ATOMIC CLOCKS

Cesium frequency standard !

!!

!!

!!The standard of time is based on the above transition between 6s hyperfine levels of Cesium. Using the formula for the hyperfine splitting of hydrogen-like atoms !

!!!!

and fact that Z=55 for Cs, can you make sense of the frequency for the transition quoted in the figure? Why can we use the above formula for Cs?!133Cs is the only stable isotope. This means that the nuclear structure (and in particular the nuclear spin) is unique for Cs, and so is the the transition in question.!In the presence of a magnetic field defining the quantization axis and splitting the hyperfine states |F, m_F>, the transition chosen for the clock is the |F=3, m_F=0> to |F=4,m_F=0>, so that inhomogeneities in the magnetic field cannot affect the frequency of the transition. !

!Definition of the second (from the “Bureau International de Poids et Mesures”)

The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.!

!

�E(HF)nljF

Ry=

1

2gp

Z3↵2

n3

me

mp

F (F + 1)� j(j + 1)� I(I + 1)

j(j + 1)(2l + 1)

Standard Cesium clock

Volume 106, Number 1, January–February 2001Journal of Research of the National Institute of Standards and Technology

2. Cesium-Beam Frequency Standards2.1 Standards Based on Magnet State Preparation

and Detection

Figure 1 shows a schematic diagram of a conventionalcesium-beam frequency standard. The cesium oven,operated near 100 !C, creates a vapor of atoms that arecollimated and passed successively through the state-preparation region (the A magnet in Fig. 1), themicrowave cavity, and then the state-detection region(the B magnet and detection system). As they emergefrom the oven the 133Cs atoms are evenly distributed inthe 16 mF states of the 2S1/2 ground electronic state. Inthe state-preparation region, a magnet with an inhomo-geneous field (Stern-Gerlach magnet) spatially sepa-rates atoms in the various mF states, and atoms in one ofthe ground-state levels (F = 3, mF = 0 or F = 4, mF = 0,often designated as !3,0" or !4,0") are transmittedthrough the microwave cavity. Because of the velocityspread in the atomic beam, this separation is not perfect,so some atoms in other mF states are mixed in with theground-state atoms that go through the cavity. This typeof state preparation naturally involves a rejection of mostof the atoms entering the system. State detection uses anidentical Stern-Gerlach magnet arranged so that atomsare directed to the hot-wire detector only if they havebeen stimulated by the microwave field to the otherground, mF = 0 level.

The designs of the seven NBS/NIST cesium-beamfrequency standards, developed between 1950 and 1993,were influenced by a need to reduce and control system-atic frequency shifts while maintaining the highest prac-tical signal-to-noise performance. The linewidths ofthese standards were reduced by extending the length ofthe microwave cavity, which grew to 3.74 m for NBS-5.This increase in length was achieved at a cost of signalintensity. Furthermore, these long beam tubes, beinghorizontal, also suffered gravitational dispersion on theorder of 1 cm (the slower atoms fall further than thefaster ones, which spreads out the beam), thereby com-plicating the dispersion associated with the magneticfocusing produced by the state-preparation magnet.Reduction of the uncertainty for these beam standardswas achieved through a variety of incremental develop-ments, but a gradual improvement in the theory alsocontributed to improved ways of evaluating andcontrolling systematic frequency shifts.

2.2 NIST-7, An Optically Pumped Cesium-BeamStandard

The use of optical pumping to replace state-selectionmagnets was first suggested by Kastler in 1950 [12],although it was not made practical until tunable laserswere developed. There are a number of ways in whichto apply optical pumping to the cesium-beam standard,

Fig. 1. Diagram of a cesium-beam frequency standard using magnetic state selection and detection. The form ofRamsey interrogation involves a U-shaped microwave cavity, called a Ramsey cavity, where the oscillatory fields arespatially separated.

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Volume 106, Number 1, January–February 2001Journal of Research of the National Institute of Standards and Technology

2.2.1 Apparatus

The general layout of this standard is similar to that ofthe conventional, magnetic-selection standard shown inFig. 1, except that the state-preparation and state-detec-tion magnets are replaced with laser systems, and theatom detector is replaced with a fluorescence detector.NIST-7 is described in greater detail elsewhere [15,16].Briefly, it has a Ramsey cavity 1.55 m long and anatomic beam diameter of 3 mm. An axial C field isemployed for field uniformity and control of the Rabi-pedestal shape. The cavity ends are designed so that thePoynting vector vanishes at the center of the atomic-beam window [13], thus minimizing distributed-cavityphase-shift effects.

The laser system uses two distributed-Bragg-reflec-tion (DBR) lasers. One is frequency-referenced to theF = 4 → F' = 5 (F refers to a level in the 2S1/2 state andF' refers to a level in the 2P3/2 state) saturated absorptionfeature in an evacuated cesium cell. The second laser isfrequency-referenced to the F = 4 → F' = 3 transition.

The digital servo system for locating and locking tothe center of the resonance uses a microwave synthesisscheme that involves the addition of a 10.7 MHZ offsetnear the top of the multiplication chain. This frequencycomes from a computer-controlled direct-digitalsynthesizer (DDS). The entire system is frequencyreferenced to an active hydrogen maser and the outputis in the form of a table of offset values sent to the DDS.

