We experimentally demonstrate and systematically study the stimulated revival
(echo) of motional wave packet oscillations. For this purpose, we prepare wave
packets in an optical lattice by non-adiabatically shifting the potential and
stimulate their reoccurence by a second shift after a variable time delay. This
technique, analogous to spin echoes, enables one even in the presence of strong
dephasing to determine the coherence time of the wave packets. We find that for
strongly bound atoms it is comparable to the cooling time and much longer than
the inverse of the photon scattering rate

A. Bulatov

A. Görlitz

C. Cohen-Tannoudji

D. R. Meacher

E. L. Hahn

F. B. J. Buchkremer

G. Birkl

G. Raithel

H. Levsen

J. Dalibard

J. I. Cirac

L. Allen

L. Guidoni

M. Gatzke

M. Kozuma

M. Morinaga

N. A. Kurnit

O. Morsch

P. Marte

P. Rudy

P. S. Jessen

R. Dumke

R. Ejnisman

W. D. Phillips

W. Ertmer

English

arXiv

Crossref

Physical Review Letters

Wave Packet Echoes in the Motion of Trapped Atoms

A new echo technique to access the coherent and dissipative dynamics of atomic wave packet oscillations in trapping potentials was experimentally and systematically validated. By using this technique, the coherence time of wave packet oscillations in strongly confining dipole potentials was measured in a regime that was previously inaccessible

Buchkremer, F.B.J.

Dumke, R.

Levsen, H.

Birkl, G.

