def _spectrum_es(H, wlist, c_ops, a_op, b_op):
"""
Internal function for calculating the spectrum of the correlation function
:math:`\left<A(\\tau)B(0)\\right>`.
"""
if debug:
print(inspect.stack()[0][3])
# construct the Liouvillian
L = liouvillian(H, c_ops)
# find the steady state density matrix and a_op and b_op expecation values
rho0 = steadystate(L)
a_op_ss = expect(a_op, rho0)
b_op_ss = expect(b_op, rho0)
# eseries solution for (b * rho0)(t)
es = ode2es(L, b_op * rho0)
# correlation
corr_es = expect(a_op, es)
# covariance
cov_es = corr_es - np.real(np.conjugate(a_op_ss) * b_op_ss)
# spectrum
spectrum = esspec(cov_es, wlist)
return spectrum
def add_states(self, state, kind='vector', alpha=1.0):
"""Add a state vector Qobj to Bloch sphere.
Parameters
----------
state : Qobj
Input state vector.
kind : {'vector', 'point'}
Type of object to plot.
alpha : float, default=1.
Transparency value for the vectors. Values between 0 and 1.
"""
if isinstance(state, Qobj):
state = [state]
for st in state:
vec = [expect(sigmax(), st),
expect(sigmay(), st),
expect(sigmaz(), st)]
if kind == 'vector':
self.add_vectors(vec, alpha=alpha)
elif kind == 'point':
self.add_points(vec, alpha=alpha)
def add_states(self, state, kind='vector'):
"""Add a state vector Qobj to Bloch sphere.
Parameters
----------
state : qobj
Input state vector.
kind : str {'vector','point'}
Type of object to plot.
"""
if isinstance(state, Qobj):
state = [state]
for st in state:
vec = [
expect(sigmax(), st),
expect(sigmay(), st),
expect(sigmaz(), st)
]
if kind == 'vector':
self.add_vectors(vec)
elif kind == 'point':
self.add_points(vec)
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 add_annotation(self, state_or_vector, text, **kwargs):
"""Add a text or LaTeX annotation to Bloch sphere,
parametrized by a qubit state or a vector.
Parameters
----------
state_or_vector : Qobj/array/list/tuple
Position for the annotaion.
Qobj of a qubit or a vector of 3 elements.
text : str/unicode
Annotation text.
You can use LaTeX, but remember to use raw string
e.g. r"$\\langle x \\rangle$"
or escape backslashes
e.g. "$\\\\langle x \\\\rangle$".
**kwargs :
Options as for mplot3d.axes3d.text, including:
fontsize, color, horizontalalignment, verticalalignment.
"""
if isinstance(state_or_vector, Qobj):
vec = [expect(sigmax(), state_or_vector),
expect(sigmay(), state_or_vector),
expect(sigmaz(), state_or_vector)]
elif isinstance(state_or_vector, (list, ndarray, tuple)) \
and len(state_or_vector) == 3:
vec = state_or_vector
else:
raise Exception("Position needs to be specified by a qubit " +
"state or a 3D vector.")
self.annotations.append({'position': vec,
'text': text,
'opts': kwargs})
def spectrum_ss(H, wlist, c_op_list, a_op, b_op):
"""
Calculate the spectrum corresponding to a correlation function
:math:`\left<A(\\tau)B(0)\\right>`, i.e., the Fourier transform of the
correlation function:
.. math::
S(\omega) = \int_{-\infty}^{\infty} \left<A(\\tau)B(0)\\right>
e^{-i\omega\\tau} d\\tau.
Parameters
----------
H : :class:`qutip.qobj`
system Hamiltonian.
wlist : *list* / *array*
list of frequencies for :math:`\\omega`.
c_op_list : list of :class:`qutip.qobj`
list of collapse operators.
a_op : :class:`qutip.qobj`
operator A.
b_op : :class:`qutip.qobj`
operator B.
Returns
-------
spectrum: *array*
An *array* with spectrum :math:`S(\omega)` for the frequencies
specified in `wlist`.
"""
# contruct the Liouvillian
L = liouvillian(H, c_op_list)
# find the steady state density matrix and a_op and b_op expecation values
rho0 = steady(L)
a_op_ss = expect(a_op, rho0)
b_op_ss = expect(b_op, rho0)
# eseries solution for (b * rho0)(t)
es = ode2es(L, b_op * rho0)
# correlation
corr_es = expect(a_op, es)
# covarience
cov_es = corr_es - np.real(np.conjugate(a_op_ss) * b_op_ss)
# spectrum
spectrum = esspec(cov_es, wlist)
return spectrum
def _spectrum_es(H, wlist, c_ops, a_op, b_op):
"""
Internal function for calculating the spectrum of the correlation function
:math:`\left<A(\\tau)B(0)\\right>`.