This system uses slow, square-wave frequency modula-tion (≈0.5 Hz) with blanking during the signal transients[17]. Its advantage is its extreme frequency agilitythat allows interrogation of a number of features in thecesium spectrum.

2.2.2 Magnetic-Field Effects

The C field is operated under closed-loop servocontrol. This low-duty-cycle (1 %) servo maintains thefrequency of the first field-dependent transition towithin 10 mHz of a preselected value. This contributesan uncertainty in the second-order Zeeman shift on theclock transition of <1!10–16. Field measurements madeduring assembly [16] showed a fractional field variationat the position of one Ramsey cavity of 5!10–4 relativeto the mean field. This nonuniformity produces a shiftof <1!10–17 at all microwave power levels. Measure-ments of the offsets of the field-dependent Ramsey linesfrom the centers of their corresponding Rabi pedestalsconfirm the size of the inhomogeneity shift [14].

2.2.3 Second-Order Doppler Effect

The second-order Doppler shift is of order 3!10–13.To achieve the accuracy goal requires a measurement ofthe effective, ensemble-averaged velocity with an uncer-tainty of <1 %. This has been done using both a Ramseylineshape-inversion technique [18] and a pulsed optical-pumping technique [19]. The second-order Dopplercorrection was computed for several microwave powerlevels using the two methods. While the correctionsvaried by nearly 2!10–13 over the 7.5 dB power range,the shifts computed from the two methods were inexcellent agreement (within 2!10–15).

To maintain stability of the second-order Dopplershift to this level requires control of the microwavepower experienced by the atoms to much less than0.1 dB. This is achieved using a power-level servosystem with a precision power splitter and a stable powermeter. The computer determines the optimum powerlevel through measurement of the signal intensity as afunction of microwave power. The Ramsey-inversionprogram used to measure the velocity profile alsoreturns a measure of the absolute power. The valuereturned by the Ramsey-inversion program and thatdetermined by the power-level servo system are inexcellent agreement.

2.2.4 Cavity-Related Errors

The end-to-end cavity phase shift is measured byreversing the beam direction. The fractional frequencyshift on beam reversal is !1.2!10–12, with an uncer-

Fig. 2. NIST-7 Ramsey pattern (F = 3 ↔ F = 4) transition in theground state). The central fringe has a full linewidth at half maximumof about 65 Hz.

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Volume 106, Number 1, January–February 2001Journal of Research of the National Institute of Standards and Technology

but even the simplest methods produce large benefits.Basically, optical state selection is achieved through thefrequency selectivity of the exciting light; this laser lightis tuned so that atoms in a particular state efficientlyabsorb the light and are excited to higher electronicstates. In relaxing back to the ground state, the excitedatoms are restricted by quantum selection rules, so thatthey can in general relax to only a limited set of thevarious ground-state levels.

Two types of situations in cesium are particularlyinteresting. For optical state preparation in NIST-7,atoms are excited from the F = 4 ground-state level tothe F = 3 level of the 2P3/2 state from whence they canrelax to either the F = 3 or F = 4 level of the groundstate. Continued selective pumping from the ground-state F = 4 level thus depopulates that level and in-creases the population in the ground-state F = 3 level.Thus, rather than discarding atoms, as is done with themagnetic method, atoms are converted to the desiredstate. Done with the proper light polarization, thispumping leaves the atoms evenly distributed in popula-tion (and velocity) among the seven mF levels within theF = 3 ground state. While a variation on this methodusing two excitation laser frequencies can achieve con-version of all atoms to the desired mF = 0 ground-statelevel, this involves scattering of many more photons intothe microwave interrogation region, increasing the light-induced frequency shift commonly called the ac Starkshift.

State detection can also be achieved using opticalmethods. This is done in NIST-7 by pumping from theF = 4 ground state to the F = 5 level of the 2P3/2 state,where quantum selection rules restrict their relaxation toonly the F = 4 level of the ground state. This is called acycling transition. Using such a transition, a single atomcan be made to scatter a very large number of photons(to fluoresce), thus achieving 100 % detection effi-ciency. Such a state-detection scheme can readilyachieve a detection noise limited by the atom shot noise;that is, the laser-detection process adds no additionalnoise.

There are several advantages to these optical tech-niques. First, as just described, for a given flux of atoms,the signal-to-noise ratio is substantially increased, thusdecreasing the time needed to make a measurement.Second, the elimination of the state-selection and state-detection magnets removes the troublesome transversedispersion of atoms associated with the fact that slowand fast atoms in the Maxwellian distribution of atomvelocities take different paths through the magneticoptics of such systems. Finally, the asymmetries in the

cesium spectrum arising in magnetic state preparationfrom the velocity selectivity of that process can beessentially eliminated by optical pumping. This meansthat pulling effects from a line on one side of the centralfringe are balanced by an equal line pulling from theother side. These advantages, together with major im-provements in the design of the microwave cavity [13]and the servo-control design, have resulted in theachievement of a combined uncertainty for NIST-7 of4.4!10–15, a factor of 20 better than the uncertainty ofNBS-6.

