# Source code for arc.calculations_atom_pairstate

# -*- coding: utf-8 -*-

"""
Pair-state basis level diagram calculations

Calculates Rydberg spaghetti of level diagrams, as well as pertubative C6
and similar properties. It also allows calculation of Foster resonances
tuned by DC electric fields.

Example:
Calculation of the Rydberg eigenstates in pair-state basis for Rubidium
in the vicinity of the
:math:|60~S_{1/2}~m_j=1/2,~60~S_{1/2}~m_j=1/2\\rangle state. Colour
highlights coupling strength from state :math:6~P_{1/2}~m_j=1/2 with
:math:\\pi (:math:q=0) polarized light.
eigenstates::

from arc import *
calc1 = PairStateInteractions(Rubidium(), 60, 0, 0.5, 60, 0, 0.5,0.5, 0.5)
calc1.defineBasis( 0., 0., 4, 5,10e9)
# optionally we can save now results of calculation for future use
saveCalculation(calc1,"mycalculation.pkl")
calculation1.diagonalise(linspace(1,10.0,30),250,progressOutput = True,drivingFromState=[6,1,0.5,0.5,0])
calc1.plotLevelDiagram()
calc1.ax.set_xlim(1,10)
calc1.ax.set_ylim(-2,2)
calc1.showPlot()

"""

from __future__ import division, print_function, absolute_import

from arc._database import sqlite3
from arc.wigner import Wigner6j, CG, WignerDmatrix
from arc.alkali_atom_functions import (
_atomLightAtomCoupling,
singleAtomState,
compositeState,
)
from scipy.constants import physical_constants, pi
import gzip
import sys
import os
import datetime
import matplotlib
from matplotlib.colors import LinearSegmentedColormap
from arc.calculations_atom_single import StarkMap
from arc.alkali_atom_functions import (
printStateStringLatex,
printStateString,
printStateLetter,
)
from arc.divalent_atom_functions import DivalentAtom
from scipy.special import factorial
from scipy.sparse.linalg import eigsh
from scipy.sparse import csr_matrix
from scipy.optimize import curve_fit
from scipy.constants import e as C_e
from scipy.constants import h as C_h
from scipy.constants import c as C_c

import numpy as np
from math import exp, sqrt
import matplotlib.pyplot as plt
import matplotlib as mpl

mpl.rcParams["xtick.minor.visible"] = True
mpl.rcParams["ytick.minor.visible"] = True
mpl.rcParams["xtick.major.size"] = 8
mpl.rcParams["ytick.major.size"] = 8
mpl.rcParams["xtick.minor.size"] = 4
mpl.rcParams["ytick.minor.size"] = 4
mpl.rcParams["xtick.direction"] = "in"
mpl.rcParams["ytick.direction"] = "in"
mpl.rcParams["xtick.top"] = True
mpl.rcParams["ytick.right"] = True
mpl.rcParams["font.family"] = "serif"

# for matrices

if sys.version_info > (2,):
xrange = range

DPATH = os.path.join(os.path.expanduser("~"), ".arc-data")

__all__ = ["PairStateInteractions", "StarkMapResonances"]

[docs]
class PairStateInteractions:
"""
Calculates Rydberg level diagram (spaghetti) for the given pair state

Initializes Rydberg level spaghetti calculation for the given atom
species (or for two atoms of different species) in the vicinity
of the given pair state. For details of calculation see
Ref. [1]_. For a quick start point example see
interactions example snippet_.
For inter-species calculations see
inter-species interaction calculation snippet_.

.. _interactions example snippet:
./Rydberg_atoms_a_primer.html#Short-range-interactions

.. _inter-species interaction calculation snippet:
./ARC_3_0_introduction.html#Inter-species-pair-state-calculations

Parameters:
atom (:obj:arc.alkali_atom_functions.AlkaliAtom or :obj:arc.divalent_atom_functions.DivalentAtom):
= {
:obj:arc.alkali_atom_data.Lithium6,
:obj:arc.alkali_atom_data.Lithium7,
:obj:arc.alkali_atom_data.Sodium,
:obj:arc.alkali_atom_data.Potassium39,
:obj:arc.alkali_atom_data.Potassium40,
:obj:arc.alkali_atom_data.Potassium41,
:obj:arc.alkali_atom_data.Rubidium85,
:obj:arc.alkali_atom_data.Rubidium87,
:obj:arc.alkali_atom_data.Caesium,
:obj:arc.divalent_atom_data.Strontium88,
:obj:arc.divalent_atom_data.Calcium40
:obj:arc.divalent_atom_data.Ytterbium174 }
Select the alkali metal for energy level
diagram calculation
n (int):
principal quantum number for the *first* atom
l (int):
orbital angular momentum for the *first* atom
j (float):
total angular momentum for the *first* atom
nn (int):
principal quantum number for the *second* atom
ll (int):
orbital angular momentum for the *second* atom
jj (float):
total angular momentum for the *second* atom
m1 (float):
projection of the total angular momentum on z-axis
for the *first* atom
m2 (float):
projection of the total angular momentum on z-axis
for the *second* atom
interactionsUpTo (int):
Optional. If set to 1, includes only
dipole-dipole interactions. If set to 2 includes interactions
up to quadrupole-quadrupole. Default value is 1.
s (float):
optional, spin state of the first atom. Default value
of 0.5 is correct for :obj:arc.alkali_atom_functions.AlkaliAtom
but for :obj:arc.divalent_atom_functions.DivalentAtom
it has to be explicitly set to 0 or 1 for
singlet and triplet states respectively.
**If s2 is not specified, it is assumed that the second
atom is in the same spin state.**
s2 (float):
optinal, spin state of the second atom. If not
specified (left to default value None) it will assume spin
state of the first atom.
atom2 (:obj:arc.alkali_atom_functions.AlkaliAtom or :obj:arc.divalent_atom_functions.DivalentAtom):
optional,
specifies atomic species for the second atom, enabeling
calculation of **inter-species pair-state interactions**.
If not specified (left to default value None) it will assume
spin state of the first atom.

References:
.. [1] T. G Walker, M. Saffman, PRA **77**, 032723 (2008)
https://doi.org/10.1103/PhysRevA.77.032723

Examples:
**Advanced interfacing of pair-state is2=None, atom2=Nonenteractions calculations
(PairStateInteractions class).** This
is an advanced example intended for building up extensions to the
existing code. If you want to directly access the pair-state
interaction matrix, constructed by :obj:defineBasis,
you can assemble it easily from diagonal part
(stored in :obj:matDiagonal ) and off-diagonal matrices whose
spatial dependence is :math:R^{-3},R^{-4},R^{-5} stored in that
order in :obj:matR. Basis states are stored in :obj:basisStates
array.

>>> from arc import *
>>> calc = PairStateInteractions(Rubidium(), 60,0,0.5, \
60,0,0.5, 0.5,0.5,interactionsUpTo = 1)
>>> # theta=0, phi = 0, range of pqn, range of l, deltaE = 25e9
>>> calc.defineBasis(0 ,0 , 5, 5, 25e9, progressOutput=True)
>>> # now calc stores interaction matrix and relevant basis
>>> # we can access this directly and generate interaction matrix
>>> # at distance rval :
>>> rval = 4  # in mum
>>> matrix = calc.matDiagonal
>>> rX = (rval*1.e-6)**3
>>> for matRX in self.matR:
>>>     matrix = matrix + matRX/rX
>>>     rX *= (rval*1.e-6)
>>> # matrix variable now holds full interaction matrix for
>>> # interacting atoms at distance rval calculated in
>>> # pair-state basis states can be accessed as
>>> basisStates = calc.basisStates
"""

dataFolder = DPATH

# =============================== Methods ===============================

def __init__(
self,
atom,
n,
l,
j,
nn,
ll,
jj,
m1,
m2,
interactionsUpTo=1,
s=0.5,
s2=None,
atom2=None,
):
# alkali atom type, principal quantum number, orbital angular momentum,
#  total angular momentum projections of the angular momentum on z axis
self.atom1 = atom  #: the first atom type (isotope)
if atom2 is None:
self.atom2 = atom  #: the second atom type (isotope)
else:
self.atom2 = atom2  #: thge second atom type (isotope)
self.n = n  # : pair-state definition: principal quantum number of the first atom
self.l = l  # : pair-state definition: orbital angular momentum of the first atom
self.j = j  # : pair-state definition: total angular momentum of the first atom
self.nn = nn  # : pair-state definition: principal quantum number of the second atom
self.ll = ll  # : pair-state definition: orbital angular momentum of the second atom
self.jj = jj  # : pair-state definition: total angular momentum of the second atom
self.m1 = m1  # : pair-state definition: projection of the total ang. momentum for the *first* atom
self.m2 = m2  # : pair-state definition: projection of the total angular momentum for the *second* atom
self.interactionsUpTo = interactionsUpTo
""" Specifies up to which approximation we include in pair-state interactions.
By default value is 1, corresponding to pair-state interactions up to
dipole-dipole coupling. Value of 2 is also supported, corresponding
to pair-state interactions up to quadrupole-quadrupole coupling.
"""

if issubclass(type(atom), DivalentAtom) and not (s == 0 or s == 1):
raise ValueError(
"total angular spin s has to be defined explicitly "
"for calculations, and value has to be 0 or 1 "
"for singlet and tripplet states respectively."
)
self.s1 = s  #: total spin angular momentum, optional (default 0.5)

if s2 is None:
self.s2 = s
else:
self.s2 = s2

# check that values of spin states are valid for entered atomic species

if issubclass(type(self.atom1), DivalentAtom):
if abs(self.s1) > 0.1 and abs(self.s1 - 1) > 0.1:
raise ValueError(
"atom1 is DivalentAtom and its spin has to be "
"s=0 or s=1 (for singlet and triplet states "
"respectively)"
)
elif abs(self.s1 - 0.5) > 0.1:
raise ValueError(
"atom1 is AlkaliAtom and its spin has to be " "s=0.5"
)
if issubclass(type(self.atom2), DivalentAtom):
if abs(self.s2) > 0.1 and abs(self.s2 - 1) > 0.1:
raise ValueError(
"atom2 is DivalentAtom and its spin has to be "
"s=0 or s=1 (for singlet and triplet states "
"respectively)"
)
elif abs(self.s2 - 0.5) > 0.1:
# we have divalent atom
raise ValueError(
"atom2 is AlkaliAtom and its spin has to be " "s=0.5"
)
if abs((self.s1 - self.m1) % 1) > 0.1:
raise ValueError(
"atom1 with spin s = %.1d cannot have m1 = %.1d"
% (self.s1, self.m1)
)
if abs((self.s2 - self.m2) % 1) > 0.1:
raise ValueError(
"atom2 with spin s = %.1d cannot have m2 = %.1d"
% (self.s2, self.m2)
)

# ====================== J basis (not resolving mj) ===================

self.coupling = []
"""
List of matrices defineing coupling strengths between the states in
J basis (not resolving :math:m_j ). Basis is given by
:obj:PairStateInteractions.channel. Used as intermediary for full
interaction matrix calculation by
:obj:PairStateInteractions.defineBasis.
"""
self.channel = []
"""
states relevant for calculation, defined in J basis (not resolving
:math:m_j. Used as intermediary for full interaction matrix
calculation by :obj:PairStateInteractions.defineBasis.
"""

# ======================= Full basis (resolving mj) ===================

self.basisStates = []
"""
List of pair-states for calculation. In the form
[[n1,l1,j1,mj1,n2,l2,j2,mj2], ...].
Each state is an array [n1,l1,j1,mj1,n2,l2,j2,mj2] corresponding to
:math:|n_1,l_1,j_1,m_{j1},n_2,l_2,j_2,m_{j2}\\rangle state.
Calculated by :obj:PairStateInteractions.defineBasis.
"""
self.matrixElement = []
"""
matrixElement[i] gives index of state in
:obj:PairStateInteractions.channel basis
(that doesn't resolve :math:m_j states), for the given index i
of the state in :obj:PairStateInteractions.basisStates
( :math:m_j resolving) basis.
"""

# variuos parts of interaction matrix in pair-state basis
self.matDiagonal = []
"""
Part of interaction matrix in pair-state basis that doesn't depend
on inter-atomic distance. E.g. diagonal elements of the interaction
matrix, that describe energies of the pair states in unperturbed
basis, will be stored here. Basis states are stored in
:obj:PairStateInteractions.basisStates. Calculated by
:obj:PairStateInteractions.defineBasis.
"""
self.matR = []
"""
Stores interaction matrices in pair-state basis
that scale as :math:1/R^3, :math:1/R^4 and :math:1/R^5
with distance in  :obj:matR[0], :obj:matR[1] and :obj:matR[2]
respectively. These matrices correspond to dipole-dipole
( :math:C_3), dipole-quadrupole ( :math:C_4) and
quadrupole-quadrupole ( :math:C_5) interactions
coefficients. Basis states are stored in
:obj:PairStateInteractions.basisStates.
Calculated by :obj:PairStateInteractions.defineBasis.
"""
self.originalPairStateIndex = 0
"""
index of the original n,l,j,m1,nn,ll,jj,m2 pair-state in the
:obj:PairStateInteractions.basisStates basis.
"""

self.matE = []
self.matB_1 = []
self.matB_2 = []

