def g1(setup, chlsi, k0, q0, i):
(i1, i2) = chlsi
E1 = setup.eigenenergies(1)
_, Sk1 = smatrix.one_particle(setup, i1, i, np.array([k0 + q0]))
_, Sk2 = smatrix.one_particle(setup, i2, i, np.array([k0 - q0]))
fij_sqrd = np.abs(Sk1)**2 + np.abs(Sk2)**2
fij2_sqrd = 0
# EE = E1[None, :].conj() - E1[:, None]
pp, pm = [], []
nchan = len(setup.channels)
for j in xrange(nchan):
pp.append(phi(setup, k0, q0, i, j, i1, i2))
pm.append(phi(setup, k0, q0, j, i, i1, i2))
# fij2_sqrd += np.sum((pp[j].conj().T*pp[j] + pm[j].conj().T*pm[j])/EE)
for ll in xrange(len(E1)):
for lp in xrange(len(E1)):
fij2_sqrd += (pp[j][lp].conj() * pp[j][ll] + pm[j][lp].conj() *
pm[j][ll]) / (E1[lp].conj() - E1[ll])
# add a prefactor
fij2_sqrd *= 2 * np.pi * 1j
croff = 0
for j in xrange(len(setup.channels)):
croff += fijm(setup, k0, q0, i, j, i1, i2).conj() * fij2(k0 + q0, k0 - q0, pp[j], pm[j], E1) \
+ fijp(setup, k0, q0, i, j, i1, i2).conj() * fij2(k0 - q0, k0 + q0, pp[j], pm[j], E1)
return fij_sqrd + fij2_sqrd + 2 * np.real(croff)
def coherent_state_tau0_quasilocal(setup, chlsi, chlso, E=0):
"""
g2 from Mikhails formula
Parameters
----------
se : Setup
Scattering setup object
chlsi : list
list of incoming channels (must be identical!)
chlso : list
list of outgoing channels
E : float or list
Incoming two photon state energ(y/ies)
"""
(i1, i2) = chlsi
(o1, o2) = chlso
assert i1 == i2, 'Photons in an incoming coherent state **must** all belong to the same channel.'
# convert to numpy array
k0s = np.atleast_1d(E) / 2
# k0s = E / 2 if util.isarray(E) else np.array([E / 2])
_, S00 = smatrix.one_particle(setup, i1, o1, k0s)
_, S11 = smatrix.one_particle(setup, i2, o2, k0s)
g2a = 1 - (S11 - (i2 == o2)) * (S00 - (i1 == o1)) / (S11 * S00)
g2b = np.zeros(k0s.shape, np.complex128)
for i, k0 in enumerate(k0s):
setup.reset() # remove cache
# eigen-energies
E1 = setup.eigenenergies(1, phi=k0)
E2 = setup.eigenenergies(2, phi=k0)
# creation operator in n = 0-1-2 eigenbasis representation
A01 = setup.transition(1, o1, 0, k0)
A12 = setup.transition(2, o2, 1, k0)
A21 = setup.transition(1, i1, 2, k0)
A10 = setup.transition(0, i2, 1, k0)
# g2
g2b[i] = A01 \
.dot(A12) \
.dot(np.diag(1 / (2 * k0 - E2))) \
.dot(A21) \
.dot(A10 / (k0 - E1[:, None]))[0][0]
prefactor = 4 * np.pi**2 # * np.prod([setup.gs[ch] for ch in chlsi + chlso])
return {
'g2': np.abs(g2a - prefactor / (S11 * S00) * g2b)**2,
'phi2': np.abs(g2a * S11 * S00 - prefactor * g2b)**2,
'normalization': S11 * S00,
'phi2_reducible': g2a * S11 * S00,
'phi2_irreducible': -prefactor * g2b
}
def test_smatrix_oneparticle2():
m = scattering.Model([0] * 2, [[0, 1, 1]], [0] * 2)
channels = [
scattering.Channel(site=0, strength=1),
scattering.Channel(site=0, strength=1),
]
s = scattering.Setup(m, channels)
E = 0
_, S1 = smatrix.one_particle(s, 0, 0, np.array([E]))
assert (np.isclose(np.abs(S1[0])**2, 1, 1e-3))
_, S1 = smatrix.one_particle(s, 0, 1, np.array([E]))
assert np.isclose(np.abs(S1[0]) ** 2, 0, 1e-3), \
'S(1) fails for two sites which are sidecoupled. S(1) = {0} != 0.'.format(np.abs(S1[0]) ** 2)
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
def nonlinear(se, chli, chlo, Es, p=0.1):
"""
Non-linear correction to the transmittance.
