# Improved frequency measurement of a one-dimensional optical lattice clock with a spin-polarized fermionic $^{87}$Sr isotope

**ABSTRACT** 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

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- Ning Chen, Min Zhou, Hai-Qin Chen, Su Fang, Liang-Yu Huang, Xiao-Hang Zhang, Qi Gao, Yan-Yi Jiang, Zhi-Yi Bi, Long-Sheng Ma, Xin-Ye Xu[Show abstract] [Hide abstract]

**ABSTRACT:**An optical atomic clock with 171Yb atoms is devised and tested. By using a two-stage Doppler cooling technique, the 171Yb atoms are cooled down to a temperature of 6±3 μK, which is close to the Doppler limit. Then, the cold 171Yb atoms are loaded into a one-dimensional optical lattice with a wavelength of 759 nm in the Lamb—Dicke regime. Furthermore, these cold 171Yb atoms are excited from the ground-state 1S0 to the excited-state 3P0 by a clock laser with a wavelength of 578 nm. Finally, the 1S0-3P0 clock-transition spectrum of these 171Yb atoms is obtained by measuring the dependence of the population of the ground-state 1S0 upon the clock-laser detuning.Chinese Physics B 09/2013; 22(9):0601-. · 1.39 Impact Factor - Junye, Sebastianblatt, Martin M.boyd, Seth M.foreman, Eric R.hudson, Tetsuyaido, Benjaminlev, Andrew D.ludlow, Brian C.sawyer, Benjaminstuhl, Tanyazelinsky[Show abstract] [Hide abstract]

**ABSTRACT:**Ultracold atoms and molecules provide ideal stages for precision tests of fundamental physics. With microkelvin neutral strontium atoms confined in an optical lattice, we have achieved a fractional resolution of 4 × 10-15 on the 1S0–3P0 doubly forbidden 87Sr clock transition at 698 nm. Measurements of the clock line shifts as a function of experimental parameters indicate systematic errors below the 10-15 level. The ultrahigh spectral resolution permits resolving the nuclear spin states of the clock transition at small magnetic fields, leading to measurements of the 3P0 magnetic moment and metastable lifetime. In addition, photoassociation spectroscopy performed on the narrow 1S0–3P1 transition of 88Sr shows promise for efficient optical tuning of the ground state scattering length and production of ultracold ground state molecules. Lattice-confined Sr2 molecules are suitable for constraining the time variation of the proton–electron mass ratio. In a separate experiment, cold, stable, ground state polar molecules are produced from Stark decelerators. These cold samples have enabled an order-of-magnitude improvement in the measurement precision of ground state, Λ doublet microwave transitions in the OH molecule. Comparing the laboratory results to those from OH megamasers in interstellar space will allow a sensitivity of 10-6 for measuring the potential time variation of the fundamental fine structure constant Δα/α over 1010 years. These results have also led to improved understandings of the molecular structure. The study of the low magnetic field behavior of OH in its 2Π3/2 ro-vibronic ground state precisely determines a differential Landé g factor between opposite parity components of the Λ doublet.International Journal of Modern Physics D 01/2012; 16(12b). · 1.42 Impact Factor - Masami Yasuda, Hajime Inaba, Takuya Kohno, Takehiko Tanabe, Yoshiaki Nakajima, Kazumoto Hosaka, Daisuke Akamatsu, Atsushi Onae, Tomonari Suzuyama, Masaki Amemiya, Feng-Lei Hong[Show abstract] [Hide abstract]

**ABSTRACT:**We demonstrate an improved absolute frequency measurement of the 1S0--3P0 clock transition at 578 nm in 171Yb atoms in a one-dimensional optical lattice. The clock laser linewidth is reduced to ≈2 Hz by phase-locking the laser to an ultrastable neodymium-doped yttrium aluminum garnet (Nd:YAG) laser at 1064 nm through an optical frequency comb with an intracavity electrooptic modulator to achieve a high servo bandwidth. The absolute frequency is determined as 518 295 836 590 863.1(2.0) Hz relative to the SI second, and will be reported to the International Committee for Weights and Measures.Applied Physics Express 10/2012; 5(10):2401-. · 2.57 Impact Factor

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arXiv:physics/0608212v1 [physics.atom-ph] 22 Aug 2006

Typeset with jpsj2.cls <ver.1.2>

Full Paper

Improved Frequency Measurement of a One-Dimensional Optical Lattice

Clock with a Spin-Polarized Fermionic87Sr Isotope

Masao Takamoto1, Feng-Lei Hong2∗, Ryoichi Higashi1, Yasuhisa Fujii2, Michito Imae2,

and Hidetoshi Katori1†

CREST, Japan Science and Technology Agency, 4-1-8 Honcho Kawaguchi, Saitama, Japan.

