Optical lattice clocks and frequency comparison

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Optical lattice clocks and frequency comparisonTo cite this article Hidetoshi Katori et al 2011 J Phys Conf Ser 264 012011

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Optical lattice clocks and frequency comparison

Hidetoshi Katori Tetsushi Takano and Masao Takamoto

Department of Applied Physics The University of Tokyo Tokyo JapanCREST Japan Science and Technology Agency Saitama Japan

E-mail katoriamotu-tokyoacjp

Abstract We consider designs of optical lattice clocks in view of the quantum statisticsrelevant atomic spins and atom-lattice interactions The first two issues lead to two optimalconstructions for the clock a one-dimensional (1D) optical lattice loaded with spin-polarizedfermions and a 3D optical lattice loaded with bosons By taking atomic multipolar interactionswith the lattice fields into account an ldquoatomic motion insensitiverdquo wavelength is proposed toprovide a precise definition of the ldquomagic wavelengthrdquo We then present a frequency comparisonof these two optical lattice clocks spin-polarized fermionic 87Sr and bosonic 88Sr prepared in1D and 3D optical lattices respectively Synchronous interrogations of these two optical latticeclocks by the same probe laser allowed canceling out its frequency noise as a common modenoise to achieve a relative stability of 3times10minus17 for an averaging time of τ = 350 s The schemetherefore provides us with a powerful means to investigate intrinsic uncertainty of the clocksregardless of the probe laser stability We discuss prospects of the synchronous operation ofthe clocks on the measurement of the geoid height difference and on the search of constancy offundamental constants

1 IntroductionTo date optical clocks based on singly trapped ions [1] and ultracold neutral atoms trapped inoptical lattices that operate at the magic wavelength to eliminate light shift perturbation [2] areregarded as promising candidates for future atomic clocks Since 2006 Sr-based ldquooptical latticeclocksrdquo have been evaluated close to the Cs clocksrsquo uncertainty limit internationally [3 4] Tofurther reduce their uncertainty and instability there remain essential experimental challengesOne is to find out better lattice geometries as well as interrogated atom species that bring outthe potential performance of the clock scheme taking into account the collisional frequency shiftthe black body radiation shift and the lattice induced light shifts The other is to establishexperimental schemes that fully utilize the advantage of a large number N of atoms to improvethe clock stability which is currently hampered by the instability of the probe laser of typically5times 10minus16 at 1 s that is due to the thermal noise of a reference cavity [5]

In this paper we first describe designs of optical lattice clocks in view of the quantumstatistics and relevant atomic spins This leads to two optimal configurations for the clock aone-dimensional (1D) optical lattice loaded with spin-polarized fermions and a 3D optical latticeloaded with bosons [6] In particular we refer to an ldquoatomic motion insensitiverdquo wavelength thatprovides a precise definition [7] of a ldquomagic wavelengthrdquo including atomic multipolar interactions

We then present a frequency comparison of these two optical lattice clocks using fermionic87Sr and bosonic 88Sr Synchronous interrogations of the two clocks by the same probe laserallow canceling out its frequency noise as a common mode noise in evaluating the relative

22nd International Conference on Atomic Physics IOP PublishingJournal of Physics Conference Series 264 (2011) 012011 doi1010881742-65962641012011

Published under licence by IOP Publishing Ltd 1

Figure 1 (Color online) Energy levels for 88Sr and 87Sr atoms Spin-polarized ultracold 87Sratoms were prepared by optical pumping on the 1S0(F = 92) minus 3P1(F = 92) transition atλ = 689 nm with circularly polarized light The first-order Zeeman shift and the vector light shifton the clock transition at λ = 698 nm were eliminated by averaging the transition frequenciesfplusmn The lack of nuclear spin makes the energy structure of 88Sr simpler

stability of the clocks therefore the scheme enables us to explore the intrinsic uncertainty of theclock regardless of the probe laser stability On the other hand in order to overcome the clockinstability that is predominantly determined by the ldquothermal noise limitrdquo of the probe laser wemention a novel frequency stabilization scheme that does not employ the ldquoelectron shelvingrdquotechnique for the clock state detection We discuss possible impacts of these approaches

