Klein-Tunneling of a Quasirelativistic Bose-Einstein Condensate in an Optical Lattice
Tobias Salger, Sebastian Kling, Christopher Grossert, Martin Weitz
aa r X i v : . [ qu a n t - ph ] A ug Klein-Tunneling of a Quasirelativistic Bose-Einstein Condensate in an Optical Lattice
Tobias Salger, ∗ Christopher Grossert, † Sebastian Kling, and Martin Weitz
Institut f¨ur Angewandte Physik der Universit¨at Bonn, Wegelerstr. 8, 53115 Bonn (Dated: June 15, 2018)Optical lattices have proven to be powerful systems for quantum simulations of solid state physicseffects. Here we report a proof-of-principle experiment simulating effects predicted by relativisticwave equations with ultracold atoms in a bichromatic optical lattice that allows for a tailoring of thedispersion relation. We observe the analog of Klein-tunneling, the penetration of relativistic particlesthrough a potential barrier without the exponential damping that is characteristic for nonrelativisticquantum tunneling. Both linear (relativistic) and quadratic (nonrelativistic) dispersion relations areinvestigated, and significant barrier transmission is observed only for the relativistic case.
Klein tunneling, a textbook effect in which relativis-tic particles penetrate through a potential barrier with-out the exponential damping that is characteristic fornonrelativistic quantum tunnelling [1, 2], has never beenobserved for elementary particles. In this counterintu-itive consequence of relativistic quantum mechanics, astrong potential, being repulsive for particles and attrac-tive for antiparticles, results in particle- and antiparticle-like states aligning in energy across the barrier. There-fore, a high transmission probability is expected whena potential drop of the order of the particles rest energy mc is achieved over the Compton length h/mc . For elec-trons, one derives an extremely high required electric fieldstrength of ≈ V/cm, which so far has prevented anexperimental realization on this system. The observationof Klein tunnelling has however been reported in solidstate analogons [3–6], for example in graphene material.In this carbon material owing to a linear, i.e. quasirel-ativistic, dispersion relation around the Fermi edge, rel-ativistic effects can be very illustratively emulated [3].Other systems suitable for the simulation of quasirela-tivistic effects are ions with a long-lived two-componentelectronic structure in Paul traps [7], where the nonrela-tivistically forbidden entry into a high potential well hasbeen observed [8]. Experiments in photonic structures[9] and in dark state media [10, 11] have been proposed.Ultracold atoms in optical lattices [12] allow for the inves-tigation of both linear and nonlinear Hamiltonians dueto the neutral charge of the atoms, with prospects in-cluding the simulation of interacting relativistic quantumfield theories [13].Here we report a proof-of-principle quantum simula-tion of relativistic wave equation predictions with ultra-cold atoms in an optical lattice. Our experiment is basedon rubidium atoms in a Fourier-synthesized lattice poten-tial consisting of an optical standing wave with spatialperiodicity λ/
2, where λ denotes the laser wavelength,and a higher spatial harmonic with λ/ ϕ = 0 ϕ = π quasimomentum q ( ¯ hk ) PSfrag replacements ene r g y E n q ( E r ) rescaled energy E ( E r ) ∆E = eff c -1 0 1 -4 0 4E>0E<0 PSfrag replacements energy E nq ( E r ) r e sc a l edene r g y E ( E r ) ∆E = eff c FIG. 1. Band structure for an optical lattice potential for arelative phase of ϕ = 0 (left) and ϕ = π (right). The parame-ters for the lattice depths were V = 5 E r and V = 1 . E r . For ϕ = π the splitting between the first and the second excitedband vanishes, and a Dirac-point (marked by the solid bluebox) is observed. The bands relevant for the experiment areshown by solid lines, and on the rescaled energy scale shownon the right hand side, zero energy is chosen at the