Frequency comparison of two high-accuracy Al+ optical clocks

Noise characterization of an ultra-stable laser for optical clocks

We report on the development and performance evaluation of an ultra-stable laser for an 27Al+ optical clock. After a series of noise suppressions, especially the vibrational and temperature fluctuation noise, the 30 cm long cavity stabilized laser obtains a frequency instability of 1.3 × 10-16 @1 s. This result is predicted by noise summation and confirmed by the three-cornered hat method. The 27Al+ optical clock transition is also used to characterize the laser frequency noise, and consistent results are yielded. This is the first reported instance of using single ion optical clocks to measure the frequency noise of ultra-stable lasers, as far as we know. With the implementation of the ultra-stable clock laser, an ultra-narrow linewidth clock transition of 2.8 Hz is obtained.

176Lu+ clock comparison at the 10-18 level via correlation spectroscopy

The extreme precision of optical atomic clocks has led to an anticipated redefinition of the second by the International System of Units. Furthermore, accuracies pushing the boundary of 1 part in 10¹⁸ and beyond will enable new applications, such as in geodesy and tests of fundamental physics. The ¹S₀ to ³D₁ optical transition in ¹⁷⁶Lu⁺ has exceptionally low sensitivity to external perturbations, making it suitable for practical clock implementations with inaccuracy at or below 10^-18. Here, we perform high-accuracy comparisons between two ¹⁷⁶Lu⁺ references using correlation spectroscopy. A comparison at different magnetic fields is used to obtain a quadratic Zeeman coefficient of -4.89264(88) Hz/mT for the reference frequency. With a subsequent comparison at low field, we demonstrate agreement at the low 10^-18 level, statistically limited by the averaging time of 42 hours. The evaluated uncertainty in the frequency difference is 9 × 10^-19 and the lowest reported in comparing independent optical references.

Observation of an Inner-Shell Orbital Clock Transition in Neutral Ytterbium Atoms

We observe a weakly allowed optical transition of atomic ytterbium from the ground state to the metastable state 4f^{13}5d6s^{2} (J=2) for all five bosonic and two fermionic isotopes with resolved Zeeman and hyperfine structures. This inner-shell orbital transition has been proposed as a new frequency standard as well as a quantum sensor for new physics. We find magic wavelengths through the measurement of the scalar and tensor polarizabilities and reveal that the measured trap lifetime in a three-dimensional optical lattice is 1.9(1) s, which is crucial for precision measurements. We also determine the g factor by an interleaved measurement, consistent with our relativistic atomic calculation. This work opens the possibility of an optical lattice clock with improved stability and accuracy as well as novel approaches for physics beyond the standard model.

All-fiber-device-coupled compact, transportable ultra-stable laser

In response to the demand for operation in non-laboratory environments, there has been a trend toward the development of compact, transportable ultra-stable lasers. This paper reports on this sort of laser system assembled in a cabinet. The whole optical part utilizes fiber-coupled devices to simplify the integration. In addition, spatial beam collimation and alignment into the high-finesse cavity are realized by a five-axis positioner and a focus-adjustable fiber collimator, which significantly relax the alignment and adjustment. A theoretical analysis is performed on how the collimator adjusts the beam profile and coupling efficiency. The support structure of the system is specially designed as well so that it features robustness and transportation without performance degradation. The observed linewidth is 1.4 Hz within a duration of 1 s. After subtracting the linear drift of 70 mHz/s, the fractional frequency instability is better than 4 × 10^-15, for the averaging time ranging from 1 to 100 s, which is close to the thermal noise limit of the high-finesse cavity.

Phase-stabilized UV light at 267 nm through twofold second harmonic generation

Providing phase stable laser light is important to extend the interrogation time of optical clocks towards many seconds and thus achieve small statistical uncertainties. We report a laser system providing more than 50 µW phase-stabilized UV light at 267.4 nm for an aluminium ion optical clock. The light is generated by frequency-quadrupling a fibre laser at 1069.6 nm in two cascaded non-linear crystals, both in single-pass configuration. In the first stage, a 10 mm long PPLN waveguide crystal converts 1 W fundamental light to more than 0.2 W at 534.8 nm. In the following 50 mm long DKDP crystal, more than 50 µW of light at 267.4 nm are generated. An upper limit for the passive short-term phase stability has been measured by a beat-node measurement with an existing phase-stabilized quadrupling system employing the same source laser. The resulting fractional frequency instability of less than 5×10^-17 after 1 s supports lifetime-limited probing of the ²⁷Al⁺ clock transition, given a sufficiently stable laser source. A further improved stability of the fourth harmonic light is expected through interferometric path length stabilisation of the pump light by back-reflecting it through the entire setup and correcting for frequency deviations. The in-loop error signal indicates an electronically limited instability of 1 × 10^-18 at 1 s.

Magnetic Sublevel Independent Magic and Tune-Out Wavelengths of the Alkaline-Earth Ions

Light shift in a state due to the applied laser in an atomic system vanishes at tune-out wavelengths (λTs). Similarly, differential light shift in a transition vanishes at the magic wavelengths (λmagics). In many of the earlier studies, values of the electric dipole (E1) matrix elements were inferred precisely by combining measurements and calculations of λmagic. Similarly, the λT values of an atomic state can be used to infer the E1 matrix element, as it involves dynamic electric dipole (α) values of only one state whereas the λmagic values require evaluation of α values for two states. However, both the λmagic and λT values depend on angular momenta and their magnetic components (M) of states. Here, we report the λmagic and λT values of many S1/2 and D3/2,5/2 states, and transitions among these states of the Mg+, Ca+, Sr+ and Ba+ ions that are independent of M values. It is possible to infer a large number of E1 matrix elements of the above ions accurately by measuring these values and combining with our calculations.