2 Designing optical lattice clocks Quantum statistics and optical lattice geometryIn addition to the removal of light shift perturbations the control and prevention of atomicinteractions are a serious concern in designing optical lattice clocks The collisional frequencyshift of atomic clocks operated with ultracold atoms is related to the mean field energy shiftE = 4πh2ang(2)(0)m of the relevant electronic state with a the s-wave scattering length natomic density and m atomic mass Here g(2)(0) is the two-particle correlation function atzero distance which is zero for identical fermions and 1 le g(2)(0) le 2 for distinguishable orbosonic atoms Hence collisional shifts are suppressed for ultracold fermions while they areintrinsically unavoidable for bosons The quantum statistical nature of atoms is determined bytheir total spins that is bosons have zero or integer spins and fermions have half-integer spinsIn particular for atoms with an even number of electrons that have a J = 0 state suitable foroptical lattice clocks their nuclear spins I may be zero for bosons and I ge 12 for fermionsConsequently the total angular momentum F = J+I of the clock states can be zero for bosonicatoms but not for fermions which causes coupling to the light polarization of the lattice fieldFigure 1 shows the relevant energy levels for fermionic 87Sr and bosonic 88Sr isotopes of Sr thatwe employ in the following experiments

We consider two lattice geometries A one-dimensional (1D) (see Fig 2(a)) or 2D latticecomposed of a single electric field vector has spatially uniform light polarization In contrast a3D lattice (see Fig 2(b)) requires at least two electric field vectors therefore the synthesized


2

B

E

E

Electric field

amplitude zLattice Probe

Lattice laser(a)

(c)

Bm mixing fieldE

(b)B

B

E

Magnetic field

amplitude

y

x

Lattice laser

Lattice laser

Probe laserx

yz

O

gravity

Ep

EL

E3

E1

E2

Figure 2 (Color online) (a) A one-dimensional optical clock is realized by a standing waveof light tuned to the magic wavelength Multiply trapped spin-polarized fermions which areindicated by allows in a single pancake potential may be protected from collisions by the Pauliblocking (b) Single occupancy 3D lattice (c) Electric and magnetic field amplitudes in astanding wave

field exhibits a polarization gradient that varies in space depending on the intensity profile andrelative phases of the lattice lasers In 1D optical lattice clocks with multiple atoms in each latticesite the application of spin-polarized fermions [8 9] may minimize the collisional frequencyshift owing to their quantum-statistical properties Figure 2(a) shows a schematic diagramfor the ldquospin-polarizedrdquo 1D optical lattice clock [10] where the upward arrows correspond tospin-polarized fermionic atoms The spatially uniform light field polarization of the 1D opticallattice allows us to cancel out the vector light shift [10] by alternately interrogating the transitionfrequencies fplusmn corresponding to two outer Zeeman components 1S0(FplusmnmF )minus 3P0(FplusmnmF ) ofthe clock transition (see Fig 1) This vector light shift cancellation technique simultaneouslycancels out the Zeeman shift thereby realizing virtual spin-zero atoms

The application of 3D optical lattices with a single (or less) atom in each lattice site is effectiveto suppress atomic collisions However light polarization inhomogeneity inevitable in 3D opticallattices makes a vector light shift for atoms with its angular momentum F = 0 problematic asthe ldquovector light shift cancellationrdquo technique is no longer applicable From this viewpoint the3D optical lattice clock is suitable for bosonic atoms with scalar clock states (J = 0) and hasbeen demonstrated with bosonic 88Sr atoms [6] Its systematic uncertainties were investigatedat 3times 10minus15 by referencing a spin-polarized 1D optical lattice clock with fermionic 87Sr atoms

Finally in order to moderate the hyperpolarizability effects that cannot be eliminated inparticular at the red-detuned magic wavelength where atoms are trapped at lattice intensitymaxima we investigated a blue-detuned lattice that confines atoms in the intensity minima ofthe electric field For Sr such wavelengths are found on the blue side of the 5s2 1S0 minus 5s5p 1P1

transition at 461 nm one such wavelength is found at λL asymp 390 nm on the blue side of the5s5p 3P0 minus 5s6d 3D1 transition at 394 nm For this magic wavelength a laser intensity ofIL = 10 kWcm2 gives a trap depth of about 200 kHz As the atoms are trapped near the


3

nodes of the standing wave the effective light intensity that atoms experience is about one-tenth of the intensity at the anti-nodes Assuming a trap depth of 10 microK the 4th-order lightshift is estimated to be 01 mHz corresponding to a fractional uncertainty of 2 times 10minus19 [11]The blue magic wavelength was experimentally determined to be 389889(9) nm [11] employing87Sr atoms trapped in a 1D optical lattice operated at the (red-detuned) magic wavelength ofλL = 8134 nm