Parameters
----------
se : scattering.Setup object
Scattering setup object.
chli : int
Incoming channel index.
chlo : int
Outgoing channel index.
Es : ndarray
Single particle energies.
p : float
The power = alpha**2/L (number of photons per time/pulse length)
"""
tr = np.zeros((len(Es), ), dtype=np.complex128)
t2 = np.zeros((len(Es), ), dtype=np.complex128)
for chl in xrange(len(se.channels)):
# single particle contribution
s1ij = smatrix.one_particle(se, chli, chl, Es)[1]
# The t2 contribution
for i, E in enumerate(Es):
t2[i] = tmatrix.two_particle(se, (chli, chli), (chlo, chl),
E=2 * E,
dE=0,
qs=np.array([E]))[1][0, 0]
# result
tr += 8 * 1j * (np.pi**2) * p * np.conjugate(s1ij) * t2
s1 = smatrix.one_particle(se, chli, chlo, Es)[1]
return np.abs(s1 - tr)**2, s1, tr
def linear(se, chli, chlo, Es):
"""
Linear component of the single particle transmittance.
Parameters
----------
se : scattering.Setup object
Scattering setup object.
chli : int
Incoming channel.
chlo : int
Outgoing channel.
Es : ndarray
Single particle energies.
"""
return np.abs(smatrix.one_particle(se, chli, chlo, Es)[1])**2
def g1(se, chli, chlo, Es):
"""
Plots g1-correlation function.
Parameters
----------
se : scattering.Setup object
The scattering setup
chli : int
Incoming channel
chlo : int
Outgoing channel
Es : array-like
List of single-photon energies
"""
return np.abs(smatrix.one_particle(se, chli, chlo, Es)[1])**2
def test_smatrix_oneparticle():
"""
Test the S(1) matrix for one example system
"""
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
_, S1 = smatrix.one_particle(s, 0, 1, np.array([E]))
S10num = np.abs(S1[0])**2
S10ana = (2 * np.pi / (1 + (np.pi)**2))**2
assert np.isclose(S10num, S10ana, 1e-3), \
'S(1) fails for two-sites coupled in series. S(1) = {0} != {1}.'.format(S10num, S01ana)
def fijp(setup, k0, q0, i, j, i1, i2):
_, Si = smatrix.one_particle(setup, i1, j, np.array([k0 + q0]))
_, Sj = smatrix.one_particle(setup, i2, i, np.array([k0 - q0]))
return Si * Sj
def fock_state_local(setup, chlsi, chlso, E, dE, tau=0):
"""
Compute g^2 correlation function for a general two-photon state.
Parameters
----------
se : scattering.Model
A given scattering model
chlsi : list
List of incoming two-photon state channel indices.
chlso : list
List of outgoing two-photon state channel indices.
E : float
Total two-photon energy
dE : float
Energy difference between the two photons.
