def test_jmm1_dictionary(self):
"""
PIQS: Test the function to generate the mapping from jmm1 to ik matrix.
"""
d1, d2, d3, d4 = jmm1_dictionary(1)
d1_correct = {(0, 0): (0.5, 0.5, 0.5), (0, 1): (0.5, 0.5, -0.5),
(1, 0): (0.5, -0.5, 0.5), (1, 1): (0.5, -0.5, -0.5)}
d2_correct = {(0.5, -0.5, -0.5): (1, 1), (0.5, -0.5, 0.5): (1, 0),
(0.5, 0.5, -0.5): (0, 1),
(0.5, 0.5, 0.5): (0, 0)}
d3_correct = {0: (0.5, 0.5, 0.5), 1: (0.5, 0.5, -0.5),
2: (0.5, -0.5, 0.5),
3: (0.5, -0.5, -0.5)}
d4_correct = {(0.5, -0.5, -0.5): 3, (0.5, -0.5, 0.5): 2,
(0.5, 0.5, -0.5): 1, (0.5, 0.5, 0.5): 0}
assert_equal(d1, d1_correct)
assert_equal(d2, d2_correct)
assert_equal(d3, d3_correct)
assert_equal(d4, d4_correct)
d1, d2, d3, d4 = jmm1_dictionary(2)
d1_correct = {(3, 3): (0.0, -0.0, -0.0), (2, 2): (1.0, -1.0, -1.0),
(2, 1): (1.0, -1.0, 0.0), (2, 0): (1.0, -1.0, 1.0),
(1, 2): (1.0, 0.0, -1.0), (1, 1): (1.0, 0.0, 0.0),
(1, 0): (1.0, 0.0, 1.0), (0, 2): (1.0, 1.0, -1.0),
(0, 1): (1.0, 1.0, 0.0), (0, 0): (1.0, 1.0, 1.0)}
d2_correct = {(0.0, -0.0, -0.0): (3, 3), (1.0, -1.0, -1.0): (2, 2),
(1.0, -1.0, 0.0): (2, 1), (1.0, -1.0, 1.0): (2, 0),
(1.0, 0.0, -1.0): (1, 2), (1.0, 0.0, 0.0): (1, 1),
(1.0, 0.0, 1.0): (1, 0), (1.0, 1.0, -1.0): (0, 2),
(1.0, 1.0, 0.0): (0, 1), (1.0, 1.0, 1.0): (0, 0)}
d3_correct = {15: (0.0, -0.0, -0.0), 10: (1.0, -1.0, -1.0),
9: (1.0, -1.0, 0.0), 8: (1.0, -1.0, 1.0),
6: (1.0, 0.0, -1.0), 5: (1.0, 0.0, 0.0),
4: (1.0, 0.0, 1.0), 2: (1.0, 1.0, -1.0),
1: (1.0, 1.0, 0.0), 0: (1.0, 1.0, 1.0)}
d4_correct = {(0.0, -0.0, -0.0): 15, (1.0, -1.0, -1.0): 10,
(1.0, -1.0, 0.0): 9, (1.0, -1.0, 1.0): 8,
(1.0, 0.0, -1.0): 6, (1.0, 0.0, 0.0): 5,
(1.0, 0.0, 1.0): 4, (1.0, 1.0, -1.0): 2,
(1.0, 1.0, 0.0): 1, (1.0, 1.0, 1.0): 0}
assert_equal(d1, d1_correct)
assert_equal(d2, d2_correct)
assert_equal(d3, d3_correct)
assert_equal(d4, d4_correct)
def test_jmm1_dictionary(self):
"""
PIQS: Test the function to generate the mapping from jmm1 to ik matrix.