"""
if debug:
print(inspect.stack()[0][3])
# construct the Liouvillian
L = liouvillian(H, c_ops)
# find the steady state density matrix and a_op and b_op expecation values
rho0 = steadystate(L)
a_op_ss = expect(a_op, rho0)
b_op_ss = expect(b_op, rho0)
# eseries solution for (b * rho0)(t)
es = ode2es(L, b_op * rho0)
# correlation
corr_es = expect(a_op, es)
# covariance
cov_es = corr_es - np.real(np.conjugate(a_op_ss) * b_op_ss)
# spectrum
spectrum = esspec(cov_es, wlist)
return spectrum
def add_annotation(self, state_or_vector, text, **kwargs):
"""Add a text or LaTeX annotation to Bloch sphere,
parametrized by a qubit state or a vector.
Parameters
----------
state_or_vector : Qobj/array/list/tuple
Position for the annotaion.
Qobj of a qubit or a vector of 3 elements.
text : str/unicode
Annotation text.
You can use LaTeX, but remember to use raw string
e.g. r"$\\langle x \\rangle$"
or escape backslashes
e.g. "$\\\\langle x \\\\rangle$".
**kwargs :
Options as for mplot3d.axes3d.text, including:
fontsize, color, horizontalalignment, verticalalignment.
"""
if isinstance(state_or_vector, Qobj):
vec = [expect(sigmax(), state_or_vector),
expect(sigmay(), state_or_vector),
expect(sigmaz(), state_or_vector)]
elif isinstance(state_or_vector, (list, ndarray, tuple)) \
and len(state_or_vector) == 3:
vec = state_or_vector
else:
raise Exception("Position needs to be specified by a qubit " +
"state or a 3D vector.")
self.annotations.append({'position': vec,
'text': text,
'opts': kwargs})
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 _spectrum_es(H, wlist, c_ops, a_op, b_op):
"""
Internal function for calculating the spectrum of the correlation function
:math:`\left<A(\\tau)B(0)\\right>`.
"""
if debug:
print(inspect.stack()[0][3])
# construct the Liouvillian
L = liouvillian(H, c_ops)
# find the steady state density matrix and a_op and b_op expecation values
rho0 = steadystate(L)
a_op_ss = expect(a_op, rho0)
b_op_ss = expect(b_op, rho0)
# eseries solution for (b * rho0)(t)
es = ode2es(L, b_op * rho0)
# correlation
corr_es = expect(a_op, es)
# covariance
cov_es = corr_es - a_op_ss * b_op_ss
# tidy up covariance (to combine, e.g., zero-frequency components that cancel)
cov_es.tidyup()
# spectrum
spectrum = esspec(cov_es, wlist)
return spectrum
def spectrum_ss(H, wlist, c_op_list, a_op, b_op):
"""
Calculate the spectrum corresponding to a correlation function
:math:`\left<A(\\tau)B(0)\\right>`, i.e., the Fourier transform of the
correlation function:
.. math::
S(\omega) = \int_{-\infty}^{\infty} \left<A(\\tau)B(0)\\right> e^{-i\omega\\tau} d\\tau.
Parameters
----------
H : :class:`qutip.Qobj`
system Hamiltonian.
wlist : *list* / *array*
list of frequencies for :math:`\\omega`.
c_op_list : list of :class:`qutip.Qobj`
list of collapse operators.
a_op : :class:`qutip.Qobj`
operator A.
b_op : :class:`qutip.Qobj`
operator B.
Returns
-------
spectrum: *array*
An *array* with spectrum :math:`S(\omega)` for the frequencies specified
in `wlist`.
"""
# contruct the Liouvillian
L = liouvillian(H, c_op_list)
# find the steady state density matrix and a_op and b_op expecation values
rho0 = steady(L)
a_op_ss = expect(a_op, rho0)
b_op_ss = expect(b_op, rho0)
# eseries solution for (b * rho0)(t)
es = ode2es(L, b_op * rho0)
# correlation
corr_es = expect(a_op, es)
# covarience
cov_es = corr_es - np.real(np.conjugate(a_op_ss) * b_op_ss)
# spectrum
spectrum = esspec(cov_es, wlist)
return spectrum
def _smepdpsolve_single_trajectory(data, L, dt, times, N_store, N_substeps,
rho_t, dims, c_ops, e_ops):
"""
Internal function. See smepdpsolve.
