Prospects of a thousand-ion Sn^2+ Coulomb-crystal clock with sub-10^−19 inaccuracy

Domain: earth_science
Published: 2024-07-05
Editorial outcome: major_revision

Abstract
Optical atomic clocks are the most accurate and precise measurement devices of any kind, enabling advances in international timekeeping, Earth science, fundamental physics, and more. However, there is a fundamental trade off between accuracy and precision, where higher precision is achieved by using more atoms, but this comes at the cost of larger interactions between the atoms that limit the accuracy. Here, we propose a many-ion optical atomic clock based on three-dimensional Coulomb crystals of order one thousand Sn2+ ions confined in a linear RF Paul trap with the potential to overcome this limitation. Sn2+ has a unique combination of features that is not available in previously considered ions: a 1S0 ↔ 3P0 clock transition between two states with zero electronic and nuclear angular momentum (I=J=F=0) making it immune to nonscalar perturbations, a negative differential polarizability making it possible to operate the trap in a manner such that the two dominant shifts for three-dimensional ion crystals cancel each other, and a laser-accessible transition suitable for direct laser cooling and state readout. We present calculations of the differential polarizability, other relevant atomic properties, and the motion of ions in large Coulomb crystals, in order to estimate the achievable accuracy and precision of Sn2+ Coulomb-crystal clocks.

Introduction
Optical atomic clocks based on single trapped ions and on ensembles of thousands of optical-lattice-trapped neutral atoms have achieved 10−18 fractional frequency accuracy and precision1, surpassing the microwave atomic clocks that underpin international atomic time (TAI) by three orders of magnitude, and enabling new technological applications such as relativistic geodesy2 as well as tests of the fundamental laws of physics3. For any particular application, trapped-ion and optical-lattice clocks based on optical atomic transitions have complimentary advantages and limitations. Optical lattice clocks use simultaneous measurements of thousands of atoms to quickly average down quantum projection noise (QPN), a fundamental limit to clock precision, speeding up both characterizations of systematic effects and clock measurements, and increasing the bandwidth of sensing applications. However, they suffer from larger systematic frequency shifts due to blackbody radiation and interactions between atoms that are difficult to control. Clocks based on one or a few trapped ions suffer from larger QPN, but offer exquisite control of environmental perturbations and interactions, high-fidelity universal quantum control at the individual ion level, and a wide variety of atomic, molecular, and highly charged ions to choose from with different sensitivities to environmental perturbations and proposed extensions of the Standard Model of particle physics.

Recently, a wide variety of new optical clock platforms, which overcome some of the limitations of conventional lattice and ion clocks, have been proposed and demonstrated. Composite optical clocks combining an ensemble of lattice-trapped atoms that provides high stability, together with a single trapped ion that provides high accuracy have been proposed4,5, and the fundamental building blocks have been demonstrated6. Ion trap arrays have been built and shown to be capable of supporting clock operation with around 100 ions and sub-10−18 accuracy7. Clocks based on arrays of optical-tweezer-trapped atoms offer the prospect of individual-atom-resolved quantum control, and have demonstrated coherence times exceeding 40s and operation with up to 150 atoms8. Three-dimensional optical lattice clocks greatly suppress the interactions between atoms present in current lattice clocks based on one and two-dimensional optical lattice traps and have demonstrated operation with up to 105 atoms9. Finally, recent demonstration has shown that it is possible to build clocks based on highly charged ions which can have lower sensitivity to environmental perturbations and higher sensitivity to physics beyond the Standard Model10–13.

Three-dimensional Coulomb crystals of thousands of ions in linear RF Paul traps have been studied for many years for other applications14–19; however they were thought to be impractical for high accuracy atomic clocks because ions located off of the trap axis, along which the RF trapping electric field is zero, suffer from driven motion called micromotion that leads to large time-dilation systematic frequency shifts. Berkeland et al.20 pointed out that for clock transitions with a negative differential polarizability, the negative micromotion time-dilation shift could be almost perfectly canceled out by a positive differential Stark-shift caused by the same RF trapping field that drives the micromotion, provided that the trap is driven at a “magic” value of the RF frequency. Building on this, Arnold et al.21 proposed that many-ion clocks could be built based on three-dimensional Coulomb crystals of ions with negative-differential-polarizability clock transitions, which at the time was known to include B+, Ca+, Sr+, Ba+, Ra+, Er2+, Tm3+, and Lu+. Lu2+22 and Pb2+23 were later added to this list, and Kazakov et al.24 performed a detailed analysis of micromotion shifts in these systems and proposed active optical clocks based on negative-differential-polarizibility transitions. In contrast with earlier theoretical predictions, it was later experimentally determined that the 1S0 ↔ 3D1 clock transition of Lu+ actually has a very small but positive differential polarizability. The alternative 1S0 ↔ 3D2 clock transition of Lu+ was determined to have a negative but larger differential polarizability25. Of the remaining candidates for Coulomb-crystal clocks, Ca+, Sr+, Ba+, Ra+, Er2+, Tm3+, Lu+ 1S0 ↔ 3D2, and Lu2+ have comparatively short excited-state lifetimes and suffer from inhomogenous broadening due to quadrupole and tensor polarizibility shifts22. Beloy et al.23 proposed doubly ionized group-14 elements with I=J=F=0 such as Pb2+ as ideal trapped-ion clock candidates because their 1S0 ↔ 3P0 transitions are completely immune to these nonscalar perturbations.

B+ and Pb2+ have doubly-forbidden intercombination clock transitions with >103 s lifetimes, but do not have transitions suitable for direct state readout. In B+ the Γ=2π×2 Hz decay rate of the 1S0 ↔ 3P1 transition is prohibitively slow, and in Pb2+ the wavelength of this transition is 155 nm. The latter is below the low-wavelength cutoffs for generation of UV laser light by nonlinear frequency conversion in workhorse materials β-BaB2O4 (BBO, 182 nm phase matching cutoff26), CsLiB6O10 (CLBO, 180 nm transparency cutoff27), and LiB3O5 (LBO, 160 nm transparency cutoff28). New candidate materials such as KBe2BO3F2 (KBBF, 150 nm transparency cutoff29) and BaMgF4 (BMF, 125 nm transparency cutoff30) are difficult to grow and therefore unavailable commercially with sufficiently high quality. Scaling indirect quantum-logic readout31 to large Coulomb crystals remains an open challenge32.

In this article, we propose Coulomb-crystal clocks based on of the order of one thousand Sn2+ ions (see Fig. 1). We present a theoretical calculation of the differential polarizability of the 1S0 ↔ 3P0 clock transition, showing it to be both small and negative, meaning that the systematic shift due to blackbody radiation is small, and there is a magic trap frequency of 225(5) MHz at which the trap Stark shift cancels the micromotion time-dilation shift. Although this is a high trap frequency, previous experiments have operated in this range33. Similar to Pb2+, in addition to having zero electronic angular momentum in the two clock states, there are several isotopes of Sn2+ that have zero nuclear spin, and thus the clock transition is completely immune to nonscalar perturbations23. Furthermore, for these nuclear-spin-zero isotopes, the clock transition is an extremely forbidden intercombination transition with a zero-magnetic-field excited state lifetime on the order of years, but the lifetime can be tuned to an experimentally convenient value by applying a magnetic field34,35. However, unlike Pb2+, Sn2+ has a laser-accessible 181 nm 1S0 ↔ 3P1 transition suitable for direct laser cooling and state readout, enabling many-ion clock operation. We present calculations of the Sn2+ atomic properties relevant for the evaluation of the systematic frequency shifts as well as calculations of the motion of ions in large Coulomb crystals that lead to imperfect cancellation of the micromotion time-dilation and trap Stark shift, and imperfections in the spectroscopic lineshape. It may be possible to simultaneously achieve 9.0×10−20 total fractional inaccuracy and 4.4×10−18/τ1/2 fractional imprecision, where τ is the measurement duration in seconds. We conclude by discussing prospects for fifth force searches using isotope shift measurements, as well as other tests of fundamental physics with Sn2+.

Results
Sn2+ atomic properties
Figure 2a shows an energy level diagram of Sn2+. We carried out calculations of Sn2+ atomic properties using a high-accuracy relativistic method combining configuration interaction (CI) with the linearized coupled-cluster approach (CI+all-order method)36. To evaluate uncertainties of the results, we also carried out the same calculations using a combination of CI and many-body perturbation theory (CI+MBPT method37). Since CI is essentially complete for a divalent system, the main source of the uncertainty is the treatment of the core, which is estimated as the difference between the results obtained by these two methods38. The results of the calculations given in Tables 1 and 2 are used to design and evaluate the performance of the Sn2+ clock schemes. The 5s5p 3P1 state mostly decays through the 3P1 ↔ 1S0 transition, which will be used for cooling and readout. We found this transition rate to be 7.04×106 s−1 and, respectively, the lifetime of the 3P1 state to be τ3P1 ≈ 142 ns. This lifetime scales like the transition wavelength cubed and is significantly shorter than other divalent atoms used for clocks primarily because of the shorter wavelength.

The 3P0 ↔ 1S0 clock transition is forbidden by single-photon transition selection rules, but it can be opened in the presence of the magnetic field due to the admixture of the 3,1P1 states to 3P0 by the μ⋅B operator (where μ is the magnetic dipole moment operator and B is the static magnetic field).

Restricting ourselves to the admixture of the two nearest to 3P0 states of the same parity, 5s5p 3P1 and 5s5p 1P1, we arrive at the following expression for the 3P0 ↔ 1S0 transition rate:

W ≈ 27 2 ε0 ω3 c3 × |∑n=3P1,1P1 〈1S0 | d | n 〉 〈n | μ | 3P0 〉 E n − E 3P0 |2 B2, (1)

where ω is the 3P0 ↔ 1S0 transition frequency, d is the electric dipole operator, E is a state energy, and ε0, h=2πℏ, and c are the vacuum permittivity, Planck’s constant, and speed of light, respectively. The matrix elements of the electric and magnetic dipole operators expressed in a0 and μ0 (where e is the elementary charge and a0 and μ0 are the Bohr radius and magneton) are given in Table 1. Using Eq. (1), we find the 3P0 lifetime to be τ3P0 ≈ 3.4×108 (sG2)/B2, where 1 G = 10−4 T. The CI+all-order values of the matrix elements and experimental energies39 are used to compute all transition rates.

Static and dynamic polarizabilities at the clock wavelengths λ0 ≈ 186.75 nm and λ = 2λ0 ≈ 373.5 nm, needed for the evaluation of the blackbody radiation shift and the ac Stark shifts, are listed in Table 2. The blackbody radiation shift is mostly determined by the differential static polarizability of the 5s5p 3P0 and 5s2 1S0 clock states, Δα = α(3P0) − α(1S0) ≈ −0.96(4) a03. Based on the difference between the CI+all-order and CI+MBPT values, we determined the uncertainty of Δα. For the static polarizabilities it was found at the level of 4%.

Micromotion driven by the rf-trapping field leads to ac Stark and second-order Doppler shifts. As predicted by Berkeland et al.20 and demonstrated by Dubé et al.40, if the differential static polarizability Δα for the clock transition is negative, there is a “magic” trap drive frequency Ωm given by

Ωm = Q i M i c 2 Δα (2)

(Qi and Mi are the ion charge and mass), at which the micromotion shift vanishes. All even isotopes of Sn2+ are equally suitable for clock operation, and different choices will have slightly different magic drive frequencies and laser cooling dynamics. The calculations throughout this article use Mi ≈ Amp with A=118 (where mp is the proton mass), resulting in Ωm/(2π) = 225(5) MHz.

Calculation of other atomic properties relevant for clock operation are described in the Methods.

Sn2+ clock operation
Each cycle of basic Sn2+ clock operation consists of four steps. First, the ions are laser cooled on the 181 nm 1S0 ↔ 3P1 transition. The Γ/(2π) = 1.1 MHz natural linewidth of this transition is sufficiently narrow for Doppler cooling to sub-mK temperatures, at which the fractional time-dilation shift due to secular (i.e., thermal) motion of the ions is at or below the 10−18 level, and can be characterized such that the uncertainty is significantly better. It is also likely to be sufficiently broad to achieve the high cooling rates necessary for the crystallization of ions initially loaded into the trap at above room temperature in a delocalized cloud phase41. For many-ion crystals with a spectrum of secular-motion normal mode frequencies spanning from tens of kHz to above 1 MHz, efficient Doppler cooling of all modes will require driving above saturation to power broaden the transition to significantly more than the motional mode bandwidth, and the use of three laser beams with k-vectors that have significant overlap with all spatial directions. In the calculations below, we assume a saturation parameter of 20, such that the transition is power broadened to a linewidth of 5 MHz.

