Coherent population trapping in a Raman atom interferometer
Bing Cheng, Pierre Gillot, Sébastien Merlet, Franck Pereira dos Santos
CCoherent population trapping in a Raman atom interferometer
B. Cheng, P. Gillot, S. Merlet, F. Pereira Dos Santos
LNE-SYRTE, Observatoire de Paris, PSL Research University,CNRS, Sorbonne Universit´es, UPMC Univ. Paris 06,61 avenue de l’Observatoire, 75014 Paris, France
We investigate the effect of coherent population trapping (CPT) in an atom inter-ferometer gravimeter based on the use of stimulated Raman transitions. We find thatCPT leads to significant phase shifts, of order of a few mrad, which may compromisethe accuracy of inertial measurements. We show that this effect is rejected by thek-reversal technique, which consists in averaging inertial measurements performedwith two opposite orientations of the Raman wavevector k, provided that internalstates at the input of the interferometer are kept identical for both configurations.PACS 37.25.+k; 37.10.Vz; 03.75.Dg; 42.60.By
I. INTRODUCTION
Gravimeters based on Mach Zehnder type atom interferometer reach nowadays long termstabilities in the low 10 − g range [1, 2] or better [3] and accuracies of a few 10 − g [3–5],comparable to classical corner cube gravimeters [6]. On going efforts to improve the stabilityof cold atom gravimeters focus on strategies to accurately determine or reject interferometerphase fluctuations arising from changes of the experimental parameters (such as due to lightshifts and Doppler shifts fluctuations [7]) or from environmental effects (via for instance thedirect comparison of two gravimeters, eventually based on different technologies [3]).A common and very efficient method consists in alternating the direction of the Ramanwavevector, which allows rejecting the phase shifts which are independent of the Raman laserwavevector direction. This rejection is in practice limited by the difference of the trajectoriesof the atoms between these two interferometer configurations, due to the change in thedirection of the momentum kick imparted to the atoms by the lasers. To be quantitative,the maximum position shift between these trajectories reaches, for our total interferometerduration of 160 ms, up to 2 mm in the vertical direction.It is thus of interest to find methods that maximize the trajectories overlap when changing a r X i v : . [ phy s i c s . a t o m - ph ] J un the direction of the Raman wevector. As already pointed out in [8], this can be realized forinstance by changing the internal state of the atom at the input of the interferometer. Themomentum kick then occurs in the same direction, despite the change of the direction ofthe Raman wavevector. We show here that this technique has a drawback, and leads to abias in the measurement of gravity, arising from a phase shift linked to coherent populationtrapping (CPT). The effect of CPT was put into evidence in [9] by measuring dark-statecoherences and population differences induced in cold cesium atoms by velocity-sensitiveand velocity-insensitive Raman pulses. It was also claimed in [9] that CPT effects shouldlead to spurious phase shifts of order of a few mrad in Mach Zehnder interferometer, whichthe measurements we present here confirm.In this article, we perform a detailed evaluation of the phase shift induced by CPTeffects. We first investigate this effect theoretically following the formalism developed in [9]and extending it to the case of a Raman interferometer. We show results of measurementswhere we exchange internal states at the input of the interferometer to put this effect inevidence. We study in particular its dependence on relevant parameters of the Raman laser,such as one-photon Raman laser detuning, Raman pulses and interferometer duration. II. THEORY
