Time Variation of Alpha

J.R. Torgerson, S.K. Lamoreaux, T. Fortier, E. Gershgoren, F.G. Omenetto, M.M. Schauer (P-23), D. Budker, T. Nguyen (University of California, Berkeley), S.A. Diddams (National Institute of Standards and Technology), V.V. Flambaum (University of New South Wales)
Excerpted from LA-14202-PR

Introduction

If we accept the idea that our universe formed from nothing with the “big bang,” it must also seem reasonable that, at some point in the past, fundamental constants of physics were varying with time. Moreover, events that likely occurred immediately following the big bang, such as inflation and later acceleration, suggest a modification of the values of fundamental constants. A smooth evolution of the values of such parameters would thus lead to the conclusion that they are still changing and that this change may be measurable even today. Because our most widely accepted theories of nature do not allow for such variation, such evidence would be proof of physics beyond our current understanding. It is important to point out that there is little basis to assert that many of the parameters in our theories of nature are constants independent of space and time.

The question of whether the fundamental constants are still varying has been of interest at least since Dirac put forward his “large number hypothesis.”¹ Dirac noticed that certain dimensionless combinations of physical constants fell into three groups of order: unity, 10⁴⁰, and 10⁸⁰. The 10⁴⁰ group, in particular, was thought to depend on the size of the universe. Although this hypothesis was based on coincidental observations, the formalism of modern string theory allows for the possibility of a dependence of the fundamental constants on the gravitational potential, or possibly on the dark energy responsible for the nonzero cosmological constant.

Of particular interest for variation of fundamental constants are the dimensionless gauge coupling constants, such as the fine-structure constant (α = 1/137 = e²/hc) whose value determines the strength of electromagnetic interactions. Table 1 summarizes the results of some of the recent searches for time variation of α. Recent astronomical observations have indicated that over the last 10 billion or so years the value of α has changed and that the average rate of fractional variation is α•/α ≈ 10^-16/yr,² although this result remains controversial³. Analysis of the ancient nuclear fission reactor in Gabon, West Africa, can also be used to search for ancient variation of α, and the two most recent results are listed in Table 1 and discussed below.

Recent precision laboratory clock-comparison experiments, where different atomic transitions that depend differently on α are used as frequency references and are measured over a period of years, have limited the rate of variation to α•/α < 10^-15/yr.^4–6 The purpose of our studies is to further improve the limit on (or observe!) time variations at the level of 10-18/yr. For this purpose, we have started efforts which include two laboratory measurements with atomic references and a study of the Oklo natural-reactor phenomena.

Reanalysis of the Oklo Natural Reactor

Two billion years ago, the relative isotopic fraction of uranium-235 in natural uranium was 3.7% compared to the present value of 0.7%. It is not possible to have a self-sustained nuclear chain reaction with a homogeneous mixture of water and uranium with isotopic composition at the 0.7% level. However, with 3.7% enrichment, it is possible to attain a neutron multiplication factor of about 1.38 with 2.4 water molecules per uranium atom. As suggested by P. Kuroda in 1956, if an ancient uranium ore deposit was sufficiently concentrated, saturated with water, and had a low concentration of neutron absorbers, an ore deposit could become a natural nuclear reactor.

The remnants of such a natural reactor were discovered in Gabon, Africa, in 1972 during routine analysis of uranium samples. (With the exception of lead and helium, the isotopic composition of the chemical elements in the Earth’s crust is so constant that deviations are used to identify, for example, rocks from Mars that have ended up on Earth.) This natural reactor ran for about 100,000 years, two billion years ago, and the ore deposit and reactor products were preserved in an extremely stable geological formation. Subsequent isotopic analysis, after correcting for influx of natural isotopic-abundance chemical elements, matches the isotopic composition of spent fuel from modern nuclear reactors.

Shlyachter⁷ pointed out that it is possible to test for a variation in α (or other parameters that determine nuclear energy levels) by determining fission fragment concentrations for isotopes that have a low-energy neutron absorption resonance due to the strong energy dependence of the low-energy neutron population.

Two analyses of samples from the Oklo deposit indicated no variation of the resonance energy and yielded an upper limit for the fractional variation of α at the 10^-17/yr level.^8,9 However, in addition to the total neutron flux and the location of the resonance, the specific spectrum of the neutron flux is important. The previous analyses assumed a Maxwell-Boltzmann spectrum, known to be incorrect in the presence of absorbers (which include uranium and water, which formed the basis of the reactor). We repeated the analysis with a more realistic neutron spectrum and found α•/α ≥ 2.3 × 10^-17/yr with 8σ confidence.¹⁰ A result of our analysis is shown in Figure 1. This result is about ten times smaller than the astrophysical result² and of opposite sign; both the magnitude and sign of the variation can be accommodated in modern theories of the universe.¹¹ Our result was covered as a feature article in New Scientist.¹²

Comparison of Optical Frequency References

Although laboratory measurements do not have the advantage of integrating over 10⁹–10¹⁰ years, they provide significant conveniences—they can be made in a carefully prepared environment and are reproducible. Furthermore, the resolution on parameters of interest can be increased by a large factor which can allow a measurement with higher precision but on a laboratory time scale. Laboratory measurements are also sensitive to short-time variations and to possible spatial and gravitational dependencies.

