from .dimensions import Space
from .states import state_number_enumerate
from . import data as _data
from . import Qobj, qdiags
import numpy as np
import scipy.sparse
from .. import settings
__all__ = ['enr_state_dictionaries', 'enr_fock',
'enr_thermal_dm', 'enr_destroy', 'enr_identity']
[docs]def enr_state_dictionaries(dims, excitations):
"""
Return the number of states, and lookup-dictionaries for translating
a state tuple to a state index, and vice versa, for a system with a given
number of components and maximum number of excitations.
Parameters
----------
dims: list
A list with the number of states in each sub-system.
excitations : integer
The maximum numbers of dimension
Returns
-------
nstates, state2idx, idx2state: integer, dict, dict
The number of states `nstates`, a dictionary for looking up state
indices from a state tuple, and a dictionary for looking up state
state tuples from state indices. state2idx and idx2state are reverses
of each other, i.e., ``state2idx[idx2state[idx]] = idx`` and
``idx2state[state2idx[state]] = state``.
"""
nstates = 0
state2idx = {}
idx2state = {}
for state in state_number_enumerate(dims, excitations):
state2idx[state] = nstates
idx2state[nstates] = state
nstates += 1
return nstates, state2idx, idx2state
class EnrSpace(Space):
_stored_dims = {}
def __init__(self, dims, excitations):
self.dims = tuple(dims)
self.n_excitations = excitations
enr_dicts = enr_state_dictionaries(dims, excitations)
self.size, self.state2idx, self.idx2state = enr_dicts
self.issuper = False
self.superrep = None
self._pure_dims = False
def __eq__(self, other):
return (
self is other
or (
type(other) is type(self)
and self.dims == other.dims
and self.n_excitations == other.n_excitations
)
)
def __hash__(self):
return hash((self.dims, self.n_excitations))
def __repr__(self):
return f"EnrSpace({self.dims}, {self.n_excitations})"
def as_list(self):
return list(self.dims)
def dims2idx(self, dims):
return self.state2idx[tuple(dims)]
def idx2dims(self, idx):
return self.idx2state[idx]
[docs]def enr_fock(dims, excitations, state, *, dtype=None):
"""
Generate the Fock state representation in a excitation-number restricted
state space. The `dims` argument is a list of integers that define the
number of quantums states of each component of a composite quantum system,
and the `excitations` specifies the maximum number of excitations for
the basis states that are to be included in the state space. The `state`
argument is a tuple of integers that specifies the state (in the number
basis representation) for which to generate the Fock state representation.
Parameters
----------
dims : list
A list of the dimensions of each subsystem of a composite quantum
system.
excitations : integer
The maximum number of excitations that are to be included in the
state space.
state : list of integers
The state in the number basis representation.
dtype : type or str, optional
Storage representation. Any data-layer known to `qutip.data.to` is
accepted.
Returns
-------
ket : Qobj
A Qobj instance that represent a Fock state in the exication-number-
restricted state space defined by `dims` and `exciations`.
"""
dtype = dtype or settings.core["default_dtype"] or _data.Dense
nstates, state2idx, _ = enr_state_dictionaries(dims, excitations)
try:
data = _data.one_element[dtype](
(nstates, 1),
(state2idx[tuple(state)], 0),
1
)
except KeyError:
msg = (
"state tuple " + str(tuple(state))
+ " is not in the restricted state space."
)
raise ValueError(msg) from None
return Qobj(data, dims=[EnrSpace(dims, excitations), [1]*len(dims)],
copy=False)
[docs]def enr_thermal_dm(dims, excitations, n, *, dtype=None):
"""
Generate the density operator for a thermal state in the excitation-number-
restricted state space defined by the `dims` and `exciations` arguments.
See the documentation for enr_fock for a more detailed description of
these arguments. The temperature of each mode in dims is specified by
the average number of excitatons `n`.