High-Precision Time-Frequency Signal Simultaneous Transfer System via a WDM-Based Fiber Link

In this paper, we demonstrate a wavelength division multiplexing (WDM)-based system for simultaneously delivering ultra-stable optical frequency reference, 10 GHz microwave frequency reference, and a one pulse per second (1 PPS) time signal via a 50 km fiber network. For each signal, a unique noise cancellation technique is used to maintain their precision. After being compensated, the transfer frequency instability in terms of the overlapping Allan deviation (OADEV) for the optical frequency achieves 2 × 10−17/s and scales down to 2 × 10−20/10,000 s, which for the 10 GHz microwave reference, approaches 4 × 10−15/s and decreases to 1.4 × 10−17/10,000 s, and the time uncertainty of the 1 PPS time signal along the system is 2.08 ps. In this scheme, specific channels of WDM are, respectively, occupied for different signals to avoid the possible crosstalk interference effect between the transmitted reference signals. To estimate the performance of the above scheme, which is also demonstrated in this 50 km link independent of these signals, the results are similar to that in the case of simultaneous delivery. This work shows that the WDM-based system is a promising method for building a nationwide time and frequency fiber transfer system with a communication optical network.

Identity Test of Single NV^{-} Centers in Diamond at Hz-Precision Level

Atomiclike defects in solids are not considered to be identical owing to the imperfections of host lattice. Here, we found that even under ambient conditions, negatively charged nitrogen-vacancy (NV^{-}) centers in diamond could still manifest identical at Hz-precision level, corresponding to a 10^{-7}-level relative precision, while the lattice strain can destroy the identity by tens of Hz. All parameters involved in the NV^{-}-^{14}N Hamiltonian are determined by formulating six nuclear frequencies at 10-mHz-level precision and measuring them at Hz-level precision. The most precisely measured parameter, the ^{14}N quadrupole coupling P, is given by -494 575 4.9(8)  Hz, whose precision is improved by nearly 4 orders of magnitude compared with previous measurements. We offer an approach for performing precision measurements in solids and deepening our understandings of NV centers as well as other solid-state defects. Besides, these high-precision results imply a potential application of a robust and integrated atomiclike clock based on ensemble NV centers.

Radial Two-Dimensional Ion Crystals in a Linear Paul Trap

We experimentally study two-dimensional (2D) Coulomb crystals in the "radial-2D" phase of a linear Paul trap. This phase is identified by a 2D ion lattice aligned entirely with the radial plane and is created by imposing a large ratio of axial to radial trapping potentials. Using arrays of up to 19 ^{171}Yb^{+} ions, we demonstrate that the structural phase boundaries of such crystals are well described by the pseudopotential approximation, despite the time-dependent ion positions driven by intrinsic micromotion. We further observe that micromotion-induced heating of the radial-2D crystal is confined to the radial plane. Finally, we verify that the transverse motional modes, which are used in most ion-trap quantum simulation schemes, are well-predictable numerically and remain decoupled and cold in this geometry. Our results establish radial-2D ion crystals as a robust experimental platform for realizing a variety of theoretical proposals in quantum simulation and computation.

Enhancement of UV-visible transmission characteristics in wet-etched hollow core anti-resonant fibers

We report on the feasibility of short-wavelength transmission window modification in anti-resonant hollow core fibers using post-processing by hydrofluoric (HF) acid etching. Direct drawing of stacked anti-resonant hollow core fibers with sub-micron thin cladding capillary membranes is technologically challenging, but so far this has been the only proven method of assuring over an octave-spanning transmission windows across the visible and UV wavelengths. In this study we revealed that low HF concentration allows us to reduce the thickness of the cladding capillary membranes from the initial 760 nm down to 180 nm in a controlled process. The glass etching rates have been established for different HF concentrations within a range non-destructive to the anti-resonant cladding structure. Etching resulted in spectral blue-shifting and broadening of anti-resonant transmission windows in all tested fiber samples with lengths between 15 cm and 75 cm. Spectrally continuous transmission, extending from around 200 nm to 650 nm was recorded in 75 cm long fibers with cladding membranes etched down to thickness of 180 nm. The experiment allowed us to verify the applicability and feasibility of controlling a silica fiber post-processing technique, aimed at broadening of anti-resonant transmission windows in hollow core fibers. A practical application of the processed fiber samples is demonstrated with their simple butt-coupling to light-emitting diodes centered at various ultraviolet wavelengths between 265 nm and 365 nm.

Development of a deep-ultraviolet pulse laser source operating at 234 nm for direct cooling of Al+ ion clocks

We report on the development of a 250-MHz 234 nm deep-ultraviolet pulse source based on a flexible wavelength-conversion scheme. The scheme is based on a frequency-doubled optical parametric oscillator (FD-OPO) together with a cascaded frequency conversion process. We use a χ⁽²⁾ nonlinear envelope equation to guide the design of an intra-cavity OPO crystal, demonstrating a flexible broadband tunable feature and providing as high as watt-level of a frequency-doubled signal output centered at 850 nm, which is served as an input wave for the cascaded frequency conversion process. As much as 3.0 mW of an average power at 234 nm is obtained, with an rms power stability of better than 1% over 20 minutes. This deep-ultraviolet pulse laser source can be used for many applications in quantum optics and for direct laser cooling of Al⁺ ion clocks.

