We report the first direct observation of Brillouin-like propagation modes in a dissipative periodic optical lattice. This has been done by observing a resonant behavior of the spatial diffusion coefficient in the direction corresponding to the propagation mode with the phase velocity of the moving intensity modulation used to excite these propagation modes. Furthermore, we show theoretically that the amplitude of the Brillouin mode is a nonmonotonic function of the strength of the noise corresponding to the optical pumping, and discuss this behavior in terms of nonconventional stochastic resonance

C. Mennerat-Robilliard

F. Renzoni

F.-R. Carminati

G. Grynberg

H. J. Metcalf

I. Hidaka

J. Dalibard

J.-Y. Courtois

K. I. Petsas

L. Fronzoni

L. Gammaitoni

L. Sanchez-Palencia

L. Viola

M. G. Raizen

M. I. Dykman

M. Schiavoni

P. J. Ungar

P. S. Jessen

R. Bartussek

R. Grimm

R. Löfstedt

R. W. Boyd

T. Wellens

Y. Castin

Y. R. Shen

English

arXiv

Crossref

Physical Review Letters

Brillouin Propagation Modes in Optical Lattices: Interpretation in Terms of Nonconventional Stochastic Resonance

Sanchez-Palencia, L

Carminati, FR

Schiavoni, M

Renzoni, F

Grynberg, G

UCL Discovery

VOLUME 88, NUMBER 13 PHYSICAL REVIEW LETTERS 1A PRIL 2002Brillouin Propagation Modes in Optical Lattices: Interpretationin Terms of Nonconventional Stochastic ResonanceL. Sanchez-Palencia, F.-R. Carminati, M. Schiavoni, F. Renzoni, and G. GrynbergLaboratoire Kastler-Brossel, Département de Physique de l’Ecole Normale Supérieure, 24 rue Lhomond,75231, Paris Cedex 05, France(Received 15 October 2001; published 19 March 2002)We report the ﬁrst direct observation of Brillouin-like propagation modes in a dissipative periodicoptical lattice. This has been done by observing a resonant behavior of the spatial diffusion coefﬁcientin the direction corresponding to the propagation mode with the phase velocity of the moving intensitymodulation used to excite these propagation modes. Furthermore, we show theoretically that the ampli-tude of the Brillouin mode is a nonmonotonic function of the strength of the noise corresponding to theoptical pumping, and discuss this behavior in terms of nonconventional stochastic resonance.DOI: 10.1103/PhysRevLett.88.133903 PACS numbers: 42.65.Es, 05.45.–a, 32.80.PjThe last decade has witnessed dramatic progress in lasercooling techniques and nowadays in several laboratoriesaround the world atoms are routinely trapped and cooledat very low temperatures and high densities [1]. Most ofthe current efforts within the cold atoms community aredirected to reaching the regime of quantum degeneracy inboth bosonic and fermionic samples, in order to investigatethe properties of the quantum gases thus obtained, andrealizing an atom laser, the matter wave analog of the laser.Cold atomic samples also constitute an ideal system for thestudy of complex nonlinear phenomena. This turns out tobe especially true if the cold atoms are ordered by the lightﬁelds in periodic structures, so-called optical lattices [2,3].These are obtained by the interference of two or more laserﬁelds: the light imposes its order on the matter via thedipole force [4], creating a periodic structure of atoms.Among the most signiﬁcant studies of nonlinear dy-namics in optical lattices, we recall here the observationof mechanical bistability in a strongly driven dissipativeoptical lattice [5] and the realization of the kicked rotorand corresponding detection of chaotic motion in a fardetuned lattice [6]. Furthermore the macroscopic transportof atoms in an asymmetric optical lattice without theapplication of external forces has been observed [7].This corresponds to the realization of an optical motor,i.e., a ratchet for cold atoms, a well-controllable modelsystem for the molecular combustion motor [8]. Brillouin-like propagation modes of atoms in a dissipative opticallattice have also been theoretically studied to explain thenonlinear optical properties of optical lattices [9,10]. Inthis Letter, we report on the ﬁrst direct observation ofthese modes. Furthermore, we discuss the propagationmechanism associated with these modes, completelydifferent from the one encountered in dense ﬂuids or