def setUp(self):
# Set up rotation angles
self.alpha = 2.67
self.beta = 1.32
self.gamma = 0.83
# Get geometric interpretation of the rotation angles
# The coordinate system is rotated by R_z(alpha)*R_y(beta)*R_z(gamma) where e.g. R_z is the elementary rotation
# R_z(alpha) = [[cos(alpha), -sin(alpha), 0], [sin(alpha), cos(alpha), 0], [0,0,1]]. In the following, we calculate
# the coordinate representations of the rotated z-axis and y-axis in the unrotated system.
s, c = np.sin(self.gamma), np.cos(self.gamma)
self.rotator = np.array([[c, -s, 0], [s, c, 0], [0, 0, 1]])
s, c = np.sin(self.beta), np.cos(self.beta)
self.rotator = np.dot(np.array([[c, 0, s], [0, 1, 0], [-s, 0, c]]), self.rotator)
s, c = np.sin(self.alpha), np.cos(self.alpha)
self.rotator = np.dot(np.array([[c, -s, 0], [s, c, 0], [0, 0, 1]]), self.rotator)
self.to_z_axis = np.dot(self.rotator, [0, 0, 1])
self.to_y_axis = np.dot(self.rotator, [0, 1, 0])
# Set up cache
self.cache = pi.MatrixElementCache()
# Setup states
self.state_one = pi.StateOne("Rb", 61, 2, 1.5, 1.5)
self.state_two = pi.StateTwo(self.state_one, self.state_one)
# Build one-atom system
self.system_one = pi.SystemOne(self.state_one.getSpecies(), self.cache)
self.system_one.restrictEnergy(self.state_one.getEnergy() - 30, self.state_one.getEnergy() + 30)
self.system_one.restrictN(self.state_one.getN() - 1, self.state_one.getN() + 1)
self.system_one.restrictL(self.state_one.getL() - 1, self.state_one.getL() + 1)
def test_permutation(self):
#######################################################
### Check permutation symmetry of two atom systems ####
#######################################################
# Remark: calling restrictEnergy() would cause a small
# deviation # TODO figure out exact reason (in case of
# symmetrized basis states, the atom-atom interaction would
# add energy to the diagonal)
# Define states
state_one = pi.StateOne("Rb", 61, 1, 0.5, 0.5)
state_two = pi.StateTwo(state_one, state_one)
# Build one atom system
system_one = pi.SystemOne(state_one.element, self.path_cache)
system_one.restrictEnergy(state_one.energy-20, state_one.energy+20)
system_one.restrictN(state_one.n-1, state_one.n+1)
system_one.restrictL(state_one.l-1, state_one.l+1)
system_one.restrictJ(state_one.j-1, state_one.j+1)
system_one.setBfield([100, 200, 300])
system_one.setEfield([1, 2, 3])
# Build two atom system
system_two = pi.SystemTwo(system_one, system_one, self.path_cache)
#system_two.restrictEnergy(state_two.energy-2, state_two.energy+2) # TODO
system_two.setDistance(1)
system_two.setOrder(3)
# Diagonalize blockwise
system_two_even = pi.SystemTwo(system_two)
system_two_even.setConservedParityUnderPermutation(pi.EVEN)
system_two_even.diagonalize()
system_two_odd = pi.SystemTwo(system_two)
system_two_odd.setConservedParityUnderPermutation(pi.ODD)
system_two_odd.diagonalize()
system_two_combined = system_two_even
system_two_combined.add(system_two_odd)
# Diagonalize altogether
system_two.setConservedParityUnderPermutation(pi.NA)
system_two.diagonalize()
# Compare results
w1 = np.sort(system_two_combined.diagonal)
w2 = np.sort(system_two.diagonal)
maxdiff = np.abs((w1-w2)/(np.max([w1,w2],axis=0)))
maxdiff[np.abs(w1-w2)<1e-14] = np.abs(w1-w2)[np.abs(w1-w2)<1e-14]
maxdiff = np.max(maxdiff)
print("Two-atom system with permutation symmetry, relative maximum deviation: ", maxdiff)
self.assertAlmostEqual(maxdiff, 0, places=9)
def setUp(self):
self.dump_new_reference_data = False
self.tolerance = 1e-6
# Set up cache
self.cache_path = tempfile.mkdtemp()
self.cache = pi.MatrixElementCache(self.cache_path)
# Setup states
self.state_one = pi.StateOne("Rb", 61, 2, 1.5, 1.5)
self.state_two = pi.StateTwo(self.state_one, self.state_one)
def test_combined(self):
#######################################################