3 Multipolar interactions of atoms with lattice field andatomic-motion-insensitive wavelengthOptical lattice clocks operated at the blue magic wavelength seem to be ideal in view of theelectric dipole (E1) interaction of atoms with the lattice laser field However when takingatomic multipolar interactions into account things are not that simple Consider for examplethe linearly polarized (||ez) standing wave electric field E = ezE0 sin ky cosωt with wave numberk and frequency ω as shown in Fig 2(c) Following the Maxwell equation nabla times E = minus1

cpartBpartt

with c the speed of light the corresponding magnetic field is given by B = minusexE0 cos ky sinωtThis indicates that the electric and magnetic field amplitudes are one quarter of the wavelengthλ4 = πc(2ω) out of phase in space Consequently the magnetic dipole (M1) interaction islargest at the nodes of the electric field Furthermore as the electric quadrupole (E2) interactionis proportional to the gradient of the electric field the E2 interaction is also largest at the nodesof the electric field Hence the blue magic wavelength is not necessarily free of light shift due toatomic multipolar interactions with the lattice field The energy shift of atoms in optical latticesis obtained by the second-order perturbation in the E1 M1 and E2 interactions that vary asVE1 sin

2 ky VM1 cos2 ky and VE2 cos

2 ky Consequently it is no longer possible to perfectlymatch the total light shift in two clock states as the differential light shifts due to the M1and E2 interactions introduce an atomic-motion-dependent light shift because of their spatialmismatch with the E1 interaction [12]

Although the contributions of the M1 and E2 interactions are 6-7 orders of magnitude smallerthan that of the E1 interaction in optical lattice clocks in Sr [2] they have a non-negligiblecontribution in pursuing a 1 times 10minus18 level uncertainty Therefore a more precise definition ofthe magic wavelength including multipolar interactions is necessary Assuming the differentialpolarizabilities of the E1 M1 and E2 interactions in the clock transition to be ∆αE1(λL)∆αM1(λL) and ∆αE2(λL) and the corresponding spatial distributions to be qE1(r) qM1(r) andqE2(r) the transition frequency of atoms in the optical lattices can be given by

ν(λL) = ν0 minus1

2h[∆αE1(λL)qE1(r) + ∆αM1(λL)qM1(r) + ∆αE2(λL)qE2(r)]E

2 (1)

where the 4th- and higher-order terms and light polarization dependences are omitted We haveshown that one can eliminate the atomic-motion-dependent light shift caused by the multipolarinteractions by choosing particular 3D optical lattice geometries that make qM1(r) or qE2(r)terms in phase or out of phase with respect to the spatial dependence of qE1(r) [7] For examplein the case of a 1D lattice with the E1 spatial dependence qE1(r) = sin2 ky

(= 1minus cos2 ky

)

the corresponding M1 and E2 interactions can be expressed as qM1(r) = qE2(r) = cos2 ky =∆q minus qE1(r) with ∆q = 1 Therefore by taking ∆αEM equiv ∆αE1 minus ∆αM1 minus ∆αE2 and∆α0 equiv ∆αM1 +∆αE2 Eq (1) can be rewritten as


2h∆αEM(λL)qE1(r)E

2 minus 1

2h∆α0(λL)∆qE2 (2)

where the second term on the right side varies in phase with the E1 interaction This equationsuggests the precise definition of the magic wavelength to be an ldquoatomic-motion insensitiverdquowavelength that satisfies ∆αEM(λL) = 0 The last term provides a spatially constant offset of


4

1D(f87)

f+92

f+B

f-92

f-B

f+92

f+92

f87=(f92+f-92)2

f88=(f+B+f-B)23D(f88)

time0 4Tc

probed synchronouslyCounter

88Sr in 3D Lattice

int

int

Probe laser

(698 nm)

Spin-polarized87Sr in 1D Lattice

AOM

(b)(a)

AOM

Shoulders of the Rabi spectrum

Tc 2Tc

Figure 3 (Color online) (a) Experimental setup for frequency comparison of 1D and 3Doptical lattice clocks A single laser with an RF frequency offset of asymp 62 MHz probes the clocktransitions of the two isotopes simultaneously (b) Timing chart for the frequency comparisonA cycle time of Tc = 14 s is used to probe a shoulder of the Rabi excitation Cancellation ofthe slowly varying residual magnetic fields is completed in 4Tc Thick vertical lines indicate aclock probing period of 100 ms

typically 10 mHz or below and is solely dependent on the total laser intensity prop ∆qE2 used toform the lattice This offset frequency can be accurately determined by measuring the atomicvibrational frequencies in the lattice [7]

