Alkali atom functions¶
Overview¶
Classes and global methods
AlkaliAtom ([preferQuantumDefects, cpp_numerov]) 
Implements general calculations for alkali atoms. 
NumerovBack (innerLimit, outerLimit, kfun, …) 
Full Python implementation of Numerov integration 
saveCalculation (calculation, fileName) 
Saves calculation for future use. 
loadSavedCalculation (fileName) 
Loads previously saved calculation. 
printState (n, l, j) 
Prints state spectroscopic label for numeric \(n\), \(l\), \(s\) label of the state 
printStateString (n, l, j) 
Returns state spectroscopic label for numeric \(n\), \(l\), \(s\) label of the state 
AlkaliAtom Methods
AlkaliAtom.getDipoleMatrixElement (n1, l1, …) 
Dipole matrix element \(\langle n_1 l_1 j_1 m_{j_1} e\mathbf{r}n_2 l_2 j_2 m_{j_2}\rangle\) in units of \(a_0 e\) 
AlkaliAtom.getTransitionWavelength (n1, l1, …) 
Calculated transition wavelength (in vacuum) in m. 
AlkaliAtom.getTransitionFrequency (n1, l1, …) 
Calculated transition frequency in Hz 
AlkaliAtom.getRabiFrequency (n1, l1, j1, mj1, …) 
Returns a Rabi frequency for resonantly driven atom in a center of TEM00 mode of a driving field 
AlkaliAtom.getRabiFrequency2 (n1, l1, j1, …) 
Returns a Rabi frequency for resonant excitation with a given electric field amplitude 
AlkaliAtom.getStateLifetime (n, l, j[, …]) 
Returns the lifetime of the state (in s) 
AlkaliAtom.getTransitionRate (n1, l1, j1, n2, …) 
Transition rate due to coupling to vacuum modes (black body included) 
AlkaliAtom.getReducedMatrixElementJ_asymmetric (n1, …) 
Reduced matrix element in \(J\) basis, defined in asymmetric notation. 
AlkaliAtom.getReducedMatrixElementJ (n1, l1, …) 
Reduced matrix element in \(J\) basis (symmetric notation) 
AlkaliAtom.getReducedMatrixElementL (n1, l1, …) 
Reduced matrix element in \(L\) basis (symmetric notation) 
AlkaliAtom.getRadialMatrixElement (n1, l1, …) 
Radial part of the dipole matrix element 
AlkaliAtom.getQuadrupoleMatrixElement (n1, …) 
Radial part of the quadrupole matrix element 
AlkaliAtom.getPressure (temperature) 
Vapour pressure (in Pa) at given temperature 
AlkaliAtom.getNumberDensity (temperature) 
Atom number density at given temperature 
AlkaliAtom.getAverageInteratomicSpacing (…) 
Returns average interatomic spacing in atomic vapour 
AlkaliAtom.corePotential (l, r) 
core potential felt by valence electron 
AlkaliAtom.effectiveCharge (l, r) 
effective charge of the core felt by valence electron 
AlkaliAtom.potential (l, s, j, r) 
returns total potential that electron feels 
AlkaliAtom.radialWavefunction (l, s, j, …) 
Radial part of electron wavefunction 
AlkaliAtom.getEnergy (n, l, j) 
Energy of the level relative to the ionisation level (in eV) 
AlkaliAtom.getQuantumDefect (n, l, j) 
Quantum defect of the level. 
AlkaliAtom.getC6term (n, l, j, n1, l1, j1, …) 
C6 interaction term for the given two pairstates 
AlkaliAtom.getC3term (n, l, j, n1, l1, j1, …) 
C3 interaction term for the given two pairstates 
AlkaliAtom.getEnergyDefect (n, l, j, n1, l1, …) 
Energy defect for the given two pairstates (one of the state has two atoms in the same state) 
AlkaliAtom.getEnergyDefect2 (n, l, j, nn, ll, …) 
Energy defect for the given two pairstates 
AlkaliAtom.updateDipoleMatrixElementsFile () 
Updates the file with precalculated dipole matrix elements. 
AlkaliAtom.getRadialCoupling (n, l, j, n1, l1, j1) 
Returns radial part of the coupling between two states (dipole and quadrupole interactions only) 
AlkaliAtom.getAverageSpeed (temperature) 
Average (mean) speed at a given temperature 
AlkaliAtom.getLiteratureDME (n1, l1, j1, n2, …) 
Returns literature information on requested transition. 
Detailed documentation¶
Implements general singleatom calculations
This module calculates single (isolated) atom properties of all alkali metals in general. For example, it calculates dipole matrix elements, quandrupole matrix elements, etc. Also, some helpful general functions are here, e.g. for saving and loading calculations (singleatom and pairstate based), printing state labels etc.

