Inducing Transport in a Dissipation-Free Lattice with Super Bloch Oscillations
Elmar Haller, Russell Hart, Manfred J. Mark, Johann G. Danzl, Lukas Reichsöllner, Hanns-Christoph Nägerl
IInducing Transport in a Dissipation-Free Lattice with Super Bloch Oscillations
Elmar Haller, Russell Hart, Manfred J. Mark, Johann G. Danzl, Lukas Reichs¨ollner, and Hanns-Christoph N¨agerl Institut f¨ur Experimentalphysik and Zentrum f¨ur Quantenphysik,Universit¨at Innsbruck, Technikerstraße 25, 6020 Innsbruck, Austria (Dated: October 28, 2018)Particles in a perfect lattice potential perform Bloch oscillations when subject to a constant force, leadingto localization and preventing conductivity. For a weakly-interacting Bose-Einstein condensate (BEC) of Csatoms, we observe giant center-of-mass oscillations in position space with a displacement across hundreds oflattice sites when we add a periodic modulation to the force near the Bloch frequency. We study the dependenceof these “super” Bloch oscillations on lattice depth, modulation amplitude, and modulation frequency and showthat they provide a means to induce linear transport in a dissipation-free lattice. Surprisingly, we find that, for aninteracting quantum system, super Bloch oscillations strongly suppress the appearance of dynamical instabilitiesand, for our parameters, increase the phase-coherence time by more than a factor of hundred.
PACS numbers: 03.75.Dg, 03.75.Lm, 05.30.Jp, 37.10.Jk
Understanding the conduction of electrons through solidsis of fundamental concern within the physical sciences. Thesimplified situation of an electron under a constant force F within a perfect, non-dissipative, periodic lattice was origi-nally studied by Bloch and Zener [1] over 70 years ago. Theirand subsequent studies revealed that the particle would un-dergo so-called Bloch oscillations (BOs), a periodic oscil-lation in position and momentum space, thereby quenchingtransport and hence resulting in zero conductivity. BOs canbe viewed as periodic motion through the first Brillouin zone,resulting in a Bloch period T B = hk / F , where k = π / d is thelattice wave vector for a lattice spacing d . They result fromthe interference of the particle’s matter wave in the presenceof the periodic lattice structure, requiring a coherent evolutionof the wave during the time T B . Generally, it is believed thatconductance is restored via dissipative effects such as scatter-ing from lattice defects or lattice phonons [2, 3]. In bulk crys-tals, relaxation processes destroy the coherence of the systemeven before a single Bloch cycle is completed. These systemsthus exhibit conductivity but prevent the observation of BOs.To observe BOs, the BO frequency ν B = / T B must be largecompared to the rate of decoherence. In semiconductor super-lattices, where the Bloch frequency is enhanced, a few cycleshave been observed [4].A recent approach to observe and study BOs is to use sys-tems of ultracold atoms in optical lattice potentials with aforce that is provided by gravity or by acceleration of the lat-tice potential. In these engineered potentials, generated by in-terfering laser waves, dissipation is essentially absent, and de-coherence can be well-controlled [5]. Essentially all relevantsystem parameters are tunable, e.g. lattice depth and spacing,particle interaction strength, and external force, i.e. latticetilt. For sufficiently low temperatures, a well-defined, suf-ficiently narrow momentum distribution can initially be pre-pared. BOs have been observed for ultracold thermal samples[6–8], for atoms in weakly-interacting Bose-Einstein conden-sates (BECs) [5, 9, 10], and for ensembles of non-interactingquantum-degenerate fermions [11]. Non-interacting BECs[12, 13] are ideally suited to study BOs as interaction-induced dephasing effects are absent, allowing for the observation ofmore than 20000 Bloch cycles [12].As for any oscillator, classical or quantum, it is naturalthat one investigates the properties of the oscillator underforced harmonic driving. In the context of ultracold atomicgases, this is readily possible, as ν B is in the range of Hz tokHz. The dynamics of a harmonically driven Bloch oscilla-tor has recently been the subject of several theoretical [14–18]and experimental studies [19–22]. For example, modulation-enhanced tunneling between lattice