def basis(N, n=0, offset=0):
"""Generates the vector representation of a Fock state.
Parameters
----------
N : int
Number of Fock states in Hilbert space.
n : int
Integer corresponding to desired number state, defaults
to 0 if omitted.
offset : int (default 0)
The lowest number state that is included in the finite number state
representation of the state.
Returns
-------
state : :class:`qutip.Qobj`
Qobj representing the requested number state ``|n>``.
Examples
--------
>>> basis(5,2)
Quantum object: dims = [[5], [1]], shape = [5, 1], type = ket
Qobj data =
[[ 0.+0.j]
[ 0.+0.j]
[ 1.+0.j]
[ 0.+0.j]
[ 0.+0.j]]
Notes
-----
A subtle incompatibility with the quantum optics toolbox: In QuTiP::
basis(N, 0) = ground state
but in the qotoolbox::
basis(N, 1) = ground state
"""
if (not isinstance(N, (int, np.integer))) or N < 0:
raise ValueError("N must be integer N >= 0")
if (not isinstance(n, (int, np.integer))) or n < offset:
raise ValueError("n must be integer n >= 0")
if n - offset > (N - 1): # check if n is within bounds
raise ValueError("basis vector index need to be in n <= N-1")
data = np.array([1], dtype=complex)
ind = np.array([0], dtype=np.int32)
ptr = np.array([0]*((n - offset)+1)+[1]*(N-(n-offset)),dtype=np.int32)
return Qobj(fast_csr_matrix((data,ind,ptr), shape=(N,1)), isherm=False)
def _unsorted_csr(N, density=0.5):
N = np.arange(N)
M = sp.diags(N,0, dtype=complex, format='csr')
N = N.shape[0]
nvals = N**2*density
while M.nnz < 0.95*nvals:
M = rand_jacobi_rotation(M)
M = M.tocsr()
return fast_csr_matrix((M.data,M.indices,M.indptr),
shape = M.shape)
def sp_reverse_permute(A, rperm=(), cperm=(), safe=True):
"""
Performs a reverse permutations of the rows and columns of a sparse CSR/CSC
matrix according to the permutation arrays rperm and cperm, respectively.
Here, the permutation arrays specify the order of the rows and columns used
to permute the original array.
Parameters
----------
A : csr_matrix, csc_matrix
Input matrix.
rperm : array_like of integers
Array of row permutations.
cperm : array_like of integers
Array of column permutations.
safe : bool
Check structure of permutation arrays.
Returns
-------
perm_csr : csr_matrix, csc_matrix
CSR or CSC matrix with permuted rows/columns.
"""
rperm = np.asarray(rperm, dtype=np.int32)
cperm = np.asarray(cperm, dtype=np.int32)
nrows = A.shape[0]
ncols = A.shape[1]
if len(rperm) == 0:
rperm = np.arange(nrows, dtype=np.int32)
if len(cperm) == 0:
cperm = np.arange(ncols, dtype=np.int32)
if safe:
if len(np.setdiff1d(rperm, np.arange(nrows))) != 0:
raise Exception('Invalid row permutation array.')
if len(np.setdiff1d(cperm, np.arange(ncols))) != 0:
raise Exception('Invalid column permutation array.')
shp = A.shape
kind = A.getformat()
if kind == 'csr':
flag = 0
elif kind == 'csc':
flag = 1
else:
raise Exception('Input must be Qobj, CSR, or CSC matrix.')
data, ind, ptr = _sparse_reverse_permute(A.data, A.indices, A.indptr,
nrows, ncols, rperm, cperm, flag)
if kind == 'csr':
return fast_csr_matrix((data, ind, ptr), shape=shp)
elif kind == 'csc':
return sp.csc_matrix((data, ind, ptr), shape=shp, dtype=data.dtype)
def _jplus(j):
"""
Internal functions for generating the data representing the J-plus
operator.
"""
m = np.arange(j, -j - 1, -1, dtype=complex)
data = (np.sqrt(j * (j + 1.0) - (m + 1.0) * m))[1:]
N = m.shape[0]
ind = np.arange(1, N, dtype=np.int32)
ptr = np.array(list(range(N-1))+[N-1]*2, dtype=np.int32)
ptr[-1] = N-1
return fast_csr_matrix((data,ind,ptr), shape=(N,N))
def _jz(j):
"""
Internal functions for generating the data representing the J-z operator.
"""
N = int(2*j+1)
data = np.array([j-k for k in range(N) if (j-k)!=0], dtype=complex)
# Even shaped matrix
if (N % 2 == 0):
ind = np.arange(N, dtype=np.int32)
ptr = np.arange(N+1,dtype=np.int32)
ptr[-1] = N
# Odd shaped matrix
else:
j = int(j)
ind = np.array(list(range(j))+list(range(j+1,N)), dtype=np.int32)
ptr = np.array(list(range(j+1))+list(range(j,N)), dtype=np.int32)
ptr[-1] = N-1
return fast_csr_matrix((data,ind,ptr), shape=(N,N))
def num(N, offset=0):
"""Quantum object for number operator.