class
arc.alkali_atom_functions.
AlkaliAtom
(preferQuantumDefects=True, cpp_numerov=True)[source]¶ Implements general calculations for alkali atoms.
This abstract class implements general calculations methods.
Parameters:  preferQuantumDefects (bool) – Use quantum defects for energy level calculations. If False,
uses NIST ASD values
where available. If True, uses quantum defects for energy calculations
for principal quantum numbers equal or above
minQuantumDefectN
which is specified for each element separately. For principal quantum numbers below this value, NIST ASD values are used, since quantum defects don’t reproduce well lowlying states. Default is True.  cpp_numerov (bool) – should the wavefunction be calculated with Numerov algorithm implemented in C++; if False, it uses pure Python implementation that is much slower. Default is True.

Z
= 0.0¶ Atomic number

a1
= [0]¶ Model potential parameters fitted from experimental observations for different l (electron angular momentum)

a2
= [0]¶ Model potential parameters fitted from experimental observations for different l (electron angular momentum)

a3
= [0]¶ Model potential parameters fitted from experimental observations for different l (electron angular momentum)

a4
= [0]¶ Model potential parameters fitted from experimental observations for different l (electron angular momentum)

abundance
= 1.0¶ relative isotope abundance

alphaC
= 0.0¶ Core polarizability

corePotential
(l, r)[source]¶ core potential felt by valence electron
For more details about derivation of model potential see Ref. [2].
Parameters: Returns: core potential felt by valence electron (in a.u. ???)
Return type: References
[2] (1, 2) M. Marinescu, H. R. Sadeghpour, and A. Dalgarno PRA 49, 982 (1994), https://doi.org/10.1103/PhysRevA.49.982

cpp_numerov
= True¶ swich  should the wavefunction be calculated with Numerov algorithm implemented in C++

dipoleMatrixElementFile
= ''¶ location of harddisk stored dipole matrix elements

effectiveCharge
(l, r)[source]¶ effective charge of the core felt by valence electron
For more details about derivation of model potential see Ref. [2].
Parameters: Returns: effective charge (in a.u.)
Return type:

elementName
= 'elementName'¶ Humanreadable element name

extraLevels
= []¶ levels that are for smaller principal quantum number (n) than ground level, but are above in energy due to angular part

getAverageInteratomicSpacing
(temperature)[source]¶ Returns average interatomic spacing in atomic vapour
See calculation of basic properties example snippet.
Parameters: temperature (float) – temperature of the atomic vapour Returns: average interatomic spacing in m Return type: float

getAverageSpeed
(temperature)[source]¶ Average (mean) speed at a given temperature
Parameters: temperature (float) – temperature (K) Returns: mean speed (m/s) Return type: float

getC3term
(n, l, j, n1, l1, j1, n2, l2, j2)[source]¶ C3 interaction term for the given two pairstates
Calculates \(C_3\) intaraction term for \(n,l,j,n,l,j\rangle \leftrightarrow n_1,l_1,j_1,n_2,l_2,j_2\rangle\)
Parameters:  n (int) – principal quantum number
 l (int) – orbital angular momenutum
 j (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: \(C_3 = \frac{\langle n,l,j ern_1,l_1,j_1\rangle \langle n,l,j ern_2,l_2,j_2\rangle}{4\pi\varepsilon_0}\) (\(h\) Hz m \({}^3\)).
Return type:

