Aided by ultra-high resolution spectroscopy, the overall systematic uncertainty of the $^{1}S_{0}$-$^{3}P_{0}$ clock resonance for lattice-confined $^{87}$Sr has been characterized to $9\times10^{-16}$. This uncertainty is at a level similar to the Cs-fountain primary standard, while the potential stability for the lattice clocks exceeds that of Cs. The absolute frequency of the clock transition has been measured to be 429,228,004,229,874.0(1.1) Hz, where the $2.5\times10^{-15}$ fractional uncertainty represents the most accurate measurement of a neutral-atom-based optical transition frequency to date.
We have measured the absolute frequency of the optical lattice clock based on $^{87}$Sr at PTB with an uncertainty of $3.9\times 10^{-16}$ using two caesium fountain clocks. This is close to the accuracy of today's best realizations of the SI second. The absolute frequency of the 5s$^2$ $^1$S$_0$-5s5p $^3$P$_0$ transition in $^{87}$Sr is 429,228,004,229,873.13(17) Hz. Our result is in excellent agreement with recent measurements performed in different laboratories worldwide. We improved the total systematic uncertainty of our Sr frequency standard by a factor of five and reach $3\times 10^{-17…
It is shown that the dynamic differential scalar polarisability of the $S_{1/2}-D_{5/2}$ transition in $^{138}$Ba$^+$ can be determined to an inaccuracy below $0.5\%$ across a wide wavelength range ($λ>700\,\mathrm{nm}$). This can be achieved using measurements for which accurate determination of laser intensity is not required, and most of the required measurements are already in the literature. Measurement of a laser-induced ac-stark shift of the clock transition would then provide an \emph{in situ} measurement of the laser's intensity to the same $0.5\%$ level of inaccuracy, which is not ea…
The 7.6(5) eV nuclear magnetic-dipole transition in a single 229Th3+ ion may provide the foundation for an optical clock of superb accuracy. A virtual clock transition composed of stretched states within the 5F5/2 electronic ground level of both nuclear ground and isomeric manifolds is proposed. It is shown to offer unprecedented systematic shift suppression, allowing for clock performance with a total fractional inaccuracy approaching 1 x 10^-19.
We have carried out calculations towards the goal of reducing the inaccuracy of the Sr optical atomic clock to 1$\times10^{-17}$ and below. We calculated a.c. polarizabilities of the $5s^2 ^1S_0$ and $5s5p ^3P_0^o$ clock states that are important for reducing the uncertainty of blackbody radiation-induced frequency shifts for the $^1S_0 - ^3P_0^o$ clock transition. We determined four low-lying even-parity states whose total contribution to the static polarizability of the $^3P_0^o$ clock state is at the level of 90%. We show that if the contribution of these states is experimentally known with…
Frequency shifts from background gas collisions currently contribute significantly to the inaccuracy of atomic clocks. Because nearly all collisions with room-temperature background gases that transfer momentum eject the cold atoms from the clock, the interference between the scattered and unscattered waves in the forward direction dominates these frequency shifts. We show they are ~10 times smaller than in room-temperature clocks and that van der Waals interactions produce the cold-atom background-gas shift. General considerations allow the loss of Ramsey fringe amplitude to bound this freque…
The use of ultra-precise optical clocks in space ("master clocks") will allow for a range of new applications in the fields of fundamental physics (tests of Einstein's theory of General Relativity, time and frequency metrology by means of the comparison of distant terrestrial clocks), geophysics (mapping of the gravitational potential of Earth), and astronomy (providing local oscillators for radio ranging and interferometry in space). Within the ELIPS-3 program of ESA, the "Space Optical Clocks" (SOC) project aims to install and to operate an optical lattice clock on the ISS towards the end of…
We propose a combination of Electromagnetically Induced Transparency (EIT)/Raman and pulsed spectroscopy techniques to accurately cancel frequency shifts arising from EIT fields in forbidden optical lattice clock transitions of alkaline earth atoms. Time-separated laser pulses are designed to trap atoms in coherent superpositions while eliminating off-resonance ac Stark contributions at particular laser detunings from the intermediate excited state. The scheme achieves efficient population transfer up to 60% with potential inaccuracy $<$ $10^{-17}$. Cancellation of external light shifts determ…
Progress in atomic optical clocks with total uncertainty of $10^{-18}$ or below requires a precise estimation of multipolar and higher-order effects due to atom-field interactions. Magnesium is an attractive candidate for optical lattice clocks because it is insensitive to blackbody radiation and has a large quality factor. We employ a combined method of the Dirac-Fock plus core polarization and the relativistic configuration interaction to calculate the dynamic multipolar polarizabilities and the hyperpolarizabilities of the atomic Mg clock. The lattice light shift against variation of the la…
