Experimental study of the role of trap symmetry in an atom-chip interferometer above the Bose-Einstein condensation threshold
Matthieu Dupont-Nivet, Romain Demur, Christoph I. Westbrook, Sylvain Schwartz
EExperimental study of the role of trap symmetry inan atom-chip interferometer above theBose-Einstein condensation threshold
M. Dupont-Nivet ‡ , R. Demur , C. I. Westbrook and S.Schwartz Thales Research and Technology France, 1 av. Augustin Fresnel, 91767 Palaiseau,France Laboratoire Charles Fabry de l’Institut d’Optique, 2 av. Augustin Fresnel, 91127Palaiseau, France Laboratoire Kastler Brossel de l’Ecole Normale Sup´erieure, 24 rue Lhomond, 75231Paris Cedex 05, France
Abstract.
We report the experimental study of an atom-chip interferometer usingultracold rubidium 87 atoms above the Bose-Einstein condensation threshold. Theobserved dependence of the contrast decay time with temperature and with the degreeof symmetry of the traps during the interferometer sequence is in good agreement withtheoretical predictions published in [Dupont-Nivet et al. , NJP 18, 113012 (2016)].These results pave the way for precision measurements with trapped thermal atoms.
Keywords : Atomic interferometry, Ultra-cold thermal atoms, Contrast decay, Ramseyinterferometer ‡ Corresponding author: [email protected] a r X i v : . [ phy s i c s . a t o m - ph ] J a n xperimental study of the role of trap symmetry...
1. Introduction
Atom interferometers [1, 2] have demonstrated excellent performance in measuringgravity [3, 4, 5, 6], gravity gradients [7, 8, 9] and rotations [10, 11, 12], using atomsin ballistic flight. In spite of being less well developped, trapped atom interferometers,for example using atom chips, [13, 14], would render the interrogation time independentof the atom’s flight, permitting miniaturization and possibly longer measurement times.To date, atom-chip-based interferometers have been successfully demonstrated usingBose-Einstein condensates [15, 16], but are subject to dephasing mechanisms resultingfrom atom-atom interactions [17, 18, 19, 20]. Recently, we proposed a trapped atominterferometer using an ensemble of cold atoms above the Bose-Einstein condensationthreshold (referred to as thermal atoms in the following) [21, 22]. This proposal isreminiscent of optical white light interferometry because of the necessity of keepingthe path length difference between the two arms of the interferometer smaller thanthe coherence length. Similarly, the contrast decay of an atom chip interferometer usingthermal atoms is related to the degree of asymmetry between the two arms [22]. Relatedcontrast decay effects have been described in optical traps, for example in references[23, 24], where the asymmetry results from state-dependent light shifts, and reference[25], where the asymmetry is induced by spatial separation along the axis of a Gaussianbeam.In our experiment, the two arms of the interferometer correspond to two differentinternal states trapped in magnetic potentials, and we are able to control the effectof asymmetry without spatially separating the paths. The asymmetry can be tunedby adjusting the bias field, which results in slightly different magnetic momentsas described by the Breit-Rabi formula. Using this technique in combination withevaporative cooling, we are able to measure the contrast decay time in an internal stateinterferometer (a Ramsey interferometer [26]) for different values of the temperatureand asymmetry, and compare our results with the theoretical predictions from reference[22]. The focus of this paper is inhomogeneous dephasing, as manifested by the contrastdecay of Ramey fringes [24]. On the other hand, homogenous dephasing, caused forexample by fluctuating magnetic fields, and probed by spin echo measurements [24, 27],is beyond the scope of this paper. We first give a brief review of the theoreticalpredictions in section 2. Then, we describe in section 3 our experimental protocoland results. We finally compare the results to a simple model in section 4. We alsodiscuss the identical spin rotation effect (ISRE) that was previously observed in similarexperiments [28, 29, 30, 31, 32].