Ultra-stable 1064-nm neodymium-doped yttrium aluminum garnet lasers with 2.5 × 10-16 frequency instability

Cavity-stabilized ultra-stable optical oscillators are one of the core ingredients in the ground-based or spaceborne precision measurements such as optical frequency metrology, test of special relativity, and gravitational wave observation. We report in detail the development of two ultra-stable systems based on 1064-nm neodymium-doped yttrium aluminum garnet lasers and 20-cm optical cavities. The optical cavities adopt ultra-low-loss silica mirrors with compensating rings. An electro-optic crystal with a wedged angle is used to reduce the residual amplitude modulation. Using two-stage thermal control, long-term stabilities of 100 µK are achieved for the outer wall of the vacuum chamber housing the optical cavity. Two additional thermal shields increased the time constant of the optical cavities to 70 h. By operating the optical cavity at the temperature of zero coefficient of thermal expansion, the frequency stability reaches 2.5 × 10^-16 at 10 s averaging time and remains below 5 × 10^-16 with an extended time of 1000 s after removing the first- and second-order drifts. The dependence of the laser linewidth on the measurement time is tested against a simplified theoretical model.

Half-minute-scale atomic coherence and high relative stability in a tweezer clock

The preparation of large, low-entropy, highly coherent ensembles of identical quantum systems is fundamental for many studies in quantum metrology¹, simulation² and information³. However, the simultaneous realization of these properties remains a central challenge in quantum science across atomic and condensed-matter systems^2,4-7. Here we leverage the favourable properties of tweezer-trapped alkaline-earth (strontium-88) atoms^8-10, and introduce a hybrid approach to tailoring optical potentials that balances scalability, high-fidelity state preparation, site-resolved readout and preservation of atomic coherence. With this approach, we achieve trapping and optical-clock excited-state lifetimes exceeding 40 seconds in ensembles of approximately 150 atoms. This leads to half-minute-scale atomic coherence on an optical-clock transition, corresponding to quality factors well in excess of 10¹⁶. These coherence times and atom numbers reduce the effect of quantum projection noise to a level that is comparable with that of leading atomic systems, which use optical lattices to interrogate many thousands of atoms in parallel^11,12. The result is a relative fractional frequency stability of 5.2(3) × 10^-17τ^-1/2 (where τ is the averaging time in seconds) for synchronous clock comparisons between sub-ensembles within the tweezer array. When further combined with the microscopic control and readout that are available in this system, these results pave the way towards long-lived engineered entanglement on an optical-clock transition¹³ in tailored atom arrays.

Direct measurement of the frequency ratio for Hg and Yb optical lattice clocks and closure of the Hg/Yb/Sr loop

We performed the first direct measurement of the frequency ratio between a mercury (¹⁹⁹Hg) and an ytterbium (¹⁷¹Yb) optical lattice clock to find ν_Hg/ν_Yb = 2.177 473 194 134 565 07(19) with the fractional uncertainty of 8.8 × 10^-17. The ratio is in excellent agreement with expectations from the ratios ν_Hg/ν_Sr and ν_Yb/ν_Sr obtained previously in comparisons against a strontium (⁸⁷Sr) optical lattice clock. The completed closure (ν_Hg/ν_Yb)(ν_Yb/ν_Sr)(ν_Sr/ν_Hg) - 1 = 0.4(1.3) × 10^-16 tests the frequency reproducibility of the optical lattice clocks beyond what is achievable in comparison against the current realization of the second in the International System of Units (SI).

Optical frequency distribution using laser repeater stations with planar lightwave circuits

We report a cascaded optical fiber link which connects laboratories in RIKEN, the University of Tokyo, and NTT within a 100-km region using a transfer light at 1397 nm, a subharmonic of the Sr clock frequency. The multiple cascaded link employing several laser repeater stations benefits from a wide feedback bandwidth for fiber noise compensation, which allows constructing optical lattice clock networks based on the master-slave configuration. We developed the laser repeater stations based on planar lightwave circuits to significantly reduce the interferometer noise for improved link stability. We implemented a 240-km-long cascaded link in a UTokyo-NTT-UTokyo loop using light sent from RIKEN via a 30-km-long link. In environments with large fiber noise, the link instability is 3 × 10^-16 at an averaging time of 1 s and reaches 1 × 10^-18 at 2,600 s.

Hyperfine Averaging by Dynamic Decoupling in a Multi-Ion Lutetium Clock

We propose and experimentally demonstrate a scheme that realizes hyperfine averaging during a Ramsey interrogation of a clock transition. The method eliminates the need to average over multiple optical transitions, reduces the sensitivity of the clock to its environment, and reduces inhomogeneous broadening in a multi-ion clock. The method is compatible with autobalanced Ramsey spectroscopy, which facilitates the elimination of residual shifts due to imperfect implementation and ac stark shifts from the optical probe. We demonstrate the scheme using correlation spectroscopy of the ^{1}S_{0}↔^{3}D_{1} clock transition in a three-ion Lu^{+} clock. From the demonstration we are able to provide a measurement of the ^{3}D_{1} quadrupole moment, Θ(^{3}D_{1})=0.634(9)ea_{0}^{2}.

