PHYSICAL REVIEW A 90, 033829 (2014)

Initial atomic coherences and Ramsey frequency pulling in fountain clocks

Vladislav Gerginov,* Nils Nemitz,† and Stefan WeyersPhysikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany

(Received 15 May 2014; published 18 September 2014)

In the uncertainty budget of primary atomic cesium fountain clocks, evaluations of frequency-pulling shiftsof the hyperfine clock transition caused by unintentional excitation of its nearby transitions (Rabi and Ramseypulling) have been based so far on an approach developed for cesium beam clocks. We re-evaluate this type offrequency pulling in fountain clocks and pay particular attention to the effect of initial coherent atomic states. Wefind significantly enhanced frequency shifts caused by Ramsey pulling due to sublevel population imbalance andcorresponding coherences within the state-selected hyperfine component of the initial atom ground state. Suchshifts are experimentally investigated in an atomic fountain clock and quantitative agreement with the predictionsof the model is demonstrated.

DOI: 10.1103/PhysRevA.90.033829 PACS number(s): 42.50.Gy, 06.30.Ft

I. INTRODUCTION

Many high-resolution spectroscopy experiments such ascoherent population trapping [1] and electromagneticallyinduced transparency [2] are based on intentionally createdinitial state coherences. The operating principle of cesiumatomic clocks themselves relies on the Ramsey scheme [3],where the phase of an intentionally induced hyperfine statecoherence is compared to the phase of the driving mi-crowave radiation field after a free propagation time inthe absence of microwave radiation. Here we consider theconsequences of creating unintentional initial coherent super-positions between the magnetic sublevels belonging to thesame ground-state hyperfine component (Zeeman coherences)before the first Ramsey interaction in cesium fountain clocks[4].

This type of cesium clock provides the most preciserealization of the SI unit of time, the second, which isdefined by the unperturbed transition frequency between thetwo hyperfine ground-state components of the cesium atom.The systematic uncertainty of fountain clocks reaches a fewparts in 10−16 [5–8] and cryogenic sapphire oscillators [9]or optically stabilized microwave oscillators [10,11] allowstatistical uncertainties also at the level of a few parts in 10−16

after 1 day of operation. These developments necessitate aprogressively careful re-evaluation of effects that previouslycontributed only marginally to the overall uncertainty budget,such as frequency shifts due to unwanted hyperfine transitionsneighboring the clock transition (namely, Rabi and Ramseypulling).

Ramsey-pulling frequency shifts appear when the mi-crowave field driving the clock transition couples magneticallysensitive states to the clock states via �mF = ±1 (or σ )transitions, which change the projection quantum number mF

of the atom’s total angular momentum F [12]. The shift isqualitatively different from the Rabi-pulling shift, which is dueto the microwave field driving neighboring �mF = 0 (or π )transitions between magnetically sensitive states without

*Vladislav.Gerginov@ptb.de†Present address: Quantum Metrology Laboratory, RIKEN, Wako,

Japan.

coupling to a clock state [13]. Work on Ramsey pulling hasbeen published only for thermal beam clocks [12,14,15]. Sofar its contribution to the uncertainty budget of fountain clockshas been evaluated based on the theory specifically developedfor thermal beam clocks by Cutler et al. [12] and is generallyestimated to be below 10−16 when atomic state selection isemployed. However, the underlying theory for this estimateassumes a broad atom velocity distribution, constant amplitudeand direction of the magnetic microwave field componentsduring the Ramsey interactions, low microwave amplitudes,and incoherent initial atomic states. In contrast, in atomicfountains a narrow velocity distribution is achieved due tothe laser cooling, and the amplitude and direction of the mag-netic microwave field components are significantly positiondependent in fountain cavities. The state selection is performedby atom interaction with a microwave field, which typicallygenerates atomic coherences. Also, experimental uncertaintyevaluations often use frequency shift measurements at elevatedmicrowave amplitudes [5,16–18].

This work extends the Ramsey-pulling theory for beamclocks presented in Ref. [12], to better represent the case ofatomic fountains and to include the effects of initial Zeemancoherences. For beam clocks large frequency shifts have beendemonstrated and explained, when initial Zeeman coherencesare caused by Majorana transitions and undesired, but unavoid-able, σ transitions occur in the Ramsey cavity [19]. Indepen-dent of this work, Ramsey pulling in the presence of initialcoherences caused by optical pumping has been discussed forthermal beam clocks [15]. In fountain clocks initial Zeemancoherences are generally created during the state preparationprocess itself, before the clock transition takes place. Wedemonstrate that, as a result, in the case of broken symmetrybetween the initially coherently populated mF = +1 andmF = −1 states, correspondingly large frequency shifts canoccur. This is confirmed by direct experimental observation, ingood agreement with the theory. The Rabi-pulling effect [13],causing a frequency shift smaller in magnitude than theRamsey-pulling shift, is naturally included in the theoreticalmodel. The theory gives an improved uncertainty estimatefor the Physikalisch-Technische Bundesanstalt (PTB) fountainclocks CSF1 and CSF2.

Our discussion presents the perspective that frequencyshifts due to Majorana transitions before the first Ramsey

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VLADISLAV GERGINOV, NILS NEMITZ, AND STEFAN WEYERS PHYSICAL REVIEW A 90, 033829 (2014)

interaction [19] should be considered as Ramsey pullinginvolving initial asymmetric Zeeman coherences. On the otherhand, the effect of Majorana transitions occurring after thesecond Ramsey interaction [20] is to increase the frequencyshifting effect of the Ramsey pulling in combination withinhomogeneous atom state detection. We, finally, also notethat interfering asymmetric initial Zeeman coherences may bepresent in optical frequency standards too, depending on theemployed state preparation techniques [21].

The physical mechanisms of Rabi and Ramsey pulling andthe employed theoretical model are described in Section II.Section III presents basic symmetries and the calculatedmagnetic field dependence of Ramsey pulling, and Sec. IV thecalculated microwave amplitude and population dependences.The measured Ramsey-pulling shifts in the presence of initialZeeman coherences are described in Sec. V and compared withthe results of the calculation. Finally, in Sec. VI the contribu-tion of the combined Rabi- and Ramsey-pulling shifts to theuncertainty budget of the PTB fountain clocks CSF1 and CSF2is discussed. Appendix A gives the detailed derivation and theexplicit form of the total Hamiltonian. Appendix B gives theexplicit position dependence of the magnetic microwave fieldcomponents assumed for the microwave cavities of CSF1 andCSF2. Appendix C describes details of the calculation relatedto the use of two-dimensional (2D) atom clouds.

II. THEORETICAL MODELING OF RABIAND RAMSEY PULLING

We focus only on the six most relevant atomic states for thecase of cesium clocks, namely, states with quantum numbersF = 3, F = 4, and mF = −1,0,+1. The energy diagram ofthese states together with the clock transition between the |3,0〉and the |4,0〉 states is shown in Fig. 1, which also indicates theenergy shift given by the Breit-Rabi formula [22]. The depictedoff-resonant �mF = 0 π transitions (caused by microwavemagnetic field components along the vertical homogeneousmagnetic quantization field) with mF = ±1 are responsi-ble for the Rabi-pulling frequency shift: An asymmetricmF = ±1 state population of incoming F = 3 atoms leads toan asymmetric incoherent sum of the transition probabilitiesfor mF = ±1 atoms and the transition probability for mF = 0atoms and, thus, to a frequency shift.

