Search

The proposal for the development of a nuclear optical clock has triggered a multitude of experimental and theoretical studies. In particular the prediction of an unprecedented systematic frequency uncertainty of about $10^{-19}$ has rendered a nuclear clock an interesting tool for many applications, potentially even for a re-definition of the second. The focus of the corresponding research is a nuclear transition of the $^{229}$Th nucleus, which possesses a uniquely low nuclear excitation energy of only $8.12\pm0.11$ eV ($152.7\pm2.1$ nm). This energy is sufficiently low to allow for nuclear laser spectroscopy, an inherent requirement for a nuclear clock. Recently, some significant progress toward the development of a nuclear frequency standard has been made and by today there is no doubt that a nuclear clock will become reality, most likely not even in the too far future. Here we present a comprehensive review of the current status of nuclear clock development with the objective of providing a rather complete list of literature related to the topic, which could serve as a reference for future investigations.

A comprehensive theoretical study of direct laser excitation of a nuclear state based on the density matrix formalism is presented. The nuclear clock isomer $^{229\text{m}}$Th is discussed in detail, as it could allow for direct laser excitation using existing technology and provides the motivation for this work. The optical Bloch equations are derived for the simplest case of a pure nuclear two-level system and for the more complex cases taking into account the presence of magnetic sub-states, hyperfine-structure and Zeeman splitting in external fields. Nuclear level splitting for free atoms and ions as well as for nuclei in a solid-state environment is discussed individually. Based on the obtained equations, nuclear population transfer in the low-saturation limit is reviewed. Further, nuclear Rabi oscillations, power broadening and nuclear two-photon excitation are considered. Finally, the theory is applied to the special cases of $^{229\text{m}}$Th and $^{235\text{m}}$U, being the nuclear excited states of lowest known excitation energies. The paper aims to be a didactic review with many calculations given explicitly.

The excitation of the 8 eV $^{229m}$Th isomer through the electronic bridge mechanism in highly charged ions is investigated theoretically. By exploiting the rich level scheme of open $4f$ orbitals and the robustness of highly charged ions against photoionization, a pulsed high-intensity optical laser can be used to efficiently drive the nuclear transition by coupling it to the electronic shell. We show how to implement a promising electronic bridge scheme in an electron beam ion trap starting from a metastable electronic state. This setup would avoid the need for a tunable vacuum ultraviolet laser. Based on our theoretical predictions, determining the isomer energy with an uncertainty of $10^{-5}$ eV could be achieved in one day of measurement time using realistic laser parameters., Comment: 6 pages, 4 figures

The first nuclear excited state of $^{229}$Th offers the unique opportunity for laser-based optical control of a nucleus. Its exceptional properties allow for the development of a nuclear optical clock which offers a complementary technology and is expected to outperform current electronic-shell based atomic clocks. The development of a nuclear clock was so far impeded by an imprecise knowledge of the energy of the $^{229}$Th nuclear excited state. In this letter we report a direct excitation energy measurement of this elusive state and constrain this to 8.28$\pm$0.17 eV. The energy is determined by spectroscopy of the internal conversion electrons emitted in-flight during the decay of the excited nucleus in neutral $^{229}$Th atoms. The nuclear excitation energy is measured via the valence electronic shell, thereby merging the fields of nuclear- and atomic physics to advance precision metrology. The transition energy between ground and excited state corresponds to a wavelength of 149.7$\pm$3.1 nm. These findings set the starting point for high-resolution nuclear laser spectroscopy and thus the development of a nuclear optical clock of unprecedented accuracy. A nuclear clock is expected to have a large variety of applications, ranging from relativistic geodesy over dark matter research to the observation of potential temporal variation of fundamental constants.