We measure gravity using an atom interferometer realized by counterpropagating Ramantransitions. Raman transitions are two-photon transitions which couple two states | g (cid:105) and | e (cid:105) (in our case two hyperfine ground states of an alkali atom) via the off-resonant excitationof an excited state | i (cid:105) . CPT effects arise from the dynamics of this 3 level system ( | g (cid:105) , | e (cid:105) , | i (cid:105) )interacting with the Raman lasers, when taking into account the influence of spontaneousemission from the excited level. In [9], the evolution of a three level system in the field oftwo lasers is developed in the interaction picture taking into account spontaneous emission.The density matrix R int of the three states is given by:d R int d t = [ 1 i ¯ h ( ˆ V int − ˆ H int ) , R int ] + R SE (1)where ˆ H int is the laser energy, ˆ V int is the coupling in the interaction picture and R SE is thespontaneous decay of the density matrix.Adiabatic elimination of the excited state | i (cid:105) allows to derive differential equations gov-erning the dynamics of the system in the basis restricted to the two states | g (cid:105) and | e (cid:105) [9].These are given by eq. 2, where Γ is the linewidth of the excited state and Ω eff is theeffective 2-photon Rabi frequency. δ ( t ) − δ AC is the two-photon Raman detuning, ∆ isthe one-photon Raman laser detuning from the excited state and δ AC is (the one-photon)differential light shift. ρ ee (cid:48) ( t ) + Im(Ω eff r eg ( t )) + Γ Re(Ω eff r eg ( t ))2∆ + ΓΩ eAC ρ ee ( t )∆ = 0 ρ gg (cid:48) ( t ) − Im(Ω eff r eg ( t )) + Γ Re(Ω eff r eg ( t ))2∆ + ΓΩ gAC ρ gg ( t )∆ = 0r eg (cid:48) ( t ) − i Ω eff ∗ ( ρ ee ( t ) − ρ gg ( t )) + Γ (Ω eAC + Ω gAC ) r eg ( t )2∆ − i r eg ( t ) ( δ ( t ) − δ AC ) + ΓΩ eff ∗ ( ρ ee ( t ) + ρ gg ( t ))4∆ = 0 (2)A detailed analysis of the evolution of the system is done in [9], where spontaneous emis-sion is shown to lead to coherent population trapping. For on resonance driving, the systemasymptotically evolves towards a dark state, uncoupled to the Raman lasers. Representingthe quantum state as a vector in the Bloch sphere helps understanding the phase shift in-troduced by the CPT effect in our situation, where the duration of Raman pulses is morethan two orders of magnitude shorter than the characteristic time of evolution into the darkstate. In this picture, the atomic state is depicted by the pseudo spin ( (cid:126)P ). While ( (cid:126)P ) rotatesin a plane perpendicular to the Raman vector ( (cid:126) Ω) during the Raman pulse, spontaneousemission makes the pseudo-spin move off this plane. For short Raman pulse ( ΓΩ eff τ (cid:28) ΓΩ eff , independent of the onephoton transition couplings (Ω gAC , Ω eAC ). This dynamic is illustrated for a π/ (cid:126) Ω) induces in the absence of spontaneous emission a rotation of the vectorstate by π/ φ CP T in the equatorial plane, as displayed in c).The CPT phase at resonance (we do not consider any detuning from the Raman resonancecondition here) is found to be approximately given by:∆ φ CP T = Γ τ Ω eff
2∆ (3) ~ ⌦ ~ ⌦ ~ ⌦ ~p ~p ~pa ) b ) c ) CP T
FIG. 1: The evolution of the pseudo-spin during a π/ π/ where τ is the pulse duration of the Raman pulse.To evaluate the amplitude of the effect, we consider the case of Rb atoms, with Ramanlasers at one-photon Raman detuning ∆ = − .
