Nuclear clock

Nuclear Clock
Concept of a thorium-229 based nuclear optical clock.

A nuclear clock or nuclear optical clock is a notional clock that would use the frequency of a nuclear transition as its reference frequency,^[1] in the same manner as an atomic clock uses the frequency of an electronic transition in an atom's shell. Such a clock is expected to be more accurate than the best current atomic clocks by a factor of about 10, with an achievable accuracy approaching the 10⁻¹⁹ level.^[2] The only nuclear state suitable for the development of a nuclear clock using existing technology is

Modern optical atomic clocks are today the most accurate time-keeping devices. Their underlying principle of operation is based on the fact that the energy of an atomic transition (the energy difference between two atomic states) is independent of space and time. The atomic transition energy corresponds (via the Planck constant) to a particular frequency of a light wave which is required to drive the transition. Therefore, an atomic transition can be excited with laser light, if the light's frequency exactly corresponds to the energy of the atomic transition. A laser's frequency can be stabilized to match the corresponding atomic transition energy by continuous verification of a successful laser excitation of the atomic transition. Once thus stabilized, the frequency of the laser light will always be the same (independent of space and time).

It is technologically possible to measure the frequency of laser light to extraordinarily high accuracy by counting the oscillations of the light wave with the help of a

• radiatively, by direct photon (gamma ray) emission, or
• by internal conversion, transferring the energy to a shell electron which is ejected from the atom.

For most nuclear isomers, the available energy is sufficient to eject any electron, and the inner-shell electrons are the most frequently ejected. In the special case of ^229m
Th
, the energy is sufficient only to eject an outer electron (thorium's first ionization energy is 6.3 eV), and once the atom is ionized, there is not enough energy remaining to eject a second (thorium's second ionization energy is 11.5 eV).

The two decay paths have different

Thus, in the typical case that the clock is designed to measure radiated photons, it is necessary to hold the thorium in an ionized state. This can be done in an ion trap, or by embedding it in an ionic crystal. In the latter case, the atoms are not 100% ionized, and a small amount of internal conversion is possible (reducing the half-life to approximately 630 s)^[4], but the loss is tolerable.

Different nuclear clock concepts

Two different concepts for nuclear optical clocks have been discussed in the literature: trap-based nuclear clocks and solid-state nuclear clocks.

Trap-based nuclear clocks

For a trap-based nuclear clock either a single ²²⁹Th³⁺ ion is trapped in a Paul trap, known as the single-ion nuclear clock,^[1][2] or a chain of multiple ions is trapped, considered as the multiple-ion nuclear clock.^[5] Such clocks are expected to achieve the highest time accuracy, as the ions are to a large extent isolated from their environment. A multiple-ion nuclear clock could have a significant advantage over the single-ion nuclear clock in terms of stability performance.

Solid-state nuclear clocks

As the nucleus is largely unaffected by the atomic shell, it is also intriguing to embed many nuclei into a crystal lattice environment. This concept is known as the crystal-lattice nuclear clock.^[1] Due to the high density of embedded nuclei of up to 10¹⁸ per cm³, this concept would allow irradiating a huge number of nuclei in parallel, thereby drastically increasing the achievable signal-to-noise ratio,^[13] but at the cost of potentially higher external perturbations.^[14] It has also been proposed to irradiate a metallic ²²⁹Th surface and to probe the isomer’s excitation in the internal conversion channel, which is known as the internal-conversion nuclear clock.^[15] Both types of solid-state nuclear clocks were shown to offer the potential for comparable performance.

