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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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-400-2000 200400
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
3.1 Stability 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.
3.2Frequency 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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0200400600
-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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020004000
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)
0200040006000
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 101001000
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
4.1Frequency 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. #MeasurementCorrectionUncertainty
time (s)(Hz)(Hz)
1910− 10.8
− 10.8
− 10.8
+ 2.7
10.7
291404.7
340206.4
434605.1
54530+ 3.85.1
66580+ 3.8 4.7
75480+ 5.73.9
86470+ 5.73.4
99320+ 10.73.0
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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