FIG. 1. (Color online) Energy diagram of the relevant atomicstates of the cesium atom. Transitions due to the microwave fieldcomponent in the direction of the quantization axis (π transitions) areshown by solid lines; those due to the microwave field component inthe transverse direction (σ transitions), by dotted lines. The first-orderZeeman effect is also shown. The g factor of the cesium ground stateF = 4 component is gF ≈ 1/4.

The off-resonant �mF = ±1 σ transitions in Fig. 1, re-sponsible for the Ramsey-pulling shift, are of primary interestin this work. Starting from the three F = 3 states, thistype of transition, caused by transverse microwave magneticfield components (perpendicular to the vertical homogeneousmagnetic quantization field) in the Ramsey cavity of cesiumclocks, together with the π transitions leads to a coherentexcitation of the three F = 4 states. As a result, measured clocktransition frequency shifts occur when there is an asymmetricmF = ±1 state population of incoming F = 3 atoms or ifatoms in |4,±1〉 states are detected with different efficiencies.

To calculate frequency shifts caused by Rabi and Ramseypulling in an atomic fountain, we find the time evolution of acesium atomic ensemble represented by the density matrix ρ

by solving the von Neumann equation,

i�dρ(t)

dt= [(H0 + HRF),ρ(t)], (1)

where the total Hamiltonian of the cesium atomic system inthe presence of a static magnetic field �B pointing along thez direction and a microwave magnetic field �BRF is composedof the atomic Hamiltonian H0 describing both the hyperfineand the static magnetic field interactions and the HamiltonianHRF describing the microwave interaction (see Appendix Afor details).

For the modeling of Rabi and Ramsey pulling we considera fountain with a spatial dependence of the microwave fieldcomponents inside the state selection and Ramsey cavitiesgiven in Appendix B. In the presence of the microwave fieldin the state selection and Ramsey cavities the von Neumannequation is solved numerically, while during periods withoutmicrowave fields it is solved analytically. We calculated thetrajectory of each atom ensemble from a given initial positionand velocity of that ensemble. Each considered trajectorypasses twice through the lower aperture of the state selectioncavity. The velocity of the atom ensemble during a cavitypassage is assumed constant and is calculated from theduration of the passage and the cavity dimensions. The relevantspatial microwave field components BRF

k (k = x,y,z) andcorresponding Rabi frequencies bk = μBBRF

k (gJ − gI )/2�

(see Appendix A) are calculated from the horizontal positionsusing Eqs. (B1) and (B2), respectively. The time sequenceof the atom-microwave interactions and free propagations isshown in Fig. 2, together with a sketch of the correspondingamplitudes of the magnetic microwave field components.

Before the state selection interaction all elements of theinitial density matrix are 0, except for the diagonal elementsdescribing the populations of the |4,0〉 and |4,±1〉 states. Thisdescribes an atomic ensemble created at the end of the launchphase through optical pumping by the molasses lasers. Aftersolving Eq. (1) for the state selection microwave interactionfor a given trajectory, the atom ensemble is projected on theF = 3 ground-state component. This is the equivalent of theexperimental state selection process, where (in the ideal case)atoms are selectively transferred to the |3,0〉 state and anyatoms remaining in the initially populated F = 4 ground-statecomponent are removed by radiation pressure from a clearinglaser pulse.

The further internal evolution of the atom ensemble isdetermined from the solution of Eq. (1) for the subsequent

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FIG. 2. Time sequence of the fountain clock cycle (not to scale),indicating the time intervals for the state selection cavity traversal τS ,the free propagation between state selection and the Ramsey cavityTSR , the Ramsey cavity traversal τR , and the free propagation betweenthe two Ramsey cavity traversals TR . The BRF

x , BRFy , and BRF

z magneticcomponents of the microwave field are schematically representedby the shaded areas. The BRF

y field component in the consideredrectangular state selection cavities is 0 (see Appendix B).

free propagations and Ramsey microwave interactions. Forthe complete fountain cycle shown in Fig. 2, the final densitymatrix is used to determine the probability that the atomis in the F = 4 ground-state component. To simulate thebehavior of a real fountain clock, we define the clock transitionprobability as p = ∑1

mF =−1 ρ ′4(mF ,mF ), averaged for all atomic

ensembles, each with a different trajectory. Here, ρ ′F(mF ,m′

F ) isthe corresponding element of the 6×6 density matrix for statesbelonging to the F ground-state component, obtained by theunitary transformation given in Appendix A. The resonanteffective Rabi frequencies bs

0 in the state selection cavity andb0 in the Ramsey cavity (as defined in Appendix B) werenumerically adjusted in the simulation to yield a maximumaverage transition probability after resonant state selection andRamsey microwave interactions.

Finally, a set of four probabilities p0, pπ , and p±π/2 iscalculated for phase changes of 0, π , and ±π/2 of theresonant microwave field between the two Ramsey interac-tions. The probabilities p0 and pπ are used to calculate thecontrast of the Ramsey fringe. The probabilities p±π/2 thengive the frequency shift (assumed to be much smaller thanthe fringe width of approximately 1 Hz) of the transitionfrequency from the unperturbed clock transition frequency as(pπ/2 − p−π/2)/(2πTR(p0 − pπ )), with TR the Ramsey timecorresponding to the center of the atom cloud.

A typical clock transition probability as a function of themicrowave detuning, measured with CSF2 [18], is shown inFig. 3 together with the result of a simulation based on thesix-level model described here. The contributions of detectioncross-talk and hyperfine pumping in the detection zone,estimated through measurements, were taken into accountin the simulation by appropriate rescaling of the transitionprobability.

III. BASIC SYMMETRIES AND MAGNETIC FIELDDEPENDENCE OF RAMSEY PULLING

To better understand the basic dependences of Ramseypulling and to avoid additional complexity from incorporatingthe fountain state selection process, first, we performed

FIG. 3. (Color online) Measured (gray curve) and calculated (redcurve) clock transition probability as a function of the microwavefrequency detuning for the fountain clock CSF2 in normal operation.The �mF = 0 transitions are labeled “π”; the �mF = ±1 transitionsare labeled “σ .” Due to the large frequency span, the Ramsey fringestructures appear as filled areas.

calculations which included only the two microwave Ramseyinteractions and the free propagation between them. Thereforein the calculations in this section and in Sec. IV the initialatomic state of the atom ensemble at the beginning of thefirst Ramsey interaction was defined by setting the appropriateelements of the density matrix as described below. We considera hypothetical situation where an atom cloud consists ofmultiple atom ensembles with positions distributed along ahorizontal 2D grid with a radius rc equal to that of thecavity aperture and identical vertical trajectories. Such a 2Ddistribution of atom ensembles saves computation time byallowing the use of small numbers of ensembles and allowsus to better check the basic symmetries of Ramsey pulling.However, some adjustments of the model are needed to ensurea realistic fringe contrast decrease and the correct dependenceof Rabi and Ramsey pulling on the static magnetic fieldstrength (see Appendix C).