Parameters
----------
N : int
The dimension of the Hilbert space.
offset : int (default 0)
The lowest number state that is included in the finite number state
representation of the operator.
Returns
-------
oper: qobj
Qobj for number operator.
Examples
--------
>>> num(4)
Quantum object: dims = [[4], [4]], \
shape = [4, 4], type = oper, isHerm = True
Qobj data =
[[0 0 0 0]
[0 1 0 0]
[0 0 2 0]
[0 0 0 3]]
"""
if offset == 0:
data = np.arange(1,N, dtype=complex)
ind = np.arange(1,N, dtype=np.int32)
ptr = np.array([0]+list(range(0,N)), dtype=np.int32)
ptr[-1] = N-1
else:
data = np.arange(offset, offset + N, dtype=complex)
ind = np.arange(N, dtype=np.int32)
ptr = np.arange(N+1,dtype=np.int32)
ptr[-1] = N
return Qobj(fast_csr_matrix((data,ind,ptr), shape=(N,N)), isherm=True)
def qzero(dimensions):
"""
Zero operator.
Parameters
----------
dimensions : (int) or (list of int) or (list of list of int)
Dimension of Hilbert space. If provided as a list of ints, then the
dimension is the product over this list, but the ``dims`` property of
the new Qobj are set to this list. This can produce either `oper` or
`super` depending on the passed `dimensions`.
Returns
-------
qzero : qobj
Zero operator Qobj.
"""
size, dimensions = _implicit_tensor_dimensions(dimensions)
# A sparse matrix with no data is equal to a zero matrix.
return Qobj(fast_csr_matrix(shape=(size, size), dtype=complex),
dims=dimensions,
isherm=True)
def destroy(N, offset=0):
'''Destruction (lowering) operator.
Parameters
----------
N : int
Dimension of Hilbert space.
offset : int (default 0)
The lowest number state that is included in the finite number state
representation of the operator.
Returns
-------
oper : qobj
Qobj for lowering operator.
Examples
--------
>>> destroy(4)
Quantum object: dims = [[4], [4]], \
shape = [4, 4], type = oper, isHerm = False
Qobj data =
[[ 0.00000000+0.j 1.00000000+0.j 0.00000000+0.j 0.00000000+0.j]
[ 0.00000000+0.j 0.00000000+0.j 1.41421356+0.j 0.00000000+0.j]
[ 0.00000000+0.j 0.00000000+0.j 0.00000000+0.j 1.73205081+0.j]
[ 0.00000000+0.j 0.00000000+0.j 0.00000000+0.j 0.00000000+0.j]]
'''
if not isinstance(N, (int, np.integer)): # raise error if N not integer
raise ValueError("Hilbert space dimension must be integer value")
data = np.sqrt(np.arange(offset+1, N+offset, dtype=complex))
ind = np.arange(1,N, dtype=np.int32)
ptr = np.arange(N+1, dtype=np.int32)
ptr[-1] = N-1
return Qobj(fast_csr_matrix((data,ind,ptr),shape=(N,N)), isherm=False)
def destroy(N, offset=0):
'''Destruction (lowering) operator.
Parameters
----------
N : int
Dimension of Hilbert space.
offset : int (default 0)
The lowest number state that is included in the finite number state
representation of the operator.
Returns
-------
oper : qobj
Qobj for lowering operator.
Examples
--------
>>> destroy(4) # doctest: +SKIP
Quantum object: dims = [[4], [4]], \
shape = [4, 4], type = oper, isHerm = False
Qobj data =
[[ 0.00000000+0.j 1.00000000+0.j 0.00000000+0.j 0.00000000+0.j]
[ 0.00000000+0.j 0.00000000+0.j 1.41421356+0.j 0.00000000+0.j]
[ 0.00000000+0.j 0.00000000+0.j 0.00000000+0.j 1.73205081+0.j]
[ 0.00000000+0.j 0.00000000+0.j 0.00000000+0.j 0.00000000+0.j]]
'''
if not isinstance(N, (int, np.integer)): # raise error if N not integer
raise ValueError("Hilbert space dimension must be integer value")
data = np.sqrt(np.arange(offset + 1, N + offset, dtype=complex))
ind = np.arange(1, N, dtype=np.int32)
ptr = np.arange(N + 1, dtype=np.int32)
ptr[-1] = N - 1
return Qobj(fast_csr_matrix((data, ind, ptr), shape=(N, N)), isherm=False)
def liouvillian(H, c_ops=[], data_only=False, chi=None):
"""Assembles the Liouvillian superoperator from a Hamiltonian
and a ``list`` of collapse operators. Like liouvillian, but with an
experimental implementation which avoids creating extra Qobj instances,
which can be advantageous for large systems.