solidmedia. Indeed in dilute optical lattices the interactionbetween the different atoms is completely negligible;therefore, the mechanism for the propagation of atomscannot be ascribed to any sound-wave-like mechanism.While a sound wave corresponds to a propagating densitywave without a net transport of atoms, the Brillouin-like resonances analyzed in this work consist of a netmotion of the atoms. In fact, the propagation of atomsthrough the lattice are determined here by the interactionwith the light. The light ﬁelds determine both the potentialwells where the atoms can oscillate, whose vibrational fre-quency determines the velocity of the propagation modes,and the escape from the potential wells, which allows thepropagation of atoms. We also show theoretically thatthe amplitude of the Brillouin mode is a nonmonotonicfunction of the strength of the noise corresponding to theoptical pumping, and we discuss this behavior in terms ofnonconventional stochastic resonance [11,12].We consider a three dimensional (3D) lin  lin nearresonant optical lattice, as in previous work [13]. Theperiodic structure is determined by the interference of fourlinearly polarized laser beams, arranged as in Fig. 1. Thisarrangement results in a periodic modulation of the lightpolarization and light intensity, which produces a periodic4xprobezyO2θ2θ123FIG. 1. Sketch of the experimental setup.133903-1 0031-90070288(13)133903(4)$20.00 © 2002 The American Physical Society 133903-1VOLUME 88, NUMBER 13 PHYSICAL REVIEW LETTERS 1A PRIL 2002modulation of the light shifts of the different ground statesof the atoms (optical potentials) [3]. The optical pumpingbetween the different atomic ground states combined withthe spatial modulation of the light shifts leads then to thecooling of the atoms [14] and to their localization [15]at the minima of the optical potentials, thus producing aperiodic array of atoms.After the cooling phase, characterized by a signiﬁcantreduction of the atomic kinetic temperature and by thecreation of a periodic spatial order, the atoms keep inter-acting with the light undergoing optical pumping cycles.The optical pumping may transfer an atom from a po-tential well to a neighboring one, giving rise to a varietyof transport phenomena [13,16]. Among these, there aremodes which correspond to the propagation of atomsthrough the optical lattice in a given direction. They con-sist of a sequence in which a one-half oscillation in apotential well is followed by an optical pumping processto the neighboring well, and so on. One can estimatetheir velocity by yi  liVi2p, where li is the latticeconstant and Vi2p is the vibrational frequency inthe i direction [10]. These modes were ﬁrst identiﬁedthrough Monte Carlo simulations in Ref. [10] and shownto produce resonance lines in the nonlinear optical re-sponse of optical lattices. However, up to now no directobservation of these modes has been reported. This isachieved in the present work by observing a resonantbehavior of the spatial diffusion coefﬁcient in the directioncorresponding to the propagation mode with the phasevelocity of the moving intensity modulation used to excitethese propagation modes.The modulation scheme for the excitation of the propa-gation modes is completely analogous to the one used inprevious investigations of the nonlinear optical response ofoptical lattices [9]. An additional laser ﬁeld linearly polar-ized along the y axis is introduced with the z axis as thepropagation direction. This probe ﬁeld interferes with thecopropagating lattice beams, creating an intensity modula-tion. The interference pattern consists of two propagatingintensity waves moving with phase velocities  yj   njdjD kjj(j  1,2) with  nj  D kjjD kjj, and D kj   kj 2  kpis the difference between the jth lattice beam and theprobe (p) wave vectors [17]. Here d  vp 2v L isthe detuning between the probe (vp) and the lattice (vL)frequencies. According to the numerical simulations forthe atomic trajectories presented in Ref. [10], for d 6Vx, the propagation modes along x are excited by thedriving ﬁeld, with the atoms effectively dragged by themoving intensity modulation [18]. Intuitively, the drag-ging of atoms by the two propagating