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Unravelling superradiance, also known as superfluorescence, relies on an ensemble of phase-matched dipole oscillators and the suppression of inhomogeneous broadening. Here we report on a novel superradiance platform that combines an optical lattice free from the ac Stark shift and a hollow-core photonic crystal fibre, enabling an extended atom-light interaction over $2 \mathrm{mm}$ free from the Doppler effect. This system allows controlling the atom spatial distribution and spectral homogeneity whilst efficiently coupling the radiation field to an optical fibre. The experimentally-observed and theoretically-corroborated temporal, spectral and spatial dynamic behaviours of the superradiance, e.g., superradiance ringing and density-dependent frequency shift, demonstrate a unique interplay between the trapped atoms and the fibre-guided field with multiple transverse modes. Our theory indicates the resulting temporal evolution of the guided light shows a minimal beam radius of $3.1 \mathrm{\mu m}$ that is three times smaller than that of the lowest-loss fibre mode., Comment: 41 pages, 8 figures in total, the first 30 pages and 6 figures are main article and the remaining pages and figures are supplemental material

We present an extended model for the lattice-induced light shifts of the clock frequency in optical lattice clocks, applicable to a wide range of operating conditions. The model extensions cover radial motional states with sufficient energies to invalidate the harmonic approximation of the confining potential. We reevaluate lattice-induced light shifts in our Yb optical lattice clock with an uncertainty of 6.1E-18 under typical clock operating conditions., Comment: 12 pages, 10 figures

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., Comment: 9 pages, 9 figures, 1 table, and Supplemental Material

The new technique of measuring frequency by optical lattice clocks now approaches to the relative precision of $(\Delta f/f)=O(10^{-18})$. We propose to place such precise clocks in space and to use Doppler tracking method for detecting low-frequency gravitational wave below 1 Hz. Our idea is to locate three spacecrafts at one A.U. distance (say at L1, L4 & L5 of the Sun-Earth orbit), and apply the Doppler tracking method by communicating "the time" each other. Applying the current available technologies, we obtain the sensitivity for gravitational wave with three or four-order improvement ($h_{\rm n}\sim 10^{-17}$ or $10^{-18}$ level in $10^{-5}$Hz -- $1$ Hz) than that of Cassini spacecraft in 2001. This sensitivity enables us to observe black-hole mergers of their mass greater than $10^5 M_\odot$ in the cosmological scale. Based on the hierarchical growth model of black-holes in galaxies, we estimate the event rate of detection will be 20-50 a year. We nickname "INO" (Interplanetary Network of Optical Lattice Clocks) for this system, named after Tadataka Ino (1745--1818), a Japanese astronomer, cartographer, and geodesist., Comment: 11 pages, 4 figures; added references; to be published in Int. J. Modern Physics D

We introduce a description of Ramsey spectra under atomic interactions as a sum of decomposed components with differing dependence on interaction parameters. This description enables intuitive understanding of the loss of contrast and asymmetry of Ramsey spectra. We derive a quantitative relationship between the asymmetry and atomic interaction parameters, which enables their characterization without changing atom density. The model is confirmed through experiments with a Yb optical lattice clock.

We demonstrate an atom interferometer operating on the $^1S_0-{}^3P_0$ clock transition of $^{87}$Sr atoms in a "magic" optical guide, where the light shift perturbations of the guiding potential are canceled. As a proof-of-principle demonstration, a Mach-Zehnder interferometer is set horizontally to map the acceleration introduced by the focused optical guide. This magic guide interferometer on the clock transition is applicable to atomic elements where magic wavelengths can be found. Possible applications of the magic guide interferometer, including a hollow-core fiber interferometer and gradiometer, are discussed., Comment: 5 pages, 4 figures