Parameters
----------
H : Qobj or QobjEvo
System Hamiltonian.
c_ops : array_like of Qobj or QobjEvo
A ``list`` or ``array`` of collapse operators.
Returns
-------
L : Qobj or QobjEvo
Liouvillian superoperator.
"""
if isinstance(c_ops, (Qobj, QobjEvo)):
c_ops = [c_ops]
if chi and len(chi) != len(c_ops):
raise ValueError('chi must be a list with same length as c_ops')
h = None
if H is not None:
if isinstance(H, QobjEvo):
h = H.cte
else:
h = H
if h.isoper:
op_dims = h.dims
op_shape = h.shape
elif h.issuper:
op_dims = h.dims[0]
op_shape = [np.prod(op_dims[0]), np.prod(op_dims[0])]
else:
raise TypeError("Invalid type for Hamiltonian.")
else:
# no hamiltonian given, pick system size from a collapse operator
if isinstance(c_ops, list) and len(c_ops) > 0:
if isinstance(c_ops[0], QobjEvo):
c = c_ops[0].cte
else:
c = c_ops[0]
if c.isoper:
op_dims = c.dims
op_shape = c.shape
elif c.issuper:
op_dims = c.dims[0]
op_shape = [np.prod(op_dims[0]), np.prod(op_dims[0])]
else:
raise TypeError("Invalid type for collapse operator.")
else:
raise TypeError("Either H or c_ops must be given.")
sop_dims = [[op_dims[0], op_dims[0]], [op_dims[1], op_dims[1]]]
sop_shape = [np.prod(op_dims), np.prod(op_dims)]
spI = fast_identity(op_shape[0])
td = False
L = None
if isinstance(H, QobjEvo):
td = True
def H2L(H):
if H.isoper:
return -1.0j * (spre(H) - spost(H))
else:
return H
L = H.apply(H2L)
data = L.cte.data
elif isinstance(H, Qobj):
if H.isoper:
Ht = H.data.T
data = -1j * zcsr_kron(spI, H.data)
data += 1j * zcsr_kron(Ht, spI)
else:
data = H.data
else:
data = fast_csr_matrix(shape=(sop_shape[0], sop_shape[1]))
td_c_ops = []
for idx, c_op in enumerate(c_ops):
if isinstance(c_op, QobjEvo):
td = True
if c_op.const:
c_ = c_op.cte
elif chi:
td_c_ops.append(lindblad_dissipator(c_op, chi=chi[idx]))
continue
else:
td_c_ops.append(lindblad_dissipator(c_op))
continue
else:
c_ = c_op
if c_.issuper:
data = data + c_.data
else:
cd = c_.data.H
c = c_.data
if chi:
data = data + np.exp(1j * chi[idx]) * \
zcsr_kron(c.conj(), c)
else:
data = data + zcsr_kron(c.conj(), c)
cdc = cd * c
cdct = cdc.T
data = data - 0.5 * zcsr_kron(spI, cdc)
data = data - 0.5 * zcsr_kron(cdct, spI)
if not td:
if data_only:
return data
else:
L = Qobj()
L.dims = sop_dims
L.data = data
L.superrep = 'super'
return L
else:
if not L:
l = Qobj()
l.dims = sop_dims
l.data = data
l.superrep = 'super'
L = QobjEvo(l)
else:
L.cte.data = data
for c_op in td_c_ops:
L += c_op
return L
def test_zcsr_isherm_compare_implicit_zero():
"""
Regression test for gh-1350, comparing explicitly stored values in the
matrix (but below the tolerance for allowable Hermicity) to implicit zeros.
"""
tol = 1e-12
n = 10
base = sp.csr_matrix(np.array([[1, tol * 1e-3j], [0, 1]]))
base = fast_csr_matrix((base.data, base.indices, base.indptr), base.shape)
# If this first line fails, the zero has been stored explicitly and so the
# test is invalid.
assert base.data.size == 3
assert zcsr_isherm(base, tol=tol)
assert zcsr_isherm(base.T, tol=tol)
# A similar test if the structures are different, but it's not
# Hermitian.
base = sp.csr_matrix(np.array([[1, 1j], [0, 1]]))
base = fast_csr_matrix((base.data, base.indices, base.indptr), base.shape)
assert base.data.size == 3
assert not zcsr_isherm(base, tol=tol)
assert not zcsr_isherm(base.T, tol=tol)