2. Theoretical model
In this section, we briefly recall the simple model of reference [22] describing theinfluence of the asymmetry on the contrast decay time. We consider a Ramsey xperimental study of the role of trap symmetry... Rb ground state manifold, namely | F = 1 , m F = − (cid:105) ≡ | a (cid:105) and | F = 2 , m F = 1 (cid:105) ≡ | b (cid:105) , coupled by a two photon transition.During the whole interferometer sequence both states are maintained trapped, but notnecessarily in identical potentials [16, 21], leading to inhomogeneous dephasing. Wesuppose that the traps are harmonic but with slightly different frequencies along oneof the trapping axes, namely ω a = ω for state | a (cid:105) and ω b = ω + δω for state | b (cid:105) with | δω | (cid:28) ω . We also assume that the gas is at sufficiently high temperature T to beaccurately described by a Boltzmann distribution. The relative asymmetry | δω | /ω thenimplies an upper bound on the contrast decay time t c given (up to a numerical factoron the order of unity) by [22]: t c (cid:39) ω | δω | (cid:126) kT . (1)Two differences between the experiment considered here and the model of reference [22]should be pointed out. First, in the model of reference [22], the relative asymmetrywas assumed to grow linearly from zero to some finite value | δω | /ω where it was heldfor some interrogation time, before being ramped back to zero. It was furthermoreassumed that the ramp was slow enough that the initial population of the eigenstateswas conserved throughout the sequence. By contrast, the splitting and recombinationare very fast in our experiment (on the order of the π/ (cid:126) ω/kT (cid:38) − in our experiment). Second, the model ofreference [22] is one-dimensional, while in the experiment described here the asymmetryoccurs along all three trapping axes (with identical relative asymmetry, as will be seenbelow). Again, we expect this not to change the results of [22] up to a numerical factoron the order of unity.
3. Experiment
The two interferometer states | a (cid:105) ≡ | F = 1 , m F = − (cid:105) and | b (cid:105) ≡ | F = 2 , m F = 1 (cid:105) aretrapped by the same DC magnetic field. Because of the coupling between the nuclearangular momentum and the magnetic field, the magnetic moments (defined as the partialderivative of the energy with respect to the magnetic field) of the two states are slightlydifferent, as described by the Breit-Rabi formula [33]. As pointed out in reference [34],there is a “magic” magnetic field B m for which the effective magnetic moments of thetwo states | a (cid:105) and | b (cid:105) are identical. By changing the value of the field at the trapminimum around this value, we can go from a situation where the two traps are almostperfectly symmetric to a situation where they have significantly different frequencies.Let us consider a static magnetic trapping field of the form B ( x ) = B +( ∂ B/∂x ) x where ∂ B/∂x > x is a linear coordinate along one trapping axis. xperimental study of the role of trap symmetry... − − − − − Figure 1. (Color online). Relative asymmetry along one axis as a function ofthe magnetic field at the trap minimum. The blue circles are the simulated valuescomputed with our current distribution, and the orange solid line is the model ofequation (5).
One can show [35], expanding the Breit-Rabi formula [33] up to the second order in themagnetic field, that the resulting trap frequencies are: ω a (cid:39) (cid:114) µ B g J m ∂ B∂x (cid:115) g I g J (cid:18) − B B m (cid:19) (2)for the state | a (cid:105) , and: ω b (cid:39) (cid:114) µ B g J m ∂ B∂x (cid:115) g I g J (cid:18) − B B m (cid:19) (3)for the state | b (cid:105) , where the magic magnetic field reads [34]: B m (cid:39) − g I E hfs µ B ( g J − g I ) (cid:39) .
23 G . (4)In the above formulas, m is the atomic mass, µ B is the Bohr magneton, g J (cid:39) . g I (cid:39) − . · − are the electron and nuclear spin g factors [33] respectively, and E hfs is the energy splitting between the two hyperfine ground states of Rb. The twodifferent curvatures for the two traps result in a non-zero value for the asymmetry alongeach trapping axis: δωω = 2 | ω b − ω a || ω b + ω a | (cid:39) − g I / (2 g J ) (cid:12)(cid:12)(cid:12)(cid:12) g I g J (cid:18) − B B m (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . (5)As expected, the asymmetry depends on the magnetic field B at the trap minimumand vanishes when the latter is equal to the magic field ( B = B m ). The model canbe easily generalized to 3 dimensions: the relative asymmetry is the same for all threetrapping axes as can be seen in equation (5).On the chip, a dimple trap is created by two crossing wires and an external biasfield. We change the trap field minimum by changing the magnitude and direction ofthis bias field. In the whole span of B in figure 1, the trap position changes by less xperimental study of the role of trap symmetry... B Simulated asymmetry δωω (cid:12)(cid:12) s Inferred asymmetry δωω (cid:12)(cid:12) i . ± ,