positionof the band crossing. of atoms through a potential barrier for the case of alinear dispersion relation, i.e. Klein-tunneling, and theusual reflection of atoms by the barrier for the case of aquadratic, i.e. Schr¨odinger-like, dispersion in an excitedBloch band.The periodic potential used to taylor the dispersionof ultracold rubidium atoms is of the form V ( z ) = V / kz ) + V / kz + ϕ ), where k = 2 π/λ isthe photon wavevector, V denotes the potential depthof the usual standing wave potential and V that of thehigher spatial harmonic, generated by the dispersion ofmultiphoton Raman transitions [14, 15]. Fig. 1 showsthe band structure for such a Fourier-synthesized lat-tice for V = 5 E r and V = 1 . E r used in our exper-iment, where E r = ~ k / m denotes the recoil energy,for two different values of the relative phase ϕ betweenlattice harmonics. For the shown parameters, while thesplitting between the first and the second excited Blochband exhibits a nonzero value for ϕ = 0, it vanishes fora phase of ϕ = π . The critical dependence on the rel-ative phase between lattice harmonics is understood interms of the splitting arising from both contributions ofsecond order Bragg scattering of the usual lattice andof first order Bragg scattering of the higher spatial har-monic, where the contributions interfere constructivelyor destructively depending on ϕ [15]. Of special inter-est is the case of destructive interference of the Bragg-scattering amplitudes, realized with ϕ = π (see Fig. 1right), for which at a suitable choice of lattice ampli-tudes the dispersion relation near the resulting crossingpoint becomes linear, i.e. relativistic, with an effectivelight speed c eff = 2 ~ k/m ≃ . λ c,eff = h/m eff c eff = 2 c eff h/ ∆ E ,with m eff = ∆ E/ c eff as the effective mass and ∆ E asthe size of the splitting, is possible by appropriate choiceof amplitude and phase values of the lattice harmonics.In the limit of ∆ E → H = m eff c eff σ z + c eff ˆ qσ x + V slow ( z ) , (1)where σ x and σ z are Pauli matrices, ˆ q = − i ~ δ z is themomentum operator and V slow ( z ) is an external poten-tial varying much slower than the lattice periodicity.The two-component Hamiltonian acts on spinors ψ =( ψ , ψ ), with ψ and ψ corresponding to course-grainatomic wave-functions in the upper and lower bands,respectively. Equation 1 becomes exact in the limit | q | ≪ ~ k and m eff c eff ≪ E r . One of the hallmark-effectsof a relativistic dispersion is Klein-tunneling, which is asingle-particle effect, so it can be equally well observedfor bosons (as used in our experiment) and fermions [17].In our experiment Klein-tunneling is investigated bymonitoring the transmission through an external poten-tial barrier that stands against the outcoupling of atomsfrom a far detuned optical dipole potential of depth V by the earth’s gravitational acceleration g . The spa-tial distribution of the combined potential V slow ( z ) = − V exp (cid:16) − z/ω ) (cid:17) − mgz is shown in Fig. 2c, wherethe height V b of the potential barrier relatively to theminimum of the trapping potential can be adjusted byvariation of the dipole trap beam power. The width of (a)(b)(c) p V b PSfrag replacements position z po t en t i a l V ( z ) p q V b E p
PSfrag replacements position zpotential V(z) pE PSfrag replacements position zpotential V(z)
FIG. 2. Klein-tunneling of atoms through a potential barrier.