We present a new measurement of the positive muon magnetic anomaly, a µ ≡ (gµ − 2)/2, from the Fermilab Muon g −2 Experiment based on data collected in 2019 and 2020. We have analyzed more than four times the number of positrons from muon decay than in our previous result from 2018 data. The systematic error is reduced by more than a factor of two due to better running conditions, a more stable beam, and improved knowledge of the magnetic field weighted by the muon distribution, ω′ p , and of the anomalous precession frequency corrected for beam dynamics effects, ωa. From the ratio ωa/ω ′ p , together with precisely determined external parameters, we determine a µ = 116 592 057(25) × 10 −11 (0.21 ppm). Combining this result with our previous result from the 2018 data, we obtain a µ (FNAL) = 116 592 055(24) × 10 −11 (0.20 ppm). The new experimental world average is aµ(Exp) = 116 592 059(22) × 10 −11 (0.19 ppm), which represents a factor of two improvement in precision.

The discrete quantum properties of matter are manifest in a variety of phenomena. Any particle that is trapped in a sufficiently deep and wide potential well is settled in quantum bound states. For example, the existence of quantum states of electrons in an electromagnetic field is responsible for the structure of atoms, and quantum states of nucleons in a strong nuclear field give rise to the structure of atomic nuclei. In an analogous way, the gravitational field should lead to the formation of quantum states. But the gravitational force is extremely weak compared to the electromagnetic and nuclear force, so the observation of quantum states of matter in a gravitational field is extremely challenging. Because of their charge neutrality and long lifetime, neutrons are promising candidates with which to observe such an effect. Here we report experimental evidence for gravitational quantum bound states of neutrons. The particles are allowed to fall towards a horizontal mirror which, together with the Earth's gravitational field, provides the necessary confining potential well. Under such conditions, the falling neutrons do not move continuously along the vertical direction, but rather jump from one height to another, as predicted by quantum theory.

The lowest stationary quantum state of neutrons in the Earth's gravitational field is identified in the measurement of neutron transmission between a horizontal mirror on the bottom and an absorber/scatterer on top. Such an assembly is not transparent for neutrons if the absorber height is smaller than the ''height'' of the lowest quantum state.

The lunar-ranging test of the equivalence principle for gravitational self-energy is ambiguous. Although the Earth has more gravitational self-energy than the Moon, its sizable Fe͞Ni core also gives it a different composition than the Moon. We removed this ambiguity by comparing, in effect, the accelerations of "miniature" earths and moons toward the Sun. Our composition-dependent Earth-Moon acceleration, Da CD ͞a s ͑10.1 6 2.7 6 1.7͒ 3 10 213 , and lunar-ranging data provide an unambiguous test at the 1.3 3 10 23 level.PACS numbers: 04.80.Cc Although general relativity, Einstein's elegant theory of gravity, has passed all experimental tests [1,2], it is a classical theory that cannot be quantized, and much theoretical effort is now devoted to developing a realistic quantum theory that would reduce to Einstein's theory in the appropriate limit [3]. General relativity has an exact symmetry, the equivalence principle (EP), that is expected to be violated to some degree by any quantum theory of gravity and by many alternative classical theories as well. The violation is linked to the fact that these theories have scalar gravitational fields in addition to the usual tensor field. Precision tests of the EP could therefore provide direct evidence for new gravitational phenomena. Recent cosmological data on distant Type Ia supernovae [4,5] that suggest a repulsive gravitational effect provide additional motivation for subjecting the theory to stringent experimental tests.The most precisely tested manifestation of the EP is the universality of free fall (UFF), the prediction that all bodies in a uniform gravitational field have exactly the same gravitational acceleration. The UFF has been tested to roughly 1 part in 10 12 [6][7][8] using laboratory test bodies in the gravitational fields of the Earth and the Sun. But, because gravitational self-energy is negligible for any laboratory-scale object, these experiments cannot address a crucial issue, whether gravitational self-energy obeys the EP. The importance of this issue was emphasized by Nordtvedt [9], who showed that theories with more than one metric field (which naturally respect the EP for laboratory-size bodies) nevertheless predict violations of the EP for gravitational self-energy. Nordtvedt noted that this could be tested by using lunar laser-ranging (LLR) to compare the accelerations of the Earth and Moon toward the Sun; the Earth and Moon are sufficiently massive that gravitational self-energy reduces their masses by 4.6 and 0.2 parts in 10 10 , respectively.However, as noted by Nordtvedt, the LLR measurement of Da LLR a e 2 a m (a e and a m are the accelerations of Earth and Moon toward the Sun) is ambiguous because, from the point of view of the EP test, the Earth and Moon "test bodies" differ in two significant ways. The Earth has a greater fraction of gravitational self-energy than the Moon, and it also has a different composition (its core gives the Earth a larger Fe͞Ni content than the Moon). The first difference allows LLR to probe the EP for gravitati...

We have studied neutron quantum states in the potential well formed by the earth's gravitational field and a horizontal mirror. The estimated characteristic sizes of the neutron wave functions in the two lowest quantum states correspond to expectations with an experimental accuracy. A position-sensitive neutron detector with an extra-high spatial resolution of ~2 µm was developed and tested for this particular experiment, to be used to measure the spatial density distribution in a standing neutron wave above a mirror for a set of some of the lowest quantum states. The present experiment can be used to set an upper limit for an additional short-range fundamental force. We studied methodological uncertainties as well as the feasibility of improving further the accuracy of this experiment.PACs Codes: 03.65, 28.20