012 G (4 . ± . · − (3 . ± . · − . ± ,
006 G (2 . ± . · − (9 . ± . · − . ± ,
009 G (2 . ± . · − (3 . ± . · − Table 1.
Trap asymmetry for different magnetic field minima B . We show thecalculated values and those inferred from the contrast decay time curves of figure 3. than 1 µ m and the trap frequencies, which are ∼
85 Hz and ∼
148 Hz in the horizontalplane and ∼
161 Hz in the vertical plane, by less than 5 Hz (numbers are given for state | b (cid:105) ). A simulation of our magnetic field geometry (performed using the known wiregeometry on our atom chip) allows us to compute the potentials for the two states | a (cid:105) and | b (cid:105) and thus the asymmetry. In figure 1, the asymmetry is shown as a function ofthe value of the magnetic field at the trap minimum. We use the experimental setup described in references [35, 36] to trap a cloud of a fewtens of thousands of Rb atoms in state | F = 2 , m F = 2 (cid:105) on an atom chip, using forcedradio-frequency evaporation to control the final temperature, which is typically chosenbetween 150 nK and 800 nK (with the Bose-Einstein condensation threshold around110 nK temperature is given for state | b (cid:105) ). After evaporative cooling in state | , (cid:105) , thecloud is transferred to state | b (cid:105) by microwave-stimulated Raman adiabatic passage asdescribed in reference [36]. We then perform Ramsey spectroscopy between | a (cid:105) and | b (cid:105) for different values of the temperature and the asymmetry.As described in the previous section, the bias field of the magnetic trap is tuned tothree different values to investigate different values of the asymmetry. For these threebiais fields, the magnetic field at the trap minimum is measured (see table 1) and used tocalculate the asymmetry using the simulation shown in figure 1 (values are given in thecolumn labeled “simulated asymmetry” δωω (cid:12)(cid:12) s in table 1). For each of these three valuesof the asymmetry, the Ramsey signal is recorded for different values of the temperature.The local oscillator used for the excitation has a short term stability below 5 · − atone second and long term drifts are corrected using a GPS clock signal. Examples ofmeasured signals are shown in figure 2 for two different values of the temperature. Thetwo outputs of the interferometer are used to normalize the atom number in state | b (cid:105) bythe total atom number [37] (although only one is shown in figure 2). The interrogationtime is limited to 600 ms so as not to overheat the atom chip. The envelopes of thefringes are extracted using a Hilbert transform [38] and then adjusted by a function ofthe form exp ( − t/t c ) to infer the value of t c in equation (1) assuming asymmetry is themain source of contrast decay. The contrast decay time is plotted for the three valuesof the asymmetry as a function of the temperature in figure 3. Each of these curves is xperimental study of the role of trap symmetry... d ec r ea s e decrease Figure 2. (Color online). Ramsey fringes as a function of the interrogation timeshowing the behaviour of the contrast as a function of the cloud temperature and theasymmetry. Note that the horizontal scale (interrogation time t R ) is different in eachplot. (a), (c) and (e) correspond to an atom cloud at a temperature of ∼
500 nK, while(b), (d) and (f) correspond to ∼
150 nK. For (a) and (b), B = 2 .
456 G, correspondingto δωω (cid:12)(cid:12) s = 4 . · − . For (c) and (d), B = 2 .
859 G, corresponding to δωω (cid:12)(cid:12) s = 2 . · − .For (e) and (f), B = 3 .
264 G, corresponding to δωω (cid:12)(cid:12) s = 2 . · − . The open bluecircles stand for experimental data, the solid light blue lines are the fit of the Ramseyfringes and the solid red lines are the envelope of the Ramsey fringes. The insets showthat for fields far from the magic field, the phase coherence is lost more rapidly thanthe contrast (see text). xperimental study of the role of trap symmetry...