(a) Relevant part of the dispersion relation in the lattice for arelative phase between lattice harmonics of ϕ = 0, for whichthe large splitting suppresses a tunneling between bands. (b)For ϕ = π the dispersion relation is linear and the two Blochbands touch each other. Shown in the three diagrams is thevariation of the atomic energy during passage of a potentialbarrier of height V b . (c) Spatial distribution of the combinedpotential formed by gravitational and dipole trapping poten-tial. Atoms can pass the potential well in the case of ϕ = π due to their possibility to drop to below the band crossingin the Bloch spectrum when loosing potential energy, see themiddle graph in (b). the potential barrier is of the same order as the usedbeam diameter 2 ω ≃ µ m. For a typical atomic en-ergy of one recoil energy below the maximum of the po-tential barrier, the estimated probability for usual non-relativistic quantum tunneling is of the order P nr =exp (cid:0) − √ mE r z/ ~ (cid:1) ≃ − , i.e. completely negligi-ble.The situation however changes when quasirelativisticKlein-tunneling occurs, because the tunneling rate forthis process does not decay exponentially with the widthof the potential barrier. A quasirelativistic dispersion re-lation for ultracold rubidium atoms is induced using theFourier-synthesized optical lattice, and Figs. 2a and 2b(left) indicate the relevant part of the atomic dispersionrelation near the crossing between the first and the sec-ond excited Bloch-band for a relative phase of ϕ = 0 and ϕ = π . The atoms are loaded at a quasimomentum q well above the crossing region, but when proceeding to-wards the potential well on its rising edge loose kineticenergy, i.e. their momentum reduces. For a phase ϕ = 0,the splitting between the first and second excited Bloch-band is comparatively large (see Fig.2a). This resultsin a small effective Compton wavelength, λ c,eff ≈ µ m,which is below the length of the rising edge of the bar-rier, and we expect no Landau-Zener tunneling into thelower band. The dispersion relation for atoms in the up-per band then is Schr¨odinger-like. When the height ofthe potential barrier is larger than q / m eff , the particlecannot pass through the barrier.On the other hand, for a relative phase ϕ = π betweenlattice harmonics, cf. Fig. 2b, the dispersion relation inthe vicinity of the crossing point between the first andthe second excited Bloch-band becomes ultrarelativis-tic. The effective Compton wavelength becomes largerthan the widths of the edges of the barrier, and particlesthat approach the potential barrier and loose kinetic en-ergy on the rising edge can be accelerated to below thecrossing point between the second and the first excitedBloch-band (the Dirac-point), i.e. to states of negativeenergy of the energy scale shown in Fig.2c. Correspond-ingly, they can surpass higher potential barriers than inthe nonrelativistic case of Fig. 2a. This corresponds tothe case of Klein-tunneling. Our experiment simulatesthe conversion of a particle into a spatially backwardspropagating antiparticle during the transmission of thebarrier, whereafter the crossing point of bands again ispassed on the tailing edge, so that particle-like states areagain observed beyond the well. As in the case of theKlein-tunneling of electrons, this is equivalent to a dou-ble Landau-Zener passage between states of positive andnegative energy respectively [2, 3].The experiment proceeds by initially producing a Bose-Einstein condensate of rubidium atoms in an m F = 0spin projection of the F = 1 ground state within a dipoletrapping potential formed by a focused CO -laser beam.To prepare atoms in the second excited Bloch-band, weinitially leave the atoms in ballistic free fall by extin-guishing the CO -laser dipole potential until the atomshave reached a momentum of 0.9 ~ k by the earths gravi-tational force, and then apply a Doppler-sensitive Ramanpulse transferring atoms to the m F = − . ~ k . Thisinitial momentum allows to load atoms into the secondexcited Bloch-band of the lattice potential (at the po-sition q = 0 . ~ k ), and the CO -laser dipole potentialrequired for the shaping of the desired slowly spatiallyvarying potential barrier is again activated. The ballisticfree fall during momentum preparation is about 2 µ m, i.e.well below the focal diameter of the trapping beam.Fig.3a shows typical experimental data for the spa-tial atomic distribution, as recorded by absorption imag-ing