def floquet_master_equation_tensor(Alist, f_energies):
"""
Construct a tensor that represents the master equation in the floquet
basis (with constant Hamiltonian and collapse operators).
Simplest RWA approximation [Grifoni et al, Phys.Rep. 304 229 (1998)]
Parameters
----------
Alist : list
A list of Floquet-Markov master equation rate matrices.
f_energies : array
The Floquet energies.
Returns
-------
output : array
The Floquet-Markov master equation tensor `R`.
"""
if isinstance(Alist, list):
# Alist can be a list of rate matrices corresponding
# to different operators that couple to the environment
N, M = np.shape(Alist[0])
else:
# or a simple rate matrix, in which case we put it in a list
Alist = [Alist]
N, M = np.shape(Alist[0])
R = Qobj(scipy.sparse.csr_matrix((N * N, N * N)), [[N, N], [N, N]],
[N * N, N * N])
R.data = R.data.tolil()
for I in range(N * N):
a, b = vec2mat_index(N, I)
for J in range(N * N):
c, d = vec2mat_index(N, J)
R.data[I, J] = -1.0j * (f_energies[a] - f_energies[b]) * \
(a == c) * (b == d)
for A in Alist:
s1 = s2 = 0
for n in range(N):
s1 += A[a, n] * (n == c) * (n == d) - A[n, a] * \
(a == c) * (a == d)
s2 += (A[n, a] + A[n, b]) * (a == c) * (b == d)
dR = (a == b) * s1 - 0.5 * (1 - (a == b)) * s2
if dR != 0.0:
R.data[I, J] += dR
R.data = R.data.tocsr()
return R
def floquet_master_equation_tensor(Alist, f_energies):
"""
Construct a tensor that represents the master equation in the floquet
basis (with constant Hamiltonian and collapse operators).
Simplest RWA approximation [Grifoni et al, Phys.Rep. 304 229 (1998)]
Parameters
----------
Alist : list
A list of Floquet-Markov master equation rate matrices.
f_energies : array
The Floquet energies.
Returns
-------
output : array
The Floquet-Markov master equation tensor `R`.
"""
if isinstance(Alist, list):
# Alist can be a list of rate matrices corresponding
# to different operators that couple to the environment
N, M = np.shape(Alist[0])
else:
# or a simple rate matrix, in which case we put it in a list
Alist = [Alist]
N, M = np.shape(Alist[0])
R = Qobj(scipy.sparse.csr_matrix((N * N, N * N)), [[N, N], [N, N]],
[N * N, N * N])
R.data = R.data.tolil()
for I in range(N * N):
a, b = vec2mat_index(N, I)
for J in range(N * N):
c, d = vec2mat_index(N, J)
R.data[I, J] = -1.0j * (f_energies[a] - f_energies[b]) * \
(a == c) * (b == d)
for A in Alist:
s1 = s2 = 0
for n in range(N):
s1 += A[a, n] * (n == c) * (n == d) - A[n, a] * \
(a == c) * (a == d)
s2 += (A[n, a] + A[n, b]) * (a == c) * (b == d)
dR = (a == b) * s1 - 0.5 * (1 - (a == b)) * s2
if dR != 0.0:
R.data[I, J] += dR
R.data = R.data.tocsr()
return R
def bloch_redfield_tensor(H, a_ops, spectra_cb, c_ops=[], use_secular=True):
"""
Calculate the Bloch-Redfield tensor for a system given a set of operators
and corresponding spectral functions that describes the system's coupling
to its environment.
.. note::
This tensor generation requires a time-independent Hamiltonian.
Parameters
----------
H : :class:`qutip.qobj`
System Hamiltonian.
a_ops : list of :class:`qutip.qobj`
List of system operators that couple to the environment.
spectra_cb : list of callback functions
List of callback functions that evaluate the noise power spectrum
at a given frequency.
c_ops : list of :class:`qutip.qobj`
List of system collapse operators.
use_secular : bool
Flag (True of False) that indicates if the secular approximation should
be used.
Returns
-------
R, kets: :class:`qutip.Qobj`, list of :class:`qutip.Qobj`
R is the Bloch-Redfield tensor and kets is a list eigenstates of the
Hamiltonian.
"""
# Sanity checks for input parameters
if not isinstance(H, Qobj):
raise TypeError("H must be an instance of Qobj")
for a in a_ops:
if not isinstance(a, Qobj) or not a.isherm:
raise TypeError("Operators in a_ops must be Hermitian Qobj.")