Second, spectroscopy is performed on the 1S0 ↔ 3P0 clock transition. This transition can be driven either as a single-photon transition using a laser at 187 nm or as a two-photon transition using a laser at 374 nm (see Fig. 2b and c). The two-photon option has the advantages that it can be driven at or near zero applied magnetic field, which minimizes the second-order Zeeman shift, and using a photon from each of two counter-propagating beams, which eliminates motional sideband transitions, but the disadvantage that it requires high laser intensity with an associated Stark shift of order 10−13, and it may not be possible to reduce the uncertainty to below 10−18. The single-photon option has the advantage that the required laser intensity can be reduced by increasing the magnetic field, but this comes at the cost of an increased second-order Zeeman shift. We estimate that at magnetic fields around 15 G, the quadrature sum of the uncertainties due to the probe-laser Stark shift and the second-order Zeeman shift can be minimized as shown in Fig. 2c. With 15 μW of laser power focused to a 100 μm 1/e2 beam diameter polarized parallel to the magnetic field direction, the single-photon Rabi frequency Ω/(2π) = 1 Hz and the Stark shift is 2.0×10−16. We conservatively estimate that this can be characterized with an uncertainty of 2.0×10−20 by using hyper-Ramsey42 or auto-balanced Ramsey43 interrogation protocols. The second-order Zeeman shift is −3.7×10−15, and by measuring the clock transition frequency at a significantly higher magnetic field, it should be possible to characterize this shift to an uncertainty of 2.3×10−20. For 1000 Sn2+ ions in a trap with single-ion secular frequencies 1.15 MHz, 0.85 MHz, and 0.10 MHz, magnetic field inhomogeneity of 1 mG over this volume is sufficient to suppress inhomogeneous broadening due to the second-order Zeeman shift to a negligible level. Although it would be difficult to apply a 15 G quantization field with such a 10−4/mm fractional magnetic field gradient using standard Helmholtz coils, gradients as small as 10−6/mm are routinely achieved using solenoids together with shim coils44. The ion trap should be designed to minimize the capacitance between electrodes in order to to minimize the contribution to the second-order Zeeman shift due to the ac magnetic field at the trap drive frequency45 (see Methods).

Third, the number of ions in the ground and excited states is read out by driving the 1S0 ↔ 3P1 transition and counting the total number of photons scattered into a photomultiplier tube (PMT) or other detector. During a τdet = 1 ms detection period and assuming a total detection efficiency ϵdet = 0.3%, up to ϵdetΓτdet/2 = 10 photons will be detected per ion that remains in the ground state after clock interrogation. This is sufficient to suppress detection-photon shot noise below the QPN limit. The fraction of excited state ions is used to determine the frequency offset between the clock laser and the atomic transition frequency, and frequency feedback is applied to the clock laser to keep it on resonance.

Fourth, the ions that are in the excited state after clock interrogation are repumped to the ground state by driving the 379 nm 3P0 ↔ 1P1 transition. The 1P1 state quickly decays to 1S0, with negligible branching ratios for decay to 3P1 and 3P2. Optionally, the total number of ions can be determined by performing readout again after repumping. Subsequently, the next cycle of clock operation can begin.

Many-ion clock considerations
Three-dimensional Coulomb crystals of many ions can be confined in a single linear RF Paul trap14. For large Coulomb crystals of a fixed number of ions, there are multiple local minima of the total zero-temperature energy consisting of the Coulomb energy due to repulsion of the ions from each other, the time-averaged Coulomb energy of the ions interacting with the trapping fields, and the time-averaged kinetic energy of the micromotion (i.e., the pseudopotential). These local minima correspond to stable geometric crystal configurations, in which each ion is spatially localized and oscillates about its equilibrium position due to micromotion and secular motion. Distinct crystal configurations have different sets of equilibrium ion positions. Different configurations are observed both theoretically46 and experimentally16,47 when ions are laser cooled to a crystalline state starting from a non-localized cloud state with identical initial thermal properties, for example after loading the trap or a collision with a background gas molecule, and the number of configurations grows exponentially with the number of ions48. We perform molecular dynamics calculations of the ion motion in two steps. First, similar to Refs. 21, 24, we calculate the equilibrium ion positions by time-evolving the equations of motion in the pseudopotential approximation starting from random initial positions and including a viscous damping force that qualitatively represents laser cooling. For each set of trap parameters and number of ions investigated, we generate up to 36 configurations in parallel using a high-performance computer cluster. For ion numbers up to 1000, this takes <1 week. The strength of the damping force is such that the cooling time constant is roughly 100 μs, which is a typical experimental cooling time constant, and we have verified that reducing this strength by an order of magnitude does not reduce the number of distinct configurations. Once the equilibrium ion positions are known, it is straightforward to compute the frequencies and eigenvectors of the normal modes of secular motion and their Lamb-Dicke parameters49,50, and the spectrum of the clock transition51. Second, we calculate the amplitude and direction of the micromotion of each ion and verify the stability of the ion crystal by time evolving the full equations of motion (i.e., without taking the pseudopotential approximation, similar to refs. 52, 53), starting from the ion positions computed in the first step.

Figure 3a, b show the fractional frequency shifts due to secular motion, micromotion, and the Stark shift due to the trapping fields, averaged over all of the ions, and estimates of the uncertainties with which these shifts can be characterized. For these calculations, the trap is assumed to be driven at the magic RF frequency with voltages such that the secular frequencies of a single ion in the trap are 1.15 MHz, 0.85 MHz, and 0.10 MHz. The number of ions varies from 50–1000. The magnitude of the average micromotion time-dilation shift increases with the number of ions as ions are pushed further away from the trap axis, along which the AC electric field is zero, reaching 3.7×10−14 for 1000 ions. The time-dilation shift due to micromotion at exactly the magic trap drive frequency is perfectly canceled by the Stark shift due to the electric field that drives the motion, as indicated in Fig. 3d. There is also a time-dilation shift due to micromotion at twice the trap drive frequency, which is not canceled by the corresponding Stark shift, and this motion prevents perfect cancellation of micromotion time dilation40. We define the net micromotion shift to be the sum of the time dilation shift and Stark shift caused by the trap electric field in the reference frame of the ion due to ion motion at integer multiples of the trap drive frequency. The net micromotion shift after this imperfect cancellation for 1000 ions is 2.8×10−19, and it should be possible to characterize this shift to achieve an uncertainty that is at least one order of magnitude smaller. Figure 3c shows a histogram of the net micromotion shifts of each ion individually; the sub-10−18 inhomogeneous broadening of this distribution will not limit spectroscopic coherence within the foreseeable future. The net secular motion shift (defined as the sum of the time dilation shift and Stark shift due to ion motion at the secular frequencies) for 1000 ions is −2.4×10−19, and we conservatively estimate that this can be characterized with an uncertainty of 7.2×10−20. Importantly, the differences of these shifts for different ion crystal configurations are smaller than our estimated uncertainties, so Sn2+ clocks will be robust to ion loss and background gas collisions that cause the crystal to melt and recrystallize in a different configuration.

Ion motion leads to motional sidebands on the clock transition and an inhomogeneous reduction of the Rabi frequency for driving the carrier transition due to the Debye-Waller factors of each ion51,54. Micromotion is dominantly along the radial trap directions and its effect on the clock transition spectrum can be nearly eliminated by probing along the axial trap direction. Figure 4a shows the spectrum of the secular motional mode frequencies for a 1000 ion Coulomb crystal with the same trap parameters as above. The distribution of Debye-Waller factors of each ion for axial probing due to secular motion at a temperature of 120 μK, corresponding to the Doppler cooling limit with a 5 MHz power broadened linewidth, is shown in Fig. 4b. The contrast for Rabi spectroscopy of the clock transition is maximized by setting the laser intensity and pulse duration such that ions with the median value of the Rabi frequency undergo a π-pulse when the laser is on resonance. Figure 4c shows the spectrum of the clock transition under these conditions. The contrast is limited to 79% due to the inhomogeneous reduction of the Rabi frequency by the Debye-Waller effect, and a broad band of overlapping motional sideband transitions surrounds the carrier transition which is at zero detuning. The lowest frequency motional sideband transitions are above 14 kHz for any of the generated crystal configurations, and should not lead to significant line-pulling due to off-resonant excitation for probe times longer than about 10 ms.

The trap parameters can be optimized to minimize the net micromotion shift, minimize the size of the ion crystal and thus inhomogeneous broadening due to magnetic field gradients as well as required laser beam diameters, maximize the lowest secular frequency, maximize the spectroscopy contrast, or for other goals. Furthermore, the number of ions can be selected to balance systematic shifts and inhomogeneous broadening that increase with the number of ions versus the QPN stability limit that decreases with increasing numbers of ions. Figure 5 shows a limited subset of this trade space. The lowest normal mode secular frequency for trap parameters such that the single-ion secular frequencies are 1.15 MHz, 0.85 MHz, and 0.10 MHz decreases as the number of ions increases, reaching a mean value of 32 kHz for 1000 ions with a distribution spanning from 15 kHz–56 kHz for different crystal configurations. For axial single-ion secular frequencies that are too low or too high relative to the radial single-ion secular frequencies, the spectroscopy contrast is reduced. The two radial single-ion secular frequencies must be different in order to avoid a zero-frequency mode in which the ion crystal rotates about the trap axis, and more generally low frequency motional modes should be avoided becuase they have higher heating rates due to electric field noise55. A more complete molecular dynamics model incorporating the details of the laser cooling can be used to optimize the cooling parameters and is likely necessary to provide a rigorous constraint on the corresponding Doppler shift uncertainty of the clock. Single-ion secular frequencies of 1.15 MHz, 0.85 MHz, and 0.10 MHz represent a trade-off of the aforementioned optimization goals that enable both high accuracy and high stability clock performance.

These secular frequencies correspond to Mathieu parameters of |qx,y| = 0.0127, ax = 2.33×10−5, and ay = −2.41×10−5, and can be achieved by differentially driving a trap with 500 μm ion-to-electrode distance with 970 V amplitude of rf. This smaller-than-typical q parameter offers several advantages for Coulomb crystal clocks. Molecular dynamics simulations show that rf heating of large ion crystals, which arises due to the breakdown of the pseudopotential approximation q≪1, scales roughly like q6 at low temperatures56. Care must be taken, however, to minimize anharmonicity of the trap potential as this can lead to additional rf heating57. Ion heating due to background gas collisions is also reduced for smaller q parameters58.

Methods
Quadrupole moment
The quadrupole moment Θ of an atomic state |J is given by

Θ = 〈J, MJ=J | Q0 | J, MJ=J 〉 = s 2J+3 J (2J+3) (2J+1) (2J−1) 1 2J+1 〈J||Q||J〉, (3)

where 〈J||Q||J〉 is the reduced matrix element of the electric quadrupole operator.

For the 5s5p 3P2 state, we find

〈3P2 ||Q||3P2 〉 ≈ −6.9 ea02 (4)

and

Θ(3P2) ≈ −1.7 ea02. (5)

Zeeman shift
Since we consider the ion with J=I=0, there is no linear Zeeman shift. The second-order Zeeman shift, ΔE, is given by81

ΔE ≈ ΔE(1) + ΔE(2) = − 1 2 αM1 B2 + μ0 2 2m ∑i=1 Z 〈0 | (B×ri)2 | 0 〉, (6)

where αM1 is the magnetic-dipole static polarizability, μ0 is the Bohr magneton, m is the electron mass, and ri is the position vector of the ith electron. For a state |J=0, the polarizability can be written as

αM1 = 2 3 ∑n |〈n | μ | J=0 〉|2 En − E0 . (7)

To estimate the shift for the clock transition due to this term, we note that the αM1(3P0) and αM1(1S0) polarizability is negligibly small compared to ΔE(3P0) − ΔE(1S0)

Δν(1) ≈ ΔE(3P0) − ΔE(1S0) h ≈ ΔE(3P0) h = − 1 2 αM1(3P0) B2 h. (8)

For an estimate of αM1(3P0), we take into account that the main contribution to this polarizability comes from the intermediate state 5s5p 3P1. Then, from Eq. (7), we obtain

αM1(3P0) ≈ 2 3 |〈3P1 ||μ||3P0 〉|2 E3P1 − E3P0 . (9)

We found the matrix element |〈3P1 ||μ||3P0 〉| to be 1.4064 μ0 and 1.4053 μ0 in the CI+all-order and CI+MBPT approximations, respectively. These values differ by 0.07%. According to our calculation, the next (after 3P1) intermediate state in Eq. (7), 1P1, contributes to αM1(3P0) at the level of 0.06%. Adding the contributions of the 3P1 and 1P1 states as 2×0.06%, we arrive at the uncertainty of Δν(1) at the level of 0.2%.

Using it and the experimental value of the energy difference E3P1 − E3P0 ≈ 1648 cm−1, we arrive at

Δν(1) = −26.16(5) mHz G2 B2, (10)

where the magnetic field B is expressed in G.

One can show81 that the contribution of the second term in Eq. (6) to the second-order Zeeman shift of the clock transition can be written as

Δν(2) ≈ (2πμ0)2m B2 × ∑i=1 Z 〈3P0 | r2i | 3P0 〉 − 〈1S0 | r2i | 1S0 〉 3h , (11)

where the matrix elements are expressed in a02.

The calculation, carried out in the framework of our approach, leads to

∑i=1 Z 〈3P0 | r2i | 3P0 〉 − 〈1S0 | r2i | 1S0 〉 ≈ 1.7 a02. (12)

To check this result, we made another calculation, reconstructing the basis set and using the pure CI method, where all 48 electrons were placed in the valence field, and several of the most important configurations were included into consideration. We obtained the same value 1.7 a02.

Using this result, we find

Δν(2) = 0.17(2) mHz G2 B2. (13)

We estimate the uncertainty of this term at the level of 10%.

Finally, we find the quadratic Zeeman shift as

|Δν| = |Δν(1) + Δν(2)| = 25.98(5) mHz G2 B2. (14)

The Rabi frequency of the one-photon 1S0 − 3P0 transition is determined as

Ω = |〈3P0 | d⋅E | 1S0 〉| ℏ, (15)

where E is the electric field of the laser wave and we designate

〈3P0 | d⋅E | 1S0 〉 = ∑n 〈3P0 | μ | n 〉 〈n | d | 1S0 〉 E n − E 3P0 B. (16)

Substituting Eq. (16) to Eq. (15), we arrive after simple transformation at

ℏΩ = B ∑n 〈3P0 ||μ||n 〉 〈n ||d||1S0 〉 E n − E 3P0 . (17)

Leaving (for an estimate) in the sum of Eq. (17) only two intermediate states, 5s5p 3P1 and 5s5p 1P1, and using our calculated values |〈3P1 ||μ||3P0 〉| ≈ 1.406 μ0 and |〈3P1 ||d||1S0 〉| ≈ 0.249 ea0, we find

|Ω|/(2π) ≈ 3.4 Hz G m W cm2 I B cosθ, (18)

where I is the laser intensity and θ is the angle between B and E.

