence_function_g2(H, rho0, taulist, c_ops, a_op, solver="me",
args=None, options=Odeoptions()):
"""
Calculate the second-order quantum coherence function:
.. math::
g^{(2)}(\\tau) =
\\frac{\\langle a^\\dagger(0)a^\\dagger(\\tau)a(\\tau)a(0)\\rangle}
{\\langle a^\\dagger(\\tau)a(\\tau)\\rangle
\\langle a^\\dagger(0)a(0)\\rangle}
Parameters
----------
H : :class:`qutip.qobj.Qobj`
system Hamiltonian.
rho0 : :class:`qutip.qobj.Qobj`
Initial state density matrix (or state vector). If 'rho0' is
'None', then the steady state will be used as initial state.
taulist : *list* / *array*
list of times for :math:`\\tau`.
c_ops : list of :class:`qutip.qobj.Qobj`
list of collapse operators.
a_op : :class:`qutip.qobj.Qobj`
The annihilation operator of the mode.
solver : str
choice of solver (currently only 'me')
Returns
-------
g2, G2: tuble of *array*
The normalized and unnormalized second-order coherence function.
"""
# first calculate the photon number
if rho0 is None:
rho0 = steadystate(H, c_ops)
n = np.array([expect(rho0, a_op.dag() * a_op)])
else:
n = mesolve(
H, rho0, taulist, c_ops, [a_op.dag() * a_op],
args=args, options=options).expect[0]
# calculate the correlation function G2 and normalize with n to obtain g2
G2 = correlation_4op_1t(H, rho0, taulist, c_ops,
a_op.dag(), a_op.dag(), a_op, a_op,
solver=solver, args=args, options=options)
g2 = G2 / (n[0] * n)
return g2, G2
def correlation_ode(H, rho0, tlist, taulist, c_op_list, a_op, b_op):
"""
Internal function for calculating correlation functions using the master
equation solver. See :func:`correlation` usage.
"""
if rho0 == None:
rho0 = steadystate(H, co_op_list)
C_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex)
rho_t = mesolve(H, rho0, tlist, c_op_list, []).states
for t_idx in range(len(tlist)):
C_mat[t_idx, :] = mesolve(H, b_op * rho_t[t_idx], taulist, c_op_list,
[a_op]).expect[0]
return C_mat
def solve_system_unpack_qutip(pack):
"""
These functions are used for the parallel computing, where we need to call the function with just one variable. Here we extract the index of the
process and unpack the parameters given to solve the system. By default, the value for hbar is 1, and the absolute and relative errors are
10^{-8} and 10^{-6} respectively.
:param pack: (list) List with the following parameters: [index of the parallel computation, time, density0, parameters, hamiltonian]. To pass the
values of default parameters, all of them must be in the sixth element of the list as a dictionary, e.g {'hbar': 1, 'atol': 1e-8}.
:return: (list) list with the index of the computation al the solution of the system.
"""
i, H, psi0, times = pack[:4]
if len(pack) > 4:
extra_params = pack[-1]
else:
extra_params = {}
if 'args' not in extra_params:
extra_params['args'] = {}
if 'options' not in extra_params:
extra_params['options'] = Options()
if 'dim' not in extra_params:
extra_params['dim'] = 3
if 'only_final' not in extra_params:
extra_params['only_final'] = False
if 'c_ops' not in extra_params:
extra_params['c_ops'] = None
result = mesolve(H,
psi0,
times,
args=extra_params['args'],
options=extra_params['options'],
c_ops=extra_params['c_ops'])
if extra_params['only_final']:
states = np.zeros([extra_params['dim']], dtype=complex)
if extra_params['c_ops'] is None:
states[:] = result.states[-1].full()[:, 0]
else:
prob = np.diag(result.states[-1].full())
phases = np.exp(1j * np.angle(result.states[-1].full())[0, :])
states[:] = np.sqrt(prob) * phases
else:
states = np.zeros([extra_params['dim'], len(times)], dtype=complex)
for j in range(0, len(times)):
if extra_params['c_ops'] is None:
states[:, j] = result.states[j].full()[:, 0]
else:
prob = np.diag(result.states[j].full())
phases = np.exp(1j * np.angle(result.states[j].full())[0, :])
states[:, j] = np.sqrt(prob) * phases
return [i, states]
def correlation_ss_ode(H, tlist, c_op_list, a_op, b_op, rho0=None):
"""
Internal function for calculating correlation functions using the master
equation solver. See :func:`correlation_ss` usage.
"""
L = liouvillian(H, c_op_list)
if rho0 is None:
rho0 = steady(L)
return mesolve(H, b_op * rho0, tlist, c_op_list, [a_op]).expect[0]
def correlation_ss_ode(H, tlist, c_op_list, a_op, b_op, rho0=None):
"""
Internal function for calculating correlation functions using the master
equation solver. See :func:`correlation_ss` usage.
"""
L = liouvillian(H, c_op_list)
if rho0 == None:
rho0 = steady(L)
return mesolve(H, b_op * rho0, tlist, c_op_list, [a_op]).expect[0]
def _correlation_me_2op_1t(H, rho0, tlist, c_ops, a_op, b_op, reverse=False, args=None, options=Options()):
"""
Internal function for calculating correlation functions using the master
equation solver. See :func:`correlation_ss` for usage.
"""
if debug:
print(inspect.stack()[0][3])
if rho0 is None:
rho0 = steadystate(H, c_ops)
elif rho0 and isket(rho0):
rho0 = ket2dm(rho0)
if reverse:
# <A(t)B(t+tau)>
return mesolve(H, rho0 * a_op, tlist, c_ops, [b_op], args=args, options=options).expect[0]
else:
# <A(t+tau)B(t)>
return mesolve(H, b_op * rho0, tlist, c_ops, [a_op], args=args, options=options).expect[0]
def _correlation_me_4op_2t(H,
rho0,
tlist,
taulist,
c_ops,
a_op,
b_op,
c_op,
d_op,
reverse=False,
args=None,
options=Odeoptions()):
"""
Calculate the four-operator two-time correlation function on the form
<A(t)B(t+tau)C(t+tau)D(t)>.
