Accuracy Evaluation of an Optical Lattice Clock with Bosonic Atoms
Xavier Baillard, Mathilde Fouché, Rodolphe Le Targat, Philip G. Westergaard, Arnaud Lecallier, Yann Le Coq, Giovanni D. Rovera, Sebastien Bize, Pierre Lemonde
We report the first accuracy evaluation of an optical lattice clock based on the 1S0 - 3P0 transition of an alkaline earth boson, namely 88Sr atoms. This transition has been enabled using a static coupling magnetic field. The clock frequency is determined to be 429 228 066 418 009(32) Hz. The isotopic shift between 87Sr and 88Sr is 62 188 135 Hz with fractional uncertainty 5.10^{-7}. We discuss the conditions necessary to reach a clock accuracy of 10^{-17} or less using this scheme.
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…
An optical atomic clock scheme is developed that utilizes two lasers to establish coherent coupling between the $5s^2 \phantom{}^1S_0$ ground state of $^{88}$Sr and the first excited state, $5s5p \phantom{}^3P_0$. The coupling is mediated by the broad $5s5p \phantom{}^1P_1$ state, exploiting the phenomenon of electromagnetically induced transparency. The effective linewidth of the clock transition can be chosen at will by adjusting the laser intensity. By trapping the $^{88}$Sr atoms in an optical lattice, long interaction times with the two lasers are ensured; Doppler and recoil effects are e…
We study a wide range of neutral atoms and ions suitable for ultra-precise atomic optical clocks with naturally suppressed black body radiation shift of clock transition frequency. Calculations show that scalar polarizabilities of clock states cancel each other for at least one order of magnitude for considered systems. Results for calculations of frequencies, quadrupole moments of the states, clock transition amplitudes and natural widths of upper clock states are presented.
Optical atomic clocks represent the state-of-the-art in the frontier of modern measurement science. In this article we provide a detailed review on the development of optical atomic clocks that are based on trapped single ions and many neutral atoms. We discuss important technical ingredients for optical clocks, and we present measurement precision and systematic uncertainty associated with some of the best clocks to date. We conclude with an outlook on the exciting prospect for clock applications.
Collisions with background gas particles can shift the resonance frequencies of atoms in atomic clocks. The internal quantum states of atoms can also become entangled with their motional states due to the recoil imparted by a collision, which leads to a further shift of the clock frequency through the relativistic Doppler shift. It can be complicated to evaluate the Doppler and collisional frequency shifts for clock atoms in such entangled states, but estimates of these shifts are essential in order to improve the accuracy of optical atomic clocks. We present a formalism that describes collisi…
We explore a feasibility of measuring atom-wall interaction using atomic clocks based on atoms trapped in engineered optical lattices. Optical lattice is normal to the wall. By monitoring the wall-induced clock shift at individual wells of the lattice, one would measure a dependence of the atom-wall interaction on the atom-wall separation. We rigorously evaluate the relevant clock shifts and show that the proposed scheme may uniquely probe the long-range atom-wall interaction in all three qualitatively-distinct regimes of the interaction: van der Waals (image-charge interaction), Casimir-Polde…
We propose a new class of atomic microwave clocks based on the hyperfine transitions in the ground state of aluminum or gallium atoms trapped in optical lattices. For these elements magic wavelengths exist at which both levels of the hyperfine doublet are shifted at the same rate by the lattice laser field, cancelling its effect on the clock transition. Our analysis of various systematic effects shows that, while offering an improved control over systematic errors, the accuracy of the proposed microwave clock is competitive to that of the state-of-the-art primary frequency standard.
Dark Matter could potentially manifest itself in the form of asymmetric dark stars. In this paper we entertain the possibility of probing such asymmetric bosonic dark matter stars by the use of atomic clocks. If the dark sector connects to the standard model sector via a Higgs or photon portal, the interior of boson stars that are in a Bose-Einstein condensate state can change the values of physical constants that control the timing of atomic clock devices. Dilute asymmetric dark matter boson stars passing through the Earth can induce frequency shifts that can be observed in separated Earth ba…
Black-body radiation (BBR) shifts of $^3\!P_0-^1\!S_0$ clock transition in divalent atoms Cd and Zn are evaluated using accurate relativistic many-body techniques of atomic structure. Static polarizabilities of the clock levels and relevant electric-dipole matrix elements are computed. We also present a comparative overview of the BBR shifts in optical clocks based on neutral divalent atoms trapped in optical lattices.