sites [20, 21] and spatialbreathing of incoherent atomic samples [22] have been ob-served. Here, for a weakly-interacting atomic BEC in a tiltedlattice potential, we demonstrate that harmonic driving canlead to directed center-of-mass motion and hence to transport.In this dissipationless system we thereby recover transportand conductivity. More strikingly, for slightly off-resonantdriving, we observe giant matter-wave oscillations that ex-tend over hundreds of lattice sites. These “super Bloch os-cillations” result from a beat between the usual BOs and thedrive. They are rescaled BOs in position space and can also beused, by appropriate switching of the detuning or the phase,to engineer transport. Interestingly, forced driving leads tostrongly reduced interaction-induced dephasing and greatlyextends the time over which ordinary BOs can be observed.The experimental starting point is a tunable BEC of 1 . × Cs atoms in a crossed beam dipole trap [23] adiabaticallyloaded within 400 ms into a vertically oriented 1D opticallattice [12] as illustrated in Fig. 1(a). The lattice spacing is d = λ /
2, where λ = . ( ) nm is the wavelength of thelight. Unless stated otherwise, we work with a shallow lat-tice with depth V = . ( ) E R , where E R = h / ( m λ ) is thephoton recoil energy for particles with mass m . The atoms areinitially levitated against gravity by means of a magnetic fieldgradient and spread across approximately 50 lattice sites withan average density near 5 × cm − in the central region ofthe sample. We control the strength of the interaction as mea-sured by the s-wave scattering length a near a Feshbach reso-nance [23]. Throughout this work, unless stated otherwise, wework at a = ( ) a , where a is Bohr’s radius. We initiate a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n (a) lattice beam magnetic levitationgravity x yz ν ( ν B ) 0.1 0.2 0.40.3 0.6 2.0 w i d t h W ( µ m ) (b) d i p o l e t r a p atoms FIG. 1. (color online) Experimental setup (a) and excitation spectrum(b) for atoms in a tilted periodic potential. The width W is plotted asa function of the drive frequency ν . The resonances correspond to adrastic spreading of the atomic wave packet as a result of modulation-assisted tunneling [21] when ν ≈ i / j × ν B , where i , i are integers. Theparameters are F = . ( ) mg , ∆ F = . ( ) mg , V = . ( ) E R ,and τ = BOs by removing the dipole trap confinement in the verticaldirection and by reducing the levitation in 1 ms to cause aforce that is a small fraction of the gravitational force mg , forwhich ν B is near 100 Hz. An additional harmonic modulationof the levitation gradient then results in an oscillating drivingforce F ( t ) = F + ∆ F sin ( πν t + φ ) , where F is the constantforce offset, ∆ F is the amplitude of the modulation, ν is themodulation frequency, and φ is a phase difference between theBOs and the drive. After a given hold time τ we switch off alloptical beams and magnetic fields and take in-situ absorptionimages after a short delay time of 800 µ s.We first determine the excitation spectrum. Fig. 1(b) showsthe 1 / √ e -width W of the matter wave after τ = ν . A series of narrow resonances at rational multi-ples of ν B can clearly be identified. In agreement with re-cent experiments [20, 21], we attribute these resonances tomodulation-enhanced tunneling between lattice sites, leadingto dramatic spreading of the atomic wave packet. Tunnelingbetween nearest neighbor lattice sites is enhanced when ν B isan integer multiple j of ν via a j -phonon process [24], whiletunneling between lattice sites i lattice units apart is enhancedwhen ν is an integer multiple i of ν B . Even combinationsthereof, e.g. i / j = / /
