Fourier analysis of Ramsey fringes observed in a continuous atomic fountain for in situ magnetometry
Gianni Di Domenico, Laurent Devenoges, André Stefanov, Alain Joyet, Pierre Thomann
FFourier analysis of Ramsey fringes observed in a continuousatomic fountain for in situ magnetometry
Gianni Di Domenico, Laurent Devenoges, Andr´e Stefanov, Alain Joyet, and Pierre Thomann Laboratoire Temps-Fr´equence, Universit´e de Neuchˆatel,Avenue de Bellevaux 51, CH-2000 Neuchˆatel, Switzerland Swiss federal office of metrology, METASLindenweg 50, CH-3003 Bern-Wabern, Switzerland
Ramsey fringes observed in an atomic fountain are formed by the superposition of the individualatomic signals. Due to the atomic beam residual temperature, the atoms have slightly differenttrajectories and thus are exposed to a different average magnetic field, and a velocity dependentRamsey interaction time. As a consequence, both the velocity distribution and magnetic field profileare imprinted in the Ramsey fringes observed on Zeeman sensitive microwave transitions. In thiswork, we perform a Fourier analysis of the measured Ramsey signals to retrieve both the timeaveraged magnetic field associated with different trajectories and the velocity distribution of theatomic beam. We use this information to reconstruct Ramsey fringes and establish an analyticalexpression for the position of the overall observed Ramsey pattern.
I. INTRODUCTION
Since its invention in 1950, the method of separated os-cillatory fields [1] has been widely used in physics exper-iments. Particularly in the field of atomic clocks whereit made possible successive improvements of the perfor-mances by many orders of magnitude. Indeed, by excit-ing the atomic transition with separated oscillatory fields,the width of the resonance is inversely proportional to thefree evolution time between the two excitation pulses. Asa consequence, any method allowing to increase the freeevolution time results in an improvement of the clockperformance.The separated oscillatory fields method was first ap-plied to thermal atomic beams where the free evolutiontime is of the order of 10 − -10 − s, limited by the lengthof the resonator. With the advent of laser cooling, itbecame possible to produce fountains of cold atoms [2],and thereby to increase the free evolution time to ap-proximately 0 . − in relative units.Our alternative approach to atomic fountain clocks isbased on a continuous beam of laser-cooled atoms. Be-sides making the intermodulation effect negligible [4, 5],a continuous beam is also interesting from the metrolog-ical point of view. Indeed, the relative importance of thecontributions to the error budget is different for a contin-uous fountain than for a pulsed one, notably for densityrelated effects (collisional shift), which are an important issue if high stability and high accuracy are to be achievedsimultaneously.As a motivation for our work, evaluation of the secondorder Zeeman shift in atomic fountain clocks requires aprecise knowledge of the magnetic field in the free evo-lution zone. The methods developed in pulsed fountainsto map the magnetic field in the resonator are based onthrowing balls of atoms at different altitudes. This is notapplicable to our continuous fountain since the atomictrajectory is not vertical and therefore the launching ve-locity range is limited by geometrical constraints. More-over, the atomic beam longitudinal temperature is higherin our continuous fountain (75 µ K) than in pulsed foun-tains (1 µ K) and as a consequence the distribution ofapogees is wider. The effect of this large distribution oftransit time is to modify significantly the Ramsey pat-tern, reducing the number of fringes but also increasingits dependance on magnetic field inhomogeneities.The work presented in this article is devoted to devel-opping a new method to investigate the magnetic field inthe atomic resonator where the free evolution takes place.In thermal beams standards, the shape of the Ramseysignal has been used as a diagnostic tool to measure thedistribution of transit times and thus the atomic beamlongitudinal velocity distribution [6], [7], [8]. In this ar-ticle we will show that, in a continuous atomic fountain,the Fourier analysis of Zeeman sensitive Ramsey fringesallows one to measure the time-averaged magnetic fieldseen by the atoms during their free evolution. Moreover,it gives a better understanding of the shape of the Ram-sey pattern that we observe in our continuous atomicfountain. More precisely, it helps one to understand thedifference between the position of the central fringe (forwhich the microwave is in phase with the atomic dipole)and the position of the fringe which shows the highestcontrast.In section II we will give a brief description of our con-tinuous atomic fountain FOCS-2. Then we will explainthe principle of our analysis in section III and present the a r X i v : . [ phy s i c s . a t o m - ph ] A p r experimental procedure and results in section IV. Finallywe will discuss and interpret the experimental results insection V and conclude in section VI. II. CONTINUOUS ATOMIC FOUNTAINCLOCK FOCS-2