# ===================== Eigen states and plotting =====================

# finding perturbed energy levels
self.r = []  # detuning scale
self.y = []  # energy levels
self.highlight = []

# pointers towards figure
self.fig = 0
self.ax = 0

# for normalization of the maximum coupling later
self.maxCoupling = 0.0

# n,l,j,mj, drive polarization q
self.drivingFromState = [0, 0, 0, 0, 0]

# sam = saved angular matrix metadata
self.angularMatrixFile = "angularMatrix.npy"
self.angularMatrixFile_meta = "angularMatrix_meta.npy"
# self.sam = []
self.savedAngularMatrix_matrix = []

# intialize precalculated values for factorial term
# in __getAngularMatrix_M
def fcoef(l1, l2, m):
return (
factorial(l1 + l2)
/ (
factorial(l1 + m)
* factorial(l1 - m)
* factorial(l2 + m)
* factorial(l2 - m)
)
** 0.5
)

x = self.interactionsUpTo
self.fcp = np.zeros((x + 1, x + 1, 2 * x + 1))
for c1 in range(1, x + 1):
for c2 in range(1, x + 1):
for p in range(-min(c1, c2), min(c1, c2) + 1):
self.fcp[c1, c2, p + x] = fcoef(c1, c2, p)

self.conn = False

def __getAngularMatrix_M(self, l, j, ll, jj, l1, j1, l2, j2):
# did we already calculated this matrix?
c = self.conn.cursor()
c.execute(
"""SELECT ind FROM pair_angularMatrix WHERE
l1 = ? AND j1_x2 = ? AND
l2 = ? AND j2_x2 = ? AND
l3 = ? AND j3_x2 = ? AND
l4 = ? AND j4_x2 = ?
""",
(l, j * 2, ll, jj * 2, l1, j1 * 2, l2, j2 * 2),
)

index = c.fetchone()
if index:
return self.savedAngularMatrix_matrix[index[0]]

# determine coupling
dl = abs(l - l1)
dj = abs(j - j1)
c1 = 0
if dl == 1 and (dj < 1.1):
c1 = 1  # dipole coupling
elif dl == 0 or dl == 2 or dl == 1:
c1 = 2  # quadrupole coupling
else:
raise ValueError("error in __getAngularMatrix_M")

dl = abs(ll - l2)
dj = abs(jj - j2)
c2 = 0
if dl == 1 and (dj < 1.1):
c2 = 1  # dipole coupling
elif dl == 0 or dl == 2 or dl == 1:
c2 = 2  # quadrupole coupling
else:
raise ValueError("error in __getAngularMatrix_M")

am = np.zeros(
(
round((2 * j1 + 1) * (2 * j2 + 1)),
round((2 * j + 1) * (2 * jj + 1)),
),
dtype=np.float64,
)

if (c1 > self.interactionsUpTo) or (c2 > self.interactionsUpTo):
return am

j1range = np.linspace(-j1, j1, round(2 * j1) + 1)
j2range = np.linspace(-j2, j2, round(2 * j2) + 1)
jrange = np.linspace(-j, j, round(2 * j) + 1)
jjrange = np.linspace(-jj, jj, round(2 * jj) + 1)

for m1 in j1range:
for m2 in j2range:
# we have chosen the first index
index1 = round(
m1 * (2.0 * j2 + 1.0) + m2 + (j1 * (2.0 * j2 + 1.0) + j2)
)
for m in jrange:
for mm in jjrange:
# we have chosen the second index
index2 = round(
m * (2.0 * jj + 1.0)
+ mm
+ (j * (2.0 * jj + 1.0) + jj)
)

# angular matrix element from Sa??mannshausen, Heiner,
# Merkt, Fr??d??ric, Deiglmayr, Johannes
# PRA 92: 032505 (2015)
elem = (
(-1.0) ** (j + jj + self.s1 + self.s2 + l1 + l2)
* CG(l, 0, c1, 0, l1, 0)
* CG(ll, 0, c2, 0, l2, 0)
)
elem = (
elem
* sqrt((2.0 * l + 1.0) * (2.0 * ll + 1.0))
* sqrt((2.0 * j + 1.0) * (2.0 * jj + 1.0))
)
elem = (
elem
* Wigner6j(l, self.s1, j, j1, c1, l1)
* Wigner6j(ll, self.s2, jj, j2, c2, l2)
)

sumPol = 0.0  # sum over polarisations
limit = min(c1, c2)
for p in xrange(-limit, limit + 1):
sumPol = sumPol + self.fcp[
c1, c2, p + self.interactionsUpTo
] * CG(j, m, c1, p, j1, m1) * CG(
jj, mm, c2, -p, j2, m2
)
am[index1, index2] = elem * sumPol

index = len(self.savedAngularMatrix_matrix)

c.execute(
""" INSERT INTO pair_angularMatrix
VALUES (?,?, ?,?, ?,?, ?,?, ?)""",
(l, j * 2, ll, jj * 2, l1, j1 * 2, l2, j2 * 2, index),
)
self.conn.commit()

self.savedAngularMatrix_matrix.append(am)
self.savedAngularMatrixChanged = True

return am

def __updateAngularMatrixElementsFile(self):
if not (self.savedAngularMatrixChanged):
return

try:
c = self.conn.cursor()
c.execute("""SELECT * FROM pair_angularMatrix """)
data = []
for v in c.fetchall():
data.append(v)

data = np.array(data, dtype=np.float32)

data[:, 1] /= 2.0  # 2 r j1 -> j1
data[:, 3] /= 2.0  # 2 r j2 -> j2
data[:, 5] /= 2.0  # 2 r j3 -> j3
data[:, 7] /= 2.0  # 2 r j4 -> j4

fileHandle = gzip.GzipFile(
os.path.join(self.dataFolder, self.angularMatrixFile_meta), "wb"
)
np.save(fileHandle, data)
fileHandle.close()
except IOError:
print(
"Error while updating angularMatrix \
data meta (description) File "
+ self.angularMatrixFile_meta
)

try:
fileHandle = gzip.GzipFile(
os.path.join(self.dataFolder, self.angularMatrixFile), "wb"
)
np.save(
fileHandle,
np.array(self.savedAngularMatrix_matrix, dtype=object),
)
fileHandle.close()
except (IOError, ValueError) as e:
print(
"Error while updating angularMatrix \
data File "
+ self.angularMatrixFile
)
print(e)

try:
fileHandle = gzip.GzipFile(
os.path.join(self.dataFolder, self.angularMatrixFile_meta), "rb"
)
data = np.load(fileHandle, encoding="latin1", allow_pickle=True)
fileHandle.close()
except Exception as ex:
print(ex)
print("Note: No saved angular matrix metadata files to be loaded.")
print(sys.exc_info())
return

data[:, 1] *= 2  # j1 -> 2 r j1
data[:, 3] *= 2  # j2 -> 2 r j2
data[:, 5] *= 2  # j3 -> 2 r j3
data[:, 7] *= 2  # j4 -> 2 r j4

data = np.array(np.rint(data), dtype=int)

try:
c = self.conn.cursor()
c.executemany(
"""INSERT INTO pair_angularMatrix
(l1, j1_x2 ,
l2 , j2_x2 ,
l3, j3_x2,
l4 , j4_x2 ,
ind)
VALUES (?,?,?,?,?,?,?,?,?)""",
data,
)

self.conn.commit()

except sqlite3.Error as e:
print("Error while loading precalculated values into the database!")
print(e)
exit()
if len(data) == 0:
print("error")
return

try:
fileHandle = gzip.GzipFile(
os.path.join(self.dataFolder, self.angularMatrixFile), "rb"
)
fileHandle, encoding="latin1", allow_pickle=True
).tolist()
fileHandle.close()
except Exception as ex:
print(ex)
print("Note: No saved angular matrix files to be loaded.")
print(sys.exc_info())

def __isCoupled(self, n, l, j, nn, ll, jj, n1, l1, j1, n2, l2, j2, limit):
if (
(
abs(
self.__getEnergyDefect(
n, l, j, nn, ll, jj, n1, l1, j1, n2, l2, j2
)
)
/ C_h
< limit
)
and not (
n == n1
and nn == n2
and l == l1
and ll == l2
and j == j1
and jj == j2
)
and not (
(
abs(l1 - l) != 1
and (
(
abs(j - 0.5) < 0.1 and abs(j1 - 0.5) < 0.1
)  # j = 1/2 and j'=1/2 forbidden
or (
abs(j) < 0.1 and abs(j1 - 1) < 0.1
)  # j = 0 and j'=1 forbidden
or (
abs(j - 1) < 0.1 and abs(j1) < 0.1
)  # j = 1 and j'=0 forbidden
)
)
or (
abs(l2 - ll) != 1
and (
(
abs(jj - 0.5) < 0.1 and abs(j2 - 0.5) < 0.1
)  # j = 1/2 and j'=1/2 forbidden
or (
abs(jj) < 0.1 and abs(j2 - 1) < 0.1
)  # j = 0 and j'=1 forbidden
or (
abs(jj - 1) < 0.1 and abs(j2) < 0.1
)  # j = 1 and j'=0 forbidden
)
)
)
and not (abs(j) < 0.1 and abs(j1) < 0.1)  # j = 0 and j'=0 forbiden
and not (abs(jj) < 0.1 and abs(j2) < 0.1)
and not (
abs(l) < 0.1 and abs(l1) < 0.1
)  # l = 0 and l' = 0 is forbiden
and not (abs(ll) < 0.1 and abs(l2) < 0.1)
):
# determine coupling
dl = abs(l - l1)
dj = abs(j - j1)
c1 = 0
if dl == 1 and (dj < 1.1):
c1 = 1  # dipole coupling
elif (
(dl == 0 or dl == 2 or dl == 1)
and (dj < 2.1)
and (2 <= self.interactionsUpTo)
):
c1 = 2  # quadrupole coupling
else:
return False
dl = abs(ll - l2)
dj = abs(jj - j2)
c2 = 0
if dl == 1 and (dj < 1.1):
c2 = 1  # dipole coupling
elif (
(dl == 0 or dl == 2 or dl == 1)
and (dj < 2.1)
and (2 <= self.interactionsUpTo)
):
c2 = 2  # quadrupole coupling
else:
return False
return c1 + c2
else:
return False

def __getEnergyDefect(self, n, l, j, nn, ll, jj, n1, l1, j1, n2, l2, j2):
"""
Energy defect between |n,l,j>x|nn,ll,jj> state and |n1,l1,j1>x|n1,l1,j1>
state of atom1 and atom2 in respective spins states s1 and s2

Takes spin vales s1 and s2 as the one defined when defining calculation.

Parameters:
n (int): principal quantum number
l (int): orbital angular momenutum
j (float): total angular momentum
nn (int): principal quantum number
ll (int): orbital angular momenutum
jj (float): total angular momentum
n1 (int): principal quantum number
l1 (int): orbital angular momentum
j1 (float): total angular momentum
n2 (int): principal quantum number
l2 (int): orbital angular momentum
j2 (float): total angular momentum

Returns:
float:  energy defect (SI units: J)
"""
return C_e * (
self.atom1.getEnergy(n1, l1, j1, s=self.s1)
+ self.atom2.getEnergy(n2, l2, j2, s=self.s2)
- self.atom1.getEnergy(n, l, j, s=self.s1)
- self.atom2.getEnergy(nn, ll, jj, s=self.s2)
)

def __makeRawMatrix2(
self,
n,
l,
j,
nn,
ll,
jj,
k,
lrange,
limit,
limitBasisToMj,
progressOutput=False,
debugOutput=False,
):
# limit = limit in Hz on energy defect
# k defines range of n' = [n-k, n+k]
dimension = 0

# which states/channels contribute significantly in the second order perturbation?
states = []

# original pairstate index
opi = 0

# this numbers are conserved if we use only dipole-dipole interactions
Lmod2 = (l + ll) % 2

l1start = max(l - self.interactionsUpTo, 0)
l2start = max(ll - self.interactionsUpTo, 0)

if debugOutput:
print("\n ======= Relevant states =======\n")

for n1 in xrange(max(n - k, 1), n + k + 1):
for n2 in xrange(max(nn - k, 1), nn + k + 1):
l1max = max(l + self.interactionsUpTo, lrange) + 1
l1max = min(l1max, n1 - 1)
for l1 in xrange(l1start, l1max):
l2max = max(ll + self.interactionsUpTo, lrange) + 1
l2max = min(l2max, n2 - 1)
for l2 in xrange(l2start, l2max):
j1 = l1 - self.s1
while j1 < -0.1:
j1 += 2 * self.s1
while j1 <= l1 + self.s1 + 0.1:
j2 = l2 - self.s2
while j2 < -0.1:
j2 += 2 * self.s2

while j2 <= l2 + self.s2 + 0.1:
ed = (
self.__getEnergyDefect(
n,
l,
j,
nn,
ll,
jj,
n1,
l1,
j1,
n2,
l2,
j2,
)
/ C_h
)
if (
abs(ed) < limit
and (
not (self.interactionsUpTo == 1)
or (Lmod2 == ((l1 + l2) % 2))
)
and (
(not limitBasisToMj)
or (j1 + j2 + 0.1 > self.m1 + self.m2)
)
and (
n1 >= self.atom1.groundStateN
or [n1, l1, j1]
in self.atom1.extraLevels
)
and (
n2 >= self.atom2.groundStateN
or [n2, l2, j2]
in self.atom2.extraLevels
)
):
if debugOutput:
pairState = (
"|"
+ printStateString(
n1, l1, j1, s=self.s1
)
+ ","
+ printStateString(
n2, l2, j2, s=self.s2
)
+ ">"
)
print(
pairState
+ (
"\t EnergyDefect = %.3f GHz"
% (ed * 1.0e-9)
)
)

states.append([n1, l1, j1, n2, l2, j2])

if (
n == n1
and nn == n2
and l == l1
and ll == l2
and j == j1
and jj == j2
):
opi = dimension

dimension = dimension + 1
j2 = j2 + 1.0
j1 = j1 + 1.0

if debugOutput:
print("\tMatrix dimension\t=\t", dimension)