# Catch possible edge case where it shouldn't be Hermitian, but faulty loop
# logic doesn't fully compare all rows.
base = sp.csr_matrix(
np.array([
[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 0, 0],
],
dtype=np.complex128))
base = fast_csr_matrix((base.data, base.indices, base.indptr), base.shape)
assert base.data.size == 1
assert not zcsr_isherm(base, tol=tol)
assert not zcsr_isherm(base.T, tol=tol)
# Pure diagonal matrix.
base = fast_identity(n)
base.data *= np.random.rand(n)
assert zcsr_isherm(base, tol=tol)
assert not zcsr_isherm(base * 1j, tol=tol)
# Larger matrices where all off-diagonal elements are below the absolute
# tolerance, so everything should always appear Hermitian, but with random
# patterns of non-zero elements. It doesn't matter that it isn't Hermitian
# if scaled up; everything is below absolute tolerance, so it should appear
# so. We also set the diagonal to be larger to the tolerance to ensure
# isherm can't just compare everything to zero.
for density in np.linspace(0.2, 1, 21):
base = tol * 1e-2 * (np.random.rand(n, n) + 1j * np.random.rand(n, n))
# Mask some values out to zero.
base[np.random.rand(n, n) > density] = 0
np.fill_diagonal(base, tol * 1000)
nnz = np.count_nonzero(base)
base = sp.csr_matrix(base)
base = fast_csr_matrix((base.data, base.indices, base.indptr), (n, n))
assert base.data.size == nnz
assert zcsr_isherm(base, tol=tol)
assert zcsr_isherm(base.T, tol=tol)
# Similar test when it must be non-Hermitian. We set the diagonal to
# be real because we want to test off-diagonal implicit zeros, and
# having an imaginary first element would automatically fail.
nnz = 0
while nnz <= n:
# Ensure that we don't just have the real diagonal.
base = tol * 1000j * np.random.rand(n, n)
# Mask some values out to zero.
base[np.random.rand(n, n) > density] = 0
np.fill_diagonal(base, tol * 1000)
nnz = np.count_nonzero(base)
base = sp.csr_matrix(base)
base = fast_csr_matrix((base.data, base.indices, base.indptr), (n, n))
assert base.data.size == nnz
assert not zcsr_isherm(base, tol=tol)
assert not zcsr_isherm(base.T, tol=tol)
def configure(self, H_sys, coup_op, coup_strength, temperature,
N_cut, N_exp, cut_freq, planck=None, boltzmann=None,
renorm=None, bnd_cut_approx=None,
options=None, progress_bar=None, stats=None):
"""
Calls configure from :class:`HEOMSolver` and sets any attributes
that are specific to this subclass
"""
start_config = timeit.default_timer()
HEOMSolver.configure(self, H_sys, coup_op, coup_strength,
temperature, N_cut, N_exp,
planck=planck, boltzmann=boltzmann,
options=options, progress_bar=progress_bar, stats=stats)
self.cut_freq = cut_freq
if renorm is not None: self.renorm = renorm
if bnd_cut_approx is not None: self.bnd_cut_approx = bnd_cut_approx
# Load local values for optional parameters
# Constants and Hamiltonian.
hbar = self.planck
options = self.options
progress_bar = self.progress_bar
stats = self.stats
if stats:
ss_conf = stats.sections.get('config')
if ss_conf is None:
ss_conf = stats.add_section('config')
c, nu = self._calc_matsubara_params()
if renorm:
norm_plus, norm_minus = self._calc_renorm_factors()
if stats:
stats.add_message('options', 'renormalisation', ss_conf)
# Dimensions et by system
N_temp = 1
for i in H_sys.dims[0]:
N_temp *= i
sup_dim = N_temp**2
unit_sys = qeye(N_temp)
# Use shorthands (mainly as in referenced PRL)
lam0 = self.coup_strength
gam = self.cut_freq
N_c = self.N_cut
N_m = self.N_exp
Q = coup_op # Q as shorthand for coupling operator
beta = 1.0/(self.boltzmann*self.temperature)
# Ntot is the total number of ancillary elements in the hierarchy
# Ntot = factorial(N_c + N_m) / (factorial(N_c)*factorial(N_m))
# Turns out to be the same as nstates from state_number_enumerate
N_he, he2idx, idx2he = enr_state_dictionaries([N_c + 1]*N_m , N_c)
unit_helems = fast_identity(N_he)
if self.bnd_cut_approx:
# the Tanimura boundary cut off operator
if stats:
stats.add_message('options', 'boundary cutoff approx', ss_conf)
op = -2*spre(Q)*spost(Q.dag()) + spre(Q.dag()*Q) + spost(Q.dag()*Q)
approx_factr = ((2*lam0 / (beta*gam*hbar)) - 1j*lam0) / hbar
for k in range(N_m):
approx_factr -= (c[k] / nu[k])
L_bnd = -approx_factr*op.data
L_helems = zcsr_kron(unit_helems, L_bnd)
else:
L_helems = fast_csr_matrix(shape=(N_he*sup_dim, N_he*sup_dim))