Transition requirements

From the principle of operation of a nuclear optical clock, it is evident that direct laser excitation of a nuclear state is a central requirement for the development of such a clock. This is impossible for most nuclear transitions, as the typical energy range of nuclear transitions (keV to MeV) is orders of magnitude above the maximum energy which is accessible with significant intensity by today's narrow-bandwidth laser technology (a few eV). There are only two nuclear excited states known which possess a sufficiently low excitation energy (below 100 eV). These are

• ^229m
  Th
  , a metastable nuclear excited state of the isotope thorium-229 with an excitation energy of only about 8 eV,^[16][17] and
• ^235m
  U
  , a metastable excited state of uranium-235 with an energy of 76.7 eV.^[18]

However, ^235m
U
has such an extraordinarily long radiative half-life (on the order of 10²² s, 20,000 times the age of the universe, and far longer than its internal conversion half-life of 26 minutes) that it is not practical to use for a clock.^[19][20] This leaves only ^229mTh with a realistic chance of direct nuclear laser excitation.

Further requirements for the development of a nuclear clock are that

• the lifetime of the nuclear excited state is relatively long, thereby leading to a resonance of narrow bandwidth (a high
  quality factor
  ) and
• the ground-state nucleus is easily available and sufficiently long-lived to allow one to work with moderate quantities of the material.

Fortunately, with ^229m
Th⁺
having a radiative half-life (time to decay to ²²⁹
Th⁺
) of around 10³ s,^[4][21][22] and ²²⁹
Th
having a half-life (time to decay to ²²⁵
Ra
) of 7917±48 years,^[23] both conditions are fulfilled for ^229m
Th⁺
, making it an ideal candidate for the development of a nuclear clock.

History

The history of the nuclear clock

As early as 1996 it was proposed by Eugene V. Tkalya to use the nuclear excitation as a "highly stable source of light for metrology".^[24]

With the development (around 2000) of the frequency comb for measuring optical frequencies exactly, a nuclear optical clock based on ^229m
Th
was first proposed in 2003 by Ekkehard Peik and Christian Tamm, who developed an idea of Uwe Sterr.^[1] The paper contains both concepts, the single-ion nuclear clock, as well as the solid-state nuclear clock.

In their pioneering work, Peik and Tamm proposed to use individual laser-cooled ²²⁹
Th³⁺
ions in a Paul trap to perform nuclear laser spectroscopy.^[1] Here the 3+ charge state is advantageous, as it possesses a shell structure suitable for direct laser cooling. It was further proposed to excite an electronic shell state, to achieve 'good' quantum numbers of the total system of the shell plus nucleus that will lead to a reduction of the influence induced by external perturbing fields. A central idea is to probe the successful laser excitation of the nuclear state via the hyperfine-structure shift induced into the electronic shell due to the different nuclear spins of ground- and excited state. This method is known as the double-resonance method.

The expected performance of a single-ion nuclear clock was further investigated in 2012 by Corey Campbell et al. with the result that a systematic frequency uncertainty (accuracy) of the clock of 1.5×10⁻¹⁹ could be achieved, which would be by about an order of magnitude better than the accuracy achieved by the best optical atomic clocks today.^[2] The nuclear clock approach proposed by Campbell et al. slightly differs from the original one proposed by Peik and Tamm. Instead of exciting an electronic shell state in order to obtain the highest insensitivity against external perturbing fields, the nuclear clock proposed by Campbell et al. uses a stretched pair of nuclear hyperfine states in the electronic ground-state configuration, which appears to be advantageous in terms of the achievable quality factor and an improved suppression of the quadratic Zeeman shift.

In 2010, Eugene V. Tkalya showed that it was theoretically possible to use ^229m
Th
as a lasing medium to generate an ultraviolet laser.^[25][26][27]

The solid-state nuclear clock approach was further developed in 2010 by W.G. Rellergert et al.^[14] with the result of an expected long-term accuracy of about 2×10⁻¹⁶. Although expected to be less accurate than the single-ion nuclear clock approach due to line-broadening effects and temperature shifts in the crystal lattice environment, this approach may have advantages in terms of compactness, robustness and power consumption. The expected stability performance was investigated by G. Kazakov et al. in 2012.^[13] In 2020, the development of an internal conversion nuclear clock was proposed.^[15]

Important steps on the road towards a nuclear clock the successful direct laser cooling of ²²⁹
Th³⁺
ions in a Paul trap achieved in 2011,^[28] and a first detection of the isomer-induced hyperfine-structure shift, enabling the double-resonance method to probe a successful nuclear excitation in 2018.^[29]