The clock transition can also be driven as the two-photon E1+M1 transition. The respective formalism was developed in Ref. 23. The E1+M1 Rabi frequency, Ω2, is given by

Ω2 = 4π 2 3ε0 hc2 I1 I2 Λ, (19)

where Λ can be written in terms of dominant contributions as

Λ ≈ |∑n=3P1,1P1 〈3P0 | d | n 〉 〈n | μ | 1S0 〉 En − E3P0 | (20)

and I1 and I2 are the intensities of two probe laser waves. Using the experimental energies and the values of the matrix elements given in Table 1, we find

Ω2 2π ≈ 0.27 Hz kW/cm2 I1 I2. (21)

Second-order Zeeman shift due to the ac magnetic field at the trap drive frequency
We estimate the size of this effect with the following simple model. We assume that the ion trap has a 500 μm ion-to-electrode distance and is driven differentially with an RF voltage amplitude of 970 V, as is necessary to achieve the secular frequencies proposed here. Standard linear RF Paul traps constructed with cylindrical rods or blade electrodes that are electrically driven from one end have current running down the electrodes due which is sunk by a capacitance at the far ends of the electrodes of order 100 fF. At the center of the trap, symmetry dictates that the RF magnetic field should be zero. At the edge of the ion crystal, 22 μm away from the trap center, this model predicts that there is an AC magnetic field of 50 mG, which causes an inhomogeneous frequency shift of 4 mHz. It should be possible to characterize the average over all ions of this shift sufficiently well as described in the text, and this is compatible with probe durations up to about 100 s. To achieve longer probe durations it will be necessary to design traps with less capacitance or more symmetric current distributions.

Background gas collision shift
A detailed analysis of the background gas collision shift and uncertainty for many ion clocks requires computationally intensive Monte Carlo simulations and is beyond the scope of this article, but in the following we provide a simple estimate based on scaling an analysis of the uncertainty for single-ion clocks82. The total rate of Langevin collisions between a Coulomb crystal and background gas molecules scales linearly with the number of ions and linearly with the charge of the ions, so for a 1000 ion Sn2+ clock the background gas collision rate will be ~2000 times larger than in single-ion clocks. As a conservative upper bound, we suppose that the systematic uncertainty due to background gas collisions scales linearly with the collision rate. A more detailed model would take into account that only the Doppler shift component of the uncertainty scales with the total collision rate while the phase shift component of the uncertainty (which dominated the collision shift uncertainty in Ref. 82) scales with the collision rate per ion. Current state-of-the-art single-ion clocks operating in room temperature ultrahigh vacuum (UHV) chambers with background gas pressures of roughly 10−10 torr have systematic uncertainties due to background gas collisions of order 10−19 59,83. Although according to this conservative estimate background gas collisions may contribute of order 10−18 uncertainty for 1000 ion Sn2+ clocks in room temperature UHV chambers with pressures of at best 10−12 torr, Monte Carlo simulations will likely be able to constrain the uncertainty to be much smaller.