1Department of Applied Physics, Graduate School of Engineering, The University of Tokyo,

Bunkyo-ku, 113-8656 Tokyo, Japan.

2National Metrology Institute of Japan (NMIJ), National Institute of Advanced Industrial Science

and Technology (AIST)

Tsukuba, 305-8563 Ibaraki, Japan.

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 the1S0(F = 9/2) −3P0(F = 9/2) clock

transition of the87Sr 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.

KEYWORDS: atomic clock, spin polarization, fermionic isotope, collision-shift, optical lattice,

Global Positioning System (GPS), carrier phase link, absolute frequency mea-

surement, light shift, redefinition of the “second”

1.Introduction

The rapid development of research on optical frequency measurement based on femtosec-

ond combs1),2)has stimulated the field of frequency metrology, especially research on high-

performance optical frequency standards. Optical frequency standards based on single trapped

ions3)–6)and ultracold neutral atoms in free fall7),8)have provided record levels of performance

that approach those of the best Cs fountain clocks with a fractional frequency uncertainty of

below 1 × 10−15.9)

We have proposed a novel approach named an “optical lattice clock”, in which atoms

trapped in an optical lattice potential serve as quantum references.10)–12)The sub-wavelength

localization of a single atom in each lattice site suppresses the first-order Doppler-shift and

collisional-frequency-shift while it provides a long interrogation time of over 1 s. The light shift

∗E-mail address: f.hong@aist.go.jp

†E-mail address: katori@amo.t.u-tokyo.ac.jp

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induced by the trapping field can be precisely canceled out by carefully tuning the lattice laser

wavelength to the “magic wavelength”.10),11),13)–15)

Recently, a higher order light shift, which is not canceled out at the “magic wavelength”

and that imposes an uncertainty limit on the lattice clock scheme, was observed in a Sr lattice

clock and revealed to affect only the accuracy at a level below 10−18.16)Optical lattice clocks

have already achieved a linewidth that is one order of magnitude narrower12),17)than that

observed for conventional neutral-atom optical clocks.7),8)The very high potential stability

as well as accuracy of this scheme would permit the measurement of a fractional frequency

difference at the 10−18level in a minute,11)which may open up new applications for ultra-

precise metrology, such as the search for the time variation of fundamental constants,4)the

real time monitoring of the gravitational frequency shift and the redefinition of the “second”.

Until now, optical lattice clocks have only been demonstrated with one-dimensional (1D)

optical lattices employing spin-unpolarized fermions12),15),18),19)or bosons.17)While these

experiments have clearly demonstrated the advantage of Lamb-Dicke confinement20)provided

by an optical lattice, collisional-frequency shifts should exist because of the relatively high

atomic densities of up to 1011/cm3trapped in a single lattice site.15),21)This collision shift

would ultimately be a fatal accuracy problem for 1D optical lattice clocks as witnessed for

such neutral atom based clocks as Cs fountain clocks9)or Ca optical clocks,7),8)in which

collision shifts dominate their uncertainty budgets. Even in the presence of other particles,

it has been predicted22)and demonstrated23),24)in the RF transition that the collisional

frequency shifts can be suppressed through the Pauli exclusion principle25)–27)by employing

ultracold spin-polarized fermionic atoms.

In this paper, we demonstrate, for the first time, a 1D optical lattice clock with ultracold

spin-polarized fermionic atoms, which, in principle, would realize collisional-shift-free atomic

clocks. In addition, the Zeeman shift and the vector light shift cancellation technique have

been introduced to further improve the clock accuracy. Furthermore, an improved frequency

measurement based on an H-maser and a Global Positioning System (GPS) carrier phase link

is discussed in detail.

The absolute frequency of the transition for the Sr lattice clock was first determined to

be 429,228,004,229,952(15) Hz using a Cs clock referenced to the SI second.12),28)Later the

JILA group measured the frequency and found it to be 429,228,004,229,869(19) Hz.18)The

measurement results obtained by the two groups were in poor agreement at a level of three

times the combined uncertainties.