4 Frequency comparison of 1D and 3D optical lattice clocksOptical lattice clocks [2] are expected to be highly stable eg σy(τ) sim 10minus18

radicτ if they operate

at the quantum projection noise (QPN) limit of σy(τ) ≃ (∆ff0)radicNτ by taking advantage of

the large number N sim 106s of atoms and moderately narrow linewidth ∆f = 1 Hz Howeverthe frequency comparison of two optical lattice clocks [6] shows that their actual stabilities areessentially limited by those of the probe lasers due to the thermal noise of the reference cavityeg σy sim 1times 10minus15 for a 75-cm-long cavity made of ultralow expansion (ULE) glass [5] and bythe Dick effect [13] which is attributed to the down-conversion of the high frequency (nTc withn an integer) components of the probe laser noise δω(t) introduced by the discrete interrogationprocess with a cycle time Tc of the clock operation

When exciting the clock transition the population fluctuation δP of the excited state causedby the frequency noise δω(t) is expressed using the sensitivity function g(t) as in [13]

δP =1

2

int Tc

0g(t)δω(t)dt (3)

where Tc is a clock cycle time The frequency servo translates a population fluctuation δP into afrequency fluctuation of the probe laser which degrades the stability of an atomic clock knownas the Dick effect

We consider a frequency comparison of two optical lattice clocks that are simultaneouslyprobed by a single laser which allows the two clocks to be probed by the same frequency noiseδω(t) As long as the sensitivity functions g(t) are the same for both clocks their populationfluctuations δP should be equal therefore the stability degradation due to δP may be rejectedas a common mode noise in evaluating the frequency difference of the two clocks This allowsan investigation of the relative stability of the two clocks that is only limited by the quantumprojection noise of atoms regardless of the probe laser instability Such a synchronous frequencycomparison was previously demonstrated in the microwave frequency domain using Cs and Rbfountain clocks [14]


5

Figure 3(a) shows an experimental setup for frequency comparison of the 1D and 3D latticeclocks loaded with 87Sr and 88Sr respectively as described in Sec 2 After cooling and loadingatoms in each optical lattice we interrogated the two clock transitions synchronously with a100 ms-long Rabi pulse A cycle time of Tc = 14 s included the time spent for cooling andtrapping of atoms in the lattices and for measuring the excitation probability by observing ashoulder of the Rabi spectrum as depicted in Fig 3(b) The clock laser frequencies f87 and f88were stabilized to the clock transitions of 87Sr and 88Sr by steering the frequencies of acousto-optic modulators (AOMs) by respective digital servos The beat note δf = f88 minus f87 whichcorresponds to their isotope shift δf ≃ 62 MHz was investigated to evaluate the stabilities anduncertainties of the two optical lattice clocks

In a 1D lattice we trapped 3 times 103 fermionic 87Sr atoms They were spin-polarized inthe mF = 92 (or minus92) stretched state in the 1S0 ground state (see Fig 1) to avoid atomiccollisions via Pauli blocking [6 8] A bias magnetic field of 022 mT was applied to define thequantization axis We excited the 1S0(F = 92mF = plusmn92) rarr 3P0(F = 92mF = plusmn92)transitions with frequencies of fplusmn successively and calculated f87 = (f++fminus)2 to eliminate thefirst order Zeeman and vector light shifts For a 3D optical lattice clock 2 times 103 bosonic 88Sratoms were loaded into a single occupancy 3D lattice (see Fig 2(b)) A bias magnetic field of|Bm| = 236 mT was applied to magnetically induce the 1S0 rarr 3P0 clock transition A probelaser intensity of Ip = 86 mWcm2 was applied to compensate the relatively small magnetically-induced transition moment These external fields introduced rather large frequency shifts thesecond order Zeeman shift of ∆B ≃ minus130 Hz and the probe laser light shift of ∆p ≃ minus15Hz whose stabilities were evaluated with fractional uncertainties of 2 times 10minus17 and 1 times 10minus17respectively To eliminate the influence of a slowly varying environmental magnetic field Bext

that affected the second order Zeeman shift as prop plusmnBm middotBext we reversed the direction of thebias magnetic field as plusmnBm in successive measurements and calculated f88 = (f+B + fminusB)2where we note that the contribution of prop |Bext|2 was negligibly small Moreover we installedfiber noise cancellation systems referencing the end mirrors of the 1D and 3D lattices to reducethe Doppler shifts arising from the relative motion between the probe lasers and atoms in thelattices

Figure 4(a) shows the temporal response of the frequency difference δf = f88minusf87 of the 1D-3D lattice clocks in the synchronous measurement Thanks to the common mode noise rejectionof the laser frequency noise atomic frequency differences of a few times 10minus16 are visible in tensof seconds This will find useful applications in investigating possible perturbations on probedatomic transitions nearly in real time

