Single atom calculations¶
Note
Some of the functions (Wavefunction, AtomSurfaceVdW, OpticalLattice1D, DynamicPolarizability, and optical materials properties) in this module are part of new ARC 3.0.0 version. See more at E. J. Robertson, N. Šibalić, R. M. Potvliege and M. P. A. Jones, *Computer Physics Communications* **261**, 107814 (2021) `https://doi.org/10.1016/j.cpc.2020.107814 .
Overview¶
Wavefunction Methods
Wavefunction.getRtimesPsiSpherical (theta, phi, r) 
Calculates list of \(r \cdot \psi_{m_s} (\theta, \phi, r)\) 
Wavefunction.getRtimesPsi (x, y, z) 
Calculates list of \(r \cdot \psi_{m_s} (x, y, z)\) 
Wavefunction.getPsi (x, y, z) 
Calculates list of \(\psi_{m_s} (x,y,z)\) 
Wavefunction.getRtimesPsiSquaredInPlane ([…]) 
Calculates \(r \cdot \psi^2\) on a mesh in a given plane. 
Wavefunction.plot2D ([plane, pointsPerAxis, …]) 
2D colour plot of \(r \cdot \psi^2\) wavefunction in a requested plane. 
Wavefunction.plot3D ([plane, pointsPerAxis, …]) 
3D colour surface plot of \(r \cdot \psi^2\) wavefunction in a requested plane. 
StarkMap Methods
StarkMap.defineBasis (n, l, j, mj, nMin, …) 
Initializes basis of states around state of interest 
StarkMap.diagonalise (eFieldList[, …]) 
Finds atom eigenstates in a given electric field 
StarkMap.plotLevelDiagram ([units, …]) 
Makes a plot of a stark map of energy levels 
StarkMap.showPlot ([interactive]) 
Shows plot made by plotLevelDiagram 
StarkMap.savePlot ([filename]) 
Saves plot made by plotLevelDiagram 
StarkMap.exportData (fileBase[, exportFormat]) 
Exports StarkMap calculation data. 
StarkMap.getPolarizability ([maxField, …]) 
Returns the polarizability of the state (set during the initalization process). 
StarkMap.getState (state, electricField, …) 
Returns basis states and coefficients that make up for a given electric field the eigenstate with largest contribution of the original state. 
LevelPlot Methods
LevelPlot is also called Grotrian diagram, or term diagram.
LevelPlot.makeLevels (nFrom, nTo, lFrom, lTo) 
Constructs energy level diagram in a given range 
LevelPlot.drawLevels ([units]) 
Draws a level diagram plot 
LevelPlot.showPlot () 
Shows a level diagram plot 
AtomSurfaceVdW Methods
AtomSurfaceVdW.getC3contribution (n1, l1, j1, …) 
Contribution to \(C_3\) of \(n_1, \ell_1, j_1\rangle\) state due to dipole coupling to \(n_2, \ell_2, j_2\rangle\) state. 
AtomSurfaceVdW.getStateC3 (n, l, j, …[, s, …]) 
Van der Waals atomsurface interaction coefficient for a given state (\(C_3\) in units of \(\mathrm{J}\cdot\mathrm{m}^3\) ) 
OpticalLattice1D Methods
OpticalLattice1D.getRecoilEnergy () 
Recoil energy for atoms in given optical lattice 
OpticalLattice1D.getTrappingFrequency (…) 
Atom’s trapping frequecy for given trapth depth 
OpticalLattice1D.defineBasis ([lLimit]) 
Define basis for Bloch band calculations 
OpticalLattice1D.diagonalise (…[, …]) 
Calculates energy levels (Bloch bands) for given quasimomentumList 
OpticalLattice1D.plotLevelDiagram () 
Plots energy level diagram (Bloch bands). 
OpticalLattice1D.BlochWavefunction (…) 
Bloch wavefunction as a function of 1D coordinate. 
OpticalLattice1D.getWannierFunction (x[, …]) 
Gives value at cooridnate x of a Wannier function localized at given lattice index. 
DynamicPolarizability Methods
DynamicPolarizability.defineBasis (nMin, nMax) 
Defines basis for calculation of dynamic polarizability 
DynamicPolarizability.getPolarizability (…) 
Calculates of scalar, vector, tensor, core and pondermotive 
DynamicPolarizability.plotPolarizability (…) 
Plots of polarisability for a range of wavelengths. 
Optical material properties
OpticalMaterial () 
Abstract class implementing calculation of basic properties for optical materials. 
Air () 
Air as an optical material at normal conditions 
Sapphire () 
Sapphire as optical material. 
Detailed documentation¶
This module provides calculations of singleatom properties.
Included calculations are Stark maps, level plot visualisations, lifetimes and radiative decays.

