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The fine structure constant, defined by α = e2/[4πε0]ħc in the S.I., is the dimensionless constant that quantifies the electromagnetic interaction between fundamental particles.

Since 1987 the most precise value for α has been obtained by comparing intricate Quantum Electrodynamics calculations of the g-factor (magnetic moment) of the free electron, with measurements of the same quantity, that use a single electron occupying the lowest quantized levels of its cyclotron motion in a Penning trap. (The theory expresses the so-called g-factor anomaly, ae = (ge2)/2, as a power series in α.) Following a series of major developments in both the theory [1] and the experiment [2], this now gives α−1 = 137.035 999 172 7(68)(46)(19)(331). Here the uncertainties in parentheses are from the eighth and tenth order QED terms, the combined hadronic and electroweak terms, and the experiment, for a combined total error of 0.25 × 10−9 (0.25 ppb). Further improvements in the experimental precision can be expected which should result in a total uncertainty for α below 0.1 ppb.

However, if the goal is to test the theory for ge2 with the possibility of uncovering new physics, an independent value for α is required. Since 2002, as pioneered by S. Chu at Stanford, the method producing the second-most precise value for α has involved measuring h/matom (Planck's constant divided by the mass of an atom), by measuring the recoil velocity of an atom absorbing a photon. This was enabled by the development of sophisticated atomic physics methods of laser cooling and atom interferometry. Chu's group, working with Cs, published a value for α with uncertainty of 8 ppb [3]. Since then, the group of F. Biraben at Paris has carried out a series of experiments on 87Rb. Their latest result, which motivates this commentary, yields the value α−1 = 137.035 999 044(90) [4]. This is in good agreement with that from ge2, and the uncertainty, 0.66 ppb, is of comparable order, see Fig. 1. This result is sufficiently precise that it allows the hadronic corrections in the ge2 theory to be tested for the first time.

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Figure 1. The most precise recent values for the fine structure constant. The point labeled ge2 corresponds to theory published in 2012 [1] combined with an experimental result published in 2008 [2]. The Cs results are from Stanford [3] (2002), and from Berkeley [9] (2013). For the Rb references, see R. Bouchendira et al., Ann. Phys. 525, 484 (2013), ref. [4].

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Measurements of h/matom actually produce a value for α through the relation

  • display math(1)

where R is the Rydberg constant and Ar(atom), Ar(e) are the atomic masses (relative to 12C) of the atom involved, and of the electron, respectively. Equation (1) follows from the relation of the Rydberg constant to the atomic unit of energy α2mec2 (me is the mass of the electron):

  • display math(2)

together with the experimental situation that h/matom, Ar(atom) and Ar(e) can all be measured more precisely than h/me. Hence this method for α, though often called “the photon-recoil method”, in fact requires the measurement of four different quantities. However, R is already known to 5 × 10−12 (from precision laser spectroscopy of atomic hydrogen combined with evaluation of QED corrections [5]), while Ar(atom) is typically known to 1 × 10−10 (from precision cyclotron frequency measurements of single ions in a Penning trap [6]), and Ar(e) is known to 4 × 10−10 (mainly from precision measurement of the electronic g-factor of a single, one-electron ion, C5+, also in a Penning trap, combined with extensive QED calculations [5, 7]). So the precision limiting step is the measurement of h/matom.

The essence of photon-recoil measurements of h/matom is that when an atom absorbs a laser photon of frequency fl, and so momentum hfl/c, it recoils with velocity vr = hfl/(cmatom). Starting with laser-cooled atoms, and by using counter-propagating laser beams to produce stimulated Raman transitions − in which absorption from the ground state is driven by one beam, and stimulated emission is driven by the opposing beam − a series of n absorption-emission cycles produces a velocity change of 2nvr. This velocity change can then be measured as a Doppler shift, Δf = 2nfr, where the “recoil frequency” fr is related to the laser frequency fl by fr = (vr/c)fl = (h/matomc2)fl2. In the latest Paris experiment, by making use of counter-propagating beams with a linearly-swept frequency difference between them, up to 500 coherent pairs of absorption-emission cycles were produced, increasing the Doppler shift to be measured. (This process, which occurs by transitions between discrete momentum states differing by 2hfl/c, can also be interpreted in terms of Bloch oscillations of atoms trapped in a continuously accelerating optical lattice). The resulting Doppler shift was then measured by arranging a Ramsey-Bordé interferometer [8]. Considering that the recoil velocity vr for 87Rb is only 6 mm/s, it is indeed remarkable that it can now be measured to a fractional precision close to 1ppb.

Fig. 1 also shows a value for α from improved photon-recoil measurements on Cs by the group of H. Müller at Berkeley [9]. Although less precise than the Paris result, this work has gained attention through the claim of a direct measurement of the Compton frequency of the cesium atom. Instead of referring the frequency of the recoil-inducing laser fl to the (usual) cesium atomic frequency standard, Müller's innovation was to lock its frequency to a multiple of the recoil frequency fr. This then implies fl is a sub-multiple of the Compton frequency of the atom, matomc2/h, so linking the mass of the atom to the frequency of the laser. But since c is a defined constant in the S.I. (assuming the constancy of the speed of light), there is really no distinction between a measurement of h/m, as usually reported, and of the Compton frequency mc2/h. In any case, as the above discussion shows, the Paris experiment also measures the Compton frequency of 87Rb. And although the Paris experiment made use of an intermediate reference for the short-term stabilization of fl, this was not a limitation.

Both the Paris and Berkeley groups propose to reduce their uncertainties, which are currently limited by issues such as laser alignment and wavefront curvature. Significant improvements in h/matom would then require improved data for the other quantities in Equation (1). In the case of the atomic mass of the electron, this has already been anticipated and improved measurements of the g-factor of C5+ have been carried out, leading to an order-of-magnitude reduction in the uncertainty of Ar(e) [10]. This now increases the motivation for further improvements in the atomic masses of 87Rb and 133Cs, but also of other atoms, such as the even isotopes of Yb, which are also candidates for measurements of h/matom.

Values for α have also been derived from the solid-state phenomena of the quantum Hall effect and the Josephson effect, the hyperfine structure of muonium, and a measurement of h/mneutron using neutron diffraction. Although important for unifying different branches of physics, so far, none of these methods have produced results with uncertainties below 10 ppb. Despite considerable effort by several groups of theorists and experimentalists, a program to obtain α from comparison of theory and experiment for the fine structure intervals of helium has so far reached an uncertainty of 20 ppb, limited by difficulties in calculating higher-order terms in the bound-state QED theory of the two-electron atom [11, 12]. Ambitiously, in view of the difficulties with helium fine structure, a proposal also exists to obtain α from measurement and theory for the g-factors of high-Z hydrogen-like and boron-like ions [13]. But in the near future at least, it seems likely that cold-atom interferometry and the supporting mass measurements will provide the best alternative α to that from ge2.

Acknowledgements

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  2. Acknowledgements
  3. References

The author acknowledges support from the US National Science Foundation.

References

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  2. Acknowledgements
  3. References