# Build the hierarchy element interaction matrix
if stats: start_helem_constr = timeit.default_timer()
unit_sup = spre(unit_sys).data
spreQ = spre(Q).data
spostQ = spost(Q).data
commQ = (spre(Q) - spost(Q)).data
N_he_interact = 0
for he_idx in range(N_he):
he_state = list(idx2he[he_idx])
n_excite = sum(he_state)
# The diagonal elements for the hierarchy operator
# coeff for diagonal elements
sum_n_m_freq = 0.0
for k in range(N_m):
sum_n_m_freq += he_state[k]*nu[k]
op = -sum_n_m_freq*unit_sup
L_he = cy_pad_csr(op, N_he, N_he, he_idx, he_idx)
L_helems += L_he
# Add the neighour interations
he_state_neigh = copy(he_state)
for k in range(N_m):
n_k = he_state[k]
if n_k >= 1:
# find the hierarchy element index of the neighbour before
# this element, for this Matsubara term
he_state_neigh[k] = n_k - 1
he_idx_neigh = he2idx[tuple(he_state_neigh)]
op = c[k]*spreQ - np.conj(c[k])*spostQ
if renorm:
op = -1j*norm_minus[n_k, k]*op
else:
op = -1j*n_k*op
L_he = cy_pad_csr(op, N_he, N_he, he_idx, he_idx_neigh)
L_helems += L_he
N_he_interact += 1
he_state_neigh[k] = n_k
if n_excite <= N_c - 1:
# find the hierarchy element index of the neighbour after
# this element, for this Matsubara term
he_state_neigh[k] = n_k + 1
he_idx_neigh = he2idx[tuple(he_state_neigh)]
op = commQ
if renorm:
op = -1j*norm_plus[n_k, k]*op
else:
op = -1j*op
L_he = cy_pad_csr(op, N_he, N_he, he_idx, he_idx_neigh)
L_helems += L_he
N_he_interact += 1
he_state_neigh[k] = n_k
if stats:
stats.add_timing('hierarchy contruct',
timeit.default_timer() - start_helem_constr,
ss_conf)
stats.add_count('Num hierarchy elements', N_he, ss_conf)
stats.add_count('Num he interactions', N_he_interact, ss_conf)
# Setup Liouvillian
if stats:
start_louvillian = timeit.default_timer()
H_he = zcsr_kron(unit_helems, liouvillian(H_sys).data)
L_helems += H_he
if stats:
stats.add_timing('Liouvillian contruct',
timeit.default_timer() - start_louvillian,
ss_conf)
if stats: start_integ_conf = timeit.default_timer()
r = scipy.integrate.ode(cy_ode_rhs)
r.set_f_params(L_helems.data, L_helems.indices, L_helems.indptr)
r.set_integrator('zvode', method=options.method, order=options.order,
atol=options.atol, rtol=options.rtol,
nsteps=options.nsteps, first_step=options.first_step,
min_step=options.min_step, max_step=options.max_step)
if stats:
time_now = timeit.default_timer()
stats.add_timing('Liouvillian contruct',
time_now - start_integ_conf,
ss_conf)
if ss_conf.total_time is None:
ss_conf.total_time = time_now - start_config
else:
ss_conf.total_time += time_now - start_config
self._ode = r
self._N_he = N_he
self._sup_dim = sup_dim
self._configured = True
def gather(self):
""" Create the HEOM liouvillian from a sorted list of smaller (fast) CSR
matrices.
.. note::
The list of operators contains tuples of the form
``(row_idx, col_idx, op)``. The row_idx and col_idx give the
*block* row and column for each op. An operator with
block indices ``(N, M)`` is placed at position
``[N * block: (N + 1) * block, M * block: (M + 1) * block]``
in the output matrix.
Returns
-------
rhs : :obj:`qutip.fastsparse.fast_csr_matrix`
A combined matrix of shape ``(block * nhe, block * ne)``.
"""
block = self._block
nhe = self._nhe
ops = self._ops
shape = (block * nhe, block * nhe)
if not ops:
return sp.csr_matrix(shape, dtype=np.complex128)
ops.sort()
nnz = sum(op.nnz for _, _, op in ops)
indptr = np.zeros(shape[0] + 1, dtype=np.int32)
indices = np.zeros(nnz, dtype=np.int32)
data = np.zeros(nnz, dtype=np.complex128)
end = 0
op_idx = 0
op_len = len(ops)
for row_idx in range(nhe):
prev_op_idx = op_idx
while op_idx < op_len:
if ops[op_idx][0] != row_idx:
break
op_idx += 1
row_ops = ops[prev_op_idx:op_idx]
rowpos = row_idx * block
for op_row in range(block):
for _, col_idx, op in row_ops:
colpos = col_idx * block
op_row_start = op.indptr[op_row]
op_row_end = op.indptr[op_row + 1]
op_row_len = op_row_end - op_row_start
if op_row_len == 0:
continue
indices[end:end + op_row_len] = (
op.indices[op_row_start:op_row_end] + colpos)
data[end:end +
op_row_len] = (op.data[op_row_start:op_row_end])
end += op_row_len
indptr[rowpos + op_row + 1] = end
return fast_csr_matrix(
(data, indices, indptr),
shape=shape,
dtype=np.complex128,
)
def basis(dimensions, n=None, offset=None):
"""Generates the vector representation of a Fock state.