getC6term
(n, l, j, n1, l1, j1, n2, l2, j2)[source]¶ C6 interaction term for the given two pairstates
Calculates \(C_6\) intaraction term for \(n,l,j,n,l,j\rangle \leftrightarrow n_1,l_1,j_1,n_2,l_2,j_2\rangle\). For details of calculation see Ref. [3].
Parameters:  n (int) – principal quantum number
 l (int) – orbital angular momenutum
 j (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: \(C_6 = \frac{1}{4\pi\varepsilon_0} \frac{\langle n,l,j ern_1,l_1,j_1\rangle^2 \langle n,l,j ern_2,l_2,j_2\rangle^2} {E(n_1,l_1,j_2,n_2,j_2,j_2)E(n,l,j,n,l,j)}\) (\(h\) Hz m \({}^6\)).
Return type: Example
We can reproduce values from Ref. [3] for C3 coupling to particular channels. Taking for example channels described by the Eq. (50ac) we can get the values:
from arc import * channels = [[70,0,0.5, 70, 1,1.5, 69,1, 1.5],\ [70,0,0.5, 70, 1,1.5, 69,1, 0.5],\ [70,0,0.5, 69, 1,1.5, 70,1, 0.5],\ [70,0,0.5, 70, 1,0.5, 69,1, 0.5]] print(" = = = Caesium = = = ") atom = Caesium() for channel in channels: print("%.0f GHz (mu m)^6" % ( atom.getC6term(*channel) / C_h * 1.e27 )) print("\n = = = Rubidium = = =") atom = Rubidium() for channel in channels: print("%.0f GHz (mu m)^6" % ( atom.getC6term(*channel) / C_h * 1.e27 ))
Returns:
= = = Caesium = = = 722 GHz (mu m)^6 316 GHz (mu m)^6 383 GHz (mu m)^6 228 GHz (mu m)^6 = = = Rubidium = = = 799 GHz (mu m)^6 543 GHz (mu m)^6 589 GHz (mu m)^6 437 GHz (mu m)^6
which is in good agreement with the values cited in the Ref. [3]. Small discrepancies for Caesium originate from slightly different quantum defects used in calculations.
References
[3] (1, 2, 3) T. G. Walker, M. Saffman, PRA 77, 032723 (2008) https://doi.org/10.1103/PhysRevA.77.032723

getDipoleMatrixElement
(n1, l1, j1, mj1, n2, l2, j2, mj2, q)[source]¶ Dipole matrix element \(\langle n_1 l_1 j_1 m_{j_1} e\mathbf{r}n_2 l_2 j_2 m_{j_2}\rangle\) in units of \(a_0 e\)
Returns: dipole matrix element( \(a_0 e\)) Return type: float Example
For example, calculation of \(5 S_{1/2}m_j=\frac{1}{2} \rightarrow 5 P_{3/2}m_j=\frac{3}{2}\) transition dipole matrix element for laser driving \(\sigma^\) transition:
from arc import * atom = Rubidium() # transition 5 S_{1/2} m_j=0.5 > 5 P_{3/2} m_j=1.5 for laser # driving sigma transition print(atom.getDipoleMatrixElement(5,0,0.5,0.5,5,1,1.5,1.5,1))

getEnergy
(n, l, j)[source]¶ Energy of the level relative to the ionisation level (in eV)
Returned energies are with respect to the center of gravity of the hyperfinesplit states. If preferQuantumDefects =False (set during initialization) program will try use NIST energy value, if such exists, falling back to energy calculation with quantum defects if the measured value doesn’t exist. For preferQuantumDefects =True, program will always calculate energies from quantum defects (useful for comparing quantum defect calculations with measured energy level values).
Parameters: Returns: state energy (eV)
Return type:

getEnergyDefect
(n, l, j, n1, l1, j1, n2, l2, j2)[source]¶ Energy defect for the given two pairstates (one of the state has two atoms in the same state)
Energy difference between the states \(E(n_1,l_1,j_1,n_2,l_2,j_2)  E(n,l,j,n,l,j)\)
Parameters:  n (int) – principal quantum number
 l (int) – orbital angular momenutum
 j (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: energy defect (SI units: J)
Return type:

getEnergyDefect2
(n, l, j, nn, ll, jj, n1, l1, j1, n2, l2, j2)[source]¶ Energy defect for the given two pairstates
Energy difference between the states \(E(n_1,l_1,j_1,n_2,l_2,j_2)  E(n,l,j,nn,ll,jj)\)
See pairstate energy defects example snippet.
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: energy defect (SI units: J)
Return type:

getLiteratureDME
(n1, l1, j1, n2, l2, j2)[source]¶ Returns literature information on requested transition.
Parameters:  n1,l1,j1 – one of the states we are coupling
 n2,l2,j2 – the other state to which we are coupling
Returns: hasLiteratureValue?, dme, referenceInformation
If Boolean value is True, a literature value for dipole matrix element was found and reduced DME in J basis is returned as the number. Third returned argument (array) contains additional information about the literature value in the following order [ typeOfSource, errorEstimate , comment , reference, reference DOI] upon success to find a literature value for dipole matrix element:
 typeOfSource=1 if the value is theoretical calculation; otherwise, if it is experimentally obtained value typeOfSource=0
 comment details where within the publication the value can be found
 errorEstimate is absolute error estimate
 reference is humanreadable formatted reference
 reference DOI provides link to the publication.
Boolean value is False, followed by zero and an empty array if no literature value for dipole matrix element is found.
Return type: Note
The literature values are stored in /data folder in <element name>_literature_dme.csv files as a ; separated values. Each row in the file consists of one literature entry, that has information in the following order:
 n1
 l1
 j1
 n2
 l2
 j2
 dipole matrix element reduced l basis (a.u.)
 comment (e.g. where in the paper value appears?)
 value origin: 1 for theoretical; 0 for experimental values
 accuracy
 source (human readable formatted citation)
 doi number (e.g. 10.1103/RevModPhys.82.2313 )
If there are several values for a given transition, program will output the value that has smallest error (under column accuracy). The list of values can be expanded  every time program runs this file is read and the list is parsed again for use in calculations.

getNumberDensity
(temperature)[source]¶ Atom number density at given temperature
See calculation of basic properties example snippet.
Parameters: temperature (float) – temperature in K Returns: atom concentration in \(1/m^3\) Return type: float

getPressure
(temperature)[source]¶ Vapour pressure (in Pa) at given temperature
Parameters: temperature (float) – temperature in K Returns: vapour pressure in Pa Return type: float

getQuadrupoleMatrixElement
(n1, l1, j1, n2, l2, j2)[source]¶ Radial part of the quadrupole matrix element
Calculates \(\int \mathbf{d}r~R_{n_1,l_1,j_1}(r)\cdot R_{n_1,l_1,j_1}(r) \cdot r^4\). See Quadrupole calculation example snippet .
Parameters: Returns: quadrupole matrix element (\(a_0^2 e\)).
Return type:

getQuantumDefect
(n, l, j)[source]¶ Quantum defect of the level.
For an example, see Rydberg energy levels example snippet.
Parameters: Returns: quantum defect
Return type:

getRabiFrequency
(n1, l1, j1, mj1, n2, l2, j2, q, laserPower, laserWaist)[source]¶ Returns a Rabi frequency for resonantly driven atom in a center of TEM00 mode of a driving field
Parameters:  n1,l1,j1,mj1 – state from which we are driving transition
 n2,l2,j2 – state to which we are driving transition
 q – laser polarization (1,0,1 correspond to \(\sigma^\), \(\pi\) and \(\sigma^+\) respectively)
 laserPower – laser power in units of W
 laserWaist – laser \(1/e^2\) waist (radius) in units of m
Returns: Frequency in rad \(^{1}\). If you want frequency in Hz, divide by returned value by \(2\pi\)
Return type:

getRabiFrequency2
(n1, l1, j1, mj1, n2, l2, j2, q, electricFieldAmplitude)[source]¶ Returns a Rabi frequency for resonant excitation with a given electric field amplitude
Parameters:  n1,l1,j1,mj1 – state from which we are driving transition
 n2,l2,j2 – state to which we are driving transition
 q – laser polarization (1,0,1 correspond to \(\sigma^\), \(\pi\) and \(\sigma^+\) respectively)
 electricFieldAmplitude – amplitude of electric field driving (V/m)
Returns: Frequency in rad \(^{1}\). If you want frequency in Hz, divide by returned value by \(2\pi\)
Return type:

getRadialCoupling
(n, l, j, n1, l1, j1)[source]¶ Returns radial part of the coupling between two states (dipole and quadrupole interactions only)
Parameters: Returns: radial coupling strength (in a.u.), or zero for forbidden transitions in dipole and quadrupole approximation.
Return type:

getRadialMatrixElement
(n1, l1, j1, n2, l2, j2, useLiterature=True)[source]¶ Radial part of the dipole matrix element
Calculates \(\int \mathbf{d}r~R_{n_1,l_1,j_1}(r)\cdot R_{n_1,l_1,j_1}(r) \cdot r^3\).
Parameters: Returns: dipole matrix element (\(a_0 e\)).
Return type:

getReducedMatrixElementJ
(n1, l1, j1, n2, l2, j2)[source]¶ Reduced matrix element in \(J\) basis (symmetric notation)
Parameters: Returns: reduced dipole matrix element in \(J\) basis \(\langle j  er  j' \rangle\) (\(a_0 e\)).
Return type:

getReducedMatrixElementJ_asymmetric
(n1, l1, j1, n2, l2, j2)[source]¶ Reduced matrix element in \(J\) basis, defined in asymmetric notation.
Note that notation for symmetric and asymmetricly defined reduced matrix element is not consistent in the literature. For example, notation is used e.g. in Steck [1] is precisely the oposite.
Note
Note that this notation is asymmetric: \(( je r j' ) \neq ( j'e r j )\). Relation between the two notation is \(\langle jerj'\rangle= \sqrt{2j+1} ( j er j')\). This function always returns value for transition from lower to higher energy state, independent of the order of states entered in the function call.
Parameters: Returns: reduced dipole matrix element in Steck notation \(( j  er  j' )\) (\(a_0 e\)).
Return type: [1] Daniel A. Steck, “Cesium D Line Data,” (revision 2.0.1, 2 May 2008). http://steck.us/alkalidata

getReducedMatrixElementL
(n1, l1, j1, n2, l2, j2)[source]¶ Reduced matrix element in \(L\) basis (symmetric notation)
Parameters: Returns: reduced dipole matrix element in \(L\) basis \(\langle l  er  l' \rangle\) (\(a_0 e\)).
Return type:

getStateLifetime
(n, l, j, temperature=0, includeLevelsUpTo=0)[source]¶ Returns the lifetime of the state (in s)
For nonzero temperatures, user must specify up to which principal quantum number levels, that is above the initial state, should be included in order to account for blackbody induced transitions to higher lying states. See Rydberg lifetimes example snippet.
Parameters:  l, j (n,) – specifies state whose lifetime we are calculating
 temperature – optional. Temperature at which the atom environment is, measured in K. If this parameter is nonzero, user has to specify transitions up to which state (due to blackbody decay) should be included in calculation.
 includeLevelsUpTo (int) – optional and not needed for atom lifetimes calculated at zero temperature. At non zero temperatures, this specify maximum principal quantum number of the state to which blackbody induced transitions will be included. Minimal value of the parameter in that case is \(n+1\)
Returns: State lifetime in units of s (seconds)
Return type: See also
getTransitionRate
for calculating rates of individual transition rates between the two states

getTransitionFrequency
(n1, l1, j1, n2, l2, j2)[source]¶ Calculated transition frequency in Hz
Returned values is given relative to the centre of gravity of the hyperfinesplit states.
Parameters:  n1 (int) – principal quantum number of the state from which we are going
 l1 (int) – orbital angular momentum of the state from which we are going
 j1 (float) – total angular momentum of the state from which we are going
 n2 (int) – principal quantum number of the state to which we are going
 l2 (int) – orbital angular momentum of the state to which we are going
 j2 (float) – total angular momentum of the state to which we are going
Returns: transition frequency (in Hz). If the returned value is negative, level from which we are going is above the level to which we are going.
Return type:

getTransitionRate
(n1, l1, j1, n2, l2, j2, temperature=0.0)[source]¶ Transition rate due to coupling to vacuum modes (black body included)
Calculates transition rate from the first given state to the second given state \(n_1,l_1,j_1\rangle \rightarrow n_2,j_2,j_2\rangle\) at given temperature due to interaction with the vacuum field. For zero temperature this returns Einstein A coefficient. For details of calculation see Ref. [4] and Ref. [5]. See Blackbody induced population transfer example snippet.
Parameters: Returns: transition rate in s \({}^{1}\) (SI)
Return type: References
[4] C. E. Theodosiou, PRA 30, 2881 (1984) https://doi.org/10.1103/PhysRevA.30.2881 [5] I. I. Beterov, I. I. Ryabtsev, D. B. Tretyakov, and V. M. Entin, PRA 79, 052504 (2009) https://doi.org/10.1103/PhysRevA.79.052504

