def test_g2_fock_state():
N, U, = 2, 0
gs = (.2, .1)
model = scattering.Model(omegas=[0] * N, links=[(0, 1, 1)], U=[2 * U] * N)
channels = []
channels.append(scattering.Channel(site=0, strength=gs[0]))
channels.append(scattering.Channel(site=N - 1, strength=gs[1]))
setup = scattering.Setup(model, channels)
Es = np.linspace(-3, 12, 1024)
dE = 0
g2s = np.zeros(Es.shape, dtype=np.complex128)
g2n = np.zeros(Es.shape, dtype=np.complex128)
g2d = np.zeros(Es.shape, dtype=np.complex128)
for i, E in enumerate(Es):
g2s[i], g2n[i], g2d[i] = g2.fock_state(setup, (0, 0), (1, 1), E, dE)
plt.semilogy(Es, g2s, label='g2')
plt.semilogy(Es, g2n, label='g2n')
plt.semilogy(Es, g2d, label='g1g1')
plt.legend()
plt.show()
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_ttmatrix():
m = scattering.Model([0] * 2, [[0, 1, 1]], [0] * 2)
channels = [
scattering.Channel(site=0, strength=.1),
scattering.Channel(site=1, strength=.1),
]
se = scattering.Setup(m, channels)
E = 0
dE = 0
chls = [[0, 0], [0, 1], [1, 0], [1, 1]]
qs = np.linspace(-5, 5, 512)
# T2as = np.zeros((4, 4, len(qs)), np.complex128)
# T2bs = np.zeros((4, 4, len(qs)), np.complex128)
for ci in xrange(4):
for co in xrange(4):
_, T2a, _, T2b = ttmatrix.two_particle(se,
chlsi=chls[ci],
chlso=chls[co],
E=E,
dE=dE,
qs=qs)
plt.subplot(4, 4, 1 + ci + co * 4)
plt.plot(qs,
np.real(T2a + T2b),
label='real'.format(chls[ci], chls[co]))
plt.plot(qs, np.imag(T2a + T2b))
plt.title('{}-{}'.format(chls[ci], chls[co]))
plt.legend()
plt.show()
def test_g2_unity_in_noninteracting_system():
# model with two sites and no interactiong
N, U = 2, 0
model = scattering.Model(omegas=[0] * N,
links=[[i, i + 1, 1] for i in xrange(N - 1)],
U=[2 * U] * N)
# channels
channels = []
channels.append(scattering.Channel(site=0, strength=.4))
channels.append(scattering.Channel(site=N - 1, strength=1.1))
setup = scattering.Setup(model, channels)
Es = np.linspace(-6, 20, 128)
chlso = [1, 1]
g12 = np.zeros((len(Es), ), dtype=np.float64)
g22 = np.zeros((len(Es), ), dtype=np.float64)
for i, E in enumerate(Es):
g12[i] = np.real(g2.coherent_state_tau0(setup, [0, 0], chlso, E)[0][0])
g22[i] = np.real(g2.general_state(setup, [0, 0], chlso, E, 0, 0)[0])
assert np.allclose(g22, 1), \
'g2 != 1 for a weakly coherent state in a non-interacting system'
assert np.allclose(g22, 1), \
'g2 != 1 for a general two-photon state in a non-interacting system'
def test_Setup_transitions():
"""
Test the scattering transitions
"""
m = scat.Model([0] * 6, links=[[c, c + 1] for c in xrange(5)], U=[2] * 6)
c1 = scat.Channel(site=0, strength=1.)
c2 = scat.Channel(site=4, strength=1.)