In order to resolve this inconsistency, we have improved the absolute frequency measure-

ments based on an H-maser linked to UTC (NMIJ) using GPS carrier phase signals. The UTC

(NMIJ) is in turn linked to international atomic time (TAI). The Allan standard deviation

is obtained for the Sr lattice clock and is found to reach 2 × 10−15at an averaging time of

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1S0(F=9/2)

Cooling/Detection

λ = 461 nm

mF = -9/2

mF = +9/2

Clock

λ = 698 nm

f+9/2

3P0(F=9/2)

mF = -9/2

mF = +9/2

1P1(F=9/2)

1P1(F=7/2)

1P1(F=11/2)

3P1(F=11/2)

3P1(F=7/2)

3P1(F=9/2)

Cooling/Trapping

Optical pumping

λ = 689 nm

f-9/2

Fig. 1.Energy levels for87Sr atoms. Spin-polarized ultracold87Sr atoms were prepared by using the

1S0−1P1transition at λ = 461 nm and the1S0−3P1transition at λ = 689 nm. The first-order

Zeeman shift on the clock transition at λ = 698 nm was eliminated by averaging the transition

frequencies f±9/2, corresponding to the1S0(F = 9/2,mF = ±9/2) −3P0(F = 9/2,mF = ±9/2)

clock transitions, respectively.

1300 s. The newly obtained absolute frequency in this work is 429,228,004,229,875 Hz, with

an uncertainty of 4 Hz. This frequency value differs from that of our previous measurement

by five times the combined uncertainty but falls within the uncertainty of the JILA value.

We reported the preliminary results of our improved frequency measurement at CLEO/QELS

2006.29)Later we learned that the SYRTE group had also reported a measured frequency value

for the Sr lattice clock of 429,228,004,229,879(5) Hz on the arXiv19)during the CLEO/QELS

conference. There is good agreement between the measurement results obtained by the three

groups.

2.Method

2.1 Experimental setup

Figure 1 shows relevant energy levels for87Sr atoms. Strontium atomic beams effused from

an oven heated at 800 K were decelerated and magneto-optically trapped on the1S0−1P1

transition at λ = 461 nm down to a few mK. The atoms were further cooled and trapped using

the Dynamic Magneto-Optical Trapping (DMOT) scheme21)on the1S0−3P1transition at λ =

689 nm. Ultracold Sr atoms at a few µK were then loaded into a 1D optical lattice consisting

of a standing wave laser operated at the “magic wavelength” of λL= 813.428 nm.12),16)The

lattice laser was focused into an e−2beam radius of 30 µm. At the anti-node of the standing

wave a peak power density of IL=10 kW/cm2was obtained, which gave axial and radial trap

frequencies of νx= 40 kHz and νy(≈ νz) = 350 Hz, respectively.

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Optical frequency comb

f

fc

fn

fCEO

fb

frep

fAOM=ferr+ 40 MHz

PC+Synthesizer

fAOM±γ/2

AOM

Optical lattice with Sr

Counter

800 MHz

Synthesizer

H-maser

(Tokyo)

Carrier phase

common view

GPS

satellite

TAI

UTC(NMIJ)

(Tsukuba)

Clock laser

(698 nm)

ULE cavity

f’rep

Optical fiber with

noise canceller

DBM

f’b

Fig. 2.Schematic diagram of the experimental setup. AOM, acousto-optic modulator; PC, personal

computer; DBM, double balanced mixer; ULE, ultra-low expansion; GPS, Global Positioning

System; UTC, coordinated universal time; NMIJ, National Metrology Institute of Japan; TAI,

international atomic time. A beat signal f′

bwas used to stabilize the n-th comb component to the

Sr transition frequency.

Figure 2 shows a schematic of the experiment. A clock laser operating at λ0= 698 nm

was frequency-stabilized to a high-finesse ULE cavity with its finesse of 430,000 to reduce

the laser linewidth to about 20 Hz. The clock laser was steered into a Sr optical lattice15),30)

through a polarization-maintaining single-mode optical fiber, where a fiber noise canceller31)

was installed. An acousto-optic modulator (AOM) operating at about 40 MHz was used for

both frequency control and intensity switching to produce an excitation π-pulse. The beam

waist of the clock laser was 300 µm to reduce intensity inhomogeneity.

2.2Spectroscopy of trapped atoms

2.2.1 Spin polarization and cold collision suppression

In 1D optical lattice clocks, collisional frequency shifts should occur in each lattice site,

where typically a few tens of atoms were trapped, corresponding to an atomic density of

1011/cm3. Its suppression, therefore, is an important matter to be resolved if we are to realize

accurate atomic clocks.