Alternatively, pressures below 10−16 torr have been achieved for ion traps operated in cryogenic vacuum chambers84. An even more conservative upper bound can be obtained by considering the rate of collisions as follows. For 10−16 torr of He background gas at 10 K, the total rate of Langevin collisions with any of the ions in a 1000 Sn2+ ion crystal is 1×10−4/s. For a 1 s probe time, this amounts to one collision every 104 probes. At most, this could result in a fractional shift of the measured transition frequency of 10−4 relative to the spectroscopic linewidth, which in this case is 1 Hz. Thus, the background gas collision shift can be straightforwardly upper bounded to be below 10−19 for Sn2+ clocks operating in cryogenic vacuum chambers. In practice, collisions that deposit a significant amount of kinetic energy to the ion crystal will not contribute to the clock spectroscopy signal due to the Debye-Waller reduction of the Rabi frequency82, so the actual shift and its uncertainty will be much smaller.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In this manuscript, the authors have proposed Sn2+ ion as a new candidate for multi-ions clock. The\nSn2+ ion has many advantages suitable for multi-ions clock operation, for example, the total\nelectronic angular momentum of both clock states is zero (J=0), makes it immune to electric\nquadrupole shift. The differential static polarizability Δα<0, there is a “magic” trap drive frequency\nΩm, in which the time-dilation shift and the Stark shift cancel each other. In the manuscript, the\nauthors give the detailed analysis of Sn2+ atomic properties, like energy level, lifetime, magnetic\nfield, second-order Zeeman shift and uncertainty, stark shift and uncertainty, et al,. And also analyzed\nthe net micromotion shift at “magic” trapping frequency, secular motion shift, blackbody radiation\nshift. Prospects of a thousand-ion Sn2+ Coulomb-crystal clock with sub-10-19 inaccuracy. Although\nthis manuscript only gives the proposal and the calculation, I think the Sn2+ ion can be very\npromising for building an advanced ion optical clock in the near future, with the development of\ntechniques for highly changed ion related experiments, this makes the ion optical clock competitive to\nthe optical lattice clocks on frequency stabilities, thus I think the manuscript reaches the criterion of\nnature communication, in the aspect of innovation in physics. I recommend the publication of this\nmanuscript in nature communication.\nBesides, I have a few comments that need to be explained:\n1. Highly changed ion (HCI) candidates for optical clocks are recently a hot topic. However, it seems to\nbe very difficult to make an HCI-based clock until the improvements on both the experimental\ntechniques and the theoretical calculations. I think the authors should mention the recent progress on\nHCI-based experiments in the instruction, eg. PRA 103, L040801 (2021), PRA 103, 022804 (2021),\nPRX 11, 041049 (2021) , arXiv:2207.10926.\n2. In this manuscript, the author proposed using 1000 Sn2+ ion as multi-ion clock. Except the\nconsideration of secular motion, blackbody radiation shift, micromotion, quadratic Zeeman shift, probe\nlaser Stark shift, the author also should consider he background gas collision, collisions between ions\nand ions lose rate, and give their shifts and uncertainties.\n3. How can one make sure the loaded Sn2+ ions are pure, and avoid dark ions like Sn+ co-trapped?\nBecause the dark ions also bring huge heating during clock interrogation and affect micromotion and\nsecular motion.\n4. Considering of this large crystal size, the author should not only care about the magnetic field\nfluctuation but also care about magnetic field gradient. Further, when increases the number of ions,\nthe ions are pushed further away from the trap axis, the RF trap frequency maybe cause large AC\nmagnetic field. The author should give the shifts and uncertainty budget of magnetic field gradient and\nAC magnetic field.\n5. Fig2(c) and line 266: When discussing the uncertainty of the probe laser Stark shift, the uncertainty\nis a few orders of magnitude smaller than the shift, how is this achieved? By measuring the shift by\ncomparing the transition frequency between the increased power situation and the normal power\nsituation? Or by hyper Ramsey scheme?\n6. Line 221: As far as I know, the Doppler cooling efficiency is very sensitive to the power of the\ncooling laser, for such a large ion crystal, the laser power on each every ion should be different, thus\nthe saturation parameter is not always 20, I don’t know if the authors have already considered this in\ntheir calculations.\n7. Line 275: I’m not sure if I have some misunderstanding on this: for single photon approach, the\nexperiment requires a magnetic field of 15 G, and have a field inhomogeneity of 1 mG over the size of\n~ 1 mm. It seems to me that this is very difficult to be achieved by simply adding a pair of Helmholtz\ncoils.\n8. Line 428: As mentioned earlier, I think it is very difficult to be achieved for making a very strong\nmagnetic field with a very small gradient like 1 mG over the size of ~1 mm, 10 micro-G is even more\ndifficult.\n9. Line 460: “assuming a conservative 1 s probe duration and 80 % duty cycle.” Although the lifetime\nof 3P0 clock state is quite long ~ 106 s at 15 G magnetic field, ideally one can get very long coherence\ntime, but actually the coherence time is determined by high performance of clock laser and ion\nheating during interrogation time. What is the evidence for 1 s, is it based on the experience of the\nAl+ ion clock or have you done the calculation and analysis? It seems to me that the probe time can\nbe even longer.\n10. Line 472: “It may be possible…maintaining sub-1019 inaccuracy.” The author proposal using Sr+\nion to co-trapping Sn2+ ions, the outside trapped Sr+ can push the Sn2+ ions towards the trap axis\nand make the micromotion shift smaller. The issue is, it seems not have a “magic wavelength”, cooling\nSr+ ion during clock interrogation can lead into ac Stark shifts. Is the cooling laser off when probing?\nIf so, the authors should consider the heating effect.", "reviewer_2": "dear editor,\nthe paper proposed by Leibrandt et al is a proposal for an optical atomic clock that could be operated\nwith a cloud of one thousand ions as a base, whereas so far, trapped ion based optical clock are\noperated with a single ion. The idea is to make a more stable clock by interrogation of a thousand ions\nat the same time and then reduce the quantum projection noise which is the limiting factor for short\nterm stability. The issue is to do that without degrading the accuracy and precision of the clock as the\ninhomogeneity of the ion environment can induce non uniform shifts responsible for a line broadening\nand an uncertainty in the shift estimation.\nThis idea is not new and several ions have been identified that share some characteristics that make\nthem good candidates for this kind of set-up. The main figure of merit is the differential polarizability\nof the transition. If it is negative, two major effects cancel each-other, provided the RF-trap is\noperated at a proper frequency, which depends on this difference. This is covered by ref 17-21.\nThe new results of this paper is to add an ion to this list : Sn2+, which have some technical advantage\nover recent proposals, like the accessibility of the transition wavelength, but requires a strong\nmagnetic field to mix different atomic fine-structure states and make the clock transition accessible to\nlaser excitation. The best performances are reached by a compromise between the lowest possible\nmagnetic field to control Zeeman shifts and the lowest possible laser power to control light induced\nshifts. If an experiment is able to follow these prescriptions without excess inhomogeneity, the\nexpected clock performance are very good compared to the state of art in terms of uncertainty and\ninaccuracy. At this stage of the paper, the reader still do not know what isotope is considered. We can\nguess that an even isotope is assumed but nothing is said about their abundance and possible choices.\nThe calculations used for atomic structure calculations is now well established and one of the author is\na reference in the optical clock community for the relevance of her methods.\nThe discussion regarding the clock operation in the context of a thousand trapped ion is far less\nconvincing. There are a lot of statements regarding the cooling efficiency that look like speculations\nand are not supported by comparison with published and cited work. If the statements are based on\nmolecular dynamics simulations, it is not written clearly and the fact that 3 laser beams with different\ndirections are required is not justified anywhere. Indeed, as far as I know, no 3D Coulomb crystal\nexperiments use this set-up as the micro-motion induced Doppler effect make Doppler cooling less\nefficient (see url = {https://link.aps.org/doi/10.1103/PhysRevA.105.023101} for the 2D case where\ndifferent laser detunings are needed). A single one propagating along the axis is often the best\nsolution for large 3D system. Other estimations lack an established support like the requirement for\nthe magnetic field homogeneity. It is assumed to be better than 0.01% over a mm scale. It should be\nsupported by results coming from similar experiments proving the feasibility of this requirement.\nA lot of discussion regarding the cloud shape and ion dynamics assume operating parameters that are\njustified a lot later without any notice. This is very detrimental. The RF optimum frequency is so large\nand the chosen single-ion secular frequencies are such that they correspond to very low Mathieu\nparameter, of the order of 10^-2. This is at the edge of the stability diagram for an RF-trap and to\nmake it convincing, this proposal should cite experimental results where such a low parameter has\nbeen operated with success. If these works do not exist yet, it is because it is really a challenge and\nthis should not be considered as obvious in the article.\nThe part concerning the different spatial configuration of an ion ensemble is very confusing as the\ncited works all set the ions in a static pseudo-potential (I suggest the authors to read also Calvo and\nYurtsever, EPJD (2007) DOI: 10.1140/epjd/e2007-00137-2 ), and the authors also use the same\nstrategy, so why mentioning time-averaged kinetic energy of the micro-motion? This is the source of\nthe pseudo-potential and it has nothing to do in a pseudo-potential description, it is one or the other,\nnot both. How micro-motion is obtained in the pseudo-potential approximation? This point deserve\nclarification. The description of the forces taken into account should be more rigourous and corrected.\nIndeed, more details about the numerical simulations should be given (like in a methods part) to\nappreciate the relevance of the results and more earlier works should be cited to support the chosen\nstrategy (see DOI: 10.1103/PhysRevA.82.033406 and https://doi.org/10.1103/PhysRevLett.70.818 to\nstart with). What about the choice of the damping constant simulating laser cooling? If it is too strong,\nthe system is frozen in a non-equilibrium state. How the authors proceed to make sure this is not the\ncase and that the motional temperature is 120 microK (cf caption of fig 4) ?\nThe paragraph starting at line 393 is completely speculative as nothing proves that the listed\nrequirements are compatible with each other. It is surprising to read that “maximize the lowest secular\nfrequency” is in the list whereas a so small q_x parameter has been chosen. So this is certainly not\nthe case. Line 413 mention a trade off to reach so small values, so the most important conditions\nshould be enlightened and this need for a maximization should be justified.\nOn the step of line 458, the requirements concerning the stability and the precision/accuracy of the\nproposed clock should be clearly identified and separated as some technical choices can be good for\none figure of merit but bad for another.\nFrom line 472, another list of speculations appear, concerning the best side ion to be trapped with\nSn2+, to compress the ion cloud around the trap axis. The obvious solution would be to increase the\ncloud density, but once again, the reason why such a low Mathieu parameter is chosen is not given.\nFurthermore, this idea of using an outer cloud shell to compress the inner cloud is not obvious at all\nand is not supported by any calculations or previous work. This naive view seems in contradiction with\nthe observation reported in PRL.86.1994. As for the idea of using a multipole trap, I’d like to point to\nthe authors that ref 54 and 56 concern room temperature large ion clouds and in the cold limit, the\nself organization of ions is very different (see for example J. Phys. B: At. Mol. Opt. Phys. 42 (2009)\n154002 ). For smaller sample, the use of multipole trap is described here : DOI:\n10.1103/PhysRevA.81.043410 . \nContrary to what is stated on line 489-494, ions will not gather closer to the axis, because of Coulomb\nrepulsion. (read again section V of ref 17) Anyway, and unfortunately, using a multipole trap with laser\ncooled ions may be a wrong good idea as stated in this series of paper :\nhttps://doi.org/10.1080/09500340.2017.1408866 and M Marchenay et al 2021 Quantum Sci. Technol.\n6 024016 .\nAt this stage, I think the reader is kind of lost with the come back of spectroscopic considerations from\nline 495. This would be better placed at the beginning of the paper or it could be suppressed as\nnothing new is brought here.\nAs a conclusion, I think this paper is not ready yet for publication. Furthermore, I also think that the\npresented results are not noteworthy neither original enough for publication by nature communication.", "reviewer_3": "The manuscript ”Prospects of a thousand-ion Sn^{2+} Coulomb-crystal clock with sub-10^{-19}\ninaccuracy”, gives a detailed account on the possibility of building a highly accurate atomic clock based\non a large three dimensional ion Coulomb crystal. By considering a clock based on up to 1000 ions,\nthe authors’ goal is to produce an atomic clock based on ions, which not only can compete with\nneutral atom lattice clocks with respect to total achievable inaccuracy, but also with respect to needed\nexperimental integration time. The authors have made a great effort to assess many of the potential\nlimiting issues related to the specific ion and trapping configuration in mind. The manuscript is\norganized in a logical way by first describing the reason to consider the Sn^{2+} ion species and the\nexpected properties of the relevant electronic levels, followed by presenting how the trapping fields\nand properties of and ion motion within a Coulomb crystal influence the accuracy of the Coulomb\ncrystal clock.\nFinally, before a summary and outlook, a discussion on various limiting factors and potential further\nimprovements are briefly presented.\nAll in all, after a revision of the manuscript with respect to the comments and suggestions stated\nbelow, I find the manuscript suitable for publication in Nature Communication, since the work\nrepresents a promising scenario for a future new improved time standard.\nComments/questions:\n1) In Fig. 1, a Coulomb crystal of 1000 ions is presented. It is obvious that a very elongated crystal\nalong the rf field-free axis is preferable, but it is less clear why trap parameters giving rise to a\nrelatively large difference in the crystal dimensions in the perpendicular plane is chosen. Perhaps the\nauthors could elaborate a bit on that.\n2) In Fig. 1, the colors of the clock and cooling beams are very close to each other. Choosing a\ndifferent color for the cooling beams would be helpful also for the readability of the text on the\nillustration.\n3) Since the “magic” rf frequency of 225 MHZ is rather high, a reference that shows this should be\nachievable with the trap dimensions in mind would be needed.\n4) In fig. 2.c, the lines indicating the uncertainties of\nthe second order Zeeman shifts and Stark uncertainties translates into a magnetic and laser field\ninstabilities of 10^{-5} and 10^{-4}, respectively. It would be needed with some references that such\ninstabilities should be achievable over the total volume of the Coulomb crystal.\n5) With respect to the assessment of time dilation and rf field induced Stark shifts, the authors have\nconsidered 36 stationary configurations of the Coulomb crystals which seems reasonable. A single\nbackground gas collisions may though lead to dynamical changes in the crystal structure or even lead\nto crystal melting. Only very few experimental investigations have been made so far to address these\nissues quantitatively, but they could potentially constitute limitations to use larger crystals in\nparticularly in room temperature trapping environments. Please comment on this in a revised\nmanuscript.