Since our calculations confirmed the assumption thatnormally a symmetry between the |3,±1〉 states does not leadto a frequency shift in the absence or presence of coherences,we considered asymmetric initial internal atomic states thatcorrespond to 0 population in the |3,−1〉, 99% in the |3,0〉, and1% in the |3,+1〉 state, respectively, set through the diagonalelements of the initial density matrix. To explore the coherenteffects, we included Zeeman coherences through the relevantoff-diagonal matrix elements set according to

ρ3(1,0) ≡ ρ

3†(0,1) = e+iϕ

√ρ3

(0,0)ρ3(1,1),

ρ3(−1,0) ≡ ρ

3†(0,−1) = e+i(π−ϕ)

√ρ3

(0,0)ρ3(−1,−1), (2)

which describes a maximally coherent initial state with phaseϕ.

The Ramsey-pulling shift depends on the direction of thetransverse microwave field component driving the �mF = ±1transitions causing the shift and on the phase of the initial

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FIG. 4. Diagram of the initial horizontal cloud positions (shadedarea) and atomic states. The circle represents a Ramsey cavity apertureof radius rc. Top labels: the absence (light gray) or presence (darkgray or gray gradient) of coherences and their phase dependenceon the azimuthal angle (gray gradient). Bottom labels: maximummagnitudes of the Ramsey-pulling frequency shift for magnetic fieldBz = BR

z values between 50 and 250 nT.

coherences. We considered two specific spatial phase depen-dences of the coherences.

In the first case, we assumed that the phase ϕ is the same forall atom ensembles (chosen to be ϕ = 0) and the relationshipof the density matrix elements is given by Eq. (2). This mightoccur due to Majorana transitions from the |3,0〉 state tothe |3,±1〉 states [23] or due to interaction with a homoge-neous microwave field on-resonance with the clock transitionfrequency and with a position-independent direction of themicrowave field component. In the latter situation the twomagnetically sensitive transitions are symmetrically detunedfrom resonance, and their corresponding matrix elements haveequal magnitude and opposite sign (see Appendix A). In thesecond case, we chose the phase ϕ in Eq. (2) to correspondto the azimuthal angle of the horizontal position of the atomensemble. This situation is similar to the result from the stateselection process in a cylindrical cavity with the microwavefield on-resonance, where the transverse field components havethis symmetry.

The cases considered are summarized in Fig. 4, includingcases with horizontal cloud offset, obtained by shifting theentire distribution by 2 mm and then discarding the trajectorieswhich are more than rc away from the fountain axis beforethe first Ramsey interaction. Also indicated are the calculatedmaximum magnitudes of the Ramsey-pulling frequency shiftfor static magnetic field Bz = BR

z values in and above theRamsey cavity between 50 and 250 nT. The calculatedfrequency shifts as a function of Bz are shown in Fig. 5, whereexperimental parameters listed in Table I were used in thecalculations.

Case A is the typically considered Ramsey-pulling shiftdue to a population difference between the ensembles in the|3,±1〉 states without initial Zeeman coherences. As shown inFig. 5 (left scale), the shift is less than 4×10−17 for magneticfield strengths between 50 and 250 nT and a 1% populationimbalance.

When the cloud is horizontally offset from the cavityaxis as in case B, the shift is slightly reduced, due to theremoval of atom ensembles from the edge regions of the cavityaperture, where the transverse field components have a higheramplitude. It is worth mentioning here that the single-atom

FIG. 5. (Color online) Ramsey-pulling shift as a function of thestatic magnetic field Bz = BR

z in and above the Ramsey cavity for a1% population imbalance of the |3, −1〉 and |3, +1〉 states. The labelsof the individual curves represent the initial conditions from Fig. 4:without coherences (cases A and B, left scale) and with coherences(case C, left scale; cases D–F, right scale).

Rabi- and Ramsey-pulling shifts for cases A and B wouldbe above the 10−16 level for field strengths between 50 and250 nT.

With a 1% population imbalance in the presence of coher-ences, but neither spatial dependence of their phase nor atomcloud horizontal offset (case C), the Ramsey-pulling shift isexactly the same as in case A. The potential enhancement of thefrequency shift due to the coherences is canceled completelybecause of the symmetry of the atom ensemble positions:for two ensembles which for both Ramsey interactions havediametrically opposite horizontal positions in the Ramseycavity with respect to the axis, the shifts due to the coherenceshave the same magnitudes but opposite signs. This is a resultof the phase of the coherence ρ3

(1,0) being the same for the

TABLE I. Parameters corresponding to the fountain CSF1 at PTB.The values τS , τR , TSR , and TR correspond to the center of the atomcloud. BS

z , BSRz , BR

z , and Bz are the average magnetic field strengths inthe state selection cavity, between the state selection and the Ramseycavity, in and above the Ramsey cavity. The effective Rabi frequenciesbs

0 and b0 (defined in Appendix B) correspond to a π pulse area (stateselection cavity) and a π/2 pulse area (Ramsey cavity).

Parameter Value

τS 7.2 msTSR 24.2 msτR 10.3 msTR 0.5539 sBS

z 96.7 nTBSR

z 97.2 nTBR

z 97.3 nTBz 99.8 nTbs

0 1377 rad/sb0 493 rad/sb0 (2D)a 511 rad/s

aThe value is for a uniform 2D cloud.

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two ensembles and the transverse microwave field componentpointing in opposite directions.

If a cloud offset from the cavity axis is included in thecoherent case, the cancellation of the Ramsey-pulling shift isnot complete, as only some of the atom ensembles have adiametrically opposite partner for both Ramsey interactions.The Ramsey-pulling shift in case D can be above the 10−15

level due to the presence of initial Zeeman coherences. It isshown in Fig. 5 (right scale) and is orders of magnitude strongerthan the shift with a cloud offset but without coherences(case B).

The coherent enhancement is also present for the importantcase E, where for each atom ensemble position the phase of thecoherences is chosen to depend on the azimuthal angle just asthe direction of the transverse microwave field component inthe Ramsey cavity. This leads to a frequency shift that has thesame sign and amplitude for all ensembles with the same radialdistance from the cavity axis. This prevents the cancellationeven for a cloud with no horizontal offset from the cavity axis,and the shift is above the 10−15 level. The magnitude of thefrequency shift depends only slightly on the cloud offset asdemonstrated in Fig. 5 (right scale) by comparison of casesE and F, because of the different amplitude of the microwavefield components. Due to the removal of ensembles from theedge regions, the shift in case F is reduced compared to caseE, similar to cases A and B.

We note that the magnitude of the Ramsey-pulling shiftdecreases with the inverse square of the value Bz of the staticmagnetic field without coherences [12] (Fig. 5; shaded area forpositive shifts) and with the inverse of Bz with coherences [15](Fig. 5; shaded area for negative shifts).

