We demonstrate the controlled coherent transport and splitting of atomic wave
packets in spin-dependent optical lattice potentials. Such experiments open
intriguing possibilities for quantum state engineering of many body states.
After first preparing localized atomic wave functions in an optical lattice
through a Mott insulating phase, we place each atom in a superposition of two
internal spin states. Then state selective optical potentials are used to split
the wave function of a single atom and transport the corresponding wave packets
in two opposite directions. Coherence between the wave packets of an atom
delocalized over up to 7 lattice sites is demonstrated.Comment: 4 pages, 6 figure

Artur Widera

Immanuel Bloch

Markus Greiner

Olaf Mandel

P. S. Jessen

R. Grimm

Theodor W. Hänsch

Tim Rom

English

arXiv

Mandel, O.

Greiner, M.

Widera, A.

Rom, T.

Hänsch, T.

Bloch, I.

MPG.PuRe

Coherent transport of neutral atoms in spin-dependent optical lattice potentials

arXiv:cond-mat/0301169v1  [cond-mat.soft]  11 Jan 2003Coherent transport of neutral atoms in spin-dependent optical lattice potentialsOlaf Mandel, Markus Greiner, Artur Widera, Tim Rom, Theodor W. Hänsch, and Immanuel Bloch∗Ludwig-Maximilians-Universität, Schellingstr. 4/III, 80799 Munich, GermanyMax-Planck-Institut für Quantenoptik, 85748 Garching, Germany(Dated: September 3, 2018)We demonstrate the controlled coherent transport and splitting of atomic wave packets in spin-dependent optical lattice potentials. Such experiments open intriguing possibilities for quantumstate engineering of many body states. After first preparing localized atomic wave functions inan optical lattice through a Mott insulating phase, we place each atom in a superposition of twointernal spin states. Then state selective optical potentials are used to split the wave function ofa single atom and transport the corresponding wave packets in two opposite directions. Coherencebetween the wave packets of an atom delocalized over up to 7 lattice sites is demonstrated.Over the last years Bose-Einstein condensates in op-tical lattices have opened fascinating new experimentalpossibilities in condensed matter physics, atomic physics,quantum optics and quantum information processing.Already now the study of Josephson junction like ef-fects [1, 2], the formation of strongly correlated quantumphases [3, 4, 5] and the observation of the collapse andrevival of the matter wave field of a BEC [6] have shownsome of these diverse applications. In an optical lattice,neutral atoms can be trapped in the intensity maxima (orminima) of a standing wave light field due to the opticaldipole force [7]. So far the optical potentials used havebeen mostly independent of the internal ground state ofthe atom. However, it has been suggested that by usingspin-dependent periodic potentials one could bring atomson different lattice sites into contact and thereby realizefundamental quantum gates [8, 9, 10, 11], create largescale entanglement [12, 13], excite spin waves [14], studyquantum random walks [15] or form a universal quantumsimulator to simulate fundamental complex condensedmatter physics hamiltonians [16]. Here we report on therealization of a coherent spin-dependent transport of neu-tral atoms in optical lattices [17]. We show how the wavepacket of an atom that is initially localized to a single lat-tice site can be split and delocalized in a controlled andcoherent way over a defined number of lattice sites.In order to realize a spin dependent transport for neu-tral atoms in optical lattices, a standing wave config-uration formed by two counterpropagating laser beamswith linear polarization vectors enclosing an angle θhas been proposed [8, 12]. Such a standing wave lightfield can be decomposed into a superposition of a σ+and σ− polarized standing wave laser field, giving riseto lattice potentials V+(x, θ) = V0 cos2(kx + θ/2) andV−(x, θ) = V0 cos2(kx − θ/2). Here k is the wave vec-tor of the laser light used for the standing wave and V0is the potential depth of the lattice. By changing thepolarization angle θ one can thereby control the sepa-ration between the two potentials ∆x = θ/180◦ · λx/2.When increasing θ, both potentials shift in opposite di-rections and overlap again when θ = n · 180◦, with nbeing an integer. For a spin-dependent transfer two in-ternal spin states of the atom should be used, whereone spin state dominantly experiences the V+(x, θ) po-tential and the other spin state mainly experiences theV−(x, θ) dipole force potential. Such a situation can berealized in rubidium by tuning the wavelength of the op-tical lattice laser to a value of λx = 785nm betweenthe fine structure splitting of the rubidium D1 and D2transition. Then the dipole potential experienced byan atom in e.g. the |1〉 ≡ |F = 2,mF = −2〉 stateis given by V1(x, θ) = V−(x, θ) and that for an atomin the |0〉 ≡ |F = 1,mF = −1〉 state is given byV0(x, θ) = 3/4V+(x, θ) + 1/4V−(x, θ). If an atom is nowfirst placed in a coherent superposition of both internalstates 