\n6) With respect to computing the frequencies and eigenvectors of the normal modes of the secular\nmotion and Lamb-Dicke parameters reference is given to [41] in the manuscript. Since this paper\ndeals only with linear chains of ions, a more suitable reference for the current proposal could, e.g., be\nNew J. Phys. 14 093023 (2012).\n7)\nFig. 3, which is central to the manuscript, contains a lot of information presented in a condensation\nfashion. For the non-expert it can be very hard to be sure to have understood it fully as it is.\nIn fig. 3.b, it would be great to have a clearer physical description of what is meant by “net secular\nmotion shift”. As it looks like it is independent of the number of ions, I assume it accounts only for the\ntime dilation related to the secular motion? Could there also be a secular motion dependent Stark shift\nwhich should be accounted for?\nFig. 3.d is very challenging. If I am right that the green bars are just the difference between the red\nand blue, I believe one could communicate that clearer. The two first panels deal with the secular\nmotion, while the rest of the panels deal with rf induced micromotion. For the sake of clarity, it would\nbe good to separate Fig. 3.d. into three separate sub-figures (d.,e., and f.) and not for all panels use\nthe term “net micromotion shift”.\nIn the last four panels the fractional frequency shifts exhibit large increases at the magical rf\nfrequency. Why is this the case? This should be explained more carefully in the text.\nThe assumed temperature for the secular contributions should be stated in the figure caption.\n8) Somewhere in the manuscript it would be great to have a short discussion of what would the\ninfluence of a radial temperature gradient in the crystal. A priori one cannot expect to be able to cool\nall ions in the crystal to the same final temperature.\n9) With respect to uncertainty in the shifts due to BBR, is an effective temperature with an uncertainty\nof 500 mK reasonable over the whole crystal lengths (~1 mm)?\n10) With respect to the two species Coulomb crystal discussion on p. 8 and 9, it should be noted that\nvolume of the Sn^{2+} ions in this case will be a nearly perfect cylinder, and hence for a given crystal\nlength and number of ions the outermost ions will be positioned closes to the trap axis when\ncompared to the case of a single Sn^{2+} species ellipsoidal crystal. Consequently, this would lead to\na smaller average time dilation and Stark shifts. A potential reference in this relation could, e.g., be\nPhys. Rev. Lett. 86, 1994 (2001)."}, "author_response": "Response to referee comments on manuscript “Prospects of a thousand-ion Sn2+\nCoulomb-crystal clock with sub-10−19 inaccuracy”\nDavid R. Leibrandt,1,2,3,∗ Sergey G. Porsev,4 Charles Cheung,4 and Marianna S. Safronova4\n1Department of Physics and Astronomy, University of California, Los Angeles, California, 90095, USA\n2Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado, 80309, USA\n3Department of Physics, University of Colorado, Boulder, Colorado, 80309, USA\n4Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, USA\n(Dated: July 17, 2023)\nI. INTRODUCTION\nWe thank the editor and referees for their thorough and insightful comments, which have helped us to improve the\nmanuscript considerably.\nFurthermore, we apologize for the unusually long delay between receiving the referee’s comments and this reply.\nOne of us (Leibrandt) has moved to a new position (from NIST in Boulder, CO to UCLA in Los Angeles, CA) and\nduring this period of time was not able to devote sufficient time to address the referee’s comments until now.\nInthefollowing,wereplytothereferee’scommentspointbypoint. Therefereecommentsarereproducedverbatim\nin gray text, followed by our responses in black text. We have appended a copy of the revised manuscript with the\nchanges highlighted.\nII. REVIEWER # 1\nIn this manuscript, the authors have proposed Sn2+ ion as a new candidate for multi-ions clock. The Sn2+ ion\nhas many advantages suitable for multi-ions clock operation, for example, the total electronic angular momentum of\nboth clock states is zero (J=0), makes it immune to electric quadrupole shift. The differential static polarizability\n∆α<0, there is a “magic” trap drive frequency Ωm, in which the time-dilation shift and the Stark shift cancel each\nother. In the manuscript, the authors give the detailed analysis of Sn2+ atomic properties, like energy level, lifetime,\nmagnetic field, second-order Zeeman shift and uncertainty, stark shift and uncertainty, et al,. And also analyzed\nthe net micromotion shift at “magic” trapping frequency, secular motion shift, blackbody radiation shift. Prospects\nof a thousand-ion Sn2+ Coulomb-crystal clock with sub-10-19 inaccuracy. Although this manuscript only gives the\nproposal and the calculation, I think the Sn2+ ion can be very promising for building an advanced ion optical clock\nin the near future, with the development of techniques for highly changed ion related experiments, this makes the\nion optical clock competitive to the optical lattice clocks on frequency stabilities, thus I think the manuscript reaches\nthe criterion of nature communication, in the aspect of innovation in physics. I recommend the publication of this\nmanuscript in nature communication.\nWethankthereviewerfortheirconstructivefeedbackandforsupportingthepublicationofourmanuscriptinNature\nCommunications.\nBesides, I have a few comments that need to be explained:\n1. Highly changed ion (HCI) candidates for optical clocks are recently a hot topic. However, it seems to be very\ndifficult to make an HCI-based clock until the improvements on both the experimental techniques and the theoretical\ncalculations. IthinktheauthorsshouldmentiontherecentprogressonHCI-basedexperimentsintheinstruction, eg.\nPRA 103, L040801 (2021), PRA 103, 022804 (2021), PRX 11, 041049 (2021) , arXiv:2207.10926.\nYes,therehasbeensignificantprogressonHCIclocksinthepastfewyearswiththedemonstrationoftheAr13+ clock,\nand some of the experimental techniques will also be useful for Sn2+ clocks. We have added the sentence “Finally,\nrecent demonstrations have shown that it is possible to build clocks based on highly charged ions which can have\nlowersensitivitytoenvironmentalperturbationsandhighersensitivitytophysicsbeyondtheStandardModel[1–4].”\n∗ leibrandt@physics.ucla.edu\n2\n2. In this manuscript, the author proposed using 1000 Sn2+ ion as multi-ion clock. Except the consideration of\nsecular motion, blackbody radiation shift, micromotion, quadratic Zeeman shift, probe laser Stark shift, the author\nalso should consider the background gas collision, collisions between ions and ions lose rate, and give their shifts and\nuncertainties.\nWe have added the following paragraph to the methods:\n“Adetailedanalysisofthebackgroundgascollisionshiftanduncertaintyformanyionclocksisbeyondthescopeofthis\narticle, but in the following we provide a simple estimate based on scaling an analysis of the shift for single-ion clocks\n[5]. ThetotalrateofLangevincollisionsbetweenaCoulombcrystalandbackgroundgasmoleculesscaleslinearlywith\nthenumberofionsandquadraticallywiththechargeoftheions,sofora1000ionSn2+ clockthebackgroundgascolli-\nsionratewillbeapproximately4000timeslargerthaninsingle-ionclocks. Asaconservativeupperbound,wesuppose\nthatthesystematicuncertaintyduetobackgroundgascollisionsscaleslinearlywiththecollisionrate. Amoredetailed\nmodelwouldtakeintoaccountthatonlytheDopplershiftcomponentoftheuncertaintyscaleswiththetotalcollision\nratewhilethephaseshiftcomponentoftheuncertaintyscaleswiththecollisionrateperion. Currentstate-of-the-art\nsingle-ionclocksoperatinginroomtemperatureultrahighvacuum(UHV)chamberswithbackgroundgaspressuresof\nroughly10−10 torrhavesystematicuncertaintiesduetobackgroundgascollisionsoforder10−19 [6,7]. Althoughback-\ngroundgascollisionswouldlikelycontributesignificantuncertaintyfor1000ionSn2+clocksinroomtemperatureUHV\nchambers,pressuresbelow10−16 torrhavebeenachievedincryogenicvacuumchambers[8]. Thus,byoperatingSn2+\nclocksincryogenicvacuumchambersitshouldbepossibletosuppressthecollisionshiftuncertaintytobelow10−21.”\nAs ions are lost from the trap or if they undergo chemical reactions to become dark ions, the crystal configura-\ntion will change. However, one important result of our simulations is that the systematic shifts do not change at\na level that would increase the overall uncertainty of the clock even as the ion crystal configuration changes. We\nhave added the following sentence to the text to highlight this feature: “Importantly, the differences of these shifts\nfor different ion crystal configurations are smaller than our estimated uncertainties, so Sn2+ clocks will be robust to\nion loss and background gas collisions that cause the crystal to melt and recrystallize in a different configuration.”\n3. How can one make sure the loaded Sn2+ ions are pure, and avoid dark ions like Sn+ co-trapped? Because the\ndark ions also bring huge heating during clock interrogation and affect micromotion and secular motion.\nDuring ion loading, the ion trap can be operated in a way such that Sn2+ is stably trapped but Sn+ is not\ntrapped, either by setting the Mathieu parameters to make Sn+ unstable as in a residual gas analyzer or by\napplying an oscillating electric field resonant with the motional frequency of uncrystallized Sn+ to heat it out\nof the trap. These techniques to kick undesired ions out of the trap become more difficult for impurity ions\nwith a charge-to-mass ratio very close to the desired isotope of Sn2+ , such as undesired isotopes of Sn2+ .\nFor these, however, it is possible to use a laser that is blue detuned with respect to the 1S ↔ 3P tran-\n0 1\nsition in the undesired isotope to heat the ions out of the trap. This laser would be far detuned from the\n1S ↔ 3P transition of the desired isotope and would not cause significant heating of the desired isotope.\n0 1\nDespite use of these or other methods, the ion crystal may still contain a small fraction of impurity ions. How-\never, these impurity ions will not significantly affect the clock performance. Impurity ions will change the spec-\ntrum of the secular motional modes [9], but all of the mode frequencies will still be within the Doppler cool-\ning bandwidth, so the Doppler shift and uncertainty will be unchanged. The micromotion of each individual\nion in a mixed species ion crystal is primarily determined by the charge-to-mass ratio of that ion, so the mi-\ncromotion Doppler shift and uncertainty of the clock ions will not be significantly changed by impurity ions.\nIn regard to the heating comment, we assume that the comment refers to RF heating, which is large when ion\nmotion is a significant fraction of the distance between ions and the ions experience the anharmonicity of the poten-\ntial due to neighboring ions. This heating needs to be overcome with a large laser cooling rate in order to crystallize\ninitially hot and decrystallized ions [10], but is negligible for ion crystals near the Doppler temperature limit [11],\neven for mixed species ion crystals where only one species is laser cooled [12]. Thus, heating due to impurity ions\nduring clock operation will be negligible.\n4. Considering of this large crystal size, the author should not only care about the magnetic field fluctuation but\nalsocareaboutmagneticfieldgradient. Further, whenincreasesthenumberofions, theionsarepushedfurtheraway\nfrom the trap axis, the RF trap frequency maybe cause large AC magnetic field. The author should give the shifts\nand uncertainty budget of magnetic field gradient and AC magnetic field.\ncharacterizedwithanuncertaintyof10µG,andthatthemagneticfieldgradientacrossthesizeoftheioncrystalshould\nbe less than 1 mG, in order to avoid degrading the clock performance. The latter requirement corresponds to a frac-\ntional gradient smaller than 10−4/mm. We have added the following sentence to the manuscript: “Although it would\nbe difficult to apply a 15 G quantization field with such a 10−4/mm fractional magnetic field gradient using standard\nHelmholtz coils, gradients as small as 10−6/mm are routinely achieved using solenoids together with shim coils [13].”\nWe thank the reviewer for drawing our attention to the inhomogeneity of the AC magnetic field due to the ion\ntrap. We have added the following sentence to the main text: “The ion trap should be designed to minimize the\ncapacitance between electrodes in order to minimize the contribution to the second-order Zeeman shift due to the ac\nmagnetic field at the trap drive frequency [14] (see Methods)”. We also added following paragraph to the Methods\naddressing this: “We estimate the size of this effect with the following simple model. We assume that the ion\ntrap has a 500 µm ion-to-electrode distance and is driven differentially with an RF voltage amplitude of 970 V,\nas is necessary to achieve the secular frequencies proposed here. Standard linear RF Paul traps constructed with\ncylindricalrodsorbladeelectrodesthatareelectricallydrivenfromoneendhavecurrentrunningdowntheelectrodes\ndue which is sunk by a capacitance at the far ends of the electrodes of order 100 fF. At the center of the trap,\nsymmetry dictates that the RF magnetic field should be zero. At the edge of the ion crystal, 22 µm away from\nthe trap center, this model predicts that there is an AC magnetic field of 50 mG, which causes an inhomogeneous\nfrequency shift of 4 mHz. It should be possible to characterize the average over all ions of this shift sufficiently\nwell as described in the text, and this is compatible with probe durations up to about 100 s. To achieve longer\nprobe durations it will be necessary to design traps with less capacitance or more symmetric current distributions.”\n5. Fig2(c) and line 266: When discussing the uncertainty of the probe laser Stark shift, the uncertainty is a few\norders of magnitude smaller than the shift, how is this achieved? By measuring the shift by comparing the transition\nfrequency between the increased power situation and the normal power situation? Or by hyper Ramsey scheme?\nWe anticipate using hyper-Ramsey or other protocols, and have modified the sentence to read: “We conservatively\nestimatethatthiscanbecharacterizedbyanuncertaintyof2.0×10−20 byusinghyper-Ramsey[15]orauto-balanced\nRamsey [16] interrogation protocols.”\n6. Line 221: As far as I know, the Doppler cooling efficiency is very sensitive to the power of the cooling laser, for\nsuch a large ion crystal, the laser power on each every ion should be different, thus the saturation parameter is not\nalways 20, I don’t know if the authors have already considered this in their calculations.\nAlthough the Doppler temperature limit and cooling rate do depend somewhat on the laser power, this is a rel-\nativelyminoreffect, especiallyinthepowerbroadenedregimewhereweproposeoperating. ForanellipticalGaussian\nbeam with waists of 100 µm and 2 mm propagating perpendicular to the trap axis, the laser intensity varies by 10 %\nacross the size of the ion crystal. This intensity variation would cause the power broadened linewidth of the cooling\ntransition to vary by 5 % across the ion crystal, with a similar variation in temperature limit. This is much smaller\nthan the 30 % uncertainty we assume for the characterization of the secular motion temperature.\n7. Line 275: I’m not sure if I have some misunderstanding on this: for single photon approach, the experiment\nrequires a magnetic field of 15 G, and have a field inhomogeneity of 1 mG over the size of 1 mm. It seems to me that\nthis is very difficult to be achieved by simply adding a pair of Helmholtz coils.\nSee response to comment 4.\n8. Line 428: As mentioned earlier, I think it is very difficult to be achieved for making a very strong magnetic field\nwith a very small gradient like 1 mG over the size of 1 mm, 10 micro-G is even more difficult.\nSee response to comment 4. Note in particular that the field does not need to be uniform to 10 µG, but rather\nthe average field over all of the ion locations needs to be characterized at this level.\n9. Line 460: “assuming a conservative 1 s probe duration and 80 % duty cycle.” Although the lifetime of 3P0\nclock state is quite long 106 s at 15 G magnetic field, ideally one can get very long coherence time, but actually the\ncoherence time is determined by high performance of clock laser and ion heating during interrogation time. What is\nthe evidence for 1 s, is it based on the experience of the Al+ ion clock or have you done the calculation and analysis?\nWhilewehavenotdonearigorouscalculationoftheachievablelasercoherencetime,1satthetransitionfrequencyof\nSn2+ iswellwithinthecapabilitiesofstate-of-the-artclocklasers[17],whichhaveachievedsignificantlyhigherQspec-\ntroscopy. Furthermore, newspectroscopyprotocolssuchascorrelationspectroscopy[18]anddifferentialspectroscopy\n[19] enable probing beyond the laser coherence time, and may support probe times of many seconds with Sn2+ .\n10. Line 472: “It may be possible...maintaining sub-10−19 inaccuracy.” The author proposal using Sr+ ion to co-\ntrappingSn2+ions,theoutsidetrappedSr+canpushtheSn2+ionstowardsthetrapaxisandmakethemicromotion\nshift smaller. The issue is, it seems not have a “magic wavelength”, cooling Sr+ ion during clock interrogation can\nlead into ac Stark shifts. Is the cooling laser off when probing? If so, the authors should consider the heating effect.