The major sources of uncertainty in frequency biasesin thermal-beam, cesium frequency standards aresecond-order Zeeman shift, second-order Doppler shift,end-to-end cavity phase shift and possibly cavitypulling, fluorescence light shift, and line-overlap shift.Because of their sizes, these effects must be consideredvariable on a level significant to the overall accuracy ofthe standard. Therefore, they must be evaluated often,and NIST-7 incorporates a number of servo-controlsystems that allow major portions of this evaluation to beautomated. Furthermore, a measurement technique hasbeen developed to allow the very small effects of cavitypulling, line-overlap shift and magnetic-field inhomo-geneity to be measured quickly with low-precisionmeasurements [14]. This technique relies on the fact thatthe shift in the Rabi pedestal part of the line shape islarge compared to the shift in the Ramsey fringe part ofthe line shape. Because of this leverage, the offsetbetween the Ramsey line and its Rabi pedestal must bemeasured to only about 0.1 Hz to obtain corrections of<1!10–15. This requires fractional frequency measure-ments no better than 1!10–12. Combining the digitalservo-system and this new measurement technique, themajor systematic errors can be evaluated in just a fewdays. The short-term stability of NIST-7 is typically"y (#) = 7!10–13 # –1/2. This means that frequency mea-surements with uncertainties near 10–15 (one standarddeviation) can be made in about 10 hours. Figure 2shows the Ramsey pattern for the central spectralfeature (F = 3↔F = 4). The linewidth $% is about65 Hz corresponding to Q ≡ %0/$% ! 1.5!108. Toachieve an uncertainty of 5!10–15, the line center mustbe located with an uncertainty of <1!10–6, a difficulttask. Furthermore, the very broad velocity distributionfound in an optically pumped standard leads to arelatively strong dependence of the frequency onmicrowave power and modulation parameters. Toachieve confidence in measurements to such exactingaccuracy, two or more independent techniques havebeen used to evaluate most of the error sources.

50

Atomic fountain From Wikipedia, the free encyclopedia!!An atomic fountain is a cloud of atoms that is tossed upwards by lasers in the Earth's gravitational field. If it were visible, it would resemble the water in a fountain. While weightless in the toss, the atoms are measured to set the frequency of an atomic clock.[1]!

The primary motivation behind the development of the atomic fountain derives from the Ramsey method of measuring the frequency of atomic transitions.[2] In broad strokes, the Ramsey method involves exposing a cloud of atoms to a brief radiofrequency (rf) electromagnetic field; waiting a time T; briefly exposing the cloud to the rf field again; and then measuring what fraction of the atoms in the cloud have transitioned.[2] If the frequency of the rf field is identical to the atomic transition frequency, 100% of the atoms will have transitioned; if the frequency of the field differs slightly from the transition frequency, some of the atoms will not have transitioned.[2] By repeatedly sending clouds of atoms through such an apparatus, the frequency of the field can be adjusted to match the atomic transition frequency.!

The precision of the Ramsey method can be increased by increasing the wait time T of the cloud.[2] The use of an atomic fountain with a cooled atomic cloud allows for wait times on the order of one second, which is vastly greater than what can be achieved by performing the Ramsey method on a hot atomic beam.[2] This is one reason why NIST-F1, a cesium fountain clock, can keep time more precisely than NIST-7, a cesium beam clock.[1]!

!!!!!!!!!!!!!!!!!

Volume 106, Number 1, January–February 2001Journal of Research of the National Institute of Standards and Technology

3. Cesium-Fountain Frequency StandardThe fountain concept for extending atom observation

time was introduced by Zacharias in 1954 [23]. Lasercooling of atoms was unknown at the time, but Zachar-ias believed that it might be possible to direct a thermalbeam of atoms upward and then depend on finding thata small number of slower atoms in the Maxwellian ve-locity distribution would reach apogee within the deviceand return to the source. While there would be a dra-matic loss of signal, the time of flight for atoms goingup 1 m and returning would be on the order of 1 s,resulting in a large reduction in resonance linewidth.Furthermore, atoms could traverse the same microwavecavity twice (once on the way up and once on the waydown), and this would provide for Ramsey interrogation(temporally separated rather than spatially separated re-gions) without the end-to-end cavity phase shift found inbeam standards. The experiment was attempted shortlyafter the proposal was made, but no return signal wasobserved. It was later found that scattering processeswithin the beam have the effect of removing the slowestof atoms, so it is clear now that the original experimentwas doomed to failure. The concept was not to beproven until 35 years later, after laser cooling techniqueswere developed.

The first demonstration of the fountain concept was atStanford University [24] in 1989 and the first primarystandard based on the fountain concept was developedshortly thereafter by a group at the Laboratoire Primairedu Temps et Fr!quences (LPTF) [25]. The laser-cooledfountain concept is shown in Fig. 3. The atom ball in theLPTF cesium fountain is formed either using a mag-neto-optical trap (MOT) or optical-molasses [26]. TheNIST fountain collects atoms only in optical molasses.The MOT can achieve higher atom density, but the ballof atoms is typically converted to optical molasses priorto launching. For fountain frequency standards thetransverse temperature of the atoms is a key parameter.During flight of the atoms through the devices, a largefraction (of order 90 %) of the atoms with higher trans-verse velocities are lost before they return to the detec-tion region. Additional transverse cooling would allowincreased utilization of source atoms, and a better trade-off between signal strength and spin-exchange shift.