We report on the isotope shift between ${}^{88}$Sr and ${}^{87}$Sr on the ${}^1S_0 - {}^3P_0$ clock transitions. The interleaved operation of an optical lattice clock with two isotopes allows the canceling out of common perturbations, such as the quadratic Zeeman shift, the clock-light shift, and the blackbody radiation shift. The isotope shift is determined to be 62 188 134.004(10) Hz, where the major uncertainty is introduced by the collisional shift that is distinct for each isotope. Our result allows us to determine the frequency of $^{88}$Sr-$^{87}$Sr optical lattice clocks with a fractional uncertainty of $2\times 10^{-17}$. The scheme is generally applicable for measuring the isotope shift with significantly reduced uncertainty., Comment: 15 pages, 4 figures

All-polarization-maintaining, single-port Er:fiber combs offer long-term robust operation as well as high stability. We have built two such combs and evaluated the transfer noise for linking optical clocks. A uniformly broadened spectrum over 135-285 THz with a high signal-to-noise ratio enables the optical frequency measurement of the subharmonics of strontium, ytterbium, and mercury optical lattice clocks with the fractional frequency-noise power spectral density of $(1-2)\times 10^{-17}$ Hz$^{-1/2}$ at 1 Hz. By applying a synchronous clock comparison, the comb enables clock ratio measurements with $10^{-17}$ instability at 1 s, which is one order of magnitude smaller than the best instability of the frequency ratio of optical lattice clocks., Comment: 14 pages, 4 figures

According to the Einstein's theory of relativity, the passage of time changes in a gravitational field. On earth, raising a clock by one centimetre increases its tick rate by 1.1 parts in 10$^{18}$, enabling optical clocks to perform precision geodesy. Here, we demonstrate geopotentiometry by determining the height difference of master and slave clocks separated by 15 km with uncertainty of 5 cm. The subharmonic of the master clock is delivered through a telecom fibre to phase-lock and synchronously interrogate the slave clock. This protocol rejects laser noise in the comparison of two clocks, which improves the stability of measuring the gravitational red shift. Such phase-coherently operated clocks facilitate proposals for linking clocks and interferometers. Over half a year, 11 measurements determine the fractional frequency difference between the two clocks to be $1,652.9(5.9)\times 10^{-18}$, or a height difference of 1,516(5) cm, consistent with an independent measurement by levelling and gravimetry. Our system is as a building block of an internet of clocks, consisting of a master and a number of slave clocks, which will provide "quantum benchmarks" that are height references with dynamic response., Comment: 25 pages, 4 figures, published online 15 August 2016 in Nature Photonics

Continuous-wave output at 229 nm for the application of laser cooling of Cd atoms was generated by the 4th harmonic using two successive second harmonic generation stages. Employing a single-frequency optically pumped semiconductor laser as a fundamental source, 0.56 W of output at 229 nm was observed with a 10-mm long, Brewster-cut BBO crystal in an external cavity with 1.62 W of 458 nm input. Conversion efficiency from 458 nm to 229 nm was more than 34%. By applying a tapered amplifier as a fundamental source, we demonstrated magneto-optical trapping of all stable Cd isotopes including isotopes $^{111}$Cd and $^{113}$Cd, which are applicable to optical lattice clocks.

Transition frequencies of atoms and ions are among the most accurately accessible quantities in nature, playing important roles in pushing the frontiers of science by testing fundamental laws of physics, in addition to a wide range of applications such as satellite navigation systems. Atomic clocks based on optical transitions approach uncertainties of $10^{-18}$, where full frequency descriptions are far beyond the reach of the SI second. Frequency ratios of such super clocks, on the other hand, are not subject to this limitation. They can therefore verify consistency and overall accuracy for an ensemble of super clocks, an essential step towards a redefinition of the second. However, with the measurement stabilities so far reported for such frequency ratios, a confirmation to $1 \times 10^{-18}$ uncertainty would require an averaging time $\tau$ of multiple months. Here we report a measurement of the frequency ratio of neutral ytterbium and strontium clocks with a much improved stability of $4 \times 10^{-16} ({\tau}/s)^{-1/2}$. Enabled by the high stability of optical lattice clocks interrogating hundreds of atoms, this marks a 90-fold reduction in the required averaging time over a previous record-setting experiment that determined the ratio of Al+ and Hg+ single-ion clocks to an uncertainty of $5.2 \times 10^{-17}$. For the Yb/Sr ratio, we find R = 1.207 507 039 343 337 749(55), with a fractional uncertainty of $4.6 \times 10^{-17}$., Comment: 19 pages, 5 figures