Strong Coupling of a Single Ion to an Optical Cavity

Strong coupling between an atom and an electromagnetic resonator is an important condition in cavity quantum electrodynamics. While strong coupling in various physical systems has been achieved so far, it remained elusive for single atomic ions. Here, we achieve a coupling strength of 2π×(12.3±0.1)  MHz between a single ^{40}Ca^{+} ion and an optical cavity, exceeding both atomic and cavity decay rates which are 2π×11.5 and 2π×(4.1±0.1)  MHz, respectively. We use cavity assisted Raman spectroscopy to precisely characterize the ion-cavity coupling strength and observe a spectrum featuring the normal mode splitting in the cavity transmission due to the ion-cavity interaction. Our work paves the way towards new applications of cavity quantum electrodynamics utilizing single trapped ions in the strong coupling regime for quantum optics and quantum technologies.

Quantum Logic Spectroscopy with Ions in Thermal Motion

A mixed-species geometric phase gate has been proposed for implementing quantum logic spectroscopy on trapped ions, which combines probe and information transfer from the spectroscopy to the logic ion in a single pulse. We experimentally realize this method, show how it can be applied as a technique for identifying transitions in currently intractable atoms or molecules, demonstrate its reduced temperature sensitivity, and observe quantum-enhanced frequency sensitivity when it is applied to multi-ion chains. Potential applications include improved readout of trapped-ion clocks and simplified error syndrome measurements for quantum error correction.

Detection of the 5p - 4f orbital crossing and its optical clock transition in Pr9

Recent theoretical works have proposed atomic clocks based on narrow optical transitions in highly charged ions. The most interesting candidates for searches of physics beyond the Standard Model are those which occur at rare orbital crossings where the shell structure of the periodic table is reordered. There are only three such crossings expected to be accessible in highly charged ions, and hitherto none have been observed as both experiment and theory have proven difficult. In this work we observe an orbital crossing in a system chosen to be tractable from both sides: Pr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^{9+}$$\end{document}9+. We present electron beam ion trap measurements of its spectra, including the inter-configuration lines that reveal the sought-after crossing. With state-of-the-art calculations we show that the proposed nHz-wide clock line has a very high sensitivity to variation of the fine-structure constant, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}α, and violation of local Lorentz invariance; and has extremely low sensitivity to external perturbations.

Atomic clocks are based on the frequency of optical transitions and offer high precision. Here the authors demonstrate a configuration crossing in the highly charged ion praseodymium (Pr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^{9+}$$\end{document}9+) and determine the frequency of a potential reference transition for a highly charged ion clock.

Comparison of VLBI and GNSS common view for time transfer

With the rapid development of optical clock, the stability and system uncertainty of optical clocks has reached a 1.0e–18 level. Optical clocks will likely constitute the next generation of time-frequency standards for redefining the SI second. Because time and frequency transfer services that rely on satellite systems are not always reliable and currently available technologies are insufficient for comparing the next generation of frequency standards, high-precision time and transfer techniques are strongly desired. Very Long Baseline Interferometry (VLBI) is one of the space geodetic techniques that measure the arrival time delays between multiple stations utilizing radio signals from distant celestial radio sources. Not only can VLBI obtain the angle position measurement of the radio source with sub-millisecond accuracy and the station coordinate measurement with millimeter accuracy, but also, it can provide high-precision information regarding inter-station atomic clock differences. Therefore, it is theoretically feasible to use the VLBI technology to do the remote time transfer. Because of this characteristic of VLBI technology, VLBI has significant application potential in the field of remote time transfer. To confirm the suitability of VLBI to time-frequency transfer for future practical applications, the results of VLBI and GPS common view time transfer were compared using a Kunming-Urumqi baseline. The performance characteristics of time transfer based on VLBI are then analyzed. Experimental results show that VLBI technology can accurately measure the variation of clock differences between stations as same as the GPS common view time comparison technology. It briefly describes the challenges of future VLBI technology for practical applications of time transfer.

Ground-to-satellite time and frequency synchronization link with active carrier phase compensation

In this paper, a synchronization link between one ground station and one geostationary satellite is established. The ground station receives retransmitted signals from the satellite, measures phase delay along the propagation route, and actively compensates back to its sending signals, realizing real-time phase fluctuation compensation. The transmitted signal contains two frequencies to eliminate common-mode phase noise. The difference between their carrier phase delays is measured. Different modes of carrier phase variation are separated and compensated, achieving a remaining time jitter of ±200 ps. Major sources of error are analyzed, and potential methods for improvement are discussed. The proposed ground-to-satellite link and active compensation method has potential applications in frequency standard dissemination to remote receivers (including ground stations or satellites). These potential applications justify further study of this system.

Highly Polarizable Rydberg Ion in a Paul Trap

Usually the influence of the quadratic Stark effect on an ion's trapping potential is minuscule and only needs to be considered in atomic clock experiments. In this work we excite a trapped ion to a Rydberg state with polarizability ∼8 orders of magnitude higher than a low-lying electronic state; we find that the highly polarizable ion experiences a vastly different trapping potential owing to the Stark effect. We observe changes in trap stiffness, equilibrium position, and minimum potential, which can be tuned using the trapping electric fields. These effects lie at the heart of several proposed studies, including a high-fidelity submicrosecond entangling operation; in addition we demonstrate these effects may be used to minimize ion micromotion. Mitigation of Stark effects is important for coherent control of Rydberg ions; we illustrate this by carrying out the first Rabi oscillations between a low-lying electronic state and a Rydberg state of an ion.