1/√2(|0〉 + i|1〉) and the polarization angle θ iscontinuously increased, the spatial wave packet of theatom is split with both components moving in oppositedirections. In a fixed polarization configuration, spin-dependent tunnelling between two lattice sites has beenobserved in [18].As in our previous experiments, Bose-Einstein con-densates of up to 3 × 105 atoms are created in the|F = 1,mF = −1〉 hyperfine state in a harmonic mag-netic trap with almost isotropic oscillation frequencies ofω = 2π × 16Hz. A three dimensional lattice potential isthen superimposed on the Bose-Einstein condensate andthe intensity raised in order to drive the system into aMott insulating phase [5] and thereby localize the atomsto individual lattice sites. Two of the three orthogonalstanding wave light fields forming the lattice potentialare operated at a wavelength of λy,z = 840 nm. For thethird standing wave field along the horizontal x-directiona laser at a wavelength of λx = 785 nm is used. Along thisaxis a quarter wave plate and an electro-optical modula-tor (EOM) allow us to dynamically rotate the polariza-tion vector of the retro-reflected laser beam through anangle θ by applying an appropriate voltage to the EOM(see Fig. 1). Initially the polarization angle θ is set toa lin‖lin polarization configuration. After reaching theMott insulating phase we completely turn off the har-monic magnetic trapping potential but maintain a 1Ghomogeneous magnetic field along the x-direction in or-der to preserve the spin polarization of the atoms. This2homogeneous field is actively stabilized to an accuracyof ≈ 1mG. Shortly before moving the atoms along thisstanding wave direction we adiabatically turn off the lat-tice potentials along the y- and z-direction. This is donein order to reduce the interaction energy, which stronglydepends on the confinement of the atoms at a single lat-tice site. We can thereby study the transport process it-self, without having to take into account the phase shiftsin the many body state that result from a coherent col-lisional interaction between atoms.By using microwave radiation around 6.8GHz we areable to drive Rabi oscillations between the |0〉 and the |1〉state with resonant Rabi frequencies of Ω = 2π× 40kHz,such that e.g. a π-pulse can be achieved in a time of12.5µs. The microwave field therefore allows us to placethe atom in any of the two internal states |0〉 or |1〉 or anarbitrary superposition of both states.During the shifting process of the atoms it is crucialto avoid unwanted vibrational excitations, especially ifthe shifting process would be repeated frequently. Wetherefore analyze the timescale for such a movement pro-cess in the following way. First the atom is placed eitherin state |0〉 or state |1〉 by using microwave pulses in astanding wave lin‖lin polarization configuration. Thenwe rotate the polarization to an angle θ = 180◦ in alinear ramp within a time τ , such that again a lin‖linpolarization configuration is achieved. However, duringthis process the atoms will have moved by a distance±λx/4 depending on their internal state. In order to de-termine whether any higher lying vibrational states havebeen populated, we adiabatically turn off the lattice po-tential within a time of 500µs. The population of theenergy bands is then mapped onto the population of thecorresponding Brillouin zones [19, 20]. By counting thenumber of atoms outside of the first Brillouin zone of thesystem relative to the total number of atoms we are ableto determine the fraction of vibrationally excited atomsafter the shifting of the lattice potential (see Fig. 2). Fora perfectly linear ramp with infinite acceleration at thebeginning and ending of the ramp one would expect thefraction of atoms in the first vibrational state to be givenby |c1(τ)|2 = 2v2/(a0ω)2 sin2(ωτ/2), where v = λx/(4τ)is the shift velocity, a0 the size of the ground state har-monic oscillator wave function and ω the vibrational fre-quency on each lattice site.We have measured the vibrational frequencies on alattice site for different polarization angles θ by slightlymodulating the lattice position and observing a resonanttransfer of atoms to the first excited vibrational state.For atoms in the |1〉 state the vibrational frequencies re-main constant for different polarization angles θ as thelattice potential depth V1(x, θ) remains constant. How-ever, for atoms in the |0〉 state the lattice potential depthV0(x, θ) decreases to 50% in a lin⊥lin configuration. Inorder to reduce this effect we tilt the EOM by 3◦ andthereby decrease the strength of the σ− standing wavebut increase the strength of the σ+ standing wave insuch a polarization configuration. Then both trappingfrequencies for the |0〉 and the |1〉 state decrease to ap-prox. 85% in a lin⊥lin configuration relative to their ini-tial value of ω = 2π × 45kHz in a lin‖lin standing waveconfiguration. For such trapping frequencies of ≈45kHzduring the transport process, the excitation probabilityshould remain below 5% for shifting times longer than≈ 2π/ωx, taking