intensity modula-tions should result in an increase of the spatial diffusioncoefﬁcient Dx in the x direction. Therefore, it should bepossible to detect these Brillouin propagation modes bymonitoring Dx as a function of the detuning d. The propa-gation modes are then revealed by a resonance in Dxaround d  6Vx. We tested the validity of this reason-ing with the help of semiclassical Monte Carlo simula-tions [19]. Taking advantage of the symmetry between thex and y directions (see Fig. 1), we restricted the atomicdynamics in the xOz plane. Our calculations are for aJg  12 ! Je  32 transition, as is customary in thenumerical analysisof Sisyphuscooling, ofan atom ofmassM. We expect our 2D calculations to reproduce the de-pendencies of the different quantities associated with thereal 3D atomic dynamics to within a scaling factor corre-sponding to the difference in dimensionality [16]. In thenumerical simulations, we monitored the variance of theatomic position distribution at a given value of the probeﬁeld detuning. We veriﬁed that the spatial diffusion is nor-mal; i.e., the atomic square displacements Dx2 and Dz2increase linearly with time. Accordingly, we derived thespatial diffusion coefﬁcients Dx and Dz by ﬁtting the nu-merical data with Dx2i   2Dxit (xi  x,z). Results forthe spatial diffusion coefﬁcients as functions of the probedetuning d are shown in Fig. 2. Two narrow resonances,centered approximately at d  6Vx, appear clearly in thespectrum of the diffusion coefﬁcient along the x axis. Incontrast, Dz does not show any resonant behavior withthe driving ﬁeld detuning. This demonstrates the validityof the detection scheme based on the measurement of thediffusion coefﬁcients.Inthe experiment 85Rb atoms are cooled and trappedin amagneto-optical trap (MOT). The MOT laser beams andmagnetic ﬁeld are then suddenly turned off. Simultane-ously the four lattice beams are turned on and after 10 msof thermalization of the atoms in the lattice the probe laserﬁeld is introduced along the z axis. We studied the trans-port of atoms in the optical lattice by observing the atomiccloud expansion with a charge coupled device camera. Theprocedure to derive the diffusion coefﬁcients has been de-scribed in detail in previous work [13], and we recall hereonly the basic idea. For a given detuning of the probeﬁeld we took images of the expanding cloud at different   0 250 500 750−5 −4 −3 −2 −1 0 1 2 3 4 5Dx, Dzδ/Ωx2πMDx/h2πMDz/hFIG. 2. Numerical results for the spatial diffusion coefﬁcientsin the x and z directions as functions of the probe ﬁeld detuning.The lattice beam angle is u  30±, the lattice detuning D 210G, and the light shift per beam D00  2200vr. Here Gand vr are the width of the excited state and the atomic recoilfrequency, respectively. The amplitude of the probe beam is0.4 times that of each lattice beam.133903-2 133903-2VOLUME 88, NUMBER 13 PHYSICAL REVIEW LETTERS 1A PRIL 2002instants after the atoms have been loaded into the opticallattice. Fromthe images ofthe atomiccloud we derivedtheatomic mean square displacement along the x and z axes.We veriﬁed that the cloud expansion corresponds to nor-mal diffusion and derived the diffusion coefﬁcients Dx andDz. The procedure has been repeated for several differentvalues of the detuning d of the probe ﬁeld. Results for Dxand Dz as functions of d are shown in Fig. 3. The probetransmission spectrum is also reported to allow the com-parison of the position of the resonances in the spectrum ofthe diffusion coefﬁcients and in that of the probe transmis-sion. We observe two narrow resonances in the diffusioncoefﬁcient along the x axis centered at d  6Vx. In con-trast, no resonant behavior of Dz with d is observed. Thisis in agreement with the numerical simulations and consti-tutes the ﬁrst direct observation of Brillouin-like propaga-tion modes in an optical lattice.We turn now to the analysis of the mechanism be-hind these propagation modes. Brillouin-like propagation 50010001500−3 −2 −1 0 1 2 3Dx, Dzδ/Ωx2πMDx/h2πMDz/h − 5   0  5−3 −2 −1 0 1 2 3100 x (T−T0)/T0δ/ΩxFIG. 