Concepts for Direct Frequency-Comb Spectroscopy of $^{229\Text {M

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 Mössbauer 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 Mössbauerspectroscopy, 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. Mössbauer 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. [51–62]). Importantly, any direct nuclear laser ex- which is subsequently ejected. Importantly, the nucleus citation will most likely be accompanied by a process of couples directly to the atomic shell and the process does nuclear excitation via EB. In general, the EB process is ex- not correspond to a γ decay followed by photo-electron pected to have larger excitation rates compared to direct emission. For this reason the lifetime of the nuclear excited laser excitation, as the electronic shell acts as a resonator, state shortens by the amount of the conversion coefficient thereby enhancing the electric field at the site of the nu- αic, if IC is energetically permitted. Based on the observa- 4 229m cleus. Enhancement factors between 10 and 10 were pre- tion of the IC electrons, the Th IC-lifetime in neutral, dicted in Ref. [56], depending on the isomer’s exact en- surface-bound atoms was determined to be about 10 µs ergy value and a potential overlap with an electronic shell [39], in agreement with theoretical expectations [26, 40, transition. Electronic bridge excitation has not yet been 41]. Further, in 2018, collinear laser-spectroscopy of the 229 2+ experimentally observed, however the reverse process, EB electronic shell of Th ions with 2% in the isomeric decay, was observed for 93Nb [63] as well as 193Ir [64] and state led to the observation of the isomer-induced change the closely related process of nuclear excitation by elec- of the hyperfine-structure [42]. Independently, also a new tron transition (NEET) has been detected for 197Au [65] Lars von der Wense, Chuankun Zhang: Concepts for direct frequency-comb spectroscopy of 229mTh 3 and 193Ir [66]. with significant intensity in non-linear crystals. The only available non-linear crystal that allows for tunable cw- In 2012 it was proposed to excite 229mTh with a two- light generation in the deep VUV is KBe BO F (KBBF) photon EB excitation scheme using a frequency comb [67]. 2 3 2 [71]. Although transparent down to 147 nm [71] and al- This concept is intriguing, as in this way all comb modes ready being used for the generation of pulsed laser light at would contribute to the nuclear excitation. Further, a laser below 150 nm via sum frequency mixing [72], KBBF fulfills system operating at the technologically easier accessible the phase-matching condition required for SHG only down wavelength of about 300 nm could be used. The proposed to 164 nm [73]. Alternative crystals for cw-light generation excitation scheme makes important use of the EB mech- at 150 nm might be BaMgF (BMF) [74] and BPO [75]. anism, as the probability for direct nuclear two-photon 4 4 Both crystals provide good transparency at 150 nm, how- excitation can generally be considered as very low [60]. ever, they do not offer the possibility for phase matching. The investigation of the EB effect and the potential for Quasi-phase-matching (QPM) might, however, be possi- an EB-based nuclear clock, that may even include nuclear ble [76]. Recently, improved computational methods also transitions other than 229mTh [68], is an interesting field allowed the prediction of a couple of new materials that of continued research. may offer the possibility for SHG at 150 nm, these are In this paper, purely direct nuclear excitation is consid- AlCO3F, PB3O6F2 and SiCO3F2 [77]. For these reasons ered, leading to a conservative estimate for the nuclear there is hope, that cw laser technology for nuclear laser excitation rates. This is mainly owed to the complexity excitation of 229mTh will be available in the future. How- of the EB process, which will drastically depend on the ever, as long as this is not the case, other methods have electronic shell and the details of the excitation scheme. to be investigated. Direct frequency-comb spectroscopy [78, 79] could be a promising alternative way for narrow-band nuclear laser excitation. During direct frequency-comb spectroscopy, a 2.2 Direct nuclear laser spectroscopy single comb line is in resonance with the nuclear transition and the energy is determined from knowledge of the rep- Direct nuclear laser spectroscopy of 229Th was first con- etition frequency frep and the carrier envelope offset fre- sidered in the 1990s [50, 53, 55], followed by the proposal quency fceo via the comb equation fnucl = fceo +Nnuclfrep for a nuclear clock in 2003 [16]. At that time, it was found [80]. The comb mode number Nnucl can be obtained from to be unfavorable compared to other nuclear laser excita- multiple measurements with different repetition rates [81]. tion schemes that make use of the EB mechanism. Direct In recent years, the frequency-comb technology has been laser excitation of 229mTh is challenging for three reasons: drastically extended to higher energies and now also cov- (1) The natural linewidth corresponding to the radiative ers wavelengths in the deep to extreme ultra-violet region −4 (see, e.g., Refs. [69, 78, 82–97]). The method that is usu- decay channel is extraordinary narrow, Γγ = 10 Hz 4 ally applied for VUV frequency-comb generation is cavity- when assuming a radiative lifetime τγ of 10 s. There- fore, the fraction of laser light that contributes to the nu- enhanced high-harmonic generation (HHG) in a noble gas. clear excitation can generally considered to be small. (2) In particular, the 7th harmonic of an Yb-doped fiber laser [78, 79, 88, 97] or the 5th harmonic of a Ti:Sapphire laser The isomer’s excitation energy is only weakly constrained, 229m which leads to a large energy range that has to be scanned [96] are matching with the Th nuclear transition en- when searching for the nuclear excitation. (3) The energy ergy. is located in the vacuum-ultra-violet (VUV) energy range, In this paper, the potential to perform direct frequency- where laser technology is not easily available. comb spectroscopy of 229mTh is discussed. It is assumed Despite these challenges, direct laser spectroscopy of 229mTh that the the 7th harmonic of an Yb-doped fiber laser, gen- has become a viable option within the recent years, due erated via cavity-enhanced harmonic generation, is used to technological advances in laser technology (see, e.g., for nuclear excitation. Parameters of an existing laser sys- Ref. [97] and references therein), an improved knowledge tem operational at the JILA laboratory in Boulder, Col- about the isomer’s transition energy [25] and new concepts orado [69, 78, 91, 97] are used for the quantitative es- for nuclear laser excitation [70]. timates. Two different approaches to use this laser for nuclear spectroscopy are discussed: In the first concept Due to compactness, robustness and stability, it would be the irradiation of more than 1013 229Th atoms on a sur- most advantageous to use a continuous-wave (cw) laser face is analyzed, which can be considered as the simplest for 229mTh nuclear excitation. The 229mTh energy value method and would allow for a high-precision 229mTh en- of 8.28±0.17 eV, corresponding to a wavelength of 149.7± ergy determination and the development of an IC-based 3.1 nm, makes this, however, challenging. Unfortunately, solid-state nuclear clock. In the second concept, nuclear no cw laser at about 150 nm wavelength is available by to- frequency-comb spectroscopy of laser-cooled 229Th3+ ions day. Due to lack of a broadband gain medium in the VUV in a Paul trap is considered. It is found, that the laser region, building a laser oscillator in the VUV is not possi- intensity would be sufficient to drive nuclear Rabi oscil- ble right now. Therefore, nonlinear conversion like second lations in case that the bandwidth of an individual comb harmonic generation (SHG) would be required for obtain- ing a cw laser in the VUV. This can only be achieved 4 Lars von der Wense, Chuankun Zhang: Concepts for direct frequency-comb spectroscopy of 229mTh mode is narrowed down to the Hz-range as would be ideal level. In fact, the only reason why the system cannot be for clock operation. modeled via the Einstein rate equations, as was done in Ref. [70], is, that the bandwidth of the laser light used for excitation is not significantly broader than the natu- 3 Theoretical Background ral linewidth of the IC-broadened nuclear transition. For the case of laser excitation of 229Th ions in a Paul trap, In the following, the nuclear excitation probability as a as considered in Sec. 7, the laser-induced depopulation of function of time under resonant laser irradiation will be the excited state is significantly larger than the natural modeled. For this purpose, a closed nuclear two-level sys- decay rate, again validating the two-level approximation. tem consisting of ground-level sub-state g and excited level Under this assumption, the time-dependent nuclear exci- sub-state e is assumed. In case of laser excitation of a large tation probability ρ (t) for a single nucleus under reso- 229 exc number of Th atoms on a surface, as discussed in Sec. 4, nant irradiation is given by Torrey’s solution of the optical a closed nuclear two-level system is a valid assumption, as Bloch equations [98, 99]: the nuclear de-excitation rate via the IC-channel is large, not leading to any significant depopulation of the ground-

2 " ˜ !# Ωeg − 1 (Γ +Γ˜)t Γ + Γ ρexc(t) = 1 − e 2 cos(λt) + sin(λt) . (1) ˜ 2 2λ 2 Γ Γ + Ωeg