Figure 4(b) shows the relative stabilities of the 1D and 3D optical lattice clocks The trianglesshow the relative stability of the two clocks measured for an asynchronous interrogation wherethe 1D and 3D clocks operated alternately every 1 s [6] The short-term stability is limited bythat of the probe laser operated at the thermal noise limit of a reference cavity made of ULEglass After an averaging time of a few tens of seconds the Allan deviation starts to decreasedue to the feedback control with a Nyquist frequency of fN = (4 s)minus1 The Allan deviationdecreased with σy(τ) = 6times10minus15

radicτ (green dashed line) and reached 1times10minus16 at τ = 2times103 s

In contrast the relative stability measured for the synchronous interrogations is shown by thecircles The Allan deviation decreased with σy(τ) = 6times 10minus16

radicτ and reached 3times 10minus17 for an

averaging time of τ = 350 s which shows 10 times better stability than that of the asynchronousinterrogation It is noted that the stability in the synchronous measurement is approaching theQPN limit for N = 1times 103 and ∆f = 8 Hz (blue dashed line) These parameters are similar tothose used in the measurements


6

10-14

Rel

ativ

e s

tabili

ty o

f 1D

-3D

lat

tice

clocks

0 200 400 600 800 1000 1200 1400 1600-04

-0 2

00

02

04

Fre

qu

en

cy f

88-f

87 (

Hz)

T im e (s)

5x10-16

-5x10-16

60 sec averaging

Time (s)

(a)

(b)

100 101 102 103 10410-17

10-16

10-15

Rel

ativ

e s

tabili

ty o

f 1D

-3D

lat

tice

clocks

Averaging time (s)

Figure 4 (Color online) (a) Frequency difference δf = f88 minus f87 of the 1D-3D lattice clocksmeasured for the synchronous clock operation where the isotope shift of asymp 62 MHz is subtracted(b) Allan standard deviation calculated for δf = f88minusf87 for sequential interrogations (triangles)and for synchronous interrogations (circles) In the synchronous interrogations the Allanstandard deviation decreased with σy(τ) = 6times10minus16τminus12 and reached 3times10minus17 for an averagingtime of τ = 350 s The blue dashed line shows the QPN limit for N = 1times 103 and ∆f = 8 Hz

5 Summary and outlookWe have discussed possible realizations of optical lattice clocks taking quantum statistics thehigher order and the multipolar atom-field interactions into account Two configurations 1Doptical lattices loaded with spin-polarized fermions and 3D lattices loaded with bosons with asingle atom in each lattice site will allow an investigation of the clock scheme free from thecollision shifts and the vector light shift In addition the concept of ldquoatomic motion insensitiverdquowavelength is introduced to redefine the ldquomagic wavelengthrdquo including the multipolar atom-field interactions The ldquoblue atomic-motion-insensitiverdquo wavelength may provide neutral atomswith ideal traps for spectroscopy as it closely simulates Paul traps for a single ion We havedemonstrated the advantage of employing a ldquolarge number of atomsrdquo in optical lattice clocks bythe synchronous operation of the 1D and 3D optical lattice clocks which allowed us to explore ahitherto uninvestigated clock stability of 3times10minus17 with an averaging time as short as τ = 350 s

The synchronous frequency comparison scheme provides a convenient means to investigatesystematic uncertainties of optical lattice clocks such as the collision shift the blackbodyradiation shift the higher order and the multipolar light shifts at fractional uncertaintiesof 1 times 10minus17 and beyond The scheme is readily applicable to the frequency comparisonof two clocks located at remote sites by sharing a local oscillator with the help of preciseoptical frequency link technologies [4 15] Assuming the achieved frequency stability shownin Fig 4(b) the remote frequency comparison will allow an investigation of the geoid heightdifference of two sites with an uncertainty of 30 cm in minutesrsquo averaging time On the otherhand the synchronous interrogation may significantly increase the stability in the frequency