"""
states_list = []
rho_t = np.copy(rho_t)
sigma_t = np.copy(rho_t)
prng = RandomState() # todo: seed it
r_jump, r_op = prng.rand(2)
jump_times = []
jump_op_idx = []
for t_idx, t in enumerate(times):
if e_ops:
for e_idx, e in enumerate(e_ops):
data.expect[e_idx, t_idx] += expect_rho_vec(e, rho_t)
else:
states_list.append(Qobj(vec2mat(rho_t), dims=dims))
for j in range(N_substeps):
if sigma_t.norm() < r_jump:
# jump occurs
p = np.array([expect(c.dag() * c, rho_t) for c in c_ops])
p = np.cumsum(p / np.sum(p))
n = np.where(p >= r_op)[0][0]
# apply jump
rho_t = c_ops[n] * rho_t * c_ops[n].dag()
rho_t /= expect(c_ops[n].dag() * c_ops[n], rho_t)
sigma_t = np.copy(rho_t)
# store info about jump
jump_times.append(times[t_idx] + dt * j)
jump_op_idx.append(n)
# get new random numbers for next jump
r_jump, r_op = prng.rand(2)
# deterministic evolution wihtout correction for norm decay
dsigma_t = spmv(L.data, sigma_t) * dt
# deterministic evolution with correction for norm decay
drho_t = spmv(L.data, rho_t) * dt
rho_t += drho_t
# increment density matrices
sigma_t += dsigma_t
rho_t += drho_t
return states_list, jump_times, jump_op_idx
def _smepdpsolve_single_trajectory(data, L, dt, times, N_store, N_substeps, rho_t, dims, c_ops, e_ops):
"""
Internal function. See smepdpsolve.
"""
states_list = []
rho_t = np.copy(rho_t)
sigma_t = np.copy(rho_t)
prng = RandomState() # todo: seed it
r_jump, r_op = prng.rand(2)
jump_times = []
jump_op_idx = []
for t_idx, t in enumerate(times):
if e_ops:
for e_idx, e in enumerate(e_ops):
data.expect[e_idx, t_idx] += expect_rho_vec(e, rho_t)
else:
states_list.append(Qobj(vec2mat(rho_t), dims=dims))
for j in range(N_substeps):
if sigma_t.norm() < r_jump:
# jump occurs
p = np.array([expect(c.dag() * c, rho_t) for c in c_ops])
p = np.cumsum(p / np.sum(p))
n = np.where(p >= r_op)[0][0]
# apply jump
rho_t = c_ops[n] * rho_t * c_ops[n].dag()
rho_t /= expect(c_ops[n].dag() * c_ops[n], rho_t)
sigma_t = np.copy(rho_t)
# store info about jump
jump_times.append(times[t_idx] + dt * j)
jump_op_idx.append(n)
# get new random numbers for next jump
r_jump, r_op = prng.rand(2)
# deterministic evolution wihtout correction for norm decay
dsigma_t = spmv(L.data, sigma_t) * dt
# deterministic evolution with correction for norm decay
drho_t = spmv(L.data, rho_t) * dt
rho_t += drho_t
# increment density matrices
sigma_t += dsigma_t
rho_t += drho_t
return states_list, jump_times, jump_op_idx
def covariance_matrix(basis, rho):
"""
The covariance matrix given a basis of operators.
.. note::
Experimental.
"""
return np.array([[expect(op1*op2+op2*op1, rho)-expect(op1, rho)*expect(op2, rho) for op1 in basis] for op2 in basis])
def testOperatorListState(self):
"""
expect: operator list and state
"""
res = expect([sigmax(), sigmay(), sigmaz()], fock(2, 0))
assert_(len(res) == 3)
assert_(all(abs(res - [0, 0, 1]) < 1e-12))
res = expect([sigmax(), sigmay(), sigmaz()], fock_dm(2, 1))
assert_(len(res) == 3)
assert_(all(abs(res - [0, 0, -1]) < 1e-12))
def testOperatorListState(self):
"""
expect: operator list and state
"""
res = expect([sigmax(), sigmay(), sigmaz()], fock(2, 0))
assert_(len(res) == 3)
assert_(all(abs(res - [0, 0, 1]) < 1e-12))
res = expect([sigmax(), sigmay(), sigmaz()], fock_dm(2, 1))
assert_(len(res) == 3)
assert_(all(abs(res - [0, 0, -1]) < 1e-12))
def testOperatorDensityMatrix(self):
"""
expect: operator and density matrix
"""
N = 10
op_N = num(N)
op_a = destroy(N)
for n in range(N):
e = expect(op_N, fock_dm(N, n))
assert_(e == n)
assert_(type(e) == float)
e = expect(op_a, fock_dm(N, n))
assert_(e == 0)
assert_(type(e) == complex)
def testOperatorDensityMatrix(self):
"""
expect: operator and density matrix
"""
N = 10
op_N = num(N)
op_a = destroy(N)
for n in range(N):
e = expect(op_N, fock_dm(N, n))
assert_(e == n)
assert_(type(e) == float)
e = expect(op_a, fock_dm(N, n))
assert_(e == 0)
assert_(type(e) == complex)
def testOperatorKet(self):
"""
expect: operator and ket
"""
N = 10
op_N = num(N)
op_a = destroy(N)
for n in range(N):
e = expect(op_N, fock(N, n))
assert_(e == n)
assert_(type(e) == float)
e = expect(op_a, fock(N, n))
assert_(e == 0)
assert_(type(e) == complex)
def testOperatorKet(self):
"""
expect: operator and ket
"""
N = 10
op_N = num(N)
op_a = destroy(N)
for n in range(N):
e = expect(op_N, fock(N, n))
assert_(e == n)
assert_(type(e) == float)
e = expect(op_a, fock(N, n))
assert_(e == 0)
assert_(type(e) == complex)
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 _ssesolve_single_trajectory(H, dt, tlist, N_store, N_substeps, psi_t,
A_ops, e_ops, data, rhs, d1, d2):
"""
Internal function. See ssesolve.