In our experiment, we use the separated oscillatoryfields method [1] to measure the transition probabilityof cesium atoms between | F = 3 , m F (cid:105) and | F = 4 , m F (cid:105) for m F = − , · · · ,
3. A scheme of the continuous atomicfountain clock FOCS-2 is shown in Fig. 1. The atomicbeam is produced with a two-dimensional magneto-optical trap [9]. The atoms are further cooled andlaunched at a speed of 4 m/s with the moving molassestechnique [10]. The longitudinal temperature at the exitof the moving molasses is 75 µ K. Before entering the mi-crowave cavity, the atomic beam is collimated by trans-verse Sisyphus cooling and the atoms are pumped into F = 3 with a state preparation scheme combining opticalpumping with laser cooling [11]. After these two steps,the transverse temperature is decreased to approximately3 µ K. During the first passage into the microwave cav-ity, we apply a π/ . π/ | F = 3 , m F (cid:105) and | F = 4 , m F (cid:105) is measuredby fluorescence detection of the atoms in F = 4.Ramsey fringes are obtained by measuring the transi-tion probability as a function of the microwave frequency,which is scanned around each of the hyperfine transitionsbetween the states | F = 3 , m F (cid:105) and | F = 4 , m F (cid:105) . A mag-netic field is used in the interogation zone to lift the de-generacy of the Zeeman sub-levels. The Ramsey fringesare formed by the superposition of individual atomic sig-nals. Because of the residual atomic beam temperaturein the longitudinal direction, every atom has a slightlydifferent trajectory with a different transit time. More-over, the magnetic field in the free evolution zone hassmall inevitable spatial variations and therefore the av-erage magnetic field seen by the atoms depends on thetrajectory. As a consequence, information on both theatomic velocity distribution and the magnetic field pro-file is contained in the experimental Ramsey fringes for m F (cid:54) = 0. Our objective is to retrieve this informationfrom a measurement of Ramsey fringes. III. PRINCIPLE
As explained in the previous section, the Ramseyfringes measured in our continuous atomic fountain areformed by the superposition of Ramsey signals comingfrom individual atoms. The contribution of an individ-ual atom to the Ramsey fringes observed on the Zeeman xzB
Microwave cavity2D (cid:45)
MOTMoving MolassesCollimationState preparation Detection
FIG. 1: Scheme of the continuous atomic fountain clockFOCS-2. An intense atomic beam of pre-cooled cesium atomsis produced in the two-dimensional magneto-optical trap (2D-MOT). The atoms are then captured by the 3D moving mo-lasses which further cools and launches the atoms at a speedof 4 m/s. Then the atomic beam is collimated with Sisyphuscooling in the transverse directions, and before entering themicrowave cavity, the atoms are pumped in | F = 3 , m = 0 (cid:105) by state preparation. Finally, after the second passage in themicrowave cavity, the transition probability is measured byfluorescence detection of the atoms in F = 4. The transittime between the two π/ T ≈ . component m F is given by: I m F ( ω rf , T ) = 12 I sin ( b τ ) [1 + cos ( ϕ m F ( ω rf , T ))] (1)where b is the Rabi angular frequency, τ is the microwaveinteraction time, ω rf is the microwave frequency, T is theeffective transit time between the first and second π/ I is a global amplitude factor, and the phase ϕ m F ( ω rf , T ) is given by : ϕ m F ( ω rf , T ) = (cid:90) T [ ω rf − ω − m F πK z B ( z ( t ))] dt (2)In this last equation, ω is the frequency of the magneticfield-insensitive clock transition, the linear Zeeman shiftsensitivity constant is K z = 7 Hz/nT, and B ( z ( t )) isthe magnetic field seen by the atom during its free evo-lution. The magnetic field in the interrogation regioncan be separated in a constant value plus residual