at a time 5ms after preparation, for a relative phase ϕ = 0 and ϕ = π between lattice harmonics respec-tively. The height of the potential barrier was V b =5 E r , while the average initial energy of the particle was E = E kin + m eff c eff = 3 . E r , with m eff c eff of 0 . E r and0 for ϕ = 0 and ϕ = π , respectively (see also Fig. 1a).The insets are false-colour shadow images of the atomic f r a c t i ono f a t o m s beh i ndba rr i e r effective Compton wavelength λ c,eff ( µm ) (a) (b) ϕ = 0 : ”nonrelativistic” PSfrag replacements π/ π position z ( µm )phase shift ϕ (rad) ∞ PSfrag replacements π/ π position z ( µm )phase shift ϕ (rad) ∞ ϕ = π : ”relativistic” −50 0 50 100 150 PSfrag replacements π/ π position z ( µm )phase shift ϕ (rad) ∞ PSfrag replacements π/ π position z ( µm )phase shift ϕ (rad) ∞ position z ( µm ) . PSfrag replacements π/ π position z ( µm ) phase shift ϕ (rad) ∞ op t i c a l c o l u m nden s i t y ( a . u . ) FIG. 3. (a) Cuts (solid red) through the measured spa-tial atomic distribution (insets) for a 5ms experiment time.Atoms proceeding from the center of the dipole trapping po-tential towards the potential barrier that detains against anoutcoupling by the earths gravitational field for a relativephase between lattice harmonics of (top) ϕ = 0 and (bottom) ϕ = π . The dotted black line indicates the potential V slow ( z ).(b) Relative atomic population beyond the potential barrierversus phase ϕ . The corresponding effective Compton wave-length λ c, eff = 2 c eff h/ ∆ E is shown on the top scale. Theshown horizontal error bars refer to the latter scale and aredominated by the uncertainty in ∆ E , while the estimated un-certainty for the phase ϕ (lower scale) is below the drawingsize of the dots. The solid line is the result of a numerical in-tegration of the relativistic wave equation (see eq. (1)). Theonly free fit parameters were amplitude and offset. cloud, with the spot on the left-hand side correspondingto atoms near the trap location (i.e. before the barrier)and atoms on the right-hand side to particles that havetransmitted the barrier. The solid red lines in the mainimages are cuts through the center, with the external po-tential indicated by dotted black lines. The data showsthat for a relative phase of ϕ = 0, almost all atoms re-main in front of the barrier, as expected. On the otherhand, for ϕ = π , our experimental data show that mostof the atomic population can be found beyond the poten-tial barrier, equivalent to Klein-tunneling in this opticallattice system.Note that for Klein-tunneling perfect transmissionthrough the barrier is expected, when the energy split-ting ∆ E would be zero. In our experiment, some 30% ofthe population remains in front of the barrier, which wemainly attribute to atoms that did not take part in theDoppler-sensitive Raman transfer, and remain trapped inthe CO -laser beam focus. We have measured the atomicpopulation found beyond the barrier for variable valuesof the phase ϕ between lattice harmonics, to investigatethe case of intermediate values of the splitting ∆ E be-tween Bloch-bands. Corresponding data is displayed bythe dots in Fig. 3b. The effective Compton wavelengthbecomes sufficiently large for Klein-tunneling only in anarrow region near ϕ = π , corresponding to the ultra-relativistic case, while the atoms remain in front of the PSfrag replacements barrier height V b ( E r ) f r a c t i ono f un r e fl e c t eda t o m s ϕ = πϕ = 0 FIG. 4. Fraction of unreflected atoms versus the height ofthe potential barrier V b for a phase shift (solid red) ϕ = 0and (dashed blue) ϕ = π . The solid lines show numericalsimulations. The grey shaded region corresponds to barrierheight values where the atomic velocity reaches the edge ofthe Brillouin zone at | q | = ~ k . The desired dispersion rela-tion is reached only in the white region. The experimentalparameters are the same as in Fig. 3a except for the