Parameters
----------
dims : list
A list of the dimensions of each subsystem of a composite quantum
system.
excitations : integer
The maximum number of excitations that are to be included in the
state space.
n : integer
The average number of exciations in the thermal state. `n` can be
a float (which then applies to each mode), or a list/array of the same
length as dims, in which each element corresponds specifies the
temperature of the corresponding mode.
dtype : type or str, optional
Storage representation. Any data-layer known to `qutip.data.to` is
accepted.
Returns
-------
dm : Qobj
Thermal state density matrix.
"""
dtype = dtype or settings.core["default_dtype"] or _data.CSR
nstates, _, idx2state = enr_state_dictionaries(dims, excitations)
enr_dims = [EnrSpace(dims, excitations)] * 2
if not isinstance(n, (list, np.ndarray)):
n = np.ones(len(dims)) * n
else:
n = np.asarray(n)
diags = [np.prod((n / (n + 1)) ** np.array(state))
for idx, state in idx2state.items()]
diags /= np.sum(diags)
out = qdiags(diags, 0, dims=enr_dims,
shape=(nstates, nstates), dtype=dtype)
out._isherm = True
return out
[docs]def enr_destroy(dims, excitations, *, dtype=None):
"""
Generate annilation operators for modes in a excitation-number-restricted
state space. For example, consider a system consisting of 4 modes, each
with 5 states. The total hilbert space size is 5**4 = 625. If we are
only interested in states that contain up to 2 excitations, we only need
to include states such as
(0, 0, 0, 0)
(0, 0, 0, 1)
(0, 0, 0, 2)
(0, 0, 1, 0)
(0, 0, 1, 1)
(0, 0, 2, 0)
...
This function creates annihilation operators for the 4 modes that act
within this state space:
a1, a2, a3, a4 = enr_destroy([5, 5, 5, 5], excitations=2)
From this point onwards, the annihiltion operators a1, ..., a4 can be
used to setup a Hamiltonian, collapse operators and expectation-value
operators, etc., following the usual pattern.
Parameters
----------
dims : list
A list of the dimensions of each subsystem of a composite quantum
system.
excitations : integer
The maximum number of excitations that are to be included in the
state space.
dtype : type or str, optional
Storage representation. Any data-layer known to `qutip.data.to` is
accepted.
Returns
-------
a_ops : list of qobj
A list of annihilation operators for each mode in the composite
quantum system described by dims.
"""
dtype = dtype or settings.core["default_dtype"] or _data.CSR
nstates, state2idx, idx2state = enr_state_dictionaries(dims, excitations)
enr_dims = [EnrSpace(dims, excitations)] * 2
a_ops = [scipy.sparse.lil_matrix((nstates, nstates), dtype=np.complex128)
for _ in dims]
for n1, state1 in idx2state.items():
for idx, s in enumerate(state1):
# if s > 0, the annihilation operator of mode idx has a non-zero
# entry with one less excitation in mode idx in the final state
if s > 0:
state2 = state1[:idx] + (s-1,) + state1[idx+1:]
n2 = state2idx[state2]
a_ops[idx][n2, n1] = np.sqrt(s)
return [Qobj(a, dims=enr_dims).to(dtype) for a in a_ops]
[docs]def enr_identity(dims, excitations, *, dtype=None):
"""
Generate the identity operator for the excitation-number restricted
state space defined by the `dims` and `exciations` arguments. See the
docstring for enr_fock for a more detailed description of these arguments.
Parameters
----------
dims : list
A list of the dimensions of each subsystem of a composite quantum
system.
excitations : integer
The maximum number of excitations that are to be included in the
state space.
dtype : type or str, optional
Storage representation. Any data-layer known to `qutip.data.to` is
accepted.
Returns
-------
op : Qobj
A Qobj instance that represent the identity operator in the
exication-number-restricted state space defined by `dims` and
`exciations`.
"""
dtype = dtype or settings.core["default_dtype"] or _data.Dia
dims = EnrSpace(dims, excitations)
return Qobj(_data.identity[dtype](dims.size),
dims=[dims, dims],
isherm=True,
isunitary=True,
copy=False)