Here t denotes the time since the start of the irradiation, [98] Ωeg is the Rabi frequency specific to the considered sub- s 2 2 2πc ICgeΓγ states, Γ = Γγ + Γnr is the total decay rate of the nu- Ωeg = 3 , (3) clear excited state, including γ decay of rate Γγ as well ~ω0 as potential non-radiative decay channels Γnr. In case of a non-vanishing IC decay channel, which is expressed by a with I the intensity of the laser light, ω0 the angular fre- non-zero IC coefficient α = Γ /Γ , one obtains Γ = Γ quency of the nuclear transition, Γγ the radiative decay ic ic γ nr ic rate of the nuclear excited level, c the speed of light and and therefore Γ = (1+αic)Γγ . The parameter λ is defined 2 ˜ 2 1/2 ˜ ~ the reduced Planck constant. as λ = |Ωeg − (Γ − Γ ) /4| . For Ωeg < |Γ − Γ |/2, the sin and cos functions in Eq. (1) have to be exchanged by The factor Cge is a constant that depends on the spe- sinh and cosh, respectively. Further, cific sub-states of the nuclear ground- and excited level. In case of nuclei embedded in a solid-state environment, Cge is simply the Clebsch-Gordan coefficient correspond- Γ + ΓL ing to the transition. For nuclei of isolated atoms or ions Γ˜ = + Γãdd (2) 2 the constant has a more complicated form that is given in Sec. 7. As we do not want to consider any particular sub-state right in the beginning and are only interested in denotes the total decay rate of the coherences, with ΓL the an order-of-magnitude estimate for the number of excited bandwidth of the laser light used for excitation and Γãdd corresponding to potential additional decoherence. Addi- nuclei, we set Cge = 1 for the following calculations. It tional decoherence can arise due to homogeneous broad- will be detailed in Sec. 5 that the actual Clebsch-Gordan 229 p p ening in a solid-state environment as will be discussed in coefficients for Th are 4/15 or 2/5. This will either Sec. 5 or due to, e.g., phonon coupling of ions in a Coulomb lead to a reduction of the actually observed excitation rate 229 3+ p crystal. For laser-cooled Th ions, Γãdd will in general by up to a factor of 4 (for the case of Cge = 4/15) or, be small and for the considered experimental conditions in case of degeneracy of all sub-states, to an increase by a we assume Γãdd ΓL, which allows one to neglect addi- factor of 1.3. tional decoherence. For a solid-state environment a value Note, that Eq. (1) remains valid also if the bandwidth for Γãdd between 2π · 1 kHz and 2π · 10 kHz due to nuclear of the laser light used for irradiation, ΓL, is significantly spin coupling was estimated in Ref. [18], about 2π ·150 Hz broader than the total linewidth Γ of the nuclear transi- was obtained in Ref. [19]. Here it is assumed that addition, as the definition of the decay rate of the coherences Γ˜ tional decoherence in the solid-state environment is sig- takes respect for that. In case of the low-saturation limit, nificantly smaller than the IC-enhanced natural 229mTh ˜ 5 which is fulfilled for Γ Γ Ωeg, Eq. (1) transforms to decay rate of Γ ≈ 10 Hz. This allows one to omit Γãdd in all discussed calculations. Experimentally, this assump- 2 Ωeg −Γ t tion is supported by the fact, that the 93.3 keV Mössbauer ρexc(t) = 1 − e . (4) state of 67Zn with 9 µs half-life could be resolved down to 2ΓΓ˜ its natural linewidth of 2π · 12.4 kHz [100]. This result is consistent with the one obtained from the The Rabi frequency of the system can be calculated as Einstein rate equations used in Ref. [70]. Under this ap- Lars von der Wense, Chuankun Zhang: Concepts for direct frequency-comb spectroscopy of 229mTh 5

proximation, the nuclear excitation rate Γexc can be esti- 1 3rd 2nd 1st Yb-fiber mated asρ ˙exc(0) to be ampl. ampl. ampl. frequency stage stage stage comb 2 2 2 Ωeg πc ICgeΓγ Γexc ≈ = . (5) 2Γ˜ ω3Γ˜ ~ 0 1070nm 10kW The number of excited nuclei Nexc is obtained from Eq. (1) 77MHz 120fs by multiplication with the number of irradiated nuclei N0. 100µs gatepulses Throughout the paper Eq. (1) will be used to model the electrons time-dependent nuclear excitation probability. 7thharmonic 150nm,1.2mW

cavity 229m 4 Laser excitation of Th atoms on a 229 HHGingasjet shutter Thtarget surface MCP detector

In this section, it is proposed to use the 7th harmonic of an brewsterplate Yb-doped fiber frequency comb, to irradiate a solid sample of 229Th atoms deposited as a thin layer on a surface. The presented concept is based on a proposal presented in Refs. [70, 102], where it was shown that broadband VUV Fig. 1: Schematic of the experimental setup proposed laser light, generated via four-wave mixing in a noble gas, for frequency-comb-based direct nuclear laser excitation of 229mTh. A vacuum ultra-violet (VUV) frequency comb is gen- could be used for 229mTh laser excitation if the isomer’s erated via cavity-enhanced harmonic generation from an Yb- IC decay channel is used to probe the nuclear resonance. doped-fiber frequency comb [69, 78, 91, 97]. Gate pulses of 100 Here, the concept is adapted to a source of narrow-band µs duration with 5 kHz gate frequency are produced with the laser light as offered by a VUV frequency comb. The ad- help of a mechanical shutter [104]. A solid sample surface of vantages of the presented scheme compared to the scheme 229Th is irradiated with the laser light. In case of resonance, the proposed in Refs. [70, 102] are that the isomeric energy nuclear isomeric state is excited and will decay within about could be constrained to higher precision and that the same 10 µs lifetime via internal conversion (IC) under emission of concept could be used to develop a new type of nuclear- an electron [39]. The electrons can be guided with the help of clock, which is based on the isomer’s IC decay channel. electric fields and detected by an MCP detector. Further, the same laser system could also be used for the excitation of 229Th3+ ions in a Paul trap and for the development of a single-ion nuclear clock. with the nuclear transition and therefore also the absolute frequency can be obtained by making several mea- 4.1 General concept surements with different repetition rates (mode spacings) of the frequency comb [78, 81]. A schematic of the experimental setup is shown in Fig. 1. Important aspects of the The VUV frequency comb light consists of femtosecond scheme are: a huge number of irradiated atoms of more pulses whose repetition rate determines the comb mode than 1013; an IC-broadened nuclear transition linewidth spacing. For the purpose of laser excitation of the 229Th of 15.9 kHz and a 9 orders of magnitude shortened life- nuclei, it is assumed that the laser light is gated, thereby time of about 10 µs, which allows for laser-triggered iso- generating fs pulse trains of 100 µs duration. In case that meric decay detection. The concept requires three main an individual mode of the frequency comb is tuned to the components as visualized in Fig. 1: A target consisting nuclear resonance energy, a fraction of the nuclei is excited of a thin surface of 229Th, a gated laser system used for into the isomeric state during a single gate pulse. After target irradiation and a detection system that will allow the end of the laser gate pulse the excited nuclei will de- for the detection of the low-energy IC electrons with high cay with about 10 µs lifetime via internal conversion (IC) efficiency. Each part will be discussed individually in the [38, 39]. The low-energy electrons emitted in the IC decay following. will be guided by electric fields and detected with the help of a micro-channel plate (MCP) detector [103] in order to probe the nuclear excitation. In this way it can be inferred if the laser light was tuned to the nuclear resonance. The absolute number of the comb mode that is in resonance 4.2 The 229Th target 1 Considering the relation between the absorption cross sec- 229 tion σ and the excitation rate, σ = ~ω0Γexc/I [101], for ΓL Γ It is assumed, that the Th target consists of a ∼ 10 nm 2 2 229 one has σ = λ /(2π)CgeΓγ /ΓL, with λ being the wavelength thin ThO2 layer deposited as a round surface with corresponding to the nuclear transition. This is in complete 0.3 mm diameter onto a metal substrate. Technically it agreement with the expressions given in Refs. [50, 53, 55]. might be advantageous to start with a larger surface area 6 Lars von der Wense, Chuankun Zhang: Concepts for direct frequency-comb spectroscopy of 229mTh and later cover the outer part of the target with a thin metal plate. ThO2 is chosen as a target material, as it is chemically very stable and possesses a cubic lattice structure, which results in a vanishing electric quadrupole splitting of the nuclear states (see Sec. 5). The thorium oxide band gap was measured to about 5.8 eV [105], which is significantly below the isomer’s energy, therefore decay of the isomeric state will securely occur by inter- 229 nal conversion. Thin ThO2 layers could be produced, e.g., by spray pyrolysis [106], by photochemical deposition [107, 108], by vapor deposition [109], by the sol-gel technique [110], by the drop-on-demand technique [111], by hydrothermal synthesis [112, 113] or from non-aqueous solution [114]. The useful thickness is limited by the mean- free-path length of electrons in the target material [115] and the large absorption coefficient for VUV light [116]. 3 229 With a ThO2 density of 9.86 g/cm , the number of Th atoms in the target amounts to 1.6 · 1013. As the 229Th half-life is about 7900 years, this corresponds to an activ- Fig. 2: Sketch of the ThO2 band structure together with ity of ∼ 45 Bq. Even if the target material was chemically parameters used for the escape efficiency estimate: E purified prior to its deposition, within about 100 days, the F Fermi energy, Evac vacuum energy, Ewf work function, Ecbm 229Th decay chain approaches a radioactive equilibrium, conduction-band minimum, Evbm valence-band maximum, leading to an activity increase by a factor of 8. The tar- Egap band gap, Eiso isomeric energy. get activity in equilibrium will therefore amount to about 360 Bq. This assumes that no other thorium isotopes are contained in the target material. In case of a potential contamination with 228Th, the 228Th activity may even mum. All parameters are taken with respect to the Fermi 229 exceed the one of Th due to the comparatively short level EF and are shown in Fig. 2. ηtrans(E) denotes the 228Th half-life of 1.9 years. A 228Th contamination may quantum mechanical probability for the electron to be 229 easily occur, as Th is usually obtained by chemical transmitted at the surface potential well. ηsurf(E, h) is the separation from 233U material, which contains spurious probability for an electron to reach the surface when it is amounts of 232U. Even in case that the fractional content emitted from an energy E below the Fermi level at depth of 228Th in the target material amounts to 10−3 only, its h in the material and DOS(E) is the density of states activity would be by a factor of 4.2 larger compared to functional. “norm” denotes a normalization factor defined that one of 229Th. Further, 228Th approaches the radioac- as Z −Evbm tive equilibrium with its daughters after about 20 days. norm = DOS(E)dE, (7) The factor of activity enhancement is 7. In this case, the −Eiso+Ecbm total target activity in equilibrium would amount to about with E being the energy of the conduction-band mini- 1.7 kBq. Fortunately, the 228Th activity would fade away cbm mum. The lower limit of the integral takes respect for the a few years after chemical separation from its mother nu- fact that there are no electronic levels in the band gap that clei. can be populated during the IC decay. Except for the ad- An important aspect for the proposed detection method ditional transmission efficiency, Eq. (6) transforms to the is how efficient the low-energy IC electrons can leave the expression given in Ref. [117] for Evbm = Ecbm = 0 as ThO2 target. An analysis scheme for the escape efficiency would be the case in a metal. The transmission efficiency from a metal surface was presented in Ref. [117]. Here we can be calculated as [118] adapt the procedure to an insulator and take additionally √ √ also the quantum mechanical reflection of the electrons at 4 Ekin Ekin − Ewf ηtrans(E) = √ √ 2 . (8) the material surface into account. Input parameters are E + E − E the inelastic mean-free-path (IMFP) length of the low- kin kin wf energy IC electrons in the target material, as well as the Here the electron’s kinetic energy with respect to the Fermi density-of-state (DOS) distribution of the electrons. For level Ekin = E + Eiso was introduced. Further, the prob- electrons being emitted at a certain depth of the target h, ability for the electron to reach the surface is obtained as the escape efficiency can be estimated as [117] [117]