class
arc.calculations_atom_single.
AtomSurfaceVdW
(atom, surfaceMaterial=None)[source]¶ Calculates atomsurface Van der Waals interaction.
Energy of atom state \(i\rangle\) at distance \(z\) from the surface of material is offseted in energy by \(V_{\rm VdW}\) at small distances \(z\ll\rm{min}(\lambda_{i,j})\) , where \(\lambda_{i,j}\) are the wavelengths from atom state \(i \rangle\) to all stronglycoupled states \(j\) , due to (unretarded) atomsurface interaction, also called Van der Waals interaction. The interaction potential can be expressed as
\(V_{\rm VdW} =  \frac{C_3}{z^3}\)
This class calculates \(C_3\) for individual states \(i\rangle\).
See example atomsurface calculation snippet.
Parameters:  atom (
AlkaliAtom
orDivalentAtom
) – specified Alkali or Alkaline Earth atom whose interaction with surface we want to explore  material (from
arc.materials
) – specified surface material
Note
To find frequecy shift of a transition \(\rm a \rangle\rightarrow \rm b \rangle\), one needs to calculate difference in \(C_3\) coefficients obtained for the two states \(\rm a\rangle\) and \(\rm b\rangle\) respectively. See example TODO (TODO)

getC3contribution
(n1, l1, j1, n2, l2, j2, s=0.5)[source]¶ Contribution to \(C_3\) of \(n_1, \ell_1, j_1\rangle\) state due to dipole coupling to \(n_2, \ell_2, j_2\rangle\) state.
Calculates \(\frac{1}{4\pi\varepsilon_0}\ \frac{ n(\omega_{\rm ab})^2  1}{ n(\omega_{\rm ab})^2 + 1}\ \frac{ \left \langle a D_x  b \rangle \right^2 \ + \left \langle a  D_y  b \rangle \right^2 + \ 2 \cdot \left\langle a D_z b \rangle \right^2}{16}\)
where \({\rm a}\rangle \equiv n_1, \ell_1, j_1\rangle\) , \({\rm b}\rangle \equiv n_2, \ell_2, j_2\rangle\), \(\mathbf{D} \equiv e \cdot \mathbf{r} \ \equiv \hat{x} D_x + \hat{y} D_y\ + \hat{z} D_z\) is atomic dipole operator and \(n(\omega_{\rm ab})\) is refractive index of the considered surface at transition frequency \(\omega_{\rm ab}\) .
Parameters:  n1 (int) – principal quantum number of state 1
 l1 (int) – orbital angular momentum of state 1
 j1 (float) – total angular momentum of state 1
 n2 (int) – principal quantum number od state 2
 l2 (int) – orbital angular momentum of state 2
 j2 (float) – total angular momentum of state 2
 s (float) – optional, spin angular momentum of states. Default value of 0.5 is correct for AlkaliAtoms. For DivalentAtom it has to be explicitly stated
Returns: contribution to VdW coefficient \(C_3\) ,estimated error \(\delta C_3\) (in units of \({\rm J}\cdot{\rm m}^3\)), and refractive index \(n\) of the surface material for the given transition.
Return type: Warning
This is just contribution of one transition to the level shift of a particular state. To calculate total level shift, check
AtomSurfaceVdW.getStateC3

getStateC3
(n, l, j, coupledStatesList, s=0.5, debugOutput=False)[source]¶ Van der Waals atomsurface interaction coefficient for a given state (\(C_3\) in units of \(\mathrm{J}\cdot\mathrm{m}^3\) )
Parameters:  n (int) – principal quantum number of the state
 l (int) – orbital angular momentum of the state
 j (int) – total angular momentum of state
 coupledStatesList (array) – array of states that are strongly dipolecoupled to the initial state, whose contribution to \(C_3\) will be take into account. Format [[n1,l1,j1],…]
 s (float, optional) – total spin angular momentum for the considered state. Default value of 0.5 is correct for AlkaliAtoms, but it has to be explicitly specifiied for DivalentAtom.
 debugOutput (bool, optional) – prints additional output information, False by default.
Returns: \(C_3\) (in units of \({\rm J}\cdot {\rm m}^3\) ), estimated error \(\delta C_3\)
Return type:
 atom (

class
arc.calculations_atom_single.
DynamicPolarizability
(atom, n, l, j, s=0.5)[source]¶ Calculations of magic wavelengths and dynamic polarizability (scalar and tensor).
Parameters:  atom – alkali or alkaline element of choice
 n (int) – principal quantum number of the selected stated
 l (int) – orbital angular momentum of the selected state
 j (float) – total angular momentum of selected state
 s (float) – optional, spin state of the atom. Default value of 0.5 is correct for Alkali atoms, but it has to be explicitly specified for DivalentAtom.