See, Gardiner, Quantum Noise, Section 5.2.1
"""
if debug:
print(inspect.stack()[0][3])
if rho0 is None:
rho0 = steadystate(H, c_ops)
elif rho0 and isket(rho0):
rho0 = ket2dm(rho0)
C_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex)
rho_t = mesolve(H, rho0, tlist, c_ops, [], args=args,
options=options).states
for t_idx, rho in enumerate(rho_t):
C_mat[t_idx, :] = mesolve(H,
d_op * rho * a_op,
taulist,
c_ops, [b_op * c_op],
args=args,
options=options).expect[0]
return C_mat
def _parallel_mesolve(n, N, H, tlist, c_op_list, args, options, dims=None):
col_idx, row_idx = np.unravel_index(n, (N, N))
rho0 = projection(N, row_idx, col_idx)
rho0.dims = dims
output = mesolve(H,
rho0,
tlist,
c_ops=c_op_list,
args=args,
options=options,
_safe_mode=False)
return output
def testExpectSolverCompatibility(self):
"""
expect: operator list and state list
"""
c_ops = [0.0001 * sigmaz()]
e_ops = [sigmax(), sigmay(), sigmaz(), sigmam(), sigmap()]
times = np.linspace(0, 10, 100)
res1 = mesolve(sigmax(), fock(2, 0), times, c_ops, e_ops)
res2 = mesolve(sigmax(), fock(2, 0), times, c_ops, [])
e1 = res1.expect
e2 = expect(e_ops, res2.states)
assert_(len(e1) == len(e2))
for n in range(len(e1)):
assert_(len(e1[n]) == len(e2[n]))
assert_(isinstance(e1[n], np.ndarray))
assert_(isinstance(e2[n], np.ndarray))
assert_(e1[n].dtype == e2[n].dtype)
assert_(all(abs(e1[n] - e2[n]) < 1e-12))
def testExpectSolverCompatibility(self):
"""
expect: operator list and state list
"""
c_ops = [0.0001 * sigmaz()]
e_ops = [sigmax(), sigmay(), sigmaz(), sigmam(), sigmap()]
times = np.linspace(0, 10, 100)
res1 = mesolve(sigmax(), fock(2, 0), times, c_ops, e_ops)
res2 = mesolve(sigmax(), fock(2, 0), times, c_ops, [])
e1 = res1.expect
e2 = expect(e_ops, res2.states)
assert_(len(e1) == len(e2))
for n in range(len(e1)):
assert_(len(e1[n]) == len(e2[n]))
assert_(isinstance(e1[n], np.ndarray))
assert_(isinstance(e2[n], np.ndarray))
assert_(e1[n].dtype == e2[n].dtype)
assert_(all(abs(e1[n] - e2[n]) < 1e-12))
def floquet_modes_table(f_modes_0, f_energies, tlist, H, T, args=None):
"""
Pre-calculate the Floquet modes for a range of times spanning the floquet
period. Can later be used as a table to look up the floquet modes for
any time.
Parameters
----------
f_modes_0 : list of :class:`qutip.qobj` (kets)
Floquet modes at :math:`t`
f_energies : list
Floquet energies.
tlist : array
The list of times at which to evaluate the floquet modes.
H : :class:`qutip.qobj`
system Hamiltonian, time-dependent with period `T`
T : float
The period of the time-dependence of the hamiltonian.
args : dictionary
dictionary with variables required to evaluate H
Returns
-------
output : nested list
A nested list of Floquet modes as kets for each time in `tlist`
"""
# truncate tlist to the driving period
tlist_period = tlist[np.where(tlist <= T)]
f_modes_table_t = [[] for t in tlist_period]
opt = Options()
opt.rhs_reuse = True
for n, f_mode in enumerate(f_modes_0):
output = mesolve(H, f_mode, tlist_period, [], [], args, opt)
for t_idx, f_state_t in enumerate(output.states):
f_modes_table_t[t_idx].append(
f_state_t * exp(1j * f_energies[n] * tlist_period[t_idx]))
return f_modes_table_t
def floquet_modes_table(f_modes_0, f_energies, tlist, H, T, args=None):
"""
Pre-calculate the Floquet modes for a range of times spanning the floquet
period. Can later be used as a table to look up the floquet modes for
any time.
Parameters
----------
f_modes_0 : list of :class:`qutip.qobj` (kets)
Floquet modes at :math:`t`
f_energies : list
Floquet energies.
tlist : array
The list of times at which to evaluate the floquet modes.
H : :class:`qutip.qobj`
system Hamiltonian, time-dependent with period `T`
T : float
The period of the time-dependence of the hamiltonian.
args : dictionary
dictionary with variables required to evaluate H
Returns
-------
output : nested list
A nested list of Floquet modes as kets for each time in `tlist`
"""
# truncate tlist to the driving period
tlist_period = tlist[np.where(tlist <= T)]
f_modes_table_t = [[] for t in tlist_period]
opt = Options()
opt.rhs_reuse = True
for n, f_mode in enumerate(f_modes_0):
output = mesolve(H, f_mode, tlist_period, [], [], args, opt)
for t_idx, f_state_t in enumerate(output.states):
f_modes_table_t[t_idx].append(
f_state_t * exp(1j * f_energies[n] * tlist_period[t_idx]))
return f_modes_table_t
def _correlation_me_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 master
equation solver. See :func:`correlation_ss` for usage.
"""
if debug:
print(inspect.stack()[0][3])
if rho0 is None:
rho0 = steadystate(H, c_ops)
elif rho0 and isket(rho0):
rho0 = ket2dm(rho0)
if reverse:
# <A(t)B(t+tau)>
return mesolve(H,
rho0 * a_op,
tlist,
c_ops, [b_op],
args=args,
options=options).expect[0]
else:
# <A(t+tau)B(t)>
return mesolve(H,
b_op * rho0,
tlist,
c_ops, [a_op],
args=args,
options=options).expect[0]
def _correlation_me_gtt(H, rho0, tlist, taulist, c_ops, a_op, b_op,
c_op, d_op):
"""
Calculate the correlation function <A(t)B(t+tau)C(t+tau)D(t)>
(gtt = general two-time)
See, Gardiner, Quantum Noise, Section 5.2.1
.. note::
Experimental.
"""
if rho0 is None:
rho0 = steadystate(H, c_ops)
C_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex)
rho_t = mesolve(H, rho0, tlist, c_op_list, []).states
for t_idx, rho in enumerate(rho_t):
C_mat[t_idx, :] = mesolve(H, d_op * rho * a_op, taulist,
c_ops, [b_op * c_op]).expect[0]
return C_mat
def _correlation_me_ss_gtt(H, tlist, c_ops, a_op, b_op, c_op, d_op, rho0=None):
"""
Calculate the correlation function <A(0)B(tau)C(tau)D(0)>
(ss_gtt = steadystate general two-time)
See, Gardiner, Quantum Noise, Section 5.2.1
.. note::
Experimental.
"""
if rho0 is None:
rho0 = steadystate(H, c_ops)
return mesolve(H, d_op * rho0 * a_op, tlist,
c_ops, [b_op * c_op]).expect[0]
def _correlation_me_4op_1t(H, rho0, tlist, c_ops, a_op, b_op, c_op, d_op, args=None, options=Options()):
"""
Calculate the four-operator two-time correlation function on the form
<A(0)B(tau)C(tau)D(0)>.
See, Gardiner, Quantum Noise, Section 5.2.1
"""
if debug:
print(inspect.stack()[0][3])
if rho0 is None:
rho0 = steadystate(H, c_ops)
elif rho0 and isket(rho0):
rho0 = ket2dm(rho0)
return mesolve(H, d_op * rho0 * a_op, tlist, c_ops, [b_op * c_op], args=args, options=options).expect[0]
def propagator(H,
t,
c_op_list=[],
args={},
options=None,
unitary_mode='batch',
parallel=False,
progress_bar=None,
_safe_mode=True,
**kwargs):
r"""
Calculate the propagator U(t) for the density matrix or wave function such
that :math:`\psi(t) = U(t)\psi(0)` or
:math:`\rho_{\mathrm vec}(t) = U(t) \rho_{\mathrm vec}(0)`
where :math:`\rho_{\mathrm vec}` is the vector representation of the
density matrix.