We have developed an optical lattice clock that can operate in dual modes: a strontium (Sr) clock mode and an ytterbium (Yb) clock mode. Dual-mode operation of the Sr-Yb optical lattice clock is achieved by alternately cooling and trapping $^{87}$Sr and $^{171}$Yb atoms inside the vacuum chamber of the clock. Optical lattices for Sr and Yb atoms were arranged with horizontal and vertical configurations, respectively, resulting in a small distance of the order of 100 $μ$m between the trapped Sr and Yb atoms. The $^{1}$S$_{0}$-$^{3}$P$_{0}$ clock transitions in the trapped atoms were interrogate…
We present an improvement of short term frequency stability of the integrating sphere cold atom clock after increasing the intensities of clock signals and optimizing the feedback loop of the clock. A short term frequency stability of $5.0\times10^{-13}τ^{-1/2}$ has been achieved and the limiting factors have been analyzed.
The rapid increase in accuracy and stability of optical atomic clocks compared to the caesium atomic clock as primary standard of time and frequency asks for a future re-definition of the second in the International System of Units (SI). The status of the optical clocks based on either single ions in radio-frequency traps or on neutral atoms stored in an optical lattice is described with special emphasis of the current work at the Physikalisch-Technische Bundesanstalt (PTB). Besides the development and operation of different optical clocks with estimated fractional uncertainties in the 10^-18 …
Spherically-symmetric ground states of alkali-metal atoms do not posses electric quadrupole moments. However, the hyperfine interaction between nuclear moments and atomic electrons distorts the spherical symmetry of electronic clouds and leads to non-vanishing atomic quadrupole moments. We evaluate these hyperfine-induced quadrupole moments using techniques of relativistic many-body theory and compile results for Li, Na, K, Rb, and Cs atoms. For heavy atoms we find that the hyperfine-induced quadrupole moments are strongly (two orders of magnitude) enhanced by correlation effects. We further a…
We demonstrate how to realize an optical clock with neutral atoms that is competitive to the currently best single ion optical clocks in accuracy and superior in stability. Using ultracold atoms in a Ca optical frequency standard we show how to reduce the relative uncertainty to below 1E-15. We observed atom interferences for stabilization of the laser to the clock transition with a visibility of 0.36, which is 70 % of the ultimate limit achievable with atoms at rest. A novel scheme was applied to detect these atom interferences with the prospect to reach the quantum projection noise limit at …
We report on design and construction of a Zeeman slower for strontium atoms which will be used in an optical atomic clock experiment. The paper describes briefly required specifications of the device, possible solutions, and concentrates on the chosen design. The magnetic field produced by the built Zeeman slower has been measured and compared with the simulations. The system consisting of an oven and Zeeman slower are designed to produce an atomic beam of 10-12 s-1 flux and final velocity of ~30 m/s.
Theories unifying gravity and other interactions suggest the possibility of spatial and temporal variation of physical ``constants''. Accuracy achieved for the atomic optical frequency standards (optical clocks) approaches the level when possible time evolution of the fine structure constant $α$ can be studied by comparisons of rates between clocks based on different atomic transitions in different atoms. The sensitivity to variation of $α$ is due to relativistic corrections which are different in different atoms ($\sim Z^2α^2$). We have calculated the values of the relativistic energy shifts …
The medium-long term frequency stability of the integrating sphere cold atom clock was improved.During the clock operation, Rb atoms were cooled and manipulated using cooling light diffusely reflected by the inner surface of a microwave cavity in the clock. This light heated the cavity and caused a frequency drift from the resonant frequency of the cavity. Power fluctuations of the cooling light led to atomic density variations in the cavity's central area, which increased the clock frequency instability through a cavity pulling effect. We overcame these limitations with appropriate solutions.…
We analyze the AC Stark shift of the Cs microwave atomic clock transition theoretically and experimentally. Theoretical and experimental data are in a good agreement with each other. Results indicate the absence of a magic wavelength at which there would be no differential shift of the clock states having zero projections of the total angular momentum.
An ultrastable optical clock based on neutral atoms trapped in an optical lattice is proposed. Complete control over the light shift is achieved by employing the $5s^2 {}^1S_0 \to 5s5p {}^3P_0$ transition of ${}^{87}{\rm Sr}$ atoms as a "clock transition". Calculations of ac multipole polarizabilities and dipole hyperpolarizabilities for the clock transition indicate that the contribution of the higher-order light shifts can be reduced to less than 1 mHz, allowing for a projected accuracy of better than $ 10^{-17}$.
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…
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 $1\times 10^{-18}$.