$^{229}$Th is the only nucleus currently under investigation for the development of a nuclear optical clock (NOC) of ultra-high accuracy. The insufficient knowledge of the first nuclear excitation energy of $^{229}$Th has so far hindered direct nuclear laser spectroscopy of thorium ions and thus the development of a NOC. Here, a nuclear laser excitation scheme is detailed, which makes use of thorium atoms instead of ions. This concept, besides potentially leading to the first nuclear laser spectroscopy, would determine the isomeric energy to 40 $\mu$eV resolution, corresponding to 10 GHz, which is a $10^4$ times improvement compared to the current best energy constraint. This would determine the nuclear isomeric energy to a sufficient accuracy to allow for nuclear laser spectroscopy of individual thorium ions in a Paul trap and thus the development of a single-ion nuclear optical clock.

$^{229}$Th is the only known nucleus with an excited state that offers the possibility for a direct laser excitation using existing laser technology. Its excitation energy has been measured indirectly to be 7.8(5) eV ($\approx$160 nm). The energy and lifetime of the isomeric state make it the presently only suitable candidate for a nuclear optical clock, the uncertainty of the excitation energy is, however, still too large to allow for a direct laser excitation in a Paul trap. Therefore, a major goal during the past years has been an improved energy determination. One possible approach is to measure the kinetic energy of electrons which are emitted in the internal conversion decay of the first isomeric state in $^{229}$Th. For this reason an electron spectrometer based on a magnetic bottle combined with electrical retarding fields has been built. Its design, as well as first test measurements are presented, which reveal a relative energy resolution of 3 % and thus enable to measure the electrons' expected kinetic energy to better than 0.1 eV. This is sufficiently precise to specify a laser system able to drive the nuclear clock transition in $^{229}$Th.

An overview over the current status of the development of a nuclear clock based on the state of lowest known nuclear excitation energy in $^{229}$Th is presented. The text is especially written for the interested reader without any particular knowledge in this field of research. It is thus ideal as an introductory reading to get a broad overview over the various different aspects of the field, in addition it can serve as a guideline for future research. An introductory part is provided, giving a historic context and explaining the fundamental concept of clocks. Finally, potential candidates for nuclear clocks other than $^{229}$Th are discussed.

The first excited isomeric state of $^{229}$Th possesses the lowest energy among all known excited nuclear states. The expected energy is accessible with today's laser technology and in principle allows for a direct optical laser excitation of the nucleus. The isomer decays via three channels to its ground-state (internal conversion, $\gamma$ decay and bound internal conversion), whose strengths depend on the charge state of $^{229m}$Th. We report on the measurement of the internal-conversion decay half-life of neutral $^{229m}$Th. A half-life of 7 $\pm$ 1 $\mu$s has been measured, which is in the range of theoretical predictions and, based on the theoretically expected lifetime of $\approx10^4$ s of the photonic decay channel, gives further support for an internal conversion coefficient of $\approx10^9$, thus constraining the strength of a radiative branch in the presence of IC., Comment: 8 pages, 4 Figures

Today's most precise time and frequency measurements are performed with optical atomic clocks. However, it has been proposed that they could potentially be outperformed by a nuclear clock, which employs a nuclear transition instead of the atomic shell transitions used so far. By today there is only one nuclear state known which could serve for a nuclear clock using currently available technology, which is the isomeric first excited state in $^{229}$Th. Here we report the direct detection of this nuclear state, which is a further confirmation of the isomer's existence and lays the foundation for precise studies of the isomer's decay parameters. Based on this direct detection the isomeric energy is constrained to lie between 6.3 and 18.3 eV, and the half-life is found to be longer than 60 s for $^{229\mathrm{m}}$Th$^{2+}$. More precise determinations appear in reach and will pave the way for the development of a nuclear frequency standard.

In the whole landscape of atomic nuclei, $^{229}$Th is currently the only known nucleus which could allow for the development of a nuclear-based frequency standard, as it possesses an isomeric state of just 7.6 eV energy above the ground state. The 3+ charge state is of special importance in this context, as Th$^{3+}$ allows for a simple laser-cooling scheme. Here we emphasize the direct extraction of triply-charged $^{229}$Th from a buffer-gas stopping cell. This finding will not only simplify any future approach of $^{229}$Th ion cooling, but is also used for thorium-beam purification and in this way provides a powerful tool for the direct identification of the $^{229}$Th isomer to ground state nuclear transition.