Parameters
----------
H : qobj or list
Hamiltonian as a Qobj instance of a nested list of Qobjs and
coefficients in the list-string or list-function format for
time-dependent Hamiltonians (see description in :func:`qutip.mesolve`).
t : float or array-like
Time or list of times for which to evaluate the propagator.
c_op_list : list
List of qobj collapse operators.
args : list/array/dictionary
Parameters to callback functions for time-dependent Hamiltonians and
collapse operators.
options : :class:`qutip.Options`
with options for the ODE solver.
unitary_mode = str ('batch', 'single')
Solve all basis vectors simulaneously ('batch') or individually
('single').
parallel : bool {False, True}
Run the propagator in parallel mode. This will override the
unitary_mode settings if set to True.
progress_bar: BaseProgressBar
Optional instance of BaseProgressBar, or a subclass thereof, for
showing the progress of the simulation. By default no progress bar
is used, and if set to True a TextProgressBar will be used.
Returns
-------
a : qobj
Instance representing the propagator :math:`U(t)`.
"""
kw = _default_kwargs()
if 'num_cpus' in kwargs:
num_cpus = kwargs['num_cpus']
else:
num_cpus = kw['num_cpus']
if progress_bar is None:
progress_bar = BaseProgressBar()
elif progress_bar is True:
progress_bar = TextProgressBar()
if options is None:
options = Options()
options.rhs_reuse = True
rhs_clear()
if isinstance(t, (int, float, np.integer, np.floating)):
tlist = [0, t]
else:
tlist = t
if _safe_mode:
_solver_safety_check(H, None, c_ops=c_op_list, e_ops=[], args=args)
td_type = _td_format_check(H, c_op_list, solver='me')
if isinstance(
H,
(types.FunctionType, types.BuiltinFunctionType, functools.partial)):
H0 = H(0.0, args)
if unitary_mode == 'batch':
# batch don't work with function Hamiltonian
unitary_mode = 'single'
elif isinstance(H, list):
H0 = H[0][0] if isinstance(H[0], list) else H[0]
else:
H0 = H
if len(c_op_list) == 0 and H0.isoper:
# calculate propagator for the wave function
N = H0.shape[0]
dims = H0.dims
if parallel:
unitary_mode = 'single'
u = np.zeros([N, N, len(tlist)], dtype=complex)
output = parallel_map(_parallel_sesolve,
range(N),
task_args=(N, H, tlist, args, options),
progress_bar=progress_bar,
num_cpus=num_cpus)
for n in range(N):
for k, t in enumerate(tlist):
u[:, n, k] = output[n].states[k].full().T
else:
if unitary_mode == 'single':
output = sesolve(H,
qeye(dims[0]),
tlist, [],
args,
options,
_safe_mode=False)
if len(tlist) == 2:
return output.states[-1]
else:
return output.states
elif unitary_mode == 'batch':
u = np.zeros(len(tlist), dtype=object)
_rows = np.array([(N + 1) * m for m in range(N)])
_cols = np.zeros_like(_rows)
_data = np.ones_like(_rows, dtype=complex)
psi0 = Qobj(sp.coo_matrix((_data, (_rows, _cols))).tocsr())
if td_type[1] > 0 or td_type[2] > 0:
H2 = []
for k in range(len(H)):
if isinstance(H[k], list):
H2.append([tensor(qeye(N), H[k][0]), H[k][1]])
else:
H2.append(tensor(qeye(N), H[k]))
else:
H2 = tensor(qeye(N), H)
options.normalize_output = False
output = sesolve(H2,
psi0,
tlist, [],
args=args,
options=options,
_safe_mode=False)
for k, t in enumerate(tlist):
u[k] = sp_reshape(output.states[k].data, (N, N))
unit_row_norm(u[k].data, u[k].indptr, u[k].shape[0])
u[k] = u[k].T.tocsr()
else:
raise Exception('Invalid unitary mode.')
elif len(c_op_list) == 0 and H0.issuper:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
unitary_mode = 'single'
N = H0.shape[0]
sqrt_N = int(np.sqrt(N))
dims = H0.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
if parallel:
output = parallel_map(_parallel_mesolve,
range(N * N),
task_args=(sqrt_N, H, tlist, c_op_list, args,
options),
progress_bar=progress_bar,
num_cpus=num_cpus)
for n in range(N * N):
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output[n].states[k].full()).T
else:
rho0 = qeye(N, N)
rho0.dims = [[sqrt_N, sqrt_N], [sqrt_N, sqrt_N]]
output = mesolve(H,
psi0,
tlist, [],
args,
options,
_safe_mode=False)
if len(tlist) == 2:
return output.states[-1]
else:
return output.states
else:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
unitary_mode = 'single'
N = H0.shape[0]
dims = [H0.dims, H0.dims]
u = np.zeros([N * N, N * N, len(tlist)], dtype=complex)
if parallel:
output = parallel_map(_parallel_mesolve,
range(N * N),
task_args=(N, H, tlist, c_op_list, args,
options),
progress_bar=progress_bar,
num_cpus=num_cpus)
for n in range(N * N):
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output[n].states[k].full()).T
else:
progress_bar.start(N * N)
for n in range(N * N):
progress_bar.update(n)
col_idx, row_idx = np.unravel_index(n, (N, N))
rho0 = Qobj(
sp.csr_matrix(([1], ([row_idx], [col_idx])),
shape=(N, N),
dtype=complex))
output = mesolve(H,
rho0,
tlist,
c_op_list, [],
args,
options,
_safe_mode=False)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
progress_bar.finished()
if len(tlist) == 2:
if unitary_mode == 'batch':
return Qobj(u[-1], dims=dims)
else:
return Qobj(u[:, :, 1], dims=dims)
else:
if unitary_mode == 'batch':
return np.array([Qobj(u[k], dims=dims) for k in range(len(tlist))],
dtype=object)
else:
return np.array(
[Qobj(u[:, :, k], dims=dims) for k in range(len(tlist))],
dtype=object)
def propagator(H,
t,
c_op_list=[],
args={},
options=None,
parallel=False,
progress_bar=None,
**kwargs):
"""
Calculate the propagator U(t) for the density matrix or wave function such
that :math:`\psi(t) = U(t)\psi(0)` or
:math:`\\rho_{\mathrm vec}(t) = U(t) \\rho_{\mathrm vec}(0)`
where :math:`\\rho_{\mathrm vec}` is the vector representation of the
density matrix.