3. Experimental results for the spatial diffusion coef-ﬁcients in the x and z directions as functions of the probeﬁeld detuning. The experimental parameters are lattice de-tuning D2p  242 MHz, intensity per lattice beam IL 3.5 mWcm2, and lattice angle u  30±. These parameters cor-respond to a vibrational frequency in the x direction Vx2p55 kHz. A probe transmission spectrum is reported for com-parison, T and T0 being the intensity of the transmitted probebeam with and without the atomic cloud. The two resonancesat d  6Vx correspond to stimulated light scattering by theBrillouin propagation modes. The two resonances at larger de-tuning are Raman lines [3] in the z direction (d  6Vz), whichdo not correspond to propagation modes. For the measurementsof the diffusion coefﬁcients, the probe beam intensity is Ip 0.3 mWcm2; in the transmission spectrum Ip  0.1 mWcm2.modes have been widely studied in condensed matter anddense ﬂuids [20]. However, in the present case the mecha-nism associated with these modes is clearly of a differentnature, as in dilute optical lattices the interaction betweenatoms is negligible and therefore sound-wave-like propa-gation modes cannot be supported. On the contrary, in adilute optical lattice the propagation of the atoms is deter-mined by the synchronization of the oscillation within apotential well with the optical pumping from a well to aneighboring one, as ﬁrst identiﬁed in the numerical analy-sis of Ref. [10]. This dynamics can be interpreted in termsof noise-induced resonances: the probe ﬁeld induces alarge scale moving modulation of the periodic potentialof the four-beam optical lattice, with the optical pumpingconstituting the noise source which allows transfer from awell to a neighboring one. It is then natural to investigatethe dependence of the amplitude of the Brillouin mode onthe strength of the noise, i.e., on the optical pumping rate.We studied, via semiclassical Monte Carlo simulations, theatomic cloud expansion for a given depth of the potentialwell at different values of the optical pumping rate G00,proportional to the rate G0esc of escape from the well [21].This has been done by varying the lattice intensity I anddetuning D so as to keep the depth of the potential wellsU0 ~ ID constant while varying G00 ~ ID2. The diffu-sion coefﬁcient in the x direction has been calculated bothfor a probe ﬁeld at resonance (jdj  Vx) and for a probeﬁeld far off-resonance (jdj¿Vx). The two diffusion co-efﬁcients will be indicated by Dx and D0x, respectively.To characterize quantitatively the response of the atomicsystem to a noise strength variation, we introduce theenhancement factor j deﬁned asj Dx 2 D0xD0x. (1)Numerical results for the enhancement factor j as a func-tion of the optical pumping rate at a given value of thepotential well depth (i.e., for ﬁxed light shift per beamD00) are shown in Fig. 4. At small pumping rates, jincreases abruptly with G00; then a maximum is reached,corresponding to the synchronization of the oscillation ofthe atoms within a well with the escape from a well tothe neighboring one; ﬁnally at larger pumping rates thissynchronization is lost and j decreases. This depen-dence recalls the typical behavior of stochastic resonance[11], with the noise enhancing the response of the at-omic system to the weak moving modulation. It shouldbe noted that the system analyzed here has one impor-tant peculiarity with respect to the model usually con-sidered in the analysis of stochastic resonance. Stochasticresonance is in general understood as the noise-inducedenhancement of a weak periodic signal with a frequencymuch smaller than the intrawell relaxation frequencywithin a single metastable state. In contrast, in the presentcase, the noise synchronizes precisely with the intrawell133903-3 133903-3VOLUME 88, NUMBER 13 PHYSICAL REVIEW LETTERS 1A PRIL 20020.00.20.40.60.81.00 10 20 30 40ξΓ0’/ωrIp/IL=2.25 %Ip/IL=16 %FIG. 4. Numerical results for the enhancement factor j as afunction of the optical pumping rate, for a given depth of theoptical potential wells. Parameters for the calculations are D00 250vr and u  30±. The two data sets correspond to differentintensities of the probe beam. For comparison, we recall that inthe experiment (Fig. 3) D00  260vr and G00  8.5vr.motion of the atoms. This corresponds to a nonconven-tional stochastic resonance scenario [12].In