Focus | Review articleS
Published online: 31 March 2011|doi: 10.1038/nphoton.2011.45
Optical lattice clocks and quantum metrology

Hidetoshi Katori

The magic wavelength protocol has made it possible to design atomic clocks based on well-engineered perturbations. Such optical lattice clocks will allow extremely high stability using a large number of atoms and fractional uncertainties of ~1018 by sharing particular magic wavelengths. This Review covers the experimental realizations of such clocks, the optimal design of optical lattices and recent demonstrations of improved stability for large numbers of atoms. Possible impacts and future applications of optical lattice clocks are also discussed, such as testing the fundamental laws of physics and developing relativistic geodesy.
recise time standard1 forms the basis of sciences such as precision spectroscopy 2,3, the determination of fundamental constants4 , relativity 5 and astronomy 6,7. It is also a fundamental technology with a significant impact on applications found throughout modern society, such as spacetime measurement with global positioning systems or the synchronization of high-speed communications network. Cs atomic clocks were developed by Louis Essen8 half a century ago and have provided the definition of the second since 1967. The international atomic time (TAI) is currently kept by an ensemble of Cs clocks with a fractional uncertainty of /0 1015, and some Cs fountain clocks9,10 operate at even smaller uncertainties. Although Cs atomic clocks operate at the microwave transition of 09.2GHz, using optical transition frequencies of several hundred terahertz should improve the fractional uncertainty /0 by 45 orders of magnitude, as most measurement uncertainties are independent of 0 (except the Doppler shift D=(v/c)0, where v is the atomic velocity and c is the speed of light). The invention of the laser and the subsequent improvement of laser frequency stabilities11 has allowed the realization of optical clocks to become a reality 2,3,12. In the 1980s, Hans Dehmelt suggested13 and established the experimental basics14 for an optical clock that used singly trapped ions in a Paul trap, which soon became a prime candidate for next-generation time/frequency standards. Around the same time, researchers proposed15,16 and demonstrated a laser cooling technique that reduced the thermal motion of atoms and the associated Doppler shift. Finally, at the end of the twentieth century, Theodor Hnsch2, John Hall3 and others developed the optical frequency comb technique17,18, which allows the direct counting of optical frequencies using electronics. Ultraprecise optical clocks12 are therefore considered to be practical devices. Such research into atomic clocks has been the highlight 19 of atomic physics, a field that has seen large numbers of Nobel laureates since the magnetic deflection experiments of Otto Stern and the atomic beam resonance experiments of Isidor Rabi in the 1930s.
frequency 0=(EbEa)/h, where Ea and Eb are the energy of the |a and |b states, respectively, and h is Plancks constant. State-of-theart atomic clocks operate periodically with a clock cycle time of TC. A frequency fluctuation of C(tn)=C(tn)0' in the clock laser from the reference frequency 0' is detected in the nth clock cycle (occurring at tn=nTC) and the laser frequency is corrected through a feedback loop such that C(tn+1) approaches 0'. The frequency fluctuation C(t) of the laser is most sensitively detected by tuning C(t) to the shoulders of a spectrum 0'=0/2, where the excitation probability is around 0.5 (Fig.1a). The excitation probability changes as KC(t)/, where is the excitation linewidth and K is the slope coefficient (which is around 1). Quantum mechanically, the laser excites an atom to a superposition state of |=ca|a+cb|b with probability amplitudes cacb1/2. The excitation probability =|cb|2 is determined by projecting the state | onto |b. This measurement is accompanied by the quantum fluctuation QPN=1/N for N uncorrelated atoms, which is referred to as quantum projection noise20 (QPN). State measurement at the limit is achieved by the shelved electron amplifier technique14, which employs an auxiliary state |c for the projection measurement. The QPN limit of the

Detector Ti Energy Tc

Excitation probability 0 0.
g(t) Servo tn= nTC tn+1 = (n + 1)TC Time
Servo ULE cavity Clock laser (local oscillator)