# default spectrum
if not spectra_cb:
spectra_cb = [lambda w: 1.0 for _ in a_ops]
if c_ops is None:
c_ops = []
# use the eigenbasis
evals, ekets = H.eigenstates()
N = len(evals)
K = len(a_ops)
#only Lindblad collapse terms
if K == 0:
Heb = H.transform(ekets)
L = liouvillian(Heb, c_ops=[c_op.transform(ekets) for c_op in c_ops])
return L, ekets
A = np.array([a_ops[k].transform(ekets).full() for k in range(K)])
Jw = np.zeros((K, N, N), dtype=complex)
# pre-calculate matrix elements and spectral densities
# W[m,n] = real(evals[m] - evals[n])
W = np.real(evals[:, np.newaxis] - evals[np.newaxis, :])
for k in range(K):
# do explicit loops here in case spectra_cb[k] can not deal with array arguments
for n in range(N):
for m in range(N):
Jw[k, n, m] = spectra_cb[k](W[n, m])
dw_min = np.abs(W[W.nonzero()]).min()
# pre-calculate mapping between global index I and system indices a,b
Iabs = np.empty((N * N, 3), dtype=int)
for I, Iab in enumerate(Iabs):
# important: use [:] to change array values, instead of creating new variable Iab
Iab[0] = I
Iab[1:] = vec2mat_index(N, I)
# unitary part + dissipation from c_ops (if given):
Heb = H.transform(ekets)
L = liouvillian(Heb, c_ops=[c_op.transform(ekets) for c_op in c_ops])
# dissipative part:
rows = []
cols = []
data = []
for I, a, b in Iabs:
# only check use_secular once per I
if use_secular:
# only loop over those indices J which actually contribute
Jcds = Iabs[np.where(
np.abs(W[a, b] - W[Iabs[:, 1], Iabs[:, 2]]) < dw_min / 10.0)]
else:
Jcds = Iabs
for J, c, d in Jcds:
elem = 0 + 0j
# summed over k, i.e., each operator coupling the system to the environment
elem += 0.5 * (A[:, a, c] * A[:, d, b] *
(Jw[:, c, a] + Jw[:, d, b]))[0]
if b == d:
# sum_{k,n} A[k, a, n] * A[k, n, c] * Jw[k, c, n])
elem -= 0.5 * np.sum(A[:, a, :] * A[:, :, c] * Jw[:, c, :])
if a == c:
# sum_{k,n} A[k, d, n] * A[k, n, b] * Jw[k, d, n])
elem -= 0.5 * np.sum(A[:, d, :] * A[:, :, b] * Jw[:, d, :])
if elem != 0:
rows.append(I)
cols.append(J)
data.append(elem)
R = sp.coo_matrix((np.array(data), (np.array(rows), np.array(cols))),
shape=(N**2, N**2),
dtype=complex).tocsr()
L.data = L.data + R
return L, ekets
def bloch_redfield_tensor(H, a_ops, spectra_cb=None, c_ops=[], use_secular=True, sec_cutoff=0.1):
"""
Calculate the Bloch-Redfield tensor for a system given a set of operators
and corresponding spectral functions that describes the system's coupling
to its environment.
.. note::
This tensor generation requires a time-independent Hamiltonian.
Parameters
----------
H : :class:`qutip.qobj`
System Hamiltonian.
a_ops : list of :class:`qutip.qobj`
List of system operators that couple to the environment.
spectra_cb : list of callback functions
List of callback functions that evaluate the noise power spectrum
at a given frequency.
c_ops : list of :class:`qutip.qobj`
List of system collapse operators.
use_secular : bool
Flag (True of False) that indicates if the secular approximation should
be used.
sec_cutoff : float {0.1}
Threshold for secular approximation.
Returns
-------
R, kets: :class:`qutip.Qobj`, list of :class:`qutip.Qobj`
R is the Bloch-Redfield tensor and kets is a list eigenstates of the
Hamiltonian.
"""
if not (spectra_cb is None):
warnings.warn("The use of spectra_cb is depreciated.", DeprecationWarning)
_a_ops = []
for kk, a in enumerate(a_ops):
_a_ops.append([a,spectra_cb[kk]])
a_ops = _a_ops
# Sanity checks for input parameters
if not isinstance(H, Qobj):
raise TypeError("H must be an instance of Qobj")
for a in a_ops:
if not isinstance(a[0], Qobj) or not a[0].isherm:
raise TypeError("Operators in a_ops must be Hermitian Qobj.")