5, are detectable.We now investigate the dynamics of the wave packet inmore detail. For this, we use the resonance with i = j = ν = ν B + ∆ ν , where ∆ ν is the detuning. In Fig. 2(a)-(d) we present absorption images and spatial profiles for theweakly-interacting BEC. The time evolution for the width,shape, and center position of the BEC is dramatic. On res-onance ( ∆ ν = φ . Offresonance, (a) and (b), for small detuning ∆ ν = − time τ (T B ) c e n t e r - o f - m a ss p o s i t i o n ( d ) (e) -200 0 200 400 d e n s i t y ( a r b . ) -400 0 400 position (d) position (d) t i m e force (a) (b) (c) (d) t i m e d e n s i t y ( a r b . ) FIG. 2. (color online) Observation of super Bloch oscillations andmodulation-driven wave packet spreading. (a) and (b) In-situ ab-sorption images and density profiles for off-resonant modulation( ∆ ν = − ∆ ν = F = . ( ) mg , ∆ F = . ( ) mg , V = . ( ) E R , a = ( ) a . (e) Center-of-mass motion for a = ( ) a (squares), a = ( ) a (diamonds), a = ( ) a (squares). for ordinary BOs corresponds to about 4 d = . µ m. Alsothe width and higher moments of the distribution show oscil-latory behavior. In Fig. 2(e) we plot the center-of-mass posi-tion as a function of time for ∆ ν = − a = ( ) a we typically observe sBOs over the course of several seconds.The dynamics of sBOs strongly depends upon the site-to-sitephase evolution of the matter-wave. In fact, stronger inter-actions, e.g. a = ( ) a , distort the density profile of thedriven BEC and alter the BEC’s oscillation frequency and am-plitude. For sufficiently strong interactions, no sBOs are ob-served. We also attribute the wave-packet spreading as seenafter one cycle in Fig. 2(b) mostly to interactions. For themeasurements above, we intentionally use a large modulationamplitude ∆ F to enhance the amplitude of sBOs. However, alleffects equally exist for ∆ F (cid:28) F , as we will also demonstratebelow in Fig. 4(b).It is useful to develop a simple semi-classical model to ob-tain a qualitative understanding of the origin of sBOs. Theonly elements of this model are that the wave packet is ac-celerated by the applied force and that, once the wave packet (e) -150-100-50050100150 p o s i t i o n ( d ) time (T B ) -101 time (T B ) m o m e n t u m ( ħ k / m ) (a) time (T B ) -101 (c) (b) -1000100 time (T B ) (d) p o s i t i o n ( d ) time (T B ) (i)(ii) (iii) (i)(ii) FIG. 3. (color online) Results from a semi-classical model for sBOs.(a) For a constant force, here F = . mg , the velocity (in units of¯ hk / m ) exhibits a symmetric, saw-tooth-like time evolution, typicalfor BOs. (b) Resonant modulation, here with ∆ F = . F , altersthe symmetric periodic velocity excursions of normal BOs ( φ = φ = π , dashed line), leading to a net-movement, (c), with φ = φ = π / φ = π (iii). An additional detuning ∆ ν = ± . ν B results in a periodically changing phase difference andhence in giant oscillations in position space, (i) and (ii) in (d). Ontop of the motion, normal BOs can clearly be seen. The phase ofsBOs depends on the sign of ∆ ν , as shown by experimental data in(e), where F = . ( ) mg , ∆ F = . ( ) mg , ∆ ν = − reaches the edge of the first Brillouin zone, it is Bragg re-flected. This model does not include an effective mass andcannot be used to predict quantitative results. Fig. 3(a)-(d) shows the result of a numerical integration of the time-dependent acceleration a ( t ) = F / m + ∆ F / m sin ( π ( ν B + ∆ ν ) t + φ ) with periodic Bragg reflection. For a constant ac-celeration ∆ F =
0, the wave packet’s velocity shows the well-known saw-tooth-like time evolution that corresponds to BOs.The curve in (a) is symmetric, hence, there is no net move-ment, as indicated by the shaded regions of equal area. If,however, there is additional harmonic modulation at ν = ν B ,the velocity excursions will not be symmetric about zero, (b),and result in a net movement for each period, leading to lin-ear motion, (c). Only for φ = π / φ = π / φ . Off-resonant modulation with ∆ ν (cid:28) ν B induces a slowly-varying phase mismatch between the drive and the originalBloch period. This results in a slow oscillation of the netmovement for each Bloch cycle, which finally sums up to agiant oscillation in position space, (d). Evidently, this oscilla-tion is the result of a beat between the drive and the originalBO. The initial direction of the motion depends on φ and ∆ ν . time (ms) p o s i t i o n ( µ m ) detuning (Hz) f r e q u e n c y ( H z ) p o s i t i o n ( µ m ) p o s i t i o n ( µ m ) time (ms) a m p t li t u d e ( µ m ) p o s i t i o n ( µ m ) time (ms) lattice depth (E R ) a m p t li t u d e ( µ m ) (a)(b)(c) modultion amptlitude ( ∆ F/F ) a m p li t u d e ( µ m ) FIG. 4. (color online) Quantitative analysis of sBOs. (a) The effectof the detuning ∆ ν on the oscillation frequency and the amplitude ofsBOs, with ∆ ν = . ∆ ν − -dependence, re-spectively. (b) Dependence of the amplitude of sBOs on ∆ F / F . Thedata sets correspond to ∆ F / F = .