into account the finite bandwidth of ourhigh voltage amplifier. This finite bandwidth smoothesthe edges of our linear voltage ramp and thereby effi-ciently suppresses the oscillatory structure in the calcu-lated excitation probability (see Fig. 2).In order to verify the coherence of the spin-dependenttransport we use the interferometer sequence of Fig. 3.Let us first consider the case of a single atom being ini-tially localized to the jth lattice site. First, the atomis placed in a coherent superposition of the two internalstates |0〉j and |1〉j with a π/2 microwave pulse (herethe index denotes the position in the lattice). Then thepolarization angle θ is rotated to 180◦, such that thespatial wave packet of an atom in the |0〉 and the |1〉state are transported in opposite directions. The finalstate after such a movement process is then given by1/√2(|0〉j + i exp(iβ)|1〉j+1), where the wave function ofan atom has been delocalized over the jth and the (j+1)thlattice site. The phase β between the two wave packetsdepends on the accumulated kinetic and potential en-ergy phases in the transport process and in general willbe nonzero. In order to reveal the coherence betweenthe two wave packets, we apply a final π/2 microwavepulse that erases the which-way information encoded inthe hyperfine states. We then release the atoms from theconfining potential by suddenly turning off the stand-ing wave optical potential and observe the momentumdistribution of the trapped atoms in the |1〉 state withabsorption imaging after a time of flight period. As aresult of the above sequence, the spatial wave packet ofan atom in the |0〉 (|1〉) state is delocalized over two lat-tice sites resulting in a double slit momentum distribu-tion w(p) ∝ exp(−p2/(h̄/σx)2) ·cos2(p δx0/2h̄+β/2) (seeFig. 4a) where δx0 denotes the separation between thetwo wave packets and σx is the spatial extension of thegaussian ground state wave function on each lattice site.In order to increase the separation between the two wavepackets further, one could increase the polarization angleθ to further integer multiples of 180◦. In practice, suchan approach is however limited by the finite maximumvoltage that can be applied to the EOM. In order to cir-cumvent this limitation we apply a microwave π pulseafter the polarization has been rotated to θ = +180◦,thereby swapping the role of the two hyperfine states. Bythen returning the polarization vector to θ = 0◦, we donot bring the two wave packets of an atom back to theiroriginal site but rather further increase the separationbetween the wave packets (see Fig. 3). The interlaced π3pulse provides a further advantage of cancelling inhomo-geneous phase shifts acquired in the single particle phaseβ in a photon echo like sequence. With increasing sepa-ration between the two wave packets the fringe spacingof the interference pattern further decreases (see Fig. 4).We have been able to observe such interference patternsfor two wave packets delocalized over up to 7 lattice sites(see Fig. 4f). When moving the atoms over up to threelattice sites, the visibility of the interference pattern re-mains rather high with up to 60% (see Fig. 5). Thesehigh contrast interference patterns directly prove the co-herence of the transport process but also show that thesingle particle phase β acquired for each atom is almostconstant throughout the cloud of atoms in our system.If the movement process is repeated more often, inhomo-geneously acquired phase shifts over the cloud of atomssignificantly decrease the visibility.For many further applications of the coherent spin-dependent transport it will also be crucial that the sin-gle particle phase β is not only constant throughout thecloud of atoms within a single run of the experiment, butis also reproducible between different sets of experiments.We have verified this by varying the phase α of the finalmicrowave π/2 pulse in a sequence where an atom is de-localized over three lattice sites. In Figure 6 we plot theexperimentally measured phase of the interference pat-tern vs the phase α of the final microwave pulse obtainedin different runs of the experiment. We find a high corre-lation between the detected phase of the interference pat-tern vs the phase of the applied microwave pulse whichproves that indeed the single particle phase is constantbetween different experiments and can be cancelled viathe phase of the final microwave pulse.In conclusion we have demonstrated the coherent spin-dependent transport of neutral atoms in optical lattices,thereby showing an essential level of coherent control formany future applications. The method demonstratedhere e.g. provides a simple way to continuously tunethe interspecies interactions by controlling the overlap ofthe the two ground state wave functions for the two spinstates. Furthermore, if such a transport is carried outin a three dimensional lattice, where the on-site inter-action energy between atoms is large, one could induceinteractions between almost any two atoms on differentlattice sites in a controlled way. Such controlled interac-tions of Ising or Heisenberg type could then be used tosimulate the behavior of quantum magnets [14], to realizequantum gates between different atoms [8, 9, 10, 11] or togenerate highly entangled cluster states [9, 12] that couldform the basis of a one-way quantum computer [10].We would like to thank Ignacio Cirac and Hans Briegelfor stimulating discussions, Alexander Altmeyer for helpduring the initial phase of the experiment and An-ton Scheich for assistance with the electronics. Wealso acknowledge financial support from the BayerischeForschungsstiftung.angle θatomcloudlinlinEOMλ/4mirrorFIG. 