The possibility of using neutral and double ionized erbium for atomic clocks of high precision is investigated. In both cases the narrow electric quadrupole clock transition between the ground and first exited state of the same configuration lies in optical region. The estimated ratio of decay width to transition energy is less then 10 20. We demonstrate that this transitions are not sensitive to black body radiation and if other perturbations are also considered the relative accuracy of the clocks can probably reach the level of 10 (pow minus 18) or better.
We report a frequency measurement of the 1S0-3P0 transition of 87Sr atoms in an optical lattice clock. The frequency is determined to be 429 228 004 229 879 (5) Hz with a fractional uncertainty that is comparable to state-of-the-art optical clocks with neutral atoms in free fall. Two previous measurements of this transition were found to disagree by about 2x10^{-13}, i.e. almost four times the combined error bar, instilling doubt on the potential of optical lattice clocks to perform at a high accuracy level. In perfect agreement with one of these two values, our measurement essentially dissipa…
A black-body radiation (BBR) shifts of (nsnp ^3P_0) - (ns^2 ^1S_0) clock transition in divalent atoms Mg, Ca, Sr, and Yb are evaluated. A theory of multipolar BBR shifts is developed and its implications are discussed. At room temperatures, the resulting uncertainties in the BBR shifts are relatively large and substantially affect the projected 10^{-18} fractional accuracy of the optical-lattice-based clocks.
Since the atomic clock was invented, its performance has been improved for one digit every decade until 90s of last century when the traditional atomic clock almost reached its limit. With laser cooled atoms, the performance can be further improved, and nowadays the cold atom based clocks are widely used as primary frequency standards. Such a kind of cold atom clocks has great applications in space. This paper presents the design and tests of a cold atom clock (CAC) operating in space. In microgravity, the atoms are cooled, trapped, launched and finally detected after being interrogated by mic…
A method for highly accurate calculations of atomic electric quadrupole moments (EQM) is presented, using relativistic general-excitation-rank configuration interaction wavefunctions based on Dirac spinors. Application to the clock transition states of the thulium atom employing up to full Quadruple excitations for the atomic wavefunction yields a final value of $Q_{zz}({^2F}_{7/2}) = 0.07 \pm 0.07$ a.u., establishing that the thulium electronic ground state has an exceptionally small EQM. A detailed analysis of this result is presented which has implications for EQMs of other atoms with unpai…
We report a vapor-cell magneto-optical trapping of Hg isotopes on the ${}^1S_0-{}^3P_1$ intercombination transition. Six abundant isotopes, including four bosons and two fermions, were trapped. Hg is the heaviest non-radioactive atom trapped so far, which enables sensitive atomic searches for ``new physics'' beyond the standard model. We propose an accurate optical lattice clock based on Hg and evaluate its systematic accuracy to be better than $10^{-18}$. Highly accurate and stable Hg-based clocks will provide a new avenue for the research of optical lattice clocks and the time variation of t…
A caesium fountain clock is operated utilizing a microwave oscillator that derives its frequency stability from a stable laser by means of a fiber-laser femtosecond frequency comb. This oscillator is based on the technology developed for optical clocks and replaces the quartz based microwave oscillator commonly used in fountain clocks. As a result, a significant decrease of the frequency instability of the fountain clock is obtained, reaching 0.74E-14 at 100 s averaging time. We could demonstrate that for a significant range of detected atom numbers the instability is limited by quantum projec…
The pursuit of better atomic clocks has advanced many research areas, providing better quantum state control, new insights in quantum science, tighter limits on fundamental constant variation, and improved tests of relativity. The record for the best stability and accuracy is currently held by optical lattice clocks. This work takes an important step towards realizing the full potential of a many-particle clock with a state-of-the-art stable laser. Our 87Sr optical lattice clock now achieves fractional stability of 2.2e-16 at 1 s. With this improved stability, we perform a new accuracy evaluat…
We describe a method to stabilize the amplitude of the interrogating microwave field in compact atomic clocks working in a Ramsey approach. In this technique, we take advantage of the pulsed regime to use the atoms themselves as microwave amplitude discriminators. Specifically, in addition to the dependence on the microwave detuning, the atomic signal after the Ramsey interrogation acquires a dependence on the microwave pulse area (amplitude times duration) that can be exploited to implement an active stabilization of the microwave field amplitude, in a similar way in which the Ramsey clock si…