"""
dW = np.sqrt(dt) * scipy.randn(len(A_ops), N_store, N_substeps)
states_list = []
for t_idx, t in enumerate(tlist):
if e_ops:
for e_idx, e in enumerate(e_ops):
data.expect[e_idx, t_idx] += expect(e, Qobj(psi_t))
else:
states_list.append(Qobj(psi_t))
for j in range(N_substeps):
dpsi_t = (-1.0j * dt) * (H.data * psi_t)
for a_idx, A in enumerate(A_ops):
dpsi_t += rhs(
H.data, psi_t, A, dt, dW[a_idx, t_idx, j], d1, d2)
# increment and renormalize the wave function
psi_t += dpsi_t
psi_t /= norm(psi_t, 2)
return states_list
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_matrix(basis, rho=None):
"""
Given a basis set of operators :math:`\\{a\\}_n`, calculate the correlation
matrix:
.. math::
C_{mn} = \\langle a_m a_n \\rangle
Parameters
----------
basis : list of :class:`qutip.qobj.Qobj`
List of operators that defines the basis for the correlation matrix.
rho : :class:`qutip.qobj.Qobj`
Density matrix for which to calculate the correlation matrix. If
`rho` is `None`, then a matrix of correlation matrix operators is
returned instead of expectation values of those operators.
Returns
-------
corr_mat: *array*
A 2-dimensional *array* of correlation values or operators.
"""
if rho is None:
# return array of operators
return np.array([[op1 * op2 for op1 in basis] for op2 in basis], dtype=object)
else:
# return array of expectation values
return np.array([[expect(op1 * op2, rho) for op1 in basis] for op2 in basis], dtype=object)
def testOperatorKetRand(self):
"""
expect: rand operator & rand ket
"""
for kk in range(20):
N = 20
H = rand_herm(N, 0.2)
psi = rand_ket(N, 0.3)
out = expect(H, psi)
ans = (psi.dag() * H * psi).tr()
assert_(np.abs(out - ans) < 1e-14)
G = rand_herm(N, 0.1)
out = expect(H + 1j * G, psi)
ans = (psi.dag() * (H + 1j * G) * psi).tr()
assert_(np.abs(out - ans) < 1e-14)
def _smesolve_single_trajectory(L, dt, tlist, N_store, N_substeps, rho_t,
A_ops, e_ops, data, rhs, d1, d2):
"""
Internal function. See smesolve.
"""
dW = np.sqrt(dt) * scipy.randn(len(A_ops), N_store, N_substeps)
states_list = []
for t_idx, t in enumerate(tlist):
if e_ops:
for e_idx, e in enumerate(e_ops):
# XXX: need to keep hilbert space structure
data.expect[e_idx, t_idx] += expect(e, Qobj(vec2mat(rho_t)))
else:
states_list.append(Qobj(rho_t)) # dito
for j in range(N_substeps):
drho_t = spmv(
L.data.data, L.data.indices, L.data.indptr, rho_t) * dt
for a_idx, A in enumerate(A_ops):
drho_t += rhs(
L.data, rho_t, A, dt, dW[a_idx, t_idx, j], d1, d2)
rho_t += drho_t
return states_list
def _smesolve_single_trajectory(L, dt, tlist, N_store, N_substeps, rho_t, A_ops, e_ops, data, rhs, d1, d2):
"""
Internal function. See smesolve.
"""
dW = np.sqrt(dt) * scipy.randn(len(A_ops), N_store, N_substeps)
states_list = []
for t_idx, t in enumerate(tlist):
if e_ops:
for e_idx, e in enumerate(e_ops):
# XXX: need to keep hilbert space structure
data.expect[e_idx, t_idx] += expect(e, Qobj(vec2mat(rho_t)))
else:
states_list.append(Qobj(rho_t)) # dito
for j in range(N_substeps):
drho_t = spmv(L.data.data, L.data.indices, L.data.indptr, rho_t) * dt
for a_idx, A in enumerate(A_ops):
drho_t += rhs(L.data, rho_t, A, dt, dW[a_idx, t_idx, j], d1, d2)
rho_t += drho_t
return states_list
def wigner_covariance_matrix(a1=None, a2=None, R=None, rho=None):
"""
calculate the wigner covariance matrix given the quadrature correlation
matrix (R) and a state.