barrierheight. barrier for smaller phase values. The solid line is theresult of a numerical simulation of the relativistic waveequation (see eq.1), in good agreement with the experi-mental data.A striking prediction for Klein-tunneling is that thetransmission through the barrier is expected to be inde-pendent of the barrier height, an issue in clear contrastto the expectations for nonrelativistic quantum mechan-ics. For a corresponding measurement in our system amomentum resolved time-of-flight measurement was em-ployed to allow for a Stern-Gerlach separation of atomsthat did not take part in the Raman transfer (see Supple-mentary Material). Fig.4 shows the relative signal of un-reflected atoms, corresponding to Klein-tunneled atoms,versus the barrier height V b . The grey shaded regioncorresponds to barrier height values where the atomicvelocity reaches the edge of the lattice Brillouin zone at | q | = ~ k , for which the bottom of the first exited bandcan be reached, i.e. only in the left white region the de-sired dispersion relation is achieved, with larger accuracywhen remaining far from the shaded region. The bluesquares are data recorded for ϕ = π , corresponding to aquasirelativistic dispersion, for which this signal remainsat a high value within nearly the complete white region,illustrating the prediction of Klein-tunneling being inde-pendent of the barrier height with good accuracy. Onthe other hand, a pronounced loss of this signal at largebarrier heights is observed for ϕ = 0 (red dots), corre-sponding to a nonrelativistic dispersion. Quantum tun-nelling remains negligible due to the large spatial width of the barrier. The finite width of the kinetic atomicenergy distribution here softens the otherwise expectedsharp decay for E kin < V b . The shown solid and dashedlines are the result of a numerical simulation, which forthe relativistic case are in good agreement with the dataand also qualitatively reproduce the nonrelativistic case.To conclude, we report an experiment demonstratingthe analog of Klein-tunneling, as a proof-of-principle ex-periment testing relativistic wave equation predictions,with ultracold atoms in a bichromatic optical lattice. Bytuning the relative phase between lattice harmonics theatomic dispersion can be tuned continuously from thenonrelativistic to the ultrarelativistic case, though theatoms move at a velocity 10 orders of magnitude belowthe speed of light.For the future, we expect that ultracold atoms in opti-cal lattices allow quantum simulations of a wide range ofeffects of both linear and nonlinear Dirac-dynamics. Per-spectives include the verification of theoretical high en-ergy physics predictions [18], as chiral confinement, andother results of massive Thirring models [13, 19, 20].Financial support of the DFG is acknowledged. Wethank A. Rosch, L. Santos, D. Witthaut, H. Kroha, andK. Ziegler for discussions. ∗ [email protected] † [email protected][1] 0.Klein, Z.Phys , 157 (1929).[2] See, e.g.: J. Bjorken, S. Drell, Relativistic Quantum Me-chanics (Mc Graw-Hill, New York, 1964).[3] M.I. Katsnelson, K.S. Novoselov, A.K. Geim, NaturePhys. , 620 (2006).[4] A.F. Young, P. Kim, Nature Phys. , 222 (2009).[5] N. Stander, B. Huard, D. Goldhaber-Gordon, Phys. Rev.Lett , 026807 (2009).[6] G.A. Steele, G. Gotz, L.P. Kouwenhoven, Nature Nan-otechnology , 363 (2009).[7] R. Gerritsma et al., Nature , 68 (2010).[8] R. Gerritsma et al., Phys. Rev. Lett , 060503 (2011).[9] S. Longhi, Phys. Rev. B , 075102 (2010).[10] G. Juzeliunas et al. Phys. Rev. A , 011802 (2008).[11] J.Y. Vaishnav, C.W. Clark, Phys. Rev. Lett , 153002(2008).[12] I. Bloch, J. Dalibard, W. Zwerger, Rev. Mod. Phys. ,885 (2008).[13] J.I. Cirac, P. Maraner, J.K. Pachos, Phys. Rev. Lett. , 190403 (2010).[14] G. Ritt et al., Phys. Rev. A , 063622 (2006).[15] T. Salger, C. Geckeler, S. Kling, and M. Weitz, Phys.Rev. Lett. , 190405 (2007).[16] D. Witthaut et al., Phys. Rev. A (in press).[17] H. Feshbach, F. Villars, Rev. Mod. Phys. , 24 (1958).[18] M. Merkl et al., Phys. Rev. Lett. , 073603 (2010).[19] S.-J. Chang, S.D. Ellis, B.W. Lee, Phys. Rev. D , 3572(1975).[20] S.Y. Lee, T.K. Kuo, A. Gavrielides, Phys. Rev. D ,2249 (1975). Supplementary Material
