EPJ manuscript No. (will be inserted by the editor) Concepts for direct frequency-comb spectroscopy of 229mTh and an internal-conversion-based solid-state nuclear clock Lars von der Wense1,2 and Chuankun Zhang2 1 Ludwig-Maximilians-Universit¨atM¨unchen, 85748 Garching, Germany 2 JILA and Department of Physics, University of Colorado, Boulder, CO 80309-0440, USA June 23, 2020 Abstract. A new concept for narrow-band direct nuclear laser spectroscopy of 229mTh is proposed, using a single comb mode of a vacuum ultraviolet frequency comb generated from the 7th harmonic of an Yb-doped fiber laser system. In this concept more than 1013 229Th atoms on a surface are irradiated in parallel and a successful nuclear excitation is probed via the internal-conversion (IC) decay channel. A net scanning time of 15 minutes for the most recent 1 σ energy uncertainty interval of 0.34 eV appears to be achievable when searching for the nuclear transition. In case of successful observation, the isomer's energy value would be constrained to an uncertainty of about 100 MHz, which is a factor of 106 of improvement compared to today's knowledge. Further, the comb mode could be stabilized to the nuclear transition using the same detection method, allowing for the development of an IC-based solid-state nuclear clock, which is shown to achieve the same performance as a crystal-lattice nuclear clock, however, with the advantage of a drastically simpler detection scheme. Finally, it is shown that the same laser system could be used to narrow down the isomer's transition energy by further six orders of magnitude during laser excitation of 229Th3+ ions in a Paul trap and to drive nuclear Rabi oscillations, as required for the development of a nuclear clock based on a single 229Th3+ ion. 1 Introduction clocks operational today was estimated for a nuclear clock based on a single 229Th ion [17]. In a pictorial understand- The accuracy of time measurements has considerably evolveding, the reason for this expected improved accuracy is that within the past decades, starting with the development of the atomic nucleus is several orders of magnitude smaller the first atomic clock by Louis Essen and Jack Parry in than the atomic shell and therefore, due to the related 1955 [1], followed by the legal definition of the second by smallness of the nuclear moments, drastically more stable means of the Cesium standard in 1967 [2] and the in- against external influences [20]. vention of the fountain clock in the late 1980s [3]. With the invention of the optical frequency comb [4{6] a new Historically, the idea of a nuclear clock dates back to the technological leap has occurred, allowing for the develop- late 1950s, shortly after the discovery of the M¨ossbauer ment of optical atomic clocks [7], which by today pose effect [21]. At that time it was noticed by Pound and Re- the most precise timekeepers. The achieved accuracies of bka that the concept of M¨ossbauerspectroscopy, due to these clocks vary around 10−18, corresponding to an error its potential to measure extremely small relative energy of 1 second in 30 billion years, significantly longer than differences, could be used to observe the gravitational red- the age of the universe [8{11]. Such accuracy in frequency shift [22]. Here, a 93.3 keV nuclear excited state of 67Zn, measurement opens up a plethora of applications, e.g., in which exhibits an extraordinary narrow relative linewidth satellite-based navigation [12], geodesy [13] and the search of ∆f=f = 5:5 · 10−16, is of particular interest. M¨ossbauer for temporal variation of fundamental constants [14, 15]. spectroscopy with 67Zn was first achieved in 1960 together with the proposal for a 67Zn-based nuclear clock [23, 24]. arXiv:1905.08060v4 [physics.ins-det] 22 Jun 2020 As an alternative to the existing optical atomic clocks, the However, in this context, the expression \nuclear clock" idea of a nuclear optical clock was developed [16{19]. Com- refers to a relative frequency measurement rather than to pared to an optical atomic clock, the operational principle an absolute frequency determination. remains unchanged, however, with the important differ- ence that a nuclear transition instead of an atomic shell A nuclear optical clock has not yet been experimentally transition is used for time measurement. A nuclear op- realized. One challenge is that most nuclear transitions tical clock is expected to outperform even today's best exhibit an excitation energy which is several orders of atomic clocks due to conceptual advantages. An accuracy magnitude above what is accessible with today's narrow- approaching 10−19 and therefore about an order of magni- bandwidth laser technology. However, there is one excep- tude of improvement compared to the best optical atomic tion, which is