.. note::
Experimental.
"""
if R != None:
if rho is None:
return np.array([[np.real(R[i,j]+R[j,i]) for i in range(4)] for j in range(4)])
else:
return np.array([[np.real(expect(R[i,j]+R[j,i], rho)) for i in range(4)] for j in range(4)])
elif a1 != None and a2 != None:
if rho != None:
x1 = (a1 + a1.dag()) / np.sqrt(2)
p1 = 1j * (a1 - a1.dag()) / np.sqrt(2)
x2 = (a2 + a2.dag()) / np.sqrt(2)
p2 = 1j * (a2 - a2.dag()) / np.sqrt(2)
return covariance_matrix([x1, p1, x2, p2], rho)
else:
raise ValueError("Must give rho if using field operators (a1 and a2)")
else:
raise ValueError("Must give either field operators (a1 and a2) or a precomputed correlation matrix (R)")
def _ssesolve_single_trajectory(H, dt, tlist, N_store, N_substeps, psi_t, A_ops, e_ops, data, rhs, d1, d2):
"""
Internal function. See ssesolve.
"""
dW = np.sqrt(dt) * scipy.randn(len(A_ops), N_store, N_substeps)
states_list = []
for t_idx, t in enumerate(tlist):
if e_ops:
for e_idx, e in enumerate(e_ops):
data.expect[e_idx, t_idx] += expect(e, Qobj(psi_t))
else:
states_list.append(Qobj(psi_t))
for j in range(N_substeps):
dpsi_t = (-1.0j * dt) * (H.data * psi_t)
for a_idx, A in enumerate(A_ops):
dpsi_t += rhs(H.data, psi_t, A, dt, dW[a_idx, t_idx, j], d1, d2)
# increment and renormalize the wave function
psi_t += dpsi_t
psi_t /= norm(psi_t, 2)
return states_list
def _expectation_values(
e_ops,
n_expt_op,
expt_callback,
res,
evolved_states,
partitions,
idx,
t_idx,
options,
):
if options.store_states:
res.states += evolved_states
for t, state in zip(range(t_idx, t_idx + len(partitions[idx][1:-1])),
evolved_states):
if expt_callback:
# use callback method
res.expect.append(e_ops(t, state))
for m in range(n_expt_op):
op = e_ops[m]
if not isinstance(op, Qobj) and callable(op):
res.expect[m][t] = op(t, state)
continue
res.expect[m][t] = expect(op, state)
if (idx == len(partitions) - 1 and options.store_final_state
and not options.store_states):
res.states = [evolved_states[-1]]
return res
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_matrix(basis, rho=None):
r"""
Given a basis set of operators :math:`\{a\}_n`, calculate the correlation
matrix:
.. math::
C_{mn} = \langle a_m a_n \rangle
Parameters
----------
basis : list
List of operators that defines the basis for the correlation matrix.
rho : Qobj
Density matrix for which to calculate the correlation matrix. If
`rho` is `None`, then a matrix of correlation matrix operators is
returned instead of expectation values of those operators.
Returns
-------
corr_mat : ndarray
A 2-dimensional *array* of correlation values or operators.
"""
if rho is None:
# return array of operators
out = np.empty((len(basis), len(basis)), dtype=object)
for i, op2 in enumerate(basis):
out[i, :] = [op1 * op2 for op1 in basis]
return out
else:
# return array of expectation values
return np.array([[expect(op1 * op2, rho) for op1 in basis]
for op2 in basis])
def coherence_function_g1(H, taulist, c_ops, a_op, solver="me", args=None,
options=Options(ntraj=[20, 100])):
"""
Calculate the normalized first-order quantum coherence function:
.. math::
g^{(1)}(\\tau) = \lim_{t \to \infty}
\\frac{\\langle a^\\dagger(t+\\tau)a(t)\\rangle}
{\\langle a^\\dagger(t)a(t)\\rangle}
using the quantum regression theorem and the evolution solver indicated by
the `solver` parameter. Note: g1 is only defined for stationary
statistics (uses steady state).
Parameters
----------
H : :class:`qutip.qobj.Qobj`
system Hamiltonian.
taulist : *list* / *array*
list of times for :math:`\\tau`. taulist must be positive and contain
the element `0`.
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 (`me` for master-equation and
`es` for exponential series)
options : :class:`qutip.solver.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
-------
g1: *array*
The normalized first-order coherence function.