932 GHz, and for a Raman pulse durationcorresponding to a π/ − − µ s, which correspond to a π/ − π − π/ σ v ∼ hk L /m Rb , where m Rb is the mass of a Rbatom and k L is the photon momentum at 780 nm. In addition, we consider that the atomsare velocity selected with a Raman π pulse of duration 44 µ s before entering the interferom-eter (as we will do later in the experiment). We find for these parameters a phase shift ∆ φ of 5.35 mrad. This differs from the result of eq. 3 by about 6% only, which indicates thatthe average over the velocity distribution has a limited influence on the result. Moreover,with the simulation, we confirm that the effect on the interferometer phase is given by theCPT phase of the first pulse. Finally, the calculated phase shift corresponds to a bias onthe g measurement of ∆ g = ∆ φ/kT = 5 . µ Gal , where k (cid:39) k L is the effective Ramanwavevector, and 1 Gal = 1 cm/s .Hopefully, this phase shift is independent of the Raman wavevector direction. It is thusin principle well rejected by the k-reversal technique, which consists in averaging the mea-surements performed using two opposite directions of the Raman effective wavevector k .Yet, as a remarkable feature, we find that this phase shift changes sign when the internalstate at the input of the interferometer is changed. For the k-reversal rejection to hold, itis thus mandatory that the internal state at the input of the interferometer is the same forboth directions of k . III. EXPERIMENTS
To put the CPT effect into evidence and evaluate its influence, we exploit its dependenceon the internal state at the input of the interferometer. We will thus perform differentialmeasurements of the gravity acceleration g for given directions of the Raman wavevector,but with different internal states of the atom at the input of the interferometer.The experimental setup is described in detail in [10]. We briefly recall here the mainphases of the experimental sequence. We start by trapping a few 10 atoms in a 3D Magneto-Optical Trap for 80 ms. A subsequent molasses phase cools the atoms down to a temperatureof 2 µ K. The molasses beams are then switched off and the atomic cloud is let to fall. Aftera preparation phase detailed below, we drive a three pulse Mach-Zehnder type Raman inter-ferometer, with a total interferometer time of 2 T = 160 ms, where T is the separation timebetween consecutive pulses. The populations in the two output ports of the interferometerare finally measured via a state selective fluorescence detection setup at the bottom of thevacuum chamber.For the preparation of the atomic state at the input of the interferometer, we normallyapply 2 microwave pulses. The first one is used for the sub- m F state selection into the state | F = 1 , m F = 0 (cid:105) . It transfers atoms in the | F = 2 , m F = 0 (cid:105) into the | F = 1 , m F = 0 (cid:105) , andis followed by a pulse of a pusher beam that removes atoms remaining in the | F = 2 (cid:105) state.The second one is used to retransfer the atoms into the | F = 1 (cid:105) internal state before thevelocity selection occurs. This selection is realized with a Raman pulse (that transfers thecentre of the velocity distribution back into the state | F = 1 , m F = 0 (cid:105) ) and a subsequentsecond pulse of the pusher beam. The use of a second microwave pulse is required as we donot have a pusher beam resonant with | F = 1 (cid:105) → | F (cid:48) (cid:105) transition. The final internal stateat the input of the interferometer is thus | F = 1 , m F = 0 (cid:105) . To prepare the atoms into the | F = 2 , m F = 0 (cid:105) state at the input of the interferometer, a possibility would be to simplyapply a third microwave pulse after the normal sequence. In this way, though, the velocitykicks imparted by the selection and Raman pulses would occur in the same direction, whichwould modify the trajectories of the interferometer paths. As an alternative, we remove thesecond microwave pulse, so that the velocity selection is performed from | F = 1 (cid:105) to | F = 2 (cid:105) .Then, we get rid of the atoms that are not velocity selected with a sequence comprised oftwo microwave π pulses and a pulse of pusher beam in between them.The different preparation sequences and the corresponding interferometer configurationswe use for the gravity measurements