spatialvariations[15]: B ( z ) = B + B res ( z ) (3)Therefore, by introducing the following definitions, firstlyfor the microwave detuning with respect to the atomictransition: Ω = ω rf − ω − m F πK z B (4)and secondly for the residual phase: ϕ res ( T ) = 2 πK z (cid:90) T B res ( z ( t )) dt (5)one can write: ϕ m F (Ω , T ) = Ω T − m F ϕ res ( T ) (6)The total signal is given by adding the contributions fromdifferent velocity classes: I m F (Ω) = I (cid:90) ∞ ˆ ρ ( T ) [1 + cos ( ϕ m F (Ω , T ))] dT (7)= I (cid:20) (cid:90) ∞ ˆ ρ ( T ) e iϕ m F (Ω ,T ) dT (cid:21) (8)= I I (cid:90) ∞−∞ ˆ ρ ( T ) e − im F ϕ res ( T ) e i Ω T dT (9)where ˆ ρ ( T ) = ρ ( T ) sin ( b τ ), ρ ( T ) is the transit time dis-tribution, and the last equality is valid if one extendsthe definition of both ˆ ρ ( T ) and ϕ res ( T ) to negative val-ues of the transit time as follows ˆ ρ ( − T ) = ˆ ρ ( T ) and ϕ res ( − T ) = − ϕ res ( T ). From equation (9), one can seethat the Fourier transform of I m F (Ω) is given by:FT { I m F (Ω) } = I (cid:104) δ ( T ) + ˆ ρ ( T ) e − im F ϕ res ( T ) (cid:105) (10)where δ is the Dirac delta function. In other words, theFourier transform of the Ramsey signal gives access to thedistribution of transit times and to the dephasing inducedby residual magnetic field spatial variations when m F (cid:54) =0. IV. EXPERIMENTAL RESULTS
In order to verify the Fourier relation presented inEq. (10), we measured the Ramsey fringes in our contin-uous atomic fountain for all m F values and for differentlaunching velocities of the moving molasses. Due to geo-metrical constraints (the atoms have to pass through thetwo holes of the microwave cavity) the launching veloc-ity can be changed between 3 .
74 m/s and 4 .
22 m/s. Theatomic flux decreases to zero outside of this range, andit is maximum for 4 . m F = − .
74 m/s, 3 .
80 m/s, 3 .
86 m/s,3 .
92 m/s, 3 .
98 m/s, 4 .
04 m/s, 4 . .
16 m/s,4 .
22 m/s). The first column shows the measured Ramseysignals I − (Ω) as a function of the microwave detuningΩ. The second and third columns display the moduleand the phase of the Fourier transform FT { I − (Ω) } re-spectively. According to Eq. (10): • | FT { I − (Ω) }| , displayed in the second column, isproportional to the distribution of transit time ρ ( T ). The relation between the Fourier trans-form of the Ramsey fringes pattern pattern andthe velocity distribution has already been studiedfor thermal beams [12], [13]. However because ofthe broad velocity distribution of thermal beams,the interaction time in the cavity cannot be consid-ered as constant, which makes the analysis of theRamsey fringes more complicated. On the contraryin FOCS-2, the velocity distribution is sufficientlynarrow (see section V E) such that ˆ ρ ( T ) ≈ ρ ( T ) atoptimum power, the difference being smaller than1%. • Arg [FT { I − (Ω) } ], displayed in the third column,is equal to the residual phase change ϕ res ( T ) dueto magnetic field spatial variations.In Fig. 3 we show the same series of measurements butsuperposed on the same graphs. We observe that themodules of the Fourier transforms are bell-shaped curveswhose centre of gravity shifts to the right for increasinglaunching velocities, in agreement with their interpreta-tion as distribution of transit times. On the other hand,the phases of the Fourier transforms superpose to eachother in the domains of T values where the module isdifferent from zero. It is thus possible to measure theresidual phase ϕ res ( T ) in the range of T values accessiblein the experiment i.e. between 0 .
44 s and 0 .
57 s in ourcase.To determine the residual phase, and thus the mag-netic field spatial variations, it would be useful to finda method allowing to glue together the different phasecurves shown in Fig. 3. This would be immediate witha measurement of the Ramsey fringes pattern for a verywide atomic beam velocity distribution. Thanks to thelinearity of the Fourier transform, one can simulate sucha wide velocity distribution by superposing all the Ram-sey signals corresponding to the same m F value but withdifferent launching velocities. The Fourier analysis canbe performed on these sums of experimental signals. In-deed, the sum of the different signals are shown in Fig. 4and the phases obtained from their Fourier transformsin Fig. 5. One observes in Fig. 5 that the phases mea-sured on the different Zeeman components are equal to − m F ϕ res ( T ) where ϕ res ( T ) is the phase measured for m F = −
1, as expected from Eq. (10).