Because α determines the strength of electromagnetic interactions, the energies of atomic states depend upon α. Moreover, different states have different dependencies on α primarily due to relativistic corrections to the energies. That these corrections can be calculated accurately is the basis for nearly all ideas regarding laboratory measurements of α variation.

Twenty-five years ago, Hans Dehmelt recognized that group-IIIA ions, in particular, indium and thallium (In⁺ and Tl⁺), would be excellent choices on which to base atomic clocks of unprecedented stability. By the same fundamental considerations, we recently identified a ytterbium ion (Yb²⁺) as another promising candidate. These atomic ions possess metastable energy states that decay back to the ground state. This transition is insensitive to fluctuating ambient fields and the transition frequency is ~ 10¹⁵ Hz (in the optical regime), which improves the short-term stability of a reference based upon it. In addition, atomic ions can be trapped and cooled until they are nearly motionless unlike clouds of neutral atoms that suffer from systematic collisional effects and are not trapped. These ions were chosen as candidates for this effort because these “clock” transitions are some of the most α sensitive (Figure 2). We are currently constructing an ion-trapping/ -imaging apparatus for spectroscopy of both indium(+) and ytterbium(2+) as the first step towards our goal of comparing all three. A measurement with these three ions has the potential to achieve a sensitivity of δ(α•/α) ≈ 10^-18/yr with a measurement time of one year. This is over 1000 times more sensitive than current laboratory measurements.

We are constructing narrow-band laser sources to probe the metastable transitions, thereby creating accurate ion-stabilized optical frequency references. Other transitions from the ground state are strong enough to be used to optically cool the ions and serve as “readout” transitions to determine the electronic states of the ions.

One of the main obstacles in performing an experiment to compare frequency references in which at least one of the references is optical is that the appropriate high-speed electronics do not exist to directly measure the 10¹⁴–10¹⁵ Hz frequency. However, our experiment does not require accurate knowledge of the frequencies of the references, or even the difference frequency. It is sufficient to measure changes in the difference of frequencies of relatively stable atomic references to measure changes in α. It is this fact that makes this experiment feasible with existing experimental methods.

We will use two existing technologies—the optical comparison resonator (OCR) and the femtosecond optical frequency comb (OFC)—to make redundant measurements to verify our results. The underlying concept is illustrated in Figure 3 and is similar for both the OCR and OFC although its implementation differs. For the OCR, the ion-stabilized frequency references are shifted into resonance with passive optical resonator modes, and the frequencies shift is measured. For the OFC, the stable references beat with populated modes emitted from the OFC laser source and the beat frequency is measured.

The OFC relies on a broadband, mode-locked laser. When operated in the mode-locked regime, the output of the laser consists of well-defined pulses whose repetition frequency is the free spectral range (FSR) of the laser resonator. The spectral components of the output are then separated by the FSR and span the fraction of the gain bandwidth of the laser over which dispersion is compensated. With our collaborators at the National Institute of Standards and Technology (NIST), we have constructed a titanium-doped sapphire laser with an FSR of 500 MHz and a spectrum covering more than an octave centered at about 800 nm (Figure 4). This spectral region easily includes wavelengths of interest for indium(+), thallium(+), and ytterbium(2+) optical frequency references and is sufficient to allow accurate stabilization without additional noisy measurements. We are currently working to stabilize the spectrum of this laser and understand its subtleties.

Atomic Dysprosium

We are also conducting an experiment at the University of California (UC), Berkeley, which takes advantage of a fortunate near degeneracy of two energy levels in atomic dysprosium.¹³ As shown in Figure 5, two states of opposite parity are separated by as little as 300 MHz for a particular isotope (and by as much a 1 GHz for other isotopes). Careful mathematical modeling of these levels has been conducted due to interest in these states for measurements of parity nonconservation, and the states have been determined to be nearly as sensitive to variations of α as ytterbium(2+). Although optical transitions offer advantages for frequency references, a measurement of α variation does not require the use of two separated references.

The advantage of this system over others is that the frequency measurements have been reduced from optical frequencies to radio frequencies, which can be monitored with well-developed techniques. A determination of a fractional shift of α•/α ≈ 10^-18 requires an uncertainty δν ≈ 1 mHz. This is a fractional uncertainty of only δν/ν ≈ 10^-12, which can be achieved with a commercially available radio-frequency reference, such as that derived from a cesium beam standard.

A schematic of the apparatus currently in use at UC-Berkeley is shown in Figure 6. The apparatus is essentially a simple radio-frequency atomic “clock” based upon a dysprosium beam and possessing a single interaction region. The transition between states A and B can be addressed with an applied electric field at the appropriate frequency. Spectroscopy on this transition is performed by monitoring the fluorescence of the atoms at 564 nm. We are currently studying several systematics that can affect the sensitivity of our measurements for tests of α variation.

References

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Acknowledgment

We would like to thank our LANL management for their ongoing support of our efforts. This work is currently funded through the LANL Laboratory Directed Research and Development program, the UC/LANL Campus-Laboratory Collaborations Program and by a NIST Precision Measurements Grant.

For further information, contact Justin Torgerson, 505-665-3365, torgerson@lanl.gov.