"""
# first calculate the steady state photon number
rho0 = steadystate(H, c_ops)
n = np.array([expect(rho0, a_op.dag() * a_op)])
# calculate the correlation function G1 and normalize with n to obtain g1
G1 = correlation_2op_1t(H, None, taulist, c_ops, a_op.dag(), a_op,
args=args, solver=solver, options=options)
g1 = G1 / n
return g1
def wigner_covariance_matrix(a1=None, a2=None, R=None, rho=None):
"""
Calculate the Wigner covariance matrix
:math:`V_{ij} = \\frac{1}{2}(R_{ij} + R_{ji})`, given
the quadrature correlation matrix
:math:`R_{ij} = \\langle R_{i} R_{j}\\rangle -
\\langle R_{i}\\rangle \\langle R_{j}\\rangle`, where
:math:`R = (q_1, p_1, q_2, p_2)^T` is the vector with quadrature operators
for the two modes.
Alternatively, if `R = None`, and if annihilation operators `a1` and `a2`
for the two modes are supplied instead, the quadrature correlation matrix
is constructed from the annihilation operators before then the covariance
matrix is calculated.
Parameters
----------
a1 : :class:`qutip.qobj.Qobj`
Field operator for mode 1.
a2 : :class:`qutip.qobj.Qobj`
Field operator for mode 2.
R : *array*
The quadrature correlation matrix.
rho : :class:`qutip.qobj.Qobj`
Density matrix for which to calculate the covariance matrix.
Returns
-------
cov_mat: *array*
A 2-dimensional *array* of covariance values.
"""
if R is not None:
if rho is None:
return np.array([[0.5 * np.real(R[i, j] + R[j, i]) for i in range(4)] for j in range(4)], dtype=object)
else:
return np.array(
[[0.5 * np.real(expect(R[i, j] + R[j, i], rho)) for i in range(4)] for j in range(4)], dtype=object
)
elif a1 is not None and a2 is not None:
if rho is not None:
x1 = (a1 + a1.dag()) / np.sqrt(2)
p1 = -1j * (a1 - a1.dag()) / np.sqrt(2)
x2 = (a2 + a2.dag()) / np.sqrt(2)
p2 = -1j * (a2 - a2.dag()) / np.sqrt(2)
return covariance_matrix([x1, p1, x2, p2], rho)
else:
raise ValueError("Must give rho if using field operators " + "(a1 and a2)")
else:
raise ValueError("Must give either field operators (a1 and a2) " + "or a precomputed correlation matrix (R)")
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 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 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 testOperatorListStateList(self):
"""
expect: operator list and state list
"""
operators = [sigmax(), sigmay(), sigmaz(), sigmam(), sigmap()]
states = [fock(2, 0), fock(2, 1), fock_dm(2, 0), fock_dm(2, 1)]
res = expect(operators, states)
assert_(len(res) == len(operators))
for r_idx, r in enumerate(res):
assert_(isinstance(r, np.ndarray))
if operators[r_idx].isherm:
assert_(r.dtype == np.float64)
else:
assert_(r.dtype == np.complex128)
for s_idx, s in enumerate(states):
assert_(r[s_idx] == expect(operators[r_idx], states[s_idx]))
def testOperatorListStateList(self):
"""
expect: operator list and state list
"""
operators = [sigmax(), sigmay(), sigmaz(), sigmam(), sigmap()]
states = [fock(2, 0), fock(2, 1), fock_dm(2, 0), fock_dm(2, 1)]
res = expect(operators, states)
assert_(len(res) == len(operators))
for r_idx, r in enumerate(res):
assert_(isinstance(r, np.ndarray))
if operators[r_idx].isherm:
assert_(r.dtype == np.float64)
else:
assert_(r.dtype == np.complex128)
for s_idx, s in enumerate(states):
assert_(r[s_idx] == expect(operators[r_idx], states[s_idx]))
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 add_states(self, state, kind='vector'):
"""Add a state vector Qobj to Bloch sphere.
Parameters
----------
state : qobj
Input state vector.
kind : str {'vector','point'}
Type of object to plot.
"""
if isinstance(state, Qobj):
state = [state]
for st in state:
vec = [expect(sigmax(), st),
expect(sigmay(), st),
expect(sigmaz(), st)]
if kind == 'vector':
self.add_vectors(vec)
elif kind == 'point':
self.add_points(vec)
def testOperatorStateList(self):
"""
expect: operator and state list
"""
N = 10
op = num(N)
res = expect(op, [fock(N, n) for n in range(N)])
assert_(all(res == range(N)))
assert_(isinstance(res, np.ndarray) and res.dtype == np.float64)
res = expect(op, [fock_dm(N, n) for n in range(N)])
assert_(all(res == range(N)))
assert_(isinstance(res, np.ndarray) and res.dtype == np.float64)
op = destroy(N)
res = expect(op, [fock(N, n) for n in range(N)])
assert_(all(res == np.zeros(N)))
assert_(isinstance(res, np.ndarray) and res.dtype == np.complex128)
res = expect(op, [fock_dm(N, n) for n in range(N)])
assert_(all(res == np.zeros(N)))
assert_(isinstance(res, np.ndarray) and res.dtype == np.complex128)
def _correlation_matrix(basis, rho=None):
"""
The correlation matrix given a basis of operators.