V. DISCUSSION OF RESULTSA. In situ magnetometry
According to our theoretical analysis (section III), thephase of the Fourier transform obtained for m F = − ϕ res ( T ) defined by Eq. (5). As a consequence,one obtains the time average of the magnetic field along (cid:45) (cid:45)
50 0 50 100 (cid:87) (cid:144) Π (cid:64) Hz (cid:68) I (cid:45) (cid:72) (cid:87) (cid:76) T (cid:64) s (cid:68) (cid:200) F T (cid:56) I (cid:45) (cid:60) (cid:200) (cid:45)Π Π T (cid:64) s (cid:68) A r g (cid:64) F T (cid:56) I (cid:45) (cid:60) (cid:68) (cid:45) (cid:45)
50 0 50 100 (cid:87) (cid:144) Π (cid:64) Hz (cid:68) I (cid:45) (cid:72) (cid:87) (cid:76) T (cid:64) s (cid:68) (cid:200) F T (cid:56) I (cid:45) (cid:60) (cid:200) (cid:45)Π Π T (cid:64) s (cid:68) A r g (cid:64) F T (cid:56) I (cid:45) (cid:60) (cid:68) (cid:45) (cid:45)
50 0 50 100 (cid:87) (cid:144) Π (cid:64) Hz (cid:68) I (cid:45) (cid:72) (cid:87) (cid:76) T (cid:64) s (cid:68) (cid:200) F T (cid:56) I (cid:45) (cid:60) (cid:200) (cid:45)Π Π T (cid:64) s (cid:68) A r g (cid:64) F T (cid:56) I (cid:45) (cid:60) (cid:68) (cid:45) (cid:45)
50 0 50 100 (cid:87) (cid:144) Π (cid:64) Hz (cid:68) I (cid:45) (cid:72) (cid:87) (cid:76) T (cid:64) s (cid:68) (cid:200) F T (cid:56) I (cid:45) (cid:60) (cid:200) (cid:45)Π Π T (cid:64) s (cid:68) A r g (cid:64) F T (cid:56) I (cid:45) (cid:60) (cid:68) (cid:45) (cid:45)
50 0 50 100 (cid:87) (cid:144) Π (cid:64) Hz (cid:68) I (cid:45) (cid:72) (cid:87) (cid:76) T (cid:64) s (cid:68) (cid:200) F T (cid:56) I (cid:45) (cid:60) (cid:200) (cid:45)Π Π T (cid:64) s (cid:68) A r g (cid:64) F T (cid:56) I (cid:45) (cid:60) (cid:68) (cid:45) (cid:45)
50 0 50 100 (cid:87) (cid:144) Π (cid:64) Hz (cid:68) I (cid:45) (cid:72) (cid:87) (cid:76) T (cid:64) s (cid:68) (cid:200) F T (cid:56) I (cid:45) (cid:60) (cid:200) (cid:45)Π Π T (cid:64) s (cid:68) A r g (cid:64) F T (cid:56) I (cid:45) (cid:60) (cid:68) (cid:45) (cid:45)
50 0 50 100 (cid:87) (cid:144) Π (cid:64) Hz (cid:68) I (cid:45) (cid:72) (cid:87) (cid:76) T (cid:64) s (cid:68) (cid:200) F T (cid:56) I (cid:45) (cid:60) (cid:200) (cid:45)Π Π T (cid:64) s (cid:68) A r g (cid:64) F T (cid:56) I (cid:45) (cid:60) (cid:68) (cid:45) (cid:45)
50 0 50 100 (cid:87) (cid:144) Π (cid:64) Hz (cid:68) I (cid:45) (cid:72) (cid:87) (cid:76) T (cid:64) s (cid:68) (cid:200) F T (cid:56) I (cid:45) (cid:60) (cid:200) (cid:45)Π Π T (cid:64) s (cid:68) A r g (cid:64) F T (cid:56) I (cid:45) (cid:60) (cid:68) (cid:45) (cid:45)
50 0 50 100 (cid:87) (cid:144) Π (cid:64) Hz (cid:68) I (cid:45) (cid:72) (cid:87) (cid:76) T (cid:64) s (cid:68) (cid:200) F T (cid:56) I (cid:45) (cid:60) (cid:200) (cid:45)Π Π T (cid:64) s (cid:68) A r g (cid:64) F T (cid:56) I (cid:45) (cid:60) (cid:68) FIG. 2: The first column shows the Ramsey signals I − (Ω) measured on m F = − .