Direct laser excitation of the lowest known nuclear excited state in $^{229}$Th has been a longstanding objective. It is generally assumed that reaching this goal would require a considerably reduced uncertainty of the isomer's excitation energy compared to the presently adopted value of $(7.8\pm 0.5)$ eV. Here we present a direct laser excitation scheme for $^{229\text{m}}$Th, which circumvents this requirement. The proposed excitation scheme makes use of already existing laser technology and therefore paves the way for nuclear laser spectroscopy. In this concept, the recently experimentally observed internal-conversion decay channel of the isomeric state is used for probing the isomeric population. A signal-to-background ratio of better than $10^4$ and a total measurement time of less than three days for laser scanning appear to be achievable.

The isotope $^{229}$Th is the only nucleus known to possess an excited state $^{229m}$Th in the energy range of a few electron volts, a transition energy typical for electrons in the valence shell of atoms, but about four orders of magnitude lower than common nuclear excitation energies. A number of applications of this unique nuclear system, which is accessible by optical methods, have been proposed. Most promising among them appears a highly precise nuclear clock that outperforms existing atomic timekeepers. Here we present the laser spectroscopic investigation of the hyperfine structure of $^{229m}$Th$^{2+}$, yielding values of fundamental nuclear properties, namely the magnetic dipole and electric quadrupole moments as well as the nuclear charge radius. After the recent direct detection of this long-searched-for isomer, our results now provide detailed insight into its nuclear structure and present a method for its non-destructive optical detection., Comment: 18 pages

With an expected energy of 7.8(5) eV, the isomeric first excited state in $^{229}$Th exhibits the lowest excitation energy of all known nuclei. Until today, a value for the excitation energy has been inferred only by indirect measurements. In this paper, we propose to use the internal conversion decay channel as a probe for the ground-state transition energy. MatLab-based Monte Carlo simulations have been performed to obtain an estimate of the expected statistics and to test the feasibility of the experiment. From the simulations we conclude, that with the presented methods an energy determination with a precision of better than 0.1 eV is possible., Comment: 8 Pages, 5 Figures

Following the $\alpha$ decay of $^{233}$U, $^{229}$Th recoil ions are shown to be extracted in a significant amount from the MLL buffer-gas stopping cell. The produced recoil ions and subsequent daughter nuclei are mass purified with the help of a customized quadrupole mass spectrometer. The combined extraction and mass-purification efficiency for $^{229}$Th$^{3+}$ is determined via MCP-based measurements and via the direct detection of the $^{229}$Th $\alpha$ decay. A large value of $(10\pm2)$\% for the combined extraction and mass-purification efficiency of $^{229}$Th$^{3+}$ is obtained at a mass resolution of about 1 u/e. In addition to $^{229}$Th, also other $\alpha$-recoil ions of the $^{233,232}$U decay chains are addressed.

With an expected energy of 7.6(5) eV, $^{229}$Th possesses the lowest excited nuclear state in the landscape of all presently known nuclei. The energy corresponds to a wavelength of about 160 nm and would conceptually allow for an optical laser excitation of a nuclear transition. We report on a VUV optical detection system that was designed for the direct detection of the isomeric ground-state transition of $^{229}$Th. $^{229(m)}$Th ions originating from a $^{233}$U $\alpha$-recoil source are collected on a micro electrode that is placed in the focus of an annular parabolic mirror. The latter is used to parallelize the UV fluorescence that may emerge from the isomeric ground-state transition of $^{229}$Th. The parallelized light is then focused by a second annular parabolic mirror onto a CsI-coated position-sensitive MCP detector behind the mirror exit. To achieve a high signal-to-background ratio, a small spot size on the MCP detector needs to be achieved. Besides extensive ray-tracing simulations of the optical setup, we present a procedure for its alignment, as well as test measurements using a D$_2$ lamp, where a focal-spot size of $\approx$100 $\mu$m has been achieved. Assuming a purely photonic decay, a signal-to-background ratio of $\approx$7000:1 could be achieved., Comment: 17 pages, 21 figures, 4 tables