"""
d1, d2, d3, d4 = jmm1_dictionary(1)
d1_correct = {(0, 0): (0.5, 0.5, 0.5), (0, 1): (0.5, 0.5, -0.5),
(1, 0): (0.5, -0.5, 0.5), (1, 1): (0.5, -0.5, -0.5)}
d2_correct = {(0.5, -0.5, -0.5): (1, 1), (0.5, -0.5, 0.5): (1, 0),
(0.5, 0.5, -0.5): (0, 1),
(0.5, 0.5, 0.5): (0, 0)}
d3_correct = {0: (0.5, 0.5, 0.5), 1: (0.5, 0.5, -0.5),
2: (0.5, -0.5, 0.5),
3: (0.5, -0.5, -0.5)}
d4_correct = {(0.5, -0.5, -0.5): 3, (0.5, -0.5, 0.5): 2,
(0.5, 0.5, -0.5): 1, (0.5, 0.5, 0.5): 0}
assert_equal(d1, d1_correct)
assert_equal(d2, d2_correct)
assert_equal(d3, d3_correct)
assert_equal(d4, d4_correct)
d1, d2, d3, d4 = jmm1_dictionary(2)
d1_correct = {(3, 3): (0.0, -0.0, -0.0), (2, 2): (1.0, -1.0, -1.0),
(2, 1): (1.0, -1.0, 0.0), (2, 0): (1.0, -1.0, 1.0),
(1, 2): (1.0, 0.0, -1.0), (1, 1): (1.0, 0.0, 0.0),
(1, 0): (1.0, 0.0, 1.0), (0, 2): (1.0, 1.0, -1.0),
(0, 1): (1.0, 1.0, 0.0), (0, 0): (1.0, 1.0, 1.0)}
d2_correct = {(0.0, -0.0, -0.0): (3, 3), (1.0, -1.0, -1.0): (2, 2),
(1.0, -1.0, 0.0): (2, 1), (1.0, -1.0, 1.0): (2, 0),
(1.0, 0.0, -1.0): (1, 2), (1.0, 0.0, 0.0): (1, 1),
(1.0, 0.0, 1.0): (1, 0), (1.0, 1.0, -1.0): (0, 2),
(1.0, 1.0, 0.0): (0, 1), (1.0, 1.0, 1.0): (0, 0)}
d3_correct = {15: (0.0, -0.0, -0.0), 10: (1.0, -1.0, -1.0),
9: (1.0, -1.0, 0.0), 8: (1.0, -1.0, 1.0),
6: (1.0, 0.0, -1.0), 5: (1.0, 0.0, 0.0),
4: (1.0, 0.0, 1.0), 2: (1.0, 1.0, -1.0),
1: (1.0, 1.0, 0.0), 0: (1.0, 1.0, 1.0)}
d4_correct = {(0.0, -0.0, -0.0): 15, (1.0, -1.0, -1.0): 10,
(1.0, -1.0, 0.0): 9, (1.0, -1.0, 1.0): 8,
(1.0, 0.0, -1.0): 6, (1.0, 0.0, 0.0): 5,
(1.0, 0.0, 1.0): 4, (1.0, 1.0, -1.0): 2,
(1.0, 1.0, 0.0): 1, (1.0, 1.0, 1.0): 0}
assert_equal(d1, d1_correct)
assert_equal(d2, d2_correct)
assert_equal(d3, d3_correct)
assert_equal(d4, d4_correct)
def dicke(N, j, m):
"""
Generate a Dicke state as a pure density matrix in the Dicke basis.
For instance, the superradiant state given by
:math:`|j, m\\rangle = |1, 0\\rangle` for N = 2,
and the state is represented as a density matrix of size (nds, nds) or
(4, 4), with the (1, 1) element set to 1.
Parameters
----------
N: int
The number of two-level systems.
j: float
The eigenvalue j of the Dicke state (j, m).
m: float
The eigenvalue m of the Dicke state (j, m).
Returns
-------
rho: :class: qutip.Qobj
The density matrix.
"""
nds = num_dicke_states(N)
rho = np.zeros((nds, nds))
jmm1_dict = jmm1_dictionary(N)[1]
i, k = jmm1_dict[(j, m, m)]
rho[i, k] = 1.
return Qobj(rho)
def dicke(N, j, m):
"""
Generate a Dicke state as a pure density matrix in the Dicke basis.
For instance, the superradiant state given by
:math:`|j, m\\rangle = |1, 0\\rangle` for N = 2,
and the state is represented as a density matrix of size (nds, nds) or
(4, 4), with the (1, 1) element set to 1.