Optical lattice clocks have the potential for extremely high frequency stability owing to the simultaneous interrogation of many atoms, but this precision may come at the cost of systematic inaccuracy due to atomic interactions. Density-dependent frequency shifts can occur even in a clock that uses fermionic atoms if they are subject to inhomogeneous optical excitation [1, 2]. Here we present a seemingly paradoxical solution to this problem. By dramatically increasing the strength of atomic interactions, we suppress collisional shifts in lattice sites containing $N$ > 1 atoms; strong interacti…
The ESA mission "Space Optical Clock" project aims at operating an optical lattice clock on the ISS in approximately 2023. The scientific goals of the mission are to perform tests of fundamental physics, to enable space-assisted relativistic geodesy and to intercompare optical clocks on the ground using microwave and optical links. The performance goal of the space clock is less than $1 \times 10^{-17}$ uncertainty and $1 \times 10^{-15} τ^{-1/2}$ instability. Within an EU-FP7-funded project, a strontium optical lattice clock demonstrator has been developed. Goal performances are instability b…
We describe an optical atomic clock based on quantum-logic spectroscopy of the $^1$S$_0$ $\leftrightarrow$ $^3$P$_0$ transition in $^{27}$Al$^{+}$ with a systematic uncertainty of ${9.4 \times 10^{-19}}$ and a frequency stability of ${1.2\times10^{-15}/\sqrtτ}$. 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. Op…
Ultra-precise optical clocks in space will allow new studies in fundamental physics and astronomy. Within an European Space Agency (ESA) program, the Space Optical Clocks (SOC) project aims to install and to operate an optical lattice clock on the International Space Station (ISS) towards the end of this decade. It would be a natural follow-on to the ACES mission, improving its performance by at least one order of magnitude. The payload is planned to include an optical lattice clock, as well as a frequency comb, a microwave link, and an optical link for comparisons of the ISS clock with ground…
We discuss the treatment of the systematic frequency shifts due to microwave lensing and distributed cavity phase in "First accuracy evaluation of NIST-F2" 2014 Metrologia 51 174-182. We explain that the microwave lensing frequency shift is generally non-zero and finite in the limit of no applied microwave field. This systematic error was incorrectly treated and we find that it contributes a significant frequency offset. Accounting for this shift implies that the measured microwave amplitude dependence (e.g due to microwave leakage) is comparable to the total reported inaccuracy. We also discu…
Building scalable quantum systems that demonstrate genuine performance enhancement based on entanglement is a major scientific goal for fields including computing, networking, simulations, and metrology. The tremendous challenge arises from the fragility of entanglement in increasingly larger sized quantum systems. Optical atomic clocks utilizing a large number of atoms have pushed the frontier of measurement science, building on precise engineering of quantum states and control of atomic interactions. However, today's state-of-the-art optical atomic clocks are limited by the quantum projectio…
Experimental and theoretical study of dynamic polarizabilities in the $5S_{1/2}$-$5D_{5/2}$ clock transition in rubidium-87 and determination of E1 matrix elements
An active optical clock based on highly charged Nd$^{9+}$ ion is proposed for the first time. The clock can offer ultra-narrow linewidth at the $\upmu$Hz-level which is more than two-order of magnitude below the currently recorded laser linewidth. Operating at 605(90) nm superradiation lasing transition between the $5p^2~4f$ ground state and one of the long-lived $5p~4f^2$ excited state, the proposed active clock is inherently immune against the cavity noise which provides high stability. The clock serves as a sensitive probe with high sensitivity to variation of the fine-structure constant wi…
We propose here a few selective highly charged ions (HCIs), namely Ni$^{12+}$ and Cu$^{13+}$, Pd$^{12+}$ and Ag$^{13+}$, that not only promise to be very high accurate optical clocks below $10^{-19}$ uncertainties, but also offer quality factors larger than $10^{15}$ and yet possess simple atomic structures for the experimental set-up. Moreover, these ions have strong optical magnetic-dipole (M1) transitions than the previously proposed HCI clocks. They can be used for the cooling and detection techniques. To demonstrate the projected fractional uncertainties below $10^{-19}$ level, we have es…
Recent progress in optical lattice clocks requires unprecedented precision in controlling systematic uncertainties at $10^{-18}$ level. Tuning of nonlinear light shifts is shown to reduce lattice-induced clock shift for wide range of lattice intensity. Based on theoretical multipolar, nonlinear, anharmonic and higher-order light shifts, we numerically demonstrate possible strategies for Sr, Yb, and Hg clocks to achieve lattice-induced systematic uncertainty below $1\times 10^{-18}$.