se = scat.Setup(model=m, channels=[c1, c2])
A10, A01 = [], []
for c in xrange(se.model.n):
A10.append(se.trsn(0, c, 1))
A01.append(se.trsn(1, c, 0))
A = np.zeros((se.model.n, se.model.n))
for i in xrange(se.model.n):
for j in xrange(se.model.n):
A[i, j] = np.abs(A01[i].dot(A10[j])[0][0])
# Test that A is orthogonal
assert np.allclose(A, np.eye(se.model.n))
A21, A12 = [], []
for c in xrange(se.model.n):
A21.append(se.trsn(1, c, 2))
A12.append(se.trsn(2, c, 1))
# Assert that A21 works
for c in xrange(se.model.n):
assert np.isclose(np.abs(A01[c].dot(A12[c]).dot(A21[c]).dot(A10[c])), 2), \
'Transitions 2 -> 1 not properly normalized'
def test_Setup():
"""
Test the scattering setup object
"""
m = scat.Model([0], [], [2])
c1 = scat.Channel(sites=[1], strengths=[2.1])
s = scat.Setup(model=m, channels=[c1])
assert_equals(s.strengths, s.gs)
def _test_g2_optical_theorem(N, U, gs=(1, 1)):
N, U, = 2, 2
model = scattering.Model(omegas=[0] * N, links=[(0, 1, 1)], U=[2 * U] * N)
channels = []
channels.append(scattering.Channel(site=0, strength=gs[0]))
channels.append(scattering.Channel(site=N - 1, strength=gs[1]))
setup = scattering.Setup(model, channels)
# Es = np.linspace(-6, 20, 128)
E = 10
chlsi = (0, 0)
dE = .1
i1, i2 = chlsi
k0, q0 = .5 * E, .5 * dE
E1 = setup.eigenenergies(1)
ps = []
nchan = len(setup.channels)
for i in xrange(nchan):
pj = []
for j in xrange(nchan):
pj.append(g2.phi(setup, k0, q0, i, j, i1, i2))
ps.append(pj)
# fij2_sqrd
fij2_sqrd = 0
for i, _ in enumerate(setup.channels):
for j, _ in enumerate(setup.channels):
for ll in xrange(nchan):
for lp in xrange(nchan):
fij2_sqrd += (ps[i][j][lp].conj() * ps[i][j][ll] +
ps[j][i][lp].conj() * ps[j][i][ll]) / (
E1[lp].conj() - E1[ll])
fij2_sqrd *= 2 * np.pi * 1j
# print(np.real(fij2_sqrd[0]))
# croff
croff = 0
for i in xrange(nchan):
for j in xrange(nchan):
croff += g2.fijm(setup, k0, q0, i, j, i1, i2).conj()*g2.fij2(k0 + q0, k0 - q0, ps[i][j], ps[j][i], E1) \
+ g2.fijp(setup, k0, q0, i, j, i1, i2).conj()*g2.fij2(k0 - q0, k0 + q0, ps[i][j], ps[j][i], E1)
croff = 2 * np.real(croff)
# print(croff[0])
assert np.isclose(np.real(fij2_sqrd[0]), -np.real(croff[0]), 1e-6), \
'Optical theorem broken since: {0} != {1}'.format(np.real(fij2_sqrd[0]), croff[0])
def test_eigenvector_overlaps():
""""
Test the eigenbasis
"""
m = scat.Model([0] * 6, links=[[c, c + 1] for c in xrange(5)], U=[2] * 6)
c1 = scat.Channel(sites=[0], strengths=[1.783])
c2 = scat.Channel(sites=[4], strengths=[0.85])
se = scat.Setup(model=m, channels=[c1, c2])
E2, psi2l, psi2r = se.eigenbasis(2)
assert np.allclose(np.abs(psi2l.conj().T.dot(psi2r)), np.eye(len(E2))), \
"Basis is not normalized"
def test_tmatrix_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
_, T1 = tmatrix.one_particle(s, 0, 0, np.array([E]))
assert (np.isclose(np.abs(T1[0])**2, 0, 1e-3))
_, T1 = tmatrix.one_particle(s, 0, 1, np.array([E]))
assert np.isclose(np.abs(T1[0])**2, 0, 1e-3), \
'T(1) fails for two side coupled sites. T(1) = {0} != 0.'.format(np.abs(T1[0])**2)
def kerr(E=0, U=0, gs=None):
"""
Setup the Kerr nonlinear element
"""
model = scattering.Model(omegas=[E] * 1, links=[], U=[U])
if gs is None:
gs = (0.1, 0.1)
channels = []
channels.append(scattering.Channel(site=0, strength=gs[0]))
channels.append(scattering.Channel(site=0, strength=gs[1]))
se = scattering.Setup(model, channels)
se.label = 'U={0}'.format(U)
return se
def test_tmatrix_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
_, T1 = tmatrix.one_particle(s, 0, 1, np.array([E]))
T10num = 2 * np.pi * np.abs(T1[0])
T10ana = 2 * np.pi / (1 + (np.pi)**2)
assert np.isclose(T10num, T10ana, 1e-3), \
'T(1) fails for two-sites coupled in series. T(1) = {0} != {1}.'.format(T10num, T10ana)
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 test_tmatrix_vs_ttmatrix():