Ertmer, Wolfgang

Institutional Repository of Leibniz Universität Hannover

VOLUME 85, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 9 OCTOBER 2000Wave Packet Echoes in the Motion of Trapped AtomsF. B. J. Buchkremer, R. Dumke, H. Levsen, G. Birkl, and W. ErtmerInstitut für Quantenoptik, Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany(Received 1 March 2000)We experimentally demonstrate and systematically study the stimulated revival (echo) of motionalwave packet oscillations. For this purpose, we prepare wave packets in an optical lattice by nonadiabati-cally shifting the potential and stimulate their reoccurrence by a second shift after a variable time delay.This technique, analogous to spin echoes, enables one even in the presence of strong dephasing to de-termine the coherence time of the wave packets. We find that for strongly bound atoms it is comparableto the cooling time and much longer than the inverse of the photon scattering rate.PACS numbers: 32.80.Pj, 42.50.VkThe process of decoherence, i.e., the collapse of super-position states due to the dissipative interaction with theirenvironment, is one of the basic concepts for our under-standing of the connection between classical and quantumphysics. In order to study the effect of decoherence unam-biguously, one has to be able to distinguish it from other,nondissipative effects. The macroscopic (i.e., ensemble- ortime-averaged) response of a quantum system prepared ina superposition state typically decays not only due to theloss of coherence (homogeneous decay) but also due todephasing resulting from local variations in the evolutionof the quantum system (inhomogeneous decay). In manycases decoherence cannot be studied directly because theinhomogeneous decay is by far the dominating process.This limitation has been overcome in a famous seriesof experiments by introducing the techniques of spin echofor nuclear magnetic resonance and photon echo for opticalresonance [1–3]. These techniques are based on the ob-servation that inhomogeneous decay due to dephasing is areversible process. Thus, by appropriately modifying su-perposition states at a time Dt after their preparation, thedephasing can be partially or fully reversed and a stimu-lated macroscopic response (echo) is induced at 2Dt. Thiseffect enables one to measure the coherence time evenin the presence of strong dephasing. We have, for thefirst time, applied this method to the investigation of thedecoherence of motional wave packets of trapped atoms(Fig. 1). The method can be used independent of thespecific experimental realization of the confining poten-tial (e.g., a single dipole potential, periodic dipole poten-tials, magnetic trapping potentials, inhomogeneous arraysof atom traps, etc.).The specific system investigated here consists of mo-tional wave packets of neutral atoms in a one-dimensionaloptical lattice. Optical lattices are periodic dipole poten-tials for atoms created by the interference of multiple laserbeams [4]. Atoms can be trapped and cooled at the poten-tial minima (mean position spread zrms  l18 [5]). Inoptical lattices symmetrically and asymmetrically oscillat-ing motional wave packets can be induced by nonadiabati-cally changing the lattice potential [6–11].0031-90070085(15)3121(4)$15.00In dipole potentials, the macroscopic oscillation signaldecays because of decoherence due to the spontaneousscattering of photons and because of dephasing due to theanharmonicity of the potential wells and spatial variationsof the potential depth. Typically, the effect of dephasing isdominating decoherence [6–13], so that a direct determi-nation of the coherence time is not possible. Here, we showhow these limitations can be overcome: For the case ofsymmetrical oscillations, Bulatov et al. [14] recently pro-posed and numerically simulated an echo mechanism toreverse the effect of dephasing and stimulate the revivalof the wave packet oscillations by means of two succes-sive nonadiabatic changes in the depth of the lattice po-tential. With our work we extend their proposal to thecase of asymmetrical wave packet oscillations, present thefirst experimental observation of the echo effect, and ap-ply it to determine the coherence time of the motional wavepackets.FIG. 1. Wave packet oscillations without (curve a) andwith (curves b,c) stimulated revival (echo) (U0  831ER ,d  27.8G, dz  0.10l). Depicted are the oscillationsafter the second shift (at t  0) for time delays betweenthe two shifts of Dt  108 ms (a, reference curve, completedecoherence before second shift) and Dt  32 ms (b, signalcurve), showing additional oscillations at about t  32 ms.The difference between curves b and a (c, echo curve) showsthe net effect of the wave packet echo.