A process that changes the coherences phase relationsgiven by Eq. (2) can lead to frequency shifts even when thepopulations of the two |3,±1〉 states are the same. We testedthis prediction by calculating the shift for a 1% symmetricpopulation in both |3,±1〉 states, and an additional phase ofπ added only to the phase of ρ3

(1,0) and subtracted from thephase of ρ3

(0,1), while the phases of ρ3(−1,0) and ρ3

(0,−1) wereleft as given by Eq. (2). We performed calculations using thesymmetries of cases C–F from Fig. 4. For the symmetries ofcase C and equal populations of mF = ±1 states, the coherentenhancement of the shift is canceled for the same reasonsas discussed for case C with population imbalance. Becausethe |3,±1〉 state populations are the same, the noncoherentpart of the shift is even below 10−17. In contrast, for thesymmetries corresponding to cases D–F, the shift has aninverse sign and is approximately twice as large compared tocases D–F.

The significance of coherently enhanced Ramsey-pullingshifts with symmetric populations of the |3,±1〉 states isthat the shifts can be due to the state selection process witha microwave field on-resonance. In practice, the necessarydeviation of the phase of the atomic coherences from thatgiven by Eq. (2) could result from the ac Stark shift of the|3,±1〉 states during the clearing process, when the lightfrequency detuning and polarization couple the two |3,±1〉states to the 6p 2P3/2 excited-state components differently.Thus, a coherent enhancement of the Ramsey-pulling shiftcould be present under normal operation in fountains withresonant microwave state selection. Further investigations are

required to determine the magnitude of such light-assistedRamsey-pulling enhancement.

In summary, when initial Zeeman coherences between the|3,0〉 clock state and a |3,±1〉 state are present, the Ramsey-pulling shift can be above the 10−15 level when the symmetryof the coherences involving the |3,−1〉 and |3,+1〉 states isbroken, either by unequal populations in these states or dueto a difference in the phase of the corresponding coherences.Fortunately, effects such as Majorana transitions originatingfrom the |3,0〉 state will transfer the population to the |3,−1〉and |3,+1〉 states symmetrically, with coherences that fulfillEq. (2). A broken symmetry can still result from Majoranatransitions if an asymmetric population of the |3,−1〉 and|3,+1〉 states exists beforehand or if the relative phase of thecoherences is altered later by effects such as different ac Starkshifts of the |3,−1〉 and |3,+1〉 states with respect to the|3,0〉 state. A frequency shift of the state selection microwavefield from the clock transition frequency directly leads to apopulation and coherence asymmetry. Because the phase ofthe coherences created during a microwave state selectionbecomes position dependent, a significant shift can result evenfor a cloud centered on the cavity axis.

IV. MICROWAVE AMPLITUDE AND POPULATIONDEPENDENCES OF RAMSEY PULLING

The microwave amplitude dependence is an important fea-ture of Ramsey pulling. Experimentally, elevated microwaveamplitudes are used to study distributed cavity phase frequencyshifts and microwave leakage effects [24,25], and therefore itmight be useful to put an upper limit on the possible Ramsey-pulling contribution to the observed microwave-dependentfrequency shifts. As the microwave amplitude increases, themagnitude of the Ramsey-pulling shift also increases, asunder normal operation the microwave pulse area for the�mF = ±1 transitions is far below π , and their transitionprobability increases with the microwave amplitude, whilethe contrast of the central Ramsey fringe diminishes. Incontrast to the approach in Ref. [12], the numerical modelallows the calculations to be extended to Rabi frequenciessignificantly exceeding the frequency splitting between theZeeman components.

Here we study the microwave amplitude dependence ofRamsey pulling for the cases introduced in the previoussection. The microwave amplitude dependence for the caseof Ramsey-pulling shifts without initial Zeeman coherences(cases A and B in Fig. 4) as well as with initial Zeemancoherences (cases C to F in Fig. 4) are shown in Fig. 6. Sinceindividual atom ensembles experience microwave amplitudesthat vary by up to 15% with position, the Ramsey fringecontrast never goes to 0, and the shift does not showsingularities for even-π/2 pulse areas. As shown in Fig. 6, theamplitude dependence reaches the 10−15 level even withoutinitial coherences and, due to its highly nonlinear dependence,can significantly affect frequency evaluations with elevatedmicrowave amplitude levels.

Without coherences, additional calculations show that theshift depends almost linearly on the initial population of oneof the |3,±1〉 states, with the other state not populated. Thedeviation from linearity is less than 10% for a population

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FIG. 6. (Color online) Microwave amplitude dependence of theRamsey-pulling shift for a 1% population imbalance of the |3, −1〉and |3, +1〉 states. The labels of the individual curves represent theinitial conditions from Fig. 4: without coherences (cases A and C, leftscale) and with coherences (case B, left scale; cases D–F, right scale).The microwave amplitude is given with respect to normal operationwith a microwave π/2 pulse area. Circles: odd multiples of a π/2pulse area for case A.

of either of the |3,±1〉 states reaching 10% of the entireatom ensemble population (with the population of the otherstate kept 0). The shift value changes by less than 10%when the initial population difference in the |3,±1〉 statesis kept at 1%, and the respective populations of these statesare increased to 10% and 9% of the entire atom ensemblepopulation. The shift depends quadratically on the ensembleoffset from the cavity axis and is 0 for ensembles on theaxis. For smaller atom cloud sizes, the atoms are closer tothe cavity axis (where the transverse microwave field com-ponents have a lower amplitude) than in the case of a clouduniformly distributed across the cavity aperture as done inthe calculation (with the exception of small clouds withlarge offsets from the cavity axis). The uniform distributionof the atom cloud across the cavity aperture leads to anoverestimation of the calculated Ramsey-pulling shifts by10%–15% compared to clouds smaller than the cavity apertureand with small offsets from the cavity axis.

In the presence of initial coherences with a magnitudeexpressed through Eq. (2), the frequency shift with populationimbalance deviates from linearity by less than 10% until therelative population of either of the |3,±1〉 states reaches 10%.Its deviation from linearity is less than 3% when the populationof either of the |3,±1〉 states is kept at 10%, and only thecoherence magnitude is changed from 0 to the maximumone given by Eq. (2). The frequency shift dependence onthe |3,0〉 ↔ |3,±1〉 coherence magnitudes given by Eq. (2)deviates from linearity by less than 10% when the populationdifference of the |3,±1〉 states is kept at 1%, and the respectivepopulations of these states are increased to 10% and 9%of the entire atom ensemble population. The frequency shiftdependence on the ensemble offset from the cavity axis in thepresence of coherences is nonlinear, as the coherence phasedepends strongly on the amplitude of the transverse microwavecomponents, which varies with the offset from the cavityaxis.

V. EXPERIMENTAL OBSERVATION OFRAMSEY-PULLING FREQUENCY SHIFTS

Without Zeeman coherences between the initial atomicstates, the calculated Ramsey-pulling shift is below the 10−16

level for normal fountain operation. The presence of Zeemancoherences can make the shift significant at the 10−15 leveland makes experimental observation possible. To test thepredictions of the calculation, we used the state selectionprocess in the fountain PTB CSF1 to create coherent atomicsuperpositions, which allowed us to observe for the first timea Ramsey-pulling shift in a fountain clock. In CSF1, thestate selection cavity is mounted below the Ramsey cavityin the region of the low magnetic C field. As a result, thepropagation time between the state selection and the Ramseycavity is only TSR = 24.2 ms (see Fig. 2), and the phases ofthe individual atom coherences created by the state selectionprocess do not average out. The clearing pulse is also appliedduring this travel time by a vertical, circularly polarized laserbeam.