74 m/s,3 .
80 m/s, 3 .
86 m/s, 3 .
92 m/s, 3 .
98 m/s, 4 .
04 m/s, 4 . .
16 m/s, 4 .
22 m/s from top to bottom. The second and thirdcolumns are the module and the phase of the Fourier transform FT { I − (Ω) } . They give access, respectively, to the distributionof transit time ρ ( T ) and to the residual phase change ϕ res ( T ) due to magnetic field spatial variations. Note that the phaseis meaningful only in the region where the module is different from zero (highlighted in red). The value of B in Eq. (4) is73 . T (cid:64) s (cid:68) (cid:200) F T (cid:56) I (cid:45) (cid:60) (cid:200) (cid:45)Π Π T (cid:64) s (cid:68) A r g (cid:64) F T (cid:56) I (cid:45) (cid:60) (cid:68) FIG. 3: Fourier transforms of the Ramsey fringes measured on m F = − the atomic trajectory z ( t ) according to : B ( T ) = 1 T (cid:90) T B ( z ( t )) dt (11)= 1 T (cid:90) T ( B + B res ( z ( t ))) dt (12)= B + ϕ res ( T )2 πK z T (13)In principle, this last equation allows one to calculate B ( T ) from the Fourier transform of Ramsey fringes.However, ϕ res ( T ) is obtained by calculating the phaseof the Fourier transform which inevitably results in a 2 π ambiguity. As a consequence, the average magnetic fieldmay take the following values: B ( T ) = B + 12 πK z T ( ϕ res ( T ) + n π ) (14)where n is any integer number. The resulting graphs areshown in Fig. 6, the solid line corresponding to n = 0.This ambiguity in the determination of B ( T ) deservesa few comments. Firstly, by extending the domain oftransit time values, it would be possible to distinguish thecorrect curve ( n = 0) for B ( T ) from the others ( n (cid:54) = 0).Indeed, from Eq. (14) it is clear that only the n = 0 curvedoes not diverge when T = 0. In pulsed fountains, this (cid:45) (cid:45) (cid:45)
50 0 50 100 150 Ω rf (cid:144) Π (cid:64) Hz (cid:68) m F (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) m F (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) m F (cid:61) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) m F (cid:61) (cid:45) FIG. 4: Each of these curves, corresponding to m F =0 , − , − , −
3, has been obtained by summing the Ramseyfringes patterns measured for different launching velocities(3 .
74 m/s, 3 .
80 m/s, 3 .
86 m/s, 3 .
92 m/s, 3 .
98 m/s, 4 .
04 m/s,4 . .
16 m/s, 4 .
22 m/s). ambiguity is solved by throwing the atoms at different al-titudes from T = 0 to its nominal value. However, in ourcontinuous fountain the transit time values are limitedby geometrical constraints. Secondly, this 2 π ambiguityin the phase results in a 0 . B ( T ), whichcorresponds to a 2 Hz indetermination on the microwavefrequency. This is the distance between two consecutiveRamsey fringes, therefore, determining the correct curve( n = 0) is a problem equivalent to finding the centralfringe in the Ramsey pattern (see section V C). In theevaluation process of the continuous fountain FOCS-2,we used a complementary method (time resolved Zee-man spectroscopy) to measure the magnetic field spatialprofile and thus lift the above mentioned ambiguity [14]. (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) T (cid:64) s (cid:68) A r g (cid:64) F T (cid:56) I m F (cid:60) (cid:68)(cid:64) r a d (cid:68) m F (cid:61) m F (cid:61) (cid:45) m F (cid:61) (cid:45) m F (cid:61) (cid:45) FIG. 5: Phases of the Fourier transforms measured for m F =0 , − , − , − T . Each transittime corresponds to a different height of the apogee above themicrowave cavity. T (cid:64) s (cid:68) B (cid:72) T (cid:76) (cid:64) n T (cid:68) FIG. 6: Time average of the magnetic field, calculated fromthe phase of the Fourier transform of the Ramsey fringes mea-sured on m F = −
1. Due to the 2 π ambiguity of the phase,this function is not uniquely determined. The different curvesrepresent possible realizations, they differ by n/ ( K z T ) where n is an integer. The solid line corresponds to n = 0. This technique consists in applying short pulses of an os-cillating magnetic field in order to induce Zeeman transi-tions (∆ m F = ±