© 2000 The American Physical Society 3121VOLUME 85, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 9 OCTOBER 2000In our experiment, we chirp-slow rubidium atoms85Rb from a thermal beam and trap and cool them in amagneto-optical trap (MOT), giving a sample of approxi-mately 107 atoms with a central density of 109 atomscm3.After the loading phase, the magnetic field of the MOTis switched off and the intensity of the trapping beams isreduced to achieve optimal cooling in a three-dimensionalmolasses. The MOT-laser beams are then turned offand two lattice beams are switched on, forming a one-dimensional lin  lin optical lattice [4]. The lattice beamshave intensities up to I  60 mWcm2, and detuningsd of 2–10 natural linewidths G2p  5.89 MHzbelow the 5S12F  3 ! 5P32F0  4 transition atl  780 nm. The beam waist of about 2.75 mm (1e2radius) is large compared to the 1e2 radius of the atomiccloud of 1.52 mm.After an initial cooling phase of 1 to 2 ms in the lat-tice, which localizes the atoms at the center of the poten-tial wells, we nonadiabatically change the relative phasebetween the two lattice beams with an electro-optic phaseshifter [11] (1e switching time of 0.4 ms). This causes atranslation of the lattice by a controllable amount dz 0 ,dz , l4 and induces asymmetrical coherent-state– likemotional wave packets. We observe the wave packet os-cillations by measuring the photon redistribution-inducedpower difference DPt between the two lattice beams[11,15]. No repumping light is present during the wavepacket evolution. Figure 1(a) shows a typical example ofa wave packet oscillation. Clearly visible are about fiveoscillations with a period of 5.2 6 0.1 ms. The oscilla-tion signal is damped with a decay time of t1  7 6 1 ms(exponential fit).Figure 2 shows the decay time t1 (relative uncertainty#20%) of asymmetrical wave packet oscillations as a func-FIG. 2. Decay times t1 of asymmetrical wave packet oscilla-tions as a function of the potential depth U0 for the indicatedvalues of the detuning d. The solid line shows a calculationof the decay time for anharmonicity-induced dephasing as thedecay mechanism.3122tion of the potential depth U0 for various detunings d.U0 is calculated from the measured oscillation frequencyvosc according to U0ER  10.862 vosc2vR2, tak-ing into account the anharmonicity of the potential wells[5,9]. The data show that the decay is not due to deco-herence caused by spontaneous scattering, for in that casethe decay time t1 should be proportional to the inverse ofthe photon scattering rate, tsc  1G0  1U0Gh̄jdj.For different detunings this should result in different de-cay times t1 for the same U0, which is not observed forjdj $ 7G. For a decay caused by anharmonicity, on theother hand, the decay time should be proportional to theinverse of the mean spread Dvosc of the occurring os-cillation frequencies, which can be approximated by themean shift of the oscillation frequencies from the har-monic frequency minus the common shift of vR , i.e.,Dvosc  0.14vosc 2 vR [5,9]. The line in Fig. 2 is a plotof 1Dvosc as a function of U0. Within the 20% uncer-tainty of our data the decay times t1 are in agreement withthis calculation. Thus, the decay of the wave packet oscil-lations is dominated by dephasing [16].The investigation of decoherence, on the other hand,becomes possible by nonadiabatically shifting the latticeback to its initial position after a variable delay Dt. Thissecond translation leads to a stimulated revival (echo) ofthe oscillations with a maximum amplitude at about 2Dt[14] if the coherence of the wave packets induced by thefirst shift still persists—at least partially—at 2Dt.Figure 1 shows the first experimental demonstration ofthis echo effect [17]. Figure 1(a) serves as a referencecurve showing wave packet oscillations for atoms thatmove fully incoherently at the time of the shift. For thepurpose of reducing systematic uncertainties, this curvealso has been recorded after two translations of the lat-tice. However, here the long delay time Dt  108 msguarantees a complete loss of coherence before the secondshift. Figure 1(b) depicts the wave packet oscillations af-ter the second shift with Dt  32 ms. The curve showsadditional oscillations at about t  32 ms. Figure 1(c)presents the magnified difference of curves (b) and (a).Clearly visible is the recurrence of wave packet oscillation,i.e., echo, centered around a time close to t  32 ms. Thistime corresponds to a total time of ttotal  32 ms 1 Dt 64 ms  2Dt after the first shift, as predicted.In order to gain a qualitative and quantitative under-standing of the echo mechanism, we have performed afull quantum Monte Carlo wave-function simulation (QM-CWF) [18] of the echo experiment, taking into accountthe full coherent and dissipative dynamics as well as allinternal ground states while adiabatically eliminating