Z −Evbm π/2 ηtrans(E)ηsurf(E, h)DOS(E)dE 1 Z −h ηesc(h) = . ηsurf(E, h) = exp sin(θ)dθ, −Eiso+Ewf norm π 0 cos(θ)λn(Ekin) (6) (9) Here Eiso denotes the isomer’s energy, Ewf the ThO2 work- where λn(Ekin) denotes the IMFP of the electrons in nanome- function and Evbm the energy of the valence-band maxi- ter as a function of their kinetic energy above the Fermi Lars von der Wense, Chuankun Zhang: Concepts for direct frequency-comb spectroscopy of 229mTh 7

level. Based on Ref. [115], λn is given as λn = a·λm, with a Now using the ThO2 resistivity, which was measured to 6 being the monolayer thickness of the material and λm the be about 10 Ωcm [106], the resistance of our thin layer IMFP in dimension of monolayers, which can be estimated amounts to about R = 130 Ω. This in turn allows esti- for low-energy electrons propagating in an anorganic-compoundmating the current across the layer to ∼ 0.05 A. The to- 2 material as λm = 2170/Ekin [115], where Ekin denotes the tal charge would be compensated in about 0.3 µs, which electron energy above the Fermi level in eV. For ThO2, is significantly shorter than the duration of a laser gate the monolayer thickness is obtained as a = p3 A/(ρnN) = pulse. Therefore no charge accumulation has to be ex- 0.24 nm, where A denotes the molecular mass, ρ the den- pected to occur during laser irradiation of the thin ThO2 sity, n the number of atoms in the molecule and N the layer, which is in agreement with experimental observa- Avogadro constant. For electrons with an energy of 4.3 eV tion [119]. above the Fermi level, this leads to and IMFP of 28 nm. Importantly, the IMFP is rising for electrons of lower energy, however, as soon as the energy is below the material’s 4.3 The laser system work-function, the electrons will not be able to leave the surface. For this reason, a high electron escape efficiency The laser light that is assumed to be used for irradiation is obtained for materials with a low work function and of the 229Th target comes from the 7th harmonic of an a DOS distribution that exhibits a high density of elec- ytterbium-doped fiber-based frequency comb operating at tronic states around an energy of Eiso − Ewf below the a wavelength around 1070 nm. Parameters of an exist- Fermi level, which corresponds to the lowest energy that ing frequency comb operational in the group of Jun Ye still allows the electrons to overcome the work function. at JILA in Boulder, CO [69, 78, 91, 97] are used for all Fortunately, ThO2 offers both, a very low work-function quantitative estimates. Comparable laser systems are also of only about 2.6 eV [109] and a high DOS between 4 and available in Tokyo, Japan [86, 87, 92] and at MPQ in Ger- 6 eV below the Fermi level [119]. many [88–90, 93, 94]. With the help of an enhancement A further important parameter is the attenuation length cavity, a high average power of 10 kW is produced. Fo- of the VUV light in the ThO2 target material. Exper- cusing this laser power to a helium-xenon gas-jet leads to imental studies of the VUV absorption coefficient α of an intensity of 5 · 1013 W/cm2, sufficiently large to effi- −1 ThO2 are scarce. A value of α ≈ 0.1 nm for light with ciently generate high harmonics. For the considered ex- 8.3 eV energy was found in Ref. [116]. This translates into periment, the 7th harmonic with a wavelength around a transmission efficiency of 37% for 10 nm layer thickness. 150 nm is of interest, which can be out-coupled with a The corresponding factor of decrease in nuclear excitation high efficiency of up to 75% for example with a Brew- rate over the total depth of the target is ηabs = 0.63. For ster plate [92] or with a non-collinear out-coupling scheme this reason, significantly thicker ThO2 layers are also not [84, 97, 120–122]. The seventh harmonic spans an energy advantageous due to absorption losses in the material. range of 0.038 eV (9.2 THz) and tunability between 7.9 and 8.7 eV (143 to 157 nm) is planned to be provided. The fraction of electrons that is able to leave the target It consists of about 1.2 · 105 comb modes, each of them material ηeff can be numerically estimated based on the with a bandwidth of about 490 Hz and a power of about integral 10 nW. The mode spacing is 77 MHz. In the considered Z d 1 −αh concept the laser beam is gated either with the help of ηeff = ηesc(h)e dh, (10) a fast rotating shutter system [104] or with an acousto- dηabs 0 optical modulator to provide laser gate pulses of 100 µs with d denoting the ThO2 oxide layer thickness. It is found pulse duration at 5 kHz gate frequency. Important param- that the result does not significantly depend on the details eters of the concept are visualized in Fig. 3. The values of of the DOS function, therefore a simple step function with variables used for the quantitative analysis according to a constant DOS starting at Evbm was used in the calcu- Eq. (1) are listed in Tab. 1. lation. A relatively large escape efficiency of about 15% is When using this laser system for surface irradiation, the obtained for d = 10 nm layer thickness. potential for laser ablation and surface heating requires Using an insulator as a target material requires the dis- consideration. The frequency comb has a repetition rate of cussion of charge-up effects during laser irradiation. Laser 77 MHz and a fourier-transform limited pulse duration of gate pulses of 100 µs duration with a total power of 1.2 mW about 50 fs of the 7th harmonic. Considering that all 1.2 · (1.2·105 comb modes with 10 nW average power per mode) 105 comb modes will irradiate the target in parallel with are assumed to be used for target irradiation, which cor- a well defined phase relation to create a pulse train, the responds to 9 · 1010 photons at the considered wavelength peak fluence of the light amounts to Φ = 2.0 · 10−8 J/cm2. of 150 nm. Under the pessimistic assumption that one For estimating the target heating, two different time scales electron is carried away by each photon, the total ThO2 have to be considered. On a femtosecond timescale, no sig- charge-up would amount to Q = 1.4 · 10−8 C. Treating nificant heat transport to the non-irradiated target sub- the ThO2 layer deposited on a metal surface as a plate strate will take place and the absorbed energy leads to a capacitor, the capacitance amounts to C = 2.2 nF, where local temperature rise of the target. Assuming that the the dielectric constant r = 3.17 of ThO2 was used [106]. light energy is completely transferred to heat, the maxi- Therefore the voltage across the layer would be U = 6.4 V. mum temperature rise per fs laser pulse is estimated as 8 Lars von der Wense, Chuankun Zhang: Concepts for direct frequency-comb spectroscopy of 229mTh