1) at different positions along the atomictrajectory. It allowed us to identify the n = 0 curve inFig. 6. However, its spatial resolution is limited, espe-cially at the apogee where the spread of atomic beamis maximum, and therefore the Fourier analysis methodpresented in this article provides precious informationsabout the magnetic field in the region of the apogee.Finally, in the graph of Fig. 6 we observe that B ( T )shows a local minimum T a and a local maximum T b .When the distribution of transit times is large enough tocover both extrema, the superposition of individual Ram-sey signals will be constructive when T ≈ T a and T ≈ T b .These two contributions give rise to Ramsey fringes withslightly different periods 1 / (2 T a ) and 1 / (2 T b ), which ex-plains the appearance of beat-like Ramsey patterns in Fig. 4. However, the complete reconstruction of Ram-sey fringes is more complex than that, mainly due tothe effect of the magnetic field which produces a T de-pendent Zeeman shift, and will be discussed in detail insections V B and V D. B. Reconstruction of Ramsey fringes
With the knowledge of the transit time distributions ρ ( T ) and of the measured average magnetic field B ( T ),it is possible to recalculate all the Ramsey fringes bysumming the individual Ramsey signals according to: I m F ( ω rf ) = 12 I (cid:90) ∞ ρ ( T ) [1 + cos ( ϕ m F ( ω rf , T ))] dT (15)with: ϕ m F ( ω rf , T ) = ( ω rf − ω ) T − m F πK z B ( T ) T (16)The results are presented in Fig. 7 for every Zeeman com-ponents and launching velocities used in the experiment.We should emphasize that all the Ramsey fringes shownin Fig. 7 are reconstructed using the same function B ( T )for the time average magnetic field in Eq. (16). Con-sidering the fact the frequency sampling interval of themeasured Ramsey fringes is 0 . C. Position of the central fringe
For a given transit time T , one can define the positionof the central fringe as: ω c = ω + m F πK z B ( T ) (17)In other words, it is the microwave frequency for whichthere is no dephasing between the microwave and theatomic dipole during the free evolution time T . Sincethe transit time is not unique, the same is true for theposition of the central fringe. The distribution of cen-tral fringe positions is given by ρ ( ω c ) = ρ ( T ) dT /dω c .In the situation of FOCS-2, the function B ( T ) changessmoothly on the extent of the transit time distribution.Therefore, one can estimate the parameters of the dis-tribution of the central fringes as follows. The averageposition is given by: (cid:104) ω c (cid:105) = ω + m F πK z (cid:104) B ( T ) (cid:105) (18) ≈ ω + m F πK z B ( (cid:104) T (cid:105) ) (19)and the standard deviation by: σ ( ω c ) ≈ m F πK z B (cid:48) ( (cid:104) T (cid:105) ) σ ( T ) (20)where (cid:104) T (cid:105) and σ ( T ) are the average and standard devia-tion of the transit time distribution ρ ( T ). Here we should m F (cid:61) m F (cid:61) (cid:45) m F (cid:61) (cid:45) m F (cid:61) (cid:45) T (cid:61) . s (cid:45) (cid:45)
50 0 50 100 Ω rf (cid:144) Π (cid:64) Hz (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) T (cid:61) . s (cid:45) (cid:45)
50 0 50 100 Ω rf (cid:144) Π (cid:64) Hz (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) T (cid:61) . s (cid:45) (cid:45)
50 0 50 100 Ω rf (cid:144) Π (cid:64) Hz (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) T (cid:61) . s (cid:45) (cid:45)
50 0 50 100 Ω rf (cid:144) Π (cid:64) Hz (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) T (cid:61) . s (cid:45) (cid:45)
50 0 50 100 Ω rf (cid:144) Π (cid:64) Hz (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) T (cid:61) . s (cid:45) (cid:45)
50 0 50 100 Ω rf (cid:144) Π (cid:64) Hz (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) T (cid:61) . s (cid:45) (cid:45)
50 0 50 100 Ω rf (cid:144) Π (cid:64) Hz (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) T (cid:61) . s (cid:45) (cid:45)
50 0 50 100 Ω rf (cid:144) Π (cid:64) Hz (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) T (cid:61) . s (cid:45) (cid:45)
50 0 50 100 Ω rf (cid:144) Π (cid:64) Hz (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) Ω rf (cid:144) Π (cid:64) Hz (cid:68) FIG. 7: Comparison of the measured (upper blue curve) and calculated (lower red curve) Ramsey fringes. The four columnscorrespond to m F = 0 , − , − , − .