theexcited states. Figure 3 shows the result for the experi-mental conditions of Fig. 1. In Fig. 3(a) the spontaneousscattering rate is reduced G0 ! G01000 in order to em-phasize the coherent dynamics which gives rise to the echomechanism. Clearly visible is the occurrence of the wavepacket echo at t  32 ms.VOLUME 85, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 9 OCTOBER 2000FIG. 3. QMCWF simulations of the echo experiment[I  56.5 mWcm2 and d  27.8G ) U0  823ER,dz  0.064l, Dt  32 ms (potential shifts at t  232 ms andt  0)]. (a) With a reduced rate of spontaneous scatteringprocesses G0 ! G01000, a strong wave packet echo isvisible. (b) With the regular rate of spontaneous scattering, theecho is still visible but its amplitude is significantly reduced.The underlying physics of the echo mechanism canbe understood in terms of the coherent evolution of thecontributions of different eigenstates to the wave packet:The first translation creates a coherent superposition ofeigenstates of the translated anharmonic trapping poten-tial which are all in phase (with phase  0 per definition).The phases of different eigenstates evolve with differentfrequencies in time which causes the decay of the oscil-lation signal. Shifting the lattice back after Dt creates anew superposition state. Because of the asymmetry of thetranslation, eigenstates with phases close to odd multiplesof p give the strongest and eigenstates with phases close toeven multiples of p give the weakest contributions to thenew superposition state, which leads to new wave packetoscillations which again dephase in time. However, eigen-states that were in phase at t  Dt are again in phase att  2Dt, so that the strongest contributions again realignat t  2Dt, which results in a partial revival of the oscilla-tions. Thus, the phase-dependent selection of the strengthof eigenstate contributions at the second shift leads to thewave packet echo.Spontaneous scattering leads to an incoherent evolutionof the superposition state by randomizing the phases ofthe eigenstates and thus causes a decrease in the echoamplitude. This is clearly observed in our wave-functionsimulation for a nonreduced spontaneous scattering rate[Fig. 3(b)]. The wave packet echo is still visible but itsamplitude is reduced to a value that is consistent with theone obtained in the experiment (Fig. 1).In order to experimentally study additional effects of de-phasing, we have narrowed the transverse intensity profileof our lattice beams, causing different parts of the atomcloud to experience different potential depths U0.The almost freestanding echo in the signal curve ofFig. 4 experimentally demonstrates, that the method ofFIG. 4. Wave packet oscillations and echo for increased de-phasing (U0  368ER , d  8G, dz  0.11l). The referencecurve (a, Dt  132 ms) depicts the faster signal decay. Thesignal curve (b, Dt  30 ms) shows a freestanding echo.wave packet echoes can also be applied to other systems,where the spatial variation in the coherent dynamics is thedominating effect for dephasing as, e.g., in anharmonicdipole traps based on focused laser beams, inhomoge-neous arrays of atom traps, or magnetic quadrupole traps,etc. (see, e.g., Ref. [13]).The physics behind the echo effect predicts that the timeat which the echo occurs can be varied by changing thetime delay Dt between lattice shifts. Figure 5 shows echocurves [analogous to Fig. 1(c)] for the indicated values ofthe time delay Dt. For each curve, t  0 corresponds tothe second lattice shift. The total time after the first shift isgiven by ttotal  t 1 Dt. Our measurements confirm thatthe echo moves to larger t with increasing Dt and has itsmaximum value at approximately ttotal  2Dt [19].We can use the echo technique to determine the coher-ence time t2 of the atomic motion by measuring the echoamplitude as a function of the echo time 2Dt. Figure 6(a)shows this for the data of Fig. 5. With increasing DtFIG. 5. Increase in occurrence time and decrease in amplitudeof the echoes for increasing delay Dt between translations ofthe trapping potential (parameters as in Fig. 1).3123VOLUME 85, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 9 OCTOBER 2000FIG. 6. (a) Dependence of the echo amplitude on the echotime 2Dt for the data of Fig. 5. The line is an exponentialfit to the data, giving a coherence time of t2  27 6 3 ms.