tau : float
Time
"""
# channel indices
(i1, i2) = chlsi
(o1, o2) = chlso
# single photon energies
k0, q0 = .5 * E, .5 * dE
k1, k2 = k0 + q0, k0 - q0
# times
taus = np.atleast_1d(tau)
_, S11 = smatrix.one_particle(setup, i1, o1, np.array([k1]))
_, S22 = smatrix.one_particle(setup, i2, o2, np.array([k2]))
_, S12 = smatrix.one_particle(setup, i1, o2, np.array([k1]))
_, S21 = smatrix.one_particle(setup, i2, o1, np.array([k2]))
g2a = S11 * S22 * np.exp(-1j * q0 * taus) + S21 * S12 * np.exp(
1j * q0 * taus)
# eigen energies
E1, E2 = [setup.eigenenergies(nb) for nb in (1, 2)]
# creation operator in n = 0-1 eigenbasis representation
A10, A01, A21, A12 = transitions(setup)
prefactor = (2 * np.pi)**2 * np.prod([setup.gs[c] for c in chlsi + chlso])
At = A01[o1] * np.exp(-1j * (E1[None, :] - k0) * taus[:, None])
g2b = At.dot(A12[o2]).dot(np.diag(1 / (E - E2))).dot(A21[i2]).dot(A10[i1] / (k1 - E1[:, None])) \
+ At.dot(A12[o2]).dot(np.diag(1 / (E - E2))).dot(A21[i1]).dot(A10[i2] / (k2 - E1[:, None])) \
- (At / (k2 - E1)).dot(A10[i2]).dot(A01[o2]).dot(A10[i1] / (k1 - E1[:, None])) \
- (At / (k1 - E1)).dot(A10[i1]).dot(A01[o2]).dot(A10[i2] / (k2 - E1[:, None]))
g2 = np.abs(g2a - prefactor * g2b.T)[:][0]**2
# normalization
if o2 == o1:
g1ig1j = g1(setup, chlsi, k0, q0, o1)**2
else:
g1ig1j = g1(setup, chlsi, k0, q0, o1) * g1(setup, chlsi, k0, q0, o2)
# return
return g2 / np.abs(g1ig1j[0]), g2, g1ig1j[0]
def coherent_state_quasilocal(setup, chlsi, chlso, E=0, tau=0):
"""
g2(E, tau) correlation as a function of incoming energy, E, and
delay time, tau. Here specifically for quasi-local systems.
Parameters
----------
se : Setup
Scattering setup object
chlsi : list
list of incoming channels (must be identical!)
chlso : list
list of outgoing channels
E : float or list
Two photon state energ(y/ies)
"""
# channels
# two incoming photons from same channel
assert chlsi[0] == chlsi[
1], 'Photons in an incoming coherent state must all belong to the same channel.'
(i1, _) = chlsi
(o1, o2) = chlso
# numpy arraify
k0s, taus = np.atleast_1d(E / 2, tau)
# smatrix single photon
_, S00 = smatrix.one_particle(setup, i1, o1, k0s)
_, S11 = smatrix.one_particle(setup, i1, o2, k0s)
g2 = np.zeros((len(k0s), len(taus)), dtype=np.float64)
S2 = np.zeros((len(k0s), len(taus)), dtype=np.complex128)
for i, k0 in enumerate(k0s):
# eigen energies
E1 = setup.eigenenergies(1, phi=k0)
E2 = setup.eigenenergies(2, phi=k0)
# creation operator in n = 0-1 eigenbasis representation
A01_i = setup.transition(1, o1, 0, k0)
A12_j = setup.transition(2, o2, 1, k0)
A01_j = setup.transition(1, o2, 0, k0)
A21_0 = setup.transition(1, i1, 2, k0)
A10_0 = setup.transition(0, i1, 1, k0)
prefactor = 4 * np.pi**2 * np.prod(
[setup.gs[c] for c in chlsi + chlso])
op = (A12_j.dot(np.diag(1 / (2 * k0 - E2))).dot(A21_0)
- np.diag(1 / (k0 - E1)).dot(A10_0).dot(A01_j)) \
.dot(A10_0 / (k0 - E1[:, None]))
S2[i, :] = prefactor * np.dot(
A01_i * np.exp(-1j * (E1[None, :] - k0) * taus[:, None]), op)[:, 0]
g2[i, :] = np.abs(1 - S2[i, :] / (S00[i] * S11[i]))**2
# np.abs(1 - (prefactor/(S00[i]*S11[i])
# * np.dot(A01_i*np.exp(-1j*(E1[None, :] - k0)*taus[:, None]), op))
# )[:, 0]**2
return {
'g2': g2,
'phi2': np.abs(np.tile(S00 * S11, (len(taus), 1)) - S2)**2,
'normalization': S11 * S00,
'phi2_irreducible': S2
}