getTransitionWavelength
(n1, l1, j1, n2, l2, j2)[source]¶ Calculated transition wavelength (in vacuum) in m.
Returned values is given relative to the centre of gravity of the hyperfinesplit states.
Parameters:  n1 (int) – principal quantum number of the state from which we are going
 l1 (int) – orbital angular momentum of the state from which we are going
 j1 (float) – total angular momentum of the state from which we are going
 n2 (int) – principal quantum number of the state to which we are going
 l2 (int) – orbital angular momentum of the state to which we are going
 j2 (float) – total angular momentum of the state to which we are going
Returns: transition wavelength (in m). If the returned value is negative, level from which we are going is above the level to which we are going.
Return type:

getZeemanEnergyShift
(l, j, mj, magneticFieldBz)[source]¶ Retuns linear (paramagnetic) Zeeman shift.
:math:`mathcal{H}_P=
rac{mu_B B_z}{hbar}(hat{L}_{ m z}+g_{ m S}S_{ m z})`
 Returns:
 float: energy offset of the state (in J)

groundStateN
= 0¶ principal quantum number for the ground state

levelDataFromNIST
= ''¶ location of stored NIST values of measured energy levels in eV

literatureDMEfilename
= ''¶ Filename of the additional literature source values of dipole matrix elements.
These additional values should be saved as reduced dipole matrix elements in J basis.

mass
= 0.0¶ atomic mass in kg

minQuantumDefectN
= 0¶ minimal quantum number for which quantum defects can be used; uses measured energy levels otherwise

potential
(l, s, j, r)[source]¶ returns total potential that electron feels
Total potential = core potential + SpinOrbit interaction
Parameters: Returns: potential (in a.u.)
Return type:

quadrupoleMatrixElementFile
= ''¶ location of harddisk stored dipole matrix elements

quantumDefect
= [[[0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0]], [[0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0, 0.0, 0.0]]]¶ Contains list of modified RydbergRitz coefficients for calculating quantum defects for [[ \(S_{1/2},P_{1/2},D_{3/2},F_{5/2}\)], [ \(S_{1/2},P_{3/2},D_{5/2},F_{7/2}\)]].

radialWavefunction
(l, s, j, stateEnergy, innerLimit, outerLimit, step)[source]¶ Radial part of electron wavefunction
Calculates radial function with Numerov (from outside towards the core). Note that wavefunction might not be calculated all the way to the requested innerLimit if the divergence occurs before. In that case third returned argument gives nonzero value, corresponding to the first index in the array for which wavefunction was calculated. For quick example see Rydberg wavefunction calculation snippet.
Parameters:  l (int) – orbital angular momentum
 s (float) – spin angular momentum
 j (float) – total angular momentum
 stateEnergy (float) – state energy, relative to ionization threshold, should be given in atomic units (Hatree)
 innerLimit (float) – inner limit at which wavefunction is requested
 outerLimit (float) – outer limit at which wavefunction is requested
 step (flaot) – radial step for integration mesh (a.u.)
Returns: \(r\)
\(R(r)\cdot r\)
Return type: Note
Radial wavefunction is not scaled to unity! This normalization condition means that we are using spherical harmonics which are normalized such that \(\int \mathrm{d}\theta~\mathrm{d}\psi~Y(l,m_l)^* \times Y(l',m_{l'}) = \delta (l,l') ~\delta (m_l, m_{l'})\).
Note
Alternative calculation methods can be added here (potenatial package expansion).