\nYes, the Sr+ cooling laser will cause a Stark shift on the Sn2+ clock transition if the cooling is applied si-\nmultaneously with the clock interrogation. For Sr+ Doppler cooling on the 422 nm transition with a typi-\ncal saturation parameter of 0.1, and using the value of the Sn2+ differential polarizibility at 374 nm (which\nshould be slightly larger than that at 422 nm), the fractional frequency shift of the Sn2+ clock transition is\n5 × 10−19. It should be possible to stabilize and characterize this shift significantly better, so we do not antic-\nipate that this would be a problem. If it is, the cooling can be switched off during the clock interrogation.\nIII. REVIEWER # 2\ndear editor,\nthe paper proposed by Leibrandt et al is a proposal for an optical atomic clock that could be operated with a\ncloud of one thousand ions as a base, whereas so far, trapped ion based optical clock are operated with a single\nion. The idea is to make a more stable clock by interrogation of a thousand ions at the same time and then reduce\nthe quantum projection noise which is the limiting factor for short term stability. The issue is to do that without\ndegradingtheaccuracyandprecisionoftheclockastheinhomogeneityoftheionenvironmentcaninducenonuniform\nshifts responsible for a line broadening and an uncertainty in the shift estimation.\nThe idea is not new and several ions have been identified that share some characteristics that make them good\ncandidates for this kind of set-up. The main figure of merit is the differential polarizability of the transition. If it is\nnegative, two major effects cancel each-other, provided the RF-trap is operated at a proper frequency, which depends\non this difference. This is covered by ref 17-21. The new results of this paper is to add an ion to this list : Sn2+,\nwhich have some technical advantage over recent proposals, like the accessibility of the transition wavelength, but\nrequires a strong magnetic field to mix different atomic fine-structure states and make the clock transition accessible\nto laser excitation. The best performances are reached by a compromise between the lowest possible magnetic field\nto control Zeeman shifts and the lowest possible laser power to control light induced shifts. If an experiment is able\nto follow these prescriptions without excess inhomogeneity, the expected clock performance are very good compared\nto the state of art in terms of uncertainty and inaccuracy.\nWethanktherefereeforthereportandagreecompletelythatthenoveltyofthepapersisproposingtheSn2+clockand\ndemonstratingthatithasanumberofadvantagesthatmayallowtoachievesub-10-19inaccuracyandastabilityofthe\nbestopticallatticeclocks,butinasystemwithmuchlongerlifetime. Tothebestofourknowledge,thisisalsothefirst\nfull-scale simulation of micromotion for a 1000-ion clock, which has required high-performance computing resources\nto perform a detailed modeling of the shifts under various experimental conditions. We will address all of the spe-\ncificcommentsbelowandbelievethatfollowingthereferees’sadvicehassignificantlyimprovedtheclarityofthepaper.\nWe have broken up some of reviewer 2’s paragraphs and added numbers below so that they can be easily referenced.\n1. At this stage of the paper, the reader still do not know what isotope is considered. We can guess that an even\nisotope is assumed but nothing is said about their abundance and possible choices. The calculations used for atomic\nstructure calculations is now well established and one of the author is a reference in the optical clock community for\nthe relevance of her methods.\nWe have modified the description of the magic trap drive frequency to make our choice of isotope clear: “All even\nisotopesofSn2+areequallysuitableforclockoperation,anddifferentchoiceswillhaveslightlydifferentmagicdrivefre-\nquenciesandlasercoolingdynamics. ThecalculationsthroughoutthisarticleuseM ≈Am withA=118(wherem\ni p p\n5\nis the proton mass), resulting in Ω /(2π) = 225(5) MHz.”\nm\n2. The discussion regarding the clock operation in the context of a thousand trapped ion is far less convincing.\nThere are a lot of statements regarding the cooling efficiency that look like speculations and are not supported by\ncomparison with published and cited work. If the statements are based on molecular dynamics simulations, it is not\nwrittenclearlyandthefactthat3laserbeamswithdifferentdirectionsarerequiredisnotjustifiedanywhere.\nIndeed, asfarasIknow,no3DCoulombcrystalexperimentsusethisset-upasthemicro-motioninducedDopplereffectmake\nDopplercoolinglessefficient(seeurl=https://link.aps.org/doi/10.1103/PhysRevA.105.023101forthe2Dcasewhere\ndifferent laser detunings are needed). A single one propagating along the axis is often the best solution for large 3D\nsystem.\nAs the reviewer points out, micromotion does make the Doppler cooling of large ion crystals less efficient for\nlaserbeamswithnonzerocomponentsoftheirk-vectoralongtheradialtrapdirections. However,forlow-temperature\nions in a perfectly harmonic trap, cooling with a single laser perfectly along the axial direction does not cool the\ntwo center-of-mass radial modes at all. In the presence of finite temperature and trap imperfections, cooling of the\nradial modes is still inefficient and the radial mode temperatures are higher than the Doppler limit, which would\nbe problematic for minimizing the Doppler shift. Thus, here we propose using three orthogonal cooling beams. In\nthis proposal, the high rf trapping frequency and narrow cooling transition linewidth help to make the radial cooling\nbeams effective. This is both because the micromotion modulation index scales like the reciprocal of the trap fre-\nquencyandbecausethemicromotionsidebandtransitionsareverywellspectrallyseparatedfromeachotherandfrom\nthemicromotioncarriertransition(thisisadistinctregimefromthatofPRA105,023101towhichthereviewerrefers).\nFor ions that are 22 µm away from the trap axis (i.e., the larger crystal radius), micromotion modulates the\nelectric field of the purely radial vertical cooling beam with a modulation index of 2.4. The laser intensity\nfor driving the micromotion carrier transition is reduced to 0.04 % of that for ions on the trap axis, and the\nlaser intensity for driving the first (second) micromotion sideband transition is 26 % (19 %) of the carrier for\nions on the trap axis. Cooling with a laser intensity of 0.04 % for the outermost ions will change the Doppler\nlimit for the radial modes, but may still be ok because most of the ions will be cooled with a significantly\nhigher intensity. If this turns out to be a problem, the cooling can be made more efficient by adding light\nthat is red detuned from the first micromotion sideband to the radial beams. The latter technique has been\nsuccessfully used in the NIST Ion Storage Group for single ion traps with large uncompensable micromotion.\nAs the reviewer suggests, a detailed molecular dynamics model of the laser cooling will be necessary to optimize\nthe cooling parameters and to provide a rigorous constraint on the corresponding Doppler shift uncertainty of the\nclock, but this is beyond the scope of this manuscript and will be the subject of future work. However, it is unlikely\nthat the detailed model will yield an ion temperature or Doppler shift uncertainty that is significantly larger than\nthose of the simple model.\nWe have added the following sentence to the text: “A more complete molecular dynamics model incorporating\nthe details of the laser cooling can be used to optimize the cooling parameters and is likely necessary to provide a\nrigorous constraint on the corresponding Doppler shift uncertainty of the clock.”\n3. Other estimations lack an established support like the requirement for the magnetic field homogeneity. It is\nassumedtobebetterthan0.01%overammscale. Itshouldbesupportedbyresultscomingfromsimilarexperiments\nproving the feasibility of this requirement.\nAsstatedintheoriginalmanuscript,wesuggestthatthemagneticfieldaveragedoveralloftheionlocationsshould\nbe characterized with an uncertainty of 10 µG, and that the magnetic field gradient across the size of the ion crystal\nshould be less than 1 mG, in order to avoid degrading the clock performance. The latter requirement corresponds to\na fractional gradient smaller than 10−4/mm. We have added the following sentence to the manuscript: “Although it\nwould be difficult to apply a 15 G quantization field with such a 10−4/mm fractional magnetic field gradient using\nstandard Helmholtz coils, gradients as small as 10−6/mm are routinely achieved using solenoids together with shim\ncoils [13].” For more details about the magnetic field homogeneity, see the response to reviewer 1, comment 4.\n4. A lot of discussion regarding the cloud shape and ion dynamics assume operating parameters that are jus-\ntified a lot later without any notice. This is very detrimental. The RF optimum frequency is so large and the\nchosen single-ion secular frequencies are such that they correspond to very low Mathieu parameter, of the order\nof 10−2. This is at the edge of the stability diagram for an RF-trap and to make it convincing, this proposal\n6\nshould cite experimental results where such a low parameter has been operated with success. If these works\ndo not exist yet, it is because it is really a challenge and this should not be considered as obvious in the article.\nThetrappingparametersproposedherecorrespondtoMathieuparametersof|q |=0.0127,a =2.33×10−5,and\nx,y x\na =−2.41×10−5. Whilethisisindeedasmallerq parameterthantypicalexperiments,itisnotclosertotheedgeof\nx\nthe stability region than typical experiments. For small q, the border of the stability diagram is given by |a| < q2/2\n[20]. Here, |a|/(q2/2)=0.29(cid:28)1. For comparison, in Ref. [7] |a|/(q2/2)=0.22 and in Ref. [21] |a|/(q2/2)=0.52 are\nused experimentally without problems. Furthermore, our molecular dynamics calculations in the full-time-dependent\npotential do not indicate any instability issues. We have made this point more direct by modifying the last sen-\ntence of the paragraph describing our calculations to read “Second, we calculate the amplitude and direction of\nthe micromotion of each ion and verify the stability of the ion crystal by time evolving the full equations of mo-\ntion (without taking the pseudopotential approximation), starting from the ion positions computed in the first step.”\nIn fact, smaller q offers several advantages for Coulomb crystal clocks as the ions are deeper in the pseudopotential\napproximation. First, molecular dynamics simulations show that rf heating of large ion crystals, which arises due to\nthebreakdownofthepseudopotentialapproximationq (cid:28)1, scalesroughlylikeq6 atlowtemperatures[11]. Extrapo-\nlatingtheresultsofRef.[11]toΩ=2π×225MHz,q =0.0127,andaninitialiontemperatureof120µKsuggeststhat\ntherfheatingmightbeoforder10−12 µK/s,whichiscompletelynegligibleevenforprobedurationsofmanyseconds.\nWe have added the following to the manuscript: “These secular frequencies correspond to Mathieu parameters\nof |q | = 0.0127, a = 2.33×10−5, and a = −2.41×10−5, and can be achieved by differentially driving a trap\nx,y x x\nwith 500 µm ion-to-electrode distance with 970 V amplitude of rf. This smaller-than-typical q parameter offers\nseveral advantages for Coulomb crystal clocks. Molecular dynamics simulations show that rf heating of large ion\ncrystals, whicharisesduetothebreakdownofthepseudopotentialapproximationq (cid:28)1, scalesroughlylikeq6 atlow\ntemperatures [11]. Care must be taken, however, to minimize anharmonicity of the trap potential as this can lead to\nadditionalrfheating[22]. Ionheatingduetobackgroundgascollisionsisalsoreducedforsmallerq parameters[12].”\n5. Thepartconcerningthedifferentspatialconfigurationofanionensembleisveryconfusingasthecitedworksall\nset the ions in a static pseudo-potential (I suggest the authors to read also Calvo and Yurtsever, EPJD (2007) DOI:\n10.1140/epjd/e2007-00137-2 ), and the authors also use the same strategy, so why mentioning time-averaged kinetic\nenergy of the micro-motion? This is the source of the pseudo-potential and it has nothing to do in a pseudo-potential\ndescription, it is one or the other, not both. How micro-motion is obtained in the pseudo-potential approxima-\ntion? This point deserve clarification. The description of the forces taken into account should be more rigourous\nand corrected. Indeed, more details about the numerical simulations should be given (like in a methods part) to\nappreciate the relevance of the results and more earlier works should be cited to support the chosen strategy (see\nDOI:10.1103/PhysRevA.82.033406andhttps://doi.org/10.1103/PhysRevLett.70.818tostartwith). Whataboutthe\nchoice of the damping constant simulating laser cooling? If it is too strong, the system is frozen in a non-equilibrium\nstate. How the authors proceed to make sure this is not the case and that the motional temperature is 120 microK\n(cf caption of fig 4) ?\nWethankthereviewerformakingusawareoftheinterestingresultsinRef.[23],andhaveaddedthistothemanuscript.\nWe have also added Refs. [24, 25] as examples of earlier molecular dynamics calculations following similar methods.\nWe have clarified the description of our molecular dynamics calculation method as follows: “First, similar to\nRefs. [26, 27], we calculate the equilibrium ion positions by time-evolving the equations of motion in the pseudopo-\ntential approximation starting from random initial positions and including a viscous damping force that qualitatively\nrepresents laser cooling. For each set of trap parameters and number of ions investigated, we generate up to 36\nconfigurations in parallel using a high-performance computer cluster. For ion numbers up to 1000, this takes less\nthanoneweek. Thestrengthofthedampingforceissuchthatthecoolingtimeconstantisroughly100µs, whichisa\ntypicalexperimentalcoolingtimeconstant,andwehaveverifiedthatreducingthisstrengthbyanorderofmagnitude\ndoes not reduce the number of distinct configurations. Once the equilibrium ion positions are known, it is straight-\nforward to compute the frequencies and eigenvectors of the normal modes of secular motion and their Lamb-Dicke\nparameters [28, 29], and the spectrum of the clock transition [30]. Second, we calculate the amplitude and direction\nof the micromotion of each ion and verify the stability of the ion crystal by time evolving the full equations of mo-\ntion (without taking the pseudopotential approximation), starting from the ion positions computed in the first step.”\nWe have also added a sentence about the cooling strength: “The strength of the damping force is such that the\ncooling time constant is roughly 100 µs, which is a typical experimental cooling time constant. We have veri-\n7\nfied that reducing this strength by an order of magnitude does not reduce the number of distinct configurations.”\nWe hope that added text clarifies our approach. As we use well-established methods in these numerical simula-\ntions, we feel that an additional methods section would not add significant value.\n6. The paragraph starting at line 393 is completely speculative as nothing proves that the listed requirements are\ncompatiblewitheachother. Itissurprisingtoreadthat“maximizethelowestsecularfrequency”isinthelistwhereas\na so small q x parameter has been chosen. So this is certainly not the case. Line 413 mention a trade off to reach\nso small values, so the most important conditions should be enlightened and this need for a maximization should be\njustified.\nAvoiding secular frequencies that are too low is important both because ion heating rates due to anomalous electric\nfield noise are larger for smaller motional frequencies and because off-resonant excitation of low frequency motional\nsidebands during clock spectroscopy can lead to line pulling. The latter concern is briefly discussed in the previ-\nous paragraph: “The lowest frequency motional sideband transitions are above 14 kHz for any of the generated\ncrystal configurations, and should not lead to significant line-pulling due to off-resonant excitation for probe times\nlonger than about 10 ms.” We have added a new phrase to the manuscript about the former concern: “...low\nfrequency motional modes should be avoided becuase they have higher heating rates due to electric field noise [31]”\nHowever, as has been previously discussed, operating at low Mathieu q parameter is also adventageous, and fur-\nthermoreitwouldbedifficulttoachievesignificantlyhigherMathieuq parameterswithrealisticrfvoltageamplitudes.\nOur intent with this paragraph is to motivate our choice of parameters, which are compatible with each other\nandenablebothhighaccuracyandhighstabilityclockperformance. Wehavemodifiedthefinalsentenceofthispara-\ngraph accordingly to read “Single-ion secular frequencies of 1.15 MHz, 0.85 MHz, and 0.10 MHz represent a trade-off\nof the aforementioned optimization goals that enable both high accuracy and high stability clock performance.” We\nagree with the referee that a different set of trap parameters may be chosen, and could may lead to even higher\nperformance. An exhaustive optimization of trap parameters is beyond the scope of this manuscript.