52 (circles), 0 .
76 (squares), 0 . .
08 (stars). Right: The solid line is a fit proportionalto B ( ∆ F / F ) . (c) Amplitude of sBOs as a function of lattice depth,V = E R (circles), 4 E R (squares), 5 E R (diamonds), 7 E R (stars).If not stated otherwise, the parameters for all measurements shownhere are F = . ( ) mg , ∆ F = . ( ) mg , ∆ ν = − In particular, a change in the sign of ∆ ν at a given φ can lead toopposite motion in position space, as verified experimentallyin Fig. 3(e) for ∆ ν = ± J → J eff = JB ( ∆ F / F ) and the force F → F eff = h ∆ ν / d for a stationary lattice with tilt. Here, B is the first Bessel function of the first kind. The ampli-tude of sBOs is thus given by a new Wannier-Stark localiza-tion length L eff = J eff / ( dF eff ) [18]. In this sense, sBOs arerescaled BOs. We quantitatively study the dependence of am-plitude and period of sBOs on ∆ ν , ∆ F / F , and V . The resultsare shown in Fig. 4. As expected, the period T is given by1 / ∆ ν . Also, the oscillation amplitude scales as 1 / ∆ ν , andits Bessel-function dependence on ∆ F / F is well reproduced.Given our spatial resolution, we can observe sBOs down to ∆ F / F = .
08 (Fig. 4(b)). Note that sBOs can only be ob-served with sufficient wave function coherence and for well-defined initial conditions, i.e. for sufficient wave packet local-ization in the first Brillouin zone of the lattice. Nevertheless,incoherent atomic samples exhibit a breathing of the spatialdistribution [22] as the oscillation period is insensitive to the (c) µ m30 µ m (a) B ) p o s i t i o n ( d ) switch detuning (b) time µ m30 µ m (d)(e)(f) ћ k FIG. 5. (color online) Inducing transport and suppressing inter-action-induced dephasing. (a) Linear motion for resonant modula-tion. ∆ φ = ◦ diamonds, 65 ◦ circles, 120 ◦ triangles, 190 ◦ squares. ∆ φ = ◦ and ∆ φ = ◦ were chosen to maximize the speed in op-posite directions. The solid lines are linear fits to the data pointsexcluding the first data point. (b) Directed motion for off-resonantmodulation. ∆ ν was switched from − F = . ( ) mg , ∆ F = . ( ) mg . (d)-(f) BOs in quasi-momentum space after 10BO cycles with no drive (d), after 17 BO cycles with drive (e), after750 BO cycles with drive (f). The parameters are F = . ( ) mg , ∆ F = . ( ) mg , ∆ ν = − initial conditions. In the work of Ref.[22], the breathing canbe understood in terms of an incoherent sum over localizedWannier-Stark states that individually show a breathing mo-tion with period T [17].The results above provide two mechanisms to circumventthe localization inherent in BOs and to induce coherent trans-port in an otherwise insulating context. As shown in Fig. 5(a),resonant modulation ( ∆ ν =