defineBasis
(nMin, nMax)[source]¶ Defines basis for calculation of dynamic polarizability
Parameters:

getPolarizability
(driveWavelength, units='SI', accountForStateLifetime=False, mj=None)[source]¶  Calculates of scalar, vector, tensor, core and pondermotive polarizability, and returns state corresponding to the closest transition resonance.
Note that pondermotive polarisability is calculated as \(\alpha_P = e^2 / (2 m_e \omega^2)\), i.e. assumes that the definition of the energy shift in field \(E\) is \(\frac{1}{2}\alpha_P E^2\). For more datils check the preprint arXiv:2007.12016 that introduced the update.
 Args:
 driveWavelength (float): wavelength of driving field
 (in units of m)
 units (string): optional, ‘SI’ or ‘a.u.’ (equivalently ‘au’),
 switches between SI units for returned result (\(Hz V^2 m^2\) ) and atomic units (”\(a_0^3\) “). Defaul ‘SI’
 accountForStateLifetime (bool): optional, should we account
 for finite transition linewidths caused by finite state lifetimes. By default False.
 Returns:
 scalar, vector, tensor, pondermotive polarisability of state, core polarisability and atomic state whose resonance is closest in energy. Returned units depend on units parameter (default SI).

plotPolarizability
(wavelengthList, mj=None, addToPlotAxis=None, line='b', units='SI', addCorePolarisability=True, addPondermotivePolarisability=False, accountForStateLifetime=False, debugOutput=False)[source]¶ Plots of polarisability for a range of wavelengths.
Can be combined for different states to allow finding magic wavelengths for pairs of states. Currently supports only driving with linearly polarised light. See example magic wavelength snippet.
Parameters:  wavelengthList (array) – wavelengths for which we want to calculate polarisability (in units of m).
 mj (float) – optional, mj projection of the total angular momenutum for the states for which we are calculating polarisability. By default it’s +j.
 line (string) – optional, line style short definition to be passed to matplotlib when plotting calculated polarisabilities
 units (string) – optional, ‘SI’ or ‘a.u.’ (equivalently ‘au’), switches between SI units for returned result (\(Hz V^2 m^2\) ) and atomic units (”\(a_0^3\) “). Deafault ‘SI’.
 addCorePolarisability (bool) – optional, should ionic core polarisability be taken into account. By default True.
 addPondermotivePolarisability (bool) – optional, should pondermotive polarisability (also called freeelectron polarisability) be added to the total polarisability. Default is False. It assumes that there is no significant variation of trapping field intensity over the range of the electric cloud. If this condition is not satisfied, one has to calculate total shift as average over the electron wavefunction.
 accountForStateLifetime (bool) – optional, should we account for finite transition linewidths caused by finite state lifetimes. By default False.
 debugOutput (bool) – optonal. Print additional output on resonances Default value False.

class
arc.calculations_atom_single.
LevelPlot
(atomType)[source]¶ Single atom level plots and decays (a Grotrian diagram, or term diagram)
For an example see Rydberg energy levels example snippet.
Parameters: atom ( AlkaliAtom
orDivalentAtom
) – ={arc.alkali_atom_data.Lithium6
,arc.alkali_atom_data.Lithium7
,arc.alkali_atom_data.Sodium
,arc.alkali_atom_data.Potassium39
,arc.alkali_atom_data.Potassium40
,arc.alkali_atom_data.Potassium41
,arc.alkali_atom_data.Rubidium85
,arc.alkali_atom_data.Rubidium87
,arc.alkali_atom_data.Caesium
,arc.divalent_atom_data.Strontium88
,arc.divalent_atom_data.Calcium40
arc.divalent_atom_data.Ytterbium174
} Alkali atom type whose levels we want to examine
drawLevels
(units='eV')[source]¶ Draws a level diagram plot
 Arg:
 units (
char
,optional): possible values {‘eV’,’cm’,’GHz’};  [case insensitive] if the value is ‘eV’ (default), Stark diagram will be plotted as energy in units eV; if the string contains ‘cm’ Stark diagram will be plotted in energy units cm \({}^{1}\); if value is ‘GHz’, Stark diagram will be plotted as energy \(/h\) in units of GHz;
 units (

makeLevels
(nFrom, nTo, lFrom, lTo, sList=[0.5])[source]¶ Constructs energy level diagram in a given range
Parameters:  nFrom (int) – minimal principal quantum number of the states we are interested in
 nTo (int) – maximal principal quantum number of the states we are interested in
 lFrom (int) – minimal orbital angular momentum of the states we are interested in
 lTo (int) – maximal orbital angular momentum of the states we are interested in
 sList (float) – optional, spin angular momentum. Default value of [0.5] corresponds to Alkali atoms. For Alkaline Earth it has to be specified. For divalent atoms one can plot either one spin state by setting for example sList=[0]`, or both spin states sList=[0,1]`


class
arc.calculations_atom_single.
OpticalLattice1D
(atom, trapWavenegth)[source]¶ Atom properties in optical lattices in 1D.
See example optical lattice calculations snippet.
Parameters:  atom – one of AlkaliAtom or DivalentAtom
 trapWavenegth (float) – wavelength of trapping laser light (in units of m)