Parameters
----------
dimensions : int or list of ints
Number of Fock states in Hilbert space. If a list, then the resultant
object will be a tensor product over spaces with those dimensions.
n : int or list of ints, optional (default 0 for all dimensions)
Integer corresponding to desired number state, defaults to 0 for all
dimensions if omitted. The shape must match ``dimensions``, e.g. if
``dimensions`` is a list, then ``n`` must either be omitted or a list
of equal length.
offset : int or list of ints, optional (default 0 for all dimensions)
The lowest number state that is included in the finite number state
representation of the state in the relevant dimension.
Returns
-------
state : :class:`qutip.Qobj`
Qobj representing the requested number state ``|n>``.
Examples
--------
>>> basis(5,2) # doctest: +SKIP
Quantum object: dims = [[5], [1]], shape = (5, 1), type = ket
Qobj data =
[[ 0.+0.j]
[ 0.+0.j]
[ 1.+0.j]
[ 0.+0.j]
[ 0.+0.j]]
>>> basis([2,2,2], [0,1,0]) # doctest: +SKIP
Quantum object: dims = [[2, 2, 2], [1, 1, 1]], shape = (8, 1), type = ket
Qobj data =
[[0.]
[0.]
[1.]
[0.]
[0.]
[0.]
[0.]
[0.]]
Notes
-----
A subtle incompatibility with the quantum optics toolbox: In QuTiP::
basis(N, 0) = ground state
but in the qotoolbox::
basis(N, 1) = ground state
"""
# Promote all parameters to lists to simplify later logic.
if not isinstance(dimensions, list):
dimensions = [dimensions]
n_dimensions = len(dimensions)
ns = [
m - off for m, off in zip(_promote_to_zero_list(n, n_dimensions),
_promote_to_zero_list(offset, n_dimensions))
]
if any((not isinstance(x, numbers.Integral)) or x < 0 for x in dimensions):
raise ValueError("All dimensions must be >= 0.")
if not all(0 <= n < dimension for n, dimension in zip(ns, dimensions)):
raise ValueError("All basis indices must be "
"`offset <= n < dimension+offset`.")
location, size = 0, 1
for m, dimension in zip(reversed(ns), reversed(dimensions)):
location += m * size
size *= dimension
data = np.array([1], dtype=complex)
ind = np.array([0], dtype=np.int32)
ptr = np.array([0] * (location + 1) + [1] * (size - location),
dtype=np.int32)
return Qobj(fast_csr_matrix((data, ind, ptr), shape=(size, 1)),
dims=[dimensions, [1] * n_dimensions],
isherm=False)
def liouvillian(H, c_ops=[], data_only=False, chi=None):
"""Assembles the Liouvillian superoperator from a Hamiltonian
and a ``list`` of collapse operators. Like liouvillian, but with an
experimental implementation which avoids creating extra Qobj instances,
which can be advantageous for large systems.
Parameters
----------
H : qobj
System Hamiltonian.
c_ops : array_like
A ``list`` or ``array`` of collapse operators.
Returns
-------
L : qobj
Liouvillian superoperator.
"""
if chi and len(chi) != len(c_ops):
raise ValueError('chi must be a list with same length as c_ops')
if H is not None:
if H.isoper:
op_dims = H.dims
op_shape = H.shape
elif H.issuper:
op_dims = H.dims[0]
op_shape = [np.prod(op_dims[0]), np.prod(op_dims[0])]
else:
raise TypeError("Invalid type for Hamiltonian.")
else:
# no hamiltonian given, pick system size from a collapse operator
if isinstance(c_ops, list) and len(c_ops) > 0:
c = c_ops[0]
if c.isoper:
op_dims = c.dims
op_shape = c.shape
elif c.issuper:
op_dims = c.dims[0]
op_shape = [np.prod(op_dims[0]), np.prod(op_dims[0])]
else:
raise TypeError("Invalid type for collapse operator.")
else:
raise TypeError("Either H or c_ops must be given.")