EXPERIMENTAL APPROACH AND SETUP
The experiment is based on a Bose-Einstein condensate of rubidium atoms ( Rb) in a Fourier-synthesized opticallattice used to taylor the atomic dispersion relation. Rubidium atoms are cooled to quantum degeneracy evapora-tively in an optical dipole trap realized by focusing a beam of mid-infrared radiation derived from a CO -laser withwavelength near 10.6 µ m. During the final stages of evaporation, a magnetic gradient field is switched on, resultingin a spin-polarized Bose-Einstein condensate with about 6 · atoms in the m F = 0 Zeeman sublevel of the F = 1hyperfine ground state component [S1]. (a) (b) Fig. S 5. (a) Left: Two-photon processes in a conventional optical standing wave lattice, which are responsible for a periodicpotential of λ/ λ/ λ/ ω and two counterpropagating optical fields of frequencies ω − ∆ ω and ω + ∆ ω . The Fourier-synthesized optical lattice potential has contributions of two lattice harmonics with spatial periodicities λ/ λ/ λ denotes the wavelength of the driving laser, tuned 3 nm to the red of the rubidiumD2-line. For the fundamental spatial frequency of λ/ λ/ λ/ | g − i and | g + i over an excited state | e i here allows to clearly separate the desired four-photon processfrom lower order contributions in frequency space. A set of three different optical frequencies is used to drive thetransitions, with two copropagation laser fields of frequencies ω − ∆ ω and ω +∆ ω respectively and a counterpropagatingbeam with frequency ω .We use the rubidium F = 1 electronic ground state Zeeman components m F = − | g − i and | g + i ,and the 5P / excited state manifold as state | e i . A magnetic bias field of 1.8 G removes the degeneracy of the Zeemansublevels. By combining lattice potentials of λ/ λ/ E between the first two excited Bloch bands of the variably shaped optical lattice can be controlled isroughly E r / -laser trapping beam for about 0.5 ms, after which the atomshave accelerated to a momentum of 0.9 ~ k due to the earths gravitational field, and then apply a Doppler-sensitiveRaman pulse of 40 µ s length transferring atoms from the m F = 0 to the m F = − ~ k , which completes the velocitypreparation sequence. The Raman transfer selects a typical Doppler width of ± . ~ k/m width from the initial atomicdistribution obtained after expansion of the interacting condensate atoms. Atoms are then loaded adiabatically intothe second excited Bloch-band of the Fourier-synthesized optical lattice potential. Further, the far detuned CO -laserbeam required for the shaping of the desired slowly varying potential V slow ( z ) = − V exp (cid:16) − z/ω ) (cid:17) − mgz (2)is reactivated, which imposes a potential barrier for the vertically downwards propagating atoms. SPATIAL AND MOMENTUM-RESOLVED MEASUREMENTS OF ULTRACOLD ATOMIC ENSEMBLES
During the course of the experiment, two types of measurements were used for an analysis of the atomic final state.In a first series of experiments (see Figs. 3(a) and 3(b) for results), the spatial atomic distribution was recordedby means of an absorption image recorded after a 5ms long experiment time of both CO -laser dipole potentialand the Fourier-synthesized lattice potential interacting with the atoms. Given the known geometry of the slowlyvarying potential V slow ( z ) (which is the sum of contributions from the CO -laser dipole trapping potential and of thegravitational field, see eq. 2), we can from their different spatial locations clearly distinguish between atoms, whichremain near the focus of the CO -laser beam (i.e. in front of the barrier) and atoms that have been outcoupled.However, residual m F = 0 atoms which did