### Check combined binary symmetries ##################
#######################################################
# Remark: calling restrictEnergy() would cause a small
# deviation # TODO figure out exact reason (in case of
# symmetrized basis states, the atom-atom interaction would
# add energy to the diagonal)
# Define states
state_one = pi.StateOne("Rb", 61, 1, 0.5, 0.5)
state_two = pi.StateTwo(state_one, state_one)
# Build one atom system
system_one = pi.SystemOne(state_one.element, self.path_cache)
system_one.restrictEnergy(state_one.energy-20, state_one.energy+20)
system_one.restrictN(state_one.n-1, state_one.n+1)
system_one.restrictL(state_one.l-1, state_one.l+1)
system_one.restrictJ(state_one.j-1, state_one.j+1)
system_one.setBfield([0, 100, 0])
system_one_combined = pi.SystemOne(system_one)
system_one_combined.setConservedParityUnderReflection(pi.EVEN)
system_one_combined.diagonalize()
system_one_inverse = pi.SystemOne(system_one)
system_one_inverse.setConservedParityUnderReflection(pi.ODD)
system_one_inverse.diagonalize()
system_one_combined.add(system_one_inverse)
system_one.setConservedParityUnderReflection(pi.NA)
system_one.diagonalize()
# Build two atom system
system_two = pi.SystemTwo(system_one, system_one, self.path_cache)
system_two.setDistance(1)
system_two.setOrder(3)
system_two.diagonalize()
# Note: it is important to use system_one_combined
system_two_from_combined = pi.SystemTwo(system_one_combined, system_one_combined, self.path_cache)
system_two_from_combined.setDistance(1)
system_two_from_combined.setOrder(3)
system_two_combined = None
for sym_reflection, sym_inversion, sym_permutation in product([pi.EVEN, pi.ODD], [pi.EVEN, pi.ODD], [pi.EVEN, pi.ODD]):
system_two_tmp = pi.SystemTwo(system_two_from_combined)
system_two_tmp.setConservedParityUnderReflection(sym_reflection)
system_two_tmp.setConservedParityUnderInversion(sym_inversion)
system_two_tmp.setConservedParityUnderPermutation(sym_permutation)
system_two_tmp.diagonalize()
if system_two_combined is None: system_two_combined = system_two_tmp
else: system_two_combined.add(system_two_tmp)
system_two_combined.diagonalize()
# Compare results
w1 = np.sort(system_two_combined.diagonal)
w2 = np.sort(system_two.diagonal)
maxdiff = np.abs((w1-w2)/(np.max([w1,w2],axis=0)))
maxdiff[np.abs(w1-w2)<1e-14] = np.abs(w1-w2)[np.abs(w1-w2)<1e-14]
maxdiff = np.max(maxdiff)
print("Two-atom system with combined binary symmetries, relative maximum deviation: ", maxdiff)
self.assertAlmostEqual(maxdiff, 0, places=9)
def test_rotation(self):
#######################################################
### Check rotation symmetry of one atom systems #######
#######################################################
# Define state
state_one = pi.StateOne("Rb", 61, 1, 0.5, 0.5)
# Build one atom system
system_one = pi.SystemOne(state_one.element, self.path_cache)
system_one.restrictEnergy(state_one.energy-40, state_one.energy+40)
system_one.restrictN(state_one.n-1, state_one.n+1)
system_one.restrictL(state_one.l-1, state_one.l+1)
system_one.restrictJ(state_one.j-1, state_one.j+1)
system_one.setBfield([0, 0, 100])
system_one.setEfield([0, 0, 1])
momenta_one = np.arange(-(state_one.j+1), (state_one.j+1)+1)
# Diagonalize blockwise
system_one_momentum = {}
for m in momenta_one:
system_one_momentum[m] = pi.SystemOne(system_one)
system_one_momentum[m].setConservedMomentaUnderRotation([m])
system_one_momentum[m].diagonalize()
system_one_combined = pi.SystemOne(system_one_momentum[momenta_one[0]])
for m in momenta_one[1:]:
system_one_combined.add(system_one_momentum[m])
# Diagonalize altogether
system_one_alternative = pi.SystemOne(system_one)
system_one_alternative.setConservedMomentaUnderRotation(momenta_one)
system_one_alternative.diagonalize()
system_one.setConservedMomentaUnderRotation([pi.ARB])