We present a compact FPGA-based pulse sequencer and radio-frequency (RF) generator suitable for experiments with cold trapped ions and atoms. The unit is capable of outputting a pulse sequence with at least 32 TTL channels with a timing resolution of 40 ns and contains a built-in 100 MHz frequency counter for counting electrical pulses from a photo-multiplier tube (PMT). There are 16 independent direct-digital-synthesizers (DDS) RF sources with fast (rise-time of ~60 ns) amplitude switching and sub-mHz frequency tuning from 0 to 800 MHz.

This article describes the recent progress of optical lattice clocks with neutral strontium ($^{87}$Sr), ytterbium ($^{171}$Yb) and mercury ($^{199}$Hg) atoms. In particular, we present frequency comparison between the clocks locally via an optical frequency comb and between two Sr clocks at remote sites using a phase-stabilized fibre link. We first review cryogenic Sr optical lattice clocks that reduce the room-temperature blackbody radiation shift by two orders of magnitude and serve as a reference in the following clock comparisons. Similar physical properties of Sr and Yb atoms, such as transition wavelengths and vapour pressure, have allowed our development of a compatible clock for both species. A cryogenic Yb clock is evaluated by referencing a Sr clock. We also report on a Hg clock, which shows one order of magnitude less sensitivity to blackbody radiation, while its large nuclear charge makes the clock sensitive to the variation of fine-structure constant. Connecting all three types of clocks by an optical frequency comb, the ratios of the clock frequencies are determined with uncertainties smaller than possible through absolute frequency measurements. Finally, we describe a synchronous frequency comparison between two Sr-based remote clocks over a distance of 15 km between RIKEN and the University of Tokyo, as a step towards relativistic geodesy., Comment: 11 pages, 5 figures, invited review article in Comptes Rendus de Physique 2015

We report on a frequency ratio measurement of a ${}^{199}$Hg-based optical lattice clock referencing a ${}^{87}$Sr-based clock. Evaluations of lattice light shift, including atomic-motion-dependent shift, enable us to achieve a total systematic uncertainty of $7.2 \times 10^{-17}$ for the Hg clock. The frequency ratio is measured to be $\nu_{\rm Hg}/\nu_{\rm Sr}=2.629\, 314\, 209\, 898\, 909\, 60(22)$ with a fractional uncertainty of $8.4 \times 10^{-17}$, which is smaller than the uncertainty of the realization of the SI second, i.e., the SI limit., Comment: 5 pages, 3 figures

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 $1\times 10^{-18}$., Comment: 5 pages, 4 figures

We study the Ho$^{14+}$ ion as a candidate for extremely accurate and stable optical atomic clock which is sensitive to the time variation of the fine structure constant. We demonstrate that the proposed system has all desired features including relatively strong optical electric dipole and magnetic dipole transitions which can be used for cooling and detection. Zero quadrupole moments for the relevant states in the clock transition allows interrogating multiple ions to improve the clock stability., Comment: 4 pages, 1 figure

Unlike photons, which are conveniently handled by mirrors and optical fibres without loss of coherence, atoms lose their coherence via atom-atom and atom-wall interactions. This decoherence of atoms deteriorates the performance of atomic clocks and magnetometers, and also hinders their miniaturisation. Here we report a novel platform for precision spectroscopy. Ultracold strontium atoms inside a kKagome-lattice hollow-core photonic crystal fibre (HC-PCF) are transversely confined by an optical lattice to prevent atoms from interacting with the fibre wall. By confining at most one atom in each lattice site, to avoid atom-atom interactions and Doppler effect, a 7.8-kHz-wide spectrum is observed for the $^1 S_0-{}^3P_1$ (m=0) transition. Atoms singly trapped in a magic lattice in hollow-core photonic crystal fibresHC-PCFs improve the optical depth while preserving atomic coherence time., Comment: 31 pages, 8 figures

