The Rydberg constant and proton size
from atomic hydrogen
Axel Beyer,1 Lothar Maisenbacher,1 Arthur Matveev,1 Randolf Pohl,1†
Ksenia Khabarova,2, 3 Alexey Grinin,1 Tobias Lamour,1 Dylan C. Yost,1‡
Theodor W. Hänsch,1, 4 Nikolai Kolachevsky,2, 3 Thomas Udem1, 4
At the core of the “proton radius puzzle” is a four–standard deviation discrepancy between
the proton root-mean-square charge radii (rp) determined from the regular hydrogen (H)
and the muonic hydrogen (µp) atoms. Using a cryogenic beam of H atoms, we measured
the 2S-4P transition frequency in H, yielding the values of the Rydberg constant R1 =
10973731.568076( 96) per meter and rp = 0.8335( 95) femtometer. Our rp value is 3. 3
combined standard deviations smaller than the previous H world data, but in good
agreement with the µp value. We motivate an asymmetric fit function, which eliminates line
shifts from quantum interference of neighboring atomic resonances.
The study of the hydrogen atom (H) has been at the heart of the development of modern physics. Precision laser spectros- copy of H is used today to determine fun- damental physical constants such as the
Rydberg constant R1 and the proton charge radius rp, defined as the root mean square (RMS)
of its charge distribution. Owing to the simplicity
of H, theoretical calculations can be carried out
with astonishing accuracy, reaching precision
up to the 12th decimal place. At the same time,
high-resolution laser spectroscopy experiments
deliver measurements with even higher accuracy, reaching up to the 15th decimal place in the
case of the 1S-2S transition (1, 2), the most precisely
determined transition frequency in H.
The energy levels in H can be expressed as
Enlj ¼ R1 ; 1 n2 þ fnlj a; me mp ; …
þ d‘0 CNS n3 rp2
where n, l, and j are the principal, orbital, and
total angular momentum quantum numbers, respectively. The first term describes the gross structure of H as a function of n and was first observed
in the visible H spectrum and explained empirically by Rydberg. Later, the Bohr model, in which
the electron is orbiting a pointlike and, in simplest approximation, infinitely heavy proton, provided a deeper theoretical understanding.
The Rydberg constant R1 = mea2c/2h links the
natural energy scale of atomic systems and the SI
unit system. It connects the mass of the electron
me, the fine structure constant a, Planck’s constant h, and the speed of light in vacuum c.
Precision spectroscopy of H has been used to
determine R1 by means of Eq. 1 with a relative
uncertainty of 6 parts in 1012, making it one of the
most precisely determined constants of nature to
date and a cornerstone in the global adjustment
of fundamental constants ( 3).
The second term in Eq. 1, fnljða;me mp;…Þ ¼
X20a2 þ X30a3 þ X31a3lnðaÞ þ X40a4 þ …, accounts for relativistic corrections, contributions
coming from the interactions of the bound-state
system with the quantum electrodynamics (QED)
vacuum fields, and other corrections calculated
in the framework of QED ( 3). The electron-to-proton mass ratio me/mp enters the coefficients
X20, X30, … through recoil corrections caused by
the finite proton mass.
The last term in Eq. 1 with coefficient CNS is
the leading-order correction originating from the
finite charge radius of the proton, rp ( 3). It only
affects atomic S states (with l = 0) for which the
electron’s wave function is nonzero at the origin.
Higher-order nuclear charge distribution contributions are included in fnljða; me mp ; …Þ.
The proton radius puzzle
The proton charge radius rp has been under de-
bate for some time now because the very accu-
rate value from laser spectroscopy of the exotic
muonic hydrogen atom (µp) ( 4, 5) yielded a value
that is 4%, corresponding to 5.6s, smaller than
the CODATA 2014 value of rp ( 3) [see ( 6–8) for
reviews on this issue]. The CODATA value is ob-
tained from a combination of 24 transition fre-
quency measurements in H and deuterium and
several results from elastic electron scattering
( 9–11). The accuracy of the µp result is enabled
by the fact that the muon’s orbit is ~200 times
smaller than the electron’s orbit in H, resulting
in a seven orders of magnitude larger influence
of rp on the energy levels.
Here we study the spectroscopic part of the
discrepancy, in particular the 4s discrepancy between the µp value and the global average of all
transitions measured in H ( 12) (H world data,
Fig. 1). Recently, a similar discrepancy has arisen
for the deuteron radius with a new result from
laser spectroscopy of muonic deuterium ( 13).
Considering Eq. 1 and the fact that fnlj ða; me mp ;…Þ
is known with sufficiently high accuracy, one
finds a very strong correlation between R∞ and
rp. CODATA quotes a correlation coefficient of
0.9891. Equation 1 involves two parameters, R∞
and rp, which need to be determined simultaneously from a combination of at least two measurements in H. The 1S-2S transition frequency
serves as a cornerstone in this procedure. Owing
to its small natural line width of only 1.3 Hz, experimental determinations are one thousand
times more accurate than for any other transition frequency in H, where typical line widths
amount to 1 MHz or more.
Examining previous determinations of the
value pairs [R∞, rp] from H (Fig. 1, bottom), one
notes that many of the individual measurements are in fact not in disagreement with the
µp value. The discrepancy of 4s appears when
averaging all H values (µp versus H world data;
Fig. 1, top).
Principle of the measurement
Here we report on a measurement of the 2S-4P
transition in H (Fig. 2A), yielding [R∞, rp] with
an uncertainty comparable to the aggregate H
world data and significantly smaller than the
proton radius discrepancy, which corresponds to
8. 9 kHz in terms of the 2S-4P transition frequency. This uncertainty requires a determination
of the resonance frequency to almost one part
in 10,000 of the observed line width of 20 MHz
The previous most accurate measurements
[see, e.g., ( 14–16) and references therein] were
limited by the electron-impact excitation used
to produce atoms in the metastable 2S state. This
excitation results in hot atoms with mean thermal velocities of 3000 m/s or more and an uncontrolled mixture of population in the four 2S
Zeeman sublevels. In turn, this typically leads to
corrections on the order of tens of kilohertz because of effects such as the second-order Doppler
and ac-Stark shifts or the excitation of multiple
unresolved hyperfine components.
Our measurement is essentially unaffected by
these systematic effects ( 17) because we use the
Garching 1S-2S apparatus (1, 2) (Fig. 3) as a well-controlled cryogenic source of 5.8-K cold 2S atoms.
Here, Doppler-free two-photon excitation is used
to almost exclusively populate the 2SF¼0 1=2 Zeeman
sublevel without imparting additional momentum on the atoms.
The remaining main systematic effects in our
experiment are the first-order Doppler shift and
apparent line shifts caused by quantum interference of neighboring atomic resonances, both of
1Max-Planck-Institut für Quantenoptik, 85748 Garching,
Germany. 2P.N. Lebedev Physical Institute, 119991 Moscow,
Russia. 3Russian Quantum Center, 143025 Skolkovo, Russia.
4Ludwig-Maximilians-Universität, 80539 München, Germany.
*Corresponding author. Email: email@example.com
†Present address: Johannes Gutenberg-Universität Mainz, 55122
Mainz, Germany. ‡Present address: Colorado State University, Fort
Collins, CO 80523, USA.