74 m/s, 3 .
80 m/s,3 .
86 m/s, 3 .
92 m/s, 3 .
98 m/s, 4 .
04 m/s, 4 . .
16 m/s, 4 .
22 m/s from top to bottom) and thus to different averagetransit times (indicated on the left side). The calculated fringes are obtained by summing the individual signals as explainedin section V B. Both the measured and calculated fringes are shown with the Rabi pedestal subtracted. We should emphasizethat all the Ramsey fringes are reconstructed using the same time average magnetic field B ( T ). See sections V B and V D fordetails. make an important remark: the ambiguity of B ( T ) dis-cussed in section V A results in an ambiguity of the po-sition of the central fringe. Indeed, by inserting Eq. (14)into Eq. (19) one obtains: (cid:104) ω c (cid:105) = ω + m F πK z B + m F ϕ res ( (cid:104) T (cid:105) ) (cid:104) T (cid:105) + n πm F (cid:104) T (cid:105) (21)where n is any integer number. However, let’s note thatthis ambiguity does not affect the shape and position ofthe reconstructed Ramsey fringes. This will be discussedin more detail in the next section. D. Position of the observed Ramsey pattern
Ramsey fringes appear when the individual Ramseysignals interfere constructively. Observing Eq. (8) it isclear that constructive interference can appear only ifthe phase ϕ m F (Ω , T ) exhibits small variations when ˆ ρ ( T )is maximum. Therefore, the position ω ∗ rf of the overallfringe pattern on the frequency axis is given by imposingthe condition ∂ϕ m F /∂T = 0 with ϕ m F given in Eq. (16): ∂ϕ m F ∂T = ∂∂T (cid:2) ( ω ∗ rf − ω ) T − m F πK z B ( T ) T (cid:3) = 0 (22)In order to evaluate this condition, we suppose that vari-ations of B ( T ) are small on the extent of the transit timedistribution ρ ( T ). This is only partially fulfilled in ourexperiment for a given launching velocity, but it is in-structive since it helps in understanding the role of B ( T )in the formation of the fringe pattern. With this assump-tion, Equ. (22) becomes: ω ∗ rf ≈ ω + m F πK z B ( (cid:104) T (cid:105) ) + m F πK z B (cid:48) ( (cid:104) T (cid:105) ) (cid:104) T (cid:105) (23)The first term is the position of the unperturbed atomictransition, then comes the linear Zeeman shift, and thethird shift is induced by a first order variation of B ( T ).This expression deserves a few comments. Firstly, theposition of the Ramsey pattern given in Eq. (23) differsfrom the position of the central fringe given in Eq. (19)and the difference is given by the last term proportionalto B (cid:48) ( (cid:104) T (cid:105) ). Secondly, the linear Zeeman shift, calculatedfrom the measurement of B ( T ) shown in Fig. 6, is muchtoo small to explain the shift in position of the Ram-sey patterns observed in Fig. 7. On the other hand, wecalculated the fringe pattern positions ω ∗ rf according toEq. (23) and reported them as vertical arrows in eachmeasurement of Fig. 7, and we observe that the agree-ment with the experimental fringes is good. Finally, weshould note that adding 1 / ( K z T ) to B ( T ) does not shiftthe Ramsey pattern since the second and third terms ofEq. (23) cancel each other. This explains why the posi-tion of the Ramsey fringe pattern is not affected by theambiguity of B ( T ) shown in Eq. (14). (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:64) m (cid:144) s (cid:68) a v e r a g e v e l o c it y (cid:64) m (cid:144) s (cid:68) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:64) m (cid:144) s (cid:68) l ong it ud i n a lt e m p e r a t u r e (cid:64) Μ K (cid:68) FIG. 8: Average velocity (upper graph) and longitudinal tem-perature (lower graph) of the atoms contributing to the Ram-sey signal, at the exit of the moving molasses, as a functionof the launching velocity. For comparison, the dashed line(upper graph) and the gray band (lower graph) show the val-ues measured in previous experiments using the time-of-flighttechnique [10]. See section V E for details.