200 400 600 800 10000100200300 200 400 600 800 100002004006008001000 200 400 600 800 10000100020003000
Figure 3. (Color online). Contrast decay time as a function of the atomic cloudtemperature for three different values of the (inferred) asymmetry. The vertical scale(contrast decay time) is different in each plot. (a) B = 2 .
456 G, correspondingto δωω (cid:12)(cid:12) s = 4 . · − , (b) B = 2 .
859 G, corresponding to δωω (cid:12)(cid:12) s = 2 . · − and(c) B = 3 .
264 G, corresponding to δωω (cid:12)(cid:12) s = 2 . · − . The crosses stand for theexperimental data with error bars, the solid blue lines are the fit of the data (inferredvalues and errors are given in table 1, column “inferred asymmetry”). The dashedblack lines are the model with the simulated values of the asymmetry. adjusted by a function of the form of equation (1) with δω/ω the fitting parameter. Theresulting fitted values are given in table 1 in the column “inferred asymmetry” δωω (cid:12)(cid:12) i .
4. Discussion
Our data is in reasonably good agreement with the model of equation (1), as can beseen in table 1. The discrepancy can be possibly explained by the missing numericalfactor in equation (1). On the other hand, closer inspection of figure 3 indicates thatthe data points seem to fall systematically above the fit for low temperature and belowit for high temperature. This effect may point to trap anharmonicities which have beenneglected up to this point. We can formulate a simple model of trap anharmonicity byconsidering the Hamiltonian: (cid:98) H i = (cid:98) p m + mω i (cid:98) x + 2 √ σ i (cid:126) ω i (cid:18) (cid:126) mω i (cid:19) / (cid:98) x , (6)where i labels the internal atomic state | i (cid:105) , σ i (cid:28) (cid:98) p (respectively (cid:98) x ) is the momentum (respectivelyposition) operator. The contrast decay time for such a potential is calculated in [22],leading to: 1 t c ∼ Dσ ) (cid:18) kT (cid:126) ω (cid:19) − Dσδω (cid:18) kT (cid:126) ω (cid:19) + 3( δω ) + 2( Dσ ) (cid:18) kT (cid:126) ω (cid:19) , (7)where Dσ = ωδσ + σ δω with σ = ( σ b + σ a ) / δσ = σ b − σ a , ω and δω aredefined in section 2. Perfectly harmonic potentials correspond to Dσ = 0 and werecover equation (1). A non vanishing Dσ will tend to accelerate the contrast decay athigh temperature. It was not possible to adjust the data to the above model because ofan insufficient signal to noise ratio. However we can estimate an order of magnitude for xperimental study of the role of trap symmetry... Figure 4. (Color online). The three relevant dimensionless parameters to assess theexistence of ISRE. (a) ∆ /ω ex , (b) 2 π Γ col /ω ex and (c) 2 π Γ col /ω min as a function ofthe mean atomic density n in unit of 10 cm − . The three sets of crosses stand forour experimental data, while the open circles stand for the data of reference [31] whereISRE was observed. the maximum value of the σ terms. For a temperature of 700 nK, kT / ( (cid:126) ω ) ∼
120 (for ω we took the geometric mean of the trapping frequencies). Keeping only the first term,we find Dσ/ω = δσ + σ ( δω/ω ) (cid:46) ( (cid:126) ω ) / ( √ ωt c ( kT ) ). In the most symmetric case(figure 3.c) we observe a contrast decay time around 500 ms for temperature around700 nK, thus Dσ/ω (cid:46) · − . This gives upper limits for the anharmonicity of thepotentials: σ (cid:46) − , and for the difference between the cubic terms of the potentials: δσ (cid:46) · − .It can be noticed from figure 2 that for some values of the asymmetry the phasecoherence is lost even though the contrast remains appreciable (see for example subfigureb). We attribute this to magnetic field fluctuations (estimated to δB (cid:39) ∼
60 s) and contrast revivals. One might wonder whythis effect is not visible in our experiment. The ISRE regime is characterized by threetime scales [31]: i) the inhomogeneity ∆ = | kT δω/ (cid:126) ω − γn/ | with n the mean atomicdensity and γ = 4 π (cid:126) ( a bb − a aa ) /m where a ij is the scattering length between states | i (cid:105) and | j (cid:105) , ii) the elastic collision rate Γ col = (32 / a ab n ( πkT /m ) / and iii) the exchangeenergy ω ex = 4 π (cid:126) | a ab | n/m . For ISRE to occur, the following three conditions must be xperimental study of the role of trap symmetry... /ω ex (cid:28)
1, ii) 2 π Γ col /ω ex (cid:28) π Γ col /ω min (cid:28) ω min is the smallest trapping frequency. To get a more quantitative criterion, we compareour experimental values of these parameters, for various densities, to the numbers fromreference [31] where ISRE was observed (figure 4). As can be seen in this figure, thereare some situations (for example B = 3 .