# mat_value, mat_row, mat_column for each sparce matrix describing
couplingMatConstructor = [
[[], [], []] for i in xrange(2 * self.interactionsUpTo - 1)
]

# original pair-state (i.e. target pair state) Zeeman Shift
opZeemanShift = (
(
self.atom1.getZeemanEnergyShift(
self.l, self.j, self.m1, self.Bz, s=self.s1
)
+ self.atom2.getZeemanEnergyShift(
self.ll, self.jj, self.m2, self.Bz, s=self.s2
)
)
/ C_h
* 1.0e-9
)  # in GHz

if debugOutput:
print("\n ======= Coupling strengths (radial part only) =======\n")

if self.interactionsUpTo == 1:
maxCoupling = "dipole-dipole"
if debugOutput:
print(
"Calculating coupling (up to ",
maxCoupling,
") between the pair states",
)

for i in xrange(dimension):
ed = (
self.__getEnergyDefect(
states[opi][0],
states[opi][1],
states[opi][2],
states[opi][3],
states[opi][4],
states[opi][5],
states[i][0],
states[i][1],
states[i][2],
states[i][3],
states[i][4],
states[i][5],
)
/ C_h
* 1.0e-9
- opZeemanShift
)

pairState1 = (
"|"
+ printStateString(
states[i][0], states[i][1], states[i][2], s=self.s1
)
+ ","
+ printStateString(
states[i][3], states[i][4], states[i][5], s=self.s2
)
+ ">"
)

states[i].append(ed)  # energy defect of given state

for j in xrange(i + 1, dimension):
coupled = self.__isCoupled(
states[i][0],
states[i][1],
states[i][2],
states[i][3],
states[i][4],
states[i][5],
states[j][0],
states[j][1],
states[j][2],
states[j][3],
states[j][4],
states[j][5],
limit,
)

if states[i][0] == 24 and states[j][0] == 18:
print("\n")
print(states[i])
print(states[j])
print(coupled)

if coupled and (
abs(states[i][0] - states[j][0]) <= k
and abs(states[i][3] - states[j][3]) <= k
):
if debugOutput:
pairState2 = (
"|"
+ printStateString(
states[j][0],
states[j][1],
states[j][2],
s=self.s1,
)
+ ","
+ printStateString(
states[j][3],
states[j][4],
states[j][5],
s=self.s2,
)
+ ">"
)
print(pairState1 + " <---> " + pairState2)

couplingStregth = (
_atomLightAtomCoupling(
states[i][0],
states[i][1],
states[i][2],
states[i][3],
states[i][4],
states[i][5],
states[j][0],
states[j][1],
states[j][2],
states[j][3],
states[j][4],
states[j][5],
self.atom1,
atom2=self.atom2,
s=self.s1,
s2=self.s2,
)
/ C_h
* 1.0e-9
)

couplingMatConstructor[coupled - 2][0].append(
couplingStregth
)
couplingMatConstructor[coupled - 2][1].append(i)
couplingMatConstructor[coupled - 2][2].append(j)

exponent = coupled + 1
if debugOutput:
print(
(
"\tcoupling (C_%d/R^%d) = %.5f"
% (
exponent,
exponent,
couplingStregth * (1e6) ** (exponent),
)
),
"/R^",
exponent,
" GHz  (mu m)^",
exponent,
"\n",
)

# coupling = [1,1] dipole-dipole, [2,1]  quadrupole dipole, [2,2] quadrupole quadrupole

couplingMatArray = [
csr_matrix(
(
couplingMatConstructor[i][0],
(
couplingMatConstructor[i][1],
couplingMatConstructor[i][2],
),
),
shape=(dimension, dimension),
)
for i in xrange(len(couplingMatConstructor))
]
return states, couplingMatArray

def __initializeDatabaseForMemoization(self):
# memoization of angular parts
self.conn = sqlite3.connect(
os.path.join(self.dataFolder, "precalculated_pair.db")
)
c = self.conn.cursor()

# ANGULAR PARTS
c.execute("""DROP TABLE IF EXISTS pair_angularMatrix""")
c.execute(
"""SELECT COUNT(*) FROM sqlite_master
WHERE type='table' AND name='pair_angularMatrix';"""
)
if c.fetchone()[0] == 0:
# create table
try:
c.execute(
"""CREATE TABLE IF NOT EXISTS pair_angularMatrix
(l1 TINYINT UNSIGNED, j1_x2 TINYINT UNSIGNED,
l2 TINYINT UNSIGNED, j2_x2 TINYINT UNSIGNED,
l3 TINYINT UNSIGNED, j3_x2 TINYINT UNSIGNED,
l4 TINYINT UNSIGNED, j4_x2 TINYINT UNSIGNED,
ind INTEGER,
PRIMARY KEY (l1,j1_x2, l2,j2_x2, l3,j3_x2, l4,j4_x2)
) """
)
except sqlite3.Error as e:
print(e)
self.conn.commit()
self.savedAngularMatrixChanged = False

def __closeDatabaseForMemoization(self):
self.conn.commit()
self.conn.close()
self.conn = False

[docs]
"""
Returns Le Roy radius for initial pair-state.

Le Roy radius [#leroy]_ is defined as
:math:2(\\langle r_1^2 \\rangle^{1/2} + \\langle r_2^2 \\rangle^{1/2}),
where :math:r_1 and :math:r_2 are electron coordinates for the
first and the second atom in the initial pair-state.
Below this radius, calculations are not valid since electron
wavefunctions start to overlap.

Returns:
float: LeRoy radius measured in :math:\\mu m

References:
.. [#leroy] R.J. Le Roy, Can. J. Phys. **52**, 246 (1974)
http://www.nrcresearchpress.com/doi/abs/10.1139/p74-035
"""
step = 0.001
r1, psi1_r1 = self.atom1.radialWavefunction(
self.l,
0.5,
self.j,
self.atom1.getEnergy(self.n, self.l, self.j, s=self.s1) / 27.211,
self.atom1.alphaC ** (1 / 3.0),
2.0 * self.n * (self.n + 15.0),
step,
)

sqrt_r1_on2 = np.trapz(
np.multiply(np.multiply(psi1_r1, psi1_r1), np.multiply(r1, r1)),
x=r1,
)

r2, psi2_r2 = self.atom2.radialWavefunction(
self.ll,
0.5,
self.jj,
self.atom2.getEnergy(self.nn, self.ll, self.jj, s=self.s2) / 27.211,
self.atom2.alphaC ** (1 / 3.0),
2.0 * self.nn * (self.nn + 15.0),
step,
)

sqrt_r2_on2 = np.trapz(
np.multiply(np.multiply(psi2_r2, psi2_r2), np.multiply(r2, r2)),
x=r2,
)

return (
2.0
* (sqrt(sqrt_r1_on2) + sqrt(sqrt_r2_on2))
* (physical_constants["Bohr radius"][0] * 1.0e6)
)

[docs]
def getC6perturbatively(
self, theta, phi, nRange, energyDelta, degeneratePerturbation=False
):
r"""
Calculates :math:C_6 from second order perturbation theory.

Calculates
:math:C_6=\sum_{\rm r',r''}|\langle {\rm r',r''}|V|\
{\rm r1,r2}\rangle|^2/\Delta_{\rm r',r''}, where
:math:\Delta_{\rm r',r''}\equiv E({\rm r',r''})-E({\rm r1, r2})
When second order perturbation couples to multiple energy degenerate
states, users shold use **degenerate perturbation calculations** by
setting degeneratePerturbation=True .

This calculation is faster then full diagonalization, but it is valid
only far from the so called spaghetti region that occurs when atoms
are close to each other. In that region multiple levels are strongly
coupled, and one needs to use full diagonalization. In region where
perturbative calculation is correct, energy level shift can be
obtained as :math:V(R)=-C_6/R^6

See perturbative C6 calculations example snippet_ and for
degenerate perturbation calculation see
degenerate pertubation C6 calculation example snippet_

.. _perturbative C6 calculations example snippet:
./Rydberg_atoms_a_primer.html#Dispersion-Coefficients

.. _degenerate pertubation C6 calculation example snippet:
./ARC_3_0_introduction.html#Pertubative-C6-calculation-in-the-manifold-of-degenerate-states

Parameters:
theta (float):
orientation of inter-atomic axis with respect
to quantization axis (:math:z) in Euler coordinates
(measured in units of radian)
phi (float):
orientation of inter-atomic axis with respect
to quantization axis (:math:z) in Euler coordinates
(measured in units of radian)
nRange (int):
how much below and above the given principal quantum number
of the pair state we should be looking
energyDelta (float):
what is maximum energy difference ( :math:\Delta E/h in Hz)
between the original pair state and the other pair states that we are including in
calculation
degeneratePerturbation (bool):
optional, default False. Should one
use degenerate perturbation theory. This should be used whenever
angle between quantisation and interatomic axis is non-zero,
as well as when one considers non-stretched states.

Returns:
float: if **degeneratePerturbation=False**, returns
:math:C_6 measured in :math:\text{GHz }\mu\text{m}^6;
if **degeneratePerturbation=True**, returns array of
:math:C_6 measured in :math:\text{GHz }\mu\text{m}^6
AND array of corresponding eigenvectors in
:math:\{m_{j_1}=-j_1, \ldots, m_{j_1} = +j1\}\bigotimes \
\{ m_{j_2}=-j_2, \ldots, m_{j_2} = +j2\}
basis

Example:
If we want to quickly calculate :math:C_6 for two Rubidium
atoms in state :math:62 D_{3/2} m_j=3/2, positioned in space
along the shared quantization axis::

from arc import *
calculation = PairStateInteractions(Rubidium(), 62, 2, 1.5, 62, 2, 1.5, 1.5, 1.5)
c6 = calculation.getC6perturbatively(0,0, 5, 25e9)
print "C_6 = %.0f GHz (mu m)^6" % c6

Which returns::

C_6 = 767 GHz (mu m)^6

Quick calculation of angular anisotropy of for Rubidium
:math:D_{2/5},m_j=5/2 states::

# Rb 60 D_{2/5}, mj=2.5 , 60 D_{2/5}, mj=2.5 pair state
calculation1 = PairStateInteractions(Rubidium(), 60, 2, 2.5, 60, 2, 2.5, 2.5, 2.5)
# list of atom orientations
thetaList = np.linspace(0,pi,30)
# do calculation of C6 pertubatively for all atom orientations
c6 = []
for theta in thetaList:
value = calculation1.getC6perturbatively(theta,0,5,25e9)
c6.append(value)
print ("theta = %.2f * pi \tC6 = %.2f GHz  mum^6" % (theta/pi,value))
# plot results
plot(thetaList/pi,c6,"b-")
title("Rb, pairstate  60 $D_{5/2},m_j = 5/2$, 60 $D_{5/2},m_j = 5/2$")
xlabel(r"$\Theta /\pi$")
ylabel(r"$C_6$ (GHz $\mu$m${}^6$")
show()

"""
self.__initializeDatabaseForMemoization()

# ========= START OF THE MAIN CODE ===========

# wigner D matrix allows calculations with arbitrary orientation of
# the two atoms
wgd = WignerDmatrix(theta, phi)

# any conservation?
# this numbers are conserved if we use only dipole-dipole interactions
Lmod2 = (self.l + self.ll) % 2