rc
= [0]¶ Model potential parameters fitted from experimental observations for different l (electron angular momentum)

scaledRydbergConstant
= 0¶ in eV
 preferQuantumDefects (bool) – Use quantum defects for energy level calculations. If False,
uses NIST ASD values
where available. If True, uses quantum defects for energy calculations
for principal quantum numbers equal or above

arc.alkali_atom_functions.
NumerovBack
(innerLimit, outerLimit, kfun, step, init1, init2)[source]¶ Full Python implementation of Numerov integration
Calculates solution function \(rad(r)\) with descrete step in \(r\) size of step, integrating from outerLimit towards the innerLimit (from outside, inwards) equation \(\frac{\mathrm{d}^2 rad(r)}{\mathrm{d} r^2} = kfun(r)\cdot rad(r)\).
Parameters:  innerLimit (float) – inner limit of integration
 outerLimit (flaot) – outer limit of integration
 kfun (function(double)) – pointer to function used in equation (see longer explanation above)
 step – descrete step size for integration
 init1 (float) – initial value, rad`(`outerLimit`+`step)
 init2 (float) – initial value, rad`(`outerLimit`+:math:`2cdot step)
Returns: \(r\) (a.u), \(rad(r)\);
Return type: numpy array of float , numpy array of float, int
Note
Returned function is not normalized!
Note
If
AlkaliAtom.cpp_numerov
swich is set to True (default option), much faster C implementation of the algorithm will be used instead. That is recommended option. See documentation installation instructions for more details.

arc.alkali_atom_functions.
loadSavedCalculation
(fileName)[source]¶ Loads previously saved calculation.
Loads
calculations_atom_pairstate.PairStateInteractions
andcalculations_atom_single.StarkMap
calculation instance from file named filename where it was previously saved withsaveCalculation
.Example
See example for
saveCalculation
.Parameters: fileName – name of the file where calculation will be saved Returns: saved calculation

arc.alkali_atom_functions.
printState
(n, l, j)[source]¶ Prints state spectroscopic label for numeric \(n\), \(l\), \(s\) label of the state
Parameters:

arc.alkali_atom_functions.
printStateString
(n, l, j)[source]¶ Returns state spectroscopic label for numeric \(n\), \(l\), \(s\) label of the state
Parameters: Returns: label for the state in standard spectroscopic notation
Return type: string

arc.alkali_atom_functions.
printStateStringLatex
(n, l, j)[source]¶ Returns latex code for spectroscopic label for numeric \(n\), \(l\), \(s\) label of the state
Parameters: Returns: label for the state in standard spectroscopic notation
Return type: string

arc.alkali_atom_functions.
saveCalculation
(calculation, fileName)[source]¶ Saves calculation for future use.
Saves
calculations_atom_pairstate.PairStateInteractions
andcalculations_atom_single.StarkMap
calculations in compact binary format in file named filename. It uses cPickle serialization library in Python, and also zips the final file.Calculation can be retrieved and used with
loadSavedCalculation
Parameters:  calculation – class instance of calculations (instance of
calculations_atom_pairstate.PairStateInteractions
orcalculations_atom_single.StarkMap
) to be saved.  fileName – name of the file where calculation will be saved
Example
Let’s suppose that we did the part of the
calculation_atom_pairstate.PairStateInteractions
calculation that involves generation of the interaction matrix. After that we can save the full calculation in a single file:calc = PairStateInteractions(Rubidium(), 60,0,0.5,60,0,0.5, 0.5,0.5) calc.defineBasis(0,0, 5,5, 25.e9) calc.diagonalise(np.linspace(0.5,10.0,200),150) saveCalculation(calc, "mySavedCalculation.pkl")
Then, at a later time, and even on the another machine, we can load that file and continue with calculation. We can for example explore the calculated level diagram:
calc = loadSavedCalculation("mySavedCalculation.pkl") calc.plotLevelDiagram() calc.showPlot() rvdw = calc.getVdwFromLevelDiagram(0.5,14,minStateContribution=0.5,\ showPlot = True)
Or, we can do additional matrix diagonalization, in some new range, then and find C6 by fitting the obtained level diagram:
calc = loadSavedCalculation("mySavedCalculation.pkl") calc.diagonalise(np.linspace(3,6.0,200),20) calc.getC6fromLevelDiagram(3,6.0,showPlot=True)
Note that for all loading of saved calculations we’ve been using function
loadSavedCalculation
.Note
This doesn’t save results of
plotLevelDiagram
for the corresponding calculations. Call the plot function before callingshowPlot
function for the corresponding calculation. calculation – class instance of calculations (instance of