Narrow-line Cooling and Determination of the Magic Wavelength of Cd

We experimentally and theoretically determine the magic wavelength of the (5s^{2})^{1}S_{0}-(5s5p)^{3}P_{0} clock transition of ^{111}Cd to be 419.88(14) and 420.1(7) nm. To perform Lamb-Dicke spectroscopy of the clock transition, we use narrow-line laser cooling on the ^{1}S_{0}-^{3}P_{1} transition to cool the atoms to 6  μK and load them into an optical lattice. Cadmium is an attractive candidate for optical lattice clocks because it has a small sensitivity to blackbody radiation and its efficient narrow-line cooling mitigates higher order light shifts. We calculate the blackbody shift, including the dynamic correction, to be fractionally 2.83(8)×10^{-16} at 300 K, an order of magnitude smaller than that of Sr and Yb. We also report calculations of the Cd ^{1}P_{1} lifetime and the ground state C_{6} coefficient.

Seconds-scale coherence on an optical clock transition in a tweezer array

Coherent control of high-quality factor optical transitions in atoms has revolutionized precision frequency metrology. Leading optical atomic clocks rely on the interrogation of such transitions in either single ions or ensembles of neutral atoms to stabilize a laser frequency at high precision and accuracy. We demonstrate a platform that combines the key strengths of these two approaches, based on arrays of individual strontium atoms held within optical tweezers. We report coherence times of 3.4 seconds, single-ensemble duty cycles up to 96% through repeated interrogation, and frequency stability of 4.7 × 10^-16 (τ/s)^-1/2 These results establish optical tweezer arrays as a powerful tool for coherent control of optical transitions for metrology and quantum information science.

Systematic uncertainty due to background-gas collisions in trapped-ion optical clocks

We describe a framework for calculating the frequency shift and uncertainty of trapped-ion optical atomic clocks caused by background-gas collisions, and apply this framework to an ²⁷Al⁺ clock to enable a total fractional systematic uncertainty below 10^-18. For this clock, with 38(19) nPa of room-temperature H₂ background gas, we find that collisional heating generates a non-thermal distribution of motional states with a mean time-dilation shift of order 10^-16 at the end of a 150 ms probe, which is not detected by sideband thermometry energy measurements. However, the contribution of collisional heating to the spectroscopy signal is highly suppressed and we calculate the BGC shift to be -0.6(2.4) × 10^-19, where the shift is due to collisional heating time dilation and the uncertainty is dominated by the worst case ±π/2 bound used for collisional phase shift of the ²⁷Al⁺ superposition state. We experimentally validate the framework and determine the background-gas pressure in situ using measurements of the rate of collisions that cause reordering of mixed-species ion pairs.

Suppressing Inhomogeneous Broadening in a Lutetium Multi-ion Optical Clock

We demonstrate precision measurement and control of inhomogeneous broadening in a multi-ion clock consisting of three ^{176}Lu^{+} ions. Microwave spectroscopy between hyperfine states in the ^{3}D_{1} level is used to characterize differential systematic shifts between ions, most notably those associated with the electric quadrupole moment. By appropriate alignment of the magnetic field, we demonstrate suppression of these effects to the ∼10^{-17} level relative to the ^{1}S_{0}↔^{3}D_{1} optical transition frequency. Correlation spectroscopy on the optical transition demonstrates the feasibility of a 10-s Ramsey interrogation in the three ion configuration with a corresponding projection noise limited stability of σ(τ)=8.2×10^{-17}/sqrt[τ].

Realization of phase locking in good-bad-cavity active optical clock

The residual cavity-pulling effect limits further narrowing of linewidth in dual-wavelength (DW) good-bad-cavity active optical clocks (AOCs). In this paper, we for the first time experimentally realize the cavity-length stabilization of the 1064/1470 nm DW-AOCs by utilizing the phase locking technique of two independent 1064 nm good-cavity lasers. The frequency tracking accuracy between the two main-cavities of DW-AOCs is better than 3 × 10^-16 at 1 s, and can reach 1 × 10^-17 at 1000 s. Each 1470 nm bad-cavity laser achieves a most probable linewidth of 53 Hz, which is about a quarter of that without phase locking. The influence of the asynchronous cavity-lengths variation between two DW laser systems is suppressed.

^{27}Al^{+} Quantum-Logic Clock with a Systematic Uncertainty below 10^{-18}

We describe an optical atomic clock based on quantum-logic spectroscopy of the ^{1}S_{0}↔^{3}P_{0} transition in ^{27}Al^{+} with a systematic uncertainty of 9.4×10^{-19} and a frequency stability of 1.2×10^{-15}/sqrt[τ]. A ^{25}Mg^{+} ion is simultaneously trapped with the ^{27}Al^{+} ion and used for sympathetic cooling and state readout. Improvements in a new trap have led to reduced secular motion heating, compared to previous ^{27}Al^{+} clocks, enabling clock operation with ion secular motion near the three-dimensional ground state. Operating the clock with a lower trap drive frequency has reduced excess micromotion compared to previous ^{27}Al^{+} clocks. Both of these improvements have led to a reduced time-dilation shift uncertainty. Other systematic uncertainties including those due to blackbody radiation and the second-order Zeeman effect have also been reduced.