# find nearby states

lmin1 = self.l - 1
if lmin1 < -0.1:
lmin1 = 1
lmin2 = self.ll - 1
if lmin2 < -0.1:
lmin2 = 1

interactionMatrix = np.zeros(
(
round((2 * self.j + 1) * (2 * self.jj + 1)),
round((2 * self.j + 1) * (2 * self.jj + 1)),
),
dtype=np.complex128,
)

for n1 in xrange(max(self.n - nRange, 1), self.n + nRange + 1):
for n2 in xrange(max(self.nn - nRange, 1), self.nn + nRange + 1):
lmax1 = min(self.l + 2, n1)
for l1 in xrange(lmin1, lmax1, 2):
lmax2 = min(self.ll + 2, n2)
for l2 in xrange(lmin2, lmax2, 2):
if (l1 + l2) % 2 == Lmod2:
j1 = l1 - self.s1
while j1 < -0.1:
j1 += 2 * self.s1
while j1 <= l1 + self.s1 + 0.1:
j2 = l2 - self.s2
while j2 < -0.1:
j2 += 2 * self.s2

while j2 <= l2 + self.s2 + 0.1:
coupled = self.__isCoupled(
self.n,
self.l,
self.j,
self.nn,
self.ll,
self.jj,
n1,
l1,
j1,
n2,
l2,
j2,
energyDelta,
)
if (
coupled
and (
not (self.interactionsUpTo == 1)
or (Lmod2 == ((l1 + l2) % 2))
)
and (
n1 >= self.atom1.groundStateN
or [n1, l1, j1]
in self.atom1.extraLevels
)
and (
n2 >= self.atom2.groundStateN
or [n2, l2, j2]
in self.atom2.extraLevels
)
):
energyDefect = (
self.__getEnergyDefect(
self.n,
self.l,
self.j,
self.nn,
self.ll,
self.jj,
n1,
l1,
j1,
n2,
l2,
j2,
)
/ C_h
)
energyDefect = (
energyDefect * 1.0e-9
)  # GHz
if abs(energyDefect) < 1e-10:
raise ValueError(
"The requested pair-state "
"is dipole coupled resonatly "
"(energy defect = 0)"
"to other pair-states"
"Aborting pertubative "
"calculation."
"(This usually happens for "
"high-L states for which "
"identical quantum defects give "
"raise to degeneracies, making "
"total L ultimately not "
"conserved quantum number) "
)

# calculate radial part
couplingStregth = (
_atomLightAtomCoupling(
self.n,
self.l,
self.j,
self.nn,
self.ll,
self.jj,
n1,
l1,
j1,
n2,
l2,
j2,
self.atom1,
atom2=self.atom2,
s=self.s1,
s2=self.s2,
)
* (1.0e-9 * (1.0e6) ** 3 / C_h)
)  # GHz / mum^3

d = self.__getAngularMatrix_M(
self.l,
self.j,
self.ll,
self.jj,
l1,
j1,
l2,
j2,
)

interactionMatrix += (
d.conj().T.dot(d)
* abs(couplingStregth) ** 2
/ energyDefect
)

j2 = j2 + 1.0
j1 = j1 + 1.0

rotationMatrix = np.kron(
wgd.get(self.j).toarray(), wgd.get(self.jj).toarray()
)

interactionMatrix = rotationMatrix.dot(
interactionMatrix.dot(rotationMatrix.conj().T)
)
# ========= END OF THE MAIN CODE ===========
self.__closeDatabaseForMemoization()

value, vectors = np.linalg.eigh(interactionMatrix)
vectors = vectors.T
stateCom = compositeState(
singleAtomState(self.j, self.m1), singleAtomState(self.jj, self.m2)
).T

if not degeneratePerturbation:
for i, v in enumerate(vectors):
if abs(np.vdot(v, stateCom)) > 1 - 1e-9:
return value[i]
#    else:
#        print(np.vdot(v, stateCom))
# if initial state is not eigen state print warning and return
# results for eigenstates, and eigenstate composition
"""
print("WARNING: Requested state is not eigenstate when dipole-dipole "
"interactions and/or relative position of atoms are "
"taken into account.\n"
"We will use degenerate pertubative theory to correctly "
"calculate C6.\n"
"Method will return values AND eigenvectors in basis \n"
"{mj1 = -j1, ... , mj1 = +j1} x {mj2 = -j2, ... , m2 = +j2}, "
"where x denotes Kronecker product\n"
"To not see this warning request explicitly "
"degeneratePerturbation=True in call of this method.\n")
"""
# print(stateCom.conj().dot(interactionMatrix.dot(stateCom.T)))
# print(stateCom.conj().dot(interactionMatrix.dot(stateCom.T)).shape)
return np.real(
stateCom.conj().dot(interactionMatrix.dot(stateCom.T))[0][0]
)
return np.real(value), vectors

[docs]
def defineBasis(
self,
theta,
phi,
nRange,
lrange,
energyDelta,
Bz=0,
progressOutput=False,
debugOutput=False,
):
r"""
Finds relevant states in the vicinity of the given pair-state

Finds relevant pair-state basis and calculates interaction matrix.
Pair-state basis is saved in :obj:basisStates.
Interaction matrix is saved in parts depending on the scaling with
distance. Diagonal elements :obj:matDiagonal, correponding to
relative energy defects of the pair-states, don't change with
interatomic separation. Off diagonal elements can depend
on distance as :math:R^{-3}, R^{-4} or :math:R^{-5},
corresponding to dipole-dipole (:math:C_3 ), dipole-qudrupole
(:math:C_4 ) and quadrupole-quadrupole coupling (:math:C_5 )
respectively. These parts of the matrix are stored in
:obj:PairStateInteractions.matR
in that order. I.e. :obj:matR[0]
stores dipole-dipole coupling
(:math:\propto R^{-3}),
:obj:matR[1] stores dipole-quadrupole
couplings etc.

Parameters:
theta (float):  relative orientation of the two atoms
(see figure on top of the page), range 0 to :math:\pi
phi (float): relative orientation of the two atoms (see figure
on top of the page), range 0 to :math:2\pi
nRange (int): how much below and above the given principal
quantum number of the pair state we should be looking?
lrange (int): what is the maximum angular orbital momentum
state that we are including in calculation
energyDelta (float): what is maximum energy difference (
:math:\Delta E/h in Hz)
between the original pair state and the other pair states
that we are including in calculation
Bz (float): optional, magnetic field directed along z-axis in
units of Tesla. Calculation will be correct only for weak
magnetic fields, where paramagnetic term is much stronger
then diamagnetic term. Diamagnetic term is neglected.
progressOutput (bool): optional, False by default. If true,
prints information about the progress of the calculation.
debugOutput (bool): optional, False by default. If true,
similarly to progressOutput=True, this will print
information about the progress of calculations, but with
more verbose output.

:obj:arc.alkali_atom_functions.saveCalculation and
:obj:arc.alkali_atom_functions.loadSavedCalculation for
information on saving intermediate results of calculation for
later use.
"""

self.__initializeDatabaseForMemoization()

# save call parameters
self.theta = theta
self.phi = phi
self.nRange = nRange
self.lrange = lrange
self.energyDelta = energyDelta
self.Bz = Bz

self.basisStates = []

# wignerDmatrix
wgd = WignerDmatrix(theta, phi)

limitBasisToMj = False
if theta < 0.001:
limitBasisToMj = True  # Mj will be conserved in calculations

originalMj = self.m1 + self.m2

self.channel, self.coupling = self.__makeRawMatrix2(
self.n,
self.l,
self.j,
self.nn,
self.ll,
self.jj,
nRange,
lrange,
energyDelta,
limitBasisToMj,
progressOutput=progressOutput,
debugOutput=debugOutput,
)

self.atom1.updateDipoleMatrixElementsFile()
self.atom2.updateDipoleMatrixElementsFile()

# generate all the states (with mj principal quantum number)

# opi = original pairstate index
opi = 0

# NEW FOR SPACE MATRIX
self.index = np.zeros(len(self.channel) + 1, dtype=np.int16)

for i in xrange(len(self.channel)):
self.index[i] = len(self.basisStates)

stateCoupled = self.channel[i]

for m1c in np.linspace(
stateCoupled[2],
-stateCoupled[2],
round(1 + 2 * stateCoupled[2]),
):
for m2c in np.linspace(
stateCoupled[5],
-stateCoupled[5],
round(1 + 2 * stateCoupled[5]),
):
if (not limitBasisToMj) or (
abs(originalMj - m1c - m2c) < 0.1
):
self.basisStates.append(
[
stateCoupled[0],
stateCoupled[1],
stateCoupled[2],
m1c,
stateCoupled[3],
stateCoupled[4],
stateCoupled[5],
m2c,
]
)
self.matrixElement.append(i)

if (
abs(stateCoupled[0] - self.n) < 0.1
and abs(stateCoupled[1] - self.l) < 0.1
and abs(stateCoupled[2] - self.j) < 0.1
and abs(m1c - self.m1) < 0.1
and abs(stateCoupled[3] - self.nn) < 0.1
and abs(stateCoupled[4] - self.ll) < 0.1
and abs(stateCoupled[5] - self.jj) < 0.1
and abs(m2c - self.m2) < 0.1
):
opi = len(self.basisStates) - 1
if self.index[i] == len(self.basisStates):
print(stateCoupled)
self.index[-1] = len(self.basisStates)

if progressOutput or debugOutput:
print("\nCalculating Hamiltonian matrix...\n")

dimension = len(self.basisStates)
if progressOutput or debugOutput:
print("\n\tmatrix (dimension ", dimension, ")\n")

# INITIALIZING MATICES
# all (sparce) matrices will be saved in csr format
# value, row, column
matDiagonalConstructor = [[], [], []]

matRConstructor = [
[[], [], []] for i in xrange(self.interactionsUpTo * 2 - 1)
]

matRIndex = 0
for c in self.coupling:
progress = 0.0
for ii in xrange(len(self.channel)):
if progressOutput:
dim = len(self.channel)
progress += (dim - ii) * 2 - 1
sys.stdout.write(
"\rMatrix R%d %.1f %% (state %d of %d)"
% (
matRIndex + 3,
float(progress) / float(dim**2) * 100.0,
ii + 1,
len(self.channel),
)
)
sys.stdout.flush()

ed = self.channel[ii][6]

# solves problems with exactly degenerate basisStates
degeneracyOffset = 0.00000001

i = self.index[ii]
dMatrix1 = wgd.get(self.basisStates[i][2])
dMatrix2 = wgd.get(self.basisStates[i][6])

for i in xrange(self.index[ii], self.index[ii + 1]):
statePart1 = singleAtomState(
self.basisStates[i][2], self.basisStates[i][3]
)
statePart2 = singleAtomState(
self.basisStates[i][6], self.basisStates[i][7]
)
# rotate individual states

statePart1 = dMatrix1.T.conjugate().dot(statePart1)
statePart2 = dMatrix2.T.conjugate().dot(statePart2)

stateCom = compositeState(statePart1, statePart2)

if matRIndex == 0:
zeemanShift = (
(
self.atom1.getZeemanEnergyShift(
self.basisStates[i][1],
self.basisStates[i][2],
self.basisStates[i][3],
self.Bz,
s=self.s1,
)
+ self.atom2.getZeemanEnergyShift(
self.basisStates[i][5],
self.basisStates[i][6],
self.basisStates[i][7],
self.Bz,
s=self.s2,
)
)
/ C_h
* 1.0e-9
)  # in GHz
matDiagonalConstructor[0].append(
ed + zeemanShift + degeneracyOffset
)
degeneracyOffset += 0.00000001
matDiagonalConstructor[1].append(i)
matDiagonalConstructor[2].append(i)

for dataIndex in xrange(c.indptr[ii], c.indptr[ii + 1]):
jj = c.indices[dataIndex]

j = self.index[jj]
dMatrix3 = wgd.get(self.basisStates[j][2])
dMatrix4 = wgd.get(self.basisStates[j][6])

if self.index[jj] != self.index[jj + 1]:
d = self.__getAngularMatrix_M(
self.basisStates[i][1],
self.basisStates[i][2],
self.basisStates[i][5],
self.basisStates[i][6],
self.basisStates[j][1],
self.basisStates[j][2],
self.basisStates[j][5],
self.basisStates[j][6],
)
secondPart = d.dot(stateCom)
else:
print(" - - - ", self.channel[jj])

for j in xrange(self.index[jj], self.index[jj + 1]):
statePart1 = singleAtomState(
self.basisStates[j][2], self.basisStates[j][3]
)
statePart2 = singleAtomState(
self.basisStates[j][6], self.basisStates[j][7]
)
# rotate individual states

statePart1 = dMatrix3.T.conjugate().dot(statePart1)
statePart2 = dMatrix4.T.conjugate().dot(statePart2)
# composite state of two atoms
stateCom2 = compositeState(statePart1, statePart2)

angularFactor = stateCom2.T.conjugate().dot(
secondPart
)
if abs(self.phi) < 1e-9:
angularFactor = angularFactor[0, 0].real
else:
angularFactor = angularFactor[0, 0]

if abs(angularFactor) > 1.0e-5:
matRConstructor[matRIndex][0].append(
)
matRConstructor[matRIndex][1].append(i)
matRConstructor[matRIndex][2].append(j)

matRConstructor[matRIndex][0].append(
)
matRConstructor[matRIndex][1].append(j)
matRConstructor[matRIndex][2].append(i)
matRIndex += 1
if progressOutput or debugOutput:
print("\n")

self.matDiagonal = csr_matrix(
(
matDiagonalConstructor[0],
(matDiagonalConstructor[1], matDiagonalConstructor[2]),
),
shape=(dimension, dimension),
)

self.matR = [
csr_matrix(
(
matRConstructor[i][0],
(matRConstructor[i][1], matRConstructor[i][2]),
),
shape=(dimension, dimension),
)
for i in xrange(self.interactionsUpTo * 2 - 1)
]

self.originalPairStateIndex = opi

self.__updateAngularMatrixElementsFile()
self.__closeDatabaseForMemoization()

[docs]
def diagonalise(
self,
rangeR,
noOfEigenvectors,
drivingFromState=[0, 0, 0, 0, 0],
eigenstateDetuning=0.0,
sortEigenvectors=False,
progressOutput=False,
debugOutput=False,
):
r"""
Finds eigenstates in atom pair basis.