\n7. On the step of line 458, the requirements concerning the stability and the precision/accuracy of the proposed\nclock should be clearly identified and separated as some technical choices can be good for one figure of merit but bad\nfor another.\nWe have added the sentence: “It may be possible to achieve a smaller secular motion and thus total system-\natic uncertainty by reducing the number of ions, but this would come at the cost of degraded stability.”\n8. From line 472, another list of speculations appear, concerning the best side ion to be trapped with Sn2+,\nto compress the ion cloud around the trap axis. The obvious solution would be to increase the cloud density,\nbut once again, the reason why such a low Mathieu parameter is chosen is not given. Furthermore, this idea of using\nan outer cloud shell to compress the inner cloud is not obvious at all and is not supported by any calculations or\nprevious work. This naive view seems in contradiction with the observation reported in PRL.86.1994.\nWe have addressed our choice of low Mathieu q parameter above.\nWe agree that the phrase “the Sr+ ions would reside on the outside edge of the crystal, pushing the Sn2+ ions\ntowardsthetrapaxis”inthemanuscriptmighthavebeenmisleading. Wehavechangedthislanguageto“theSr+ ions\nexperience a weaker radial restoring force from the trap and thus fill the Coulomb crystal lattice sites further from\nthe trap axis, causing the Sn2+ ions to reside in a cylindrical volume of lattice sites closer to the trap axis [32]”. In\nFig. 2 of Ref. [32] that the referee refers to above, it is clearly shown that the lower mass-to-charge ratio ion (24Mg+)\noccupies a cylindrical volume of Coulomb crystal lattice sites near the trap axis, while the higher mass-to-charge\nratioion(40Ca+)residesinthelatticesitesfurtherfromthetrapaxistofillouttheellipsoidaltwo-seciescrystalshape.\n9. As for the idea of using a multipole trap, I’d like to point to the authors that ref 54 and 56 concern room\ntemperature large ion clouds and in the cold limit, the self organization of ions is very different (see for exam-\nple J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 154002 ). For smaller sample, the use of multipole trap is\ndescribed here : DOI: 10.1103/PhysRevA.81.043410 . Contrary to what is stated on line 489-494, ions will not\ngather closer to the axis, because of Coulomb repulsion. (read again section V of ref 17) Anyway, and unfortu-\nnately, using a multipole trap with laser cooled ions may be a wrong good idea as stated in this series of paper :\nhttps://doi.org/10.1080/09500340.2017.1408866 and M Marchenay et al 2021 Quantum Sci. Technol. 6 024016 .\nWe thank the reviewer for pointing out these references. Based on the calculations in Ref. [33], the second-order\nDopplershiftduetomicromotionaveragedoveralloftheionsisstillsmallerinamultipoletrapthaninaquadrupole\ntrap, but it is true that the nonuniform radial density distribution does reduce the benefit. As the reviewer\npoints\nout, more calculations are needed to determine if using multipole traps is truly beneficial for these systems, so we\nhave removed this suggestion from the manuscript.\n10. At this stage, I think the reader is kind of lost with the come back of spectroscopic considerations from\nline 495. This would be better placed at the beginning of the paper or it could be suppressed as nothing new is brought\nhere.\nWe have carefully considered the placement of the section “Sensitivity to beyond Standard Model physics”. The\nintroduction primarily provides context in the form of an overview of various clock platforms and introduces the\npotential benefits and challenges of Coulomb crystal clocks, such as the issue of the micromotion. Adding a section\naboutclockapplicationsherewouldbesomewhatconfusingandwouldmaketheintroductionsectiontoolong. Placing\nit between “Sn2+ atomic properties” and “Sn2+ clock operation” will be confusing as it will break up the discussion\nof Sn2+ properties that are critical to clock operation and the assessment of systematics and the discussion of clock\noperation. Placing it later will break up the discussion of Sn2+ clock operation and resulting clock performance. As\nthis section deals with applications of the Sn2+ clock, the present placement after the discussion of clock operation\nand performance and before the conclusion appears to be most suitable. The sentence at the end of the introduction\n“We conclude by discussing prospects for fifth force searches using isotope shift measurements, as well as other tests\nof fundamental physics with Sn2+.” alerts the reader that this discussion is placed at the end of the paper.\nWe also believe that it is important to keep this discussion in the paper due to the present status of the\nfifth force searches with the non-linearity of King plots. Yb results already show a non-linearity which is most\nlikely from the nuclear deformation. However, the data indicate that non-linearity may come from at least two\nsources. Ca data do not show non-linearity at present. Therefore, it is important to find another suitable sys-\ntem lighter than Yb where such experiments may be performed. Removing dependence on the nuclear defor-\nmation requires the use of four or more even isotopes, and Sn2+ is unique in having so many suitable isotopes.\nAs a conclusion, I think this paper is not ready yet for publication. Furthermore, I also think that the\npresented results are not noteworthy neither original enough for publication by nature communication.\nWe have addressed in detail all of the referee’s comments. The thoughtful and constructive comments of all three\nreferees helped us improve the paper, providing more clarity on the choice of the specific clock operating parameters\nas well as assessing other potential systematic effects. Solving the question of significant improvement of trapped\nion clock statistics while keeping systematic uncertainties under 10−19 is of critical importance to metrology. Our\nwork provides a pathway towards a solution of this problem with a new system, Sn2+, and provides a detailed\ninvestigation of micromotion suppression with 1000 ions using high-performance computer simulations. The Sn2+\nion system also provide a way to resolve a present conundrum with fifth force searches with isotope shift measure-\nments. Therefore, we believe the paper satisfies the Nature Communications standards of novelty and importance.\nIV. REVIEWER # 3\nThe manuscript ”Prospects of a thousand-ion Sn2+ Coulomb-crystal clock with sub-10−19 inaccuracy”, gives a\ndetailed account on the possibility of building a highly accurate atomic clock based on a large three dimensional ion\nCoulomb crystal. By considering a clock based on up to 1000 ions, the authors’ goal is to produce an atomic clock\nbasedonions,whichnotonlycancompetewithneutralatomlatticeclockswithrespecttototalachievableinaccuracy,\nbut also with respect to needed experimental integration time. The authors have made a great effort to assess many\nof the potential limiting issues related to the specific ion and trapping configuration in mind. The manuscript is\norganized in a logical way by first describing the reason to consider the Sn2+ ion species and the expected properties\noftherelevantelectroniclevels,followedbypresentinghowthetrappingfieldsandpropertiesofandionmotionwithin\na Coulomb crystal influence the accuracy of the Coulomb crystal clock.\nFinally, before a summary and outlook, a discussion on various limiting factors and potential further\nimprovements are briefly presented.\nAll in all, after a revision of the manuscript with respect to the comments and suggestions stated\nbelow, I find the manuscript suitable for publication in Nature Communication, since the work\nrepresents a promising scenario for a future new improved time standard.\nWe thank the reviewer for supporting publication of our manuscript in Nature Communications.\nComments/questions:\n1) In Fig. 1, a Coulomb crystal of 1000 ions is presented. It is obvious that a very elongated crystal along the rf\nfield-free axis is preferable, but it is less clear why trap parameters giving rise to a relatively large difference in the\ncrystal dimensions in the perpendicular plane is chosen. Perhaps the authors could elaborate a bit on that.\nIn the limit of equal trapping strength in the two radial dimensions, which would lead to equal crystal dimen-\nsions in the two radial dimensions, there is a zero frequency motional mode in which the crystal rotates about the\ntrapaxis. Thismodewouldbedifficulttocool. Wehaveaddedthefollowingtothemanuscripttoaddressthis: “The\ntwo radial single-ion secular frequencies must be different in order to avoid a zero-frequency mode in which the ion\ncrystal rotates about the trap axis.”.\n2) In Fig. 1, the colors of the clock and cooling beams are very close to each other. Choosing a different color for\nthe cooling beams would be helpful also for the readability of the text on the illustration.\nWe thank the reviewer for pointing this out and have made the colors more distinct.\n3) Since the “magic” rf frequency of 225 MHZ is rather high, a reference that shows this should be achievable with\nthe trap dimensions in mind would be needed.\nWe have added Ref. [34], in which an ion trap with 160 µm ion-to-electrode distance is driven at 240 MHz with up\nto 1 kV amplitude of rf. The manuscript now reads: “Although this is a high trap frequency, previous experiments\nhave operated in this range [34].”\n4) In fig. 2.c, the lines indicating the uncertainties of the second order Zeeman shifts and Stark uncertainties\ntranslates into a magnetic and laser field instabilities of 10−5 and 10−4, respectively. It would be needed with some\nreferences that such instabilities should be achievable over the total volume of the Coulomb crystal.\nIn the original manuscript, we suggest that the magnetic field averaged over all of the ion locations should be charac-\nterized with an uncertainty of 10 µG, and that the magnetic field gradient across the size of the ion crystal should be\nlessthan1mG,inordertoavoiddegradingtheclockperformance. Thelatterrequirementcorrespondstoafractional\ngradient smaller than 10−4/mm. We have added the following sentence to the manuscript: “Although it would be\ndifficult to apply a 15 G quantization field with such a 10−4/mm fractional magnetic field gradient using standard\nHelmholtz coils, gradients as small as 10−6/mm are routinely achieved using solenoids together with shim coils [13].”\nReviewer 1 drew our attention to the inhomogeneity of the AC magnetic field due to the ion trap. We have\nadded the following sentence to the main text: “The ion trap should be designed to minimize the capacitance\nbetween electrodes in order to minimize the contribution to the second-order Zeeman shift due to the ac mag-\nnetic field at the trap drive frequency [14] (see Methods)”. We also added following paragraph to the Methods\naddressing this: “We estimate the size of this effect with the following simple model. We assume that the ion\ntrap has a 500 µm ion-to-electrode distance and is driven differentially with an RF voltage amplitude of 970 V,\nas is necessary to achieve the secular frequencies proposed here. Standard linear RF Paul traps constructed with\ncylindricalrodsorbladeelectrodesthatareelectricallydrivenfromoneendhavecurrentrunningdowntheelectrodes\ndue which is sunk by a capacitance at the far ends of the electrodes of order 100 fF. At the center of the trap,\nsymmetry dictates that the RF magnetic field should be zero. At the edge of the ion crystal, 22 µm away from\nthe trap center, this model predicts that there is an AC magnetic field of 50 mG, which causes an inhomogeneous\nfrequency shift of 4 mHz. It should be possible to characterize the average over all ions of this shift sufficiently\nwell as described in the text, and this is compatible with probe durations up to about 100 s. To achieve longer\nprobe durations it will be necessary to design traps with less capacitance or more symmetric current distributions.”\nThe hyper-Ramsey and auto-balanced Ramsey techniques relax the requirement for laser intensity stability and\nwe anticipate using hyper-Ramsey or other protocols. We have modified the sentence to read: “We conservatively\nestimatethatthiscanbecharacterizedbyanuncertaintyof2.0×10−20 byusinghyper-Ramsey[15]orauto-balanced\nRamsey [16] interrogation protocols.”\n10\n5) With respect to the assessment of time dilation and rf field induced Stark shifts, the authors have considered\n36 stationary configurations of the Coulomb crystals which seems reasonable. A single background gas collisions may\nthoughleadtodynamicalchangesinthecrystalstructureorevenleadtocrystalmelting. Onlyveryfewexperimental\ninvestigations have been made so far to address these issues quantitatively, but they could potentially constitute\nlimitationstouselargercrystalsinparticularlyinroomtemperaturetrappingenvironments. Pleasecommentonthis\nin a revised manuscript.\nWe agree that this experiment most likely needs to use a cryogenic ion trap in which the background gas pres-\nsure and thus collision rate can be greatly reduced. We have added the following paragraph to the methods: “One\ncanestimatethebackgroundgascollisionshiftbasedonscalinganalysisoftheshiftforsingle-ionclocks[5]. Thetotal\nrate of Langevin collisions between a Coulomb crystal and background gas molecules scales linearly with the number\nofionsandquadraticallywiththechargeoftheions,sofora1000ionSn2+ clockthebackgroundgascollisionratewill\nbeapproximately4000timeslargerthaninsingle-ionclocks. Asaconservativeupperbound,wetakethatthesystem-\nmatic uncertainty due to background gas collisions scales linearly with the collision rate. A more detailed model would\ntake into account that only the Doppler shift component of the uncertainty scales with the total collision rate while\nthe phase shift component of the uncertainty scales with the collision rate per ion. Current state-of-the-art single-ion\nclocks operating in room temperature ultrahigh vacuum (UHV) chambers with background gas pressures of roughly\n10−10 torr have systematic uncertainties due to background gas collisions of order 10−19 [6, 7]. Although background\ngas collisions would likely contribute significant uncertainty for 1000 ion Sn2+ clocks in room temperature UHV\nchambers,pressuresbelow10−16 torrhavebeenachievedincryogenicvacuumchambers[8]. Thus,byoperatingSn2+\nclocksincryogenicvacuumchambersitshouldbepossibletosuppressthecollisionshiftuncertaintytobelow10−21.”\nAs ions are lost from the trap or if they undergo chemical reactions to become dark ions, the crystal configura-\ntion will change. However, one important result of our simulations is that the systematic shifts do not change at\na level that would increase the overall uncertainty of the clock even as the ion crystal configuration changes. We\nhave added the following sentence to the text to highlight this feature: “Importantly, the differences of these shifts\nfor different ion crystal configurations are smaller than our estimated uncertainties, so Sn2+ clocks will be robust to\nion loss and background gas collisions that cause the crystal to melt and recrystallize in a different configuration.”\n6) With respect to computing the frequencies and eigenvectors of the normal modes of the secular motion and\nLamb-Dicke parameters reference is given to [41] in the manuscript. Since this paper deals only with linear chains of\nions, a more suitable reference for the current proposal could, e.g., be New J. Phys. 14 093023 (2012).\nWe thank the reviewer for making us aware of Ref. [29] and have added a citation in the manuscript.\n7)Fig. 3, whichiscentraltothemanuscript, containsalotofinformationpresentedinacondensationfashion. For\nthe non-expert it can be very hard to be sure to have understood it fully as it is.\nIn fig. 3.b, it would be great to have a clearer physical description of what is meant by “net secular motion shift”.\nAs it looks like it is independent of the number of ions, I assume it accounts only for the time dilation related to the\nsecular motion? Could there also be a secular motion dependent Stark shift which should be accounted for?\nWe thank the reviewer for pointing out that we had neglected to provide a definition of the net secular motion\nshift. We have modified the caption of Fig. 3b to include “The net secular motion (micromotion) shift is defined as\nthesumofthenegativetime-dilationshiftandthepositiveStarkshiftcausedbytheelectricfieldoftheiontrapinthe\nreferenceframeoftheionduetoionmotionatthesecularmotionalfrequencies(integermultiplesofthetrapdrivefre-\nquency).” Furthermore,wehavemodifiedthecorrespondingsentenceinthemaintexttoread“Thenetsecularmotion\nshift(definedasthesumofthetimedilationshiftandStarkshiftduetoionmotionatthesecularfrequencies)for1000\nionsis−2.4×10−19,andweconservativelyestimatethatthiscanbecharacterizedwithanuncertaintyof7.2×10−20.”