1: Schematic experimental setup. A one dimensionaloptical standing wave laser field is formed by two counter-propagating laser beams with linear polarizations. The po-larization angle of the returning laser beam can be adjustedthrough an electro-optical modulator. The dashed lines indi-cate the principal axes of the wave plate and the EOM.0 25 50 7500.20.40.60.8Fraction of atoms in excited vibrational statesτ (µs)FIG. 2: Fraction of atoms in excited vibrational states af-ter moving the lattice potential in a time τ over a distanceof λx/4. Filled (hollow) circles denote atoms in the |1〉 (|0〉)state. The images show the population of the Brillouin zoneswhen the lattice potential was adiabatically ramped down af-ter the shifting process. These absorption images correspondto the |1〉 state and were taken after a time of flight period of14ms. The white dashed lines in the images denote the bor-ders of the first Brillouin zone. Atoms within this Brillouinzone correspond to atoms in the vibrational ground state oneach lattice site.∗ Electronic address: imb@mpq.mpg.de; URL:http://www.mpq.mpg.de/~haensch/bec[1] B. P. Anderson and M. A. Kasevich, Science 282, 1686(1998).[2] F. S. Cataliotti et al., Science 293, 843 (2001).[3] D. Jaksch et al., Phys. Rev. Lett. 81, 3108 1998.[4] C. Orzel et al., Science 291, 2386 (2001).[5] M. Greiner et al., Nature 415, 39 (2002).[6] M. Greiner et al., Nature 419, 51 (2002).[7] R. Grimm, M. Weidemüller, and Y. B. Ovchinnikov, Adv.At. Mol. Opt. Phys. 42, 95 (2000).[8] G. K. Brennen et al., Phys. Rev. Lett. 82, 1060 (1999).4FIG. 3: General interferometer sequence used to delocalizean atom over an arbitrary number of lattice sites. Initiallyan atom is localized to the jth lattice site. The graph on theleft indicates the EOM voltage and the sequence of π/2 andπ microwave pulses that are applied over time (see text).(a) (d)(b) (e)(c) (f)FIG. 4: Observed interference patterns in state |1〉 after ini-tially localized atoms have been delocalized over (a) two, (b)three, (c) four, (d) five, (e) six and (f) seven lattice sites us-ing the interferometer sequence of Fig. 3. The time of flightperiod before taking the images was 14ms and the horizontalsize of each image is 880µm.0 100 200 300 400 500 600 70000.20.40.60.81.0x (µm)Column Density (a.u.)FIG. 5: Profile of the interference pattern obtained afterdelocalizing atoms over three lattice sites with a π/2-π-π/2microwave pulse sequence. The solid line is a fit to the inter-ference pattern with a sinusoidal modulation, a finite visibility(≈60%) and a gaussian envelope. The time of flight periodwas 15ms.0 90 180 270 360-180-90090180 Phase of Interference Pattern (deg)Microwave Phase (deg)FIG. 6: Phase of the interference pattern vs the phase α ofthe final microwave π/2-pulse in a π/2− π − π/2 delocaliza-tion sequence (see Fig. 3). The absorption images show themeasured interference pattern for α = 0◦ and α = 180◦ aftera time of flight period of 15ms. The solid line is a linear fit tothe data with unity slope and a variable offset. The dashedlines in the images correspond to the center of the envelopeof the interference pattern.[9] H.-J. Briegel et al., J. Mod. Opt. 47, 415 (2000).[10] R. Raussendorf and H.-J. Briegel, Phys. Rev. Lett. 86,5188 (2001).[11] G. K. Brennen, I. H. Deutsch, and C. J. Williams, Phys.Rev. A 65, 022313 (2002).[12] D. Jaksch et al., Phys. Rev. Lett. 82, 1975 (1999).[13] H.-J. Briegel and R. Raussendorf, Phys. Rev. Lett. 86,910 (2001).[14] A. Sørensen and K. Mølmer, Phys. Rev. Lett. 83, 2274(1999).[15] W. Dür et al., Phys. Rev. A 66, 052319 (2002).[16] E. Jané et al., quant-ph/0207011[17] Spin-dependent transport of neutral atoms has also beenobserved at NIST. William Phillips, private communica-tion.[18] D. L. Haycock et al., Phys. Rev. Lett. 85, 3365 (2000).[19] A. Kastberg et al., Phys. Rev. Lett. 74, 1542 (1999).[20] M. Greiner et al., Phys. Rev. Lett. 87, 160405 (2001).

Mandel, Olaf

Greiner, Markus

Widera, Artur

Rom, Tim

Hänsch, Theodor W.

Bloch, Immanuel

Max Planck Society eDoc Server

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Coherent Transport of Neutral Atoms in Spin-Dependent Optical Lattice Potentials

Coherent transport of neutral atoms in spin-dependent optical lattice
  potentials

Coherent transport of neutral atoms in spin-dependent optical lattice potentials

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