3.1 Description of NIST-F1

The NIST cesium fountain is described in greaterdetail elsewhere [27] and only a short overview is givenhere. The NIST-F1 optical-molasses source gathers ap-proximately 107 cesium atoms at <2 "K in about 0.4 s.The ball of atoms is then launched by differential detun-ing of the two vertical laser beams to make a movingoptical molasses. After the atoms have been accelerated

to their launch velocity the molasses laser beams are alldetuned to the red in frequency while simultaneouslyreducing the optical intensity to further cool thelaunched atom sample.

The atoms travel from the optical molasses sourceregion through a region that is used to detect atoms laterin the process, and into the magnetically shieldedC-field section of the fountain. All of the launchedatoms at this point are in the F = 4 state and approxi-mately evenly distributed over all possible mF -statevalues. The atoms first encounter a microwave state-preparation cavity, which moves the !4,0" atoms to the!3,0" state using a #-pulse at 9.192 GHz. Any remainingF = 4 atoms are then removed from the sample with anoptical pulse.

Fig. 3. Fountain concept. Atoms are trapped at the intersection of thesix orthogonal laser beams and are tossed vertically by slightly offset-ting the frequencies of the two vertical lasers and then turning all sixlasers off. The atoms rise and fall through the microwave cavity andundergo state interrogation below the microwave cavity.

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Volume 106, Number 1, January–February 2001Journal of Research of the National Institute of Standards and Technology

Measurement of the !4,1"→!3,1" transition deter-mines the temporal average of the magnetic field B overthe flight time. However, the temporal average of B 2 isneeded to correct the second-order Zeeman shift. If themagnetic field is modeled as seen by the atoms asH (t ) = H0[1–! f (t ) where f (t ) is a function with!f (t )! "1, and ! is a scaling factor, then the differencebetween the mean square and the square of the meanleads to a frequency shift given by

#$

$0

=(427%108 H 2

0)! 2

$0

[#f (t )"2–#f (t )2"] (3)

From the magnetic field model, ! can be shown to be oforder 0.1, and from consideration of atom ballistics,#f (t )"2 – #f (t )2" is found to be "0.01. The maximuminhomogeneity frequency shift is then less than#$ /$0 = 10–17.

The uncertainty associated with the quadraticZeeman shift is therefore dominated by problemsassociated with location of the central fringe and isconservatively assigned a value equivalent to the mis-assignment of one whole fringe in the mF = 1 manifold,that is, 3%10–16.

3.2.2 Spin-Exchange Frequency Shift

The evaluation of the spin-exchange frequency shiftrequires a measurement of the atomic density, which isdetermined using several methods. This involves care-fully calibrating the entire detection system includingthe size of the detection beams and their intensity, thesolid angle for collection of photons from the atomicsample, and finally a calibration of the photodiode andits associated amplifier. The average density is deter-mined from a measurement of the number of atoms

Fig. 4. NIST-F1 Ramsey pattern. The upper portion of the figure shows the entire ground state(F = 3 ↔ F = 4) pattern, while the lower portion of the figure shows an expanded view of the centralfringe. The full linewidth at half maximum is about 1 Hz.

55

!!!!!!!!!!!!!!!!!!!!!

!!

!!!!!!!!

Volume 106, Number 1, January–February 2001Journal of Research of the National Institute of Standards and Technology

interval between the pulses. The end-to-end cavity phaseshift of cesium-beam standards is absent, and there is nofirst-order Doppler shift.

The first stored-ion frequency standard that exhibiteda reasonably small uncertainty (1!10–13) was a Be+ ionstandard operating at 303 MHz [32]. While this stan-dard used a modest ion cloud (!104 ions), thestandards described below use only a few ions. Despitethe small number of ions in these standards, verycompetitive stabilities have been achieved for themicrowave-frequency standard, and the short-termstability achieved in experiments on the optical-frequency standard is exceptionally high. Storagemethods [33] include both radio-frequency traps (calledPaul traps), which use a combination of static and acelectric fields to achieve confinement, and Penningtraps, which confine ions using a combination of staticelectric and magnetic fields. There are a variety of ionsand trap configurations that have been used, but thisdiscussion is limited to work done at NIST on 199Hg+

ions stored in a linear rf trap to produce a microwavefrequency standard and in a similar linear trap toproduce an optical frequency standard.

199Hg+ offers a microwave clock transition at40.5 GHz and an opt ical clock transi t ion at1.06!1015 Hz. (see Fig. 5). To first order, both transi-tions are insensitive to uniform magnetic fields. Using

a linear trap, uncertainties in all systematic frequencyshifts of the 40.5 GHz transition are expected to be lessthan 1!10–16. Using a single ion, the relative uncer-tainty is expected to be <1!10–17 for the optical transi-tion.