7

comparison of optical lattice clocks made of different atomic species to investigate the constancyof fundamental constants [16] and the local position invariance [3] In the case of optical latticeclocks employing Sr (f0 = 429 THz) Yb (f0 = 518 THz) and Hg (f0 = 1129 THz) atoms thatare synchronously interrogated by probe lasers frequency-synthesized by an optical frequencycomb driven by a single thermal-noise-limited local oscillator operated at 5 times 10minus16 relativestabilities of 24times 10minus17

radicτ and 15times 10minus16

radicτ are expected for SrYb and SrHg frequency

comparisons where we assumed one million atoms and the excitation linewidth of ∆f = 8 HzIn pursuit of a stability goal for the optical lattice clocks [2] eg 1 times 10minus18

radicτ a novel

frequency stabilization scheme needs to be developed Considering the instability of the thermalnoise (at 300 K) limited probe lasers that show a minimum of (05 minus 1) times 10minus15 independentof the averaging time in the range of ms to a few second [5] a servo-loop time constant muchless than a second is beneficial to achieve better stability In this direction we are consideringto make use of the transmitted probe laser through millions of lattice-trapped atoms sincethe transmitted light carries full information of the atomic dipoles through the phase shift andabsorptionemission that can be extracted as quadrature components using the FM spectroscopytechnique A detailed discussion will be given elsewhere

This research was partially supported by the Photon Frontier Network Program of theMinistry of Education Culture Sports Science and Technology Japan

References[1] Rosenband T Hume D B Schmidt P O Chou C W Brusch A Lorini L Oskay W H Drullinger R E Fortier

T M Stalnaker J E Diddams S A Swann W C Newbury N R Itano W M Wineland D J and BergquistJ C 2008 Science 319 1808ndash1812

[2] Katori H Takamoto M Palrsquochikov V G and Ovsiannikov V D 2003 Phys Rev Lett 91 173005[3] Blatt S Ludlow A D Campbell G K Thomsen J W Zelevinsky T Boyd M M Ye J Baillard X Fouche

M Targat R L Brusch A Lemonde P Takamoto M Hong F L Katori H and Flambaum V V 2008 PhysRev Lett 100 140801

[4] Hong F L Musha M Takamoto M Inaba H Yanagimachi S Takamizawa A Watabe K Ikegami T ImaeM Fujii Y Amemiya M Nakagawa K Ueda K and Katori H 2009 Opt Lett 34 692ndash694

[5] Numata K Kemery A and Camp J 2004 Phys Rev Lett 93 250602[6] Akatsuka T Takamoto M and Katori H 2008 Nat Phys 4 954ndash959[7] Katori H Hashiguchi K Ilrsquoinova E Y and Ovsiannikov V D 2009 Phys Rev Lett 103 153004[8] Takamoto M and Katori H 2009 J Phys Soc Jpn 78 013301[9] Gibble K 2009 Phys Rev Lett 103 113202

[10] Takamoto M Hong F L Higashi R Fujii Y Imae M and Katori H 2006 J Phys Soc Jpn 75 104302[11] Takamoto M Katori H Marmo S I Ovsiannikov V D and Palrsquochikov V G 2009 Phys Rev Lett 102 063002[12] Taichenachev A V Yudin V I Ovsiannikov V D Palrsquochikov V G and Oates C W 2008 Phys Rev Lett 101

193601[13] Santarelli G Audoin C Makdissi A Laurent P Dick G J and Clairon A 1998 IEEE Trans Ultrason

Ferroelectr Freq Control 45 887ndash894[14] Bize S Sortais Y Lemonde P Zhang S Laurent P Santarelli G Salomon C and Clairon A 2000 IEEE Trans

Ultrason Ferroelectr Freq Control 47 1253ndash1255[15] Newbury N R Williams P A and Swann W C 2007 Opt Lett 32 3056ndash3058[16] Uzan J P 2003 Rev Mod Phys 75 403


8

Optical lattice clocks and frequency comparison

Hidetoshi Katori Tetsushi Takano and Masao Takamoto

Department of Applied Physics The University of Tokyo Tokyo JapanCREST Japan Science and Technology Agency Saitama Japan

E-mail katoriamotu-tokyoacjp

Abstract We consider designs of optical lattice clocks in view of the quantum statisticsrelevant atomic spins and atom-lattice interactions The first two issues lead to two optimalconstructions for the clock a one-dimensional (1D) optical lattice loaded with spin-polarizedfermions and a 3D optical lattice loaded with bosons By taking atomic multipolar interactionswith the lattice fields into account an ldquoatomic motion insensitiverdquo wavelength is proposed toprovide a precise definition of the ldquomagic wavelengthrdquo We then present a frequency comparisonof these two optical lattice clocks spin-polarized fermionic 87Sr and bosonic 88Sr prepared in1D and 3D optical lattices respectively Synchronous interrogations of these two optical latticeclocks by the same probe laser allowed canceling out its frequency noise as a common modenoise to achieve a relative stability of 3times10minus17 for an averaging time of τ = 350 s The schemetherefore provides us with a powerful means to investigate intrinsic uncertainty of the clocksregardless of the probe laser stability We discuss prospects of the synchronous operation ofthe clocks on the measurement of the geoid height difference and on the search of constancy offundamental constants