Towards a transportable aluminium ion quantum logic optical clock

With the advent of optical clocks featuring fractional frequency uncertainties on the order of 10^-17 and below, new applications such as chronometric leveling with few-centimeter height resolution emerge. We are developing a transportable optical clock based on a single trapped aluminum ion, which is interrogated via quantum logic spectroscopy. We employ singly charged calcium as the logic ion for sympathetic cooling, state preparation, and readout. Here, we present a simple and compact physics and laser package for manipulation of ⁴⁰Ca⁺. Important features are a segmented multilayer trap with separate loading and probing zones, a compact titanium vacuum chamber, a near-diffraction-limited imaging system with high numerical aperture based on a single biaspheric lens, and an all-in-fiber ⁴⁰Ca⁺ repump laser system. We present preliminary estimates of the trap-induced frequency shifts on ²⁷Al⁺, derived from measurements with a single calcium ion. The micromotion-induced second-order Doppler shift for ²⁷Al⁺ has been determined to be δν_EMMν=-0.4_-0.3 ^+0.4×10^-18 and the black-body radiation shift is δν_BBR/ν = (-4.0 ± 0.4) × 10^-18. Moreover, heating rates of 30 (7) quanta per second at trap frequencies of ω_rad,Ca+ ≈ 2π × 2.5 MHz (ω_ax,Ca+ ≈ 2π × 1.5 MHz) in radial (axial) direction have been measured, enabling interrogation times of a few hundreds of milliseconds.

Inner-shell clock transition in atomic thulium with a small blackbody radiation shift

One of the key systematic effects limiting the performance of state-of-the-art optical clocks is the blackbody radiation (BBR) shift. Here, we demonstrate unusually low sensitivity of a 1.14 μm inner-shell clock transition in neutral Tm atoms to BBR. By direct polarizability measurements, we infer a differential polarizability of the clock levels of -0.063(30) atomic units corresponding to a fractional frequency BBR shift of only 2.3(1.1) × 10^-18 at room temperature. This amount is several orders of magnitude smaller than that of the best optical clocks using neutral atoms (Sr, Yb, Hg) and is competitive with that of ion optical clocks (Al⁺, Lu⁺). Our results allow the development of lanthanide-based optical clocks with a relative uncertainty at the 10^-17 level.

Fast quantum control in dissipative systems using dissipationless solutions

We report on a systematic geometric procedure, built up on solutions designed in the absence of dissipation, to mitigate the effects of dissipation in the control of open quantum systems. Our method addresses a standard class of open quantum systems that encompasses non-Hermitian Hamiltonians. It provides the analytical expression of the extra magnetic field to be superimposed to the driving field in order to compensate the geometric distortion induced by dissipation for spin systems, and produces an exact geometric optimization of fast population transfer. Interestingly, it also preserves the robustness properties of protocols originally optimized against noise. Its extension to two interacting spins restores a fidelity close to unity for the fast generation of Bell state in the presence of dissipation.

Development of 8-branch Er:fiber frequency comb for Sr and Yb optical lattice clocks

We demonstrate an 8-branch Er:fiber frequency comb with seven application ports, which can be individually optimized for applications with different wavelengths. The beat between the comb and a cw laser has a signal-to-noise ratio exceeding 30 dB at a resolution bandwidth of 300 kHz. The 8-branch frequency comb is used to perform frequency locking for four repumping and lattice lasers, and the frequency measurement of two clock lasers of strontium and ytterbium optical lattice clocks. We have achieved reliable optical lattice clock operation, thanks to the stable frequency locking and measurement obtained by using the 8-branch frequency comb. The developed frequency comb is a powerful experimental tool for various applications, including not only optical lattice clocks, but also research on quantum optics that use many frequency-stabilized lasers.

Improvement of the intensity noise and frequency stabilization of Nd:YAP laser with an ultra-low expansion Fabry-Perot cavity

Continuous-wave, single-frequency, solid-state lasers with long-term frequency stability and low-intensity noise are an essential resource to generate squeezed and entangled states of light. In order to obtain the stable, nonclassical states of light, the frequency of the laser has to be stabilized with a stable reference. Due to the zero expansion property at a certain temperature, an ultra-low expansion (ULE) Fabry-Perot (F-P) cavity with a high finesse can be used as one of the best candidates of the frequency reference. We perform a detailed analysis of an extraordinarily high-frequency stability and ultra-low-intensity noise laser based on an improved cascade Pound-Drever-Hall frequency stabilization to a ULE F-P cavity. The frequency drift of the laser is suppressed to 7.72 MHz in 4 hours, and the noise level of the laser is simultaneously reduced to the quantum noise limit in the frequency below 300 kHz, which provides the possibility for the direct generation of a stable, high-level squeezed state in a lower-frequency region.

A compact and robust cooling laser system for an optical strontium lattice clock

We present a simple and robust laser system for two-color, narrow-line cooling on the ⁸⁷Sr (5s²)¹S₀ → (5s5p)³P₁ transition. Two hyperfine lines of this transition are addressed simultaneously with light from a single laser source, using sidebands created by an electro-optical phase modulator. A tapered amplifier system provides laser powers up to 90 mW. We show that amplification does not affect the phase modulation of the laser. This compact and robust laser system offers excellent reliability. Therefore, it is especially well suited for transportable and spaceborne optical clocks. The design offers the potential to be miniaturized into a fully integrated package.