We report on the implementation and the metrological characterization of a vapor-cell Rb frequency standard working in pulsed regime. The three main parts that compose the clock, physics package, optics and electronics, are described in detail in the paper. The prototype is designed and optimized to detect the clock transition in the optical domain. Specifically, the reference atomic transition, excited with a Ramsey scheme, is detected by observing the interference pattern on a laser absorption signal. \ The metrological analysis includes the observation and characterization of the clock sign…
We analyze both the s- and p-wave collision induced frequency shifts and propose a over-$π$ pulse scheme to cancel the shifts in optical lattice clocks interrogated by a Rabi pulse. The collisional frequency shifts are analytically solved as a function of the pulse area and the inhomogeneity of the Rabi frequencies. Experimentally measured collisional frequency shifts in an Yb optical lattice clock are in good agreement with the analytical calculations. Based on our analysis, the over-$π$ pulse combined with a small inhomogeneity below 0.1 allows a fractional uncertainty on a level of $10^{-18…
We develop a method of spectroscopy that uses a weak static magnetic field to enable direct optical excitation of forbidden electric-dipole transitions that are otherwise prohibitively weak. The power of this scheme is demonstrated using the important application of optical atomic clocks based on neutral atoms confined to an optical lattice. The simple experimental implementation of this method -- a single clock laser combined with a DC magnetic field-- relaxes stringent requirements in current lattice-based clocks (e.g., magnetic field shielding and light polarization), and could therefore ex…
Electrometry is performed using Rydberg states to evaluate the quadratic Stark shift of the $5s^2$ $^1\textrm{S}_0-5s5p$ $^3\textrm{P}_0$ clock transition in strontium. By measuring the Stark shift of the highly excited $5s75d\;^1\textrm{D}_2$ state using electromagnetically induced transparency, we characterize the electric field with sufficient precision to provide tight constraints on the systematic shift to the clock transition. Using the theoretically derived, and experimentally verified, polarizability for this Rydberg state we can measure the residual field with an uncertainty well belo…
We present a detailed experimental and theoretical study of the effect of nuclear spin on the performance of optical lattice clocks. With a state-mixing theory including spin-orbit and hyperfine interactions, we describe the origin of the $^1S_0$-$^3P_0$ clock transition and the differential g-factor between the two clock states for alkaline-earth(-like) atoms, using $^{87}$Sr as an example. Clock frequency shifts due to magnetic and optical fields are discussed with an emphasis on those relating to nuclear structure. An experimental determination of the differential g-factor in $^{87}$Sr is p…
Optical atomic clocks using highly-charged ions hold an intriguing promise of metrology at the 19th significant figure. Here we study transitions within the $4f^{12}$ ground-state electronic configuration of highly charged ions. We consider isotopes lacking hyperfine structure and show that the detrimental effects of coupling of electronic quadrupole moments to gradients of trapping electric field can be effectively reduced by using specially chosen virtual clock transitions. The estimated systematic fractional clock accuracy is shown to be below $10^{-19}$.