= 0.8/Ti

Frequency

AOM Clock out

C(t) 0 /2
Operation of atomic clocks
Assuming the constancy of fundamental constants, an atomic clock would be an ideal tool for sharing and distributing the standard of time. The performance of an atomic clock is given in terms of its accuracy, which depends on the uncertainty in its atomic transition frequency resulting from electromagnetic perturbations EM on atomic states and the Doppler shift D, and stability, which indicates how fast the clock can reduce its statistical uncertainty and depends on how the laser is stabilized to the atomic spectrum. Figure1 depicts an atomic clock in which the local oscillator is a laser operating at C(t). The laser is controlled by referencing the atomic transition
Figure1 | Optical clock operation. A clock laser stabilized to an optical cavity made of ultralow-expansion (ULE) glass is used to interrogate the clock transition |a|b with an energy difference of EbEa=h0. a, The laser frequency C is half a linewidth /2 detuned from the atomic resonance 0 so that the laser frequency fluctuation C is sensitively projected onto the excitation probability. The |b state population is measured by observing the fluorescence of the |b|c transition, with |cbeing an auxiliary state. The measurement outcome is fed back to the laser frequency through an acousto-optic modulator (AOM). b, The clock laser periodically probes the atomic transition with a cycle time TC and an interrogation time Ti, revealing the Fourier-limited linewidth of 0.8/Ti.
Department of Applied Physics, Graduate School of Engineering, The University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan; Innovative Space-Time Project, ERATO, Japan Science and Technology Agency, Bunkyo-ku, Tokyo 113-8656, Japan. e-mail: katori@amo.t.u-tokyo.ac.jp
nature photonics | VOL 5 | APRIL 2011 | www.nature.com/naturephotonics
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Review articleS | focus

Nature photonics doi: 10.1038/nphoton.2011.45
/01015 in equation(1), which corresponds to the inverse of the quality factor (Q=0/) of the clock transition, measuring a single ion per second requires an averaging time of 106s to achieve a fractional uncertainty of /01018, which corresponds to ten days of continuous measurement. This long averaging time therefore becomes a serious experimental concern when exploring clock uncertainties at this level.

where E1(L, eL)=b(L, eL)a(L, eL) is the difference between the E1 polarizabilities of the upper (|b) and lower (|a) states. By tuning the laser wavelength L and polarization eL to satisfy E1(L,eL)=0, the observed atomic transition frequency will be equal to 0 regardless of the lattice laser intensity ( E2), as long as higher-order corrections O(E4) are negligible3943. This particular wavelength is referred to as the magic wavelength, m. Early experiments31,44 on light-shift cancellation were performed on the 1S0 3P1 transition, which revealed the coupling of the J=1 state to the light-polarization eL (the vector light shift) to be problematic when pursuing high spectroscopic accuracy because of the difficulty in precisely controlling the light polarization. To define the light-shift cancellation condition solely by the laser wavelength L or its frequency L/2=c/L, which is the most precisely measurable quantity in physics, it is essential to use a J=0 state that exhibits a scalar light shift. However, because the J=0J=0 transition is completely forbidden, a slight admixture of the J=1 state is necessary to allow the clock transition. In the initial proposal27, the 5s 2 1S0(F=9/2)5s5p 3P0(F=9/2) transition of 87Sr with a nuclear spin of I = 9/2 was assumed, where a 1mHz transition rate was induced by hyperfine mixing in the upper state (Fig. 3b). Alternatively, mixing the 3P0 state with the 3 P1 state using a magnetic field45, elliptically polarized light 46 or multiphoton excitation of the clock transition4749 may allow even isotopes that have no nuclear spin and pure scalar state (J=0) to be used. The optical lattice clock scheme is generally applicable to atoms50 in groups ii and iib such as He, Be, Mg, Ca (ref.51), Yb

m= 813 nm

800 600

focus | Review ARticleS

m= 390 nm

= 679 nm 3 P03S1

= 461 nm 1 S01P1

= 394 nm 3 P03D1

D1 S1 Lattice laser

Light shift (kHz)

700 Lattice laser frequency (THz) Lattice laser
mF = 9/2 F = 9/2 mF = +9/2
Figure3 | Light shift and relevant energy levels of Sr atoms. a, Light shifts for the 1S0 (blue line) and 3P0 (red line) states are plotted as a function of laser frequency for a laser intensity of10kWcm2. The crossed points of blue and red lines indicate the magic wavelengths m. b, Relevant energy levels of fermionic 87Sr and bosonic 88Sr isotopes. The 1S03P0 transition is used as a clock transition. f88 represents the clock transition frequency for 88Sr, where an applied magnetic field Bm mixes the 3P0 and 3P1 states to allow the clock transition. 87Sr has a nuclear spin of I=9/2, giving the clock states a total angular moment of F=9/2. The outermost Zeeman substates mF=9/2 are used as clock transitions, where f represent the clock transition frequencies of 1S0 (F=9/2, mF=9/2) to 3P0 (F=9/2, mF=9/2). f87=(f++f)/2 provides the transition frequency of a virtual spin-zero atom.