Time and frequency are the most accurately measurable quantities, providing foundations for science and modern technologies. The accuracy relies on the SI (Syst\'eme International) second that refers to Cs microwave clocks with fractional uncertainties at $10^{-16}$. Recent revolutionary progress of optical clocks aims to achieve $1\times 10^{-18}$ uncertainty, which however has been hindered by long averaging-times or by systematic uncertainties. Here, we demonstrate optical lattice clocks with $^{87}$Sr atoms interrogated in a cryogenic environment to address the blackbody radiation-induced frequency-shift, which remains the primary source of clocks' uncertainties and has initiated vigorous theoretical and experimental investigations. The quantum-limited stability for $N \sim 1,000$ atoms allows investigation of the uncertainties at $2\times 10^{-18}$ in two hours of clock operation. After 11 measurements performed over a month, the two cryo-clocks agree to within $(-1.1\pm 1.6)\times 10^{-18}$. Besides its contribution to fundamental science and the redefinition of the SI second, such a speedy comparison of accurate clocks provides a means for relativistic geodesy.

Fiber-based remote comparison of $^{87}$Sr lattice clocks in 24 km distant laboratories is demonstrated. The instability of the comparison reaches $5\times10^{-16}$ over an averaging time of 1000 s, which is two orders of magnitude shorter than that of conventional satellite links and is limited by the instabilities of the optical clocks. By correcting the systematic shifts that are predominated by the differential gravitational redshift, the residual fractional difference is found to be $(1.0\pm7.3)\times10^{-16}$, confirming the coincidence between the two clocks. The accurate and speedy comparison of distant optical clocks paves the way for a future optical redefinition of the second.

Recently invented and demonstrated, optical lattice clocks hold great promise for improving the precision of modern timekeeping. These clocks aim at the 10^-18 fractional accuracy, which translates into a clock that would neither lose or gain a fraction of a second over an estimated age of the Universe. In these clocks, millions of atoms are trapped and interrogated simultaneously, dramatically improving clock stability. Here we discuss the principles of operation of these clocks and, in particular, a novel concept of "magic" trapping of atoms in optical lattices. We also highlight recently proposed microwave lattice clocks and several applications that employ the optical lattice clocks as a platform for precision measurements and quantum information processing., Comment: 18 pages, 15 figures

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., Comment: 4 pages, 1 figure

We investigated the coherence of spin-polarized ^{87}Sr atoms trapped in a light-shift-free one-dimensional optical lattice during their interaction with a clock laser on the ^1S_0-^3P_0 transition. Collapses and revivals appeared for more than 50 Rabi cycles, attributed to the thermal distribution of discrete vibrational states in the lattice potential. The population oscillation in the clock states lasted more than 1s, demonstrating high immunity from decoherence. This long atomic coherence suggests the feasibility of Pauli blocking of collisions in optical clock excitation., Comment: 10 pages, 4 figures

Precision metrology and quantum measurement often demand matter be prepared in well defined quantum states for both internal and external degrees of freedom. Laser-cooled neutral atoms localized in a deeply confining optical potential satisfy this requirement. With an appropriate choice of wavelength and polarization for the optical trap, two electronic states of an atom can experience the same trapping potential, permitting coherent control of electronic transitions independent of the atomic center-of-mass motion. We review a number of recent experiments that use this approach to investigate precision quantum metrology for optical atomic clocks and coherent control of optical interactions of single atoms and photons within the context of cavity quantum electrodynamics. We also provide a brief survey of promising prospects for future work., Comment: 23 pages of main text, 4 figures, and 12 pages of Supporting Online Material (SOM)