summary, in this Letter we introduced a schemefor the detection of Brillouin propagation modes in op-tical lattices and we reported on their direct observation.Furthermore, we studied via Monte Carlo simulations theamplitude of the Brillouin mode, as characterized by anincrease of the diffusion coefﬁcients due to the presenceof the probe ﬁeld, as a function of the rate of escape fromthe potential wells. The Brillouin modes examined in thiswork differ from their counterparts in solid state or denseﬂuids as they are sustained by a medium of noninteract-ing particles. From our analysis it turns out that in thepresence of noise the Brownian motion of a system of par-ticles in a periodic potential can be turned in a motion ata well-deﬁned velocity by the application of a weak mov-ing modulation. This represents a quite unusual situationin statistical physics and may constitute a model for manybiological phenomena, like the transmission of weak sig-nals in neuronal systems [22].We thank Franck Laloë for his continued interest inour work and David Lucas for comments on the manu-script. This work was supported by CNRS, the Euro-pean Commission (TMR network “Quantum Structures,”Contract No. FMRX-CT96-0077), and by Région Ile deFrance under Contract No. E.1220. Laboratoire KastlerBrossel is a “unité mixte de recherche de l’Ecole Nor-male Supérieure et de l’Université Pierre et Marie Curieassociée au Centre National de la Recherche Scientiﬁque(CNRS).”[1] H.J. Metcalf and P. van der Straten, Laser Cooling andTrapping (Springer-Verlag, Berlin, 1999).[2] P.S. Jessen and I.H. Deutsch, Adv. At. Mol. Opt. Phys.37, 95 (1996).[3] G. Grynberg and C. Mennerat-Robilliard, Phys. Rep. 355,335 (2001).[4] R. Grimm, M. Weidemüller, and Yu.B. Ovchinnikov,Adv. At. Mol. Opt. Phys. 42, 95 (2000).[5] G. Grynberg, C. Triché, L. Guidoni, and P.M. Visser,Europhys. Lett. 51, 506 (2000).[6] M.G. Raizen, Adv. At. Mol. Opt. Phys. 41, 43 (1999).[7] C. Mennerat-Robilliard, D. Lucas, S. Guibal, J.W.R. Ta-bosa, C. Jurczak, J.-Y. Courtois, and G. Grynberg, Phys.Rev. Lett. 82, 851 (1999).[8] R. Bartussek, P. Hänggi, and J.G. Kissner, Europhys. Lett.28, 459 (1994), and references therein.[9] J.-Y. Courtois and G. Grynberg, Adv. At. Mol. Opt. Phys.36, 87 (1996).[10] J.-Y. Courtois, S. Guibal, D.R. Meacher, P. Verkerk, andG. Grynberg, Phys. Rev. Lett. 77, 40 (1996).[11] L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, Rev.Mod. Phys. 70, 223 (1998); L. Fronzoni and R. Mannella,J. Stat. Phys. 70, 501 (1993); R. Löfstedt and S.N. Cop-persmith, Phys. Rev. Lett. 72, 1947 (1994); L. Viola, E.M.Fortunato, S. Loyd, C.-H. Tseng, and D.G. Cory, ibid. 84,5466 (2000); T. Wellens and A. Buchleitner, Chem. Phys.268, 131 (2001).[12] M.I. Dykman, D.G. Luchinsky, R. Mannella, P.V.E.McClintock, N.D. Stein, and N.G. Stocks, J. Stat. Phys.70, 479 (1993).[13] F.-R. Carminati, M. Schiavoni, L. Sanchez-Palencia, F.Renzoni, and G. Grynberg, Eur. Phys. J. D 17, 249 (2001).[14] J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6,2023 (1989); P.J. Ungar, D.S. Weiss, E. Riis, and S. Chu,ibid. 6, 2058 (1989).[15] Y. Castin and J. Dalibard, Europhys. Lett. 14, 761 (1991).[16] L. Sanchez-Palencia, P. Horak, and G. Grynberg, Eur.Phys. J. D 18, 353 (2002).[17] To induce a propagating wave in a single direction aslightly more complicated setup would be needed, with twoadditional laser beams, independent of the lattice beams,creating the moving intensity modulation.[18] We recall that in this geometry the Raman excitation of theVx vibration mode is forbidden [10].[19] K.I. Petsas, G. Grynberg, and J.-Y. Courtois, Eur.Phys. J. D 6, 29 (1999).[20] Y.R. Shen, The Principles of Nonlinear Optics (Wiley-Interscience, New York, 1984); R.W. Boyd, NonlinearOptics (Academic Press, New York, 1992).[21] J.-Y. Courtois and G. Grynberg, Phys. Rev. A 46, 7060(1992).[22] I. Hidaka, D. Nozaki, and Y. Yamamoto, Phys. Rev. Lett.85, 3740 (2000).133903-4 133903-4

Brillouin propagation modes in optical lattices: Interpretation in terms of nonconventional stochastic resonance

Brillouin propagation modes in optical lattices: Interpretation in terms of nonconventional stochastic resonance

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