system_one.diagonalize()
# Compare results
w1 = np.sort(system_one_combined.diagonal)
w2 = np.sort(system_one.diagonal)
w3 = np.sort(system_one_alternative.diagonal)
maxdiff12 = np.abs((w1-w2)/(np.max([w1,w2],axis=0)))
maxdiff23 = np.abs((w3-w2)/(np.max([w3,w2],axis=0)))
maxdiff12[np.abs(w1-w2)<1e-14] = np.abs(w1-w2)[np.abs(w1-w2)<1e-14]
maxdiff23[np.abs(w3-w2)<1e-14] = np.abs(w3-w2)[np.abs(w3-w2)<1e-14]
maxdiff12 = np.max(maxdiff12)
maxdiff23 = np.max(maxdiff23)
print("One-atom system with rotation symmetry, relative maximum deviation: ",
maxdiff12, " (between alternatives: ", maxdiff23,")")
self.assertAlmostEqual(maxdiff12, 0, places=9)
self.assertAlmostEqual(maxdiff23, 0, places=9)
#######################################################
### Check rotation symmetry of two atom systems #######
#######################################################
# Define state
state_two = pi.StateTwo(state_one, state_one)
# Diagonalize blockwise
system_two_momentum = {}
for m1 in momenta_one:
for m2 in momenta_one:
if m1+m2 in system_two_momentum:
tmp = pi.SystemTwo(system_one_momentum[m1], system_one_momentum[m2])
tmp.restrictEnergy(state_two.energy-2, state_two.energy+2)
system_two_momentum[m1+m2].add(tmp)
else:
system_two_momentum[m1+m2] = pi.SystemTwo(system_one_momentum[m1], system_one_momentum[m2])
system_two_momentum[m1+m2].restrictEnergy(state_two.energy-2, state_two.energy+2)
momenta_two = list(system_two_momentum.keys())
for m in momenta_two:
system_two_momentum[m].setDistance(1)
system_two_momentum[m].setOrder(5)
system_two_momentum[m].diagonalize()
system_two_combined = pi.SystemTwo(system_two_momentum[momenta_two[0]])
for m in momenta_two[1:]:
system_two_combined.add(system_two_momentum[m])
# Diagonalize blockwise, alternative
system_two_momentum_alternative = {}
for m in momenta_two:
system_two_momentum_alternative[m] = pi.SystemTwo(system_one, system_one)
system_two_momentum_alternative[m].restrictEnergy(state_two.energy-2, state_two.energy+2)
system_two_momentum_alternative[m].setConservedMomentaUnderRotation([int(m)])
system_two_momentum_alternative[m].setDistance(1)
system_two_momentum_alternative[m].setOrder(5)
system_two_momentum_alternative[m].diagonalize()
system_two_combined_alternative = pi.SystemTwo(system_two_momentum_alternative[momenta_two[0]])
for m in momenta_two[1:]:
system_two_combined_alternative.add(system_two_momentum_alternative[m])
# Diagonalize altogether
system_two = pi.SystemTwo(system_one, system_one, self.path_cache)
system_two.restrictEnergy(state_two.energy-2, state_two.energy+2)
system_two.setDistance(1)
system_two.setOrder(5)
system_two.diagonalize()
# Compare results
w1 = np.sort(system_two_combined.diagonal)
w2 = np.sort(system_two.diagonal)
w3 = np.sort(system_two_combined_alternative.diagonal)
maxdiff12 = np.abs((w1-w2)/(np.max([w1,w2],axis=0)))
maxdiff23 = np.abs((w3-w2)/(np.max([w3,w2],axis=0)))
maxdiff12[np.abs(w1-w2)<1e-14] = np.abs(w1-w2)[np.abs(w1-w2)<1e-14]
maxdiff23[np.abs(w3-w2)<1e-14] = np.abs(w3-w2)[np.abs(w3-w2)<1e-14]
maxdiff12 = np.max(maxdiff12)
maxdiff23 = np.max(maxdiff23)
print("Two-atom system with rotation symmetry, relative maximum deviation: ",
maxdiff12, " (between alternatives: ", maxdiff23,")")
self.assertAlmostEqual(maxdiff12, 0, places=9)
self.assertAlmostEqual(maxdiff23, 0, places=9)
def test_reflection(self):
#######################################################
### Check reflection symmetry of one atom systems #####
#######################################################
# Define state
state_one = pi.StateOne("Rb", 61, 1, 0.5, 0.5)
# Build one atom system
system_one = pi.SystemOne(state_one.element, self.path_cache)
system_one.restrictEnergy(state_one.energy-20, state_one.energy+20)
system_one.restrictN(state_one.n-1, state_one.n+1)