264 G and n > · cm ) where all threerelevant quantities are smaller in our case than in reference [31]. Yet, we do not seeISRE in these cases.In an attempt to reconcile our observations with those of [31], we developed amodel whose details will be published elsewhere [40]. This model indicates that thetrap anisotropy could significantly enhance ISRE. The identical spin rotation effectarises from forward collisions between cold atoms which tend to be situated at thebottom of the trap and hot atoms which tend to sample magnetic fields far from theminimum. In the case of a cigar shaped trap, a hot atom oscillating in the trap comesclose to the minimum of the trap at each oscillation enhancing forward collisions withcold atoms. In the case of an isotropic trap, a hot atom can oscillate without comingclose to the trap minimum thus reducing the number of forward collisions with coldatoms and also the identical spin rotation effect. Our trapping potential is closer to anisotropic trap (frequencies 85 Hz, 148 Hz, 161 Hz) than that in reference [31] (32 Hz,97 Hz, 121 Hz), which might explain the discrepancy.
5. Conclusion
We have shown that the simple model derived in reference [22] describes reasonablywell the contrast decay in our experiment, confirming the important role of symmetryin atom interferometry with thermal atoms. The next step is to introduce a spatialseparation between the two internal states, for example by using near-field microwavegradients [16, 21, 22]. In this context, equation (1) could serve as a benchmark to assessthe required degree of symmetry in the design of future atom-chip interferometers.
Acknowledgments
This work has been carried out within the OnACIS project ANR-13-ASTR-0031 fundedby the French National Research Agency (ANR) in the frame of its 2013 Astrid program.S.S. acknowledges funding from the European Union under the Marie Sklodowska CurieIndividual Fellowship Programme H2020-MSCA-IF-2014 (project number 658253).
References [1] M. Kasevich and S. Chu. Atomic interferometry using stimulated Raman transitions.
Phys. Rev.Lett. , 67:181–184, Jul 1991.[2] A. Cronin, J. Schmiedmayer, and D. Pritchard. Optics and interferometry with atoms andmolecules.
Rev. Mod. Phys. , 81:1051–1129, Jul 2009. xperimental study of the role of trap symmetry... [3] A. Peters, K. Y. Chung, and S. Chu. High-precision gravity measurements using atominterferometry. Metrologia , 38(1):25, 2001.[4] Z.-K. Hu, B.-L. Sun, X.-C. Duan, M.-K. Zhou, L.-L. Chen, S. Zhan, Q.-Z. Zhang, and J. Luo.Demonstration of an ultrahigh-sensitivity atom-interferometry absolute gravimeter.
Phys. Rev.A , 88(4):043610, 2013.[5] P. Gillot, O. Francis, A. Landragin, F. P. Dos Santos, and S. Merlet. Stability comparison of twoabsolute gravimeters: optical versus atomic interferometers.
Metrologia , 51(5):L15, 2014.[6] S. Abend, M. Gebbe, M. Gersemann, H. Ahlers, H. M¨untinga, E. Giese, N. Gaaloul, C. Schubert,C. L¨ammerzahl, W. Ertmer, W. P. Schleich, and E. M. Rasel. Atom-chip fountain gravimeter.