In ultracold collisions, in general, the partial waves that contribute to the collisions are

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Clock laser

k(//x)

σ±

Optical pumping

x

y

z

O

EC(//z)

1D Optical latttice

EL(//z)

B0(//z)

Fig. 3.Atoms loaded into a 1D optical lattice were spin-polarized by a σ±polarized optical-pumping

laser operating on the1S0(F = 9/2) −3P1(F = 9/2) transition at 689 nm in the presence of a

bias magnetic field of B0. A clock laser with a wave-vector k was introduced along the 1D lattice

axis and excited the1S0(F = 9/2) −3P0(F = 9/2) clock transition. The electric field vectors for

both the lattice laser ELand the clock laser ECwere parallel to the bias magnetic field B0.

greatly limited by the centrifugal barrier for the relative angular momentum l of collision

pairs. The effective potential is given by,

Ueff(r) = −C6

r6+?2

2µ

l(l + 1)

r2

,(1)

where r is the inter-atomic distance, C6the van der Waals constant, ? the Planck constant, and

µ the reduced mass of collision pairs. For example, the p-wave (l = 1) barrier for Sr atoms in the

1S0ground state is estimated to be about 96 µK assuming C6= 3103 a.u. (atomic units),32)

which is well above the atomic temperature of 3 µK used in this experiment. In addition,

even partial waves, such as the s-wave, are not allowed for spin-polarized fermions due to the

anti-symmetrization of the wavefunction. Therefore we expect the collisional frequency shift

to be effectively suppressed by spin-polarizing ultracold fermionic87Sr atoms. Furthermore

it has been demonstrated23),24)in an RF spectroscopy of ultracold fermions that the atoms

remain identical and cannot interact in the s-wave regime in the coherent transfer process.

Spin polarization was performed as follows: In a presence of a bias magnetic field |B0| =

50 mG applied perpendicular to the lattice beam axis as shown in Fig. 3, a circularly σ±

polarized pumping-laser resonant to the1S0(F = 9/2)−3P1(F = 9/2) transition was applied

for 50 ms to optically-pump the atomic population to the mF = ±9/2 Zeeman substates

in the1S0(F = 9/2) ground state, respectively. With this optical pumping scheme, heating

of spin-polarized atoms can be minimized, as the mF = ±9/2 final states are the “dark

states” with this pumping laser. The typical atomic temperature was about 3 µK after spin-

polarization, and more than 95 % of the atoms were transferred to the stretched mF= ±9/2

state depending on the helicity σ±of the pumping laser.

Then the bias magnetic field was increased to |B0| = 1.78 G to completely resolve adjacent

Zeeman components in the1S0(F = 9/2,mF = ±9/2) −3P0(F = 9/2,mF = ±9/2) clock π-

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transition (see Fig. 1). This clock laser with its wave-vector k(= (kx,ky,kz)) was introduced

along the lattice beam. The electric field vector of the clock laser ECand the lattice laser EL

were parallel to the bias magnetic field B0as shown in Fig. 3.

2.2.2Dephasing of Rabi oscillation

The Rabi frequency Ωn,nfor a trapped atom in the (nx,ny,nz)-th vibrational level in the

lattice potential is described as,33)

Ωn,n= Ω0

?

j=x,y,z

|?nj|exp(ikjj)|nj?|, (2)

where Ω0is the Rabi frequency of the electronic transition (clock transition), |nj? the wave

function of the harmonic oscillator state of an atom in the lattice potential along j = x,y,z

direction, and (x,y,z) the displacement of the atom. Here we assumed an elastic component

(∆nj= 0) in the vibrational transition that is relevant to the clock signal. Using Laguerre

polynomial Ln(x), each matrix element, say j = x component, can be written as,33)

?nx|exp(ikxx)|nx? = exp(−η2

x/2)Lnx(η2

x), (3)

where ηx= kxx0is the Lamb-Dicke parameter, x0=

?h/(2mνx)/(2π) the spatial extent of

the ground state wavefunction along x-axis, and m the mass of an trapped atom. The Lamb-

Dicke parameter for x-direction was ηx = 0.34. Similarly we estimated that for the radial

direction ηy= kyy0and ηz= kzz0to be less than 0.05, where we assumed the uncertainty in

aligning the clock laser with respect to the lattice laser to be less than 12 mrad.

These finite Lamb-Dicke parameters ηj gave rise to the variation of Rabi frequencies

Ωn,nthat depended on the vibrational states |nj? of the atoms through Laguerre polynomial

Lnj(η2

j). The thermal occupation of the vibrational states was typically ¯ nx∼ 1.1 and ¯ ny≈

¯ nz∼ 180 for the trapped atom temperature. The thermal distribution of the vibrational states

introduced a vibrational-state-dependent Rabi frequencies for each trapped atom, leading

to the dephasing of the Rabi oscillation as shown in Fig. 4(a). As a result, the excitation

probability for the 2-ms-long π clock pulse was degraded to about ξ ≈ 0.8, as deduced by a

fit in Fig. 4(a). In this measurement, the clock laser was tuned to the atomic resonance with

an intensity of 25 mW/cm2.