If fluctuations of the atomic signal are due only toquantum statistics, then the stability of a frequencysource servoed to the atomic transition is given by [34]

"y (# ) =2$

%0 "NTR

# –1/2 , (5)

where %0 is the frequency of the atomic transition, N isthe number of ions, TR is the Ramsey interrogation time,and # is the averaging time of the measurement. For theground-state hyperfine transition, %0 = 40.5 GHz.It appears feasible to use N = 100 ions and TR =100 s, which gives "y (# ) ! 4!10–14 # –1/2. For the282 nm 2S1/2 → 2D5/2 electric quadrupole transition,%0 ! 1015 Hz, so that using N = 1 and TR = 25 ms gives"y (# ) ! 10–15 # –1/2.

4.1 Cryogenic Linear RF Trap

Figure 6 shows the linear rf trap [35] used in the40.5 GHz microwave frequency standard. Two diago-nally opposite rods are grounded, while the potential ofthe other two rods is V0 cos(&t ), where nominallyV0 ! 150 V and & /2$ = 8.6 MHZ. The resultingpseudopotential confines the ions radially in a well witha secular frequency %r ! 230 kHz. To reduce patchfields and remove electrical charge that otherwise leavesthe rods slowly in the cryogenic environment, resistivewires are threaded through the rods to heat them duringand after loading the trap. Two cylindrical sections ateither end of the trap are held at a potential of approxi-mately +10 V, confining the ions axially. The ions arelaser-cooled using the 194 nm 2S1/2 → 2P1/2 electric-dipole transitions shown in Fig. 5. Typically, a string ofapproximately ten ions is confined near the trap axis.By minimizing the ion micromotion in all three dimen-sions, the laser-cooled ions are made to lie along the rfnodal line [36]. Here, parametric heating is essentiallyeliminated, so the cooling-laser radiation, which per-turbs the clock states, can be removed during the longprobe periods of the clock transition.

The trap is placed in a liquid-helium (4 K) cryogenicenvironment in which Hg and most background gasesare cryopumped onto the walls of the chamber. Thisessentially eliminates ion loss due to collisions with thebackground gas. In addition, collision shifts should benegligible. Operation at 4 K also reduces the frequencyshifts due to blackbody radiation of the 199Hg+ ground-state hyperfine transition. At T = 4 K, the fractionalblackbody Zeeman shift is –2!10–21, and the fractionalblackbody Stark shift is –3!10–24 [20].

4.1.2 Laser-Atom Interactions

Laser beams at 194 nm [37] are used for cooling,state preparation, and determining the internal states ofthe ions. To cool the ions, the frequency of a primarylaser is tuned slightly below resonance with transition p(see Fig. 5). Although this is a cycling transition, thelaser can off-resonantly excite an ion into the 2P1/2, F = 1level, from which the ion can decay into the 2S1/2, F = 0

Fig. 5. Partial energy diagram of 199Hg+.

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Volume 106, Number 1, January–February 2001Journal of Research of the National Institute of Standards and Technology

launched by using the detection region to measure thenumber of atoms in the cloud on the way up and againon the way down. Assuming Maxwellian thermal distri-butions and extracting the physical dimensions of thelaunched atom ball from the data, the atomic densitycan be determined as a function of time. This density isthen used, along with a spin-exchange frequency-shiftcoefficient for the mF = 0 state as measured by the LPTFgroup [29] and the Stanford group [30] to determine thetotal spin-exchange frequency shift. The process is com-plicated by the use of a pure molasses source, since theinitial cloud size is uncertain and this propagates into thefinal result both directly and as an uncertainty in theatom temperature. The atomic density derived above isnext used to predict the total spin-exchange frequencyshift as

!"

"0= (–2#10–21)!"ht

T#"$1+$3

2– $2

—#+ $2—$ (4)

where h is a factor of order unity and depends on theexcitation power, t is the Rabi time, T is the Ramseytime, $1

— and $3— are the atomic density on the first and

second pass through the microwave cavity, respectively,and $2

— is the average density over the entire Ramseytime. This formula, originally derived by Shirley [31]for Zeeman shifts over the length of a beam tube, hasbeen modified for the case at hand.

As a consistency check the frequency of the fountainwith state selection is compared to the frequency withno state selection. Without state selection the density ismuch higher, although the spin-exchange cross sectionis smaller by a factor of two. No spin-exchangefrequency shift is seen at 3#10–15 (limited by therandom uncertainty of the measurement) when this testis made. The total calculated spin-exchange shift in thefountain is typically 5#10–16 and as a result of thedifficulties associated with the density determinationmentioned above, a conservative uncertainty of 5#10–16

is assigned. A rigorous analytic solution to the density asa function of time is needed. This will allow furtherimprovement in evaluation of the spin-exchangeuncertainty.

3.2.3 Blackbody Radiation Shift

The next significant systematic frequency bias is theblackbody radiation shift. This radiation shift is thesame as that described for NIST-7, but it is somewhatmore significant, because the overall uncertainty for thisstandard is lower. The cavity and drift tube region of thefountain are temperature controlled at a temperature of41 oC. Sensors on the microwave cavities and the drifttube of the fountain monitor the temperature. The

largest temperature differential noted from thesesensors is 0.2 oC. A temperature error of 1 %C is taken tobe the worst possible error and this leads to an uncer-tainty of less than 3#10–16.