if c_ops is None:
c_ops = []
# use the eigenbasis
evals, ekets = H.eigenstates()
N = len(evals)
K = len(a_ops)
#only Lindblad collapse terms
if K==0:
Heb = qdiags(evals,0,dims=H.dims)
L = liouvillian(Heb, c_ops=[c_op.transform(ekets) for c_op in c_ops])
return L, ekets
A = np.array([a_ops[k][0].transform(ekets).full() for k in range(K)])
Jw = np.zeros((K, N, N), dtype=complex)
# pre-calculate matrix elements and spectral densities
# W[m,n] = real(evals[m] - evals[n])
W = np.real(evals[:,np.newaxis] - evals[np.newaxis,:])
for k in range(K):
# do explicit loops here in case spectra_cb[k] can not deal with array arguments
for n in range(N):
for m in range(N):
Jw[k, n, m] = a_ops[k][1](W[n, m])
dw_min = np.abs(W[W.nonzero()]).min()
# pre-calculate mapping between global index I and system indices a,b
Iabs = np.empty((N*N,3),dtype=int)
for I, Iab in enumerate(Iabs):
# important: use [:] to change array values, instead of creating new variable Iab
Iab[0] = I
Iab[1:] = vec2mat_index(N, I)
# unitary part + dissipation from c_ops (if given):
Heb = qdiags(evals,0,dims=H.dims)
L = liouvillian(Heb, c_ops=[c_op.transform(ekets) for c_op in c_ops])
# dissipative part:
rows = []
cols = []
data = []
for I, a, b in Iabs:
# only check use_secular once per I
if use_secular:
# only loop over those indices J which actually contribute
Jcds = Iabs[np.where(np.abs(W[a, b] - W[Iabs[:,1], Iabs[:,2]]) < dw_min * sec_cutoff)]
else:
Jcds = Iabs
for J, c, d in Jcds:
elem = 0+0j
# summed over k, i.e., each operator coupling the system to the environment
elem += 0.5 * np.sum(A[:, a, c] * A[:, d, b] * (Jw[:, c, a] + Jw[:, d, b]))
if b==d:
# sum_{k,n} A[k, a, n] * A[k, n, c] * Jw[k, c, n])
elem -= 0.5 * np.sum(A[:, a, :] * A[:, :, c] * Jw[:, c, :])
if a==c:
# sum_{k,n} A[k, d, n] * A[k, n, b] * Jw[k, d, n])
elem -= 0.5 * np.sum(A[:, d, :] * A[:, :, b] * Jw[:, d, :])
if elem != 0:
rows.append(I)
cols.append(J)
data.append(elem)
R = arr_coo2fast(np.array(data, dtype=complex),
np.array(rows, dtype=np.int32),
np.array(cols, dtype=np.int32), N**2, N**2)
L.data = L.data + R
return L, ekets
def bloch_redfield_tensor(H, a_ops, spectra_cb, c_ops=[], use_secular=True):
"""
Calculate the Bloch-Redfield tensor for a system given a set of operators
and corresponding spectral functions that describes the system's coupling
to its environment.
.. note::
This tensor generation requires a time-independent Hamiltonian.
Parameters
----------
H : :class:`qutip.qobj`
System Hamiltonian.
a_ops : list of :class:`qutip.qobj`
List of system operators that couple to the environment.
spectra_cb : list of callback functions
List of callback functions that evaluate the noise power spectrum
at a given frequency.
c_ops : list of :class:`qutip.qobj`
List of system collapse operators.
use_secular : bool
Flag (True of False) that indicates if the secular approximation should
be used.
Returns
-------
R, kets: :class:`qutip.Qobj`, list of :class:`qutip.Qobj`
R is the Bloch-Redfield tensor and kets is a list eigenstates of the
Hamiltonian.
"""
# Sanity checks for input parameters
if not isinstance(H, Qobj):
raise TypeError("H must be an instance of Qobj")
for a in a_ops:
if not isinstance(a, Qobj) or not a.isherm:
raise TypeError("Operators in a_ops must be Hermitian Qobj.")