This dephasing of the Rabi oscillations can be suppressed by applying sideband cooling

to the axial motion and Doppler cooling to the radial motion,21)and by improving the beam

overlap between the clock and lattice lasers.

2.2.3Normalization of the spectrum

The excitation of the clock transition was observed by measuring the laser induced fluores-

cence by driving the1S0−1P1cyclic transition as shown in Fig. 1. This fluorescence intensity

ISwas proportional to the atom number NSremaining unexcited in the1S0ground state. In

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-400-2000200400

0.0

0.2

0.4

0.6

0.8

1.0

Excited atom fraction κ

Clock laser frequency (Hz)

01234

0.0

0.2

0.4

0.6

0.8

1.0

Excited atom fraction κ

Pulse duration (ms)

(a)

(b)

Fig. 4.(a) Rabi oscillation of atoms in the 1D optical lattice. The excited atom fraction κ was

measured as a function of the duration of the clock laser resonant to the1S0(F = 9/2)−3P0(F =

9/2) transition. (b) Typical clock spectrum obtained with a 10-ms-long π-pulse clock laser. The

excited atom fraction κ was measured as a function of the detuning of the clock laser.

order to normalize the excited atom fraction, we measured the number of atoms NP in the

3P0state. After blowing out the unexcited atoms in the ground state by irradiating the laser

resonant with the1S0−1P1transition, the atoms in the3P0state were deexcited to the1S0

ground state by the 0.5-ms-long clock laser with a pulse area of π. Then we measured the laser

induced fluorescence intensity IP on the1S0−1P1transition. This fluorescence intensity was

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proportional to ξNP, where ξ ≈ 0.8 was the (de)excitation efficiency for the π clock-pulse

that was given by the inhomogeneous Rabi frequencies as discussed previously. The excited

atom fraction κ was thus calculated as,

κ =

NP

NS+ NP

=

IP

ξIS+ IP.(4)

Figure 4(b) shows the typical clock transition excited with the π-pulse clock laser with an

intensity of 1 mW/cm2. The excited atom fraction κ = κ(δν) was measured as a function of

the clock laser detuning δν from the atomic resonance. The duration of the clock excitation

pulse was 10 ms, and a nearly Fourier limited linewidth of 80 Hz was observed.

2.2.4Frequency stabilization and Zeeman shift cancellation

Frequency stabilization34)of the clock laser to the spectrum center was realized by feedback

control of the AOM frequency (see Fig. 2) using the error signal ferr(tn) obtained by a digital

servo loop as,

ferr(tn+1) = ferr(tn) + δf(tn).(5)

Here δf(tn) is the correction signal measured in the n-th interrogation period at t = tnas,

δf = γ ×κ(+γ/2) − κ(−γ/2)

where γ ≈ 80 Hz is the full width at half maximum (FWHM) linewidth of the observed

spectrum, and κ(±γ/2) is the atom excitation probability near the side slopes of the Rabi

excitation spectrum with respect to the stabilized line center at t = tn. In this servo loop,

2

,(6)

the drift rate of the clock laser frequency was evaluated in advance and was fed forward

to minimize servo errors occurring during the frequency stabilization. We estimate that the

servo-error would be of the order of 0.1 Hz. The application of a second order integrator, in

addition to the currently used first order integrator, will further reduce servo errors.

The measured spectrum was Zeeman shifted by a bias magnetic field of B0. As depicted

in Fig. 1, the Zeeman splitting is smaller in the3P0(F = 9/2) state than in the1S0(F = 9/2)

state due to hyperfine mixing in the3P0(F = 9/2) excited state. This introduces a first order

Zeeman shift of mF× 106 Hz/G11)for the π-transition excited from the mF sublevel in the

1S0(F = 9/2) ground state. This linear Zeeman shift can be eliminated by averaging the

two transition frequencies f±9/2, corresponding to the1S0(F = 9/2,mF = ±9/2) −3P0(F =

9/2,mF = ±9/2) transition frequencies, respectively. The transition frequency free from the

first order Zeeman shift is thus given by,

f0=f+9/2+ f−9/2

2

.(7)

Since a single measurement took 1 s or less, a cycle time tc = tn− tn−1 of nearly 2 s

was required to determine one of the Zeeman components f±9/2. The cycle time required for

cooling, trapping, and interrogating atoms in the lattice, was not optimized in this experiment.