sop_dims = [[op_dims[0], op_dims[0]], [op_dims[1], op_dims[1]]]
sop_shape = [np.prod(op_dims), np.prod(op_dims)]
spI = fast_identity(op_shape[0])
if H:
if H.isoper:
Ht = H.data.T
data = -1j * zcsr_kron(spI, H.data)
data += 1j * zcsr_kron(Ht, spI)
else:
data = H.data
else:
data = fast_csr_matrix(shape=(sop_shape[0], sop_shape[1]))
for idx, c_op in enumerate(c_ops):
if c_op.issuper:
data = data + c_op.data
else:
cd = c_op.data.H
c = c_op.data
if chi:
data = data + np.exp(1j * chi[idx]) * \
zcsr_kron(c.conj(), c)
else:
data = data + zcsr_kron(c.conj(), c)
cdc = cd * c
cdct = cdc.T
data = data - 0.5 * zcsr_kron(spI, cdc)
data = data - 0.5 * zcsr_kron(cdct, spI)
if data_only:
return data
else:
L = Qobj()
L.dims = sop_dims
L.data = data
L.isherm = False
L.superrep = 'super'
return L
def configure(self,
H_sys,
coup_op,
coup_strength,
temperature,
N_cut,
N_exp,
cut_freq,
planck=None,
boltzmann=None,
renorm=None,
bnd_cut_approx=None,
options=None,
progress_bar=None,
stats=None):
"""
Calls configure from :class:`HEOMSolver` and sets any attributes
that are specific to this subclass
"""
start_config = timeit.default_timer()
HEOMSolver.configure(self,
H_sys,
coup_op,
coup_strength,
temperature,
N_cut,
N_exp,
planck=planck,
boltzmann=boltzmann,
options=options,
progress_bar=progress_bar,
stats=stats)
self.cut_freq = cut_freq
if renorm is not None: self.renorm = renorm
if bnd_cut_approx is not None: self.bnd_cut_approx = bnd_cut_approx
# Load local values for optional parameters
# Constants and Hamiltonian.
hbar = self.planck
options = self.options
progress_bar = self.progress_bar
stats = self.stats
if stats:
ss_conf = stats.sections.get('config')
if ss_conf is None:
ss_conf = stats.add_section('config')
c, nu = self._calc_matsubara_params()
if renorm:
norm_plus, norm_minus = self._calc_renorm_factors()
if stats:
stats.add_message('options', 'renormalisation', ss_conf)
# Dimensions et by system
sup_dim = H_sys.dims[0][0]**2
unit_sys = qeye(H_sys.dims[0])
# Use shorthands (mainly as in referenced PRL)
lam0 = self.coup_strength
gam = self.cut_freq
N_c = self.N_cut
N_m = self.N_exp
Q = coup_op # Q as shorthand for coupling operator
beta = 1.0 / (self.boltzmann * self.temperature)
# Ntot is the total number of ancillary elements in the hierarchy
# Ntot = factorial(N_c + N_m) / (factorial(N_c)*factorial(N_m))
# Turns out to be the same as nstates from state_number_enumerate
N_he, he2idx, idx2he = enr_state_dictionaries([N_c + 1] * N_m, N_c)
unit_helems = fast_identity(N_he)
if self.bnd_cut_approx:
# the Tanimura boundary cut off operator
if stats:
stats.add_message('options', 'boundary cutoff approx', ss_conf)
op = -2 * spre(Q) * spost(Q.dag()) + spre(Q.dag() * Q) + spost(
Q.dag() * Q)
approx_factr = ((2 * lam0 /
(beta * gam * hbar)) - 1j * lam0) / hbar
for k in range(N_m):
approx_factr -= (c[k] / nu[k])
L_bnd = -approx_factr * op.data
L_helems = zcsr_kron(unit_helems, L_bnd)
else:
L_helems = fast_csr_matrix(shape=(N_he * sup_dim, N_he * sup_dim))
# Build the hierarchy element interaction matrix
if stats: start_helem_constr = timeit.default_timer()
unit_sup = spre(unit_sys).data
spreQ = spre(Q).data
spostQ = spost(Q).data
commQ = (spre(Q) - spost(Q)).data
N_he_interact = 0
for he_idx in range(N_he):
he_state = list(idx2he[he_idx])
n_excite = sum(he_state)
# The diagonal elements for the hierarchy operator
# coeff for diagonal elements
sum_n_m_freq = 0.0
for k in range(N_m):
sum_n_m_freq += he_state[k] * nu[k]
op = -sum_n_m_freq * unit_sup
L_he = cy_pad_csr(op, N_he, N_he, he_idx, he_idx)
L_helems += L_he
# Add the neighour interations
he_state_neigh = copy(he_state)
for k in range(N_m):
n_k = he_state[k]
if n_k >= 1:
# find the hierarchy element index of the neighbour before
# this element, for this Matsubara term
he_state_neigh[k] = n_k - 1