Parameters
----------
H : qobj or list
Hamiltonian as a Qobj instance of a nested list of Qobjs and
coefficients in the list-string or list-function format for
time-dependent Hamiltonians (see description in :func:`qutip.mesolve`).
t : float or array-like
Time or list of times for which to evaluate the propagator.
c_op_list : list
List of qobj collapse operators.
args : list/array/dictionary
Parameters to callback functions for time-dependent Hamiltonians and
collapse operators.
options : :class:`qutip.Options`
with options for the ODE solver.
parallel : bool {False, True}
Run the propagator in parallel mode.
progress_bar: BaseProgressBar
Optional instance of BaseProgressBar, or a subclass thereof, for
showing the progress of the simulation. By default no progress bar
is used, and if set to True a TextProgressBar will be used.
Returns
-------
a : qobj
Instance representing the propagator :math:`U(t)`.
"""
kw = _default_kwargs()
if 'num_cpus' in kwargs:
num_cpus = kwargs['num_cpus']
else:
num_cpus = kw['num_cpus']
if progress_bar is None:
progress_bar = BaseProgressBar()
elif progress_bar is True:
progress_bar = TextProgressBar()
if options is None:
options = Options()
options.rhs_reuse = True
rhs_clear()
if isinstance(t, (int, float, np.integer, np.floating)):
tlist = [0, t]
else:
tlist = t
td_type = _td_format_check(H, c_op_list, solver='me')[2]
if td_type > 0:
rhs_generate(H, c_op_list, args=args, options=options)
if isinstance(
H,
(types.FunctionType, types.BuiltinFunctionType, functools.partial)):
H0 = H(0.0, args)
elif isinstance(H, list):
H0 = H[0][0] if isinstance(H[0], list) else H[0]
else:
H0 = H
if len(c_op_list) == 0 and H0.isoper:
# calculate propagator for the wave function
N = H0.shape[0]
dims = H0.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
if parallel:
output = parallel_map(_parallel_sesolve,
range(N),
task_args=(N, H, tlist, args, options),
progress_bar=progress_bar,
num_cpus=num_cpus)
for n in range(N):
for k, t in enumerate(tlist):
u[:, n, k] = output[n].states[k].full().T
else:
progress_bar.start(N)
for n in range(0, N):
progress_bar.update(n)
psi0 = basis(N, n)
output = sesolve(H,
psi0,
tlist, [],
args,
options,
_safe_mode=False)
for k, t in enumerate(tlist):
u[:, n, k] = output.states[k].full().T
progress_bar.finished()
# todo: evolving a batch of wave functions:
# psi_0_list = [basis(N, n) for n in range(N)]
# psi_t_list = mesolve(H, psi_0_list, [0, t], [], [], args, options)
# for n in range(0, N):
# u[:,n] = psi_t_list[n][1].full().T
elif len(c_op_list) == 0 and H0.issuper:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
N = H0.shape[0]
sqrt_N = int(np.sqrt(N))
dims = H0.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
if parallel:
output = parallel_map(_parallel_mesolve,
range(N * N),
task_args=(sqrt_N, H, tlist, c_op_list, args,
options),
progress_bar=progress_bar,
num_cpus=num_cpus)
for n in range(N * N):
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output[n].states[k].full()).T
else:
progress_bar.start(N)
for n in range(0, N):
progress_bar.update(n)
col_idx, row_idx = np.unravel_index(n, (sqrt_N, sqrt_N))
rho0 = Qobj(
sp.csr_matrix(([1], ([row_idx], [col_idx])),
shape=(sqrt_N, sqrt_N),
dtype=complex))
output = mesolve(H,
rho0,
tlist, [], [],
args,
options,
_safe_mode=False)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
progress_bar.finished()
else:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
N = H0.shape[0]
dims = [H0.dims, H0.dims]
u = np.zeros([N * N, N * N, len(tlist)], dtype=complex)
if parallel:
output = parallel_map(_parallel_mesolve,
range(N * N),
task_args=(N, H, tlist, c_op_list, args,
options),
progress_bar=progress_bar,
num_cpus=num_cpus)
for n in range(N * N):
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output[n].states[k].full()).T
else:
progress_bar.start(N * N)
for n in range(N * N):
progress_bar.update(n)
col_idx, row_idx = np.unravel_index(n, (N, N))
rho0 = Qobj(
sp.csr_matrix(([1], ([row_idx], [col_idx])),
shape=(N, N),
dtype=complex))
output = mesolve(H,
rho0,
tlist,
c_op_list, [],
args,
options,
_safe_mode=False)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
progress_bar.finished()
if len(tlist) == 2:
return Qobj(u[:, :, 1], dims=dims)
else:
return np.array(
[Qobj(u[:, :, k], dims=dims) for k in range(len(tlist))],
dtype=object)
def propagator(H, t, c_op_list, args=None, options=None):
"""
Calculate the propagator U(t) for the density matrix or wave function such
that :math:`\psi(t) = U(t)\psi(0)` or
:math:`\\rho_{\mathrm vec}(t) = U(t) \\rho_{\mathrm vec}(0)`
where :math:`\\rho_{\mathrm vec}` is the vector representation of the
density matrix.
Parameters
----------
H : qobj or list
Hamiltonian as a Qobj instance of a nested list of Qobjs and
coefficients in the list-string or list-function format for
time-dependent Hamiltonians (see description in :func:`qutip.mesolve`).
t : float or array-like
Time or list of times for which to evaluate the propagator.
c_op_list : list
List of qobj collapse operators.
args : list/array/dictionary
Parameters to callback functions for time-dependent Hamiltonians and
collapse operators.
options : :class:`qutip.Odeoptions`
with options for the ODE solver.
Returns
-------
a : qobj
Instance representing the propagator :math:`U(t)`.
"""
if options is None:
options = Odeoptions()
options.rhs_reuse = True
rhs_clear()
elif options.rhs_reuse:
msg = ("propagator is using previously defined rhs " +
"function (options.rhs_reuse = True)")
warnings.warn(msg)
tlist = [0, t] if isinstance(t, (int, float, np.int64, np.float64)) else t
if len(c_op_list) == 0:
# calculate propagator for the wave function
if isinstance(H, types.FunctionType):
H0 = H(0.0, args)
N = H0.shape[0]
dims = H0.dims
elif isinstance(H, list):
H0 = H[0][0] if isinstance(H[0], list) else H[0]
N = H0.shape[0]
dims = H0.dims
else:
N = H.shape[0]
dims = H.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
for n in range(0, N):
psi0 = basis(N, n)
output = sesolve(H, psi0, tlist, [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = output.states[k].full().T
# todo: evolving a batch of wave functions:
# psi_0_list = [basis(N, n) for n in range(N)]
# psi_t_list = mesolve(H, psi_0_list, [0, t], [], [], args, options)
# for n in range(0, N):
# u[:,n] = psi_t_list[n][1].full().T
else:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
if isinstance(H, types.FunctionType):
H0 = H(0.0, args)
N = H0.shape[0]
dims = [H0.dims, H0.dims]
elif isinstance(H, list):
H0 = H[0][0] if isinstance(H[0], list) else H[0]
N = H0.shape[0]
dims = [H0.dims, H0.dims]
else:
N = H.shape[0]
dims = [H.dims, H.dims]
u = np.zeros([N * N, N * N, len(tlist)], dtype=complex)
for n in range(0, N * N):
psi0 = basis(N * N, n)
rho0 = Qobj(vec2mat(psi0.full()))
output = mesolve(H, rho0, tlist, c_op_list, [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
if len(tlist) == 2:
return Qobj(u[:, :, 1], dims=dims)
else:
return [Qobj(u[:, :, k], dims=dims) for k in range(len(tlist))]
def propagator(H, t, c_op_list, H_args=None, opt=None):
"""
Calculate the propagator U(t) for the density matrix or wave function such
that :math:`\psi(t) = U(t)\psi(0)` or
:math:`\\rho_{\mathrm vec}(t) = U(t) \\rho_{\mathrm vec}(0)`
where :math:`\\rho_{\mathrm vec}` is the vector representation of the
density matrix.