system_one.restrictL(state_one.l-1, state_one.l+1)
system_one.restrictJ(state_one.j-1, state_one.j+1)
system_one.setBfield([0, 100, 0])
system_one.setEfield([1, 0, 2])
# Diagonalize blockwise
system_one_even = pi.SystemOne(system_one)
system_one_even.setConservedParityUnderReflection(pi.EVEN)
system_one_even.diagonalize()
system_one_odd = pi.SystemOne(system_one)
system_one_odd.setConservedParityUnderReflection(pi.ODD)
system_one_odd.diagonalize()
system_one_combined = pi.SystemOne(system_one_even)
system_one_combined.add(system_one_odd)
# Diagonalize altogether
system_one.setConservedParityUnderReflection(pi.NA)
system_one.diagonalize()
# Compare results
w1 = np.sort(system_one_combined.diagonal)
w2 = np.sort(system_one.diagonal)
maxdiff = np.abs((w1-w2)/(np.max([w1,w2],axis=0)))
maxdiff[np.abs(w1-w2)<1e-14] = np.abs(w1-w2)[np.abs(w1-w2)<1e-14]
maxdiff = np.max(maxdiff)
print("One-atom system with reflection symmetry, relative maximum deviation: ", maxdiff)
self.assertAlmostEqual(maxdiff, 0, places=9)
#######################################################
### Check reflection symmetry of two atom systems #####
#######################################################
# Remark: calling restrictEnergy() would cause a small deviation
# Define state
state_two = pi.StateTwo(state_one, state_one)
# Diagonalize blockwise
# Note: it is called odd in order to fit to the notion of the paper
system_two_odd = pi.SystemTwo(system_one_even, system_one_even, self.path_cache)
system_two_odd.add(pi.SystemTwo(system_one_odd, system_one_odd, self.path_cache))
system_two_odd.setDistance(1)
system_two_odd.setOrder(5)
system_two_odd.diagonalize()
system_two_even = pi.SystemTwo(system_one_even, system_one_odd, self.path_cache)
system_two_even.add(pi.SystemTwo(system_one_odd, system_one_even, self.path_cache))
system_two_even.setDistance(1)
system_two_even.setOrder(5)
system_two_even.diagonalize()
system_two_combined = pi.SystemTwo(system_two_even)
system_two_combined.add(system_two_odd)
# Diagonalize blockwise alternative
# Note: it is important to use system_one_combined
system_two_alternative= pi.SystemTwo(system_one_combined, system_one_combined, self.path_cache)
system_two_alternative.setDistance(1)
system_two_alternative.setOrder(5)
system_two_even_alternative = pi.SystemTwo(system_two_alternative)
system_two_even_alternative.setConservedParityUnderReflection(pi.EVEN)
system_two_even_alternative.diagonalize()
system_two_odd_alternative = pi.SystemTwo(system_two_alternative)
system_two_odd_alternative.setConservedParityUnderReflection(pi.ODD)
system_two_odd_alternative.diagonalize()
# Diagonalize altogether
system_two = pi.SystemTwo(system_one, system_one, self.path_cache)
system_two.setDistance(1)
system_two.setOrder(5)
system_two.diagonalize()
# Compare results
w1 = np.sort(system_two_combined.diagonal)
w2 = np.sort(system_two.diagonal)
w3 = np.sort(system_two_even.diagonal)
w4 = np.sort(system_two_odd.diagonal)
w5 = np.sort(system_two_even_alternative.diagonal)
w6 = np.sort(system_two_odd_alternative.diagonal)
maxdiff12 = np.abs((w1-w2)/(np.max([w1,w2],axis=0)))
maxdiff35 = np.abs((w3-w5)/(np.max([w3,w5],axis=0)))
maxdiff46 = np.abs((w4-w6)/(np.max([w4,w6],axis=0)))
maxdiff12[np.abs(w1-w2)<1e-14] = np.abs(w1-w2)[np.abs(w1-w2)<1e-14]
maxdiff35[np.abs(w3-w5)<1e-14] = np.abs(w3-w5)[np.abs(w3-w5)<1e-14]
maxdiff46[np.abs(w4-w6)<1e-14] = np.abs(w4-w6)[np.abs(w4-w6)<1e-14]
maxdiff12 = np.max(maxdiff12)
maxdiff35 = np.max(maxdiff35)
maxdiff46 = np.max(maxdiff46)
print("Two-atom system with reflection symmetry, relative maximum deviation: ",
maxdiff12, " (between alternatives: ", maxdiff35,", ", maxdiff46, ")")
self.assertAlmostEqual(maxdiff12, 0, places=9)
self.assertAlmostEqual(maxdiff35, 0, places=9)
self.assertAlmostEqual(maxdiff46, 0, places=9)
