def test_smatrix_two_particle_non_interacting():
m = scattering.Model([0] * 2, [[0, 1, 1]], [0] * 2)
channels = [
scattering.Channel(site=0, strength=1),
scattering.Channel(site=1, strength=1),
]
s = scattering.Setup(m, channels)
E = 0
dE = 0
chls = [[0, 0], [0, 1], [1, 0], [1, 1]]
S2s = np.zeros((4, 4), np.float64)
for ci, chlsi in enumerate(chls):
for co, chlso in enumerate(chls):
_, _, S2, _ = smatrix.two_particle(s,
chlsi=chlsi,
chlso=chlso,
E=E,
dE=dE)
S2s[ci, co] = np.sum(np.abs(S2)**2)
# Test if S2 vanishes
assert np.allclose(np.ones((4, 4)) + S2s, np.ones((4, 4)), 1e-2), \
'S2 does not vanish for non-interacting systems, S2 = {0}.'.format(np.abs(S2s))
def _test_smatrix_optical_theorem2(chlsi):
m = scattering.Model(omegas=[0] * 2, links=[[0, 1, .5]], U=[5] * 2)
channels = []
channels.append(scattering.Channel(site=0, strength=1))
channels.append(scattering.Channel(site=1, strength=1))
s = scattering.Setup(m, channels)
# input statei
E = 0
dE = 0
chlsos = [[0, 0], [0, 1], [1, 0], [1, 1]]
qs = np.linspace(-100, 100, 8192, endpoint=False)
D2s, S2s = [], []
for c, chlso in enumerate(chlsos):
qs, D2, S2, S2in = smatrix.two_particle(s,
chlsi=chlsi,
chlso=chlso,
E=E,
dE=dE,
qs=qs)
D2s.append(D2)
S2s.append(S2)
iq0 = np.where(qs == E / 2)[0][0]
dsums = [np.sum(np.abs(D2s[c])**2) for c in xrange(4)]
ssums = [np.sum(np.abs(S2s[c])**2) * (qs[1] - qs[0]) for c in xrange(4)]
dssums = [
2 * np.real(np.sum(D2s[c]).conj() * S2s[c][iq0]) for c in xrange(4)
]
# Test single particle contribution
assert np.isclose(np.sum(dsums), 2, 1e-3), \
'S(1)S(1) is not normalized (= {0})'.format(np.sum(dsums))
# Test optical theorem
assert np.isclose(np.sum(ssums), -np.sum(dssums), 1e-2), \
'Optical theorem broken, S2.S2 = {0} != -S1.S2 = {1}.'.format(np.sum(ssums), -np.sum(dssums))
def g2(s, chlsi, chlso, E, dE):
"""
Calculate the intensity-intensity correlation function as a function of total energy,
E, and energy discrepancy, dE.
Parameters
----------
s : Setup
Scattering setup object.
chlsi : list
Input state channel indices.
chlso : list
Output state channel indices.
E : float
Total energy of input photonic state.
dE : float
Energy difference between the two photons in the input state.
"""
qs, D2, S2, S2in = smatrix.two_particle(s,
chlsi=chlsi,
chlso=chlso,
E=E,
dE=dE)
Ee = np.array([.5 * (E - dE), .5 * (E + dE)])
_, S10 = smatrix.one_particle(s, chlsi[0], chlso[0], Ee)
_, S11 = smatrix.one_particle(s, chlsi[1], chlso[1], Ee)
# denominator
dFS2 = np.sum(np.abs(S10)**2 * np.abs(S11)**2)
# Fourier transform
FS2 = np.fft.fft(S2) * (qs[1] - qs[0])
# frequencies
freqs = np.fft.fftfreq(len(qs), d=qs[1] - qs[0])
# add delta function contribution
FS2 += np.exp(np.pi * 1j * E * freqs) * D2[0] + np.exp(
np.pi * 1j * E * freqs) * D2[1]
return FS2, dFS2
def g2_coherent_s(s, chlsi, chlso, E, qs=None):
"""
g2 for a coherent state (or partially coherent)
Parameters
----------
s : Setup
Scattering setup
chlsi : list
List of incoming channel indices
chlso : list
List of outgoing channel indices
E : float
Energy of the two-particle state
"""
qs, D2, S2, _ = smatrix.two_particle(s,