A further reduction of the cycle time to less than 1 s will be feasible in a future experiment.

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2.3Frequency measurement

To measure the absolute frequency of the Sr lattice clock, a frequency comb system was

brought from NMIJ, AIST in Tsukuba to the University of Tokyo, where the optical lattice

clock was operated. Instead of the commercial Cs clock that was used as a local time base in

our previous frequency measurement,12),28)we moved an H-maser (Kvarz, Model CH1-75A)

from Tsukuba to Tokyo to reduce the measurement uncertainty.

In the frequency comb, the frequency of the n-th comb component is expressed as

fn= n × frep+ fCEO,(8)

where frep is the repetition rate of the laser pulse and fCEO is the carrier-envelope offset

frequency.2)When frepand fCEOare precisely controlled, the comb works as a “frequency

linker” that connects optical and radio frequencies. The control ports of frepand fCEOin the

Ti:sapphire laser are the lengths of the laser cavity and the pump laser power, respectively.

Our comb system is described in detail elsewhere.28),35)

In this experimental configuration, since the laser light after the AOM was a pulsed-light

used for the Sr spectroscopy, its frequency could not be directly measured with the frequency

comb. The frequency relations in the frequency measurement are shown in Fig. 2. We first

measured the beat frequency fb= |fc− fn| between the clock laser fcand the n-th tooth

of the comb fn with a photo-diode. We then electronically mixed the beat note fb with

fAOM= ferr+ 40 MHz with a double balanced mixer (DBM) and extracted the frequency

component,

f′

b= |fc+ fAOM− fn|, (9)

which corresponded to the + 1 order light diffracted by the AOM. This frequency is equal to a

beat frequency between the Sr-transition frequency fSr= fc+fAOMand the n-th tooth of the

comb fn. In our measurement scheme, f′

bwas used to phase-lock the n-th comb component

to the Sr clock transition by feedback controlling the cavity length of the mode-locked laser.

In this way, the whole comb was locked to the Sr clock transition, which means that the

stability of each comb component and frepfollows that of the Sr clock transition. frepwas

measured against the H-maser as follows. frep was observed at about 793 MHz and down

converted to a frequency of f′

rep= 800 MHz−frep≈7.3 MHz by using a DBM and a low-pass

filter. The frequency of 800 MHz was generated by using a synthesizer with a fixed frequency

of exact 800 MHz. f′

repwas measured and recorded with a universal counter (Agilent, Model

53132A). All the synthesizers and counters used in this experiment were phase locked to the

H-maser through a distribution amplifier. Finally, fSrwas calculated by using the equation:

fSr= n × (800MHz − f′

rep) + fCEO± f′

b, (10)

where f′

repwas measured by the counter, while fCEOand f′

digital phase-lock loops. The integer n was simply determined by solving eq. (8) for n and

bwere set at fixed frequencies using

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requiring n to be an integer, since we already know fSrwith an uncertainty much smaller

than frep. The sign of f′

bin eq. (10) was determined by changing frepslightly and observing

the variation of the f′

bin the experiment.

To calibrate the frequency of the H-maser, GPS carrier phase receivers (Javad, Model

#Lexon-GGD) were employed at both sites, the University of Tokyo and NMIJ. In our previ-

ous measurement,12),28)a GPS disciplined oscillator was introduced to link the local Cs clock

to the GPS time.36)To further improve the link precision, in the present experiment, the

H-maser was calibrated based on UTC (NMIJ) using the GPS carrier-phase technique with

the analysis software “GIPSY”.37)The distance between Tokyo and Tsukuba is about 51 km.

The relationship between the UTC (NMIJ) and TAI can be found in the monthly reports of

the Circular-T of the Bureau International des Poids et Mesures (BIPM).38)

3. Experimental results

3.1Stability evaluation of the lattice clock locked to the two Zeeman components

In order to evaluate the clock stability at optical frequencies, the optical lattice clock

was alternately stabilized to two Zeeman components, i.e., the1S0(F = 9/2,mF = ±9/2) −

3P0(F = 9/2,mF = ±9/2) clock transition (see Fig. 1), corresponding to the transition

frequencies f±9/2, respectively. The duration of the clock excitation pulse was 10 ms, and a

nearly Fourier limited linewidth of 80 Hz was observed. Four successive measurements were

used to lock the clock laser frequency to the f+9/2and f−9/2transition frequencies.