BlochWavefunction
(trapPotentialDepth, quasimomentum, blochBandIndex)[source]¶ Bloch wavefunction as a function of 1D coordinate.
Example
Returns Bloch wavefunction. Use as following:
trapPotentialDepth = 40 # units of recoil energy quasimomentum = 0 blochBandIndex = 0 # Bloch band lowest in energy is 0 wf = lattice.BlochWavefunction(trapPotentialDepth, quasimomentum, blochBandIndex) wf(x) # returns complex number corresponding to value of Bloch # wavefunction at point x (cooridnate given in units of # 1/k where k = 2 \pi / trapWavenegth ) # by default k=1, so one full wavelength is 2\pi
Parameters:  trapPotentialDepth (float) – (in units of recoil energy
OpticalLattice1D.getRecoilEnergy
)  quasimomentum (float) – (in units of 2 pi /
OpticalLattice1D.trapWavenegth
; note that reciprocal lattice momentum in this units is 2, and that full range of quasimomentum is from 1 to +1)
Returns: Bloch wavefunction as a function of coordinate (see call example above)
 trapPotentialDepth (float) – (in units of recoil energy

defineBasis
(lLimit=35)[source]¶ Define basis for Bloch band calculations
Bloch states are calculated suming up all relevant states with momenta in range [lLimit * 4 * pi /trapWavenegth, +lLimit * 4 * pi /trapWavenegth] Note that factor of 4 occurs since potential lattice period is twice the trapWavelength for standing wave.
Parameters: lLimit (integer) – Optional, defines maximal momentum to be taken for calculation of Bloch States as lLimit * 4 * pi / trapWavenegth . By default set to 35.

diagonalise
(trapPotentialDepth, quasimomentumList, saveBandIndex=None)[source]¶ Calculates energy levels (Bloch bands) for given quasimomentumList
Energy levels and their quasimomentum are saved in internal variables energy and quasimomentum. Energies are saved in units of recoil energy, and quasimomentum in units of The optional parameter saveBandIndex specifies index of the Bloch band for which eigenvectrors should be saved. If provided, eigenvectors for each value quasimomentumList[i] are saved in savedBlochBand[i].
Parameters:  latticePotential (float) – lattice depth formed by the standing wave of laser, with wavelength specified during initialisation of the lattice (in units of recoil energy).
 quasimomentumList (array) – array of quasimomentum values for which energy levels will be calculated (in units of \(\hbar \cdot k\), where \(k\) is trapping laser wavevector; since reciprocal lattice has twice the trapping laser wavevector due to standing wave, full range of quasimomentum is from 1 to +1)
 saveBandIndex (int) – optional, default None. If provided, specifies for which Bloch band should the eignevectors be also saved. saveBlochBand=0 corresponds to lowest energy band.

energy
= []¶ energy of states obtained by
OpticalLattice1D.diagonalise
method in format [[energies for quasimomentum1 ], [energies for quasimomentum2 ], …]

getRecoilEnergy
()[source]¶ Recoil energy for atoms in given optical lattice
Returns: recoil energy in units of J Return type: float

getTrappingFrequency
(trapPotentialDepth)[source]¶ Atom’s trapping frequecy for given trapth depth
Parameters: trapPotentialDepth (float) – lattice depth (in units of J) Returns: trapping frequency (in Hz) Return type: float

getWannierFunction
(x, latticeIndex=0, k=1)[source]¶ Gives value at cooridnate x of a Wannier function localized at given lattice index.
Parameters:  x (float) – spatial coordinate (in units of \(2\pi/k\) ; for default value of laser drivng wavevecto \(k=1\) , one trappinWavelength is \(2\pi\) ). Coordinate origin is at latticeIndex=0 .
 latticeIndex (int) – optional, lattice index at which the Wannier function is localised. By defualt 0.
 k (float) – optional; laser driving wavevector, defines unit of length. Default value is 1, making one trapping laser wavelenth equal to \(2\pi\)

plotLevelDiagram
()[source]¶ Plots energy level diagram (Bloch bands).
Based on diagonalisation of the lattice potential, plots descrete eigen energy spectra obtained for each value of the quasimomentum used in
OpticalLattice1D.diagonalise
method.Returns: matploltib figure with a Bloch bands

quasimomentum
= []¶ list of quzimomentum for which the energies of states was calculated by
OpticalLattice1D.diagonalise
method in format [quasimomentum1, quasimomentum2, …]

savedBlochBand
= []¶ list of saved eigen energy state compositions for each of the Calculated quasimomentums for the selected index of the Bloch band in
OpticalLattice1D.diagonalise
method in format [[eigen state decomposition for quasimomentum 1], [eigen state decomposition for quasimomentum 2], …]