As optical clocks are improved to reach the frequency uncertainty below the 10$^{-17}$ level, the frequency shift due to the blackbody radiation (BBR) has been one of the major systematic effects hindering further improvement. To evaluate the BBR shift of an Yb optical lattice clock at KRISS, we installed an in-vacuum BBR shield and made radiation thermometry using a black-coated-sphere thermal probe. After we quantitatively measured the conduction loss of the thermal probe and the effects of all the external radiation sources, we determined the temperature at the atom trap site with an uncert…
For the past 15 years, tremendous progress within the fields of laser stabilization, optical frequency combs and atom cooling and trapping have allowed the realization of optical atomic clocks with unrivaled performances. These instruments can perform frequency comparisons with fractional uncertainties well below $10^{-17}$, finding applications in fundamental physics tests, relativistic geodesy, and time and frequency metrology. Even though most optical clocks are currently laboratory setups, several proposals for using these clocks for field measurements or within an optical clock network ha…
We report a stability below $7\times 10{}^{-17}$ of two independent optical lattice clocks operating with bosonic ${}^{88}$Sr isotope. The value (429228066418008.3(1.9)${}_{syst}$(0.9)${}_{stat}$~Hz) of the absolute frequency of the ${}^{1}S_{0}$ - ${}^{3}P_{0}$ transition was measured with an optical frequency comb referenced to the local representation of the UTC by the 330 km-long stabilized fibre optical link. The result was verified by series of measurements on two independent optical lattice clocks and agrees with recommendation of Bureau International des Poids et Mesures.
Controlling light shifts in chip-scale atomic beam clocks
National Institute of Standards and Technology（美国国家标准与技术研究院）；University of Colorado Boulder（科罗拉多大学博尔德分校）；Infleqtion（英飞格蒂昂）；HRL Laboratories（HRL实验室）；DARPA（美国国防部高级研究计划局）
Atoms deeply trapped in magic wavelength optical lattices provide a Doppler- and collision-free dense ensemble of quantum emitters ideal for high precision spectroscopy. Thus, they are the basis of some of the best optical clock setups to date. However, despite their minute optical dipole moments the inherent long range dipole-dipole interactions in such lattices generate line shifts, dephasing and modified decay. We show that in a perfectly filled lattice these effects are resonantly enhanced depending on lattice constant, lattice geometry and excitation scheme inducing clock shifts of many a…
Recently, p-wave cold collisions were shown to dominate the density-dependent shift of the clock transition frequency in a 171Yb optical lattice clock. Here we demonstrate that by operating such a system at the proper excitation fraction, the cold collision shift is canceled below the 5x10^{-18} fractional frequency level. We report inelastic two-body loss rates for 3P0-3P0 and 1S0-3P0 scattering. We also measure interaction shifts in an unpolarized atomic sample. Collision measurements for this spin-1/2 171Yb system are relevant for high performance optical clocks as well as strongly-interact…
To operate an optical lattice clock at a fractional uncertainty below $10^{-17}$, one must typically consider not only electric-dipole (E1) interaction between an atom and the lattice light field when characterizing the resulting lattice light shift of the clock transition but also higher-order multipole contributions, such as electric-quadrupole (E2) and magnetic-dipole (M1) interactions. However, strongly incompatible values have been reported for the E2-M1 polarizability difference of the clock states $(5s5p)\,{}^{3}\mathrm{P}_{0}$ and $(5s^2)\,{}^{1}\mathrm{S}_{0}$ of strontium [Ushijima e…
Optical atomic clocks have already overcome the eighteenth decimal digit of instability and uncertainty demonstrating incredible control over external perturbations of the clock transition frequency. At the same time there is an increasing demand for atomic and ionic transitions with minimal sensitivity to external fields, with practical operational wavelengths and robust readout protocols. One of the goals is to simplify clock's operation maintaining its relative uncertainty at low 10-18 level. It is especially important for transportable and envisioned space-based optical clocks. We proved e…