Parameters
----------
N: int
The number of two-level systems.
j: float
The eigenvalue j of the Dicke state (j, m).
m: float
The eigenvalue m of the Dicke state (j, m).
Returns
-------
rho: :class: qutip.Qobj
The density matrix.
"""
nds = num_dicke_states(N)
rho = np.zeros((nds, nds))
jmm1_dict = jmm1_dictionary(N)[1]
i, k = jmm1_dict[(j, m, m)]
rho[i, k] = 1.
return Qobj(rho)
def dicke_basis(N, jmm1=None):
"""
Initialize the density matrix of a Dicke state for several (j, m, m1).
This function can be used to build arbitrary states in the Dicke basis
|j, m><j, m1|. We create coefficients for each (j, m, m1) value in the
dictionary jmm1. For instance, if we start from the most excited state for
N = 2, we have the following state represented as a density matrix of size
(nds, nds) or
(4, 4).
1 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
The mapping for the (i, k) index of the density matrix to the |j, m>
values is given by the cythonized function `jmm1_dictionary`.
Parameters
----------
N: int
The number of two-level systems.
jmm1: dict
A dictionary of {(j, m, m1): p} that gives a density p for the
(j, m, m1) matrix element.
Returns
-------
rho: :class: qutip.Qobj
The density matrix in the Dicke basis.
"""
if jmm1 is None:
msg = "Please specify the jmm1 values as a dictionary"
msg += "or use the `excited(N)` function to create an"
msg += "excited state where jmm1 = {(N/2, N/2, N/2): 1}"
raise AttributeError(msg)
nds = _num_dicke_states(N)
rho = np.zeros((nds, nds))
jmm1_dict = jmm1_dictionary(N)[1]
for key in jmm1:
i, k = jmm1_dict[key]
rho[i, k] = jmm1[key]
return Qobj(rho)
def dicke_basis(N, jmm1=None):
"""
Initialize the density matrix of a Dicke state for several (j, m, m1).
This function can be used to build arbitrary states in the Dicke basis
:math:`|j, m\\rangle \\langle j, m^{\\prime}|`. We create coefficients for each
(j, m, m1) value in the dictionary jmm1. The mapping for the (i, k)
index of the density matrix to the |j, m> values is given by the
cythonized function `jmm1_dictionary`. A density matrix is created from
the given dictionary of coefficients for each (j, m, m1).
Parameters
----------
N: int
The number of two-level systems.
jmm1: dict
A dictionary of {(j, m, m1): p} that gives a density p for the
(j, m, m1) matrix element.
Returns
-------
rho: :class: qutip.Qobj
The density matrix in the Dicke basis.
"""
if jmm1 is None:
msg = "Please specify the jmm1 values as a dictionary"
msg += "or use the `excited(N)` function to create an"
msg += "excited state where jmm1 = {(N/2, N/2, N/2): 1}"
raise AttributeError(msg)
nds = _num_dicke_states(N)
rho = np.zeros((nds, nds))
jmm1_dict = jmm1_dictionary(N)[1]
for key in jmm1:
i, k = jmm1_dict[key]
rho[i, k] = jmm1[key]
return Qobj(rho)
def dicke(N, j, m):
"""
Generate a Dicke state as a pure density matrix in the Dicke basis.
For instance, if the superradiant state is given |j, m> = |1, 0> for N = 2,
the state is represented as a density matrix of size (nds, nds) or (4, 4),
0 0 0 0
0 1 0 0
0 0 0 0
0 0 0 0
Parameters
----------
N: int
The number of two-level systems.
j: float
The eigenvalue j of the Dicke state |j, m>.
m: float
The eigenvalue m of the Dicke state |j, m>.
Returns
-------
rho: :class: qutip.Qobj
The density matrix.
"""
nds = num_dicke_states(N)
rho = np.zeros((nds, nds))
jmm1_dict = jmm1_dictionary(N)[1]
i, k = jmm1_dict[(j, m, m)]
rho[i, k] = 1.
return Qobj(rho)
def css(N,
x=1 / np.sqrt(2),
y=1 / np.sqrt(2),
basis="dicke",
coordinates="cartesian"):
"""
Generate the density matrix of the Coherent Spin State (CSS).