1 IntroductionTo date optical clocks based on singly trapped ions [1] and ultracold neutral atoms trapped inoptical lattices that operate at the magic wavelength to eliminate light shift perturbation [2] areregarded as promising candidates for future atomic clocks Since 2006 Sr-based ldquooptical latticeclocksrdquo have been evaluated close to the Cs clocksrsquo uncertainty limit internationally [3 4] Tofurther reduce their uncertainty and instability there remain essential experimental challengesOne is to find out better lattice geometries as well as interrogated atom species that bring outthe potential performance of the clock scheme taking into account the collisional frequency shiftthe black body radiation shift and the lattice induced light shifts The other is to establishexperimental schemes that fully utilize the advantage of a large number N of atoms to improvethe clock stability which is currently hampered by the instability of the probe laser of typically5times 10minus16 at 1 s that is due to the thermal noise of a reference cavity [5]

In this paper we first describe designs of optical lattice clocks in view of the quantumstatistics and relevant atomic spins This leads to two optimal configurations for the clock aone-dimensional (1D) optical lattice loaded with spin-polarized fermions and a 3D optical latticeloaded with bosons [6] In particular we refer to an ldquoatomic motion insensitiverdquo wavelength thatprovides a precise definition [7] of a ldquomagic wavelengthrdquo including atomic multipolar interactions

We then present a frequency comparison of these two optical lattice clocks using fermionic87Sr and bosonic 88Sr Synchronous interrogations of the two clocks by the same probe laserallow canceling out its frequency noise as a common mode noise in evaluating the relative


Published under licence by IOP Publishing Ltd 1






2

B

E

E

Electric field


Lattice laser(a)

(c)

Bm mixing fieldE

(b)B

B

E

Magnetic field

amplitude

y

x

Lattice laser

Lattice laser

Probe laserx

yz

O

gravity

Ep

EL

E3

E1

E2







3



cpartBpartt







2 (1)


(= 1minus cos2 ky

)




2 minus 1

2h∆α0(λL)∆qE2 (2)



4

1D(f87)

f+92

f+B

f-92

f-B

f+92

f+92

f87=(f92+f-92)2

f88=(f+B+f-B)23D(f88)

time0 4Tc


88Sr in 3D Lattice

int

int

Probe laser

(698 nm)


AOM

(b)(a)

AOM


Tc 2Tc








δP =1

2

int Tc

0g(t)δω(t)dt (3)




5











6

10-14

Rel

ativ

e s

tabili

ty o

f 1D

-3D

lat

tice

clocks

0 200 400 600 800 1000 1200 1400 1600-04

-0 2

00

02

04

Fre

qu

en

cy f

88-f

87 (

Hz)

T im e (s)

5x10-16

-5x10-16

60 sec averaging

Time (s)

(a)

(b)

100 101 102 103 10410-17

10-16

10-15

Rel

ativ

e s

tabili

ty o

f 1D

-3D

lat

tice

clocks

Averaging time (s)





7





radicτ a novel














8






2

B

E

E

Electric field


Lattice laser(a)

(c)

Bm mixing fieldE

(b)B

B

E

Magnetic field

amplitude

y

x

Lattice laser

Lattice laser

Probe laserx

yz

O

gravity

Ep

EL

E3

E1

E2







3



cpartBpartt







2 (1)


(= 1minus cos2 ky

)




2 minus 1

2h∆α0(λL)∆qE2 (2)



4

1D(f87)

f+92

f+B

f-92

f-B

f+92

f+92

f87=(f92+f-92)2

f88=(f+B+f-B)23D(f88)

time0 4Tc


88Sr in 3D Lattice

int

int

Probe laser

(698 nm)


AOM

(b)(a)

AOM


Tc 2Tc








δP =1

2

int Tc

0g(t)δω(t)dt (3)




5











6

10-14

Rel

ativ

e s

tabili

ty o

f 1D

-3D

lat

tice

clocks

0 200 400 600 800 1000 1200 1400 1600-04

-0 2

00

02

04

Fre

qu

en

cy f

88-f

87 (

Hz)

T im e (s)

5x10-16

-5x10-16

60 sec averaging

Time (s)

(a)

(b)

100 101 102 103 10410-17

10-16

10-15

Rel

ativ

e s

tabili

ty o

f 1D

-3D

lat

tice

clocks

Averaging time (s)





7





radicτ a novel














8

B

E

E

Electric field


Lattice laser(a)

(c)