Parameters
----------
H : qobj or list
Hamiltonian as a Qobj instance of a nested list of Qobjs and
coefficients in the list-string or list-function format for
time-dependent Hamiltonians (see description in :func:`qutip.mesolve`).
t : float or array-like
Time or list of times for which to evaluate the propagator.
c_op_list : list
List of qobj collapse operators.
H_args : list/array/dictionary
Parameters to callback functions for time-dependent Hamiltonians.
Returns
-------
a : qobj
Instance representing the propagator :math:`U(t)`.
"""
if opt is None:
opt = Odeoptions()
opt.rhs_reuse = True
tlist = [0, t] if isinstance(t, (int, float, np.int64, np.float64)) else t
if len(c_op_list) == 0:
# calculate propagator for the wave function
if isinstance(H, types.FunctionType):
H0 = H(0.0, H_args)
N = H0.shape[0]
dims = H0.dims
elif isinstance(H, list):
H0 = H[0][0] if isinstance(H[0], list) else H[0]
N = H0.shape[0]
dims = H0.dims
else:
N = H.shape[0]
dims = H.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
for n in range(0, N):
psi0 = basis(N, n)
output = mesolve(H, psi0, tlist, [], [], H_args, opt)
for k, t in enumerate(tlist):
u[:, n, k] = output.states[k].full().T
# todo: evolving a batch of wave functions:
#psi_0_list = [basis(N, n) for n in range(N)]
#psi_t_list = mesolve(H, psi_0_list, [0, t], [], [], H_args, opt)
#for n in range(0, N):
# u[:,n] = psi_t_list[n][1].full().T
else:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
if isinstance(H, types.FunctionType):
H0 = H(0.0, H_args)
N = H0.shape[0]
dims = [H0.dims, H0.dims]
elif isinstance(H, list):
H0 = H[0][0] if isinstance(H[0], list) else H[0]
N = H0.shape[0]
dims = [H0.dims, H0.dims]
else:
N = H.shape[0]
dims = [H.dims, H.dims]
u = np.zeros([N * N, N * N, len(tlist)], dtype=complex)
for n in range(0, N * N):
psi0 = basis(N * N, n)
rho0 = Qobj(vec2mat(psi0.full()))
output = mesolve(H, rho0, tlist, c_op_list, [], H_args, opt)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
if len(tlist) == 2:
return Qobj(u[:, :, 1], dims=dims)
else:
return [Qobj(u[:, :, k], dims=dims) for k in range(len(tlist))]
def run_state(self,
init_state=None,
analytical=False,
states=None,
noisy=True,
**kwargs):
"""
If `analytical` is False, use :func:`qutip.mesolve` to
calculate the time of the state evolution
and return the result. Other arguments of mesolve can be
given as keyword arguments.
If `analytical` is True, calculate the propagator
with matrix exponentiation and return a list of matrices.
Noise will be neglected in this choice.
Parameters
----------
init_state: Qobj
Initial density matrix or state vector (ket).
analytical: bool
If True, calculate the evolution with matrices exponentiation.
states: :class:`qutip.Qobj`, optional
Old API, same as init_state.
**kwargs
Keyword arguments for the qutip solver.
Returns
-------
evo_result: :class:`qutip.Result`
If ``analytical`` is False, an instance of the class
:class:`qutip.Result` will be returned.
If ``analytical`` is True, a list of matrices representation
is returned.
"""
if states is not None:
warnings.warn(
"states will be deprecated and replaced by init_state",
DeprecationWarning)
if init_state is None and states is None:
raise ValueError("Qubit state not defined.")
elif init_state is None:
# just to keep the old parameters `states`,
# it is replaced by init_state
init_state = states
if analytical:
if kwargs or self.noise:
raise warnings.warn("Analytical matrices exponentiation"
"does not process noise or"
"any keyword arguments.")
return self.run_analytically(init_state=init_state)
# kwargs can not contain H or tlist
if "H" in kwargs or "tlist" in kwargs:
raise ValueError(
"`H` and `tlist` are already specified by the processor "
"and can not be given as a keyword argument")
# construct qobjevo for unitary evolution
if "args" in kwargs:
noisy_qobjevo, sys_c_ops = self.get_qobjevo(args=kwargs["args"],
noisy=noisy)
else:
noisy_qobjevo, sys_c_ops = self.get_qobjevo(noisy=noisy)
# add collpase operators into kwargs
if "c_ops" in kwargs:
if isinstance(kwargs["c_ops"], (Qobj, QobjEvo)):
kwargs["c_ops"] += [kwargs["c_ops"]] + sys_c_ops
else:
kwargs["c_ops"] += sys_c_ops
else:
kwargs["c_ops"] = sys_c_ops
evo_result = mesolve(H=noisy_qobjevo,
rho0=init_state,
tlist=noisy_qobjevo.tlist,
**kwargs)
return evo_result
def propagator(H, t, c_op_list=[], args={}, options=None,
unitary_mode='batch', parallel=False,
progress_bar=None, **kwargs):
"""
Calculate the propagator U(t) for the density matrix or wave function such
that :math:`\psi(t) = U(t)\psi(0)` or
:math:`\\rho_{\mathrm vec}(t) = U(t) \\rho_{\mathrm vec}(0)`
where :math:`\\rho_{\mathrm vec}` is the vector representation of the
density matrix.
Parameters
----------
H : qobj or list
Hamiltonian as a Qobj instance of a nested list of Qobjs and
coefficients in the list-string or list-function format for
time-dependent Hamiltonians (see description in :func:`qutip.mesolve`).
t : float or array-like
Time or list of times for which to evaluate the propagator.
c_op_list : list
List of qobj collapse operators.
args : list/array/dictionary
Parameters to callback functions for time-dependent Hamiltonians and
collapse operators.
options : :class:`qutip.Options`
with options for the ODE solver.
unitary_mode = str ('batch', 'single')
Solve all basis vectors simulaneously ('batch') or individually
('single').
parallel : bool {False, True}
Run the propagator in parallel mode. This will override the
unitary_mode settings if set to True.
progress_bar: BaseProgressBar
Optional instance of BaseProgressBar, or a subclass thereof, for
showing the progress of the simulation. By default no progress bar
is used, and if set to True a TextProgressBar will be used.