\nFig. 3.d is very challenging. If I am right that the green bars are just the difference between the red\nand blue, I believe one could communicate that clearer. The two first panels deal with the secular\nmotion, while the rest of the panels deal with rf induced micromotion. For the sake of clarity, it would\nbe good to separate Fig. 3.d. into three separate sub-figures (d.,e., and f.) and not for all panels use\nthe term “net micromotion shift”. Inthe lastfourpanels\nthe fractional frequency shifts exhibit large increases at the magical rf frequency. Why is this the case? This should\nbe explained more carefully in the text. The assumed temperature for the secular contributions should be stated in\nthe figure caption.\nFig. 3d into three subpanels. Although we have elected not to separate Fig. 3d, we have changed the label for the\ngreen color to read “net motional shift”, referring to either the net micromotion shift or the net secular motion shift.\nThe time dilation shift due to motion at the trap drive frequency is indeed much larger than that due to mo-\ntion at other frequencies. This is because for laser cooled three-dimensional ion crystals the micromotion is much\nlarger than the secular motion. Indeed, one of the contributions of this manuscript is to carefully calculate how\nwell this very large time dilation shift due to micromotion can be cancelled by the corresponding trap Stark shift.\nWe have added a sentence to the Fig. 3b caption “The secular motion temperature here and in panel d is as-\nsumed to be 120 µK.”\n8)Somewhereinthemanuscriptitwouldbegreattohaveashortdiscussionofwhatwouldtheinfluenceofaradial\ntemperature gradient in the crystal. A priori one cannot expect to be able to cool all ions in the crystal to the same\nfinal temperature.\nWecertainlyagreethatexperimentalimperfectionswillresultinsomewhatunequaltemperaturesoftheindividualmo-\ntionalmodes. Themodelofcoolingpresentedinthemanuscriptisintendedtoserveasanestimateofthemagnitudeof\nthesecularmotionDopplershiftaveragedoveralloftheions,andtheachievableuncertaintyinthisshift,ratherthanto\nprovidearigorousevaluationofthesecularmotionDopplershiftuncertaintyforaparticularexperimentalimplemen-\ntation. ThissimplemodelindicatesthatthesecularmotionDopplershiftwillnotbeadominantlimitationtoclockun-\ncertainty.\nA detailed molecular dynamics model of the laser cooling will be necessary to optimize the cooling parameters\nand to provide a rigorous constraint on the corresponding Doppler shift uncertainty of the clock, but this is beyond\nthe scope of this manuscript and will be the subject of future work. However, it is unlikely that the detailed model\nwill yield an ion temperature or Doppler shift uncertainty that is significantly larger than those of the simple model.\nWe have added the following sentence to the text: “A more complete molecular dynamics model incorporating\nthe details of the laser cooling can be used to optimize the cooling parameters and is likely necessary to provide a\nrigorous constraint on the corresponding Doppler shift uncertainty of the clock.”\nFor more discussion of the laser cooling, see also the response to reviewer 2 comment 2.\n9)WithrespecttouncertaintyintheshiftsduetoBBR,isaneffectivetemperaturewithanuncertaintyof500mK\nreasonable over the whole crystal lengths (~1 mm)?\nWell designed ion traps (with high electrical and thermal conductivity and low dielectric loss) have achieved smaller\nthan1Kthermalgradientsacrosstheentiretrap[35,36]. Evenifthisisnotthecase,itwouldbepossibletomeasure\nthe BBR temperature as a function of position within the trap by e.g. trapping a single Sr+ ion and measuring\nthe BBR Stark shift of its clock transition (which is much larger than Sn2+ and very well known [37]) for different\npositions within the trap. Finally, most likely it will be beneficial to operate Sn2+ clocks in cryogenic ion traps where\nthe BBR shift will be negligible.\n10)WithrespecttothetwospeciesCoulombcrystaldiscussiononp. 8and9,itshouldbenotedthatvolumeofthe\nSn2+ ions in this case will be a nearly perfect cylinder, and hence for a given crystal length and number of ions the\noutermostionswillbepositionedclosestothetrapaxiswhencomparedtothecaseofasingleSn2+ speciesellipsoidal\ncrystal. Consequently, this would lead to a smaller average time dilation and Stark shifts. A potential reference in\nthis relation could, e.g., be Phys. Rev. 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Klein, P. Gill, T. Lindvall, A. E. Wallin,\nM. Merimaa, C. Tamm, C. Sanner, N. Huntemann, N. Scharnhorst, I. D. Leroux, P. O. Schmidt, T. Burgermeister, T. E.\nMehlsta¨ubler, and E. Peik, Analysis of thermal radiation in ion traps for optical frequency standards, Metrologia 52, 842\n(2015).\n[36] T. Nordmann, A. Didier, M. Doleˇzal, P. Balling, T. Burgermeister, and T. E. Mehlsta¨ubler, Sub-kelvin temperature\nmanagement in ion traps for optical clocks, Rev. Sci. Instrum. 91, 111301 (2020).\n[37] P. Dub´e, A. A. Madej, M. Tibbo, and J. E. Bernard, High-accuracy measurement of the differential scalar polarizability\nof a 88Sr+ clock using the time-dilation effect, Phys. Rev. Lett. 112, 173002 (2014)."}, {"round": 2, "reviewer_comments": {"reviewer_1": "Now, I have read the answers and revised manuscript, the authors replied to all my questions and\nconcerns. In my opinion, this new ion is a promising candidate for clock with sub-E-19 inaccuracy, the\nmanuscript is well-written and suitable for publication in NC.\nMy only concern now is the shift caused by ions loss rate: as the author said “the differences of these\nshifts for different ion crystal configurations are smaller than our estimated uncertainties”, I believe\nthat, but what I am still concerned about is how much shift is caused by ion loss during a single probe\ncycle, and thus the asymmetry of the clock line shape, and whether it will be on the order of E-19.", "reviewer_2": "dear editor,\nthe authors have addressed all the questions raised by the reviewers and are documenting their\nassumptions with a lot of relevant references. I think this paper can now be published by nature\ncommunication, with a main interest of proposing high precision frequency measurements based on a\n3D ion cloud. Nevertheless, I still have few comments for the authors:\nthe authors state that:\n\"To the best of our knowledge, this is also the first full-scale simulation of micro-motion for a 1000-ion\nclock, which has required high-performance computing resources to perform a detailed modeling of\nthe shifts under various experimental conditions. \"\nEven if the purpose of the simulations were not oriented toward frequency metrology, a full-scale\nsimulation of micromotion for a 1024-ion cloud can be found in\nhttps://journals.aps.org/pra/abstract/10.1103/PhysRevA.108.013109\nand\nhttps://doi.org/10.1063/5.0046693\nthese last results were obtained with a single laser beam propagating along the trap axis. Regarding\nthe need for different laser beams to cool a 3D sample, I refer the authors to the work done in the\ngroup of M. Drewsen where most of the time a single direction of propagation is used to cool a 3D\ncloud and it is along the trap axis when the cloud is 3D. In our group, we have tried to cool a large 3D\ncloud with a single beam having projection along the axial and radial trap axis and it was a fiasco.\nEverything went smooth when we moved the beam along the trap axis. So I think this laser beam\npropagation direction is still an open question and it is a very good thing if the authors try different\nspatial configurations in their set-up. They will bring a very useful knowledge to the community\ninterested in using large 3D ion cloud.\nCaroline Champenois, CNRS-AMU, France"}, "author_response": "Response to referee comments on manuscript “Prospects of a thousand-ion Sn2+\nCoulomb-crystal clock with sub-10−19 inaccuracy”\nDavid R. Leibrandt,1,2,3,∗ Sergey G. Porsev,4 Charles Cheung,4 and Marianna S. Safronova4\n1Department of Physics and Astronomy, University of California, Los Angeles, California, 90095, USA\n2Time and Frequency Division, National Institute of Standards and Technology, Boulder, Colorado, 80309, USA\n3Department of Physics, University of Colorado, Boulder, Colorado, 80309, USA\n4Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, USA\n(Dated: December 1, 2023)\nI. INTRODUCTION\nWe thank the referees for supporting publication of our manuscript in Nature Communications, and for their\ncommentsthathavehelpedustofurtherimprovethemanuscript. Inthefollowing,wereplytothereferees’comments\npoint by point. The referee comments are reproduced verbatim in gray text, followed by our responses in black text.\nWe have appended a copy of the revised manuscript with the changes highlighted.\nII. REVIEWER # 1\nNow, I have read the answers and revised manuscript, the authors replied to all my questions and concerns. In my\nopinion, this new ion is a promising candidate for clock with sub-E-19 inaccuracy, the manuscript is well-written and\nsuitable for publication in NC.\nWe thank the reviewer for their positive assessment of our work.\nMy only concern now is the shift caused by ions loss rate: as the author said “the differences of these shifts for\ndifferent ion crystal configurations are smaller than our estimated uncertainties”, I believe that, but what I am still\nconcerned about is how much shift is caused by ion loss during a single probe cycle, and thus the asymmetry of the\nclock line shape, and whether it will be on the order of E-19.\nWe thank the reviewer for calling attention to this important point. Previously, this point was only addressed\nbriefly in the “Background gas collision shift” section of the methods without being called out in the main\ntext. We have now added the sentence “It may, in fact, be necessary to use a cryogenic vacuum chamber to\nsuppress the uncertainty due to background gas collisions to a negligible level (see Methods).” to the main\ntext (line 514) and expanded the “Background gas collision shift” section of the methods to read as follows.\n“A detailed analysis of the background gas collision shift and uncertainty for many ion clocks requires computa-\ntionally intensive Monte Carlo simulations and is beyond the scope of this article, but in the following we provide a\nsimple estimate based on scaling an analysis of the uncertainty for single-ion clocks [1]. The total rate of Langevin\ncollisions between a Coulomb crystal and background gas molecules scales linearly with the number of ions and\nlinearly with the charge of the ions, so for a 1000 ion Sn2+ clock the background gas collision rate will be approxi-\nmately 2000 times larger than in single-ion clocks. As a conservative upper bound, we suppose that the systematic\nuncertainty due to background gas collisions scales linearly with the collision rate. A more detailed model would\ntake into account that only the Doppler shift component of the uncertainty scales with the total collision rate while\nthe phase shift component of the uncertainty (which dominated the collision shift uncertainty in Ref. [1]) scales with\nthe collision rate per ion. Current state-of-the-art single-ion clocks operating in room temperature ultrahigh vacuum\n(UHV) chambers with background gas pressures of roughly 10−10 torr have systematic uncertainties due to back-\ngroundgascollisionsoforder10−19 [2,3]. Althoughaccordingtothisconservativeestimatebackgroundgascollisions\nmay contribute of order 10−18 uncertainty for 1000 ion Sn2+ clocks in room temperature UHV chambers with pres-\nsuresofatbest10−12 torr,MonteCarlosimulationswilllikelybeabletoconstraintheuncertaintytobemuchsmaller.\n∗ leibrandt@physics.ucla.edu\nchambers [4]. An even more conservative upper bound can be obtained by considering the rate of collisions as follows.\nFor 10−16 torr of He background gas at 10 K, the total rate of Langevin collisions with any of the ions in a 1000\nSn2+ ion crystal is 1×10−4/s. For a 1 s probe time, this amounts to one collision every 104 probes. At most,\nthis could result in a fractional shift of the measured transition frequency of 10−4 relative to the spectroscopic\nlinewidth, which in this case is 1 Hz. Thus, the background gas collision shift can be straightforwardly upper\nbounded to be below 10−19 for Sn2+ clocks operating in cryogenic vacuum chambers. In practice, collisions that\ndepositasignificantamountofkineticenergytotheioncrystalwillnotcontributetotheclockspectroscopysignaldue\ntotheDebye-WallerreductionoftheRabifrequency[1],sotheactualshiftanditsuncertaintywillbemuchsmaller.”\nIII. REVIEWER # 2\ndear editor,\nthe authors have addressed all the questions raised by the reviewers and are documenting their assumptions with a\nlot of relevant references. I think this paper can now be published by nature communication, with a main interest of\nproposing high precision frequency measurements based on a 3D ion cloud. Nevertheless, I still have few comments\nfor the authors:\nWethankthereviewerforsupportingpublicationofourmanuscriptinNatureCommunications.\nthe authors state that:\n“To the best of our knowledge, this is also the first full-scale simulation of micro-motion for a 1000-ion clock,\nwhich has required high-performance computing resources to perform a detailed modeling of the shifts under various\nexperimental conditions.”\nEven if the purpose of the simulations were not oriented toward frequency metrology, a full-scale simulation of\nmicromotion for a 1024-ion cloud can be found in\nhttps://journals.aps.org/pra/abstract/10.1103/PhysRevA.108.013109\nand\nhttps://doi.org/10.1063/5.0046693\nthese last results were obtained with a single laser beam propagating along the trap axis. Regarding the need for\ndifferent laser beams to cool a 3D sample, I refer the authors to the work done in the group of M. Drewsen where\nmost of the time a single direction of propagation is used to cool a 3D cloud and it is along the trap axis when the\ncloudis3D.Inourgroup, wehavetriedtocoolalarge3Dcloudwithasinglebeamhavingprojectionalongtheaxial\nand radial trap axis and it was a fiasco. Everything went smooth when we moved the beam along the trap axis. So\nI think this laser beam propagation direction is still an open question and it is a very good thing if the authors try\ndifferent spatial configurations in their set-up. They will bring a very useful knowledge to the community\ninterested in using large 3D ion cloud.\nCaroline Champenois, CNRS-AMU, France\nPlease accept our apologies for not being aware of the simulations of large ion crystals including micromotion in\nRefs.[5,6]. Wehaveaddedcitationstothesepapersinthissentenceoftherevisedmanuscript: “Second,wecalculate\ntheamplitudeanddirectionofthemicromotionofeachionandverifythestabilityoftheioncrystalbytimeevolving\nthefullequationsofmotion(i.e.,withouttakingthepseudopotentialapproximation,similartoRefs.[5,6])”(line361).\n(Note that our statement that “To the best of our knowledge, this is also the first full-scale simulation of micro-\nmotion for a 1000-ion clock” was in our previous referee response letter but not in the manuscript itself, so\nthere was no need to correct the manuscript for this statement.)\nWe also thank the reviewer for their suggestion to try different laser beam propagation directions experimentally for\nlaser cooling, and for pointing out the potential impact of this on the broader 3D ion cloud experimental community.\nIndeed, these are complex many-body systems, and experimental data is critical for advancing theoretical under-\nstanding as well.\n3\n[1] A. M. Hankin, E. R. Clements, Y. Huang, S. M. Brewer, J.-S. Chen, C. W. Chou, D. B. Hume, and D. R. Leibrandt,\nSystematic uncertainty due to background-gas collisions in trapped-ion optical clocks, Phys. Rev. A 100, 033419 (2019).\n[2] N. Huntemann, C. Sanner, B. Lipphardt, C. Tamm, and E. Peik, Single-ion atomic clock with 3 × 10−18 systematic\nuncertainty, Phys. Rev. Lett. 116, 063001 (2016).\n[3] S. Brewer, J.-S. Chen, A. Hankin, E. Clements, C. Chou, D. Wineland, D. Hume, and D. Leibrandt, 27Al+ quantum-logic\nclock with a systematic uncertainty below 10−18, Phys. Rev. Lett. 123, 033201 (2019).\n[4] G. Gabrielse, X. Fei, L. A. Orozco, R. L. Tjoelker, J. Haas, H. Kalinowsky, T. A. Trainor, and W. Kells, Thousandfold\nimprovement in the measured antiproton mass, Phys. Rev. Lett. 65, 1317 (1990).\n[5] A.Poindron,J.Pedregosa-Gutierrez,C.Jouvet,M.Knoop,andC.Champenois,Non-destructivedetectionoflargemolecules\nwithout mass limitation, J. Chem. Phys. 154, 184203 (2021).\n[6] A. Poindron, J. Pedregosa-Gutierrez, and C. Champenois, Thermal bistability in laser-cooled trapped ions, Phys. Rev. A\n108, 013109 (2023)."}, {"round": 3, "reviewer_comments": {"reviewer_1": "I have read the answers to the questions, and the authors answered in great detail. Now the\nmanuscript is acceptable. This Sn2+ based many-ion optical atomic clock will be of great interest as a\nnew promising optical clock candidate with sub-E-19 inaccuracy.", "reviewer_2": "dear editor,\nafter this second revision, I think the paper is ready for publication,\nbest regards"}, "author_response": ""}]

Decision letter