3.2.4 Gravitational Redshift

The gravitational frequency shift (redshift) in Boulderis large, about –1.8#10–13. The gravitational potential inBoulder Colorado relative to the geoid has been reevalu-ated using an Earth potential model and the resultingclaimed uncertainty on the frequency correction is lessthan 5#10–16.

3.2.5 Summary of Uncertainty

The present values of the most-significant systematicfrequency biases in NIST-F1 are given in Table 2. Theoverall Type B uncertainty (systematic effects) is0.8#10–15, dominated by the spin-exchange shift. TheType A (statistical) uncertainty is 1.3#10–15, resultingin a combined uncertainty of 1.5#10–15. This representsthe present status of the standard. Work during thenext year should reduce the combined uncertainty to<1#10–15. However, because of the spin-exchange shift,it will be difficult to achieve an uncertainty much below5#10–16.

4. Stored-Ion Frequency Standards

The key advantage of using stored ions for frequencystandards is that they can be stored for long periods(hours to days and even weeks are common) with, insome cases, exceedingly small systematic frequencyshifts. This allows for an arbitrarily large increase inobservation time over the cesium-beam and atomic-fountain methods and results in very narrow linewidths!" /"0. Ramsey interrogation is accomplished bysubjecting the ions to pairs of microwaves pulses, andthe linewidth is then inversely proportional to the time

Table 2. Summary of the largest systematic frequency shifts inNIST-F1. The uncertainties for all items not listed are <10–16. Thetotal type B uncertainty is 0.8#10–15

Physical effect Bias Type B uncertainty(#10–15) (#10–15)

Second-order Zeeman +45.0 0.3

Spin exchange –0.9 0.5

Blackbody –20.6 0.3

Gravitation +180.54 0.1

Microwave leakage 0 0.2

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Volume 106, Number 1, January–February 2001Journal of Research of the National Institute of Standards and Technology

interval between the pulses. The end-to-end cavity phaseshift of cesium-beam standards is absent, and there is nofirst-order Doppler shift.

The first stored-ion frequency standard that exhibiteda reasonably small uncertainty (1!10–13) was a Be+ ionstandard operating at 303 MHz [32]. While this stan-dard used a modest ion cloud (!104 ions), thestandards described below use only a few ions. Despitethe small number of ions in these standards, verycompetitive stabilities have been achieved for themicrowave-frequency standard, and the short-termstability achieved in experiments on the optical-frequency standard is exceptionally high. Storagemethods [33] include both radio-frequency traps (calledPaul traps), which use a combination of static and acelectric fields to achieve confinement, and Penningtraps, which confine ions using a combination of staticelectric and magnetic fields. There are a variety of ionsand trap configurations that have been used, but thisdiscussion is limited to work done at NIST on 199Hg+

ions stored in a linear rf trap to produce a microwavefrequency standard and in a similar linear trap toproduce an optical frequency standard.

199Hg+ offers a microwave clock transition at40.5 GHz and an opt ical clock transi t ion at1.06!1015 Hz. (see Fig. 5). To first order, both transi-tions are insensitive to uniform magnetic fields. Using

a linear trap, uncertainties in all systematic frequencyshifts of the 40.5 GHz transition are expected to be lessthan 1!10–16. Using a single ion, the relative uncer-tainty is expected to be <1!10–17 for the optical transi-tion.

If fluctuations of the atomic signal are due only toquantum statistics, then the stability of a frequencysource servoed to the atomic transition is given by [34]

"y (# ) =2$

%0 "NTR

# –1/2 , (5)

where %0 is the frequency of the atomic transition, N isthe number of ions, TR is the Ramsey interrogation time,and # is the averaging time of the measurement. For theground-state hyperfine transition, %0 = 40.5 GHz.It appears feasible to use N = 100 ions and TR =100 s, which gives "y (# ) ! 4!10–14 # –1/2. For the282 nm 2S1/2 → 2D5/2 electric quadrupole transition,%0 ! 1015 Hz, so that using N = 1 and TR = 25 ms gives"y (# ) ! 10–15 # –1/2.

4.1 Cryogenic Linear RF Trap

Figure 6 shows the linear rf trap [35] used in the40.5 GHz microwave frequency standard. Two diago-nally opposite rods are grounded, while the potential ofthe other two rods is V0 cos(&t ), where nominallyV0 ! 150 V and & /2$ = 8.6 MHZ. The resultingpseudopotential confines the ions radially in a well witha secular frequency %r ! 230 kHz. To reduce patchfields and remove electrical charge that otherwise leavesthe rods slowly in the cryogenic environment, resistivewires are threaded through the rods to heat them duringand after loading the trap. Two cylindrical sections ateither end of the trap are held at a potential of approxi-mately +10 V, confining the ions axially. The ions arelaser-cooled using the 194 nm 2S1/2 → 2P1/2 electric-dipole transitions shown in Fig. 5. Typically, a string ofapproximately ten ions is confined near the trap axis.By minimizing the ion micromotion in all three dimen-sions, the laser-cooled ions are made to lie along the rfnodal line [36]. Here, parametric heating is essentiallyeliminated, so the cooling-laser radiation, which per-turbs the clock states, can be removed during the longprobe periods of the clock transition.