The state selection microwave field drives �mF = 0 and�mF = ±1 transitions simultaneously due to the presenceof vertical and transverse field components. Such excitationprepares each atom in a coherent superposition which involvesthe |3,0〉 clock state and the |3,±1〉 magnetically sensitivestates. Under normal operation, with the state selectionmicrowave field on-resonance and its amplitude correspondingto a π pulse area, the population of the |3,±1〉 states is verysmall due to the presence of the static magnetic field BS

z in thestate selection cavity, which detunes the transitions, and thelow amplitude of the transverse microwave field components.To increase the magnitude of the frequency shift, on onehand, the population of the |3,±1〉 states was increased byoperating the state selection cavity at an increased microwaveamplitude (corresponding to a 9π pulse area on-resonance).On the other hand, the Ramsey cavity was also operatedat an increased microwave amplitude (7π/2 pulse area) toenhance the �mF = ±1 transition probabilities. The fountainfrequency was measured against the frequency average of twohydrogen masers.

For zero frequency detuning in the state selection cavity,the Ramsey pulling is expected to be negligible becausethe |3,±1〉 states will be symmetrically populated, and thephase relations of Eq. (2) are valid. To cause a significantfrequency shift, we used a detuning of the state selectionmicrowave field from the clock transition frequency in therange of ±200 Hz, which creates asymmetry in the populationsof the |3,±1〉 states and, correspondingly, in the coherentstate superpositions |3,0〉 ↔ |3,±1〉. The shift in the fountainfrequency as a function of the state selection microwavefrequency is shown in Fig. 7 (symbols). The small stateselection field detuning preserved a high population of the|3,0〉 clock state and allowed us to measure the fountainfrequency without significant degradation of the frequencystability. For detunings of the order of ±100 Hz and for thechosen microwave amplitude, calculations show that after thestate selection process, roughly 2% of the atomic populationin the F = 3 component can be in either of the two statesuperpositions |3,0〉 ↔ |3,±1〉 (depending on the detuningsign), while around 8% of the atomic population in the F = 3

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FIG. 7. (Color online) Ramsey-pulling frequency shift of thefountain CSF1 as a function of the state selection microwave fielddetuning. Microwave pulse areas: 9π (state selection cavity) and7π/2 (Ramsey cavity). Data points: measurements. Dashed line:calculation based on the measured magnetic field value, BSR

z =97.2 nT. The solid line and shaded area correspond to an equivalentmagnetic field, BSR

z + BStarkz ± 0.5 nT, accounting for the clearing

pulse Stark shift. The shift is measured with respect to the fountainfrequency without state selection field detuning.

component is due to the transfer from the |4,±1〉 to the |3,±1〉states by the mF = ±1, �mF = 0 transitions.

The frequency shift due to the detuned field in the stateselection cavity was calculated using the model introducedin Sec. II. The atom cloud was approximated by a cloudwith a Gaussian radius σ = 4.0×10−4 m and temperatureTat = 1.6×10−6 K, determined from measurements. The res-onant microwave amplitudes in the simulation were adjustednumerically (for the maximum average transition probability)to correspond to 9π (state selection cavity) and 7π/2 (Ramseycavity) pulse areas. The magnetic fields were chosen to matchthe measured values as listed in Table I. An additional magneticfield offset BStark

z = −2.6 nT accounts for the ac Stark shiftresulting from the clearing laser pulse, as discussed below.The calculation results are also shown in Fig. 7. The dashedcurve corresponds to the measured magnetic field value BSR

z

between the state selection and the Ramsey cavities as listed inTable I. The solid curve corresponds to BSR

z with the additionaloffset of BStark

z = −2.6 nT, and the shaded area to BSRz with an

offset range of BStarkz ± 0.5 nT, representative of the magnetic

field uncertainties.Our model shows that the Ramsey-pulling shift is deter-

mined by both the magnitude and the phase of the Zeemancoherences with which the atoms arrive in the Ramsey cavity.First, these parameters depend on the static magnetic fieldBS

z and the microwave field parameters during the microwaveinteraction in the state selection cavity. Second, during the timethe atoms spend between the two cavities, the magnitude ofthe Zeeman coherences does not change, but their phaseevolves due to the effective static magnetic field BSR

z and dueto the occurrence of the clearing laser pulse (through the acStark shift effect). The measured and calculated dependenceof the Ramsey-pulling frequency shift on the value of the static

FIG. 8. (Color online) Plot of the Ramsey-pulling frequency shiftin the fountain CSF1 versus the static magnetic field BSR

z betweenthe state selection and the Ramsey cavities. Microwave pulseareas: 9π (state selection cavity) and 7π/2 (Ramsey cavity). Leftscale: measurements, symbols. Right scale: calculations, lines. Thestate selection microwave frequency detuning is +110 Hz [curves(a) and (c); red] or −110 Hz [curves (b) and (d); blue]. Solidlines and shaded areas correspond to an equivalent magnetic field,BSR

z + BStarkz ± 0.5 nT, accounting for the clearing pulse Stark shift.

The shift is measured with respect to the fountain frequency withoutstate selection field detuning.

magnetic field BSRz is shown in Fig. 8. The detuning of the state

selection microwave frequency was fixed to +110 Hz [(red)circles] or −110 Hz [(blue) squares]. The value of the fieldBSR

z was changed by varying the current in a compensationcoil positioned at the bottom of the innermost magnetic shieldof the fountain with its axis in the vertical direction. Thevariation of the coil current leads to different values forthe field BSR

z , the fields BSz in the state selection and BR

z

in the Ramsey cavities, and the effective field Bz above theRamsey cavity. These magnetic field values were measuredusing the atoms as a probe for several values of the compen-sation coil current. The measured values of BS

z , BSRz , BR

z , andBz were used in the calculation. We note that, as a result,the depicted frequency shifts are caused by the combination ofeffects due to the actual values of BS

z , BSRz , BR

z , and Bz, and notjust by the effect of the actual value of BSR

z , as Fig. 8 mightsuggest. The dashed curves correspond to +110 Hz [curve(a)] and −110 Hz [curve (b)]. The clearing pulse ac Stark shiftwas again included through the BSR

z magnetic field offset ofBStark

z = −2.6 nT and is shown by the solid lines [+110 Hz,curve (c); −110 Hz, curve (d)]. The shaded areas correspondto a BSR

z offset range of BStarkz ± 0.5 nT.

A disagreement between measurements and simulationexists, as indicated by the use of two vertical scales. A reasoncould be the use of interpolation to map the magnetic fieldalong the atom ensemble trajectories from a limited numberof measurements at different compensation coil currentsand atom cloud launch heights. Other possible reasons arediscussed at the end of this section.

The Ramsey-pulling shift magnitude also shows a depen-dence on the duration of the clearing laser pulse excitingthe 2S1/2(F = 4)–2P3/2(F ′ = 5) optical transition, which isapplied by a circularly polarized vertical laser beam in CSF1.