ARPACK ( :obj:scipy.sparse.linalg.eigsh) calculation of the
noOfEigenvectors eigenvectors closest to the original state. If
drivingFromState is specified as [n,l,j,mj,q] coupling between
the pair-states and the situation where one of the atoms in the
pair state basis is in :math:|n,l,j,m_j\rangle state due to
driving with a laser field that drives :math:q transition
(+1,0,-1 for :math:\sigma^-, :math:\pi and :math:\sigma^+
transitions respectively) is calculated and marked by the
colourmaping these values on the obtained eigenvectors.

Parameters:
rangeR ( :obj:array): Array of values for distance between
the atoms (in :math:\mu m) for which we want to calculate
eigenstates.
noOfEigenvectors (int): number of eigen vectors closest to the
energy of the original (unperturbed) pair state. Has to be
smaller then the total number of states.
eigenstateDetuning (float, optional): Default is 0. This
specifies detuning from the initial pair-state (in Hz)
around which we want to find noOfEigenvectors
eigenvectors. This is useful when looking only for couple
of off-resonant features.
drivingFromState ([int,int,float,float,int]): Optional. State
of one of the atoms from the original pair-state basis
from which we try to drive to the excited pair-basis
manifold, **assuming that the first of the two atoms is
already excited to the specified Rydberg state**.
By default, program will calculate just
contribution of the original pair-state in the eigenstates
obtained by diagonalization, and will highlight it's
admixure by colour mapping the obtained eigenstates plot.
State is specified as :math:[n,\ell,j,mj, d]
where :math:d is +1, 0 or
-1 for driving :math:\sigma^- , :math:\pi
and :math:\sigma^+ transitions respectively.
sortEigenvectors(bool): optional, False by default. Tries to
sort eigenvectors so that given eigen vector index
corresponds to adiabatically changing eigenstate, as
detirmined by maximising overlap between old and new
eigenvectors.
progressOutput (bool): optional, False by default. If true,
prints information about the progress of the calculation.
debugOutput (bool): optional, False by default. If true,
similarly to progressOutput=True, this will print
information about the progress of calculations, but with
more verbose output.
"""

self.r = np.sort(rangeR)
dimension = len(self.basisStates)

self.noOfEigenvectors = noOfEigenvectors

# energy of the state - to be calculated
self.y = []
# how much original state is contained in this eigenvector
self.highlight = []
# what are the dominant contributing states?
self.composition = []

if noOfEigenvectors >= dimension - 1:
noOfEigenvectors = dimension - 1
print(
"Warning: Requested number of eigenvectors >=dimension-1\n \
ARPACK can only find up to dimension-1 eigenvectors, where\
dimension is matrix dimension.\n"
)
if noOfEigenvectors < 1:
return

coupling = []
self.maxCoupling = 0.0
self.maxCoupledStateIndex = 0
if drivingFromState[0] != 0:
self.drivingFromState = drivingFromState
if progressOutput:
print("Finding coupling strengths")
# get first what was the state we are calculating coupling with
state1 = drivingFromState
n1 = round(state1[0])
l1 = round(state1[1])
j1 = state1[2]
m1 = state1[3]
q = state1[4]

for i in xrange(dimension):
thisCoupling = 0.0

if (
round(abs(self.basisStates[i][5] - l1)) == 1
and abs(
self.basisStates[i][0]
- self.basisStates[self.originalPairStateIndex][0]
)
< 0.1
and abs(
self.basisStates[i][1]
- self.basisStates[self.originalPairStateIndex][1]
)
< 0.1
and abs(
self.basisStates[i][2]
- self.basisStates[self.originalPairStateIndex][2]
)
< 0.1
and abs(
self.basisStates[i][3]
- self.basisStates[self.originalPairStateIndex][3]
)
< 0.1
):
state2 = self.basisStates[i]
n2 = round(state2[0 + 4])
l2 = round(state2[1 + 4])
j2 = state2[2 + 4]
m2 = state2[3 + 4]
if debugOutput:
print(
n1,
" ",
l1,
" ",
j1,
" ",
m1,
" ",
n2,
" ",
l2,
" ",
j2,
" ",
m2,
" q=",
q,
)
print(self.basisStates[i])
dme = self.atom2.getDipoleMatrixElement(
n1, l1, j1, m1, n2, l2, j2, m2, q, s=self.s2
)
thisCoupling += dme

thisCoupling = abs(thisCoupling) ** 2
if thisCoupling > self.maxCoupling:
self.maxCoupling = thisCoupling
self.maxCoupledStateIndex = i
if (thisCoupling > 0.000001) and debugOutput:
print(
"original pairstate index = ",
self.originalPairStateIndex,
)
print("this pairstate index = ", i)
print("state itself ", self.basisStates[i])
print("coupling = ", thisCoupling)
coupling.append(thisCoupling)

print("Maximal coupling from a state")
print("is to a state ", self.basisStates[self.maxCoupledStateIndex])
print("is equal to %.3e a_0 e" % self.maxCoupling)

if progressOutput:
print("\n\nDiagonalizing interaction matrix...\n")

rvalIndex = 0.0
previousEigenvectors = []

for rval in self.r:
if progressOutput:
sys.stdout.write(
"\r%d%%" % (rvalIndex / len(self.r - 1) * 100.0)
)
sys.stdout.flush()
rvalIndex += 1.0

# calculate interaction matrix

m = self.matDiagonal

rX = (rval * 1.0e-6) ** 3
for matRX in self.matR:
m = m + matRX / rX
rX *= rval * 1.0e-6

# uses ARPACK algorithm to find only noOfEigenvectors eigenvectors
# sigma specifies center frequency (in GHz)
ev, egvector = eigsh(
m,
noOfEigenvectors,
sigma=eigenstateDetuning * 1.0e-9,
which="LM",
tol=1e-6,
)

if sortEigenvectors:
# Find which eigenvectors overlap most with eigenvectors from
# previous diagonalisatoin, in order to find "adiabatic"
# continuation for the respective states

if previousEigenvectors == []:
previousEigenvectors = np.copy(egvector)
rowPicked = [False for i in range(len(ev))]
columnPicked = [False for i in range(len(ev))]

stateOverlap = np.zeros((len(ev), len(ev)))
for i in range(len(ev)):
for j in range(len(ev)):
stateOverlap[i, j] = (
np.vdot(egvector[:, i], previousEigenvectors[:, j])
** 2
)

sortedOverlap = np.dstack(
np.unravel_index(
np.argsort(stateOverlap.ravel()), (len(ev), len(ev))
)
)[0]

sortedEigenvaluesOrder = np.zeros(len(ev), dtype=np.int32)
j = len(ev) ** 2 - 1
for i in range(len(ev)):
while (
rowPicked[sortedOverlap[j, 0]]
or columnPicked[sortedOverlap[j, 1]]
):
j -= 1
rowPicked[sortedOverlap[j, 0]] = True
columnPicked[sortedOverlap[j, 1]] = True
sortedEigenvaluesOrder[sortedOverlap[j, 1]] = sortedOverlap[
j, 0
]

egvector = egvector[:, sortedEigenvaluesOrder]
ev = ev[sortedEigenvaluesOrder]
previousEigenvectors = np.copy(egvector)

self.y.append(ev)

if drivingFromState[0] < 0.1:
# if we've defined from which state we are driving
sh = []
comp = []
for i in xrange(len(ev)):
sh.append(
abs(egvector[self.originalPairStateIndex, i]) ** 2
)
comp.append(self._stateComposition(egvector[:, i]))
self.highlight.append(sh)
self.composition.append(comp)
else:
sh = []
comp = []
for i in xrange(len(ev)):
sumCoupledStates = 0.0
for j in xrange(dimension):
sumCoupledStates += (
abs(coupling[j] / self.maxCoupling)
* abs(egvector[j, i]) ** 2
)
comp.append(self._stateComposition(egvector[:, i]))
sh.append(sumCoupledStates)
self.highlight.append(sh)
self.composition.append(comp)

# end of FOR loop over inter-atomic dinstaces

[docs]
def exportData(self, fileBase, exportFormat="csv"):
"""
Exports PairStateInteractions calculation data.

Only supported format (selected by default) is .csv in a
human-readable form with a header that saves details of calculation.
Function saves three files: 1) filebase _r.csv;
2) filebase _energyLevels
3) filebase _highlight

For more details on the format, see header of the saved files.

Parameters:
filebase (string): filebase for the names of the saved files
without format extension. Add as a prefix a directory path
if necessary (e.g. saving outside the current working directory)
exportFormat (string): optional. Format of the exported file. Currently
only .csv is supported but this can be extended in the future.
"""
fmt = "on %Y-%m-%d @ %H:%M:%S"
ts = datetime.datetime.now().strftime(fmt)

commonHeader = "Export from Alkali Rydberg Calculator (ARC) %s.\n" % ts
"\n *** Pair State interactions for %s %s m_j = %d/2 , %s %s m_j = %d/2 pair-state. ***\n\n"
% (
self.atom1.elementName,
printStateString(self.n, self.l, self.j),
round(2.0 * self.m1),
self.atom2.elementName,
printStateString(self.nn, self.ll, self.jj),
round(2.0 * self.m2),
)
)
if self.interactionsUpTo == 1:
commonHeader += " - Pair-state interactions included up to dipole-dipole coupling.\n"
elif self.interactionsUpTo == 2:
commonHeader += " - Pair-state interactions included up to quadrupole-quadrupole coupling.\n"
" - Pair-state interactions calculated for manifold with spin angular momentum s1 = %.1d s2 = %.1d .\n"
% (self.s1, self.s2)
)

if hasattr(self, "theta"):
commonHeader += " - Atom orientation:\n"
commonHeader += "      theta (polar angle) = %.5f x pi\n" % (
self.theta / pi
)
commonHeader += "      phi (azimuthal angle) = %.5f x pi\n" % (
self.phi / pi
)
commonHeader += " - Calculation basis includes:\n"
"      States with principal quantum number in range [%d-%d]x[%d-%d],\n"
% (
self.n - self.nRange,
self.n + self.nRange,
self.nn - self.nRange,
self.nn + self.nRange,
)
)
"      AND whose orbital angular momentum (l) is in range [%d-%d] (i.e. %s-%s),\n"
% (
0,
self.lrange,
printStateLetter(0),
printStateLetter(self.lrange),
)
)
"      AND whose pair-state energy difference is at most %.3f GHz\n"
% (self.energyDelta / 1.0e9)
)
commonHeader += "      (energy difference is measured relative to original pair-state).\n"
else:
commonHeader += " ! Atom orientation and basis not yet set (this is set in defineBasis method).\n"

if hasattr(self, "noOfEigenvectors"):
" - Finding %d eigenvectors closest to the given pair-state\n"
% self.noOfEigenvectors
)

if self.drivingFromState[0] < 0.1:
" - State highlighting based on the relative contribution \n"
+ "   of the original pair-state in the eigenstates obtained by diagonalization.\n"
)
else:
" - State highlighting based on the relative driving strength \n"
+ "   to a given energy eigenstate (energy level) from state\n"
+ "   %s m_j =%d/2 with polarization q=%d.\n"
% (
printStateString(*self.drivingFromState[0:3]),
round(2.0 * self.drivingFromState[3]),
self.drivingFromState[4],
)
)

else:
commonHeader += " ! Energy levels not yet found (this is done by calling diagonalise method).\n"

if exportFormat == "csv":
print("Exporting StarkMap calculation results as .csv ...")