Returns
-------
a : qobj
Instance representing the propagator :math:`U(t)`.
"""
kw = _default_kwargs()
if 'num_cpus' in kwargs:
num_cpus = kwargs['num_cpus']
else:
num_cpus = kw['num_cpus']
if progress_bar is None:
progress_bar = BaseProgressBar()
elif progress_bar is True:
progress_bar = TextProgressBar()
if options is None:
options = Options()
options.rhs_reuse = True
rhs_clear()
if isinstance(t, (int, float, np.integer, np.floating)):
tlist = [0, t]
else:
tlist = t
td_type = _td_format_check(H, c_op_list, solver='me')
if isinstance(H, (types.FunctionType, types.BuiltinFunctionType,
functools.partial)):
H0 = H(0.0, args)
elif isinstance(H, list):
H0 = H[0][0] if isinstance(H[0], list) else H[0]
else:
H0 = H
if len(c_op_list) == 0 and H0.isoper:
# calculate propagator for the wave function
N = H0.shape[0]
dims = H0.dims
if parallel:
unitary_mode = 'single'
u = np.zeros([N, N, len(tlist)], dtype=complex)
output = parallel_map(_parallel_sesolve,range(N),
task_args=(N,H, tlist,args,options),
progress_bar=progress_bar, num_cpus=num_cpus)
for n in range(N):
for k, t in enumerate(tlist):
u[:, n, k] = output[n].states[k].full().T
else:
if unitary_mode == 'single':
u = np.zeros([N, N, len(tlist)], dtype=complex)
progress_bar.start(N)
for n in range(0, N):
progress_bar.update(n)
psi0 = basis(N, n)
output = sesolve(H, psi0, tlist, [], args, options, _safe_mode=False)
for k, t in enumerate(tlist):
u[:, n, k] = output.states[k].full().T
progress_bar.finished()
elif unitary_mode =='batch':
u = np.zeros(len(tlist), dtype=object)
_rows = np.array([(N+1)*m for m in range(N)])
_cols = np.zeros_like(_rows)
_data = np.ones_like(_rows,dtype=complex)
psi0 = Qobj(sp.coo_matrix((_data,(_rows,_cols))).tocsr())
if td_type[1] > 0 or td_type[2] > 0:
H2 = []
for k in range(len(H)):
if isinstance(H[k], list):
H2.append([tensor(qeye(N), H[k][0]), H[k][1]])
else:
H2.append(tensor(qeye(N), H[k]))
else:
H2 = tensor(qeye(N), H)
output = sesolve(H2, psi0, tlist, [] , args = args, _safe_mode=False,
options=Options(normalize_output=False))
for k, t in enumerate(tlist):
u[k] = sp_reshape(output.states[k].data, (N, N))
unit_row_norm(u[k].data, u[k].indptr, u[k].shape[0])
u[k] = u[k].T.tocsr()
else:
raise Exception('Invalid unitary mode.')
elif len(c_op_list) == 0 and H0.issuper:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
unitary_mode = 'single'
N = H0.shape[0]
sqrt_N = int(np.sqrt(N))
dims = H0.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
if parallel:
output = parallel_map(_parallel_mesolve,range(N * N),
task_args=(sqrt_N,H,tlist,c_op_list,args,options),
progress_bar=progress_bar, num_cpus=num_cpus)
for n in range(N * N):
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output[n].states[k].full()).T
else:
progress_bar.start(N)
for n in range(0, N):
progress_bar.update(n)
col_idx, row_idx = np.unravel_index(n,(sqrt_N,sqrt_N))
rho0 = Qobj(sp.csr_matrix(([1],([row_idx],[col_idx])), shape=(sqrt_N,sqrt_N), dtype=complex))
output = mesolve(H, rho0, tlist, [], [], args, options, _safe_mode=False)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
progress_bar.finished()
else:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
unitary_mode = 'single'
N = H0.shape[0]
dims = [H0.dims, H0.dims]
u = np.zeros([N * N, N * N, len(tlist)], dtype=complex)
if parallel:
output = parallel_map(_parallel_mesolve,range(N * N),
task_args=(N,H,tlist,c_op_list,args,options),
progress_bar=progress_bar, num_cpus=num_cpus)
for n in range(N * N):
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output[n].states[k].full()).T
else:
progress_bar.start(N * N)
for n in range(N * N):
progress_bar.update(n)
col_idx, row_idx = np.unravel_index(n,(N,N))
rho0 = Qobj(sp.csr_matrix(([1],([row_idx],[col_idx])), shape=(N,N), dtype=complex))
output = mesolve(H, rho0, tlist, c_op_list, [], args, options, _safe_mode=False)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
progress_bar.finished()
if len(tlist) == 2:
if unitary_mode == 'batch':
return Qobj(u[-1], dims=dims)
else:
return Qobj(u[:, :, 1], dims=dims)
else:
if unitary_mode == 'batch':
return np.array([Qobj(u[k], dims=dims) for k in range(len(tlist))], dtype=object)
else:
return np.array([Qobj(u[:, :, k], dims=dims) for k in range(len(tlist))], dtype=object)
def coherence_function_g2(H, state0, taulist, c_ops, a_op, solver="me", args={},
options=Options(ntraj=[20, 100])):
"""
Calculate the normalized second-order quantum coherence function:
.. math::
g^{(2)}(\\tau) =
\\frac{\\langle A^\\dagger(0)A^\\dagger(\\tau)A(\\tau)A(0)\\rangle}
{\\langle A^\\dagger(\\tau)A(\\tau)\\rangle
\\langle A^\\dagger(0)A(0)\\rangle}
using the quantum regression theorem and the evolution solver indicated by
the `solver` parameter.
Parameters
----------
H : Qobj
system Hamiltonian, may be time-dependent for solver choice of `me` or
`mc`.
state0 : Qobj
Initial state density matrix :math:`\\rho(t_0)` or state vector
:math:`\\psi(t_0)`. If 'state0' is 'None', then the steady state will
be used as the initial state. The 'steady-state' is only implemented
for the `me` and `es` solvers.
taulist : array_like
list of times for :math:`\\tau`. taulist must be positive and contain
the element `0`.
c_ops : list
list of collapse operators, may be time-dependent for solver choice of
`me` or `mc`.
a_op : Qobj
operator A.
solver : str
choice of solver (`me` for master-equation and
`es` for exponential series).
options : Options
solver options class. `ntraj` is taken as a two-element list because
the `mc` correlator calls `mcsolve()` recursively; by default,
`ntraj=[20, 100]`. `mc_corr_eps` prevents divide-by-zero errors in
the `mc` correlator; by default, `mc_corr_eps=1e-10`.
Returns
-------
g2, G2 : tuple
The normalized and unnormalized second-order coherence function.