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VLADISLAV GERGINOV, NILS NEMITZ, AND STEFAN WEYERS PHYSICAL REVIEW A 90, 033829 (2014)

FIG. 9. (Color online) Ramsey-pulling frequency shift as a func-tion of the clearing pulse duration. Microwave pulse areas: 9π (stateselection cavity) and 7π/2 (Ramsey cavity). The corresponding linesare sinusoidal fits to guide the eye. Detuning: curve (a), (red) circles,+120 Hz; curve (b), (blue) squares, −120 Hz. The shift is measuredwith respect to the fountain frequency without state selection fielddetuning.

The laser frequency is detuned from all transitions involvingthe F = 3 ground-state component, by the cesium hyperfinecomponent splitting of 9.2 GHz. The states of the F = 3component experience an ac Stark shift due to the presence ofthe off-resonant laser light. The ac Stark shift is different forthe |3,0〉 and the |3,±1〉 states, which leads to a phase changeof the Zeeman coherences. The phase change is proportionalto the ac Stark shift frequency differences for the |3,0〉 andthe |3,±1〉 states, the laser intensity, and the clearing pulseduration.

In the fountain CSF1 we have changed the clearing pulseduration from 0.5 to 12 ms, and the corresponding change ofthe Ramsey-pulling shift for a fixed state selection microwavefield detuning of ±120 Hz is shown in Fig. 9 [(red) circles and(blue) squares]. From the sinusoidal fits the absolute value ofthe phase change corresponding to a push pulse duration of 2ms is 1.39 rad. The magnetic field offset, BStark

z = −2.6 nT,used in preceding sections was chosen to reproduce this phaseshift after the propagation time TSR = 24.2 ms. Due to its clearsignature, the frequency shift dependence on the light pulseduration can be used as a simple test to reveal the presenceof Zeeman coherences in the initial atomic state before theRamsey interactions.

In summary, we find an antisymmetric dependence onthe state selection microwave field detuning, with no shiftpredicted for a state selection field resonant with the clocktransition (Fig. 7). In the presence of a state selection detuning,we observe a clear dependence of the resulting shift on thestatic magnetic field between the microwave cavities (Fig. 8)and on the duration of the clearing laser pulse duration (Fig. 9),both of which affect the phases of the Zeeman coherences.

The agreement between measurements and the calculationsshown in Figs. 7 and 8 is not perfect. It is important tomention that the calculations do not include free parameters.One expects a partial disagreement between measurements and

calculations because of the uncertainty in the determination ofthe different static magnetic field values and the assumptionof vertical magnetic field direction. Simulations show thatan angle of only a few degrees between the field directionand the cavity axis changes the value of the Ramsey-pullingshift appreciably. A magnetic field tilt of several degrees isobserved in CSF1 as in the fountain CSF2 [26]. Discrepancyis also expected due to the difference in the actual profile ofthe state selection microwave field and the TE 201 mode usedby the simulation. The simulation shows that the phase ofthe initial coherences depends strongly on the state selectionmicrowave pulse area and, thus, on the accuracy of the pulsearea determination. The clearing laser pulse spatial profile isnot homogeneous, which leads to a dephasing of the atomiccoherences. Finally, the calculations assumed that the cloudhad a Gaussian shape, with a center of mass on the axis for boththe state selection and the Ramsey cavities. These assumptionsare only approximately correct for CSF1.

VI. CONTRIBUTION OF THE RAMSEY-PULLING SHIFTTO THE UNCERTAINTY BUDGET OF CSF1 AND CSF2

For CSF1, the low static magnetic field in the state selectioncavity leads to coherent initial state superpositions even in thenormal mode of operation. To prevent shifts, it is important tokeep the excitation of the |3,±1〉 states low and symmetric.This can be accomplished by ensuring a zero detuning of themicrowave field in the state selection cavity. As the rapid adi-abatic passage method for collisional shift measurements [27]is not applicable to the CSF1 fountain due to the low staticmagnetic field in the state selection cavity, the method ofdetuned state selection [28] has been considered for operationwith low atomic densities to reduce systematic collisionalshift errors [29]. In this case, the Ramsey-pulling shift (atthe 10−16 level) will contribute only during low-atom-densityoperation, performed with state selection frequency detuning.The Ramsey-pulling shift could then be erroneously attributedto an additional collisional shift, and the fountain frequencywould be wrongly extrapolated to zero atom density. Such acollisional shift measurement problem can be diagnosed andcorrected for by alternating shots with negative and positivestate selection detuning during low-atom-number fountainoperation. The Ramsey-pulling shift in this case will averageto 0 as long as the detection efficiencies are identical for the|3,±1〉 states [20].

In fountain CSF2, the enhancement of the Ramsey-pullingshift due to Zeeman coherences is not expected to play arole. On the one hand, the employed rapid adiabatic passagemethod increases the probability of magnetically sensitiveπ and σ transitions due to the high microwave amplitudeand pulse spectral width, and in the case of the interruptedpulse used in low-atom-number operation, excites the |3,±1〉states asymmetrically [30]. On the other hand, however, thestate selection cavity of the fountain is outside the magneticshields, with a Zeeman component splitting of more than50 kHz. For such detunings, the π and σ transitions aredriven with a very low probability even at high microwaveamplitudes. More importantly, the free propagation of theatoms between the state selection and the Ramsey cavityoccurs in a static magnetic field above 10 μT that is strongly

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INITIAL ATOMIC COHERENCES AND RAMSEY . . . PHYSICAL REVIEW A 90, 033829 (2014)

inhomogeneous near the magnetic shield apertures, whichenables fast averaging of the phase of the Zeeman coherences.The difference in the individual atom ensembles’ propagationtimes between the cavities is considerably larger compared tothat of CSF1, leading to a significant phase decoherence. NoRamsey-pulling shifts were observed in CSF2 at the 10−15

level in experiments with operational conditions similar (withdetuning and increased microwave amplitudes of the stateselection microwave field) to those used with CSF1 to generatecoherent Ramsey-pulling shifts.

Assuming that shift enhancement due to initial coherencesdoes not play a role in either fountain, we calculated theRamsey-pulling shift without coherences for both CSF1 andCSF2 fountains for normal operation. For both fountains,the current value of the static magnetic field Bz insidethe magnetic shielding is approximately 150 nT, giving amaximum Ramsey-pulling shift of 5×10−18 for 1% populationimbalance, as shown in Fig. 5 (cases A and B). In bothCSF1 and CSF2, the measured population imbalance is 0.25%,resulting in a maximum Ramsey-pulling shift of 1.25×10−18.This value is used from now on for the uncertainty budgetcontribution of the combined Rabi- and Ramsey-pulling fre-quency shift in CSF1 and CSF2. This value naturally includesthe Rabi-pulling shifts previously investigated independently.

VII. CONCLUSIONS

Ramsey-pulling frequency shifts were calculated extendingthe theory of Ramsey pulling presented by Cutler et al. [12] tothe case of atomic fountains by taking into account coherentinitial atomic states, a narrow atomic velocity distribution, anda varying amplitude and direction of the magnetic microwavefield components during Ramsey interactions. The calculationsprovide an improved estimate of the maximum Rabi- andRamsey-pulling shift.