−3 6 −1 ∆T ≈ αΦ/Cp = 8.6 · 10 K. Here α ≈ 1 · 10 cm de- application of attractive electric fields in the time window notes the absorption coefficient of ThO2 at 150 nm [116] meant for electron detection, but repelling electric fields −3 −1 and Cp = 2.33 Jcm K is the volumetric heat capac- during the time of laser irradiation. This will prevent pho- ity of ThO2 [123]. A heat-up of a few mK is negligible toelectrons emitted at the target from reaching the detec- in terms of a potential isomer shift of the nuclear tran- tor, that would otherwise lead to detector saturation [102]. sition energy (see Sec. 5). The heat will diffuse fast (on The total efficiency of the detection system is estimated the picosecond timescale [124]) and lead to an equilibrium to be 40%. temperature rise of the entire target substrate. Assuming While the here proposed MCP detection scheme appears that the target substrate consists of a Ti-sputtered Si- to be the most straight-forward approach, one should also wafer of 0.5 mm thickness and 20 mm diameter and is consider different ways for IC electron detection in the thermally isolated from its environment, the temperature future, that do not require the electrons to leave the tar- rise would be dominated by the specific heat capacity of get material. One could imagine, for example, detection Si of C = 1.66 Jcm−3K−1, leading to ∆T = 4.6 mK p schemes used in charge-coupled devices (CCDs) [129] or per second of laser irradiation. A thermal contact with in transition-edge detectors like superconducting nanowire the environment is therefore required in order to keep the single-photon detectors (SNSPDs) [130]. In this case not temperature rise significantly below 1 K during laser ir- the amount of electrons that leaves the target surface, but radiation. Importantly, the laser fluence is eight orders of instead the inverse amount would contribute the signal, magnitude below the typical laser ablation threshold for thereby drastically increasing the expected count rate. femtosecond laser pulses [125, 126], which is in agreement with the calculated temperature rise. For this reason no significant laser ablation of the target is expected to occur. However, carbon layer growth has to be considered 4.5 Quantitative evaluation and will require ultra-high vacuum conditions in the target chamber and potentially ozone purge (see Ref. [70]). The number of nuclear excitations as a function of time is shown in Fig. 5. It was calculated based on Eq. (1) using the 229Th target- and laser parameters listed in Tab. 1 and 4.4 The detection system multiplied with the loss factor due to VUV surface absorption of ηabs = 0.63. About 36.5 nuclei can be expected to be excited into the isomeric state during each laser gate It is proposed to use an MCP detector [103] (Hamamatsu, pulse of 100 µs duration. This value is close to the maxi- type F2223-21SH, 27 mm diameter active area) for the mum as obtained in equilibrium of 37 excited nuclei. These detection of the low-energy IC electrons emitted in the nuclei will decay to the nuclear ground state with about isomeric decay. A center hole is foreseen in order to al- 10 µs lifetime via the emission of low-energy internal con- low for 229Th target irradiation in a simple geometry. version electrons [39]. Given the gate frequency of 5 kHz, High detection efficiencies of about 50% for electrons are up to 1.8 · 105 nuclear excitations are expected to occur achieved when post-accelerated to about 300 eV kinetic per second in case of resonance. It was shown that time energy [127]. The detector should be placed in some dis- gating is sufficient for the exclusion of photoelectrons as tance to the target in order to reduce the radioactive back- a background, if a time window from 5 to 10 µs after the ground in the form of high-energy particles. For this reason end of each laser pulse is chosen for IC electron detection a stack of electrostatic einzel-lenses could be used to guide [102]. However, low-energy electrons being continuously only electrons of low kinetic energy to the MCP detector. emitted from the radioactive 229ThO target in the form High energy α or β particles do not follow the electric 2 of, e.g., shake-off electrons from shell-reorganization after fields and are implanted into the chamber walls. SIMION α decay [131], Auger electrons, conversion cascades, etc., [128] simulations were performed in order to optimize the have to be considered as the main background. Due to the electric potentials of the guiding fields and estimate the complexity of these processes, a satisfactory quantitative guiding efficiency. A sketch of the electrode configuration estimate of this background will require a detailed exper- together with the voltages and electron trajectories ob- imental investigation. We here roughly estimate that, on tained from the SIMION simulations are shown in Fig. 4. average, 2 low-energy electrons per α decay are emitted The distance between the target and the detector is about from the target [131]. First preliminary experimental in- 10 cm. At the given potentials, high guiding efficiencies of vestigations indicate that this estimate is indeed valid. Of more than 80% were achieved for electrons of kinetic en- the assumed target activity of 1.7 kBq, 1.3 kBq originates ergies between 0.1 and 2.0 eV. The fraction of the unit from α decay, resulting in an expected emission of about sphere that is covered by the MCP detector at 10 cm dis- 2.6 · 103 low-energy electrons per second. On average, this tance amounts to about 0.45%. Assuming a total target corresponds to 0.013 emitted electrons in the 5 µs detec- activity of 2 kBq, about 9 high-energy α and β particles tion window. The number of expected isomeric decays in are expected to hit the detector per second. This is neg- the same time window amounts to 0.24 · N ≈ 8.9. Even ligible compared to the expected background induced by exc when assuming a factor of 10 of IC electron losses in the low-energy electrons emitted as a byproduct in α decay target, the estimated signal-to-background ratio is about (see next sub-section). 68 : 1 in case of resonance. Here the guiding and detection A grid is positioned in front of the target, which allows the efficiency for IC and background electrons was assumed Lars von der Wense, Chuankun Zhang: Concepts for direct frequency-comb spectroscopy of 229mTh 9

a b 77MHz Numberof excitednuclei combmode laseron laseroff Intensity 37 combcenter 490Hz tunablebetween 15.9kHz 7.9and8.7eV

100µ s Time 5µ s VUVfrequencycomb detectionwindow nuclearresonance

Frequency 2.0PHz 9.2 THz

Fig. 3: Sketch of the proposed experimental concept together with important parameters. a) A VUV frequency comb with 1.2 · 105 comb modes of 490 Hz mode-width (10 nW power per mode) and 77 MHz mode spacing, tunable between 7.9 and 8.7 eV, is used to search for the nuclear resonance. b) Laser excitation is achieved during gate pulses of 100 µs duration (5 kHz gate frequency). The isomeric decay is observed shortly after the end of each laser pulse. A 5 µs detection window is assumed for a signal-to-background estimate.

Table 1: Values of variables used for the calculation of the number of excited nuclei per laser gate pulse based on Eq. (1).