trapPotentialDepth
= 0¶ save slattice trap potential depth for which calculation
OpticalLattice1D.diagonalise
was done

class
arc.calculations_atom_single.
StarkMap
(atom)[source]¶ Calculates Stark maps for single atom in a field
This initializes calculation for the atom of a given type. For details of calculation see Zimmerman [1]. For a quick working example see Stark map example snippet.
Parameters: atom ( AlkaliAtom
orDivalentAtom
) – ={arc.alkali_atom_data.Lithium6
,arc.alkali_atom_data.Lithium7
,arc.alkali_atom_data.Sodium
,arc.alkali_atom_data.Potassium39
,arc.alkali_atom_data.Potassium40
,arc.alkali_atom_data.Potassium41
,arc.alkali_atom_data.Rubidium85
,arc.alkali_atom_data.Rubidium87
,arc.alkali_atom_data.Caesium
,arc.divalent_atom_data.Strontium88
,arc.divalent_atom_data.Calcium40
arc.divalent_atom_data.Ytterbium174
} Select the alkali metal for energy level diagram calculationExamples
State \(28~S_{1/2}~m_j=0.5\) polarizability calculation
>>> from arc import * >>> calc = StarkMap(Caesium()) >>> calc.defineBasis(28, 0, 0.5, 0.5, 23, 32, 20) >>> calc.diagonalise(np.linspace(00.,6000,600)) >>> print("%.5f MHz cm^2 / V^2 " % calc.getPolarizability()) 0.76705 MHz cm^2 / V^2
Stark map calculation
>>> from arc import * >>> calc = StarkMap(Caesium()) >>> calc.defineBasis(28, 0, 0.5, 0.5, 23, 32, 20) >>> calc.diagonalise(np.linspace(00.,60000,600)) >>> calc.plotLevelDiagram() >>> calc.showPlot() << matplotlib plot will open containing a Stark map >>
Examples
Advanced interfacing of Stark map calculations (StarkMap class) Here we show one easy way to obtain the Stark matrix (from diagonal
mat1
and offdiagonal partmat2
) and basis states (stored inbasisStates
), if this middleproduct of the calculation is needed for some code build on top of the existing ARC package.>>> from arc import * >>> calc = StarkMap(Caesium()) >>> calc.defineBasis(28, 0, 0.5, 0.5, 23, 32, 20) >>> # Now we have matrix and basis states, that we can used in our own code >>> # Let's say we want Stark map at electric field of 0.2 V/m >>> eField = 0.2 # V/m >>> # We can easily extract Stark matrix >>> # as diagonal matrix (state detunings) >>> # + offdiagonal matrix (propotional to electric field) >>> matrix = calc.mat1+calc.mat2*eField >>> # and the basis states as array [ [n,l,j,mj] , ...] >>> basisStates = calc.basisStates >>> # you can do your own calculation now...
References
[1] M. L. Zimmerman et.al, PRA 20:2251 (1979) https://doi.org/10.1103/PhysRevA.20.2251 
ax
= None¶ pointer towards matplotlib figure axis after
plotLevelDiagram
is called to create figure

basisStates
= None¶ List of basis states for calculation in the form [ [n,l,j,mj], …]. Calculated by
defineBasis
.

defineBasis
(n, l, j, mj, nMin, nMax, maxL, Bz=0, progressOutput=False, debugOutput=False, s=0.5)[source]¶ Initializes basis of states around state of interest
Defines basis of states for further calculation. \(n,l,j,m_j\) specify state whose neighbourhood and polarizability we want to explore. Other parameters specify basis of calculations. This method stores basis in
basisStates
, while corresponding interaction matrix is stored in two parts. First part is diagonal electricfield independent part stored inmat1
, while the second partmat2
corresponds to offdiagonal elements that are propotional to electric field. Overall interaction matrix for electric field eField can be then obtained as fullStarkMatrix =mat1
+mat2
*eFieldParameters:  n (int) – principal quantum number of the state
 l (int) – angular orbital momentum of the state
 j (flaot) – total angular momentum of the state
 mj (float) – projection of total angular momentum of the state
 nMin (int) – minimal principal quantum number of the states to be included in the basis for calculation
 nMax (int) – maximal principal quantum number of the states to be included in the basis for calculation
 maxL (int) – maximal value of orbital angular momentum for the states to be included in the basis for calculation
 Bz (float) – optional, magnetic field directed along zaxis 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) – if True prints the progress of calculation; Set to false by default.  debugOutput (
bool
, optional) – if True prints additional information usefull for debuging. Set to false by default.  s (float) – optional. Total spin angular momentum for the state. Default value of 0.5 is correct for Alkaline Atoms, but value has to be specified explicitly for divalent atoms (e.g. s=0 or s=1 for singlet and triplet states, that have total spin angular momenutum equal to 0 or 1 respectively).