It can be defined as |CSS>= Prod_i^N(a|1>_i + b|0>_i)
with a = sin(theta/2), b = exp(1j*phi) * cos(theta/2).
The default basis is that of Dicke space |j, m> < j, m'|.
The default state is the symmetric CSS, |CSS> = |+>.
Parameters
----------
N: int
The number of two-level systems.
x, y: float
The coefficients of the CSS state.
basis: str
The basis to use. Either "dicke" or "uncoupled".
coordinates: str
Either "cartesian" or "polar". If polar then the coefficients
are constructed as sin(x/2), cos(x/2)e^(iy).
Returns
-------
rho: :class: qutip.Qobj
The CSS state density matrix.
"""
if coordinates == "polar":
a = np.cos(0.5 * x) * np.exp(1j * y)
b = np.sin(0.5 * x)
else:
a = x
b = y
if basis == "uncoupled":
return _uncoupled_css(N, a, b)
nds = num_dicke_states(N)
num_ladders = num_dicke_ladders(N)
rho = dok_matrix((nds, nds))
# loop in the allowed matrix elements
jmm1_dict = jmm1_dictionary(N)[1]
j = 0.5 * N
mmax = int(2 * j + 1)
for i in range(0, mmax):
m = j - i
psi_m = np.sqrt(float(energy_degeneracy(N, m))) * \
a**(N*0.5 + m) * b**(N*0.5 - m)
for i1 in range(0, mmax):
m1 = j - i1
row_column = jmm1_dict[(j, m, m1)]
psi_m1 = np.sqrt(float(energy_degeneracy(N, m1))) * \
np.conj(a)**(N*0.5 + m1) * np.conj(b)**(N*0.5 - m1)
rho[row_column] = psi_m * psi_m1
return Qobj(rho)
def css(N, x=1/np.sqrt(2), y=1/np.sqrt(2),
basis="dicke", coordinates="cartesian"):
"""
Generate the density matrix of the Coherent Spin State (CSS).
It can be defined as,
:math:`|CSS \\rangle = \\prod_i^N(a|1\\rangle_i + b|0\\rangle_i)`
with :math:`a = sin(\\frac{\\theta}{2})`,
:math:`b = e^{i \\phi}\\cos(\\frac{\\theta}{2})`.
The default basis is that of Dicke space
:math:`|j, m\\rangle \\langle j, m'|`.
The default state is the symmetric CSS,
:math:`|CSS\\rangle = |+\\rangle`.
Parameters
----------
N: int
The number of two-level systems.
x, y: float
The coefficients of the CSS state.
basis: str
The basis to use. Either "dicke" or "uncoupled".
coordinates: str
Either "cartesian" or "polar". If polar then the coefficients
are constructed as sin(x/2), cos(x/2)e^(iy).
Returns
-------
rho: :class: qutip.Qobj
The CSS state density matrix.
"""
if coordinates == "polar":
a = np.cos(0.5 * x) * np.exp(1j * y)
b = np.sin(0.5 * x)
else:
a = x
b = y
if basis == "uncoupled":
return _uncoupled_css(N, a, b)
nds = num_dicke_states(N)
num_ladders = num_dicke_ladders(N)
rho = dok_matrix((nds, nds))
# loop in the allowed matrix elements
jmm1_dict = jmm1_dictionary(N)[1]
j = 0.5*N
mmax = int(2*j + 1)
for i in range(0, mmax):
m = j-i
psi_m = np.sqrt(float(energy_degeneracy(N, m))) * \
a**(N*0.5 + m) * b**(N*0.5 - m)
for i1 in range(0, mmax):
m1 = j - i1
row_column = jmm1_dict[(j, m, m1)]
psi_m1 = np.sqrt(float(energy_degeneracy(N, m1))) * \
np.conj(a)**(N*0.5 + m1) * np.conj(b)**(N*0.5 - m1)
rho[row_column] = psi_m*psi_m1
return Qobj(rho)