Variable Description Value Comment

−5 2 I Laser intensity 1.4 · 10 W/cm Single-mode (10 nW) focused to ∅ 0.3 mm ΓL Laser bandwidth 2π · 490 Hz Smaller than IC-broadened nuclear linewidth ω0 Angular frequency 2π · 2.0 PHz Corresponding to 8.3 eV energy [25] −4 Γγ Radiative decay rate 10 Hz Estimated from theory [27] 9 αic IC coeﬃcient 10 Based on Γγ and the IC lifetime [39] 13 N0 Irradiated atoms 1.6 · 10 For 10 nm thickness and 0.3 mm diameter ThO2

to be identical. Note that the signal-to-background ratio Table 2: Main parameters for ThO2 target irradiation. improves when the source ages, as the 228Th intrinsic activity will fade away with 1.9 years of half-life. The abso- Description Value lute number of detectable IC electrons amounts to ∼ 0.53 Number of excited nuclei per pulse 37 in the 5 µs detection window per laser gate pulse. This Number of gate pulses per second 5000 includes a 15% probability for the electrons to leave the Time per scan step 0.02 s target and a 40% total efficiency of the detection system. 3 3 Number of excited nuclei per scan step 3.7 · 10 Considering 5000 gate pulses per second, about 2.6 · 10 Number of scan steps for 0.34 eV interval 4.5 · 104 isomeric electrons could be detected per second in reso- Time required to scan 0.34 eV 15 min nance in the 5 µs detection window. Signal-to-background ratio ∼68:1 During the search for the isomeric excitation, the frequency comb has to be tuned. As the scan is performed with all 1.2 · 105 comb modes in parallel, it is sufficient to bridge the mode spacing of 77 MHz of two consecu- rameters of the proposed method are shown in Tab. 2. tive comb modes in order to probe the total bandwidth of the frequency comb of 0.038 eV (9.2 THz). Considering the IC-broadened nuclear transition linewidth of 15.9 kHz, about 5000 scan steps will therefore be required. Assum- ing that 100 gate pulses per scan step are used, the total 5 Line splitting and broadening effects in a time required for scanning of the 0.038 eV energy range solid-state environment amounts to 100 seconds. In order to cover the 0.34 eV energy range corresponding to the 1 σ uncertainty interval As is known from Mössbauerspectroscopy, several line of the currently best energy constraint of 8.28 ± 0.17 eV splitting and broadening effects have to be considered in [25], 9 of such scans would have to be performed leading a solid-state environment. A detailed discussion of these to a minimum total scanning time of 900 s. The main pa- effects for the 229Th nuclear transition can be found in Refs. [18, 19]. Here the discussion will be adapted to a 10 Lars von der Wense, Chuankun Zhang: Concepts for direct frequency-comb spectroscopy of 229mTh

intrinsic to the sample or externally applied. Quantita- tively, the line splitting is described by the equation [132] Q(s)V 3m2 − I(I + 1) ∆E = g µ m B + zz I . (11) HFS I N I 4 I(2I − 1)

Here B denotes the magnetic field and Vzz the second derivative of the electric potential at the site of the nucleus. gI is the nuclear Landég-factor, which is related to the nuclear magnetic moment via µI = gI µN I, µN is the nuclear magneton and Q(s) denotes the spectroscopic Fig. 4: Schematic drawing of the detection system pro- quadrupole moment of the nucleus. Further, I and mI are posed for low-energy IC-electron detection. Electrons the spin and magnetic quantum number, respectively, cor- emitted at the 229Th target are guided by electrostatic responding to the nuclear sub-state under consideration, lenses to an MCP detector with a center hole. SIMION where mI takes the 2I + 1 values from −I to +I. 229 [128] simulations were performed showing that guiding ef- Thorium-dioxide ( ThO2) provides a cubic crystal struc- ficiencies of more than 80% for electrons between 0.1 and ture. For this reason Vzz ≈ 0 holds and no significant 2.0 eV can easily be achieved. Potentials used for the sim- quadrupole splitting is expected to be observable at the ulations are shown in the figure together with 50 electron point of the 229Th nucleus [133]. Experimentally, Mössbauer trajectories obtained for electrons with 0.5 eV kinetic en- spectroscopy that was performed with the 49.4 keV γ- 232 232 ergy. The total distance between target and detector is ray of Th in a ThO2 crystal structure supports this about 10 cm. expectation [134]. However, the half-life of the 49.4 keV nuclear excited state of 232Th is only 345 ps, resulting in a natural linewidth of about 320 MHz, which could 50 easily cover a small quadrupole splitting. For this reason, a comparison to the long-lived Mössbauerstate at 40 93.3 keV in 67Zn may provide more useful information. This Mössbauerstate has a half-life of 9 µs, similar to 30 the 7 µs observed half-life of 229mTh under IC decay [39]. The natural linewidth of the 67Zn Mössbauertransition is 20 therefore with 12.4 kHz comparable to the one of 229mTh. Also for 67Zn no quadrupole splitting could be observed Number of excited nuclei 10 when a cubic crystal structure (67ZnS) was used for Mössbauer

0 spectroscopy [135]. This gives a clear indication, that, de- 0 0.2 0.4 0.6 0.8 1 229m −4 Time since start of irradiation [s] x 10 spite the narrow expected linewidth of the Th nuclear transition of 15.9 kHz, no quadrupole splitting of the nu- Fig. 5: Expected number of excited nuclei as a function of time clear hyperfine levels will be observable. 13 229 if 1.6 · 10 Th atoms in a ThO2 surface are irradiated in parallel. The calculation was performed based on Eq. (1) using In addition, ThO2 is paramagnetic and does not exhibit unpaired electron spins, therefore we have B ≈ 0 and no Cge = 1 and values for variables listed in Tab. 1. A loss factor of ηabs = 0.63 due to VUV absorption within the target was magnetic hyperfine splitting will arise due to target in- already taken into account. trinsic magnetic fields potentially generated by the electronic environment. It was discussed in Refs. [18, 19], that a small magnetic field will be generated due to the surrounding nuclear spins. As the nuclear spins are randomly 229 pure ThO2 environment. ThO2 is chosen as a target oriented, the corresponding Zeeman splitting will lead to material as it is chemically stable and provides a cubic a homogeneous line broadening of the transition, which lattice structure resulting in no measurable quadrupole is assumed to be the dominant line broadening effect in splitting. a solid-state environment. The corresponding broadening was estimated to be between 2π · 1 and 2π · 10 kHz in Ref. [18] and to about 2π · 150 Hz in Ref. [19]. In any case 5.1 Nuclear hyperfine splitting the broadening effect is smaller than the IC-broadened natural 229mTh transition linewidth of 2π · 15.9 kHz and A hyperfine splitting of the nuclear transition occurs due will therefore not significantly affect the considered con- to the coupling of the spins of the nuclear ground- and cept. This is in agreement with the fact that it was possible excited state to the electronic environment. This splitting to experimentally resolve the natural transition linewidth consists of two parts: (1) a quadrupole splitting that arises (2π · 12.4 kHz) of the 67Zn Mössbauerline [100]. from the nuclear quadrupole moment in an electric field 229 gradient that is generated at the point of the nucleus by In conclusion, the Mössbauer spectrum of ThO2 will the surrounding electrons and (2) a Zeemann splitting, consist of a single line as shown in Fig. 6, if no other crys- which originates from a magnetic field that can either be tal structures are present and no external magnetic field is Lars von der Wense, Chuankun Zhang: Concepts for direct frequency-comb spectroscopy of 229mTh 11