The circular polarized laser beam of the ``magic'' wavelength may be used for mixing the $^3P_1$ state into the long-living metastable state $^3P_0$, thus enabling the strictly forbidden $^1S_0$$-$$^3P_0$ ''clock'' transition in even isotopes of alkaline-earth-like atoms, without change of the transition frequency. In odd isotopes the laser beam may adjust to an optimum value the line width of the "clock" transition, originally enabled by the hyperfine mixing. We present a detailed analysis of various factors influencing resolution and uncertainty for an optical frequency standard based on atoms exposed simultaneously to the lattice standing wave and an additional "state-mixing" wave, including estimations of the "magic" wavelengths, Rabi frequencies for the ''clock'' and state-mixing transitions, ac Stark shifts for the ground and metastable states of divalent atoms., Comment: 4 pages, no figures

We demonstrate a one-dimensional optical lattice clock with a spin-polarized fermionic isotope designed to realize a collision-shift-free atomic clock with neutral atom ensembles. To reduce systematic uncertainties, we developed both Zeeman shift and vector light-shift cancellation techniques. By introducing both an H-maser and a Global Positioning System (GPS) carrier phase link, the absolute frequency of the $^1S_0(F=9/2) - {}^3P_0(F=9/2)$ clock transition of the $^{87}$Sr optical lattice clock is determined as 429,228,004,229,875(4) Hz, where the uncertainty is mainly limited by that of the frequency link. The result indicates that the Sr lattice clock will play an important role in the scope of the redefinition of the ``second'' by optical frequency standards., Comment: 10pages, 10 figures, submitted to J. Phys. Soc. Jpn

We have investigated the photoassociation (PA) spectra of ultracold $^{88}$Sr atoms near the $5s^2 ^1S_0+5s5p ^1P_1$ atomic asymptote. The intensity modulation of the PA lines was used to reconstruct the ground-state scattering wave function, whose nodal point inferred the s-wave scattering length of $a=1.1(2)$ nm. The observed PA resonances also determined a precise lifetime of the $^1P_1$ state to be 5.263(4) ns., Comment: 4 pages, 3 figures

We have measured the lifetime of the 5s5p ^3P_2 metastable state of strontium atoms by magneto-optically trapping the decayed atoms to the ground state, which allowed sensitive detection of the rare decay events. We found that the blackbody radiation-induced decay was the dominant decay channel for the state at T = 300 K. The lifetime was determined to be 500^{+280}_{-130} s in the limit of zero temperature., Comment: 4 pages, 3 figures, submitted to Physical Review Letters

We report on the spectroscopy of the $5s^2 {}^1S_0 (F=9/2) \to 5s5p {}^3P_0 (F=9/2)$ clock transition of ${}^{87}{\rm Sr}$ atoms (natural linewidth of 1 mHz) trapped in a one-dimensional optical lattice. Recoilless transitions with a linewidth of 0.7 kHz as well as the vibrational structure of the lattice potential were observed. By investigating the wavelength dependence of the carrier linewidth, we determined the magic wavelength, where the light shift in the clock transition vanishes, to be $813.5\pm0.9$ nm., Comment: 4 pages, 4 figures, submitted to Phys. Rev. Lett. (09/May/2003)

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}$., Comment: 4 pages, 2 figures, accepted for publication in Phys. Rev. Lett

We have demonstrated a recoil-free spectroscopy on the ${}^1S_0-{}^3P_1$ transition of strontium atoms confined in a one-dimensional optical lattice. By investigating the wavelength and polarization dependence of the ac Stark shift acting on the ${}^1S_0$ and ${}^3P_1(m_J=0)$ states, we determined the {\it magic wavelength} where the Stark shifts for both states coincide. The Lamb-Dicke confinement provided by this Stark-free optical lattice enabled the measurement of the atomic spectrum free from Doppler as well as recoil shifts., Comment: 5pages, 4figures