m = scattering.Model([0] * 2, [[0, 1, 1]], [0] * 2)
channels = [
scattering.Channel(site=0, strength=.1),
scattering.Channel(site=1, strength=.1),
]
se = scattering.Setup(m, channels)
E = 0
dE = 0
chlsi = (1, 1)
chlso = (1, 1)
qs = np.array([.3])
qs, T2, N2 = tmatrix.two_particle(se,
chlsi=chlsi,
chlso=chlso,
E=E,
dE=dE,
qs=qs,
verbal=True)
ks, A2, B2, C2 = ttmatrix.two_particle(se,
chlsi=chlsi,
chlso=chlso,
E=E,
dE=dE,
qs=qs,
verbal=True)
print('Result:')
print(T2)
print(A2 + C2)
print('Interacting:')
print(T2 - N2)
print(A2)
print('Non-interacting:')
print(N2)
print(C2)
def compare_g2_fock_state_to_g2_coherent_state():
N, U, = 2, 0
gs = (.2, .1)
model = scattering.Model(omegas=[0] * N, links=[(0, 1, 1)], U=[2 * U] * N)
channels = []
channels.append(scattering.Channel(site=0, strength=gs[0]))
channels.append(scattering.Channel(site=N - 1, strength=gs[1]))
setup = scattering.Setup(model, channels)
Es = np.linspace(-3, 12, 1024)
dE = 0
g2f = np.zeros(Es.shape, dtype=np.complex128)
g2c = np.zeros(Es.shape, dtype=np.complex128)
for i, E in enumerate(Es):
g2s[i], _, _ = g2.fock_state(setup, (0, 0), (1, 1), E, dE)
def test_noninteracting_dimer_eigenenergies():
"""Test"""
m = scat.Model([0] * 2, [[0, 1, 1.1]], [0] * 2)
channels = [
scat.Channel(site=0, strength=1),
scat.Channel(site=1, strength=1),
]
sp = scat.Setup(m, channels)
E1, _, _ = sp.eigenbasis(1)
E2, _, _ = sp.eigenbasis(2)
E12 = np.zeros((len(E2), ), dtype=np.complex128)
for i in xrange(len(E1)):
for j in xrange(len(E1)):
E12[i + j] = E1[i] + E1[j]
E12 = E12[np.argsort(np.real(E12))]
E2 = E2[np.argsort(np.real(E2))]
assert np.allcose(E2, E12), \
'Non-interacting dimer, single particle energies do not coincide with the two-particle energies.'
def test_g2_coherent_tau0_and_g2_coherent():
N, U, = 2, 2
gs = (0.1, 0.1)
model = scattering.Model(omegas=[0] * N, links=[(0, 1, 1)], U=[2 * U] * N)
channels = []
channels.append(scattering.Channel(site=0, strength=gs[0]))
channels.append(scattering.Channel(site=N - 1, strength=gs[1]))
setup = scattering.Setup(model, channels)
Es = np.linspace(-3, 12, 256)
g2stau0 = np.zeros((len(Es), ), dtype=np.float64)
g2s = np.zeros((len(Es), ), dtype=np.float64)
for i, E in enumerate(Es):
g2stau0[i] = g2.coherent_state_tau0(setup, (0, 0), (1, 1), E)[0]
g2s[i] = g2.coherent_state(setup, (0, 0), (1, 1), E, 0)[0]
assert np.allclose(g2stau0, g2s), \
'g2_coherent_tau0 and g2_coherent do not coincide at tau=0'
def test_noninteracting_dimer_eigenstates():
m = scat.Model([.0] * 2, [[0, 1, 1]], [0] * 2)
channels = [
scat.Channel(sites=[0], strengths=[1.]),
scat.Channel(sites=[1], strengths=[1.]),
]
sp = scat.Setup(m, channels)
# One-particle eigenbasis
E1, psi1l, psi1r = sp.eigenbasis(1)
# Two-particle eigenbasis
E2, psi2l, psi2r = sp.eigenbasis(2)
B = np.zeros((3, 3), dtype=np.complex128)
for i in xrange(2):
for j in xrange(2):
for m in xrange(2):
for n in xrange(2):
if m == n:
B[m + n, i + j] += np.sqrt(2) * psi1r[m, i] * psi1r[n, j]
else:
B[m + n, i + j] += psi1r[m, i] * psi1r[n, j]
B[:, 0] /= np.sqrt(2)
B[:, 1] /= 2
B[:, 2] /= np.sqrt(2)
# unphase
psi2r *= np.sign(np.real(psi2r[0, :]))
B *= np.sign(np.real(B[0, :]))
assert np.allclose(psi2r, B), \
'Non-interacting dimer, single particle states do not coincide with two-particle states'
import matplotlib.pyplot as plt
# model
N = 2
m = scattering.Model(
omegas=[0] * N,
links=[(0, 1, .5)],
U=[5] * N)
# channel
channels = []
channels.append(scattering.Channel(site=0, strength=1))
channels.append(scattering.Channel(site=1, strength=1))
# setup
s = scattering.Setup(m, channels)
# input state
E = 0
dE = 4.38821403174
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))
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)))