The trap is placed in a liquid-helium (4 K) cryogenicenvironment in which Hg and most background gasesare cryopumped onto the walls of the chamber. Thisessentially eliminates ion loss due to collisions with thebackground gas. In addition, collision shifts should benegligible. Operation at 4 K also reduces the frequencyshifts due to blackbody radiation of the 199Hg+ ground-state hyperfine transition. At T = 4 K, the fractionalblackbody Zeeman shift is –2!10–21, and the fractionalblackbody Stark shift is –3!10–24 [20].

4.1.2 Laser-Atom Interactions

Laser beams at 194 nm [37] are used for cooling,state preparation, and determining the internal states ofthe ions. To cool the ions, the frequency of a primarylaser is tuned slightly below resonance with transition p(see Fig. 5). Although this is a cycling transition, thelaser can off-resonantly excite an ion into the 2P1/2, F = 1level, from which the ion can decay into the 2S1/2, F = 0

Fig. 5. Partial energy diagram of 199Hg+.

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Volume 106, Number 1, January–February 2001Journal of Research of the National Institute of Standards and Technology

level. To maintain fluorescence, a repumping laserbeam, resonant with transition r in Fig. 5, is overlappedcollinearly with the primary laser beam. To preventoptical pumping into the dark states of the ground stateF = 1 level, the 194 nm radiation (containing both r andp components) is split into two beams that are made topropagate through the trap at an angle of 40! relative toeach other ("20! relative to the trap axis as shown inFig. 6). The polarization of one beam is in the plane ofthe 194 nm beams, while the polarization of the other194 nm beam is continuously modulated between leftand right circular.

4.1.3 Operation of the 40.5 GHz Microwave Clock

The ions are prepared in the 2S1/2, F = 0 state byturning the repumping beam off. The atomic state afterthe clock radiation is applied is determined by pulsingon only the primary beam p for a time comparable to thetime necessary to pump the ions from the 2S1/2, F = 1 tothe 2S1/2, F = 0 level (typically 10 ms). If the ion is foundin the 2S1/2, F = 1 level, it will scatter about 104 photonsbefore it optically pumps into the 2S1/2, F = 0 level.Approximately 150 of these photons are detected andcounted. If the ion is found in the 2S1/2, F = 0 level, it willscatter only a few photons.

For the first part of the measurement cycle, the ionsare cooled by pulsing on both the primary and repump-ing 194 nm laser beams for 300 ms. Next, the repump-ing beam is turned off for about 60 ms, so that essen-tially all of the ions are optically pumped into the 2S1/2,F = 0 level. The clock transition is probed using theRamsey technique of separated oscillatory fields [11].Both beams are blocked during the Ramsey microwaveinterrogation period, which consists of two 250 msmicrowave pulses separated by the free precession

period TR. TR is varied from 2 s to 200 s. Finally, theprimary beam is turned on for 10 ms to 20 ms whilecounting the number of detected photons. This deter-mines the ensemble average of the atomic state popula-tion and completes one measurement cycle.

The microwave frequency is derived from a synthe-sizer locked to a hydrogen maser. Stepping the synthe-sizer frequency between measurement cycles produces aset of Ramsey fringes. A digital servo locks the averagesynthesizer frequency to the central fringe. For seventrapped ions and TR = 100 s, the fractional fre-quency stability of the servoed oscillator is #y ($ ) =3%10–13 $ –1/2 for $ & 2 h. Consistently, the measured#y ($ ) is about twice the value expected from Eq. (1),primarily because of laser intensity fluctuations at thesite of the ions. The stability of the ion standard iscomparable to the stabilities of NIST-7 and NIST-F1.

4.1.4 Measurement Uncertainties

Table 3 shows the most important corrections made tothe average frequency for each run [38]. The fractionalZeeman shift due to the static magnetic flux density is

Fig. 6. Linear rf trap, and an image of a linear ion crystal. The ions are spaced approximately10 'm apart, and the gaps in the ion crystal are due to ions other than 199Hg+ that do not fluoresceat the frequencies of the 194 nm laser beams. The spatial extent of the ions is exaggerated for clarity;in reality the laser beams simultaneously overlap all the ions.

Table 3. Largest systematic shifts of the average mercury-ion-clocktransition frequency. Typical values are for an rf power of 20 mW, aRamsey interrogation time T = 100 s, and a static magnetic flux den-sity Bstatic = 3%10–7 T. Here, B is the flux-density component at fre-quency ( /2)

Physical effect Scaling Bias Type B(%10–15) uncertainty

(%10–15)

Quadratic Zeeman (static) +!Bstatic2" 20 1.4

Quadratic Zeeman (( ) +!B 2" 5 3.2

Microwave chirp, leakage, 1/TR 3 0.8and spectrum asymmetries

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Optical lattice clock !Optical lattice clocks are based on the trapping of ultra-cold neutral atoms in a deep standing wave (optical lattice). !A very recent study (B. J. Bloom et al., Nature 506, 71 (2014)) reports a precision of 6x10^-18, obtained by interrogating an ultra narrow transition (1S0 -> 3P0 clock transition, with a frequency of 429 228 004 229 873. 65 (37) Hz) for trapped 87Sr atoms via Rabi spectroscopy. !!