.. note::
Experimental.
"""
if rho is None:
# return array of operators
return np.array([[op1*op2 for op1 in basis] for op2 in basis])
else:
# return array of expectation balues
return np.array([[expect(op1*op2, rho) for op1 in basis] for op2 in basis])
def testOperatorStateList(self):
"""
expect: operator and state list
"""
N = 10
op = num(N)
res = expect(op, [fock(N, n) for n in range(N)])
assert_(all(res == range(N)))
assert_(isinstance(res, np.ndarray) and res.dtype == np.float64)
res = expect(op, [fock_dm(N, n) for n in range(N)])
assert_(all(res == range(N)))
assert_(isinstance(res, np.ndarray) and res.dtype == np.float64)
op = destroy(N)
res = expect(op, [fock(N, n) for n in range(N)])
assert_(all(res == np.zeros(N)))
assert_(isinstance(res, np.ndarray) and res.dtype == np.complex128)
res = expect(op, [fock_dm(N, n) for n in range(N)])
assert_(all(res == np.zeros(N)))
assert_(isinstance(res, np.ndarray) and res.dtype == np.complex128)
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 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 covariance_matrix(basis, rho, symmetrized=True):
"""
Given a basis set of operators :math:`\{a\}_n`, calculate the covariance
matrix:
.. math::
V_{mn} = \\frac{1}{2}\\langle a_m a_n + a_n a_m \\rangle -
\\langle a_m \\rangle \\langle a_n\\rangle
or, if of the optional argument `symmetrized=False`,
.. math::
V_{mn} = \\langle a_m a_n\\rangle -
\\langle a_m \\rangle \\langle a_n\\rangle
Parameters
----------
basis : list of :class:`qutip.qobj.Qobj`
List of operators that defines the basis for the covariance matrix.
rho : :class:`qutip.qobj.Qobj`
Density matrix for which to calculate the covariance matrix.
symmetrized : *bool*
Flag indicating whether the symmetrized (default) or non-symmetrized
correlation matrix is to be calculated.
Returns
-------
corr_mat: *array*
A 2-dimensional *array* of covariance values.
"""
if symmetrized:
return np.array(
[
[0.5 * expect(op1 * op2 + op2 * op1, rho) - expect(op1, rho) * expect(op2, rho) for op1 in basis]
for op2 in basis
],
dtype=object,
)
else:
return np.array(
[[expect(op1 * op2, rho) - expect(op1, rho) * expect(op2, rho) for op1 in basis] for op2 in basis],
dtype=object,
)
def covariance_matrix(basis, rho, symmetrized=True):
"""
Given a basis set of operators :math:`\{a\}_n`, calculate the covariance
matrix:
.. math::
V_{mn} = \\frac{1}{2}\\langle a_m a_n + a_n a_m \\rangle -
\\langle a_m \\rangle \\langle a_n\\rangle
or, if of the optional argument `symmetrized=False`,
.. math::
V_{mn} = \\langle a_m a_n\\rangle -
\\langle a_m \\rangle \\langle a_n\\rangle
Parameters
----------
basis : list of :class:`qutip.qobj.Qobj`
List of operators that defines the basis for the covariance matrix.
rho : :class:`qutip.qobj.Qobj`
Density matrix for which to calculate the covariance matrix.
symmetrized : *bool*
Flag indicating whether the symmetrized (default) or non-symmetrized
correlation matrix is to be calculated.
Returns
-------
corr_mat: *array*
A 2-dimensional *array* of covariance values.
"""
if symmetrized:
return np.array([[
0.5 * expect(op1 * op2 + op2 * op1, rho) -
expect(op1, rho) * expect(op2, rho) for op1 in basis
] for op2 in basis])
else:
return np.array([[
expect(op1 * op2, rho) - expect(op1, rho) * expect(op2, rho)
for op1 in basis
] for op2 in basis])
def add_line(self, start, end, fmt="k", **kwargs):
"""Adds a line segment connecting two points on the bloch sphere.
The line segment is set to be a black solid line by default.
Parameters
----------
start : Qobj or array-like
Array with cartesian coordinates of the first point, or a state
vector or density matrix that can be mapped to a point on or
within the Bloch sphere.
end : Qobj or array-like
Array with cartesian coordinates of the second point, or a state
vector or density matrix that can be mapped to a point on or
within the Bloch sphere.
fmt : str, default: "k"
A matplotlib format string for rendering the line.