Multipolar interactions of atoms with the lattice field

Lattice laser

(L)=0[E1(L)qE1(r)+M1(L)qM1(r) +E2(L)qE2(r)] E2/2h (3)

Lattice

E3 x z y
Bm mixing eld EP Probe laser

Gravity

where the fourth- and higher-order terms and light polarization dependence are omitted. The atomic-motion-dependent light shift caused by multipolar interactions can generally be eliminated55 by choosing a particular optical lattice geometry and electric field polarization that together force qM1(r) and/or qE2(r) terms to be either in or out of phase with the spatial dependence of qE1(r). For example, in the case of a 1D lattice with the E1 spatial dependence qE1(r) = sin2(ky), the corresponding M1 and E2 interactions are qM1(r)=qE2(r)=cos2(ky)=qqE1(r), with q=1. Thus, by taking EM=E1M1E2 and 0M1+E2, equation (3) can be rewritten as (L)=0EM(L)qE1(r)E2/2h0(L)qE2/2h, where the second term on the right-hand side varies in phase with the E1 interaction. This suggests that the magic wavelength can be redefined as an atomic-motion-insensitive wavelength satisfying EM(m)=0. The last term provides a spatially constant offset of typically 10mHz or below that is solely dependent on the total laser intensity (qE2) used to form the lattice. This offset frequency can be precisely determined by measuring the atomic vibrational frequencies in the lattice, thereby allowing all quantities to be measured in terms of frequency 55.

E /2 B z

Quantum statistics and optical lattice geometry

Electric eld amplitude

x Magnetic eld amplitude
Figure4 | Optical lattice configurations. a, A 1D optical clock is realized using a standing wave of light tuned to the magic wavelength. Multiply trapped spin-polarized fermions (indicated by arrows) in a single pancake potential may be protected from collisions by Pauli blocking. b, Singleoccupancy bosonic atoms in a 3D lattice. c, Electric and magnetic field amplitudes in a standing wave.
Consequently, it is no longer possible to perfectly cancel out the light shift in two clock states because of the spatial mismatch of the M1 and E2 interactions with the E1 interaction, which causes an atomic-motion-dependent 54 light shift. Although the contributions of the M1 and E2 interactions are predicted to be 67 orders of magnitude smaller 39 than that of the E1 interaction for Sr (at the red-detuned magic wavelength of m813nm), they have a non-negligible contribution when pursuing a 1018-level uncertainty at red and blue magic wavelengths. A more precise definition of the magic wavelength that includes multipolar interactions is therefore required. With the differential polarizabilities X(L) and corresponding spatial distributions qX(r) for X=E1, M1 and E2 interactions, the transition frequency of atoms in the optical lattice corresponding to equation (2) can be given by

The control and prevention of atomic interactions are another concern when designing optical lattice clocks. The collisional frequency shift of atomic clocks operated with ultracold atoms is related to the mean field energy shift 42ang(2)(0)/m of the relevant electronic state, where a is the s-wave scattering length, n is the atomic density, m is the atomic mass and g(2)(0) is the two-particle correlation function at zero distance (which is zero for identical fermions and 12 for distinguishable or bosonic atoms). Collisional shifts are therefore suppressed6365 for ultracold spin-polarized fermions, whereas they are intrinsically unavoidable for bosons. The quantum statistical nature of atoms is determined by their total spins; bosons have zero or integer spins and fermions have half-integer spins. Because optical lattice clocks employ atoms with J=0 electronic states (that is, an even number of electrons), isotopes with a nuclear spin of I=0 or any integer obey Bose statistics, and those with half-integer nuclear spin obey Fermi statistics. Consequently, the total angular momentum F=J+I of the clock states can be zero for bosons, but not for fermions; the latter causes a coupling of the clock states to the light polarization of the lattice field (the vector/tensor light shift). Relevant energy levels for fermionic 87Sr and bosonic 88Sr are shown in Fig.3b. Here we consider two lattice geometries. A 1D (Fig.4a,c) or 2D lattice composed of a single electric field vector allows spatially uniform light polarization. In contrast, a three-dimensional (3D) lattice (Fig.4b) requires at least two electric field vectors. The synthesized field therefore exhibits a polarization gradient that varies in space depending on the intensity profile and relative phases of the lattice lasers. For a 1D lattice loaded with multiple atoms in each lattice site, the application of spin-polarized fermions64,66,67 may minimize the collisional frequency shift owing to their quantum-statistical properties. Figure4a depicts a schematic for the spin-polarized 1D optical lattice clock68, in which the upwards arrows correspond to spin-polarized fermionic atoms. The spatially uniform light field polarization of the 1D lattice allows the vector light shift to be cancelled out by alternately interrogating the transition frequencies f corresponding to the 1S0(mF=F)3P0(mF=F) transition to obtain f87=(f++f)/2. This vector light-shift cancellation technique68 simultaneously removes the first-order Zeeman shift 69 to realize virtual spin-zero atoms.