!!!Precision of atomic clocks over time

The absolute frequency of the 87Sr optical clock transition

(d)

(c)

26

24

22

20

18

16

14

f cent

re /H

z

500040003000200010000Time/s

AOM

f1st lock

f2nd lock

f cent

re

counterfmaser

frep servo

Fibre Link to NIST DL

(5s2)1S0

(5s5p)1P1

(5s6s)3S1

(5s5p)3P2

3P13P0

λ = 461 nm

λ = 698 nm

λ = 689 nm

λ = 679 nm λ = 707 nm

(a)

(b)

0.3

0.2

0.1

0.0

3 P0 P

opul

atio

n

-150 -100 -50 0 50 100 150Detuning/Hz

flst lock

fcenter

0.4

Figure 1. Experimental setup. (a) Relevant energy levels for 87Sr used for the optical lattice clock. Transitions at 461 nm and 689 nm areused in two-stage cooling and trapping of the Sr atoms. The clock transition is at 698 nm. Lasers at 679 nm and 707 nm provide necessaryrepumping from metastable states. (b) To operate the clock, ultracold 87Sr atoms are first optically pumped to the |F = 9/2, mF = ±9/2⟩ground states. The clock centre frequency (fcentre) is found by locking the probe laser frequency to both peaks successively and taking theiraverage. The laser is locked to the centre of each transition by sampling their FWHM as illustrated in the figure by dots (f1st lock). (c)Schematic of the setup used for locking the optical local oscillator to the 87Sr transition. The clock transition is probed using a DL atλ = 698 nm which is locked to an ultrastable, high-finesse optical cavity. The laser beam is used to interrogate the Sr atoms and istransferred to the atoms using an optical fibre with active fibre noise cancellation. To steer the frequency of the laser for the lock to the Srresonance, an AOM is used to introduce a frequency offset between the cavity and the atoms. The frequency offset is steered to the lockpoints (f1st lock). The frequency offset also includes a linear feedback value (f2nd lock) to compensate for the linear drift of the high-finessecavity. The cavity-stabilized clock laser is also used to phase-lock a self-referenced octave-spanning optical frequency comb. To steer thecomb to the atomic transition the synthesizer used as the phase/frequency reference for the comb lock is updated with the atomic resonanceinformation contained in fcentre. The Sr-referenced repetition frequency of the comb (frep) is then counted relative to a H-maser located atNIST (fmaser). (d) Sample data showing the in-loop atom lock for 5000 s of data taken during the measurement of the absolute frequency.The fit gives a residual linear fractional frequency drift of <2 × 10−19 s−1.

is the lattice photon recoil energy and k = 2π/λ is thewavevector of the lattice light. At this lattice depth the atomsare longitudinally confined in the Lamb–Dicke regime and inthe resolved sideband limit [44]. Spectroscopy is performedby aligning the probe laser precisely along the axis of thelattice standing wave, and the atoms are probed free of recoil ormotional effects. The background gas pressure in the vacuumchamber of <1 × 10−9 Torr does not affect the spectroscopy,and the vacuum-limited lifetime is longer than the length ofthe experimental cycle.

Before performing spectroscopy, the atoms are firstoptically pumped to the stretched |F = 9/2, mF = ±9/2⟩states with the use of a weak optical beam resonant with the1S0(F = 9/2)–3P1(F = 7/2) transition. The beam usedfor optical pumping is aligned collinear with the lattice andis linearly polarized along the lattice polarization axis. Theoptical pumping is performed with a small magnetic bias field(∼3 µT), which is also oriented along the lattice polarization.After optical pumping, spectroscopy is performed on the 1S0–3P0 clock transition from the two spin sublevels. The clocktransition, which has a theoretical natural linewidth of ∼1 mHz

[43, 45–47], is interrogated using a diode laser (DL) at 698 nm,which is prestabilized by locking it to a high-finesse ultrastablecavity, resulting in a laser optical linewidth below 1 Hz [48].The probe beam is coaligned and copolarized with the opticallattice. To ensure that the stretched states are well resolved,the spectroscopy is performed under a magnetic bias field of25 µT, which results in a ∼250 Hz separation between the twoπ -transitions excited during the spectroscopy.

Spectroscopy is performed using an 80 ms Rabi pulse,which when on resonance transfers a fraction of the atoms intothe 3P0 state. After applying the clock pulse, atoms remainingin the 1S0 ground state are detected by measuring fluorescenceon the strong 1S0–1P1 transition. The length of the pulse islong enough both to measure the population in the 1S0 stateand to heat these atoms out of the trap. The population in the3P0 state is then measured by first pumping the atoms backto the 1S0 state through the intermediate (5s5p)3P0–(5s6s)3S1

and (5s5p)3P2–(5s6s)3S1 states and then by again measuringthe fluorescence on the 1S0–1P1 transition. Combining thesetwo measurements gives a normalized excitation fractioninsensitive to atomic number fluctuations from shot to shot.

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clock transition