commonHeader += " - Export consists of three (3) files:\n"
commonHeader += "       1) %s,\n" % (
fileBase + "_r." + exportFormat
)
commonHeader += "       2) %s,\n" % (
fileBase + "_energyLevels." + exportFormat
)
commonHeader += "       3) %s.\n\n" % (
fileBase + "_highlight." + exportFormat
)

filename = fileBase + "_r." + exportFormat
np.savetxt(
filename,
self.r,
fmt="%.18e",
delimiter=", ",
newline="\n",
+ " - - - Interatomic distance, r (\\mu m) - - -"
),
)
print("   Interatomic distances (\\mu m) saved in %s" % filename)

filename = fileBase + "_energyLevels." + exportFormat
headerDetails = " NOTE : Each row corresponds to eigenstates for a single specified interatomic distance"
np.savetxt(
filename,
self.y,
fmt="%.18e",
delimiter=", ",
newline="\n",
commonHeader + " - - - Energy (GHz) - - -\n" + headerDetails
),
)
print(
"   Lists of energies (in GHz relative to the original pair-state energy)"
+ (" saved in %s" % filename)
)

filename = fileBase + "_highlight." + exportFormat
np.savetxt(
filename,
self.highlight,
fmt="%.18e",
delimiter=", ",
newline="\n",
+ " - - - Highlight value (rel.units) - - -\n"
),
)
print("   Highlight values saved in %s" % filename)

print("... data export finished!")
else:
raise ValueError("Unsupported export format (.%s)." % format)

def _stateComposition(self, stateVector):
contribution = np.absolute(stateVector)
order = np.argsort(contribution, kind="heapsort")
index = -1
totalContribution = 0
value = "$" while (index > -5) and (totalContribution < 0.95): i = order[index] if index != -1 and ( stateVector[i].real > 0 or abs(stateVector[i].imag) > 1e-9 ): value += "+" if abs(self.phi) < 1e-9: value = ( value + ("%.2f" % stateVector[i]) + self._addState(*self.basisStates[i]) ) else: value = ( value + ( "(%.2f+i%.2f)" % (stateVector[i].real, stateVector[i].imag) ) + self._addState(*self.basisStates[i]) ) totalContribution += contribution[i] ** 2 index -= 1 if totalContribution < 0.999: value += "+\\ldots" return value + "$"

def _addState(self, n1, l1, j1, mj1, n2, l2, j2, mj2):
stateString = ""
if abs(self.s1 - 0.5) < 0.1:
# Alkali atom
stateString += "|%s %d/2" % (
printStateStringLatex(n1, l1, j1, s=self.s1),
round(2 * mj1),
)
else:
# divalent atoms
stateString += "|%s %d" % (
printStateStringLatex(n1, l1, j1, s=self.s1),
round(mj1),
)

if abs(self.s2 - 0.5) < 0.1:
# Alkali atom
stateString += ",%s %d/2\\rangle" % (
printStateStringLatex(n2, l2, j2, s=self.s2),
round(2 * mj2),
)
else:
# divalent atom
stateString += ",%s %d\\rangle" % (
printStateStringLatex(n2, l2, j2, s=self.s2),
round(mj2),
)
return stateString

[docs]
def plotLevelDiagram(
self, highlightColor="red", highlightScale="linear", units="GHz"
):
"""
Plots pair state level diagram

Call :obj:showPlot if you want to display a plot afterwards.

Parameters:
highlightColor (string): optional, specifies the colour used
for state highlighting
highlightScale (string): optional, specifies scaling of
state highlighting. Default is 'linear'. Use 'log-2' or
'log-3' for logarithmic scale going down to 1e-2 and 1e-3
respectively. Logarithmic scale is useful for spotting
units (:obj:char,optional): possible values {'*GHz*','cm','eV'};
[case insensitive] if value is 'GHz' (default), diagram will
be plotted as energy :math:/h in units of GHz; if the
string contains 'cm' diagram will be plotted in energy units
cm :math:{}^{-1}; if the value is 'eV', diagram
will be plotted as energy in units eV.
"""

rvb = LinearSegmentedColormap.from_list(
"mymap", ["0.9", highlightColor]
)

if units.lower() == "ev":
self.units = "eV"
self.scaleFactor = 1e9 * C_h / C_e
eLabel = ""
elif units.lower() == "ghz":
self.units = "GHz"
self.scaleFactor = 1
eLabel = "/h"
elif "cm" in units.lower():
self.units = "cm$^{-1}$"
self.scaleFactor = 1e9 / (C_c * 100)
eLabel = "/(h c)"

if highlightScale == "linear":
cNorm = matplotlib.colors.Normalize(vmin=0.0, vmax=1.0)
elif highlightScale == "log-2":
cNorm = matplotlib.colors.LogNorm(vmin=1e-2, vmax=1)
elif highlightScale == "log-3":
cNorm = matplotlib.colors.LogNorm(vmin=1e-3, vmax=1)
else:
raise ValueError(
"Only 'linear', 'log-2' and 'log-3' are valid "
"inputs for highlightScale"
)

print(" Now we are plotting...")
self.fig, self.ax = plt.subplots(1, 1, figsize=(11.5, 5.0))

self.y = np.array(self.y)
self.highlight = np.array(self.highlight)

colorfulX = []
colorfulY = []
colorfulState = []

for i in xrange(len(self.r)):
for j in xrange(len(self.y[i])):
colorfulX.append(self.r[i])
colorfulY.append(self.y[i][j])
colorfulState.append(self.highlight[i][j])

colorfulState = np.array(colorfulState)
sortOrder = colorfulState.argsort(kind="heapsort")
colorfulX = np.array(colorfulX)
colorfulY = np.array(colorfulY)

colorfulX = colorfulX[sortOrder]
colorfulY = colorfulY[sortOrder]
colorfulState = colorfulState[sortOrder]

self.ax.scatter(
colorfulX,
colorfulY * self.scaleFactor,
s=10,
c=colorfulState,
linewidth=0,
norm=cNorm,
cmap=rvb,
zorder=2,
picker=5,
)
cax = self.fig.add_axes([0.91, 0.1, 0.02, 0.8])
cb = matplotlib.colorbar.ColorbarBase(cax, cmap=rvb, norm=cNorm)

if self.drivingFromState[0] == 0:
# colouring is based on the contribution of the original pair state here
label = ""
if abs(self.s1 - 0.5) < 0.1:
# Alkali atom
label += r"$|\langle %s m_j=%d/2 " % ( printStateStringLatex(self.n, self.l, self.j), round(2.0 * self.m1), ) else: # divalent atom label += r"$|\langle %s m_j=%d " % (
printStateStringLatex(self.n, self.l, self.j, s=self.s1),
round(self.m1),
)

if abs(self.s2 - 0.5) < 0.1:
# Alkali atom
label += r", %s m_j=%d/2 | \mu \rangle |^2$" % ( printStateStringLatex(self.nn, self.ll, self.jj), round(2.0 * self.m2), ) else: # divalent atom label += r", %s m_j=%d | \mu \rangle |^2$" % (
printStateStringLatex(self.nn, self.ll, self.jj, s=self.s2),
round(self.m2, 0),
)

cb.set_label(label)
else:
# colouring is based on the coupling to different states
cb.set_label(r"$(\Omega_\mu/\Omega)^2$")

self.ax.set_xlabel(r"Interatomic distance, $R$ ($\mu$m)")
self.ax.set_ylabel(
r"Pair-state relative energy, $\Delta E %s$ (%s)"
% (eLabel, self.units)
)

[docs]
def savePlot(self, filename="PairStateInteractions.pdf"):
"""
Saves plot made by :obj:PairStateInteractions.plotLevelDiagram

Args:
filename (:obj:str, optional): file location where the plot
should be saved
"""
if self.fig != 0:
self.fig.savefig(filename, bbox_inches="tight")
else:
print("Error while saving a plot: nothing is plotted yet")
return 0

[docs]
def showPlot(self, interactive=True):
"""
Shows level diagram printed by
:obj:PairStateInteractions.plotLevelDiagram

By default, it will output interactive plot, which means that
clicking on the state will show the composition of the clicked
state in original basis (dominant elements)

Args:
interactive (bool): optional, by default it is True. If true,
plotted graph will be interactive, i.e. users can click
on the state to identify the state composition

Note:
interactive=True has effect if the graphs are explored in usual
matplotlib pop-up windows. It doesn't have effect on inline
plots in Jupyter (IPython) notebooks.

"""
if interactive:
self.ax.set_title("Click on state to see state composition")
self.clickedPoint = 0
self.fig.canvas.draw()
self.fig.canvas.mpl_connect("pick_event", self._onPick)

plt.show()
return 0

def _onPick(self, event):
if isinstance(event.artist, matplotlib.collections.PathCollection):
x = event.mouseevent.xdata
y = event.mouseevent.ydata / self.scaleFactor

i = np.searchsorted(self.r, x)
if i == len(self.r):
i -= 1
if (i > 0) and (abs(self.r[i - 1] - x) < abs(self.r[i] - x)):
i -= 1

j = 0
for jj in xrange(len(self.y[i])):
if abs(self.y[i][jj] - y) < abs(self.y[i][j] - y):
j = jj

# now choose the most higlighted state in this area
distance = abs(self.y[i][j] - y) * 1.5
for jj in xrange(len(self.y[i])):
if abs(self.y[i][jj] - y) < distance and (
abs(self.highlight[i][jj]) > abs(self.highlight[i][j])
):
j = jj

if self.clickedPoint != 0:
self.clickedPoint.remove()

(self.clickedPoint,) = self.ax.plot(
[self.r[i]],
[self.y[i][j] * self.scaleFactor],
"bs",
linewidth=0,
zorder=3,
)

self.ax.set_title(
"State = "
+ self.composition[i][j]
+ ("   Colourbar = %.2f" % self.highlight[i][j]),
fontsize=11,
)

event.canvas.draw()

[docs]
def getC6fromLevelDiagram(
self, rStart, rStop, showPlot=False, minStateContribution=0.0
):
"""
Finds :math:C_6 coefficient for original pair state.

Function first finds for each distance in the range
[ rStart , rStop ] the eigen state with highest contribution of
the original state. One can set optional parameter
minStateContribution to value in the range [0,1), so that function
finds only states if they have contribution of the original state
that is bigger then minStateContribution.

Once original pair-state is found in the range of interatomic
distances, from smallest rStart to the biggest rStop, function
will try to perform fitting of the corresponding state energy
:math:E(R) at distance :math:R to the function
:math:A+C_6/R^6 where :math:A is some offset.

Args:
rStart (float): smallest inter-atomic distance to be used for fitting
rStop (float): maximum inter-atomic distance to be used for fitting
showPlot (bool): If set to true, it will print the plot showing
fitted energy level and the obtained best fit. Default is
False
minStateContribution (float): valid values are in the range [0,1).
It specifies minimum amount of the original state in the given
energy state necessary for the state to be considered for
the adiabatic continuation of the original unperturbed
pair state.

Returns:
float:
:math:C_6 measured in :math:\\text{GHz }\\mu\\text{m}^6
on success; If unsuccessful returns False.

Note:
In order to use this functions, highlighting in
:obj:diagonalise should be based on the original pair
state contribution of the eigenvectors (that this,
drivingFromState parameter should not be set, which
corresponds to drivingFromState = [0,0,0,0,0]).
"""
initialStateDetuning = []
initialStateDetuningX = []

fromRindex = -1
toRindex = -1
for br in xrange(len(self.r)):
if (fromRindex == -1) and (self.r[br] >= rStart):
fromRindex = br
if self.r[br] > rStop:
toRindex = br - 1
break
if (fromRindex != -1) and (toRindex == -1):
toRindex = len(self.r) - 1
if fromRindex == -1:
print(
"\nERROR: could not find data for energy levels for interatomic"
)
print("distances between %2.f and %.2f mu m.\n\n" % (rStart, rStop))
return 0

for br in xrange(fromRindex, toRindex + 1):
index = -1
maxPortion = minStateContribution
for br2 in xrange(len(self.y[br])):
if abs(self.highlight[br][br2]) > maxPortion:
index = br2
maxPortion = abs(self.highlight[br][br2])
if index != -1:
initialStateDetuning.append(abs(self.y[br][index]))
initialStateDetuningX.append(self.r[br])

initialStateDetuning = np.log(np.array(initialStateDetuning))
initialStateDetuningX = np.array(initialStateDetuningX)

def c6fit(r, c6, offset):
return np.log(c6 / r**6 + offset)

try:
popt, pcov = curve_fit(
c6fit, initialStateDetuningX, initialStateDetuning, [1, 0]
)
except Exception as ex:
print(ex)
print("ERROR: unable to find a fit for C6.")
return False
print("c6 = ", popt[0], " GHz /R^6 (mu m)^6")
print("offset = ", popt[1])

y_fit = []
for val in initialStateDetuningX:
y_fit.append(c6fit(val, popt[0], popt[1]))
y_fit = np.array(y_fit)

if showPlot:
fig, ax = plt.subplots(1, 1, figsize=(8.0, 5.0))
ax.loglog(
initialStateDetuningX,
np.exp(initialStateDetuning),
"b-",
lw=2,
zorder=1,
)
ax.loglog(
initialStateDetuningX, np.exp(y_fit), "r--", lw=2, zorder=2
)

ax.legend(
("calculated energy level", "fitted model function"),
loc=1,
fontsize=10,
)

ax.set_xlim(np.min(self.r), np.max(self.r))
ymin = np.min(initialStateDetuning)
ymax = np.max(initialStateDetuning)
ax.set_ylim(exp(ymin), exp(ymax))

minorLocator = mpl.ticker.MultipleLocator(1)
minorFormatter = mpl.ticker.FormatStrFormatter("%d")
ax.xaxis.set_minor_locator(minorLocator)
ax.xaxis.set_minor_formatter(minorFormatter)
ax.xaxis.set_major_formatter(plt.NullFormatter())
ax.set_xlabel(r"Interatomic distance, $r$ ($\mu$m)")
ax.set_ylabel(r"Pair-state energy, $|E|$ (GHz)")
ax.set_title(r"$C_6$ fit")
plt.show()

self.fitX = initialStateDetuningX
self.fitY = initialStateDetuning
self.fittedCurveY = y_fit

return popt[0]

[docs]
def getC3fromLevelDiagram(
self,
rStart,
rStop,
showPlot=False,
minStateContribution=0.0,
resonantBranch=+1,
):
"""
Finds :math:C_3 coefficient for original pair state.