It is shown that initial asymmetric coherences betweenmagnetic sublevels of the same hyperfine ground-state com-ponent can lead to a dramatic enhancement of the Ramsey-pulling shift, by more than two orders of magnitude, and toa considerable shift under normal fountain clock operation.Initial asymmetric coherences can be due to detuning ofthe state selection microwave field, Majorana transitions, orcan be created during or after the state selection process byatom interaction with the microwave or optical fields. Whilecoherences do not lead to frequency shifts as long as theirphases fulfill the symmetry requirements discussed in Sec. III,this symmetry can easily be broken by effects such as the acStark shift resulting from the clearing pulse.

Our approach also enabled an investigation of Ramseypulling at elevated microwave amplitudes in the Ramseycavity. In this case the magnitude of the Ramsey-pullingshift increases and can reach the 10−15 level even withoutinitial Zeeman coherences, which can affect the results ofexperimental investigations at increased amplitudes, such asthose used to characterize distributed cavity phase effects.

The properly applied state selection process in fountainclocks excludes frequency shifts caused by asymmetric co-herences from Majorana transitions at the early stages ofthe fountain cycle (before the clock transition takes place).However, potential Majorana transitions and resulting Zeeman

coherences generated at the end of the clock cycle (before theatomic state detection) may enhance the Ramsey frequencypulling when the detection is inhomogeneous with regard tothe ±mF states [20].

The theoretical results are supported by experiments withthe PTB fountain CSF1, by changing the frequency of the stateselection microwave field at increased microwave amplitudein both the state selection and the Ramsey cavities. TheRamsey-pulling shift is shown to depend on the state selectionmicrowave field detuning, on the magnetic field between thestate selection and the Ramsey cavities, and on the presence oflight interacting with the cesium atoms at the end of the stateselection process.

ACKNOWLEDGMENTS

The authors thank R. Wynands, A. Taichenachev, V. Yudin,S. Jefferts, J. Shirley, and T. Heavner for their valuablecomments and suggestions.

APPENDIX A: DERIVATION AND FORMOF THE TOTAL HAMILTONIAN

The hyperfine interaction couples the nuclear spin I = 7/2and the electronic angular momentum J = 1/2 of the cesiumatomic ground state. In the absence of external fields, theatomic Hamiltonian H0 takes the form H0 = A(I · J), with A

the hyperfine structure constant of the 6s 2S1/2 cesium groundstate and I and J the nuclear spin and the electronic angularmomentum operators. The eigenstates of the atomic systemcan be written in the basis of the total angular momentumF = I + J, with values for the cesium ground state F = 3 or4 and projection quantum number mF = −F . . . F .

In the presence of an external static magnetic field �B =(0,0,Bz) pointing along the z direction, H0 is given by

H0 = A(I · J) + μBBz(gI Iz + gJ Jz), (A1)

where μB is Bohr’s magneton, gI = −0.398 853 9(5)×10−3,and gJ = 2.002 540 3(2) [31,32] are the nuclear and theelectronic g factors and Iz and Jz are the z components of thenuclear spin and the electronic angular momentum operators.

The matrix elements 〈F ′,mF ′ |H0|F,mF 〉 of H0 are cal-culated by transforming the |F,mF 〉 states into the product|I,mI 〉 ⊗ |J,mJ 〉 basis, acting on the resulting states with theoperators I and J in H0, and transforming the result backinto the |F,mF 〉 basis [33]. The resulting eigenvalues (orenergies) of the matrix 〈F ′,mF ′ |H0|F,mF 〉 are described bythe Breit-Rabi formula,

EF,mF= − 2π�ν0

2(2I + 1)± 2π�ν0

2

√(1 + 4mF xB

2I + 1+ x2

B

)

+mF gIμBBz, (A2)

where � is Planck’s constant divided by 2π, ν0 =9 192 631 770 Hz is the unperturbed hyperfine splitting fre-quency of the components F = 3 and F = 4, the (+)sign corresponds to the F = 4 and the (−) sign to theF = 3 state component, and xB = μBBz(gJ − gI )/(2π�ν0).In the low-magnetic-field approximation, the eigenstates of〈F ′,mF ′ |H0|F,mF 〉 can be replaced by the states |F,mF 〉. In

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VLADISLAV GERGINOV, NILS NEMITZ, AND STEFAN WEYERS PHYSICAL REVIEW A 90, 033829 (2014)

this approximation, the matrix 〈F ′,mF ′ |H0|F,mF 〉 is diagonal,with matrix elements given by Eq. (A2).

In the presence of a microwave field with frequencyνRF = ν0 + �ν, with frequency detuning �ν from ν0 andan amplitude of the magnetic field vector component �BRF =(BRF

x ,BRFy ,BRF

z ), the total atomic Hamiltonian becomes H =H0 + HRF, where the microwave interaction Hamiltonian HRF

is given by

HRF =∑

k=x,y,z

μBBRFk (gI Ik + gJ Jk)(ei2π(ν0+�ν)t + c.c.)/2.

(A3)

For the cesium ground state, the matrix elements of the nuclearspin operator I have the same magnitude (except for thecorresponding g factors) and the opposite sign to the matrixelements due to the electronic angular momentum operator J.Thus, we can replace the product gI Ik with −gIJk in Eq. (A3).The g factors can be incorporated in the Rabi frequencies bk =μBBRF

k (gJ − gI )/2� related to the amplitude of the magneticmicrowave field components. After these transformations, HRF

takes the form

HRF =∑

k=x,y,z

�bkJk(ei2π(ν0+�ν)t + c.c.). (A4)

The corresponding matrix elements of HRF are calculated inthe same way as those of H0. It is now convenient to usea unitary transformation which allows us to set to 0 thematrix elements of H oscillating at the hyperfine frequencyν0. The matrix elements 〈F ′,mF ′ |U |F,mF 〉 of the unitary

transformation U are

UF ′FmF ′ mF=

{δF ′F δmF ′mF

e−iπ( 78 ν0+�ν)t (for F = 4),

δF ′F δmF ′mFe+iπ( 9

8 ν0+�ν)t (for F = 3),

(A5)

where δ is the Kronecker δ. The transformation rule for thetotal Hamiltonian H ′ = H ′

0 + H ′RF is

H ′ = U †(H0 + HRF)U − i�U † dU

dt, (A6)

where U † is the Hermitian adjoint operator of U . Thetransformation rule for the density matrix is ρ ′ = U †ρ U .

The unitary transformation U causes terms to appearin H ′ which oscillate at frequencies close to 2ν0, whichwe will neglect (rotating-wave approximation). Also, thematrix elements corresponding to transitions between Zeemansublevels of the same ground-state component F have a timedependence oscillating at frequencies close to ν0. These termsare also neglected in the rotating-wave approximation, as theycorrespond to transitions well detuned for weak magneticfields Bz.