E. In situ velocimetry
From the measured distributions of transit times (seeFig. 3) we calculated the distributions of velocities atthe altitude of the microwave cavity and at the exit ofthe moving molasses which is situated 48 . . µ K using the TOF technique. As shown inFig. 8, the temperature values obtained with the Ram-sey analysis are indeed compatible with those obtainedby TOF for velocities close to 4 . VI. CONCLUSION
We applied Fourier analysis to the Ramsey fringes ob-served in a continuous atomic fountain clock. By an-alyzing the Ramsey patterns for every Zeeman compo-nent and for different transit times, we have shown thatthe phase of the Fourier transform is directly linked tothe time-averaged magnetic field B ( T ) seen by the atomsduring their free evolution of duration T . This allowedus to measure B ( T ) over the region of the apogee, withan ambiguity of n/ ( K z T ) resulting from the 2 π inde-termination of the phase. We discussed the role of thisambiguity and showed that it has no influence on theshape and position of the Ramsey pattern. Moreover,this analysis allowed us to establish an expression for thefrequency shift of the overall Ramsey pattern induced by spatial variations of the magnetic field. We showed thatthe position of the Ramsey pattern differs from the posi-tion of the so called central fringe by a term proportionalto T B (cid:48) ( T ). In our atomic fountain, the variation of thisterm induced by a change of transit time T is more im-portant than the corresponding variation of the linearZeeman shift. Finally, we also showed that the mod-ule of the Fourier transform can be interpreted as thedistribution of transit times. We used this informationto obtain the atomic beam average velocity and longi-tudinal temperature. The results are in good agreementwith previous measurements made with the time-of-flighttechnique. The method developed in this article, con-cerning the measurement of the time averaged magneticfield, will be used for the evaluation of the second orderZeeman shift in our continuous atomic fountain FOCS-2.Applicability to atom interferometers is also forseen. Acknowledgments
This work was supported by the Swiss NationalScience Foundation (grant 200020-121987/1 and Eu-roquasar project 120418), the Swiss Federal Office ofMetrology (METAS), and the University of Neuchˆatel. [1] N. F. Ramsey, Phys. Rev. , 695 (1950).[2] A. Clairon, C. Salomon, S. Guellati, and W. D. Phillips,Europhysics Letters , 165 (1991).[3] R. Wynands and S. Weyers, Metrologia , 64 (2005).[4] A. Joyet, G. Mileti, G. Dudle, and P. Thomann, IEEETransactions on Instrumentation and Measurement ,150 (2001).[5] J. Gu´ena, G. Dudle, and P. Thomann, Eur. Phys. J.Appl. Phys. , 183 (2007).[6] S. Jarvis, Metrologia , 87 (1974).[7] J. Boulanger, Metrologia , 37 (1986).[8] A. Makdissi and E. de Clercq, IEEE Transactions on Ul-trasonics, Ferroelectrics and Frequency Control , 637(1997).[9] N. Castagna, J. Gu´ena, M. D. Plimmer, and P. Thomann,Eur. Phys. J. Appl. Phys. , 21 (2006).[10] P. Berthoud, E. Fretel, and P. Thomann, Phys. Rev. A , R4241 (1999). [11] G. Di Domenico, L. Devenoges, C. Dumas, andP. Thomann, Phys. Rev. A , 053417 (2010).[12] J. Shirley, Proceedings of the 43rd Annual Symposiumon Frequency Control ?? , 162 (1989).[13] J. Shirley, IEEE Transactions on Instrumentation andMeasurement , 117 (1997).[14] G. Di Domenico, L. Devenoges, A. Stefanov, A. Joyet,and P. Thomann, Tech. Rep., LTF, Universit´e deNeuchˆatel, FOCS-2 evaluation report 2 (2011).[15] In principle, the choice of the constant part B is ar-bitrary and does not influence the present analysis. Inpractice, we will choose B such as to minimize phasevariations of the Fourier transform of Ramsey fringes (seesection IV) which is equivalent to choose B = B ( T ) = T (cid:82) T B ( z ( t )) dt where T is the average transit time and z ( tt