Function first finds for each distance in the range
[rStart , rStop] the eigen state with highest contribution of
the original state. One can set optional parameter
minStateContribution to value in the range [0,1), so that function
finds only states if they have contribution of the original state
that is bigger then minStateContribution.

Once original pair-state is found in the range of interatomic
distances, from smallest rStart to the biggest rStop, function
will try to perform fitting of the corresponding state energy
:math:E(R) at distance :math:R to the function
:math:A+C_3/R^3 where :math:A is some offset.

Args:
rStart (float): smallest inter-atomic distance to be used for fitting
rStop (float): maximum inter-atomic distance to be used for fitting
showPlot (bool): If set to true, it will print the plot showing
fitted energy level and the obtained best fit. Default is
False
minStateContribution (float): valid values are in the range [0,1).
It specifies minimum amount of the original state in the given
energy state necessary for the state to be considered for
the adiabatic continuation of the original unperturbed
pair state.
resonantBranch (int): optional, default +1. For resonant
interactions we have two branches with identical
state contributions. In this case, we will select only
positively detuned branch (for resonantBranch = +1)
or negatively detuned branch (fore resonantBranch = -1)
depending on the value of resonantBranch optional parameter
Returns:
float:
:math:C_3 measured in :math:\\text{GHz }\\mu\\text{m}^6
on success; If unsuccessful returns False.

Note:
In order to use this functions, highlighting in
:obj:diagonalise should be based on the original pair
state contribution of the eigenvectors (that this,
drivingFromState parameter should not be set, which
corresponds to drivingFromState = [0,0,0,0,0]).
"""

selectBranch = False
if abs(self.l - self.ll) == 1:
selectBranch = True
resonantBranch = float(resonantBranch)

initialStateDetuning = []
initialStateDetuningX = []

fromRindex = -1
toRindex = -1
for br in xrange(len(self.r)):
if (fromRindex == -1) and (self.r[br] >= rStart):
fromRindex = br
if self.r[br] > rStop:
toRindex = br - 1
break
if (fromRindex != -1) and (toRindex == -1):
toRindex = len(self.r) - 1
if fromRindex == -1:
print(
"\nERROR: could not find data for energy levels for interatomic"
)
print("distances between %2.f and %.2f mu m.\n\n" % (rStart, rStop))
return False

discontinuityDetected = False
for br in xrange(toRindex, fromRindex - 1, -1):
index = -1
maxPortion = minStateContribution
for br2 in xrange(len(self.y[br])):
if (abs(self.highlight[br][br2]) > maxPortion) and (
not selectBranch or (self.y[br][br2] * selectBranch > 0.0)
):
index = br2
maxPortion = abs(self.highlight[br][br2])

if len(initialStateDetuningX) > 2:
slope1 = (
initialStateDetuning[-1] - initialStateDetuning[-2]
) / (initialStateDetuningX[-1] - initialStateDetuningX[-2])
slope2 = (abs(self.y[br][index]) - initialStateDetuning[-1]) / (
self.r[br] - initialStateDetuningX[-1]
)
if abs(slope2) > 3.0 * abs(slope1):
discontinuityDetected = True
if (index != -1) and (not discontinuityDetected):
initialStateDetuning.append(abs(self.y[br][index]))
initialStateDetuningX.append(self.r[br])

initialStateDetuning = np.log(np.array(initialStateDetuning))  # *1e9
initialStateDetuningX = np.array(initialStateDetuningX)

def c3fit(r, c3, offset):
return np.log(c3 / r**3 + offset)

try:
popt, pcov = curve_fit(
c3fit, initialStateDetuningX, initialStateDetuning, [1, 0]
)
except Exception as ex:
print(ex)
print("ERROR: unable to find a fit for C3.")
return False
print("c3 = ", popt[0], " GHz /R^3 (mu m)^3")
print("offset = ", popt[1])

y_fit = []

for val in initialStateDetuningX:
y_fit.append(c3fit(val, popt[0], popt[1]))
y_fit = np.array(y_fit)

if showPlot:
fig, ax = plt.subplots(1, 1, figsize=(8.0, 5.0))
ax.loglog(
initialStateDetuningX,
np.exp(initialStateDetuning),
"b-",
lw=2,
zorder=1,
)
ax.loglog(
initialStateDetuningX, np.exp(y_fit), "r--", lw=2, zorder=2
)

ax.legend(
("calculated energy level", "fitted model function"),
loc=1,
fontsize=10,
)
ax.set_xlim(np.min(self.r), np.max(self.r))
ymin = np.min(initialStateDetuning)
ymax = np.max(initialStateDetuning)
ax.set_ylim(exp(ymin), exp(ymax))

minorLocator = mpl.ticker.MultipleLocator(1)
minorFormatter = mpl.ticker.FormatStrFormatter("%d")
ax.xaxis.set_minor_locator(minorLocator)
ax.xaxis.set_minor_formatter(minorFormatter)
ax.xaxis.set_major_formatter(plt.NullFormatter())
ax.set_xlabel(r"Interatomic distance, $r$ ($\mu$m)")
ax.set_ylabel(r"Pair-state energy, $|E|$ (GHz)")

locatorStep = 1.0
while (locatorStep > (ymax - ymin)) and locatorStep > 1.0e-4:
locatorStep /= 10.0

ax.yaxis.set_major_locator(mpl.ticker.MultipleLocator(locatorStep))
ax.yaxis.set_major_formatter(mpl.ticker.FormatStrFormatter("%.3f"))
ax.yaxis.set_minor_locator(
mpl.ticker.MultipleLocator(locatorStep / 10.0)
)
ax.yaxis.set_minor_formatter(plt.NullFormatter())
# ax.yaxis.set_minor_formatter(mpl.ticker.FormatStrFormatter('%.3f'))

ax.set_title(r"$C_3$ fit")

plt.show()

self.fitX = initialStateDetuningX
self.fitY = initialStateDetuning
self.fittedCurveY = y_fit

return popt[0]

[docs]
def getVdwFromLevelDiagram(
self, rStart, rStop, showPlot=False, minStateContribution=0.0
):
"""
Finds :math:r_{\\rm vdW} coefficient for original pair state.

Function first finds for each distance in the range [rStart,rStop]
the eigen state with highest contribution of the original state.
One can set optional parameter minStateContribution to value in
the range [0,1), so that function finds only states if they have
contribution of the original state that is bigger then
minStateContribution.

Once original pair-state is found in the range of interatomic
distances, from smallest rStart to the biggest rStop, function
will try to perform fitting of the corresponding state energy
:math:E(R) at distance :math:R to the function
:math:A+B\\frac{1-\\sqrt{1+(r_{\\rm vdW}/r)^6}}{1-\\sqrt{1+r_{\\rm vdW}^6}}
where :math:A and :math:B are some offset.

Args:
rStart (float): smallest inter-atomic distance to be used for fitting
rStop (float): maximum inter-atomic distance to be used for fitting
showPlot (bool): If set to true, it will print the plot showing
fitted energy level and the obtained best fit. Default is
False
minStateContribution (float): valid values are in the range [0,1).
It specifies minimum amount of the original state in the given
energy state necessary for the state to be considered for
the adiabatic continuation of the original unperturbed
pair state.
Returns:
float: :math:r_{\\rm vdW}  measured in :math:\\mu\\text{m}
on success; If unsuccessful returns False.

Note:
In order to use this functions, highlighting in
:obj:diagonalise should be based on the original pair
state contribution of the eigenvectors (that this,
drivingFromState parameter should not be set, which
corresponds to drivingFromState = [0,0,0,0,0]).
"""

initialStateDetuning = []
initialStateDetuningX = []

fromRindex = -1
toRindex = -1
for br in xrange(len(self.r)):
if (fromRindex == -1) and (self.r[br] >= rStart):
fromRindex = br
if self.r[br] > rStop:
toRindex = br - 1
break
if (fromRindex != -1) and (toRindex == -1):
toRindex = len(self.r) - 1
if fromRindex == -1:
print(
"\nERROR: could not find data for energy levels for interatomic"
)
print("distances between %2.f and %.2f mu m.\n\n" % (rStart, rStop))
return False

discontinuityDetected = False
for br in xrange(toRindex, fromRindex - 1, -1):
index = -1
maxPortion = minStateContribution
for br2 in xrange(len(self.y[br])):
if abs(self.highlight[br][br2]) > maxPortion:
index = br2
maxPortion = abs(self.highlight[br][br2])
if len(initialStateDetuningX) > 2:
slope1 = (
initialStateDetuning[-1] - initialStateDetuning[-2]
) / (initialStateDetuningX[-1] - initialStateDetuningX[-2])
slope2 = (abs(self.y[br][index]) - initialStateDetuning[-1]) / (
self.r[br] - initialStateDetuningX[-1]
)
if abs(slope2) > 3.0 * abs(slope1):
discontinuityDetected = True
if (index != -1) and (not discontinuityDetected):
initialStateDetuning.append(abs(self.y[br][index]))
initialStateDetuningX.append(self.r[br])

initialStateDetuning = np.log(abs(np.array(initialStateDetuning)))
initialStateDetuningX = np.array(initialStateDetuningX)

def vdwFit(r, offset, scale, vdw):
return np.log(
abs(
offset
+ scale
* (1.0 - np.sqrt(1.0 + (vdw / r) ** 6))
/ (1.0 - np.sqrt(1 + vdw**6))
)
)

noOfPoints = len(initialStateDetuningX)
print("Data points to fit = ", noOfPoints)

try:
popt, pcov = curve_fit(
vdwFit,
initialStateDetuningX,
initialStateDetuning,
[
0,
initialStateDetuning[noOfPoints // 2],
initialStateDetuningX[noOfPoints // 2],
],
)
except Exception as ex:
print(ex)
print("ERROR: unable to find a fit for van der Waals distance.")
return False

if (initialStateDetuningX[0] < popt[2]) or (
popt[2] < initialStateDetuningX[-1]
):
print("WARNING: vdw radius seems to be outside the fitting range!")
print(
"It's estimated to be around %.2f mu m from the current fit."
% popt[2]
)

print("Rvdw =  ", popt[2], " mu m")
print("offset = ", popt[0], "\n scale = ", popt[1])

y_fit = []

for val in initialStateDetuningX:
y_fit.append(vdwFit(val, popt[0], popt[1], popt[2]))
y_fit = np.array(y_fit)

if showPlot:
fig, ax = plt.subplots(1, 1, figsize=(8.0, 5.0))
ax.loglog(
initialStateDetuningX,
np.exp(initialStateDetuning),
"b-",
lw=2,
zorder=1,
)
ax.loglog(
initialStateDetuningX, np.exp(y_fit), "r--", lw=2, zorder=2
)

ax.set_xlim(np.min(self.r), np.max(self.r))
ymin = np.min(initialStateDetuning)
ymax = np.max(initialStateDetuning)
ax.set_ylim(exp(ymin), exp(ymax))

ax.axvline(x=popt[2], color="k")
ax.text(
popt[2],
exp((ymin + ymax) / 2.0),
r"$R_{vdw} = %.1f$ $\mu$m" % popt[2],
)

minorLocator = mpl.ticker.MultipleLocator(1)
minorFormatter = mpl.ticker.FormatStrFormatter("%d")
ax.xaxis.set_minor_locator(minorLocator)
ax.xaxis.set_minor_formatter(minorFormatter)
ax.xaxis.set_major_formatter(plt.NullFormatter())
ax.set_xlabel(r"Interatomic distance, $r$ ($\mu$m)")
ax.set_ylabel(r"Pair-state energy, $|E|$ (GHz)")
ax.legend(
("calculated energy level", "fitted model function"),
loc=1,
fontsize=10,
)

plt.show()

self.fitX = initialStateDetuningX
self.fitY = initialStateDetuning
self.fittedCurveY = y_fit

return popt[2]

[docs]
class StarkMapResonances:
"""
Calculates pair state Stark maps for finding resonances

Tool for finding conditions for Foster resonances. For a given pair
state, in a given range of the electric fields, looks for the pair-state
that are close in energy and coupled via dipole-dipole interactions
to the original pair-state.

See Stark resonances example snippet_.