Error signals ferrthat were fed back to the AOM (see Fig. 2) are shown in Fig. 5(a) as a

function of elapsed time. The top and bottom curves correspond to the case when the clock

laser frequency is locked to the f+9/2(top) and f−9/2(bottom), respectively. The curve in the

middle shows the average of these two Zeeman components f0(see eq. (7)), which provided

the first-order Zeeman-shift-free transition frequency as discussed previously. The slope of

these curves compensated the ULE cavity drift rate, which was found to be −0.16 Hz/s.

The two clock frequencies f+9/2and f−9/2can be regarded as output signals generated

by two independent optical clocks locked to different Zeeman components. The optical beat

note of these two clocks ∆f = f+9/2− f−9/2can be evaluated by the offset of these two error

signals, namely the top and bottom curves in Fig. 5(a), and was used to evaluate the stability

of lattice clocks. Figure 5(b) shows the Allan standard deviation, which was measured for

up to 7600 s. Until 10 s, when the servo loop started to work, the Allan standard deviation

increased. After that the deviation started to decrease with σ(τ) = 8 × 10−14/√τ. The floor

of the Allan deviation was not observed during this measurement for a time of over 2 h.

3.2 Frequency measurement and stability evaluation

Figure 6 shows the measured frequencies of the Sr clock transition against the H-maser. In

Fig. 6(a), the clock laser was stabilized to the1S0(F = 9/2,mF= −9/2)−3P0(F = 9/2,mF=

−9/2) Zeeman component with f−9/2, while in Fig. 6(b) the clock laser was stabilized to the

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0 200 400 600

-5.0

-4.5

-4.0

-3.5

Error signals ferr (kHz)

Time (s)

(a)

(b)

1101001000

10

-15

10

-14

Allan standard deviation

Averaging time (s)

Fig. 5. (a) Error signals ferrfed back to the clock laser stabilized to the ULE cavity as a function

of time. The three curves show the error signals corresponding to f+9/2(top), f0(middle), and

f−9/2(bottom), respectively. f0 provided the first-order Zeeman-shift-free clock transition fre-

quency. The slope of these curves indicated the ULE cavity drift rate of −0.16 Hz/s. (b) Allan

standard deviation evaluated by the optical lattice clock independently locked to the two Zeeman

components as shown in (a).

center of the clock transition f0by using the spectra of both Zeeman components, i.e., f+9/2

and f−9/2. The gate time of the universal counter was set at 10 s. The scatter of the measured

frequencies in both figures mainly results from the instability of the H-maser.

The measurement time for Figs. 6(a) and (b) were 9140 s and 7610 s with average frequen-

cies of 429,228,004,229,019.5 and 429,228,004,229,867.0 Hz, respectively. Note that systematic

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0 20004000

Time (s)

60008000

429,228,004,228,980

429,228,004,229,000

429,228,004,229,020

429,228,004,229,040

429,228,004,229,060

429,228,004,229,080

Frequency (Hz)

02000 40006000

429,228,004,229,820

429,228,004,229,840

429,228,004,229,860

429,228,004,229,880

429,228,004,229,900

429,228,004,229,920

Time (s)

Frequency (Hz)

(a)

(b)

Fig. 6. Variation in the Sr-stabilized laser frequency measured against the H-maser as a function

of time. The averaging time was 10 s. Systematic corrections are not applied. (a) Stabilized to

the1S0(F = 9/2,mF = −9/2) −3P0(F = 9/2,mF = −9/2) Zeeman components of f−9/2; (b)

stabilized to the line center f0.

corrections are not applied in these figures. A total of ten measurements were performed on

different days over a period of more than one month. The measurements in Figs. 6(a) and (b)

are denoted as measurement #2 and #10, respectively.

The Allan standard deviations were calculated and indicated in Fig. 7: A solid curve with

filled triangles and a dashed curve with filled circles correspond to the measured frequencies

indicated in Figs. 6(a) and (b), respectively. Both curves show similar trend for the Allan

deviation of the Sr-stabilized laser as shown in Fig. 5(b). For comparison, the Allan standard

deviation of the H-maser is also shown in Fig. 7 as a solid curve. We notice that, for the short

term (averaging time between 1 and 20 s), the measured stability was basically limited by the

H-maser. For the long term (averaging time longer than 20 s), the measured Allan deviation

shows the fractional frequency instability of the Sr lattice clock, which is consistent with the

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1 10100 1000

10

-15

10

-14

10

-13

Locked to f-9/2

Locked to f0

Allan standard deviation

Averaging time (s)

H-maser

Fig. 7. Allan standard deviation of the Sr lattice clock measured based on an H-maser. A solid curve

with filled triangles and a dashed curve with filled circles correspond to the measured frequencies

indicated in Figs. 6(a) and (b), respectively. A solid curve below indicates the Allan standard

deviation of the H-maser.