he_idx_neigh = he2idx[tuple(he_state_neigh)]
op = c[k] * spreQ - np.conj(c[k]) * spostQ
if renorm:
op = -1j * norm_minus[n_k, k] * op
else:
op = -1j * n_k * op
L_he = cy_pad_csr(op, N_he, N_he, he_idx, he_idx_neigh)
L_helems += L_he
N_he_interact += 1
he_state_neigh[k] = n_k
if n_excite <= N_c - 1:
# find the hierarchy element index of the neighbour after
# this element, for this Matsubara term
he_state_neigh[k] = n_k + 1
he_idx_neigh = he2idx[tuple(he_state_neigh)]
op = commQ
if renorm:
op = -1j * norm_plus[n_k, k] * op
else:
op = -1j * op
L_he = cy_pad_csr(op, N_he, N_he, he_idx, he_idx_neigh)
L_helems += L_he
N_he_interact += 1
he_state_neigh[k] = n_k
if stats:
stats.add_timing('hierarchy contruct',
timeit.default_timer() - start_helem_constr,
ss_conf)
stats.add_count('Num hierarchy elements', N_he, ss_conf)
stats.add_count('Num he interactions', N_he_interact, ss_conf)
# Setup Liouvillian
if stats:
start_louvillian = timeit.default_timer()
H_he = zcsr_kron(unit_helems, liouvillian(H_sys).data)
L_helems += H_he
if stats:
stats.add_timing('Liouvillian contruct',
timeit.default_timer() - start_louvillian,
ss_conf)
if stats: start_integ_conf = timeit.default_timer()
r = scipy.integrate.ode(cy_ode_rhs)
r.set_f_params(L_helems.data, L_helems.indices, L_helems.indptr)
r.set_integrator('zvode',
method=options.method,
order=options.order,
atol=options.atol,
rtol=options.rtol,
nsteps=options.nsteps,
first_step=options.first_step,
min_step=options.min_step,
max_step=options.max_step)
if stats:
time_now = timeit.default_timer()
stats.add_timing('Liouvillian contruct',
time_now - start_integ_conf, ss_conf)
if ss_conf.total_time is None:
ss_conf.total_time = time_now - start_config
else:
ss_conf.total_time += time_now - start_config
self._ode = r
self._N_he = N_he
self._sup_dim = sup_dim
self._configured = True
def liouvillian(H, c_ops=[], data_only=False, chi=None):
"""Assembles the Liouvillian superoperator from a Hamiltonian
and a ``list`` of collapse operators. Like liouvillian, but with an
experimental implementation which avoids creating extra Qobj instances,
which can be advantageous for large systems.
Parameters
----------
H : qobj
System Hamiltonian.
c_ops : array_like
A ``list`` or ``array`` of collapse operators.
Returns
-------
L : qobj
Liouvillian superoperator.
"""
if chi and len(chi) != len(c_ops):
raise ValueError('chi must be a list with same length as c_ops')
if H is not None:
if H.isoper:
op_dims = H.dims
op_shape = H.shape
elif H.issuper:
op_dims = H.dims[0]
op_shape = [np.prod(op_dims[0]), np.prod(op_dims[0])]
else:
raise TypeError("Invalid type for Hamiltonian.")
else:
# no hamiltonian given, pick system size from a collapse operator
if isinstance(c_ops, list) and len(c_ops) > 0:
c = c_ops[0]
if c.isoper:
op_dims = c.dims
op_shape = c.shape
elif c.issuper:
op_dims = c.dims[0]
op_shape = [np.prod(op_dims[0]), np.prod(op_dims[0])]
else:
raise TypeError("Invalid type for collapse operator.")
else:
raise TypeError("Either H or c_ops must be given.")
sop_dims = [[op_dims[0], op_dims[0]], [op_dims[1], op_dims[1]]]
sop_shape = [np.prod(op_dims), np.prod(op_dims)]
spI = fast_identity(op_shape[0])
if H:
if H.isoper:
Ht = H.data.T
data = -1j * zcsr_kron(spI, H.data)
data += 1j * zcsr_kron(Ht, spI)
else:
data = H.data
else:
data = fast_csr_matrix(shape=(sop_shape[0], sop_shape[1]))
for idx, c_op in enumerate(c_ops):
if c_op.issuper:
data = data + c_op.data
else:
cd = c_op.data.H
c = c_op.data
if chi:
data = data + np.exp(1j * chi[idx]) * \
zcsr_kron(c.conj(), c)
else:
data = data + zcsr_kron(c.conj(), c)
cdc = cd * c
cdct = cdc.T
data = data - 0.5 * zcsr_kron(spI, cdc)
data = data - 0.5 * zcsr_kron(cdct, spI)
if data_only:
return data
else:
L = Qobj()
L.dims = sop_dims
L.data = data
L.isherm = False
L.superrep = 'super'
return L