model, leading to [137, 138] !! 2 Z ΘD /T 6ER 1 T xdx f = exp − + x . kΘD 4 ΘD 0 e − 1 (12) 2 2 Here ER = Eγ /(2Mc ) denotes the nuclear recoil energy, k is the Boltzmann constant, ΘD the Debye temperature and T the temperature of the crystal. For high-energy γ rays the Debye-Waller factor is only large, if the crystal temperature is significantly below ΘD, which may require cooling to cryogenic temperatures. In the special 229m case of Th, however, Eγ is sufficiently small so that large values for f are obtained over a wide temperature range. Reported literature values for the Debye temperature of ThO2 vary significantly (see, e.g., Refs. [141– 145]), however, the Debye-Waller factor for 229mTh remains largely unaffected. For an assumed Debye temperature of ΘD = 400 K, a Debye Waller factor close to one Fig. 6: Nuclear magnetic sub-states of 229Th embedded in is obtained at room temperature. a ThO2 environment. ThO2 exhibits a cubic lattice struc- A second order Doppler broadening will be present due to ture, therefore no electric quadrupole shift will occur. In vibrations of the 229Th nuclei at finite temperature. This addition, no unpaired electron spins are present, therefore effect was estimated in Ref. [19] to about 2π · T Hz, where no intrinsic Zeeman splitting caused by the electronic en- T denotes the temperature in Kelvin. At room tempera- vironment will arise. Accordingly, the nuclear spectrum ture the broadening will amount to less than 2 kHz, which consists of a single line as long as no other chemical bond- is significantly below the IC-broadened nuclear transition ings are present and no external magnetic field is applied. linewidth. For this reason no temperature related second order Doppler broadening is expected to be detectable even at room temperature and the experiment does not require a cryogenic environment. applied. Importantly, a chemical (or isomer) shift, as often discussed in the context of Mössbauerspectroscopy, cor- However, two other line-broadening effects require a dis- responds to an absolute shift of the excitation energy due cussion: homogeneous broadening from rapid fluctuations to the interaction of the nucleus with its electron cloud of the electronic environment and inhomogeneous broad- [136]. It is not related to a line splitting. Although po- ening from different chemical environments in the tar- tentially being large (on the order of 100 MHz) the shift get. An inhomogeneous broadening may arise, if different 229 is constant for a constant chemical environment and does chemical bondings in the ThO2 matrix are present, as therefore not affect the proposed experimental scheme. this would potentially lead to different hyperfine splittings [146]. This can be avoided even in thin surfaces by using a pure target material. A conversion-electron Mössbauer spectroscopy (CEMS) experiment using a 25 nm thin 57Fe 5.2 Line broadening effects absorber was performed in Ref. [147]. The observed Mössbauer spectrum showed the expected unbroadened hyperfine splitting. A small absorber thickness was required, as the inelastic mean-free path of the observed electrons is only It is a central feature of Mössbauerspectroscopy, that no about 10 nm, comparable to the one expected for 229mTh. first order Doppler broadening of the transition line as well as energy shifts due to the γ-ray recoil have to be A further homogeneous line broadening will be present, considered, as the recoil momentum is transferred to the as the electronic environment can usually not be treated entire crystal if the atoms are confined in the Lamb-Dicke as static and the hyperfine structure will in general be regime [21, 137, 138]. The physical reason is, that in a crys- subject to time-dependent fluctuations [146]. Fluctuations tal lattice structure the atoms are strongly bound and are may arise in the temperature and as the electron density vibrating fast with a small amplitude around their center at the site of the nucleus is temperature dependent, this position. If this amplitude is smaller than the wavelength will lead to fluctuations in the chemical shift. The observed of the transition the Doppler effect averages out, result- energy change caused by temperature is typically on the ing in a Doppler-free linewidth (the effect is known as the order of 10 kHz/K (see Ref. [18] and references therein). Dicke effect) [137, 139, 140]. The fraction of atoms that It was shown in Sec. 4 that, despite the laser irradiation exhibit a Doppler-free linewidth is given by the Debye- of the target, temperature fluctuations can be kept signif- 2 2 2 2 Waller factor f = exp(−Eγ hx i/(~c) ), where hx i de- icantly below 1 K and the resulting homogeneous broad- notes the mean-square vibrational amplitude of the nu- ening is therefore expected to be negligible compared to cleus. Usually, hx2i is calculated in the frame of the Debye the IC-broadened natural linewidth of the transition. 12 Lars von der Wense, Chuankun Zhang: Concepts for direct frequency-comb spectroscopy of 229mTh

According to the first term of Eq. (11), fluctuations of the magnetic field at the site of the nucleus will lead to a homogeneous line-broadening. As the intrinsic magnetic fields of the target are only of minor concern, such broadening effects can be kept at a minimum by stabi- lizing the external magnetic field of the environment. A stabilization to better than 1 mT is required in order to keep any potential line broadening to below 10 kHz. Experimentally this can easily be achieved. Due to the cubic lattice structure, no temperature-dependent line- broadening effects that could potentially be caused by the coupling of a fluctuating electric field gradient to the nuclear quadrupole moment have to be considered. The above quantitative discussion shows, that during laser- 229m 229 based CEMS of Th in a thin ThO2 layer, only a single transition line with the natural linewidth of 15.9 kHz has to be expected. This situation changes if (1) a magnetic field is externally applied or (2) a different (non- cubic) crystal structure is used. Both situations will be quantitatively discussed in the following.

5.3 External magnetic ﬁelds

According to the Zeeman effect described by the first term Fig. 7: Expected Zeeman splitting of the 229mTh nu- of Eq. (11), a line splitting will arise as soon as an exter- clear transition in a CEMS experiment in case of a pure 229 nal magnetic field is applied. For the nuclear ground-state ThO2 absorber and an externally applied 100 mT con- (Ig = 5/2) a magnetic moment of µg = 0.360(7)µN was stant magnetic field. It is assumed that linearly polarized experimentally determined [148]. For the nuclear excited- laser light is used for excitation. The excitation probabil- state (Ie = 3/2) the corresponding value is µe = −0.37(6)µN ity scales proportional to the Clebsch-Gordan coefficients 2 [42]. This allows one to calculate the expected line split- squared (Cge), therefore the central lines are by a factor ting. The splitting of the magnetic sub-states as well as of 1.5 more intense than the outer lines. the transitions that can be driven with linearly polarized laser light are shown in Fig. 7 together with the expected Mössbauerspectrum under the assumption that the nuclei are exposed to a constant magnetic field of 100 mT. also a nuclear clock that makes use of a large number of In this case the splitting between the two outermost lines 229Th ions embedded in a crystal-lattice environment has is determined to be about 900 kHz. As the splitting scales attracted significant attention, as it may pose a practical proportional to B, no line splitting will be observed as tool for time measurement [16, 18, 19]. Here, a new solid- long as the magnetic field is kept below 1 mT. This value state nuclear clock concept is proposed, which is based also corresponds to the stability of the magnetic field re- on conversion-electron Mössbauerspectroscopy (CEMS) quired to suppress any line broadening due to the Zeeman [147]. For this purpose a narrow-band laser is stabilized to effect. As soon as the isomeric state has been successfully the nuclear transition of 229Th deposited as a thin layer of 229 excited, magnetic fields can be applied in order to verify ThO2 onto a substrate surface. Successful nuclear exci- the signal origin. Importantly, if no magnetic field is ap- tation is probed via detection of the IC electrons emitted plied, all four lines will be degenerate and their intensities shortly after the laser pulse, just as proposed in Sec. 4. will add up. Therefore the actual number of nuclear exci- It will be shown in the following, that this concept might tations in equilibrium should be by a factor of 1.3 larger be advantageous compared to the crystal-lattice clock ap- than the one stated in Tab. 2, where only one pair of mag- proach [18, 19]. netic sub-states was considered, however with Cge = 1. Clock performance is generally expressed via two parameters: accuracy and stability. The stability of a clock corresponds to the statistical uncertainty of a frequency mea- 6 A new solid-state nuclear clock concept surement and depends on the measurement time. Opposed to that, the accuracy (or more precisely: the systematic The development of a nuclear optical clock based on in- frequency uncertainty) of a clock also takes all systematic dividual 229Th3+ ions in a Paul trap has been extensively uncertainties of a frequency comparison between different discussed in literature [16, 17, 20]. Such a clock is expected clocks into consideration. In order to achieve a high clock to achieve an extraordinary high accuracy approaching performance, it is important that the statistical uncer- 10−19 [17]. Although expected to be less accurate [18], tainty approaches the systematic uncertainty within short Lars von der Wense, Chuankun Zhang: Concepts for direct frequency-comb spectroscopy of 229mTh 13