diagonalise
(eFieldList, drivingFromState=[0, 0, 0, 0, 0], progressOutput=False, debugOutput=False)[source]¶ Finds atom eigenstates in a given electric field
Eigenstates are calculated for a list of given electric fields. To extract polarizability of the originaly stated state see
getPolarizability
method. Results are saved ineFieldList
,y
andhighlight
.Parameters:  eFieldList (array) – array of electric field strength (in V/m) for which we want to know energy eigenstates
 progressOutput (
bool
, optional) – if True prints the progress of calculation; Set to false by default.  debugOutput (
bool
, optional) – if True prints additional information usefull for debuging. Set to false by default.

eFieldList
= None¶ Saves electric field (in units of V/m) for which energy levels are calculated
See also

exportData
(fileBase, exportFormat='csv')[source]¶ Exports StarkMap calculation data.
Only supported format (selected by default) is .csv in a humanreadable form with a header that saves details of calculation. Function saves three files: 1) filebase _eField.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.

fig
= None¶ pointer towards matplotlib figure after
plotLevelDiagram
is called to create figure

getPolarizability
(maxField=10000000000.0, showPlot=False, debugOutput=False, minStateContribution=0.0)[source]¶ Returns the polarizability of the state (set during the initalization process).
Fits offset of the energy level of the state to \(\frac{1}{2} \alpha_0 E^2\), where \(E\) is the applied static electric field, and returns fitted value \(\alpha_0\)
Parameters:  maxField (
float
, optional) – maximum field (in V/m) to be used for fitting the polarizability. By default, max field is very large, so it will use eigenvalues calculated in the whole range.  showPlot (
bool
, optional) – shows plot of calculated eigenValues of the given state (dots), and the fit (solid line) for extracting polarizability  debugOutput (
bool
, optional) – if True prints additional information usefull for debuging. Set to false by default.
Returns: scalar polarizability in units of MHz cm \(^2\) / V \(^2\)
Return type:  maxField (

getState
(state, electricField, minN, maxN, maxL, accountForAmplitude=0.95, debugOutput=False)[source]¶ Returns basis states and coefficients that make up for a given electric field the eigenstate with largest contribution of the original state.
Parameters:  state (array) – target basis state in format \([n,\ell,j,m_j]\) corresponding to the state whose composition we want to track as we apply the electric field
 electricField (float) – applied DC electric field in units of V/m.
 minN (int) – minimal principal quantum number to be taken for calculation of the Stark mixing
 maxN (int) – maximal principal quantum nunber to be take for calculation of the Start mixing
 maxL (int) – maximal orbital angular momentum of states that should be taken in calculation of the Stark mixing
 accountForAmplitude (float) – optinal, relative amplitude of state that should be reached with the subset of the eigen states returned. The returned eigen states will be sorted in the declining relative contribution to the final eigen state, and once total accounted amplitude of the state reaches 0.95, further output of additional small contribution of the other basis states to the final states will be supressed. Default value of 0.95 will force output until basis state accounts for 95% of the state amplitude.
 debugOutput (bool) – optional, prints additional debug information if True. Default False.
Returns: array of states in format [[n1, l1, j1, mj1], …] and array of complex coefficients in format [c1, c2, …] corresponding the projections of the eigenstate (thas has largest contribution of the original state in the given electric field) on the basis states, and energy of the found state in (eV)

highlight
= None¶ highlight[i] is an array of values measuring highlighted feature in the eigenstates at electric field intensity eFieldList[i]. E.g. highlight[i][j] measures highlighted feature of the state with energy y[i][j] at electric field eFieldList[i]. What will be highlighted feature is defined in the call of
diagonalise
(see that part of documentation for details).See also

indexOfCoupledState
= None¶ Index of coupled state (initial state passed to
defineBasis
) inbasisStates
list of basis states

mat1
= None¶ diagonal elements of Starkmatrix (detuning of states) calculated by
defineBasis
in the basisbasisStates
.

mat2
= None¶ offdiagonal elements of Starkmatrix divided by electric field value. To get off diagonal elemements multiply this matrix with electric field value. Full Stark matrix is obtained as fullStarkMatrix =
mat1
+mat2
*eField. Calculated bydefineBasis
in the basisbasisStates
.