"""
# first calculate the photon number
if state0 is None:
state0 = steadystate(H, c_ops)
n = np.array([expect(state0, a_op.dag() * a_op)])
else:
n = mesolve(H, state0, taulist, c_ops, [a_op.dag() * a_op]).expect[0]
# calculate the correlation function G2 and normalize with n to obtain g2
G2 = correlation_3op_1t(H, state0, taulist, c_ops,
a_op.dag(), a_op.dag()*a_op, a_op,
solver=solver, args=args, options=options)
g2 = G2 / (n[0] * n)
return g2, G2
def propagator(H, t, c_op_list=[], args={}, options=None,
parallel=False, progress_bar=None, **kwargs):
"""
Calculate the propagator U(t) for the density matrix or wave function such
that :math:`\psi(t) = U(t)\psi(0)` or
:math:`\\rho_{\mathrm vec}(t) = U(t) \\rho_{\mathrm vec}(0)`
where :math:`\\rho_{\mathrm vec}` is the vector representation of the
density matrix.
Parameters
----------
H : qobj or list
Hamiltonian as a Qobj instance of a nested list of Qobjs and
coefficients in the list-string or list-function format for
time-dependent Hamiltonians (see description in :func:`qutip.mesolve`).
t : float or array-like
Time or list of times for which to evaluate the propagator.
c_op_list : list
List of qobj collapse operators.
args : list/array/dictionary
Parameters to callback functions for time-dependent Hamiltonians and
collapse operators.
options : :class:`qutip.Options`
with options for the ODE solver.
parallel : bool {False, True}
Run the propagator in parallel mode.
progress_bar: BaseProgressBar
Optional instance of BaseProgressBar, or a subclass thereof, for
showing the progress of the simulation. By default no progress bar
is used, and if set to True a TextProgressBar will be used.
Returns
-------
a : qobj
Instance representing the propagator :math:`U(t)`.
"""
kw = _default_kwargs()
if 'num_cpus' in kwargs:
num_cpus = kwargs['num_cpus']
else:
num_cpus = kw['num_cpus']
if progress_bar is None:
progress_bar = BaseProgressBar()
elif progress_bar is True:
progress_bar = TextProgressBar()
if options is None:
options = Options()
options.rhs_reuse = True
rhs_clear()
if isinstance(t, (int, float, np.integer, np.floating)):
tlist = [0, t]
else:
tlist = t
td_type = _td_format_check(H, c_op_list, solver='me')[2]
if td_type > 0:
rhs_generate(H, c_op_list, args=args, options=options)
if isinstance(H, (types.FunctionType, types.BuiltinFunctionType,
functools.partial)):
H0 = H(0.0, args)
elif isinstance(H, list):
H0 = H[0][0] if isinstance(H[0], list) else H[0]
else:
H0 = H
if len(c_op_list) == 0 and H0.isoper:
# calculate propagator for the wave function
N = H0.shape[0]
dims = H0.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
if parallel:
output = parallel_map(_parallel_sesolve,range(N),
task_args=(N,H,tlist,args,options),
progress_bar=progress_bar, num_cpus=num_cpus)
for n in range(N):
for k, t in enumerate(tlist):
u[:, n, k] = output[n].states[k].full().T
else:
progress_bar.start(N)
for n in range(0, N):
progress_bar.update(n)
psi0 = basis(N, n)
output = sesolve(H, psi0, tlist, [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = output.states[k].full().T
progress_bar.finished()
# todo: evolving a batch of wave functions:
# psi_0_list = [basis(N, n) for n in range(N)]
# psi_t_list = mesolve(H, psi_0_list, [0, t], [], [], args, options)
# for n in range(0, N):
# u[:,n] = psi_t_list[n][1].full().T
elif len(c_op_list) == 0 and H0.issuper:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
N = H0.shape[0]
sqrt_N = int(np.sqrt(N))
dims = H0.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
if parallel:
output = parallel_map(_parallel_mesolve,range(N * N),
task_args=(sqrt_N,H,tlist,c_op_list,args,options),
progress_bar=progress_bar, num_cpus=num_cpus)
for n in range(N * N):
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output[n].states[k].full()).T
else:
progress_bar.start(N)
for n in range(0, N):
progress_bar.update(n)
col_idx, row_idx = np.unravel_index(n,(sqrt_N,sqrt_N))
rho0 = Qobj(sp.csr_matrix(([1],([row_idx],[col_idx])), shape=(sqrt_N,sqrt_N), dtype=complex))
output = mesolve(H, rho0, tlist, [], [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
progress_bar.finished()
else:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
N = H0.shape[0]
dims = [H0.dims, H0.dims]
u = np.zeros([N * N, N * N, len(tlist)], dtype=complex)
if parallel:
output = parallel_map(_parallel_mesolve,range(N * N),
task_args=(N,H,tlist,c_op_list,args,options),
progress_bar=progress_bar, num_cpus=num_cpus)
for n in range(N * N):
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output[n].states[k].full()).T
else:
progress_bar.start(N * N)
for n in range(N * N):
progress_bar.update(n)
col_idx, row_idx = np.unravel_index(n,(N,N))
rho0 = Qobj(sp.csr_matrix(([1],([row_idx],[col_idx])), shape=(N,N), dtype=complex))
output = mesolve(H, rho0, tlist, c_op_list, [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
progress_bar.finished()
if len(tlist) == 2:
return Qobj(u[:, :, 1], dims=dims)
else:
return np.array([Qobj(u[:, :, k], dims=dims) for k in range(len(tlist))], dtype=object)
def propagator(H,
t,
c_op_list,
args=None,
options=None,
sparse=False,
progress_bar=None):
"""
Calculate the propagator U(t) for the density matrix or wave function such
that :math:`\psi(t) = U(t)\psi(0)` or
:math:`\\rho_{\mathrm vec}(t) = U(t) \\rho_{\mathrm vec}(0)`
where :math:`\\rho_{\mathrm vec}` is the vector representation of the
density matrix.
Parameters
----------
H : qobj or list
Hamiltonian as a Qobj instance of a nested list of Qobjs and
coefficients in the list-string or list-function format for
time-dependent Hamiltonians (see description in :func:`qutip.mesolve`).
t : float or array-like
Time or list of times for which to evaluate the propagator.
c_op_list : list
List of qobj collapse operators.
args : list/array/dictionary
Parameters to callback functions for time-dependent Hamiltonians and
collapse operators.
options : :class:`qutip.Options`
with options for the ODE solver.
progress_bar: BaseProgressBar
Optional instance of BaseProgressBar, or a subclass thereof, for
showing the progress of the simulation. By default no progress bar
is used, and if set to True a TextProgressBar will be used.
Returns
-------
a : qobj
Instance representing the propagator :math:`U(t)`.