The history of ^229mTh

Since 1976, the ²²⁹Th nucleus has been known to possess a low energy excited state,^[30] the excitation energy of which was shown to be less than 10 eV in 1990.^[31]

This was, however, too broad an energy range to apply high-resolution spectroscopy techniques; the transition energy had to be narrowed down first. Initial efforts used the fact that, after the alpha decay of ²³³
U
, the resultant ²²⁹
Th
nucleus is in an excited state and promptly emits a gamma ray to decay to either the base state or the metastable state. Measuring the small difference in the gamma-ray energies emitted in these processes allows the metastable state energy to be found by subtraction.

An early mis-step was the (incorrect) measurement of the energy value as 3.5±1.0 eV in 1994.^[32] In particular, this energy was comfortably below the 6.3 eV ionization energy of thorium (implying that decay by internal conversion was impossible even in neutral thorium, see § Ionization) and the 6.2 eV limit of ultraviolet transmission through molecular oxygen (air). Thus, direct detection experiments were attempted which had no hope of detecting the ultraviolet light at the true, higher, energy.^[5]

The energy value remained elusive until 2003, when the nuclear clock proposal triggered a multitude of experimental efforts to pin down the excited state's parameters like energy and half-life. The detection of light emitted in the direct decay of ^229m
Th
would significantly help to determine its energy to higher precision, but all efforts to observe the light emitted in the decay of ^229m
Th
were failing.^[5] The energy level was corrected to 7.6±0.5 eV in 2007^[33] (slightly revised to 7.8±0.5 eV in 2009^[34]). Subsequent experiments continued to fail to observe any signal of light emitted in the direct decay, leading people to suspect the existence of a strong non-radiative decay channel.^{[35][36][37][38]} The detection of light emitted by the decay of ^229mTh was reported in 2012,^[39] and again in 2018,^[40] but the observed signals were the subject of controversy within the community.^[41]

A direct detection of electrons emitted by the isomer's internal conversion decay channel was achieved in 2016.^[42] This detection laid the foundation for the determination of the ^229mTh half-life in neutral, surface-bound atoms in 2017^[12] and a first laser-spectroscopic characterization in 2018.^[29]

In 2019, the isomer’s energy was measured via the detection of internal conversion electrons emitted in its direct ground-state decay to 8.28±0.17 eV.^[16] Also a first successful excitation of the 29 keV nuclear excited state of ²²⁹
Th
via synchrotron radiation was reported,^[43] enabling a clock transition energy measurement of 8.30±0.92 eV.^[44] In 2020, an energy of 8.10±0.17 eV was obtained from precision gamma-ray spectroscopy.^[17]

Finally, precise measurements were achieved in 2023 by unambiguous detection of the emitted photons (8.338(24) eV)^[45][46] and in April 2024 by two reports of excitation with a tunable laser at 8.355733(10) eV^[47] and 8.35574(3) eV.^[3][4][48] The light frequency (2020407.6±2.8 GHz) is now known with sufficient accuracy to enable future construction of a prototype clock,^[49] and determine the transition's exact frequency and its stability.

Applications

When operational, a nuclear optical clock is expected to be applicable in various fields. Obviously, it may be used wherever today's atomic clocks are in operation, such as satellite-based navigation or data transfer. Its high precision would allow new applications inaccessible to other atomic clocks, such as relativistic geodesy, the search for topological dark matter,^[50] or the determination of time variations of fundamental constants.^[51]

A nuclear clock has the potential to be highly sensitive to possible time variations of the fine-structure constant.^[52] The central idea is that a nuclear transition couples differently to the fine-structure constant than an atomic shell transition does.^[19] For this reason a comparison of the frequency of a nuclear clock with an atomic clock could lead to an extraordinary high sensitivity for potential time variations of the fine structure constant. The achievable factor of sensitivity, however, remains subject to speculation. A recent measurement is consistent with enhancement factors between 1 (no enhancement) and 10⁴.^[29]

References