0) causes directed motion of thewave packet’s center-of-mass. For longer times, we find thatthe motion is approximately linear. The mean velocity de-pends on the relative phase φ of the Bloch oscillator and thedrive. In the experiment, we varied φ via φ = φ + ∆ φ , where φ is a constant phase offset, which depends on the detailshow BOs are initiated. For off-resonant modulation, transportcan be induced by switching the sign of ∆ ν before a half-cycle of a sBO is completed. The wave packet then continuesto move in the original direction. This motion is shown inFig. 5(b), where we switch the sign after 400 ms. For compar-ison, Fig. 5(c) shows a sBO with T = ∆ ν from the Blochfrequency is introduced. Localization as a result of BOs isbroken, allowing us to engineer matter wave transport overmacroscopic distances in lattice potentials with high relevanceto atom interferometry [26]. We are now in a position to inves-tigate the effect of interactions on driven transport, for whichsubdiffusive and chaotic dynamics have been proposed [27].During the final preparation of the manuscript we becameaware of related work on non-dissipative transport in a quan-tum ratchet [28]. We thank A. R. Kolovsky, A. Zenesini,and A. Wacker for discussions and R. Grimm for generoussupport. We gratefully acknowledge funding by the AustrianMinistry of Science and Research and the Austrian ScienceFund and by the European Union within the framework of theEuroQUASAR collective research project QuDeGPM. R.H.is supported by a Marie Curie International Incoming Fellow-ship within FP7. [1] F. Bloch, Z. Phys. , 555 (1928); C. Zener, Proc. R. Soc. Lond.A , 523 (1934).[2] Y. Kanemitsu, T. Ogawa, Optical Properties of Low-Dimensional Materials (World Scientific, Singapore 1995)[3] N.W. Ashcroft, N.D. Mermin,
Solid State Physics (SaundersCollege, Philadelphia, 1976)[4] K. Leo et al. , Solid State Comm. , 943 (1992); J. Feldmann et al. , Phys. Rev. B , 7252 (1992).[5] M. Gustavsson et al. , arXiv:0812.4836 (2008).[6] M. Ben Dahan, E. Peik, J. Reichel, Y. Castin, and C. Salomon,Phys. Rev. Lett. , 4508 (1996).[7] R. Battesti et al. , Phys. Rev. Lett. , 253001 (2004).[8] G. Ferrari, N. Poli, F. Sorrentino, and G. M. Tino, Phys. Rev.Lett. , 060402 (2006).[9] B. P. Anderson and M. A. Kasevich, Science , 1686 (1998).[10] O. Morsch, J.H. M¨uller, M. Cristiani, D. Ciampini, and E. Ari-mondo, Phys. Rev. Lett. , 140402 (2001).[11] G. Roati et al. , Phys. Rev. Lett. , 230402 (2004).[12] M. Gustavsson et al. , Phys. Rev. Lett. , 080404 (2008).[13] M. Fattori et al. , Phys. Rev. Lett. , 080405 (2008).[14] H.J. Korsch and S. Mossmann, Phys. Lett. A , 54 (2003).[15] T. Hartmann, F. Keck, H.J. Korsch, and S. Mossmann, New J.Phys. (2004).[16] Q. Thommen, J. C. Garreau, and V. Zehnl´e, Phys. Rev. A ,053406 (2002).[17] Q. Thommen, J.C. Garreau, and V. Zehnl´e, J. Opt. B: QuantumSemiclass. Opt. , 301 (2004).[18] A. Kolovsky and H.J. Korsch, arXiv0912.2587 (2009).[19] S.R. Wilkinson, C.F. Bharucha, K.W. Madison, Q. Niu, and M.G. Raizen, Phys. Rev. Lett. , 4512 (1996). [20] V. V. Ivanov et al. , Phys. Rev. Lett. , 043602 (2008).[21] C. Sias et al. , Phys. Rev. Lett. , 040404 (2008).[22] A. Alberti, V.V. Ivanov, G.M. Tino, and G. Ferrari, Nature Phys. , 547 (2009).[23] T. Kraemer et al. , Appl. Phys. B , 1013 (2004).[24] A. Eckardt, T. Jinasundera, C. Weiss, and M. Holthaus, Phys. Rev. Lett. , 200401 (2005).[25] B. Wu, Q. Niu, New J. Phys. , 104 (2003).[26] A.D. Cronin, J. Schmiedmayer, and D.E. Pritchard, Rev. Mod.Phys. , 1051 (2009).[27] A.R. Kolovsky, E.A. G´omez, and H.-J.Korsch, arXiv0904.4549(2009).[28] T. Salger et al. , Science326