**kwargs : dict
Additional parameters to pass to the matplotlib .plot function
when rendering this line.
"""
if isinstance(start, Qobj):
pt1 = [
expect(sigmax(), start),
expect(sigmay(), start),
expect(sigmaz(), start),
]
else:
pt1 = start
if isinstance(end, Qobj):
pt2 = [
expect(sigmax(), end),
expect(sigmay(), end),
expect(sigmaz(), end),
]
else:
pt2 = end
pt1 = np.asarray(pt1)
pt2 = np.asarray(pt2)
x = [pt1[1], pt2[1]]
y = [-pt1[0], -pt2[0]]
z = [pt1[2], pt2[2]]
v = [x, y, z]
self._lines.append([v, fmt, kwargs])
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 _generic_ode_solve(r, psi0, tlist, e_ops, opt, progress_bar,
state_norm_func=None,dims=None):
"""
Internal function for solving ODEs.
"""
#
# prepare output array
#
n_tsteps = len(tlist)
dt = tlist[1] - tlist[0]
output = Odedata()
output.solver = "sesolve"
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:
# fallback on storing states
output.states = []
opt.store_states = True
else:
output.expect = []
output.num_expect = n_expt_op
for op in e_ops:
if op.isherm and psi0.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)
for t_idx, t in enumerate(tlist):
progress_bar.update(t_idx)
if not r.successful():
break
if state_norm_func:
data = r.y / state_norm_func(r.y)
r.set_initial_value(data, r.t)
if opt.store_states:
output.states.append(Qobj(r.y,dims=dims))
if expt_callback:
# use callback method
e_ops(t, Qobj(r.y))
for m in range(n_expt_op):
output.expect[m][t_idx] = expect(e_ops[m], Qobj(r.y)) # optimize
r.integrate(r.t + dt)
progress_bar.finished()
if not opt.rhs_reuse and odeconfig.tdname is not None:
try:
os.remove(odeconfig.tdname + ".pyx")
except:
pass
if opt.store_final_state:
output.final_state = Qobj(r.y)
return output
def floquet_markov_mesolve(R, ekets, rho0, tlist, e_ops, f_modes_table=None,
options=None, floquet_basis=True):
"""
Solve the dynamics for the system using the Floquet-Markov master equation.
"""
if options is None:
opt = Options()
else:
opt = options
if opt.tidy:
R.tidyup()
#
# check initial state
#
if isket(rho0):
# Got a wave function as initial state: convert to density matrix.
rho0 = ket2dm(rho0)
#
# prepare output array
#
n_tsteps = len(tlist)
dt = tlist[1] - tlist[0]
output = Result()
output.solver = "fmmesolve"
output.times = tlist
if isinstance(e_ops, 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:
output.states = []
else:
if not f_modes_table:
raise TypeError("The Floquet mode table has to be provided " +
"when requesting expectation values.")
output.expect = []
output.num_expect = n_expt_op
for op in e_ops:
if op.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")
#
# transform the initial density matrix to the eigenbasis: from
# computational basis to the floquet basis
#
if ekets is not None:
rho0 = rho0.transform(ekets)
#
# setup integrator
#
initial_vector = mat2vec(rho0.full())
r = scipy.integrate.ode(cy_ode_rhs)
r.set_f_params(R.data.data, R.data.indices, R.data.indptr)
r.set_integrator('zvode', method=opt.method, order=opt.order,
atol=opt.atol, rtol=opt.rtol, max_step=opt.max_step)
r.set_initial_value(initial_vector, tlist[0])
#
# start evolution
#
rho = Qobj(rho0)
t_idx = 0
for t in tlist:
if not r.successful():
break
rho.data = vec2mat(r.y)
if expt_callback:
# use callback method
if floquet_basis:
e_ops(t, Qobj(rho))
else:
f_modes_table_t, T = f_modes_table
f_modes_t = floquet_modes_t_lookup(f_modes_table_t, t, T)
e_ops(t, Qobj(rho).transform(f_modes_t, True))
else:
# calculate all the expectation values, or output rho if
# no operators
if n_expt_op == 0:
if floquet_basis:
output.states.append(Qobj(rho))
else:
f_modes_table_t, T = f_modes_table
f_modes_t = floquet_modes_t_lookup(f_modes_table_t, t, T)
output.states.append(Qobj(rho).transform(f_modes_t, True))
else:
f_modes_table_t, T = f_modes_table
f_modes_t = floquet_modes_t_lookup(f_modes_table_t, t, T)
for m in range(0, n_expt_op):
output.expect[m][t_idx] = \
expect(e_ops[m], rho.transform(f_modes_t, False))
r.integrate(r.t + dt)
t_idx += 1
return output