The atomic state is represented by a density matrix ρ

constructed from the six relevant states in the followingorder: (|4,+1〉,|4,0〉,|4,−1〉,|3,+1〉,|3,0〉,|3,−1〉). The totalHamiltonian H ′ of the six-level atomic system, after applica-tion of the unitary transformation U (t) and the rotating-waveapproximation, is

H ′ = �

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

δ1 0 0 −√

1516 bz

√5

16 (bx − iby) 0

0 δ2 0 −√

316 (bx + iby) − 1

4bz

√3

16 (bx − iby)

0 0 δ3 0 −√

516 (bx + iby) −

√15

16 bz

−√

1516 bz −

√3

16 (bx − iby) 0 δ4 0 0√

516 (bx + iby) − 1

4bz −√

516 (bx − iby) 0 δ5 0

0√

316 (bx + iby) −

√15

16 bz 0 0 δ6

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (A7)

where the Rabi frequencies bk (k = x, y, and z) depend on the atom ensemble position in the microwave cavities. The parametersδi are given below:

δ1 = −π(ν0 + �ν − ν0

√1 + xB/2 + x2

B

) + BzgIμB/�, δ2 = −π(ν0 + �ν − ν0

√1 + x2

B

),

δ3 = −π(ν0 + �ν − ν0

√1 − xB/2 + x2

B

) − BzgIμB/�, δ4 = +π(ν0 + �ν − ν0

√1 + xB/2 + x2

B

) + BzgIμB/�, (A8)

δ5 = +π(ν0 + �ν − ν0

√1 + x2

B

), δ6 = +π

(ν0 + �ν − ν0

√1 − xB/2 + x2

B

) − BzgIμB/�.

APPENDIX B: MICROWAVE FIELD DISTRIBUTIONIN THE FOUNTAIN CAVITIES

The PTB fountains CSF1 and CSF2 have different rectan-gular state selection cavities (supporting the TE 201 and TE 401

modes, respectively) and identical cylindrical Ramsey cavities(supporting the TE 011 mode). The solution of the Maxwell

equations for a real cavity with apertures and resistive lossescan be obtained using finite-element analysis methods [34]. Inthis work, we use the TEM modes of an ideal rectangular (stateselection) or cylindrical (Ramsey) cavity with no apertures asan approximation of the spatial microwave field distributions.

The expressions for the magnetic components of themicrowave field of a TE m01 mode of the rectangular state

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TABLE II. Cavity parameters for PTB fountains CSF1 and CSF2(in millimeters).

Parameter CSF1 CSF2

dsx (state selection) 48.00 84.46

dsy (state selection) 12.00 38.00

dsz (state selection) 22.00 25.67

Height (above loading zone) 346 186

d (Ramsey) 28.62 28.62r (Ramsey) 24.20 24.20rc (aperture) 5.00 5.00Height (above loading zone) 442 635

selection cavities in Cartesian coordinates are [35]

BRFx = BRF

0ds

x

mdsz

sin

(πm

dsx

x

)sin

(π

dsz

z

),

BRFy = 0,

BRFz = BRF

0 cos

(πm

dsx

x

)cos

(π

dsz

z

), (B1)

where the field components are normalized such that the z

component is equal to BRF0 in the center of the cavity. The

horizontal dimension of the cavity along the x axis is dsx , and

the cavity height is dsz . The mode number is m = 2 for CSF1

and m = 4 for CSF2. The distances along x and z are measuredfrom the cavity center.

The expressions for the magnetic components of themicrowave field of a TE 011 mode of the Ramsey cavity incylindrical coordinates are [35]

BRFρ = BRF

0πr

x0dJ1

(x0

rρ

)sin

(π

dz

),

BRFϕ = 0,

BRFz = BRF

0 J0

(x0

rρ

)cos

(π

dz

), (B2)

where, again, the vertical microwave field component is equalto BRF

0 in the center of the cavity, x0 is the first zero of the Besselfunction J1, ρ is the radial distance from the cavity axis, r isthe radius of the cavity, z is the vertical distance measuredfrom the center of the cavity, and d is the cavity height. Thecavity parameters for the fountains CSF1 and CSF2 are listedin Table II.

The value of the microwave field component BRF0 can be

more conveniently expressed in terms of an effective Rabifrequency, b0 = μBBRF

0 (gj − gI )/2�, taking a value bs0 for the

state selection and b0 for the Ramsey microwave interaction.To solve the von Neumann equation (1), we convert thespatial dependence of the microwave amplitude into a timedependence assuming a constant velocity along the z directionfor a given atom crossing the cavity on the way up or down.

APPENDIX C: FRINGE CONTRAST IN THE 2D MODEL

For a given atom ensemble i, the mF = ±1, �mF = 0,and �mF = ±1 transition probabilities as a function of themicrowave frequency have a Ramsey fringe structure with a

period TRi . The Ramsey fringe structures of these transitionsoverlap and extend all the way to the clock transitionfrequency. When the static magnetic field is changed, thefringe structure of these transitions and the correspondingfrequency shift of the clock transition change accordingly,leading to a field-dependent Rabi-pulling shift with a periodproportional to 1/TRi and a Ramsey-pulling shift with aperiod proportional to 2/TRi (see Ref. [12]). The factor-of-2difference in the field dependence is due to the twice-smallerfield dependence of the �mF = ±1 transitions compared tothe mF = ±1, �mF = 0 transitions.

In practice, the fringe contrast of the transitions involvingmagnetically sensitive states decreases in the same way as thatof the clock transition with their detuning from resonancedue to the averaging over atom ensembles with differentvertical trajectories and corresponding Ramsey times TRi .This decrease can be seen in Fig. 3 (bottom; red curve). Thediminished fringe contrast of the �mF = ±1 and mF = ±1,�mF = 0 transitions near the clock transition frequency ν0

generally leads to Rabi- and Ramsey-pulling shifts of reducedmagnitude and actual dependences on the static magneticfield strength with a period proportional to 1/τR and 2/τR ,i.e., to the microwave interaction time in the Ramsey cavitycorresponding to the cloud center [14]. Because, for the2D atom cloud used in the calculation, the atom ensembleshave identical vertical trajectories, the fringe contrast is notdiminished with detuning from resonance and adjustments ofthe model are needed.

To cancel the TR dependence of Ramsey pulling, we usedin the simulation two same-state atom ensembles at eachhorizontal grid point at the beginning of the first Ramsey in-teraction. The first ensemble had the same horizontal positionduring both Ramsey interactions, while the second ensemblehad a diametrically opposite horizontal position during thesecond Ramsey interaction. For each such pair, the Ramseyfringe structure of these transitions has an opposite phase,as, due to the cylindrical cavity symmetry, the microwavecomponents driving the �mF = ±1 transitions in each atomensemble point in opposite directions during the secondRamsey interaction. At the same time, the fringe structureof the �mF = 0 transitions for the two ensembles remainsthe same. As a result, the Ramsey-pulling shift dependenceon the Ramsey time TR is canceled out when the shiftis averaged over the cloud containing such pairs of atomensembles.

To cancel also the dependence of the Rabi-pulling shift onTR , we calculated the average of two shift values correspondingto two magnetic fields in and above the Ramsey cavity: one fora field Bz = BR

z and one for a slightly higher field, B ′z = B ′R

z ,which increases the |3,+1〉 → |4,+1〉 transition frequencyby 1/2TR . For these two magnetic field values, the fringestructures of the mF = ±1, �mF = 0 transitions have an op-posite phase around ν0, and the Rabi-pulling shift dependenceon TR is canceled. Additional calculations performed usingMonte Carlo simulated 3D clouds with temperatures of theorder of 1 μK showed that, after the cancellation, the resultsof the calculation using distribution of atom ensembles ona 2D grid resemble closely these using a 3D atom cloudsfilling uniformly the cavity apertures on the way up anddown.

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