the first nuclear excited state of 229Th, 2 Lars von der Wense, Chuankun Zhang: Concepts for direct frequency-comb spectroscopy of 229mTh known as the \thorium isomer" (229mTh). With an ex- way of isomer population via excitation of the 29 keV state citation energy that has recently been measured to be in 229Th has been experimentally realized [43]. This al- about 8.3 eV [25], 229mTh is the nuclear excited state lowed for a 229mTh energy determination to 8:30±0:92 eV of lowest known excitation energy. In fact, the energy is in [44]. In 2019, the 229mTh energy was also measured to within reach of currently available frequency comb tech- higher precision via direct spectroscopy of the IC electrons nology thereby in principle allowing for nuclear laser spec- emitted in the isomeric decay, thereby drawing a clear troscopy and the development of a nuclear optical clock. path for future nuclear laser spectroscopy experiments The state is metastable, with an estimated radiative life- [25]. The value obtained by this method is 8:28 ± 0:17 eV 4 +3:2 time of about 10 seconds [26, 27]. If non-radiative decay (149:7−3:0 nm) and is used throughout this paper. More channels are suppressed, e.g., in individual 229Th3+ ions, recently the Beck et al micro-calorimetric measurement the long radiative lifetime would lead to a narrow rela- was repeated with improved detector resolution, resulting −20 +0:14 +3:6 tive linewidth of up to ∆f=f ≈ 10 , which allows the in a value of 8:09−0:19 eV (153:3−2:7 nm) [45]. This value development of an optical atomic clock of highest stabil- would even be advantageous for the proposed experiment ity [16, 17]. In the following, the experimental history of as it is closer to the center wavelength of the here consid- 229mTh will briefly be sketched. ered 7th harmonic of an Yb-doped fiber frequency comb 229Th was first considered to possess a low-energy nuclear (152.8 nm). excited state in 1976, when it helped explaining some pe- culiarities of the 229Th nuclear γ-ray spectrum as observed in the α decay of 233U, which remained otherwise unex- plained [28]. Based on γ-ray spectroscopy of nuclear rota- 2 Concepts for nuclear laser excitation tional states of larger energies, the 229mTh energy value was constrained to −1±4 eV in 1990 [29] and then further Various concepts for nuclear laser excitation of 229mTh improved to 3:5 ± 1 eV in 1994 [30], which was until 2007 were discussed in literature. Two different principles re- the adopted energy value. In 2007, however, a different quire particular attention: (1) The isomer's excitation via energy value was measured using a metallic magnetic mi- the electronic bridge (EB) excitation mechanism and (2) crocalorimeter with an improved energy resolution leading direct laser excitation of the nucleus. Both approaches will to 7:6 ± 0:5 eV [31], later corrected to 7:8 ± 0:5 eV [32]. be discussed individually in the following. A central problem, that has hindered the development of a nuclear clock based on 229mTh within the past decade is, that its energy is not sufficiently well constrained to al- 2.1 The electronic bridge excitation mechanism low for laser spectroscopy of individual laser-cooled 229Th ions. This has led to a multitude of efforts to pin down the In the EB excitation process, the hyperfine coupling be- isomeric energy value to higher precision (a recent review tween the atomic shell and the nucleus is used for nuclear can be found in Ref. [33]). It would be ideal to measure excitation (see, e.g., Ref. [46]). This allows the excitation the energy via spectroscopy of photons directly emitted in of a (virtual) electronic shell state instead of directly excit- the isomer's ground state decay. However, despite world- ing the nucleus. During the decay of the excited electronic wide efforts, until today no unambiguous signal of pho- shell state, part of its energy is transferred to the nu- tons emitted in the isomeric decay has been observed, po- cleus with a certain probability, resulting in nuclear exci- tentially pointing toward a significant non-radiative decay tation. Historically, the process was introduced by Morita channel [34{37]. and Batkin [47, 48]. The reverse process, which is the de- excitation of a nuclear state via the excitation of an elec- In 2016, the direct observation of electrons emitted in tronic shell state, was already discussed earlier by Krutov the internal conversion (IC) decay channel of the isomeric [49]. The process was applied to the excitation of 229mTh state opened a new path for exploration of the isomer's by E.V. Tkalya in 1992 [50]. Later, the concept was con- properties [38]. In the IC decay, the nucleus couples to sidered for various different physical conditions (see, e.g., the atomic shell and transfers its energy to an electron, Refs.