"""
if progress_bar is None:
progress_bar = BaseProgressBar()
elif progress_bar is True:
progress_bar = TextProgressBar()
if options is None:
options = Options()
options.rhs_reuse = True
rhs_clear()
if isinstance(t, (int, float, np.integer, np.floating)):
tlist = [0, t]
else:
tlist = t
if isinstance(
H,
(types.FunctionType, types.BuiltinFunctionType, functools.partial)):
H0 = H(0.0, args)
elif isinstance(H, list):
H0 = H[0][0] if isinstance(H[0], list) else H[0]
else:
H0 = H
if len(c_op_list) == 0 and H0.isoper:
# calculate propagator for the wave function
N = H0.shape[0]
dims = H0.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
progress_bar.start(N)
for n in range(0, N):
progress_bar.update(n)
psi0 = basis(N, n)
output = sesolve(H, psi0, tlist, [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = output.states[k].full().T
progress_bar.finished()
# todo: evolving a batch of wave functions:
# psi_0_list = [basis(N, n) for n in range(N)]
# psi_t_list = mesolve(H, psi_0_list, [0, t], [], [], args, options)
# for n in range(0, N):
# u[:,n] = psi_t_list[n][1].full().T
elif len(c_op_list) == 0 and H0.issuper:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
N = H0.shape[0]
dims = H0.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
progress_bar.start(N)
for n in range(0, N):
progress_bar.update(n)
psi0 = basis(N, n)
rho0 = Qobj(vec2mat(psi0.full()))
output = mesolve(H, rho0, tlist, [], [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
progress_bar.finished()
else:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
N = H0.shape[0]
dims = [H0.dims, H0.dims]
u = np.zeros([N * N, N * N, len(tlist)], dtype=complex)
if sparse:
progress_bar.start(N * N)
for n in range(N * N):
progress_bar.update(n)
psi0 = basis(N * N, n)
psi0.dims = [dims[0], 1]
rho0 = vector_to_operator(psi0)
output = mesolve(H, rho0, tlist, c_op_list, [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = operator_to_vector(
output.states[k]).full(squeeze=True)
progress_bar.finished()
else:
progress_bar.start(N * N)
for n in range(N * N):
progress_bar.update(n)
psi0 = basis(N * N, n)
rho0 = Qobj(vec2mat(psi0.full()))
output = mesolve(H, rho0, tlist, c_op_list, [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
progress_bar.finished()
if len(tlist) == 2:
return Qobj(u[:, :, 1], dims=dims)
else:
return [Qobj(u[:, :, k], dims=dims) for k in range(len(tlist))]
def propagator(H, t, c_op_list, args=None, options=None, sparse=False):
"""
Calculate the propagator U(t) for the density matrix or wave function such
that :math:`\psi(t) = U(t)\psi(0)` or
:math:`\\rho_{\mathrm vec}(t) = U(t) \\rho_{\mathrm vec}(0)`
where :math:`\\rho_{\mathrm vec}` is the vector representation of the
density matrix.
Parameters
----------
H : qobj or list
Hamiltonian as a Qobj instance of a nested list of Qobjs and
coefficients in the list-string or list-function format for
time-dependent Hamiltonians (see description in :func:`qutip.mesolve`).
t : float or array-like
Time or list of times for which to evaluate the propagator.
c_op_list : list
List of qobj collapse operators.
args : list/array/dictionary
Parameters to callback functions for time-dependent Hamiltonians and
collapse operators.
options : :class:`qutip.Options`
with options for the ODE solver.
Returns
-------
a : qobj
Instance representing the propagator :math:`U(t)`.
"""
if options is None:
options = Options()
options.rhs_reuse = True
rhs_clear()
if isinstance(t, (int, float, np.integer, np.floating)):
tlist = [0, t]
else:
tlist = t
if isinstance(H, (types.FunctionType, types.BuiltinFunctionType,
functools.partial)):
H0 = H(0.0, args)
elif isinstance(H, list):
H0 = H[0][0] if isinstance(H[0], list) else H[0]
else:
H0 = H
if len(c_op_list) == 0 and H0.isoper:
# calculate propagator for the wave function
N = H0.shape[0]
dims = H0.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
for n in range(0, N):
psi0 = basis(N, n)
output = sesolve(H, psi0, tlist, [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = output.states[k].full().T
# todo: evolving a batch of wave functions:
# psi_0_list = [basis(N, n) for n in range(N)]
# psi_t_list = mesolve(H, psi_0_list, [0, t], [], [], args, options)
# for n in range(0, N):
# u[:,n] = psi_t_list[n][1].full().T
elif len(c_op_list) == 0 and H0.issuper:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
N = H0.shape[0]
dims = H0.dims
u = np.zeros([N, N, len(tlist)], dtype=complex)
for n in range(0, N):
psi0 = basis(N, n)
rho0 = Qobj(vec2mat(psi0.full()))
output = mesolve(H, rho0, tlist, [], [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
else:
# calculate the propagator for the vector representation of the
# density matrix (a superoperator propagator)
N = H0.shape[0]
dims = [H0.dims, H0.dims]
u = np.zeros([N * N, N * N, len(tlist)], dtype=complex)
if sparse:
for n in range(N * N):
psi0 = basis(N * N, n)
psi0.dims = [dims[0], 1]
rho0 = vector_to_operator(psi0)
output = mesolve(H, rho0, tlist, c_op_list, [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = operator_to_vector(
output.states[k]).full(squeeze=True)
else:
for n in range(N * N):
psi0 = basis(N * N, n)
rho0 = Qobj(vec2mat(psi0.full()))
output = mesolve(H, rho0, tlist, c_op_list, [], args, options)
for k, t in enumerate(tlist):
u[:, n, k] = mat2vec(output.states[k].full()).T
if len(tlist) == 2:
return Qobj(u[:, :, 1], dims=dims)
else:
return [Qobj(u[:, :, k], dims=dims) for k in range(len(tlist))]
def coherence_function_g2(H,
rho0,
taulist,
c_ops,
a_op,
solver="me",
args=None,
options=Odeoptions()):
"""
Calculate the second-order quantum coherence function:
.. math::
g^{(2)}(\\tau) =
\\frac{\\langle a^\\dagger(0)a^\\dagger(\\tau)a(\\tau)a(0)\\rangle}
{\\langle a^\\dagger(\\tau)a(\\tau)\\rangle
\\langle a^\\dagger(0)a(0)\\rangle}
Parameters
----------
H : :class:`qutip.qobj.Qobj`
system Hamiltonian.
rho0 : :class:`qutip.qobj.Qobj`
Initial state density matrix (or state vector). If 'rho0' is
'None', then the steady state will be used as initial state.
taulist : *list* / *array*
list of times for :math:`\\tau`.
c_ops : list of :class:`qutip.qobj.Qobj`
list of collapse operators.
a_op : :class:`qutip.qobj.Qobj`
The annihilation operator of the mode.
solver : str
choice of solver (currently only 'me')
Returns
-------
g2, G2: tuble of *array*
The normalized and unnormalized second-order coherence function.
"""
# first calculate the photon number
if rho0 is None:
rho0 = steadystate(H, c_ops)
n = np.array([expect(rho0, a_op.dag() * a_op)])
else:
n = mesolve(H,
rho0,
taulist,
c_ops, [a_op.dag() * a_op],
args=args,
options=options).expect[0]
# calculate the correlation function G2 and normalize with n to obtain g2
G2 = correlation_4op_1t(H,
rho0,
taulist,
c_ops,
a_op.dag(),
a_op.dag(),
a_op,
a_op,
solver=solver,
args=args,
options=options)
g2 = G2 / (n[0] * n)
return g2, G2