not take part in the Raman acceleration transfer also remain trappednear the CO -laser beam focus, and in this detection method contribute to lower value of the observed transmission.For the data shown in Fig.4 a second detection method, based on a far field time-of-flight technique that allows for aStern-Gerlach separation of the individual Zeeman sublevels, was used. After a typically 2 ms long experimental cyclethe optical lattice beams were switched off, and a magnetic field gradient was activated. An absorption image was thenrecorded after a 10 ms long ballistic time-of-flight period, during which atoms in different Zeeman sublevels spatiallyseparate from each other by the Stern-Gerlach force, so that residual population in m F = 0 can be distinguishedfrom atoms in m F = 1. Due to the wave nature of the atomic Bose-Einstein condensate, atomic clouds are formedin different diffraction orders during the time-of-flight. Atoms that have undergone the analog of Klein-tunneling areexpected to be in the same diffraction order as prior to the experiment (this is the signal shown on the vertical axisof Fig. 4). An analysis of the atomic diffraction order allows us to distinguish tunneled from untunneled atoms, withthe latter being reflected by the potential barrier.The shown theory curves in Fig. 3b and Fig. 4 have been obtained by numerical integration of the one-dimensionaltime-dependent relativistic wave equation (eq. 4). The used initial conditions were an initial mean momentum q = 0 . ~ k and a Gaussian wavepacket size of σ = 4 . µ m FWHM spatial width (which yields an associated uncertainlylimited width of the momentum distribution). For the case of the spatially resolved measurements of Fig. 3b,amplitude and offset were used as free fit parameters to account for residual atoms in the m F = 0 Zeeman sublevelremaining before the barrier and uncertainties in the barrier height respectively. For the momentum resolved techniqueused in the measurement with results shown in Fig.4, the Zeeman sublevels can be clearly resolved. For the theorycurve shown in this figure for the case of ϕ = 0, only the positive energy branch is relevant in the calculation, and thevisible width of the dropoff is determined by the finite spread of the atomic velocity distribution. Quantum tunnellinghere remains neglibible due to the large width of the barrier, even if one accounts for a modification of the effectivemass in the lattice (compared to the free atomic mass, the effective mass in the lattice can lighter by up to an order ofmagnitude for typical experimental parameters). A residual discrepancy between theoretical and experimental curvesis here mainly attributed to partly overlapping atomic clouds in the far field image of reflected and transmitted atoms.Within the 2 ms experiment time, transmitted atoms (i.e. atoms outcoupled from the dipole potential) during theirballistic free fall already reach the end of the Brillouin zone and perform Bloch oscillations. In the far field image, thecorresponding atomic cloud then partly overlaps with the signal of atoms reflected by the barrier. The correspondingoverlap introduces systematic uncertainties in this far field time-of-flight technique. Other (smaller) effects, whichaffect the theory curves in both of the figures, include the nonlinearity if the dispersion for larger values of q . EFFECTIVE RELATIVISTIC WAVE EQUATION