Laser cooling is a key ingredient for quantum control of atomic systems in a variety of settings. In divalent atoms, two-stage Doppler cooling is typically used to bring atoms to the uK regime. Here, we implement a pulsed radial cooling scheme using the ultranarrow 1S0-3P0 clock transition in ytterbium to realize sub-recoil temperatures, down to tens of nK. Together with sideband cooling along the one-dimensional lattice axis, we efficiently prepare atoms in shallow lattices at an energy of 6 lattice recoils. Under these conditions key limits on lattice clock accuracy and instability are reduc…
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 present a new scheme of compact Rubidium cold-atom clock which performs the diffuse light cooling, the microwave interrogation and the detection of the clock signal in a cylindrical microwave cavity. The diffuse light is produced by the reflection of the laser light at the inner surface of the microwave cavity. The pattern of injected laser beams is specially designed to make most of the cold atoms accumulate in the center of the microwave cavity. The microwave interrogation of cold atoms in the cavity leads to Ramsey fringes whose line-width is 24.5 Hz and the contrast of 95.6% when the fr…
We present experimental work for improved atom loading in the optical molasses of a caesium fountain clock, employing a low-velocity intense source of atoms (LVIS) [Lu et al., Phys. Rev. Lett. 77, 3331 (1996)], which we modified by adding a "dark" state pump laser. With this modification the atom source has a mean flux of $4 \times 10^{8}$ atoms/s at a mean atom velocity of $8.6$ m/s. Compared to fountain operation using background gas loading, we achieved a significant increase of the loaded and detected atom number by a factor of 40. Operating the fountain clock with a total number of detect…
A physics package for a compact cold atomic clock is hereby presented. The uniqueness of this package is its small dimensions that enable, for the first time, implementation of a primary cold atomic clock in a standard package of 3U height (=133mm). These dimensions are made possible by using an in-vacuum Microwave (MW) Loop Gap Resonator (LGR) whose length and diameter can be reduced from those of a typical resonator. Following our presentation of the design, we analyze the homogeneity of the MW field amplitude and phase. We find that the expected clock instability due to non-uniformity of th…
We report the evaluation of the second order Zeeman shift in the continuous atomic fountain clock FoCS-2. Because of the continuous operation and of its geometrical constraints, the methods used in pulsed fountains are not applicable. We use here time-resolved Zeeman spectroscopy to probe the magnetic field profile in the clock. The pulses of ac magnetic excitation allow us to spatially resolve the Zeeman frequency and to evaluate the Zeeman shift with a relative uncertainty smaller than 10E-16.
The frequency stability achieved by an optical atomic clock ultimately depends on the coherence of its local oscillator. Even the best ultrastable lasers only allow interrogation times of a few seconds, at present. Here we present a universal measurement protocol that overcomes this limitation. Engineered dynamic decoupling of laser phase noise allows any optical atomic clock with high signal-to-noise ratio in a single interrogation to reconstruct the laser's phase well beyond its coherence limit. A compound clock is then formed in combination with another optical clock of any type, allowing t…
In the last ten years extraordinary results in time and frequency metrology have been demonstrated. Frequency-stabilization techniques for continuous-wave lasers and femto-second optical frequency combs have enabled a rapid development of frequency standards based on optical transitions in ultra-cold neutral atoms and trapped ions. As a result, today's best performing atomic clocks tick at an optical rate and allow scientists to perform high-resolution measurements with a precision approaching a few parts in $10^{18}$. This paper reviews the history and the state of the art in optical-clock re…
We present a new evaluation of an 87Sr optical lattice clock using spin polarized atoms. The frequency of the 1S0-3P0 clock transition is found to be 429 228 004 229 873.6 Hz with a fractional accuracy of 2.6 10^{-15}, a value that is comparable to the frequency difference between the various primary standards throughout the world. This measurement is in excellent agreement with a previous one of similar accuracy.
With direct polarizability measurements we demonstrated extremely low sensitivity of the inner-shell clock transition at $1.14\,μ$m in Tm atoms to external dc electric fields and black-body radiation (BBR). We measured differential polarizabilities of clock levels in Tm at wavelengths of 810--860\,nm and at 1064\,nm and inferred the static scalar differential polarizability of the inner-shell clock transition of $-0.047(18)$ atomic units corresponding to only $2\times10^{-18}$ fractional frequency shift from BBR at the room temperature. This is a few orders of magnitude smaller compared to the…
Squeezed many-body states of atoms are a valuable resource for high precision frequency metrology and could tremendously boost the performance of atomic lattice clocks. Here, we theoretically demonstrate a viable approach to spin squeezing in lattice clocks via optical dressing of one clock state to a highly excited Rydberg state, generating switchable atomic interactions. For realistic experimental parameters, this is shown to generate over 10 dB of squeezing in a few microseconds interaction time without affecting the subsequent clock interrogation.