Suppressing residual amplitude modulation to the 10-7 level in optical phase modulation

Residual amplitude modulation is one of the major stability-degrading factors in many precision measurements. Using an electro-optic (EO) crystal with wedged input and output surfaces is an effective way to suppress residual amplitude modulation. Here the mechanism of residual amplitude modulation in this approach is investigated. The residual amplitude modulations measured in standard and wedged EO crystals bear similarities in their temperature and polarization dependences, implying that a mixture of the two orthogonal polarizations in the extraordinary light is responsible for the residual amplitude modulation in the wedged EO crystal. Similar to a standard EO crystal, a non-uniform spatial distribution of residual amplitude modulation is also observed in the extraordinary light emerging from the wedged EO crystal. The optical isolator after the EO crystal is replaced by a Faraday rotator, and an improvement in the long-term stability is observed. With the wedged-crystal approach, residual amplitude modulation as low as 2×10^-7 is observed, contributing a frequency instability of 8×10^-18 (500 s) in Pound-Drever-Hall frequency stabilization with a discrimination slope of 1×10^-4  V/Hz.

State-selective coherent motional excitation as a new approach for the manipulation, spectroscopy and state-to-state chemistry of single molecular ions

We present theoretical and experimental progress towards a new approach for the precision spectroscopy, coherent manipulation and state-to-state chemistry of single isolated molecular ions in the gas phase. Our method uses a molecular beam for creating packets of rotationally cold neutrals from which a single molecule is state-selectively ionized and trapped inside a radiofrequency ion trap. In addition to the molecular ion, a single co-trapped atomic ion is used to cool the molecular external degrees of freedom to the ground state of the trap and to detect the molecular state using state-selective coherent motional excitation from a modulated optical-dipole force acting on the molecule. We present a detailed discussion and theoretical characterization of the present approach. We simulate the molecular signal experimentally using a single atomic ion, indicating that different rovibronic molecular states can be resolved and individually detected with our method. The present approach for the coherent control and non-destructive detection of the quantum state of a single molecular ion opens up new perspectives for precision spectroscopies relevant for, e.g., tests of fundamental physical theories and the development of new types of clocks based on molecular vibrational transitions. It will also enable the observation and control of chemical reactions of single particles on the quantum level. While focusing on N2+ as a prototypical example in the present work, our method is applicable to a wide range of diatomic and polyatomic molecules.

Operational Magic Intensity for Sr Optical Lattice Clocks

We experimentally investigate the lattice-induced light shift by the electric-quadrupole (E2) and magnetic-dipole (M1) polarizabilities and the hyperpolarizability in Sr optical lattice clocks. Precise control of the axial as well as the radial motion of atoms in a one-dimensional lattice allows observing the E2-M1 polarizability difference. Measured polarizabilities determine an operational lattice depth to be 72(2)E_{R}, where the total light shift cancels to the 10^{-19} level, over a lattice-intensity variation of about 30%. This operational trap depth and its allowable intensity range conveniently coincide with experimentally feasible operating conditions for Sr optical lattice clocks.

Uncertainty Evaluation of an 171Yb Optical Lattice Clock at NMIJ

We report an uncertainty evaluation of an ¹⁷¹Yb optical lattice clock with a total fractional uncertainty of 3.6×10^-16 , which is mainly limited by the lattice-induced light shift and the blackbody radiation shift. Our evaluation of the lattice-induced light shift, the density shift, and the second-order Zeeman shift is based on an interleaved measurement where we measure the frequency shift using the alternating stabilization of a clock laser to the 6s^{2 1}S₀-6s6p ³P₀ clock transition with two different experimental parameters. In the present evaluation, the uncertainties of two sensitivity coefficients for the lattice-induced hyperpolarizability shift d incorporated in a widely used light shift model by RIKEN and the second-order Zeeman shift a_Z are improved compared with the uncertainties of previous coefficients. The hyperpolarizability coefficient d is determined by investigating the trap potential depth and the light shifts at the lattice frequencies near the two-photon transitions 6s6p³P₀-6s8p³P₀, 6s8p³P₂, and 6s5f³F₂. The obtained values are d=-1.1(4) μ Hz and a_Z=-6.6(3) Hz/mT². These improved coefficients should reduce the total systematic uncertainties of Yb lattice clocks at other institutes.

Recent Advances Concerning the 87Sr Optical Lattice Clock at the National Time Service Center

We review recent experimental progress concerning the 87Sr optical lattice clock at the National Time Service Center in China. Hertz-level spectroscopy of the 87Sr clock transition for the optical lattice clock was performed, and closed-loop operation of the optical lattice clock was realized. A fractional frequency instability of 2.8 × 10−17 was attained for an averaging time of 2000 s. The Allan deviation is found to be 1.6 × 10−15/τ1/2 and is limited mainly by white-frequency-noise. The Landé g-factors of the (5s2)1S0 and (5s5p)3P0 states in 87Sr were measured experimentally; they are important for evaluating the clock’s Zeeman shifts. We also present recent work on the miniaturization of the strontium optical lattice clock for space applications.

Ultra-low noise optical injection locking amplifier with AOM-based coherent detection scheme

A novel optical injection locking amplifier with acousto-optic modulator based phase modulation and a coherent detection scheme for optical frequency transfer applications is experimentally demonstrated in this study. A commercial distributed feedback diode laser is injection-locked to the resonant frequency of the optical signal with an optical fiber path length of hundreds of kilometers. This provides approximately 59 dB gain and ensures that the input carrier frequency fractional stability can be as good as 10-20 at 1000 s. The amplifier was tested for the transfer of a commercial narrow-linewidth laser in a 180 km fiber link to a remote site with only a single amplification step. The transferred frequency at the remote end reached 10-20 at 20000 s, which is suitable for optical frequency distribution and remote comparison between optical atomic clocks.