Photon counts (a.u.)

30 Fluorescence intensity (a.u.)
Vibrational frequency difference (kHz)

n=0 /2

1.0 0.8 0.6

1.0 0.8

0.Clock laser detuning (kHz)

0.6 0.4 0.2

700 Hz
0 -100 -50 Frequency (kHz) 100
-Lattice laser wavelength (nm) 840
0.0 -200 -150 -100 -Clock laser detuning (kHz)

0.0 -2 -1 0

Figure5 | Spectroscopy in optical lattices. a, Laser-induced fluorescence of atoms confined in a 1D optical lattice (red dots) and in free-fall (blue circles), measured on the 1S03P1 (mJ=0) transition. The lattice confinement suppresses the Doppler width of 83kHz (blue line) and the photon recoil shift of 4.8kHz, giving a Lorentzian linewidth of 11kHz (red line). b, The first determination of a magic wavelength based on spectral line broadening, which is caused by a dependence of the transition frequency on n vibrational states (shown inset). By minimizing the linewidth, the magic wavelength was found to be 813.5(9) nm. c, Clock transition measured in the magic lattice.The upper and lower sidebands at 64kHz represent the heating and the cooling sidebands, respectively. The inset shows the recoilless spectrum (the carrier component), which has a linewidth of 0.7kHz (full-width at half-maximum).
3D optical lattices with a single (or less) atom in each lattice site may be used to suppress collision shifts in bosonic isotopes observed in a 1D lattice70,71. However, the inhomogeneity of light polarization that is inevitable in 3D lattices makes vector light shifts problematic for fermionic isotopes with F0, as the vector light shift cancellation technique is no longer applicable. In this respect, a 3D clock is suitable for bosonic isotopes with purely scalar (J=0) clock states, and has been demonstrated using 88Sr atoms66.
Experimental realization and limitations
Magic lattice spectroscopy was first demonstrated44 on the 1S03P1 transition of 88Sr at 0=689nm. Red dots in Fig.5a show the laserinduced fluorescence of atoms confined in a 1D lattice, in which the tight confinement (/kHz) of atoms by the lattice absorbs the photon recoil energy ER=(h/0)2/2m< <. A recoilless spectrum close to the natural linewidth of the transition was observed44. The significance of this Mssbauer spectrum can be seen by comparing it with the spectrum in the absence of the lattice potential (blue circles), where the recoil shift of ER/h4.8kHz and the Doppler broadening of ~83 kHz that were suppressed by the lattice appear as a result of momentum and energy conservation between an atom and a photon. The highly forbidden 1S03P0 clock transition of 87Sr was observed in 2003 when the SYRTE group measured the transition frequency 72 and the University of Tokyo group demonstrated spectroscopy 73 in the magic lattice. The inset of Fig.5b shows the clock spectrum nn of the electronic-vibrational transition |1S0,n|3P0,n (Fig.2b), where n represents the vibrational states with the mean thermal occupation nth. Line broadening is caused by the vibrational frequency difference =ba in the |a|1S0 and |b|3P0 electronic states. By investigating the linewidth reduction attributed to nth, the magic wavelength was determined to be m=813.5(9)nm (ref. 73). The first clock spectrum in the magic lattice is shown in Fig. 5c, consisting of the narrow-carrier nn (Fig. 5c, inset) as well as heating and cooling sidebands nn1 at /264kHz. Absolute frequency measurements have been carried out since 2005 by groups at Tokyo-NMIJ68,74, JILA75 and SYRTE76. Based on the agreement of three independent measurements68,75,76, the 87Srbased optical lattice clock was adopted as a secondary representation of the second by the Comite International des Poids et Mesures (CIPM) in October 2006. Following recent measurements from the JILA77, SYRTE78 and Tokyo-NMIJ79 groups, CIPM adopted the clock