performed here are shown in figure 2.Usually, we use two interleaved measurements with opposite wavevectors (displayed ascase a) and b) in figure 2) with atoms entering the interferometer in the state | F = 1 (cid:105) ,which requires two microwave pulses in the preparation. The gravity measurement is thenobtained from the average of the two measurements. Case c) and d) correspond to a differentpreparation sequence, using three microwave pulses, with atoms entering the interferometerin the state | F = 2 (cid:105) . One can note that the trajectories of the atomic wavepackets along thetwo interferometer paths are the same for the k ↓ interferometer using two microwave pulses(case a)) and the k ↑ interferometer using three microwave pulse (case c)). The same holdsfor the k ↑ interferometer using two microwave pulses (case b)) and the k ↓ interferometerusing three microwave pulses (case d)). This allows to realize interleaved measurements Preparation
Interferometers F ,3 mw g k k k k F F F F F F F F m i c r o w a ve m i c r o w a ve pu s h e r pu s h e r m i c r o w a ve m i c r o w a ve m i c r o w a ve pu s h e r pu s h e r p/2 p/2 pp sel F t ,3 mw g d)c) ,2 mw g a) ,2 mw g b) FIG. 2: Different preparation sequences, with two or three microwave pulses, corresponding toinput states in | F = 1 (cid:105) or | F = 2 (cid:105) . The corresponding interferometer configurations with thetrajectories along the two interferometer paths are also displayed. with k ↑ and k ↓ interferometers while keeping the trajectories overlapped. It simply requiresto replace for instance the k ↑ interferometer of case b) by the k ↑ interferometer of case c)(or the k ↓ interferometer of case a) by the k ↓ interferometer of case d)).We show now that the change of internal state at the input of the interferometer which isassociated with this swap makes the new pair of configurations sensitive to CPT effect. Wepresent in the following measurements of the difference in the phases (and the correspondingdifferences in the measured values of g) between the k ↑ interferometers of case b) and c),and the difference between the k ↓ interferometers of case a) and d).Figure 3 displays the measured differences in the interferometer phases as a function ofthe Raman pulse spacing T . We find small variations with T of these differences, withopposite trends for k ↑ and k ↓ interferometers, which are not reproduced by the simple modelabove. We find on average a value of about 7.7(4) mrad in absolute value. As the CPTphase changes sign with the internal state, the measured difference in the interferometerphases is twice this CPT phase. We would thus expect differences of 10.7 mrad, which issignificantly larger than our measurement. This difference may be explained by the fact thatthe our model neglects the detailed structure of the energy levels of the atoms (hyperfinestructure of the excited state i , Zeeman sublevels ...). The interferometer phase differencecorresponds to a difference in the g value of 7 . µ Gal for an interferometer duration of2 T = 80 ms. As the gravity phase shift scales as T , we find, as displayed in Figure 3, thatthe lower the separation time T , the higher the effect on the gravity value.
40 50 60 70 80-10-9-8-7-6-5 ( m r a d ) T (ms)
40 50 60 70 80-40-30-20-10010203040 g ( (cid:181) G a l ) T (ms)
FIG. 3: Differences in the interferometer phases and in the measured g values for input statesin different hyperfine states as a function of T , ranging from 40 to 80 ms. Black squares: k ↓ interferometers, Red circles: k ↑ interferometers. The one-photon detuning of the Raman lasers is-0.9 GHz. We then measured the dependence of the phase shift with the one-photon laser detuningfrom the excited state ∆, keeping the Rabi frequency constant, by adjusting the Ramanlaser intensity. The results, displayed on fig. 4, confirm the expected scaling: the phaseshift decreases inversely proportionally to ∆ (see eq. 3), which we take as a strong evidencethat the measured shift originates indeed from the effect of spontaneous emission.Finally, we measured the variation of the CPT induced phase shift with the duration ofthe first Raman pulse, for a fixed Rabi frequency of 2 π × . µ s (close to the duration of 22 µ s of the perfect π/ µ s (close to a 3 π/ -1.6 -1.4 -1.2 -1.0 -0.8 -0.6-20-15-10-5051015 g ( (cid:181) G a l ) One-photon Raman detuning (GHz)