Composite laser-pulses spectroscopy for high-accuracy optical clocks: a review of recent progress and perspectives

Probing an atomic resonance without disturbing it is an ubiquitous issue in physics. This problem is critical in high-accuracy spectroscopy or for the next generation of atomic optical clocks. Ultra-high resolution frequency metrology requires sophisticated interrogation schemes and robust protocols handling pulse length errors and residual frequency detuning offsets. This review reports recent progress and perspective in such schemes, using sequences of composite laser-pulses tailored in pulse duration, frequency and phase, inspired by NMR techniques and quantum information processing. After a short presentation of Rabi technique and NMR-like composite pulses allowing efficient compensation of electromagnetic field perturbations to achieve robust population transfers, composite laser-pulses are investigated within Ramsey's method of separated oscillating fields in order to generate non-linear compensation of probe-induced frequency shifts. Laser-pulses protocols such as hyper-Ramsey, modified hyper-Ramsey, generalized hyper-Ramsey and hybrid schemes as auto-balanced Ramsey spectroscopy are reviewed. These techniques provide excellent protection against both probe induced light-shift perturbations and laser intensity variations. More sophisticated schemes generating synthetic frequency-shifts are presented. They allow to reduce or completely eliminate imperfect correction of probe-induced frequency-shifts even in presence of decoherence due to the laser line-width. Finally, two universal protocols are presented which provide complete elimination of probe-induced frequency shifts in the general case where both decoherence and relaxation dissipation effects are present by using exact analytic expressions for phase-shifts and the clock frequency detuning. These techniques might be applied to atomic, molecular and nuclear frequency metrology, Ramsey-type mass spectrometry as well as precision spectroscopy.

Direct Observation of Atom-Ion Nonequilibrium Sympathetic Cooling

Sympathetic cooling is the process of energy exchange between a system and a colder bath. We investigate this fundamental process in an atom-ion experiment where the system is composed of a single ion trapped in a radio-frequency Paul trap and prepared in a classical oscillatory motion with total energy of ∼200  K, and the bath is an ultracold cloud of atoms at μK temperature. We directly observe the sympathetic cooling dynamics with single-shot energy measurements during one to several collisions in two distinct regimes. In one, collisions predominantly cool the system with very efficient momentum transfer leading to cooling in only a few collisions. In the other, collisions can both cool and heat the system due to nonequilibrium dynamics in the presence of the ion trap's oscillating electric fields. While the bulk of our observations agree well with a molecular-dynamics simulation of hard-sphere (Langevin) collisions, a measurement of the scattering angle distribution reveals forward-scattering (glancing) collisions which are beyond the Langevin model. This work paves the way for further nonequilibrium and collision dynamics studies using the well-controlled atom-ion system.

Octave-spanning optical frequency comb based on a laser-diode pumped Kerr-lens mode-locked Yb:KYW laser for optical frequency measurement

We developed an optical frequency comb based on a Yb:KYW laser. Soft-aperture Kerr-lens mode-locking at the cavity transverse-mode degeneration enabled us to generate 360 mW from a 750 mW pump laser diode. This resulted in spectral broadening over one octave using just a photonic crystal fiber. We achieved a free-running linewidth of 15 kHz in the carrier-envelope offset frequency by optimizing the cavity group delay dispersion, crystal position, and pump laser power, which led to a residual phase noise of 0.51 rad during phase-locking. We measured the frequency drift of a cavity-stabilized laser for a clock transition in Yb171⁺.

Atomic clocks for geodesy

We review experimental progress on optical atomic clocks and frequency transfer, and consider the prospects of using these technologies for geodetic measurements. Today, optical atomic frequency standards have reached relative frequency inaccuracies below 10^-17, opening new fields of fundamental and applied research. The dependence of atomic frequencies on the gravitational potential makes atomic clocks ideal candidates for the search for deviations in the predictions of Einstein's general relativity, tests of modern unifying theories and the development of new gravity field sensors. In this review, we introduce the concepts of optical atomic clocks and present the status of international clock development and comparison. Besides further improvement in stability and accuracy of today's best clocks, a large effort is put into increasing the reliability and technological readiness for applications outside of specialized laboratories with compact, portable devices. With relative frequency uncertainties of 10^-18, comparisons of optical frequency standards are foreseen to contribute together with satellite and terrestrial data to the precise determination of fundamental height reference systems in geodesy with a resolution at the cm-level. The long-term stability of atomic standards will deliver excellent long-term height references for geodetic measurements and for the modelling and understanding of our Earth.

Systematic evaluation of a 171Yb optical clock by synchronous comparison between two lattice systems

Optical clocks are the most precise measurement devices. Here we experimentally characterize one such clock based on the ¹S₀-³P₀ transition of neutral ¹⁷¹Yb atoms confined in an optical lattice. Given that the systematic evaluation using an interleaved stabilization scheme is unable to avoid noise from the clock laser, synchronous comparisons against a second ¹⁷¹Yb lattice system were implemented to accelerate the evaluation. The fractional instability of one clock falls below 4 × 10⁻¹⁷ after an averaging over a time of 5,000 seconds. The systematic frequency shifts were corrected with a total uncertainty of 1.7 × 10⁻¹⁶. The lattice polarizability shift currently contributes the largest source. This work paves the way to measuring the absolute clock transition frequency relative to the primary Cs standard or against the International System of Units (SI) second.