Bm mixing fieldE

(b)B

B

E

Magnetic field

amplitude

y

x

Lattice laser

Lattice laser

Probe laserx

yz

O

gravity

Ep

EL

E3

E1

E2







3



cpartBpartt







2 (1)


(= 1minus cos2 ky

)




2 minus 1

2h∆α0(λL)∆qE2 (2)



4

1D(f87)

f+92

f+B

f-92

f-B

f+92

f+92

f87=(f92+f-92)2

f88=(f+B+f-B)23D(f88)

time0 4Tc


88Sr in 3D Lattice

int

int

Probe laser

(698 nm)


AOM

(b)(a)

AOM


Tc 2Tc








δP =1

2

int Tc

0g(t)δω(t)dt (3)




5











6

10-14

Rel

ativ

e s

tabili

ty o

f 1D

-3D

lat

tice

clocks

0 200 400 600 800 1000 1200 1400 1600-04

-0 2

00

02

04

Fre

qu

en

cy f

88-f

87 (

Hz)

T im e (s)

5x10-16

-5x10-16

60 sec averaging

Time (s)

(a)

(b)

100 101 102 103 10410-17

10-16

10-15

Rel

ativ

e s

tabili

ty o

f 1D

-3D

lat

tice

clocks

Averaging time (s)





7





radicτ a novel














8



cpartBpartt







2 (1)


(= 1minus cos2 ky

)




2 minus 1

2h∆α0(λL)∆qE2 (2)



4

1D(f87)

f+92

f+B

f-92

f-B

f+92

f+92

f87=(f92+f-92)2

f88=(f+B+f-B)23D(f88)

time0 4Tc


88Sr in 3D Lattice

int

int

Probe laser

(698 nm)


AOM

(b)(a)

AOM


Tc 2Tc








δP =1

2

int Tc

0g(t)δω(t)dt (3)




5











6

10-14

Rel

ativ

e s

tabili

ty o

f 1D

-3D

lat

tice

clocks

0 200 400 600 800 1000 1200 1400 1600-04

-0 2

00

02

04

Fre

qu

en

cy f

88-f

87 (

Hz)

T im e (s)

5x10-16

-5x10-16

60 sec averaging

Time (s)

(a)

(b)

100 101 102 103 10410-17

10-16

10-15

Rel

ativ

e s

tabili

ty o

f 1D

-3D

lat

tice

clocks

Averaging time (s)





7





radicτ a novel














8

1D(f87)

f+92

f+B

f-92

f-B

f+92

f+92

f87=(f92+f-92)2

f88=(f+B+f-B)23D(f88)

time0 4Tc


88Sr in 3D Lattice

int

int

Probe laser

(698 nm)


AOM

(b)(a)

AOM


Tc 2Tc








δP =1

2

int Tc

0g(t)δω(t)dt (3)




5











6

10-14

Rel

ativ

e s

tabili

ty o

f 1D

-3D

lat

tice

clocks

0 200 400 600 800 1000 1200 1400 1600-04

-0 2

00

02

04

Fre

qu

en

cy f

88-f

87 (

Hz)

T im e (s)

5x10-16

-5x10-16

60 sec averaging

Time (s)

(a)

(b)

100 101 102 103 10410-17

10-16

10-15

Rel

ativ

e s

tabili

ty o

f 1D

-3D

lat

tice

clocks

Averaging time (s)





7





radicτ a novel














8











6

10-14

Rel

ativ

e s

tabili

ty o

f 1D

-3D

lat

tice

clocks

0 200 400 600 800 1000 1200 1400 1600-04

-0 2

00

02

04

Fre

qu

en

cy f

88-f

87 (

Hz)

T im e (s)

5x10-16

-5x10-16

60 sec averaging

Time (s)

(a)

(b)

100 101 102 103 10410-17

10-16

10-15

Rel

ativ

e s

tabili

ty o

f 1D

-3D

lat

tice

clocks

Averaging time (s)





7





radicτ a novel














8

10-14

Rel

ativ

e s

tabili

ty o

f 1D

-3D

lat

tice

clocks

0 200 400 600 800 1000 1200 1400 1600-04

-0 2

00

02

04

Fre

qu

en

cy f

88-f

87 (

Hz)

T im e (s)

5x10-16

-5x10-16

60 sec averaging

Time (s)

(a)

(b)

100 101 102 103 10410-17

10-16

10-15

Rel

ativ

e s

tabili

ty o

f 1D

-3D

lat

tice

clocks

Averaging time (s)





7





radicτ a novel














8





radicτ a novel














8

Date post:	03-Feb-2022
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Optical lattice clocks and frequency comparison

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