FIG. 4: Differences in the measured g values for input states in different hyperfine states as afunction of the one-photon laser detuning, ranging from -0.6 GHz to -1.6 GHz, for T = 80ms.Black squares: k ↓ , Red circles: k ↑ . becomes longer than a π pulse. The trends we measure are in good agreement with theresults of the numerical simulation, which are displayed as lines, though the quantitativeagreement is here again not perfect. IV. CONCLUSION
We have studied the effect of CPT in an atom gravimeter, based on an Mach-Zehnder typeatom interferometer, realized with a sequence of three Raman pulses. Measurements of thephase shift induced by this effect, and thus of the corresponding bias onto the measurementof gravity, have been performed as a function of the parameters of the Raman lasers and ofthe pulse sequence, such as pulse duration, and detuning of the Raman lasers. The trendsin the measurements are found to be in good agreement with the behaviour derived fromcalculations based on a simple three level model. A better match between measured andcalculated phase shifts would certainly require a model which takes into account the realinternal structure of the atom and the polarization state of the Raman lasers.This phase shift is a drawback when alternating interferometer measurements with con-0
10 20 30 40 50 60 70 80-40-30-20-10010203040 g ( (cid:181) G a l ) Duration of first and last Raman pulses ((cid:181)s)
FIG. 5: Differences in the measured g values for input states in different hyperfine states as afunction of the duration of the first and third Raman pulses, for a Rabi frequency of 2 π × . T = 80 ms. The duration of the second pulse is kept constant at 44 µ s. Black squares: k ↓ ,Red circles: k ↑ . Lines: calculations. figurations that change not only the direction of the Raman wavevector but also the internalstate at the input of the interferometer. Indeed, it changes sign with configuration, as doesthe gravity phase shift. This finally results in a bias in the determination of g, when averag-ing the g measurements over the two configurations. However, changing the internal stateat the input of the interferometer offers a better superposition of the trajectories betweenthese two configurations. This allows for a better rejection of magnetic field gradients [8]and eventual light shift longitudinal inhomogeneities. In that case, though, the measured gvalue needs to be corrected for the phase shift induced by CPT effects.1 V. ACKNOWLEGMENTS
B. C. thanks the Labex First-TF for financial support. [1] P. Gillot, O. Francis, A. Landragin, F. Pereira Dos Santos, S. Merlet, ”Stability comparisonof two absolute gravimeters: optical versus atomic interferometers”, Metrologia , L15-L17(2014)[2] 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 , 043610 (2013)[3] C. Freier, M. Hauth, V. Schkolnik, B. Leykauf, M. Schilling, H. Wziontek, H.-G. Scherneck, J.Mller, A. Peters, ”Mobile quantum gravity sensor with unprecedented stability”, Proceedingsfor the 8th Symposium on Frequency Standards and Metrology, to be published in Journal ofPhysics: Conference Series (JPCS), arXiv:1512.05660[4] O. Francis et al., ”CCM.G-K2 key comparison”, Metrologia , 07009 (2015)[5] A. Peters, K. Y. Chung and S. Chu, ”High-precision gravity measurements using atom inter-ferometry”, Metrologia , 25 (2001)[6] T. M. Niebauer, G. S. Sasagawa, J. E. Faller, R. Hilt and F. Klopping, ”A new generation ofabsolute gravimeters”, Metrologia , 159, 1995[7] P. Gillot, B. Cheng, S. Merlet, F. Pereira Dos Santos, ”Limits to the symmetry of a MachZehnder type atom interferometer”, Phys. Rev. A , 013609 (2016)[8] T. E. Mehlst¨aubler, J. Le Gou¨et, S. Merlet, D. Holleville, A. Clairon, A. Landragin, F. PereiraDos Santos, in Proceedings of the XLIIth Rencontres de Moriond Gravitational waves andexperimental gravity, edited by J. Dumarchez and J. Trˆan Thanh Vˆan (Thˆe’ Gi´o’i Publishers,Vietnam, 2007), pp. 323-333[9] D. L. Butts, J. M. Kinast, K. Kotru, A. M. Radojevic, B. P. Timmons, and R. E. Stoner,”Coherent population trapping in Raman-pulse atom interferometry”, Phys. Rev A 84, 043613(2011)[10] A. Louchet-Chauvet, T. Farah, Q. Bodart, A. Clairon, A. Landragin, S. Merlet, F. PereiraDos Santos, ”Influence of transverse motion within an atomic gravimeter”, New J. Phys. ,2