[time-nuts] Rydberg Constant accuracy

[Peter Burbery]

Dear Time nuts,

What needs to be done to redefine the second in terms of the Rydberg
constant? The second will be redefined around 2026, 2030, or 2034. The
second will be redefined around 2034 because it will take time for
technological advancements to occur to the point where sufficient accuracy
can be achieved reliably. I have read the 2021 Consultative Committee for
the Units (CCU) meeting report
<https://www.bipm.org/documents/20126/68201438/CCU25.pdf/ab1be833-d656-ae2e-36fa-feb04948ed4c>,
and they have identified 3 options.

Dr Dimarcq highlighted the three options for redefinition. Option 1: A
definition based on a single atomic reference transition – although at the
current time there was no strong favourite transition identified. Option 2:
A definition based on an ensemble of frequencies. Option 3: A definition
based on fixing the numerical value of a fundamental constant. Option 3 was
not feasible in the near future as no fundamental physical constant was
known with sufficient accuracy. Dr Dimarcq concluded by stating that the
CCTF planned to finalize their white paper on the redefinition of the
second in 2022 and this would include the methodology for making the future
choice of the new definition

It would be awesome to redefine the second as a physical constant. One
possibility would be to redefine the second by fixing the value of the
Rydberg constant, which this paper
<https://www.researchgate.net/publication/51654829_When_should_we_change_the_definition_of_the_second>
describes.

Here is the except that interests me (the formatting is hard to read so I
paraphrased parts of it):
In this section, the different conceptual proposals for the redefinition of
the second are briefly considered. The possibilities arising include a
definition based on a fixed value for the Rydberg constant, the choice of a
single optical clock transition, the combination of a single optical
transition and a set of optical frequency ratios, a virtual frequency based
on a matrix of ‘best’ optical transitions, or a very high-frequency
transition in the vacuum ultraviolet or X-ray regions of the spectrum. Each
of these options is considered in turn. The atomic energy level structure
that can be calculated with the best accuracy is that of the hydrogen atom
because of its relatively simple combination of a single proton and single
electron.

The Dirac equation extends the analysis to a fully relativistic
description. Still, it requires further adjustment via quantum
electrodynamics (QED) theory to account properly for the Lamb shift in
order to agree with direct experimental measurements of the 1S–2S
transition frequency. Currently, the measurement of this transition by
laser spectroscopy, relative to the Cs primary standard, has an uncertainty
of 1.4 parts in 1014 [21] and, when combined with measurements of other
hydrogen transitions [22] to account for the Lamb shift, gives a Rydberg
constant value with the lowest uncertainty of 6.6 parts in 1012. This
allows redefining the second by fixing the Rydberg at its measured value
and computing the hydrogen 1S–2S transition or other hydrogen transitions.
The difficulty with this is the loss of accuracy of the 1S–2S transition
relative to current Cs performance and that between the measured value of
the 1S–2S and the Rydberg constant, which arises due to uncertainties in
QED theory corrections and knowledge of the proton size. This latter issue
is the subject of the paper by Nez [23] in this issue. As a result, this
option for redefinition does not seem competitive with the much lower
uncertainties now quoted for cold atom and ion clock frequencies. It
requires significant improvement in knowledge of the QED corrections. In
addition, hydrogen would need to be trapped and cooled, which is more
difficult because of its small mass and correspondingly large second-order
Doppler shifts. Generating the cooling radiation at 121.5 nm is also
difficult.

(PDF) When should we change the definition of the second?. Available from:
https://www.researchgate.net/publication/51654829_When_should_we_change_the_definition_of_the_second
[accessed Jan 04 2023].

My question is, what QED calculations would be necessary to fix the Rydberg
constant to the necessary number of digits, say 20, to match the most
accurate and precise optical clocks at JILA and NIST that employ
femtosecond combs to trap elements like strontium within 1 part in 10^20?
How could the QED uncertainty corrections mentioned above be improved? What
models, formulas, theories, equations, and calculations would you use to
determine the Rydberg constant accurately? Would using increased
computational power to determine the Rydberg constant to more digits for a
given model for example, with a supercomputer cluster, make a difference?
How do you account for the Lamb shift concerning the 1S-2S transition
within the context of quantum electrodynamics? How do you combine these
relativistic calculations of the 1S-2S transition with other hydrogen
transitions to make a final determination? What does the paper refer to
when it states, "The difficulty with this is loss of accuracy of the 1S–2S
transition relative to current Cs performance, and that between the
measured value of1S–2S and the Rydberg constant, which arises due to
uncertainties in QED theory corrections and knowledge of the proton size"?
I understand there were discrepancies in the CODATA measurements of the
proton radius regarding muonic hydrogen, but these experimental
uncertainties have cleared up (I think). There is a paper I've consulted on
this mentioned after this:
 This latter issue is the subject of the paper by Nez [23] in this issue.
https://royalsocietypublishing.org/doi/10.1098/rsta.2011.0233
The paper continues:
As a result, this option for redefinition does not seem competitive with
the much lower uncertainties now quoted for cold atom and ion clock
frequencies and requires significant improvement in knowledge of the QED
corrections.

Has there been any significant improvement in knowledge of the QED
corrections? If not, how could this progress be made? If possible, I would
like to contribute computational power to QED improvements, but I don't
know what equations I need to use to improve the state of the Rydberg
constant.

Thank you very much,

Peter Burbery

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