# default spectrum
if not spectra_cb:
spectra_cb = [lambda w: 1.0 for _ in a_ops]
if c_ops is None:
c_ops = []
# use the eigenbasis
evals, ekets = H.eigenstates()
N = len(evals)
K = len(a_ops)
A = np.zeros((K, N, N), dtype=complex) # TODO: use sparse here
Jw = np.zeros((K, N, N), dtype=complex)
# pre-calculate matrix elements and spectral densities
# W[m,n] = real(evals[m] - evals[n])
W = np.real(evals[:,np.newaxis] - evals[np.newaxis,:])
for k in range(K):
# A[k,n,m] = a_ops[k].matrix_element(ekets[n], ekets[m])
A[k, :, :] = a_ops[k].transform(ekets).full()
# do explicit loops here in case spectra_cb[k] can not deal with array arguments
for n in range(N):
for m in range(N):
Jw[k, n, m] = spectra_cb[k](W[n, m])
dw_min = abs(W[W.nonzero()]).min()
# pre-calculate mapping between global index I and system indices a,b
Iabs = np.empty((N*N,3),dtype=int)
for I, Iab in enumerate(Iabs):
# important: use [:] to change array values, instead of creating new variable Iab
Iab[0] = I
Iab[1:] = vec2mat_index(N, I)
# unitary part + dissipation from c_ops (if given):
Heb = H.transform(ekets)
R = liouvillian(Heb, c_ops=[c_op.transform(ekets) for c_op in c_ops])
R.data = R.data.tolil()
# dissipative part:
for I, a, b in Iabs:
# only check use_secular once per I
if use_secular:
# only loop over those indices J which actually contribute
Jcds = Iabs[np.where(abs(W[a, b] - W[Iabs[:,1], Iabs[:,2]]) < dw_min / 10.0)]
else:
Jcds = Iabs
for J, c, d in Jcds:
# summed over k, i.e., each operator coupling the system to the environment
R.data[I, J] += 0.5 * np.sum(A[:, a, c] * A[:, d, b] * (Jw[:, c, a] + Jw[:, d, b]))
if b==d:
# sum_{k,n} A[k, a, n] * A[k, n, c] * Jw[k, c, n])
R.data[I, J] -= 0.5 * np.sum(A[:, a, :] * A[:, :, c] * Jw[:, c, :])
if a==c:
# sum_{k,n} A[k, d, n] * A[k, n, b] * Jw[k, d, n])
R.data[I, J] -= 0.5 * np.sum(A[:, d, :] * A[:, :, b] * Jw[:, d, :])
R.data = R.data.tocsr()
return R, ekets
def bloch_redfield_tensor(H, a_ops, spectra_cb, use_secular=True):
"""
Calculate the Bloch-Redfield tensor for a system given a set of operators
and corresponding spectral functions that describes the system's coupling
to its environment.
Parameters
----------
H : :class:`qutip.qobj`
System Hamiltonian.
a_ops : list of :class:`qutip.qobj`
List of system operators that couple to the environment.
spectra_cb : list of callback functions
List of callback functions that evaluate the noise power spectrum
at a given frequency.
use_secular : bool
Flag (True of False) that indicates if the secular approximation should
be used.
Returns
-------
R, kets: :class:`qutip.qobj`, list of :class:`qutip.qobj`
R is the Bloch-Redfield tensor and kets is a list eigenstates of the
Hamiltonian.
"""
# Sanity checks for input parameters
if not isinstance(H, Qobj):
raise TypeError("H must be an instance of Qobj")
for a in a_ops:
if not isinstance(a, Qobj) or not a.isherm:
raise TypeError("Operators in a_ops must be Hermitian Qobj.")
# default spectrum
if not spectra_cb:
spectra_cb = [lambda w: 1.0 for _ in a_ops]
# use the eigenbasis
evals, ekets = H.eigenstates()
N = len(evals)
K = len(a_ops)
A = np.zeros((K, N, N), dtype=complex) # TODO: use sparse here
W = np.zeros((N, N))
# pre-calculate matrix elements
for n in range(N):
for m in range(N):
W[m, n] = np.real(evals[m] - evals[n])
for k in range(K):
# A[k,n,m] = a_ops[k].matrix_element(ekets[n], ekets[m])
A[k, :, :] = a_ops[k].transform(ekets).full()
dw_min = abs(W[W.nonzero()]).min()
# unitary part
Heb = H.transform(ekets)
R = -1.0j * (spre(Heb) - spost(Heb))
R.data = R.data.tolil()
for I in range(N * N):
a, b = vec2mat_index(N, I)
for J in range(N * N):
c, d = vec2mat_index(N, J)
# unitary part: use spre and spost above, same as this:
# R.data[I,J] = -1j * W[a,b] * (a == c) * (b == d)
if use_secular is False or abs(W[a, b] - W[c, d]) < dw_min / 10.0:
# dissipative part:
for k in range(K):
# for each operator coupling the system to the environment
R.data[I, J] += ((A[k, a, c] * A[k, d, b] / 2) *
(spectra_cb[k](W[c, a]) +
spectra_cb[k](W[d, b])))
s1 = s2 = 0
for n in range(N):
s1 += A[k, a, n] * A[k, n, c] * spectra_cb[k](W[c, n])
s2 += A[k, d, n] * A[k, n, b] * spectra_cb[k](W[d, n])
R.data[I, J] += - (b == d) * s1 / 2 - (a == c) * s2 / 2
R.data = R.data.tocsr()
return R, ekets