plotLevelDiagram
(units='cm', highlightState=True, progressOutput=False, debugOutput=False, highlightColour='red', addToExistingPlot=False)[source]¶ Makes a plot of a stark map of energy levels
To save this plot, see
savePlot
. To print this plot seeshowPlot
. Pointers (handles) towards matplotlib figure and axis used are saved infig
andax
variables respectively.Parameters:  units (
char
,optional) – possible values {’cm’,’GHz’,’eV’}; [case insensitive] if the string contains ‘cm’ (default) Stark diagram will be plotted in energy units cm \({}^{1}\); if value is ‘GHz’, Stark diagram will be plotted as energy \(/h\) in units of GHz; if the value is ‘eV’, Stark diagram will be plotted as energy in units eV.  highlightState (
bool
, optional) – False by default. If True, scatter plot colour map will map in red amount of original state for the given eigenState  progressOutput (
bool
, optional) – if True prints the progress of calculation; Set to False by default.  debugOutput (
bool
, optional) – if True prints additional information usefull for debuging. Set to False by default.  addToExistingPlot (
bool
, optional) – if True adds points to existing old plot. Note that then interactive plotting doesn’t work. False by default.
 units (

s
= None¶ spin manifold in which we are working default value of 0.5 is correct for Alkaline Atoms. Otherwise it has to be specified when calling defineBasis as s=0 or s=1 for singlet and triplet states respectively

savePlot
(filename='StarkMap.pdf')[source]¶ Saves plot made by
plotLevelDiagram
Parameters: filename ( str
, optional) – file location where the plot should be saved

showPlot
(interactive=True)[source]¶ Shows plot made by
plotLevelDiagram

y
= None¶ y[i] is an array of eigenValues corresponding to the energies of the atom states at the electric field eFieldList[i]. For example y[i][j] is energy of the j eigenvalue (energy of the state) measured in cm \({}^{1}\) relative to the ionization threshold.
See also


class
arc.calculations_atom_single.
Wavefunction
(atom, basisStates, coefficients)[source]¶  Calculates and plots electron wavefunctions.
For an example see wavefunction plotting example snippet.
 Args:
atom: atom type considered (for example
Rubidum87()
) basisStates (array): array of states in fine basis that contributeto the state whose wavefunction is requested. \([[n_1, \ell_1, j_1, m_{j1}], ...]\) For efficient calculation do not pass all the possible basis states, but just the once that have significant contribution to the reqested state. coefficients (array): array [c1, …] of complex coefficents
 \(c_i = \langle \psi_i \psi\rangle\) corresponding to decomposition of required state \(\psi\rangle\) on basis states \(\psi_i \rangle\) .

getPsi
(x, y, z)[source]¶ Calculates list of \(\psi_{m_s} (x,y,z)\)
At point define by Cartesian coordinates returns list of \(\psi_{m_s} (x,y,z)\) wavefunction values corresponding to different electron spin projection values \(m_s\).
Parameters:  x (float) – Cartesian coordinates of selected point, relative to the atom core. (in atomic units of Bohr radius \(a_0\))
 y (float) – Cartesian coordinates of selected point, relative to the atom core. (in atomic units of Bohr radius \(a_0\))
 z (float) – Cartesian coordinates of selected point, relative to the atom core. (in atomic units of Bohr radius \(a_0\))
Returns: list of complex values corresponding to \(\psi_{m_s} (\theta, \phi, r)\) for different spin states \(m_s\) contributing to the state in decreasing order of \(m_s\). For example, for
arc.AlkaliAtom
returns \(\psi_{m_s=+1/2} (\theta, \phi, r)\) and \(\psi_{m_s=1/2} (\theta, \phi, r)\) . )`.

getRtimesPsi
(x, y, z)[source]¶ Calculates list of \(r \cdot \psi_{m_s} (x, y, z)\)
At a point defined by Cartesian coordinates returns list of \(r \cdot \psi_{m_s} (x, y, z)\) wavefunction values for different electron spin projection values \(m_s\).
Parameters:  x (float) – Cartesian coordinates of selected point, relative to the atom core. (in atomic units of Bohr radius \(a_0\))
 y (float) – Cartesian coordinates of selected point, relative to the atom core. (in atomic units of Bohr radius \(a_0\))
 z (float) – Cartesian coordinates of selected point, relative to the atom core. (in atomic units of Bohr radius \(a_0\))
Returns: list of complex values corresponding to \(r \cdot \psi_{m_s} (\theta, \phi, r)\) for different spin states \(m_s\) contributing to the state in decreasing order of \(m_s\). For example, for
arc.AlkaliAtom
returns \(r \cdot \psi_{m_s=+1/2} (\theta, \phi, r)\) and \(r \cdot \psi_{m_s=1/2} (\theta, \phi, r)\) . )`, where \(r=\sqrt{x^2+y^2+z^2}\).