After pumped from $5s_{1/2}$ ground state to $6p_{1/2}$ state, the population inversion between $6s_{1/2}$ and $5p_{1/2,3/2}$ will be established for Rubidium four-level active optical clock. In this paper, we calculate AC Stark shift due to lattice trapping laser which dominates the frequency shift of clock transition in lattice trapped Rubidium four-level active optical clock. Several magic wavelengths are found that can form desired optical lattice trapping potential. By choosing a proper intensity and linewidth of trapping laser, the fractional frequency uncertainty of clock transition due…
We predict the possibility of "triply-magic" optical lattice trapping of neutral divalent atoms. In such a lattice, the ${^1}\!S_{0}$ and ${^3}\!P_{0}$ clock states and an additional Rydberg state experience identical optical potentials, fully mitigating detrimental effects of the motional decoherence. In particular, we show that this triply magic trapping condition can be satisfied for Yb atom at optical wavelengths and for various other divalent systems (Ca, Mg, Hg and Sr) in the UV region. We assess the quality of triple magic trapping conditions by estimating the probability of excitation …
We present an experimental study of the lattice induced light shifts on the 1S_0-3P_0 optical clock transition (v_clock~518 THz) in neutral ytterbium. The ``magic'' frequency, v_magic, for the 174Yb isotope was determined to be 394 799 475(35)MHz, which leads to a first order light shift uncertainty of 0.38 Hz on the 518 THz clock transition. Also investigated were the hyperpolarizability shifts due to the nearby 6s6p 3P_0 - 6s8p 3P_0, 6s8p 3P_2, and 6s5f 3F_2 two-photon resonances at 759.708 nm, 754.23 nm, and 764.95 nm respectively. By tuning the lattice frequency over the two-photon resonan…
A Zeeman-insensitive optical clock atomic transition is engineered when nuclear spins are dressed by a non resonant radio-frequency field. For fermionic species as $^{87}$Sr, $^{171}$Yb, and $^{199}$Hg, particular ratios between the radiofrequency driving amplitude and frequency lead to "magic" magnetic values where a net cancelation of the Zeeman clock shift and a complete reduction of first order magnetic variations are produced within a relative uncertainty below the $10^{-18}$ level. An Autler-Townes continued fraction describing a semi-classical radio-frequency dressed spin is numerically…
We present a new interrogation scheme for the next generation of quantum clocks to suppress frequency-shifts induced by laser probing fields themselves based on Generalized Hyper-Ramsey resonances. Sequences of composite laser pulses with specific selection of phases, frequency detunings and durations are combined to generate a very efficient and robust frequency locking signal with almost a perfect elimination of the light-shift from off resonant states and to decouple the unperturbed frequency measurement from the laser's intensity. The frequency lock point generated from synthesized error s…
Optical lattice clocks can now resolve the height difference below 1 mm within an atomic ensemble by means of gravitational redshift with integration over sufficient amount of time. Further improvement in the stability enables the clocks to resolve the height difference of subsystems within an atomic ensemble that is conventionally interrogated as a single coherent spin state in a single Ramsey sequence, resulting in the dephasing of the coherent spin state. This effect is observable with a clock of a stability of $\lesssim 10^{-21}$ by introducing a single-layer-resolved imaging system for a …
International Optical Clock Comparison Using the European Optical Fiber Network
Istituto Nazionale di Ricerca Metrologica (INRIM)（意大利国家计量研究所）；Laboratoire Temps Espace (LNE-OP), Observatoire de Paris, Université PSL, Sorbonne Université, Université de Lille, LNE, CNRS（巴黎天文台时间空间实验室（LNE-OP））；Laboratoire de Physique des Lasers (LPL), Université Sorbonne Paris Nord, CNRS（索邦巴黎北大学激光物理实验室（LPL））；Physikalisch-Technische Bundesanstalt (PTB)（德国联邦物理技术研究院）
We present non-standard optical Ramsey schemes that use pulses individually tailored in duration, phase, and frequency to cancel spurious frequency shifts related to the excitation itself. In particular, the field shifts and their uncertainties of Ramsey fringes can be radically suppressed (by 2-4 orders of magnitude) in comparison with the usual Ramsey method (using two equal pulses) as well as with single-pulse Rabi spectroscopy. Atom interferometers and optical clocks based on two-photon transitions, heavily forbidden transitions, or magnetically induced spectroscopy could significantly ben…
Optical lattice clocks with uncertainty and instability in the $10^{-17}$-range and below have so far been demonstrated exclusively using fermions. Here, we demonstrate a bosonic optical lattice clock with $3\times 10^{-18}$ instability and $2.0\times 10^{-17}$ accuracy, both values improving on previous work by a factor 30. This was enabled by probing the clock transition with an ultra-long interrogation time of 4 s, using the long coherence time provided by a cryogenic silicon resonator, by careful stabilization of relevant operating parameters, and by operating at low atom density. This wor…
