These changes are aimed at improving the way arc calculates the quantum defect (and hence energy) of states with high angular momentum (l>5). This will improve accuracy of Stark maps etc. arc has an edge case where quantum defects for states with l >= 5 are approximated by scaling the measured (tabled) value of the g (l=4) quantum defect.
I also include some updates to the Caesium and potassium quantum defects which will improve the calculation of energies. These are from Deiglmayr, Johannes, et al. Physical Review A 93.1 (2016): 013424. and Peper, Michael, et al. Physical Review A 100.1 (2019): 012501.
Methods
The code replaces the current method by direct calculation of the polarisation energy (effective adiabatic polarisation energy) which is more accurate than the current method. This requires two constants $\alpha{d}$ and $\alpha{q}$, the effective dipole and quadrupole polarizability of the Ion species (Cs+, Rb+... etc.) and the expectation values of the hydrogenic wavefunction (where closed forms exist). The former are taken from the available literature which are referenced in the code. The latter is calculated, (r4 and r6). I have also added hydrogenic fine structure splitting to high l states to lift degeneracy.
Examples
Transition
Old Method (GHz)
New Method (GHz)
Literature (GHz)
Cs 17g -> 17h [3][5]
6.119
6.013
6.018(3), 6.0072(7)
Cs 17h -> 17i [3][5]
1.850
1.855
1.850(1), 1.8513(7)
Cs 15g -> 15h [5]
8.814
8.708
8.7008(4)
Cs 15h -> 15i [5]
2.693
2.681
2.6751(5)
Cs 16i -> 16k [5]
0.801
0.8396
0.840(1)
Cs 17k -> 17l [5]
0.280
0.3058
0.306(1)
Rb 18g -> 18h [4]
2.964
2.8868
2.8704
Rb 18h -> 18i [4]
0.884
0.8985
0.8976
Rb 19g -> 19h [4]
2.526
2.456
2.4447
Rb 19h -> 19i [4]
0.752
0.7657
0.7659
Most improvements are to the calculation of h (l=5) levels which are over ~100MHz different from measured. The others improve from ~15 MHz to <5 MHz accuracy.
Details
The current approximation in the code is
$$\delta{\ell} = \delta{g}(4/l)^{5}$$
This approximation is from [1] and is based on the formula for the polarisation energy (for an introduction see [2]) that is
$$\delta E = \frac{\delta{l}}{n^{3}} = -\frac{1}{2}\alpha{d}\langle r^{-4}\rangle -\frac{1}{2}\alpha{q}\langle r^{-6}\rangle$$ where $\alpha{d}$ are constants associated with the Ion (e.g. Cs+).
By approximating $$\langle r^{-4} \rangle \approx \frac{3}{2n^{3}l^{5}}$$ and taking the $\langle r^{-6}\rangle$ expression to be zero, we find
which is the current approximation. Instead, the code calculates the full expression for the polarisation and adds this onto the hydrogenic energy. We also add hydrogenic energy splitting's to states with l>5 (which is set to be zero presently). This is expected and seen in [4] as there is little core penetration in high angular momentum states.
References:
[1] - Gallagher, Thomas F. "Rydberg atoms."
[2] - Freeman and Kleppner Physical Review A 14.5 (1976): 1614.
[3] - Safinya, K. A., T. F. Gallagher, and W. Sandner. Physical Review A 22.6 (1980): 2672.
[4] - Berl, S. J., et al. Physical Review A 102.6 (2020): 062818.
[5] - Unpublished own work (in prep.)
Overview
These changes are aimed at improving the way arc calculates the quantum defect (and hence energy) of states with high angular momentum (l>5). This will improve accuracy of Stark maps etc. arc has an edge case where quantum defects for states with l >= 5 are approximated by scaling the measured (tabled) value of the g (l=4) quantum defect.
I also include some updates to the Caesium and potassium quantum defects which will improve the calculation of energies. These are from Deiglmayr, Johannes, et al. Physical Review A 93.1 (2016): 013424. and Peper, Michael, et al. Physical Review A 100.1 (2019): 012501.
Methods
The code replaces the current method by direct calculation of the polarisation energy (effective adiabatic polarisation energy) which is more accurate than the current method. This requires two constants $\alpha{d}$ and $\alpha{q}$, the effective dipole and quadrupole polarizability of the Ion species (Cs+, Rb+... etc.) and the expectation values of the hydrogenic wavefunction (where closed forms exist). The former are taken from the available literature which are referenced in the code. The latter is calculated, (r4 and r6). I have also added hydrogenic fine structure splitting to high l states to lift degeneracy.
Examples
Most improvements are to the calculation of h (l=5) levels which are over ~100MHz different from measured. The others improve from ~15 MHz to <5 MHz accuracy.
Details
The current approximation in the code is
$$\delta{\ell} = \delta{g}(4/l)^{5}$$
This approximation is from [1] and is based on the formula for the polarisation energy (for an introduction see [2]) that is
$$\delta E = \frac{\delta{l}}{n^{3}} = -\frac{1}{2}\alpha{d}\langle r^{-4}\rangle -\frac{1}{2}\alpha{q}\langle r^{-6}\rangle$$ where $\alpha{d}$ are constants associated with the Ion (e.g. Cs+).
By approximating $$\langle r^{-4} \rangle \approx \frac{3}{2n^{3}l^{5}}$$ and taking the $\langle r^{-6}\rangle$ expression to be zero, we find
$$\delta{\ell} \simeq \frac{3\alpha{\rm d}}{4\ell^{5}} \Rightarrow \ell^{5}\delta_{\ell} = \textrm{const.}$$
which is the current approximation. Instead, the code calculates the full expression for the polarisation and adds this onto the hydrogenic energy. We also add hydrogenic energy splitting's to states with l>5 (which is set to be zero presently). This is expected and seen in [4] as there is little core penetration in high angular momentum states.
References:
[1] - Gallagher, Thomas F. "Rydberg atoms." [2] - Freeman and Kleppner Physical Review A 14.5 (1976): 1614. [3] - Safinya, K. A., T. F. Gallagher, and W. Sandner. Physical Review A 22.6 (1980): 2672. [4] - Berl, S. J., et al. Physical Review A 102.6 (2020): 062818. [5] - Unpublished own work (in prep.)