Science - USA (2022-04-08)

We measuredfTOðÞQA;V as a function of
the probe beam polarization parametersQA
andVand interpolated using Eq. 1 to deter-
minefTO(−1,0) (Fig. 3). We took the sign ofbT
from theory but used no other predictions in
our calculation. Thus, we determined a value
of 725,736,700 MHz for thefTO(−1,0) tune-out
with a statistical uncertainty of 40 MHz and
a systematic uncertainty of 260 MHz (SM
section 4).
The dominant systematic effect in our mea-
surement was the uncertainty in the light
polarization. The probe beam passed through
a vacuum window before it interacted with
the atoms, which may have subtly altered the
laser polarization relative to measurements
made outside the vacuum chamber. We con-
strained this error to be <200 MHz by mea-
suring the probe beam polarization before
entering, and after exiting, the vacuum sys-
tem (SM section 4.1).
Separately, we improved on the state-of-the-
art calculation ( 28 ) of the tune-out frequency
by accounting for finite nuclear mass, rela-
tivistic, QED, finite nuclear size, and finite
wavelength retardation effects ( 27 , 29 ). We
achieved a 10-fold improvement in precision
and found a theoretical value of 725,736,252
(9) MHz forfTO(−1,0). The major contribution
to the theoretical uncertainty stems from the
nonradiative QED corrections (±6 MHz) of
ordera^4 Ry, which was an order of magni-
tude less than the systematic experimental
uncertainty. We show a comparison of our
experimental and theoretical uncertainties
to the main contributions of interest to the

theoretical value in Fig. 4, to demonstrate the

contributions to which our measurement was

sensitive.

Our experimental determination is a 20-fold

improvement over the previous experimental

determination and is larger than the theoret-

ical prediction by 1.7 times the measurement

uncertainty (herein,s). Our measurement cor-

responds to a relative precision in oscillator

strength ratio of 6 parts per million (SM sec-

tion 6), which is a factor of two improvement

over the previous record ( 17 ). The combined

theoretical and experimental uncertainties

(∼260 MHz) were able to discern the contri-

bution of QED effects (∼ 30 s) and are similar

to the retardation corrections to the dipole

interaction (∼ 2 s)butmuchgreaterthan

the contribution of finite nuclear size effects

(5 MHz). Furthermore, our method for mea-

suring the dipole potential was able to dis-

cern a peak potential energy of as little as

10 −^35 J. This is, to our knowledge, the most

sensitive measurement of potential energy

reported to date.

Our measurement was sensitive to the re-

tardation corrections not normally included

in the theory of the frequency-dependent po-

larizability ( 27 , 29 ). The result was an∼1.7s

difference between experiment and theory,

which took into account the estimated un-

certainty from terms not currently included

in the theoretical calculation. It is notable

that by ignoring the retardation correction

term—proposed in ( 29 ) and included here in

tune-out frequency calculations—the differ-

ence between theory and experiment fell to

∼0.1s. If the experimental precision is in-

creased by an order of magnitude, then the

effect of the retardation contribution could

be more stringently tested.

Future experimental improvements could

include more precise laser polarization cali-

brations, likely using in-vacuum optics, and a

finer measurement of the angle between the

laser propagation and the magnetic field.

These would allow an independent compar-

ison of the predicted and measured scalar,

vector, and tensor polarizabilities, providing

further information on the structure of the

helium atom and QED theory itself.

Our method could be easily applied to other

tune-out frequencies in helium and used as an

investigative tool for other problems in QED

theory. If the precision of future measure-

ments reaches the megahertz level, the tune-

out frequency could determine the nuclear

charge radius of helium. Further improvements

and use of our method may thus continue to

challenge and elucidate QED theory.

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202 8 APRIL 2022•VOL 376 ISSUE 6589 science.orgSCIENCE

Effect Size (MHz)

-10^5 -10^4 -10^3 -10^2 -10^1 0 10^1102102102102

Theory - Exp.

Exp. Unc.

Theory Unc.

Nuclear Size

Mag. Pol.

State Retardation

Relativistic Tensor

Relativistic Scalar

QED α^3

QED α^4

Fig. 4. Experimental and theoretical sensitivity.Comparison of uncertainties in the theoretical and
experimental determinations of the 2^3 S 1 – 23 P/3^3 Ptune-out frequency and the various theoretical
contributions to the tune-out value. Exp. Unc., experimental uncertainty; Theory Unc., theoretical uncertainty;
Mag. Pol., magnetic polarizability.

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