Phys. Rev. Lett. , 117(20):203003, 2016.[7] J. McGuirk, G. Foster, J. Fixler, M. Snadden, and M. Kasevich. Sensitive absolute-gravitygradiometry using atom interferometry.
Phys. Rev. A , 65:033608, Feb 2002.[8] N. Yu, J. M. Kohel, J. R. Kellogg, and L. Maleki. Development of an atom-interferometer gravitygradiometer for gravity measurement from space.
Appl. Phys. B , 84(4):647–652, 2006.[9] G. Rosi, F. Sorrentino, L. Cacciapuoti, M. Prevedelli, and G. M. Tino. Precision measurement ofthe newtonian gravitational constant using cold atoms.
Nature , 510(7506):518–521, 2014.[10] T. Gustavson, A. Landragin, and M. Kasevich. Rotation sensing with a dual atom-interferometersagnac gyroscope.
Classical Quant. Grav. , 17(12):2385, 2000.[11] D. S. Durfee, Y. K. Shaham, and M. A. Kasevich. Long-term stability of an area-reversibleatom-interferometer sagnac gyroscope.
Phys. Rev. Lett. , 97:240801, Dec 2006.[12] I. Dutta, D. Savoie, B. Fang, B. Venon, C. L. Garrido Alzar, R. Geiger, and A. Landragin.Continuous cold-atom inertial sensor with 1 nrad/sec rotation stability.
Phys. Rev. Lett. ,116(18):183003, 2016.[13] J. Fort´agh, H. Ott, S. Kraft, A. G¨unther, and C. Zimmermann. Surface effects in magneticmicrotraps.
Phys. Rev. A , 66:041604, Oct 2002.[14] J. Reichel and V. Vuletic.
Atom Chips . John Wiley & Sons, 2010.[15] T. Schumm, S. Hofferberth, L. M. Andersson, S. Wildermuth, S. Groth, I. Bar-Joseph,J. Schmiedmayer, and P. Kruger. Matter-wave interferometry in a double well on an atomchip.
Nat. Phys. , 1:57–62, 2005.[16] P. B¨ohi, M. Riedel, J. Hoffrogge, J. Reichel, T. Hansch, and P. Treutlein. Coherent manipulationof bose-einstein condensates with state-dependent microwave potentials on an atom chip.
Nat.Phys. , 5:592–597, 2009.[17] J. Javanainen and M. Wilkens. Phase and phase diffusion of a split bose-einstein condensate.
Phys. Rev. Lett. , 78:4675–4678, Jun 1997.[18] G.-B. Jo, Y. Shin, S. Will, T. A. Pasquini, M. Saba, W. Ketterle, D. E. Pritchard, M. Vengalattore,and M. Prentiss. Long phase coherence time and number squeezing of two bose-einsteincondensates on an atom chip.
Phys. Rev. Lett. , 98:030407, Jan 2007.[19] J. Grond, U. Hohenester, I. Mazets, and J. Schmiedmayer. Atom interferometry with trappedbose-einstein condensates: impact of atom-atom interactions.
New J. Phys. , 12(6):065036, 2010.[20] T. Berrada, S. van Frank, R. B¨ucker, T. Schumm, J.-F. Schaff, and J. Schmiedmayer. IntegratedMach–Zehnder interferometer for Bose–Einstein condensates.
Nat. Commun. , 4, 2013.[21] M. Ammar, M. Dupont-Nivet, L. Huet, J.-P. Pocholle, P. Rosenbusch, I. Bouchoule, C. I.Westbrook, J. Est`eve, J. Reichel, C. Guerlin, and S. Schwartz. Symmetric microwave potentialsfor interferometry with thermal atoms on a chip.
Phys. Rev. A , 91:053623, May 2015.[22] M. Dupont-Nivet, C. I. Westbrook, and S. Schwartz. Contrast and phase-shift of a trappedatom interferometer using a thermal ensemble with internal state labelling.
New J. Phys. ,18(11):113012, 2016.[23] S. Kuhr, W. Alt, D. Schrader, I. Dotsenko, Y. Miroshnychenko, W. Rosenfeld, M. Khudaverdyan,V. Gomer, A. Rauschenbeutel, and D. Meschede. Coherence properties and quantum statetransportation in an optical conveyor belt.