In a standing wave of light, a difference in spatial distributions of multipolar atom-field interactions may alter the definition of the magic wavelength to minimize the uncertainty of optical lattice clocks. We show that the magic wavelength can be determined so as to eliminate the spatial mismatch in electric dipole, magnetic dipole, and electric quadrupole interactions for specific combinations of standing waves. Experimental prospects of such lattices used with a blue magic wavelength are discussed.
We report direct single-laser excitation of the strictly forbidden (6s^2)^1S_0 -(6s6p)^3P_0 clock transition in the even 174Yb isotope confined to a 1D optical lattice. A small (~1.2 mT) static magnetic field was used to induce a nonzero electric dipole transition probability between the clock states at 578.42 nm. Narrow resonance linewidths of 20 Hz (FHWM) with high contrast were observed, demonstrating a record neutral-atom resonance quality factor of 2.6x10^13. The previously unknown ac Stark shift-canceling (magic) wavelength was determined to be 759.35+/-0.02 nm. This method for using the…
We present a compact atomic clock using cold rubidium atoms based on an isotropic light cooling, a Ramsey microwave interrogation and an absorption detection. Its technology readiness level is suitable to industrial transfer. We use a fibre optical bench, based on a frequency-doubled telecom laser. The isotropic light cooling technique allows us to cool down the atoms in 100 ms and works with a cycle time around 200 ms. We carried out measurements in simulated microgravity and obtained the narrowest fringes ever recorded in microgravity.
The total mass density of the Universe appears to be dominated by dark matter. However, beyond its gravitational interactions at the galactic scale, little is known about its nature. Extensions of the quantum electrodynamics Lagrangian with dark-matter coupling terms may result in changes to Standard Model parameters. Recently, it was proposed that a network of atomic clocks could be used to search for transient signals of a hypothetical dark matter in the form of stable topological defects. The clocks become desynchronized when a dark-matter object sweeps through the network. This pioneering …
We report on a theoretical analysis of the phase-shift in compact atomic clocks working either with cold or thermal atoms. It is well known that in a microwave cavity with electromagnetic losses, a traveling wave adds to the standing wave of a given resonant mode. We calculate the spatial varying phase related to this travelling wave for two geometries of interest in clock applications, the cylindrical cavity and the spherical cavity. Due to their motion, the atoms probe different regions of the cavity and then experience different phases of the interrogating microwave field. We show that this…
We analyze the nature and performance of clocks formed by stabilizing an oscillator to the phase difference between two paths of an atom interferometer. The phase evolution has been modeled as being driven by the proper-time difference between the two paths, although it has an ambiguous origin in the non-relativistic limit and it requires a full quantum-field-theory treatment in the general case. We present conditions for identifying deviations from the non-relativistic limit as a way of testing the proper-time driven phase evolution model. We show that the system performance belies the premis…
We derive an expression for the magnetic blackbody shift of hyperfine transitions such as the cesium primary reference transition which defines the second. The shift is found to be a complicated function of temperature, and has a T^2 dependence only in the high-temperature limit. We also calculate the shift of ground-state p_1/2 hyperfine transitions which have been proposed as new atomic clock transitions. In this case interaction with the p_3/2 fine-structure multiplet may be the dominant effect.
Hyperfine-induced electric dipole contributions may significantly increase probabilities of otherwise very weak electric octupole and magnetic quadrupole atomic clock transitions (e.g. transitions between $s$ and $f$ electron orbitals). These transitions can be used for exceptionally accurate atomic clocks, quantum information processing and search for dark matter. They are very sensitive to new physics beyond the Standard Model, such as temporal variation of the fine structure constant, the Lorentz invariance and Einstein equivalence principle violation. We formulate conditions under which th…
A Wannier-Stark optical lattice clock has demonstrated unprecedented measurement precision for optical atomic clocks. We present a systematic evaluation of the lattice light shift, a necessary next step for establishing this system as an accurate atomic clock. With precise control of the atomic motional states in the lattice, we report accurate measurements of the multipolar and the hyperpolar contributions and the operational lattice light shift with a fractional frequency uncertainty of 3.5$\times$10${}^{-19}$.
We demonstrate state-of-the-art technique of an active clock to provide a continuous superradiant lasing signal using an ensemble of trapped Cs atoms in the optical lattice. A magic wavelength of the proposed |7S1/2; F = 4, MF = 0> - |6P3/2; F = 3, MF = 0> clock transition in Cs atom is identified at 1181 nm for constructing the optical lattice. Pertinent optical lines are also found for pumping and repumping atoms from their ground states. A fractional uncertainty about 10-15 level to the clock frequency has been predicted by carrying out rigorous calculations of several atomic properties. Th…