averaging times. On the other hand, any improvement of clear decay rate: Γ˜ ≈ Γ/2 ≈ Γic/2. Inserting these values, the statistical uncertainty of the frequency measurement together with a short interrogation time of T = 10 µs, −15 √ significantly below the value of the systematic frequency into Eq. (13), a stability of σy(τ) ≈ 9.5 · 10 / τ is 13 uncertainty will not improve the clock performance, as achieved. Here it was assumed that N0 = 10 atoms are this will be limited by other systematic effects. irradiated with a laser of intensity I = 1.4 · 10−5 W/cm2. Further, N = kS/(4π) was used, taking into account It was shown in Ref. [19] that the stability of a 229Th- eff that no competing decay channels exist, with k = 0.1 as based solid-state nuclear clock can be estimated for the before, but S = 4π/2, as the emitted electrons can be realistic case of low laser intensities and short interroga- attracted towards the detector. Obviously, the calculated tion times (T < 1/∆ω ) via a shot-noise limited Allan 0 stability performance is lower than the best that could deviation of2 be achieved with a crystal-lattice nuclear clock. Impor- 3 1 Γ˜ 1 tantly, however, after about 2.5 · 10 s of averaging the σy(τ) ≈ √ . (13) stability has surpassed the expected systematic frequency ω T Γ 0 ΓexcNeffτ uncertainty, for which a value of 2 · 10−16, like for the Here T denotes the interrogation time, which should not crystal-lattice approach, is assumed. Of course there is the exceed the isomeric lifetime, Γ˜ is the decay rate of the option to choose shorter interrogation times also in the crystal-lattice clock approach. This will, however, signifi- coherences given in Eq. (2) and Γ = Γγ + Γnr the total decay rate of the transition, including non-radiative decay cantly affect the clock’s stability performance, as it scales channels. N denotes the effective number of irradiated proportional to 1/T (e.g., using√ T = 1000 s interrogation eff time, a stability of 6.0 · 10−15/ τ is obtained, comparable nuclei defined as Neff ≈ Γγ /Γ · kS/(4π) · N0, with k being the quantum efficiency, S the effective solid angle covered to the expected stability of an IC-based nuclear clock). by the detector and N0 the actual number of irradiated From the above considerations it is evident that the stabil- nuclei. τ is the averaging time, which has to be equal or ities of both, a crystal-lattice nuclear clock as well as an larger than the time T chosen for interrogation and Γexc IC-based nuclear clock, should be sufficient to approach is the nuclear excitation rate, which is estimated for the a statistical uncertainty of the frequency measurement of considered case of low laser intensities based on Eq. (5). about 10−16 after a few thousand seconds of averaging. In the following, the crystal-lattice approach as proposed Longer measurement times are not expected to improve clock performance, which will most likely be limited to in Refs. [16, 18] is discussed. Assuming purely radiative −16 4 a level of ∼ 10 by systematic frequency uncertainties decay (Γ = Γγ ) and an interrogation time of T = 10 s, corresponding to the expected radiative lifetime, as well [18]. However, it may be seen as a big advantage of the as a decay rate of the coherences of Γ˜ ≈ 1 kHz due to cou- IC-based nuclear clock approach that no competing decay pling to the crystal-lattice environment (see Ref. [19] for channels exist next to internal conversion, which has al- details), a straight forward calculation reveals a high sta- ready been experimentally observed and is known to be −17 √ the dominant decay channel for the considered experimen- bility of σy(τ) ≈ 6.0 · 10 / τ for a solid state nuclear 13 tal conditions. Opposed to that, the crystal-lattice nuclear optical clock [19]. Here, values of N0 = 10 , k = 0.1, 229m S = 4π/10 and I = 1.4 · 10−5 W/cm2 (corresponding to clock makes important use of the Th γ decay, which 10 nW of laser power irradiating a crystal of 0.3 mm diam- requires suppression of the IC decay channel by nine or- eter) were used. Thus, after the shortest useful averaging ders of magnitude in a solid-state environment. Even when time of τ = 104 s, a stability in the 10−19 range appears achieved, non-radiative decay via bound internal conver- to be achievable. However, the systematic frequency un- sion (BIC) and electronic bridge (EB) mechanisms might certainty of the clock is expected to be limited by other be present, reducing the isomeric lifetime and the fraction effects, most importantly temperature shifts of the crys- of observable γ decays. tal, to a value of 2·10−16 [18]. For this reason only limited advantage will be gained from the high stability as long 229 3+ as other systematic shifts will not be sufficiently well con- 7 Laser spectroscopy of Th ions in a trolled. Paul trap

In the following, the same stability estimate is applied to 229 a new nuclear clock concept, which is based on the obser- Most of the Th-related research in the past decade has vation of internal conversion electrons following the laser focused on the problem of constraining the isomeric energy 229 value by about a factor of 10 to 50 meV uncertainty, cor- irradiation of ThO2 deposited on a surface. In this case, the nuclear decay rate will be dominated by the short IC responding to the total bandwidth of a frequency comb 5 in the VUV region. However, the problem of improving lifetime of about 10 µs [39]: Γ ≈ Γic ≈ 10 Hz. Further, as the laser used for irradiation is narrowband, the de- the energy uncertainty down to a few 100 Hz, as required cay rate of the coherences will be dominated by the nu- for the development of single-ion nuclear clock, has not been addressed. Importantly, laser excitation of 229Th ions 2 The equation is obtained by inserting Eq. (37) of Ref. [19] or atoms in a solid-state environment does not solve this into Eq. (48) of the same paper and using that by definition problem, as the nuclear hyperfine structure splitting is ex- t = 4T , where t denotes the time for a complete interrogation pected to deviate by several 100 MHz from the single-ion cycle used in Ref. [19]. case. In the following this problem will be addressed. As 14 Lars von der Wense, Chuankun Zhang: Concepts for direct frequency-comb spectroscopy of 229mTh soon as the isomeric energy has been constrained to suf- Table 3: List of input values used in Eq. (1) for calcula- ficient precision, a single-ion nuclear clock can be build, tion of the number of nuclear excitations in a chain of 10 which requires to drive nuclear Rabi oscillations. The po- trapped 229Th3+ ions. tential for driving nuclear Rabi oscillations will be discussed in Sec. 7.2. Variable Value Comment

2 I 0.14 W/cm Single-mode focused to ∅ 3 µm ΓL 2π · 490 Hz Larger than nuclear linewidth ω0 2π · 2.0 PHz Corresponding to 8.3 eV energy 229 3+ −4 7.1 Excitation of multiple trapped Th ions Γγ 10 Hz Estimated from theory 3+ αeb 50 EB decay for Th ions [151] It is assumed that the laser system already presented in N0 10 Multiple laser-cooled ions Sec. 4 is used to irradiate 10 laser-cooled 229Th3+ ions in a Paul trap. Laser cooling could be achieved either directly [46] or via sympathetic cooling [149]. Further, it is assumed that the laser light is focused to a spot of 3 µm di- the concept is not affected as long as αeb 2200 holds. ameter in order to irradiate the chain of ions and that the However, EB coefficients significantly larger than 100 are energy has been constrained by a different method, e.g., experimentally excluded [38]. the one described in Sec. 4, to a value that corresponds The result of the calculation is shown in Fig. 8. Here a to the total bandwidth of the frequency comb used for factor Cge = 1 was used, making no assumption on the nu- excitation (0.038 eV in the considered case)3. In this con- clear magnetic sub-system. In a real experiment, a particu- cept, the successful laser excitation is probed via exploit- lar sub-system has to be chosen, where each possible tran- ing the change of the hyperfine structure of the electronic sition is defined by the quantum numbers of the ground shell, induced by the different spins of nuclear ground- and excited nuclear state corresponding to the electronic and excited state, which is known as the double-resonance angular momentum J, the nuclear spin I, the total angular method [16]. In the following, the expected number of ex- momentum F = I + J, as well as the magnetic quantum cited nuclei as a function of time will be calculated based number m. These define the multipolarity of the transi- 2 on Eq. (1). tion L and the polarization of the light σ. In this case Cge In 229Th3+ ions the internal conversion decay channel is is obtained as [98] energetically suppressed and αic = 0 holds. However, a 2 potential electronic bridge (EB) decay channel might ex- Cge =(2Fe + 1)(2Fg + 1)(2Ie + 1)× ist and lead to an enhancement of the total decay rate 2 2 (14) Ig LIe Fg LFe according to Γ = Γγ + Γeb, where Γeb denotes the decay , Fe JFg −mFg σ mFe rate related to the EB process and replaces Γic of the previous considerations. In the EB decay the nucleus couples where the curly bracket denotes the Wigner-6J and the to the electronic shell, exciting an electron into a virtual curved bracket the Wigner-3J symbol, respectively. The shell state, which immediately decays under photon emis- 229 3+ definition of these symbols can be found, for example, in sion [40]. For Th , the EB coefficient αeb = Γeb/Γγ Ref. [152]. In case that a stretched pair of nuclear hyper- was estimated to be between 0.01 and 0.1 [150] for the fine states is chosen, as proposed in the concept of Ref. [17] electronic ground state and between 20 [150] and about 50 2 2 for a single-ion nuclear clock, one obtains Cge = 2/3. Driv- [151] for the metastable 7s S1/2 electronic excited state. ing this sub-system will however require circular polarized For the following calculations αeb = 50 is assumed, re- light. sulting in an isomeric lifetime which is reduced by a factor of 50 to τeb ≈ 200 s compared to the radiative life- Assuming Cge = 1 for simplicity, the time required to 4 time τγ = 1/Γγ ≈ 10 s. Experimentally, a lower limit reach saturation is about 20 s as shown in Fig. 8. The of 60 s for the isomeric lifetime in 229Th2+ ions was ob- excitation probability amounts to 50%, resulting in 5 ex- served [38]. The parameters used to calculate the nuclear cited nuclei in case of resonance. In the search for the excitation probability via Eq. (1) are listed in Tab. 3. As nuclear excitation the mode spacing of 77 MHz has to be the laser intensity is sufficiently strong to enter the power- bridged. With a mode bandwidth of 490 Hz, this leads to broadened regime, the result is not significantly affected 1.57·105 scan steps. Assuming that each scan step takes a by the EB decay channel. Following Eq. (1) this is correct time of 20 s, the time required to bridge the mode spacing ˜ 2 amounts to 3.14 · 106 s or 36.4 days. As all 1.2 · 105 comb as long as ΓΓ Ωeg holds. In the considered case one modes are used in parallel, the energy range scanned in has Γ ΓL, which results in combination with Eq. (2) in 2 this way equals 9.2 THz or 0.038 eV. For this reason, it ap- the condition Γ 2Ω /ΓL ≈ 0.22 Hz. For this reason, eg pears realistic to use the frequency comb in the search for 3 Importantly, any improvement of the energy uncertainty to the nuclear transition in Paul a trap, as soon as the energy a value below the total bandwidth of the frequency comb does has been constrained to an uncertainty which corresponds not lead to a shortening of the scanning time as the frequency to the total bandwidth of the comb. In case of successful spacing between two consecutive comb modes will always have excitation, the isomeric energy could be constrained to a to be scanned. few 100 Hz of precision, providing the basis for the devel- Lars von der Wense, Chuankun Zhang: Concepts for direct frequency-comb spectroscopy of 229mTh 15