We devise a perturbation-immune version of Ramsey's method of separated oscillatory fields. Spectroscopy of an atomic clock transition without compromising the clock's accuracy is accomplished by actively balancing the spectroscopic responses from phase-congruent Ramsey probe cycles of unequal durations. Our simple and universal approach eliminates a wide variety of interrogation-induced line shifts often encountered in high precision spectroscopy, among them, in particular, light shifts, phase chirps, and transient Zeeman shifts. We experimentally demonstrate auto-balanced Ramsey spectroscopy…
We report calculations designed to assess the ultimate precision of an atomic clock based on the 578 nm $6 ^1S_0 --> 6 ^3P^o_0$ transition in Yb atoms confined in an optical lattice trap. We find that this transition has a natural linewidth less than 10 mHz in the odd Yb isotopes, caused by hyperfine coupling. The shift in this transition due to the trapping light acting through the lowest order AC polarizability is found to become zero at the magic trap wavelength of about 752 nm. The effects of Rayleigh scattering, higher-order polarizabilities, vector polarizability, and hyperfine induced e…
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 $^{27}$Al$^+$ clock to enable a total fractional systematic uncertainty below $10^{-18}$. For this clock, with 38(19) nPa of room temperature H$_2$ 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 cont…
A major obstacle for optical clocks is the frequency shift due to black body radiation. We discuss how one can tackle this problem in an optical lattice clock; in our case 87-Sr: firstly, by a measurement of the dc Stark shift of the clock transition and, secondly, by interrogating the atoms in a cryogenic environment. Both approaches rely on transporting ultracold atoms over several cm within a probe cycle. We evaluate this approach of mechanically moving the optical lattice and conclude that it is feasible to transport the atoms over 50 mm within 300 ms. With this transport a dc Stark shift …
We describe Doppler spectroscopy of Bose-Einstein condensates of ytterbium atoms using a narrow optical transition. We address the optical clock transition around 578 nm between the ${^1}S_0$ and ${^3}P_0$ states with a laser system locked on a high-finesse cavity. We show how the absolute frequency of the cavity modes can be determined within a few tens of kHz using high-resolution spectroscopy on molecular iodine. We show that optical spectra reflect the velocity distribution of expanding condensates in free fall or after releasing them inside an optical waveguide. We demonstrate sub-kHz spe…
The next generation of atomic frequency standards based on an ensemble of neutral atoms or a single-ion will provide very stringent tests in metrology, applied and fundamental physics requiring a new step in very precise control of external systematic corrections. In the proceedings of the 8th Symposium on Frequency Standards and Metrology, we present a generalization of the recent Hyper-Ramsey spectroscopy with separated oscillating fields using composites pulses in order to suppress field frequency shifts induced by the interrogation laser itself. Sequences of laser pulses including specific…
In order to improve the short-term stability of trapped-ion optical clocks, we are developing a frequency standard based on ${}^{115}$In${}^+$ / ${}^{172}$Yb${}^+$ Coulomb crystals. For this purpose, we have developed scalable segmented Paul traps which allow a high level of control for multiple ion ensembles. In this article, we detail on our recent results regarding the reduction of the leading sources of frequency uncertainty introduced by the ion trap: 2nd-order Doppler shifts due to micromotion and the heating of secular motion, as well as the black-body radiation shift due to warming of …
This paper describes advances in microwave frequency standards using laser-cooled atoms at BNM-SYRTE. First, recent improvements of the $^{133}$Cs and $^{87}$Rb atomic fountains are described. Thanks to the routine use of a cryogenic sapphire oscillator as an ultra-stable local frequency reference, a fountain frequency instability of $1.6\times 10^{-14}τ^{-1/2}$ where $τ$ is the measurement time in seconds is measured. The second advance is a powerful method to control the frequency shift due to cold collisions. These two advances lead to a frequency stability of $2\times 10^{-16}$ at $50,000s…