chlsi=chlsi,
chlso=chlso,
E=E,
dE=0,
qs=qs)
# Single particle scattering
Ee = np.array([.5 * E])
_, S10 = smatrix.one_particle(s, chlsi[0], chlso[0], Ee)
_, S11 = smatrix.one_particle(s, chlsi[1], chlso[1], Ee)
dFS2 = np.abs(S10[0])**2 * np.abs(S11[0])**2
# Fourier transform
FS2 = np.fft.fft(S2) * (qs[1] - qs[0])
# times
times = np.fft.fftfreq(len(qs), d=qs[1] - qs[0])
# add delta function contribution
FS2 += np.exp(2 * np.pi * 1j * E / 2 * times) * np.sum(D2)
return np.abs(FS2[0])**2, dFS2
chlsi = [0, 0]
# output state
chlsos = [[0, 0], [0, 1], [1, 0], [1, 1]]
qs = np.linspace(-.1, .1, 8192, endpoint=False)
d = 120
# qs = np.linspace(-d, d, 8192, endpoint=False)
qs, iq0 = scattering.energies(E, dE, N=2*8192, WE=2 * d)
plt.figure(figsize=(6, 6))
D2s, S2s = [], []
for c, chlso in enumerate(chlsos):
qs, D2, S2, S2in = smatrix.two_particle(s,
chlsi=chlsi,
chlso=chlso,
E=E,
dE=dE,
qs=qs)
D2s.append(D2)
S2s.append(S2)
plt.subplot(2, 2, c + 1)
plt.plot(qs, np.real(S2))
plt.plot(qs, np.imag(S2))
for c, chlso in enumerate(chlsos):
dsums = [np.sum(np.abs(D2s[c]) ** 2) for c in xrange(4)]
ssums = [np.sum(np.abs(S2s[c]) ** 2) * (qs[1] - qs[0]) for c in xrange(4)]
dssums = [2 * np.real(np.sum(D2s[c]).conj() * S2s[c][iq0]) for c in xrange(4)]
def _test_smatrix_optical_theorem(domega, dE, chlsi, verbose=False):
E = 0
m = scattering.Model(omegas=[0] * 2, links=[[0, 1, .5]], U=[5] * 2)
channels = []
channels.append(scattering.Channel(site=0, strength=1))
channels.append(scattering.Channel(site=1, strength=1))
s = scattering.Setup(m, channels)
# input statei
chlsos = [[0, 0], [0, 1], [1, 0], [1, 1]]
d = 100
# 50 / np.max(np.abs(dE), .5)
print(d)
qs, iq0 = scattering.energies(E, dE, N=8192, WE=2 * d)
D2s, S2s = [], []
for c, chlso in enumerate(chlsos):
qs, D2, S2, S2in = smatrix.two_particle(s,
chlsi=chlsi,
chlso=chlso,
E=E,
dE=dE,
qs=qs)
D2s.append(D2)
S2s.append(S2)
# iq0s = [np.where(np.isclose(qs, (E - dE) / 2))[0][0], np.where(np.isclose(qs, (E + dE) / 2))[0][0]]
dsums = [.5 * np.sum(np.abs(D2s[c])**2) for c in xrange(4)]
ssums = [
.5 * np.sum(np.abs(S2s[c])**2) * (qs[1] - qs[0]) for c in xrange(4)
]
dssums = [
.5 * 2 * np.real(np.sum(D2s[c]).conj() * S2s[c][iq0])
for c in xrange(4)
]
# dsums = [.5 * np.sum(np.abs(D2s[c]) ** 2) for c in xrange(4)]
# ssums = [.5 * np.sum(np.abs(S2s[c]) ** 2) * (qs[1] - qs[0]) for c in xrange(4)]
# dssums = [.5 * 2 * np.real(np.sum(D2s[c].conj() * S2s[c][iq0])) for c in xrange(4)]
if verbose:
print('For d(2) = {} and dE = {}:'.format(domega, dE))
# Test single particle contribution
assert np.isclose(np.sum(dsums), 1, 1e-3), \
'S(1)S(1) is not normalized (= {0})'.format(np.sum(dsums))
if verbose:
print('S1.S1 norm is: {0}'.format(np.sum(dsums)))
# Test optical theorem
assert np.isclose(np.sum(ssums), -np.sum(dssums), 1e-2), \
'Optical theorem broken for parameters: E={0}, dE={1}, chlsi={2}.'.format(E, dE, chlsi) + \
'\n' + \
'Compare S2.S2 = {0} != -S1.S2 = {1}.'.format(np.sum(ssums), -np.sum(dssums))
if verbose:
print('Compare S2.S2 = {0} to -S1.S2 = {1}.'.format(
np.sum(ssums), -np.sum(dssums)))