Table 5: Values of variables used for the calculation of 5 nuclear Rabi oscillations.

4 Variable Value Comment 3 2 I 0.14 W/cm Single-mode focused to ∅ 3 µm 2 ΓL 2π · 1 Hz Larger than nuclear linewidth ω0 2π · 2.0 PHz Corresponding to 8.3 eV energy −4

Number of excited nuclei 1 Γγ 10 Hz Estimated from theory 3+ αeb 50 EB decay for Th ions [151] 0 N0 1 Individual laser-cooled ion 0 5 10 15 20 Time since start of irradiation [s]

Fig. 8: Expected number of excited nuclei as a function of time if 10 229Th3+ ions in a Paul trap are irradiated in parallel. The 1 curve is generated based on Eq. (1) with Cge = 1, making use of the input values listed in Tab. 3. 0.8

0.6 Table 4: Main results for laser irradiation of 10 trapped 229Th3+ ions. 0.4

0.2 Description Value Nuclear excitation probability

0 Time per scan step 20 s 0 0.5 1 1.5 2 2.5 3 Number of excited nuclei per scan step 5 Time since start of irradiation [s] Number of scan steps for 0.038 eV interval 1.57 · 105 Fig. 9: Nuclear excitation probability as a function of time for Time required to scan 0.038 eV 36.4 days a single 229Th3+ ion in a Paul trap. Nuclear Rabi oscillations with a frequency of about 3 Hz become visible. Eq. (1) was used with Cge = 1 and input values listed in Tab. 5. opment of a single-ion nuclear clock. The main parameters of the concept are listed in Tab. 4. Table 6: Main parameters for driving nuclear Rabi oscillations. 7.2 The potential for driving nuclear Rabi oscillations Description Value For the single-ion nuclear clock, it is the goal to irradiate π-pulse duration ∼170 ms an individual, laser-cooled 229Th3+ ion with a single mode Excitation probability ∼ 90% of a frequency comb with extraordinary narrow bandwidth Time per scan step 1 s [17]. Ideally, Rabi oscillations should be generated, in or- Number of scan steps for 500 Hz interval 500 der to allow for the Ramsey interrogation scheme in the Time required to scan 500 Hz 500 s nuclear clock concept [7]. This would lead to a quantum- projection-noise limited Allan deviation, which is a re- quirement for the highly stable operation of a single-ion nuclear clock. For the following it is assumed that the same excitation via the double-resonance method, the time for frequency comb already discussed in the previous sections scanning the 77 MHz mode spacing is prohibitively long. is used and focused to a diameter of 3 µm to irradiate a For a bandwidth of 1 Hz the number of scan steps amounts single ion, however, the bandwidth of an individual comb to 7.7·107. This corresponds to 2.4 years of scanning time mode is narrowed down by a factor of ∼ 500 to 1 Hz. In in order to cover the energy range of 0.038 eV. However, in this case it is shown that nuclear Rabi oscillations could the case that the energy has been previously constrained be generated. The input values for Eq. (1) are listed in by the concept described in Sec. 7.1 to a few 100 Hz, a Tab. 5. maximum number 500 scan steps would be required in the The nuclear excitation probability as a function of time search for the resonance, leading to a net scanning time is shown in Fig. 9. Again, Cge = 1 was used and the of 500 s. The main parameters of the concept are listed in result is not significantly affected by the large EB decay Tab. 6. rate as long as Γ ΓL = 2π · 1 Hz holds. This is the 4 case for αeb 6.3 · 10 , which will certainly be fulfilled. Based on Eq. (1) the π-pulse duration is determined to 8 Conclusion and outlook ≈ 170 ms. The excitation probability after the π-pulse is about 90%. Under the assumption that 1 second read- A new concept for narrow-band direct nuclear laser spec- out time per scan step is required in order to observe the troscopy using the 7th harmonic of an Yb-doped fiber laser 16 Lars von der Wense, Chuankun Zhang: Concepts for direct frequency-comb spectroscopy of 229mTh frequency comb is presented. The concept makes use of ex- ment in order to take full advantage of the high expected isting laser technology and would pave the way for narrow- accuracy of a single-ion nuclear clock. While conceptu- band laser spectroscopy of 229mTh and the development ally possible, driving nuclear Rabi oscillations will require of a solid-state nuclear optical clock. significant experimental effort and much improved con- straints on the isomeric energy value as obtained by the It is proposed to use the laser light for the irradiation of concept discussed in Sec. 7.1. These experiments can be a 10 nm thin 229Th layer deposited as ThO on a target 2 envisaged as follow-up investigations to the experiment substrate. In case of resonance, the successful excitation discussed in Sec. 4. of the nuclear state can be probed via the detection of the internal conversion electrons emitted shortly after laser irradiation. The experiment appears to be advantageous compared to direct nuclear laser spectroscopy of 229Th ions in a Paul trap in terms of measurement time and absolute number of nuclear excitations. The mode spac- 9 Acknowledgments ing of the frequency comb of 77 MHz could be scanned within 100 s, thereby effectively probing an energy range We acknowledge discussions with B. Seiferle, G. Porat, of 9.2 THz (0.038 eV), as all 1.2 · 105 comb modes are S. Schoun, C. Heyl, D. Renisch, G. Kazakov, P. Bilous, used in parallel in the search for the nuclear excitation. A. Pálffy, F. Karpeshin, J. Weitenberg, O. Pronin, I. Pu- In case of success, the isomer’s energy uncertainty would peza, T. Udem, E. Peik, S. Stellmer, T. Schumm, C.E. improve by a factor of 106 down to about 100 MHz when Düllmann,T. Schibli, P.G. Thirolf and J. Ye. C. Zhang compared with the free atom or ion. In addition, it would acknowledges the Graduate Research Assinstantship from be straight-forward to develop an IC-based solid-state nu- J. Ye’s group in JILA. Part of this work was submitted as clear clock making use of this detection scheme, by stabi- a research proposal by L. v.d.Wense for a Humboldt fel- lizing the comb mode to the nuclear transition. It is ar- lowship, which has been granted. L. v.d.Wense is grateful gued that this concept may be advantageous compared to to the Humboldt Foundation for this support. Further, he the crystal-lattice nuclear clock approach, as it does not would like to thank J. Ye for accepting him as a Hum- require the suppression of non-radiative decay channels, boldt fellow and P.G. Thirolf for eight years of successful but makes instead use of these. An IC-based solid-state joint work. This work was funded by the European Union’s nuclear clock would have the same potential for the ob- Horizon 2020 research and innovation programme under servation of time-variations of fundamental constants like grant agreement 664732 “nuClock” and by DFG grant No. a crystal-lattice nuclear clock [16, 18, 153]. Th956/3-2 as well as by the LMU Mentoring program. In addition, two experiments using the same frequency comb for excitation of laser-cooled 229mTh3+ ions in a Paul trap were discussed for reasons of their importance: References In the concept described in Sec. 7.1, multiple 229Th3+ ions in a Paul trap are irradiated in parallel. This experiment 1. L. Essen, J.V.L. Parry, Nature 176, 280 (1955). is important, as it would allow constraining the isomeric 2. Comptes Rendus de la 13e CGPM (1967/68), 103 energy value of individual 229Th3+ ions by further six or- (1969). 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