getRtimesPsiSpherical
(theta, phi, r)[source]¶ Calculates list of \(r \cdot \psi_{m_s} (\theta, \phi, r)\)
At point defined by spherical coordinates, returns list of \(r \cdot \psi_{m_s} (\theta, \phi, r)\) wavefunction values for different electron spin projection values \(m_s\).
Coordinates are defined relative to atomic core.
Parameters:  theta (float) – polar angle (angle between \(z\) axis and vector pointing towards selected point) (in units of radians).
 phi (float) – azimuthal angle (angle between \(x\) axis and projection at \(xy\) plane of vector pointing towards selected point) (in units of radians).
 r (float) – distance between coordinate origin and selected point. (in atomic units of Bohr radius \(a_0\))
Returns: list of complex values corresponding to \(\psi_{m_s} (\theta, \phi, r)\) for different spin states \(m_s\) contributing to the state in decreasing order of \(m_s\). For example, for
arc.AlkaliAtom
returns \(r \cdot \psi_{m_s=+1/2} (\theta, \phi, r)\) and \(r \cdot \psi_{m_s=1/2} (\theta, \phi, r) \)

getRtimesPsiSquaredInPlane
(plane='xz', pointsPerAxis=150, axisLength=None, units='atomic')[source]¶ Calculates \(r \cdot \psi^2\) on a mesh in a given plane.
Parameters:  plane (str) – optiona, set’s calculation plane to ‘xy’ or ‘xz’. Default value ‘xy’
 pointsPerAxis (int) – optional, a number of mesh points per Carthesian axis. Default value of 150, gives a mesh with total size of \(150 \times 150 = 22500\) points.
 axisLength (float) – optional, length of the square in the selected plane on which wavefunction will be calculated. By default it is largw enough to fit the whole wavefunction (in atomic units of Bohr radius \(a_0\)).
 units (str) – optional, units of length in which calculated mesh will be returned (note that axisLength is on the other hand always in atomi units.). Supported values are ‘atomic’ or ‘nm’. Default value ‘atomic’ .
Returns: meshCoordinate1, meshCoordinate2 and \(r \cdot \psi^2 = \sum_{m_s} r \cdot \psi_{m_s}^2\), where sum is over possible electron spin projection values \(m_s\).

plot2D
(plane='xz', pointsPerAxis=150, axisLength=None, units='atomic', colorbar=True, labels=True)[source]¶ 2D colour plot of \(r \cdot \psi^2\) wavefunction in a requested plane.
Parameters:  plane (str) – optiona, set’s calculation plane to ‘xy’ or ‘xz’. Default value ‘xy’
 pointsPerAxis (int) – optional, a number of mesh points per Carthesian axis. Default value of 150, gives a mesh with total size of \(150 \times 150 = 22500\) points.
 axisLength (float) – optional, length of the square in the selected plane on which wavefunction will be calculated. By default it is large enough to fit the whole wavefunction (in atomic units of Bohr radius \(a_0\)).
 units (str) – optional, units of length in which calculated mesh will be returned (note that axisLength is on the other hand always in atomi units.). Supported values are ‘atomic’ or ‘nm’. Default value ‘atomic’ .
 colorbar (bool) – optional, determens if the colour bar scale of should be shown. Default value is True.
 labels (bool) – optional, determines if the labels on the axis of the plot should be shown. Default value is True.
Returns: matplotlib.pyplot.figure
object with a requested plot. Use show() method to see figure.

plot3D
(plane='xz', pointsPerAxis=150, axisLength=None, units='atomic', labels=True)[source]¶ 3D colour surface plot of \(r \cdot \psi^2\) wavefunction in a requested plane.
Parameters:  plane (str) – optiona, set’s calculation plane to ‘xy’ or ‘xz’. Default value ‘xy’
 pointsPerAxis (int) – optional, a number of mesh points per Carthesian axis. Default value of 150, gives a mesh with total size of \(150 \times 150 = 22500\) points.
 axisLength (float) – optional, length of the square in the selected plane on which wavefunction will be calculated. By default it is large enough to fit the whole wavefunction (in atomic units of Bohr radius \(a_0\)).
 units (str) – optional, units of length in which calculated mesh will be returned (note that axisLength is on the other hand always in atomi units.). Supported values are ‘atomic’ or ‘nm’. Default value ‘atomic’ .
 labels (bool) – optional, determines if the labels on the axis of the plot should be shown. Default value is True.
Returns: matplotlib.figure
object with a requested plot. Use show() method to see figure.

class
arc.materials.
Air
[source]¶ Bases:
arc.materials.OpticalMaterial
Air as an optical material at normal conditions

class
arc.materials.
OpticalMaterial
[source]¶ Bases:
object
Abstract class implementing calculation of basic properties for optical materials.

name
= ''¶ Humanfriendly name of material

sources
= []¶ List of .csv files listing refractive index measurements first column in these files is wavelength (in mu m), the second refractive index

sourcesComment
= []¶ Any notes about measured values

sourcesRange
= []¶ Array of max and minimal wavelegth pairs [lambdaMin, lambdaMax] for each of the sources. Automatically loaded from sources list


class
arc.materials.
Sapphire
[source]¶ Bases:
arc.materials.OpticalMaterial
Sapphire as optical material.