Phys. Rev. Lett. , 91(21):213002, 2003.[24] S. Kuhr, W. Alt, D. Schrader, I. Dotsenko, Y. Miroshnychenko, A. Rauschenbeutel, and xperimental study of the role of trap symmetry... D. Meschede. Analysis of dephasing mechanisms in a standing-wave dipole trap.
Phys. Rev. A ,72(2):023406, 2005.[25] A. Hilico, C. Solaro, M.-K. Zhou, M. Lopez, and F. Pereira dos Santos. Contrast decay in atrapped-atom interferometer.
Phys. Rev. A , 91(5):053616, 2015.[26] N. Ramsey.
Molecular beams . Oxford University Press, 1956.[27] E. L. Hahn. Spin echoes.
Phys. Rev. , 80(4):580, 1950.[28] H. J. Lewandowski, D. M. Harber, D. L. Whitaker, and E. A. Cornell. Observation of anomalousspin-state segregation in a trapped ultracold vapor.
Phys. Rev. Lett. , 88:070403, Jan 2002.[29] X. Du, L. Luo, B. Clancy, and J. E. Thomas. Observation of anomalous spin segregation in atrapped fermi gas.
Phys. Rev. Lett. , 101:150401, Oct 2008.[30] X. Du, Y. Zhang, J. Petricka, and J. E. Thomas. Controlling spin current in a trapped fermi gas.
Phys. Rev. Lett. , 103:010401, Jul 2009.[31] C. Deutsch, F. Ramirez-Martinez, C. Lacroˆute, F. Reinhard, T. Schneider, J. N. Fuchs, F. Pi´echon,F. Lalo¨e, J. Reichel, and P. Rosenbusch. Spin self-rephasing and very long coherence times ina trapped atomic ensemble.
Phys. Rev. Lett. , 105:020401, Jul 2010.[32] G. Kleine B¨uning, J. Will, W. Ertmer, E. Rasel, J. Arlt, C. Klempt, F. Ramirez-Martinez,F. Pi´echon, and P. Rosenbusch. Extended coherence time on the clock transition of opticallytrapped rubidium.
Phys. Rev. Lett. , 106:240801, Jun 2011.[33] D. A. Steck. Rubidium 87 D line data, revision 1.6.
Source–http://steck.us/alkalidata , 2003.[34] D. M. Harber, H. J. Lewandowski, J. M. McGuirk, and E. A. Cornell. Effect of cold collisionson spin coherence and resonance shifts in a magnetically trapped ultracold gas.
Phys. Rev. A ,66:053616, Nov 2002.[35] M. Dupont-Nivet.
Vers un acc´el´erom´etre atomique sur puce . PhD thesis, Universit´e Paris Saclay,2016.[36] M. Dupont-Nivet, M. Casiulis, T. Laudat, C. I. Westbrook, and S. Schwartz. Microwave-stimulatedraman adiabatic passage in a bose-einstein condensate on an atom chip.
Phys. Rev. A , 91:053420,May 2015.[37] G. Santarelli, Ph. Laurent, P. Lemonde, A. Clairon, A. G. Mann, S. Chang, A. N. Luiten, andC. Salomon. Quantum projection noise in an atomic fountain: A high stability cesium frequencystandard.
Phys. Rev. Lett. , 82:4619–4622, Jun 1999.[38] K. G. Larkin. Efficient nonlinear algorithm for envelope detection in white light interferometry.
J. Opt. Soc. Am. A , 13(4):832–843, 1996.[39] P. Treutlein, T. W. H¨ansch, J. Reichel, A. Negretti, M. A. Cirone, and T. Calarco. Microwavepotentials and optimal control for robust quantum gates on an atom chip.
Phys. Rev. A ,74:022312, Aug 2006.[40] M. Dupont-Nivet, S. Schwartz, and C. I. Westbrook. Effect of trap symmetry and atom-atom interactions on the contrast decay and the phase-shift of a double well trapped ramseyinterferometer with internal states labelling.