Allan standard deviation derived from the optical beat note shown in Fig. 5(b). The Allan

standard deviation of the Sr lattice clock has reached a level of 2×10−15at an averaging time

of 1300 s. This means that the statistical error of each measurement is reduced to about 0.9

Hz after a 1300 s averaging time.

4.Frequency corrections and uncertainties

4.1 Frequency link

To reach the final measurement result, systematic corrections were applied to the obtained

averaged frequencies. In the present measurement, the main correction is from the H-maser,

which was calibrated based on the UTC (NMIJ) using the GPS carrier-phase technique.

Figure 8(a) shows a typical recording of the time difference between the UTC (NMIJ) re-

alized in Tsukuba and the H-maser located at the University of Tokyo. The data was recorded

every 300 s over 2 days. The slope of the curve indicates the fractional frequency offset between

the UTC (NMIJ) and the H-maser, which was calculated to be −1.33 × 10−14using a linear

fitting. The minute variations in the measured data around the fitted solid line are considered

to be mainly caused by the atmospheric fluctuation along the link path. Figure 8(b) shows

the Allan standard deviation of the GPS link calculated from the measured data shown in

Fig. 8(a). The Allan standard deviation was 6.6 × 10−14for a 300 s averaging time, which

improved to 3 × 10−15after a 30000 s averaging time.

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10

Averaging Time (s)

3

10

4

10

-15

10

-14

10

-13

Allan standard deviation

0.0

5.0x10

4

1.0x10

5

1.5x10

5

1.4782610

1.4782615

1.4782620

1.4782625

1.4782630

UTC(NMIJ) - HM_KV

Slope = -1.33 × × × × 10-14

Time difference (ms)

Time (s)

(a)

(b)

Fig. 8. (a) Time difference between the UTC (NMIJ) and the H-maser measured by the GPS link

between Tokyo and Tsukuba using carrier-phase signals. The slope indicates the fractional fre-

quency difference between the UTC (NMIJ) and the H-maser. (b) The frequency stability of the

GPS link.

During these two days, we performed measurements #7 and #8. Based on the results

of the GPS link, the correction from the H-maser was calculated to be +5.7 Hz as listed in

Table I. The uncertainty of the correction is basically limited by the stability of the GPS link

obtained within the measurement time (also listed in Table I).

The uncertainties of the H-maser correction were calculated to be 3.9 and 3.4 Hz for mea-

surements #7 and #8, respectively. The corrections and uncertainties of all ten measurements

contributed from the calibration of the H-maser are listed in Table I.

Note that the correction from the H-maser gradually varied from −10.8 to + 10.7 Hz during

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Table I. Corrections and uncertainties of the H-maser calculated based on the GPS link.

Meas. #MeasurementCorrection Uncertainty

time (s)(Hz)(Hz)

1 910− 10.8

− 10.8

− 10.8

+ 2.7

10.7

291404.7

340206.4

434605.1

54530+ 3.85.1

66580+ 3.84.7

75480+ 5.73.9

86470+ 5.73.4

99320+ 10.7 3.0

107610+ 10.73.4

the 38-day measurement period, which was due to a fractional frequency drift of 6.6×10−14for

the H-maser. The fractional frequency drift of the H-maser at the NMIJ was about 1×10−14

over one month. The increased frequency drift at the University of Tokyo is considered to be

due to the short setting time and the difference in the laboratory environment after moving

the H-maser.

The results of the frequency link between the UTC (NMIJ) and the TAI can be found in

Circular T, which is published on the BIPM web page.38)From Circular T 221 and 222, the

correction and the uncertainty were calculated to be 0 and 1.4 Hz, respectively (see Table II).

UTC (NMIJ) has been very stable during the past several months.

A frequency counter usually counts a frequency with systematic and statistical errors. The

universal counter (Agilent 53132A) used in the present experiment was measured and found

to have a fractional systematic error (correction) of +7.4 × 10−13and a fractional statistical

error (uncertainty) of 2.0×10−13. Since the counter is used to read the f′

of the frepat 793 MHz, the correction and the uncertainty applied to the frepis −6.8×10−15

and 1.8×10−15, which corresponds to a correction of −2.9 Hz and an uncertainty of 0.8 Hz in

the Sr optical frequency (see Table II). The systematic correction and statistical uncertainty

repat 7.3 MHz instead

of the counter can be further reduced by increasing the gate time in the measurement. By

observing the higher order of frepwith a fast detector and increasing the frequency of the

synthesizer, the effect of the counter errors in the frequency measurement can also be further

reduced.

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