.. _Stark resonances example snippet:
././Rydberg_atoms_a_primer.html#Tuning-the-interaction-strength-with-electric-fields

Parameters:
atom1 (:obj:arc.alkali_atom_functions.AlkaliAtom or :obj:arc.divalent_atom_functions.DivalentAtom):
={ :obj:arc.alkali_atom_data.Lithium6,
:obj:arc.alkali_atom_data.Lithium7,
:obj:arc.alkali_atom_data.Sodium,
:obj:arc.alkali_atom_data.Potassium39,
:obj:arc.alkali_atom_data.Potassium40,
:obj:arc.alkali_atom_data.Potassium41,
:obj:arc.alkali_atom_data.Rubidium85,
:obj:arc.alkali_atom_data.Rubidium87,
:obj:arc.alkali_atom_data.Caesium,
:obj:arc.divalent_atom_data.Strontium88,
:obj:arc.divalent_atom_data.Calcium40
:obj:arc.divalent_atom_data.Ytterbium174 }
the first atom in the pair-state
state1 ([int,int,float,float,(float)]): specification of the state
of the first state as an array of values :math:[n,l,j,m_j].
For :obj:arc.divalent_atom_functions.DivalentAtom and other divalent atoms, 5th value
should be added specifying total spin angular momentum s.
Full definition of state then has format
:math:[n,l,j,m_j,s].
atom2 (:obj:arc.alkali_atom_functions.AlkaliAtom or :obj:arc.divalent_atom_functions.DivalentAtom):
={ :obj:arc.alkali_atom_data.Lithium6,
:obj:arc.alkali_atom_data.Lithium7,
:obj:arc.alkali_atom_data.Sodium,
:obj:arc.alkali_atom_data.Potassium39,
:obj:arc.alkali_atom_data.Potassium40,
:obj:arc.alkali_atom_data.Potassium41,
:obj:arc.alkali_atom_data.Rubidium85,
:obj:arc.alkali_atom_data.Rubidium87,
:obj:arc.alkali_atom_data.Caesium,
:obj:arc.divalent_atom_data.Strontium88,
:obj:arc.divalent_atom_data.Calcium40
:obj:arc.divalent_atom_data.Ytterbium174 }
the second atom in the pair-state
state2 ([int,int,float,float,(float)]): specification of the state
of the first state as an array of values :math:[n,l,j,m_j],
For :obj:arc.divalent_atom_functions.DivalentAtom and other divalent atoms, 5th value
should be added specifying total spin angular momentum s.
Full definition of state then has format
:math:[n,l,j,m_j,s].

Note:
In checking if certain state is dipole coupled to the original
state, only the highest contributing state is checked for dipole
coupling. This should be fine if one is interested in resonances
in weak fields. For stronger fields, one might want to include
effect of coupling to other contributing base states.

"""

def __init__(self, atom1, state1, atom2, state2):
self.atom1 = atom1
if issubclass(type(self.atom1), DivalentAtom) and (
len(state1) != 5 or (state1[4] != 0 and state1[4] != 1)
):
raise ValueError(
"For divalent atoms state specification has to "
"include total spin angular momentum s as the last "
"number in the state specification [n,l,j,m_j,s]."
)
self.state1 = state1
# add exlicitly total spin of the state for Alkaline atoms
if len(self.state1) == 4:
self.state1.append(0.5)

self.atom2 = atom2
if issubclass(type(self.atom2), DivalentAtom) and (
len(state1) != 5 or (state1[4] != 0 and state1[4] != 1)
):
raise ValueError(
"For divalent atoms state specification has to "
"include total spin angular momentum s as the last "
"numbre in the state specification [n,l,j,m_j,s]."
)
self.state2 = state2
# add exlicitly total spin of the state for Alkaline atoms
if len(self.state2) == 4:
self.state2.append(0.5)

self.pairStateEnergy = (
(
atom1.getEnergy(
self.state1[0],
self.state1[1],
self.state1[2],
s=self.state1[4],
)
+ atom2.getEnergy(
self.state2[0],
self.state2[1],
self.state2[2],
s=self.state2[4],
)
)
* C_e
/ C_h
* 1e-9
)

[docs]
def findResonances(
self,
nMin,
nMax,
maxL,
eFieldList,
energyRange=[-5.0e9, +5.0e9],
Bz=0,
progressOutput=False,
):
r"""
Finds near-resonant dipole-coupled pair-states

For states in range of principal quantum numbers [nMin,nMax]
and orbital angular momentum [0,maxL], for a range of electric fields
given by eFieldList function will find near-resonant pair states.

Only states that are in the range given by energyRange will be
extracted from the pair-state Stark maps.

Args:
nMin (int): minimal principal quantum number of the state to be
included in the StarkMap calculation
nMax (int): maximal principal quantum number of the state to be
included in the StarkMap calculation
maxL (int): maximum value of orbital angular momentum for the states
to be included in the calculation
eFieldList ([float]): list of the electric fields (in V/m) for
which to calculate level diagram (StarkMap)
Bz (float): optional, magnetic field directed along z-axis in
units of Tesla. Calculation will be correct only for weak
magnetic fields, where paramagnetic term is much stronger
then diamagnetic term. Diamagnetic term is neglected.
energyRange ([float,float]): optinal argument. Minimal and maximal
energy of that some dipole-coupled state should have in order
to keep it in the plot (in units of Hz). By default it finds
states that are :math:\pm 5 GHz
progressOutput (:obj:bool, optional): if True prints the
progress of calculation; Set to false by default.
"""

self.eFieldList = eFieldList
self.Bz = Bz
eMin = energyRange[0] * 1.0e-9  # in GHz
eMax = energyRange[1] * 1.0e-9

# find where is the original pair state

sm1 = StarkMap(self.atom1)
sm1.defineBasis(
self.state1[0],
self.state1[1],
self.state1[2],
self.state1[3],
nMin,
nMax,
maxL,
Bz=self.Bz,
progressOutput=progressOutput,
s=self.state1[4],
)
sm1.diagonalise(eFieldList, progressOutput=progressOutput)
if (
(self.atom2 is self.atom1)
and (self.state1[0] == self.state2[0])
and (self.state1[1] == self.state2[1])
and (abs(self.state1[2] - self.state2[2]) < 0.1)
and (abs(self.state1[3] - self.state2[3]) < 0.1)
and (abs(self.state1[4] - self.state2[4]) < 0.1)
):
sm2 = sm1
else:
sm2 = StarkMap(self.atom2)
sm2.defineBasis(
self.state2[0],
self.state2[1],
self.state2[2],
self.state2[3],
nMin,
nMax,
maxL,
Bz=self.Bz,
progressOutput=progressOutput,
s=self.state2[4],
)
sm2.diagonalise(eFieldList, progressOutput=progressOutput)

self.originalStateY = []
self.originalStateContribution = []
for i in xrange(len(sm1.eFieldList)):
jmax1 = 0
jmax2 = 0
for j in xrange(len(sm1.highlight[i])):
if sm1.highlight[i][j] > sm1.highlight[i][jmax1]:
jmax1 = j
for j in xrange(len(sm2.highlight[i])):
if sm2.highlight[i][j] > sm2.highlight[i][jmax2]:
jmax2 = j

self.originalStateY.append(
sm1.y[i][jmax1] + sm2.y[i][jmax2] - self.pairStateEnergy
)
self.originalStateContribution.append(
(sm1.highlight[i][jmax1] + sm2.highlight[i][jmax2]) / 2.0
)

# M= mj1+mj2 is conserved with dipole-dipole interaction

dmlist1 = [1, 0]
if self.state1[3] != 0.5:
dmlist1.append(-1)
dmlist2 = [1, 0]
if self.state2[3] != 0.5:
dmlist2.append(-1)

n1 = self.state1[0]
l1 = self.state1[1] + 1
j1 = self.state1[2] + 1
mj1 = self.state1[3]

n2 = self.state2[0]
l2 = self.state2[1] + 1
j2 = self.state2[2] + 1
mj2 = self.state2[3]

self.fig, self.ax = plt.subplots(1, 1, figsize=(9.0, 6))
cm = LinearSegmentedColormap.from_list("mymap", ["0.9", "red", "black"])
cNorm = matplotlib.colors.Normalize(vmin=0.0, vmax=1.0)

self.r = []
self.y = []
self.composition = []

for dm1 in dmlist1:
sm1.defineBasis(
n1,
l1,
j1,
mj1 + dm1,
nMin,
nMax,
maxL,
Bz=self.Bz,
progressOutput=progressOutput,
s=self.state1[4],
)
sm1.diagonalise(eFieldList, progressOutput=progressOutput)

for dm2 in dmlist2:
sm2.defineBasis(
n2,
l2,
j2,
mj2 + dm2,
nMin,
nMax,
maxL,
Bz=self.Bz,
progressOutput=progressOutput,
s=self.state2[4],
)
sm2.diagonalise(eFieldList, progressOutput=progressOutput)

for i in xrange(len(sm1.eFieldList)):
yList = []
compositionList = []
if progressOutput:
sys.stdout.write("\rE=%.2f V/m " % sm1.eFieldList[i])
sys.stdout.flush()
for j in xrange(len(sm1.y[i])):
for jj in xrange(len(sm2.y[i])):
energy = (
sm1.y[i][j]
+ sm2.y[i][jj]
- self.pairStateEnergy
)
statec1 = sm1.basisStates[
sm1.composition[i][j][0][1]
]
statec2 = sm2.basisStates[
sm2.composition[i][jj][0][1]
]
if (
(energy > eMin)
and (energy < eMax)
and (abs(statec1[1] - self.state1[1]) == 1)
and (abs(statec2[1] - self.state2[1]) == 1)
):
# add this to PairStateMap
yList.append(energy)
compositionList.append(
[
sm1._stateComposition(
sm1.composition[i][j]
),
sm2._stateComposition(
sm2.composition[i][jj]
),
]
)

if len(self.y) <= i:
self.y.append(yList)
self.composition.append(compositionList)
else:
self.y[i].extend(yList)
self.composition[i].extend(compositionList)

if progressOutput:
print("\n")

for i in xrange(len(sm1.eFieldList)):
self.y[i] = np.array(self.y[i])
self.composition[i] = np.array(self.composition[i])
self.ax.scatter(
[sm1.eFieldList[i] / 100.0] * len(self.y[i]),
self.y[i],
c="k",
s=5,
norm=cNorm,
cmap=cm,
lw=0,
picker=5,
)
self.ax.plot(sm1.eFieldList / 100.0, self.originalStateY, "r-", lw=1)
self.ax.set_ylim(eMin, eMax)
self.ax.set_xlim(
min(self.eFieldList) / 100.0, max(self.eFieldList) / 100.0
)
self.ax.set_xlabel("Electric field (V/cm)")
self.ax.set_ylabel(r"Pair-state relative energy, $\Delta E/h$ (GHz)")

[docs]
def showPlot(self, interactive=True):
"""
Plots initial state Stark map and its dipole-coupled resonances

Args:

interactive (optional, bool): if True (by default) points on plot
will be clickable so that one can find the state labels
and their composition (if they are heavily admixed).

Note:
Zero is given by the initial states of the atom given in
initialisation of calculations, calculated **in absence of
magnetic field B_z**. In other words, for non-zero magnetic
field the inital states will have offset from zero even
for zero electric field due to Zeeman shift.
"""

if self.fig != 0:
if interactive:
self.ax.set_title("Click on state to see state composition")
self.clickedPoint = 0
self.fig.canvas.draw()
self.fig.canvas.mpl_connect("pick_event", self._onPick)
plt.show()
else:
print("Error while showing a plot: nothing is plotted yet")

def _onPick(self, event):
if isinstance(event.artist, matplotlib.collections.PathCollection):
x = event.mouseevent.xdata * 100.0
y = event.mouseevent.ydata

i = np.searchsorted(self.eFieldList, x)
if i == len(self.eFieldList):
i -= 1
if (i > 0) and (
abs(self.eFieldList[i - 1] - x) < abs(self.eFieldList[i] - x)
):
i -= 1

j = 0
for jj in xrange(len(self.y[i])):
if abs(self.y[i][jj] - y) < abs(self.y[i][j] - y):
j = jj

if self.clickedPoint != 0:
self.clickedPoint.remove()

(self.clickedPoint,) = self.ax.plot(
[self.eFieldList[i] / 100.0],
[self.y[i][j]],
"bs",
linewidth=0,
zorder=3,
)

atom1 = self.atom1.elementName
atom2 = self.atom2.elementName
composition1 = str(self.composition[i][j][0])
composition2 = str(self.composition[i][j][1])
self.ax.set_title(
("[%s,%s]=[" % (atom1, atom2))
+ composition1
+ ","
+ composition2
+ "]",
fontsize=10,
)

event.canvas.draw()

def _onPick2(self, xdata, ydata):
x = xdata * 100.0
y = ydata

i = np.searchsorted(self.eFieldList, x)
if i == len(self.eFieldList):
i -= 1
if (i > 0) and (
abs(self.eFieldList[i - 1] - x) < abs(self.eFieldList[i] - x)
):
i -= 1

j = 0
for jj in xrange(len(self.y[i])):
if abs(self.y[i][jj] - y) < abs(self.y[i][j] - y):
j = jj

if self.clickedPoint != 0:
self.clickedPoint.remove()

(self.clickedPoint,) = self.ax.plot(
[self.eFieldList[i] / 100.0],
[self.y[i][j]],
"bs",
linewidth=0,
zorder=3,
)

atom1 = self.atom1.elementName
atom2 = self.atom2.elementName
composition1 = str(self.composition[i][j][0])
composition2 = str(self.composition[i][j][1])
self.ax.set_title(
("[%s,%s]=[" % (atom1, atom2))
+ composition1
+ ","
+ composition2
+ "]",
fontsize=10,
)