The adiabatic potential created with the bichromatic lattice is of the form V ( z ) = V / kz ) + V / kz + ϕ ), where V and V are the potential depths of the lattice potentials with spatial periodicities λ/ λ/ ϕ denotes the relative phase between spatial harmonics. We here show that the equationof motion in the periodic potential can be described by an effective relativistic wave equation for particles near thecenter of the Brillouin zone in the first two excited Bloch-bands. Atoms in the bichromatic lattice potential can bedescribed by the Hamiltonian H ,SG = ˆ p m + V ( z ) , with ˆ p as the momentum operator. Solutions to the stationary one-dimensional Schr¨odinger equation with thisHamiltonian can be found in terms of a Bloch ansatz as a product of a plane wave and a function with the sameperiodicity as the lattice potential as φ nq ( z ) = exp ( iqz/ ~ ) · u nq ( z ), where the index n is an integer number and indexesthe corresponding Bloch band and q is the quasimomentum that is conventionally restricted to the reduced Brioullinzone q = [ − ~ k, ~ k ] [S4]. We obtain an eigenvalue equation H ,SG φ nq ( z ) = E nq φ nq ( z ) . (3)Since both the periodic potential and the function have the same periodicity as the lattice, they can be written as aFourier series. The Bloch wavefunction can be written in the form φ nq ( z ) = X s c s exp ( i ( q + 2 s ~ k ) z/ ~ )which is inserted into the stationary Schr¨odinger equation (1). The obtained eigenvalue equation is X s ′ M s,s ′ c s ′ = E nq c s (4)where M s,s ′ can be written in the form M = m ( q − ~ k ) V V e iϕV m q V V e − iϕ V m ( q + 2 ~ k ) , when we restrict ourselves to the ground and the first two excited Bloch bands. More specifically, we are here interestedin the region around the band splitting between the first and the second excited band. In previous work it has beendemonstrated that the size of this splitting depends critically on the relative phase (and amplitudes) of the latticeharmonics [S2]. Assuming that the quasimomentum | q | ≪ ~ k , i.e. we consider only the region close to the avoidedcrossing, adiabatic elimination of the ground band leads to a reduced 2x2 matrix M red = − ~ km q ∆ E/ E/ ~ km q = − qc eff m eff c eff m eff c eff qc eff ! , (5)with the basis states { exp ( i ( q − ~ k ) z/ ~ ) | q − ~ k i , exp ( i ( q + 2 ~ k ) z/ ~ ) | q + 2 ~ k i} , where we are only left with thefirst two excited Bloch bands. The eigenenergies here have been measured relatively to the position of the bandcrossing. Further, we have used as the effective light speed and an effective mass m eff = ∆ E c eff = c eff · (cid:12)(cid:12)(cid:12) V E r + V e iϕ (cid:12)(cid:12)(cid:12) ,with the latter depending on the amplitudes of the potential depths of the lattice harmonics and on the relative phase.Applying a rotation to eq. (3) with a unitary transformation matrix1 / √ (cid:18) − (cid:19) and replacing q by the corresponding operator finally gives an effective Hamiltonian H = m eff c eff σ z + c eff ˆ qσ x For a derivation of the full effectively relativistic wave equation Hamiltonian H = m eff c eff σ z + ˆ qc eff σ x + V slow ( z ) with anadditional external potential V slow ( z ) (see eq. 2), which acts on spinors ψ = ( ψ , ψ ) with ψ and ψ corresponding tocourse-grain atomic wave-functions in the upper and lower bands respectively, see Ref. [S5]. This derivation assumesthat the external potential varies slowly on the scale of the lattice periodicity. The corresponding time-dependentwave equation i ~ ∂∂t ψ = Hψ (6)has a Dirac-like form when using the described Hamiltonian. By forming the temporal derivative of one of the spinorwavefunction components and reinserting the two components into the obtained equation, this can readily be shownto be equivalent to the one-dimensional Klein-Gordon equation (cid:18) i ~ ∂∂t − V slow ( z ) (cid:19) Φ = m eff c eff + (cid:18) ~ i ∂∂z (cid:19) c eff ! Φ (7)where Φ is a (scalar) course-grain Klein-Gordon wavefunction. Eq. (5) is the well-known relativistic wave equationfor a bosonic particle [S6].[S1] G. Cennini, G. Ritt, C. Geckeler, M. Weitz Phys. Rev. Lett , 240408 (2003).[S2] T. Salger, C. Geckeler, S. Kling, M. Weitz Phys. Rev. Lett , 190405 (2007).[S3] P.R. Berman, B. Dubetsky, J.L. Cohen, Phys. Rev. A , 4801 (1998).[S4] See e.g.: N.W. Ashcroft, N.D. Mermin Solid State Physics (Saunders College Publishing, New York, 1976)[S5] D. Witthaut et al., Phys. Rev. A (in press).[S6] See e.g.: G. Baym,