QPN, the Dick effect becomes the major obstacle in achieving higher stability. When synchronously evaluating the frequency difference of two clocks by using a common clock laser, the stability degradation due to D may be rejected as a common-mode noise, thereby allowing the two clocks to be compared at the QPN limit. Such a synchronous frequency comparison was demonstrated in Cs and Rb fountain clocks at microwave frequencies88 and recently in optical lattice clocks89. Figure 6a shows the excitation spectra of 87Sr (red dots) and 88Sr (blue circles) measured for a Rabi pulse with interrogation time Ti = 400 ms; the well-matched excitation spectrum ensured good common-mode rejection of laser-noise-induced fluctuations in the excitation probability (D) when comparing the two clocks. Figure 6b shows the Allan standard deviation90 evaluated for f/Sr, where Sr429THz is the transition frequency and f=888762MHz is the frequency difference between the two isotopes. Orange dots represent the stability for asynchronous interrogation66 when the two clocks are subjected to different laser noise. The Allan deviation decreases as y()=61015/ (orange dashed line), which agrees with the limit set by the Dick effect calculated for a clock laser 91 operated near the thermal noise limit of around 11015 at 1s. In contrast, the Allan deviation measured for synchronous interrogation (red, grey and blue dots) decreases as y() = 4 1016/. In particular, the stability measured for Ti =100ms (red dots) approaches the red dashed line predicted by the QPN and the signal-to-noise ratio in the state detection92, demonstrating the reduction of the Dick effect 89 towards the QPN-limited stability corresponding to around 1,000 atoms. For an averaging time of > 1,000 s, however, three measurements at different Ti failed to lower the fractional uncertainty below 11017, which suggests a flicker floor resulting from ambient temperature fluctuations of around 0.1K.

Excitation

1015 Allan standard deviation

-Frequency (Hz)

Asynchronous Ti = 100 ms Ti = 200 ms Ti = 400 ms

Averaging time (s) 103

Figure6 | Frequency comparisons of 1D and 3D clocks. a, Excitation probabilities of 87Sr in a 1D lattice (red dots) and 88Sr in a 3D lattice (blue circles) as a function of clock laser detuning for a Rabi pulse with Ti=400ms. Red line shows the calculated Rabi excitation profile. b, The Allan standard deviation evaluated for f=8887 for asynchronous (orange dots) and synchronous interrogations (red, grey and blue dots). The dashed lines with corresponding colours show the calculated stabilities for each experimental condition. Departures from theoretical values for longer Ti are due to the large frequency jitter C of the clock laser compared with the excitation linewidth 0.8/Ti, which causes a reduction in frequency sensitivity.

An interesting question is whether these clocks tick the same way as the others throughout the year 100,101,103 as the gravitational potential from the sun changes. Precise comparisons of atomic clocks support such a challenge that is, testing the coupling between electromagnetic constants (such as ) and gravity which may contribute to the experimental foundations towards a unified theory. Entirely new roles for atomic clocks may appear in the future. Precise timekeeping has long been considered to be a time-consuming task; for example, achieving an uncertainty of 1018 with a single-ion clock would require an averaging time of more than ten days. In contrast, the large number of atoms N used in optical lattice clocks could significantly speed up this process by a factor of N, allowing ultrastable and highly precise timekeeping. According to the general theory of relativity, a clock placed h higher than one at the Earths surface runs faster by /0=gh/c 21.11016hm1, where g is acceleration due to gravity. Frequency comparisons of two single-ion optical clocks placed h 0.33 m apart have measured the corresponding gravitational red shift with an uncertainty of 1.71017 in an averaging time of 40,000s (half a day)104. According to Fig.6b, synchronously operated optical lattice clocks with N1,000 would allow the same uncertainty to be achieved in 1,000s. Optical lattice clocks employing 106 atoms would allow frequency shifts of 1018 to be detected in an averaging time of 1s, in theory. This sensitivity corresponds to detecting the clock height difference of h=1cm in the Earths gravitational field or to the special relativistic effect occurring at walking speeds; that is, optical lattice clocks may be capable of measuring even relativistic effects that appear in our daily lives. Optical lattice clocks would then become a tool not only for achieving time-synchronization but also for probing relativistically curved spacetime. With the help of optical frequency linking technologies79,84,105 that allow a local oscillator to be shared between two remote clocks, clock comparison at the level demonstrated in Fig.6b will enable a clock height difference of 10cm to be detected in an averaging time of tens of minutes. Such endeavours may lead to the realization of relativistic geodesy 106. Geoids equipotential surfaces of gravity have so far been the reference frame for sharing atomic time by correcting for gravitational shifts. However, the current uncertainty level of geoid mapping is not good enough to be used with atomic clocks that have uncertainties of 1017 and beyond. Moreover, the Earth itself is not a rigid reference frame but is instead deformed by astronomical bodies such as the Sun and the Moon. Ultrastable and accurate atomic clocks may shed more light on the elasticity of the Earth and the fundamental foundations of physics.

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Acknowledgements

The author thanks M. Takamoto and T. Takano for useful discussions and careful reading of the manuscript.