(b) Coherence time in units of the photon scattering time tsc 1G0 as a function of the lattice detuning d.the echo amplitude decreases exponentially as expectedfor a coherence loss due to spontaneous scattering pro-cesses occurring at a constant rate G0. An exponential fitto the data [line in Fig. 6(a)] gives a 1e decay time oft2  27 6 3 ms which is significantly longer than theobserved dephasing time of 7 ms. This shows that the co-herent motion still persists for times at which the macro-scopic oscillation signal has already disappeared.The echo technique even enables a quantitative determi-nation of the constant of proportionality between the co-herence time t2 and the photon scattering time tsc  1G0.For the data of Fig. 6(a), tsc  0.40 6 0.04 ms and wefind that, during the coherence time, one atom scatters68 6 10 photons. In a series of measurements similarto Fig. 5 for 210G # d # 24G we observe coherencetimes in the range from 40 to 68 scattering times tsc withan average value of t2  49 6 7tsc independent of thedetuning d [Fig. 6(b)]. This proves that the coherence timet2 is proportional to the scattering time tsc, and that forthe decay of motional coherence a large number of pho-tons has to be scattered spontaneously.The quantitative determination of the coherence timeenables us to prove an important prediction given for thedissipative coupling of the motion of atoms in (nearly)harmonic potentials to their environment: It has beenpointed out that for (nearly) harmonic oscillators the co-herence time t2 should be twice as long as the dampingtime of the oscillator’s energy, i.e., the cooling time tcoolof the atoms [20–22]. In a recent experiment, tcool in one-dimensional optical lattices was found to be the time tospontaneously scatter 30 photons: tcool  30tsc [7]. Ourdirect measurement of the coherence time t2  49 67tsc is the first experimental confirmation of the predictedrelation between the coherence time and the cooling time.We expect the relation t2  2tcool to hold universally for(nearly) harmonic trapping potentials in which the motionof atoms is coupled to a dissipative reservoir [20].In summary, we have experimentally demonstratedand systematically investigated a new echo technique to3124access the coherent and dissipative dynamics of atomicwave packet oscillations in trapping potentials. With thistechnique, we have measured the coherence time of wavepacket oscillations in strongly confining dipole potentials,in a regime, which was previously inaccessible. Weshowed that the coherence time is directly connected tothe energy dissipation time. We also demonstrated thatthe echo technique works for atomic motion exhibitingvarious causes of dephasing, which shows that it can beused to access the external dynamics in a broad range ofdifferent atom traps or inhomogeneous arrays of atomtraps.We thank A. Pahl and M. Wilken for their support in theearly stages of the project. This work is supported by theSFB 407 of the Deutsche Forschungsgemeinschaft.[1] E. L. Hahn, Phys. Rev. 80, 580 (1950).[2] N. A. Kurnit, I. D. Abella, and S. R. Hartmann, Phys. Rev.Lett. 13, 567 (1964).[3] L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1987).[4] P. S. Jessen and I. H. Deutsch, Adv. At. Mol. Opt. Phys. 37,95 (1996); D. R. Meacher, Contemp. Phys. 39, 329 (1998);L. Guidoni and P. Verkerk, J. Opt. B 1, R23 (1999).[5] M. Gatzke et al., Phys. Rev. A 55, R3987 (1997).[6] M. Kozuma et al., Phys. Rev. Lett. 76, 2428 (1996).[7] G. Raithel et al., Phys. Rev. Lett. 78, 630 (1997).[8] A. Görlitz et al., Phys. Rev. Lett. 78, 2096 (1997).[9] G. Raithel et al., Phys. Rev. Lett. 78, 2928 (1997).[10] P. Rudy, R. Ejnisman, and N. P. Bigelow, Phys. Rev. Lett.78, 4906 (1997); R. Ejnisman et al., Phys. Rev. A 56, 4331(1997).[11] G. Raithel, W. D. Phillips, and S. L. Rolston, Phys. Rev.Lett. 81, 3615 (1998).[12] O. Morsch, P. H. Jones, and D. R. Meacher, Phys. Rev. A61, 023410 (2000).[13] M. Morinaga et al., Phys. Rev. Lett. 83, 4037 (1999).[14] A. Bulatov et al., Phys. Rev. A 57, 3788 (1998).[15] The maximum power difference DP is about 1% of thetotal lattice power for all data presented here.[16] The remaining deviations may be due to the additionalinfluence of decoherence and local variations in U0.[17] In order to compensate fluctuations in atom number, theinitial amplitudes of all wave packet oscillations are nor-malized. For clarity some curves are offset vertically.[18] J. Dalibard, Y. Castin, and K. Mølmer, Phys. Rev. Lett. 68,580 (1992); P. Marte et al., Phys. Rev. A 47, 1378 (1993);P. Marte et al., Phys. Rev. Lett. 71, 1335 (1993).[19] The fact that the maxima occur at times slightly earlierthan 2Dt is caused by decoherence, which decreases thesignal strength during the echo, thus shifting the apparentecho maximum to earlier times.[20] W. D. Phillips and C. I. Westbrook, Phys. Rev. Lett. 78,2676 (1997).[21] J. I. Cirac et al., Phys. Rev. A. 48, 2169 (1993).[22] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg,Atom-Photon Interactions (Wiley, New York, 1992).

Wave packet echoes in the motion of trapped atoms

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Wave Packet Echoes in the Motion of Trapped Atoms

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