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Veselago Lensing with Ultracold Atoms in an Optical Lattice

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Veselago Lensing with Ultracold Atoms in an Optical Lattice ARTICLE Received 9 Dec 2013 | Accepted 27 Jan 2014 | Published 14 Feb 2014 DOI: 10.1038/ncomms4327 Veselago lensing with ultracold atoms in an optical lattice Martin Leder1, Christopher Grossert1 & Martin Weitz1 Veselago pointed out that electromagnetic wave theory allows for materials with a negative index of refraction, in which most known optical phenomena would be reversed. A slab of such a material can focus light by negative refraction, an imaging technique strikingly different from conventional positive refractive index optics, where curved surfaces bend the rays to form an image of an object. Here we demonstrate Veselago lensing for matter waves, using ultracold atoms in an optical lattice. A relativistic, that is, photon-like, dispersion relation for rubidium atoms is realized with a bichromatic optical lattice potential. We rely on a Raman p-pulse technique to transfer atoms between two different branches of the dispersion relation, resulting in a focusing that is completely analogous to the effect described by Veselago for light waves. Future prospects of the demonstrated effects include novel sub-de Broglie wavelength imaging applications. 1 Institut fu¨r Angewandte Physik der Universita¨t Bonn, Wegelerstrae 8, 53115 Bonn, Germany. Correspondence and requests for materials should be addressed to M.W. (email: [email protected]). NATURE COMMUNICATIONS | 5:3327 | DOI: 10.1038/ncomms4327 | www.nature.com/naturecommunications 1 & 2014 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms4327 eselago lensing is a concept based on negative of spatial periodicity l/4, generated by the dispersion of multi- refraction1–3, where a spatially diverging pencil of rays photon Raman transitions17. Here, l denotes the laser wavelength Vis focused to a spatially converging one during the with wavevector k ¼ 2p/l, and j the relative phase between transition from a medium with positive refractive index to a lattice harmonics. For a relative phase j ¼ 180° and a suitable negative index material, as depicted in Fig. 1. Pendry4 has shown choice of the amplitude of lattice harmonics, the splitting between that the spatial resolution of a negative index lens is not limited the first and the second excited Bloch band vanishes and the by the usual diffraction limit, because it restores the evanescent corresponding dispersion relation near the crossing becomes waves. Using a thin silver film, a spatial resolution below the linear, that is, relativistic, see the solid line in Fig. 2. This is diffraction limit has indeed been experimentally observed5, and understood from the destructive interference of the contributions other experiments have demonstrated negative refractive index from first order Bragg scattering of the higher periodicity lattice optics with optical metamaterials6. Veselago lensing has also been and second-order Bragg scattering of the standing wave lattice. In proposed for electrons in graphene material7 and cold atoms in general, if we set the zero of the energy scale to the position of the dark state media8, but so far no experimental realization has been crossing,q theffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi energy of atomic eigenstates near the crossing is reported. The refractive action of a Veselago lens differs clearly 2 2 E ¼Æ ðc hk Þ þðDE=2Þ , where hk denotes the atomic from usual lenses, for the latter see also earlier work on the eff x x quasimomentum along the lattice axis, and the splitting between lensing of atomic de Broglie waves using light potentials with band is DE ¼ðV2=32E ÞþðV =2Þeij in the limit |k |ook and quadratic dispersion9,10. The essential prerequisite to transfer 1 r 2 x DEooE , with E ¼ (hk)2/(2m) as the recoil energy. Further, such novel light optics effects towards matter waves is a r r c ¼ 2hk/m is an effective light speed, which for rubidium atoms relativistic, that is, photon-like, dispersion relation, an issue that eff and a 783 nm lattice laser wavelength is 1.1 cm s À 1, ten orders of has been achieved in some recent experiments with cold atoms in magnitude smaller than the speed of light in vacuum. The optical lattices, including the observation of Klein tunnelling, the dynamics of atoms in the bichromatic lattice near the crossing is moving and creation of Dirac points, and Zitterbewegung11–13, well described by a one-dimensional Dirac equation, which allows see also Cheianov et al.7, Soltan-Panahi et al.14, Lamata et al.15 us to study relativistic wave equation predictions with ultracold and Gerritsma et al.16 atoms18. By tuning of the splitting DE, moreover different Here we demonstrate Veselago lensing for matter waves. Our projections of a two-dimensional Dirac cone can be realized. experiment uses ultracold rubidium atoms in a bichromatic optical lattice with a linear, that is, relativistic, dispersion relation. A Raman p-pulse transfers the atomic de Broglie waves between One-dimensional Veselago lensing of matter waves. We have different branches of the dispersion relation, and the relativistic studied the one-dimensional analogue of Veselago lensing with lensing occurs by a backwards propagation of atomic wavepackets cold atoms in the optical lattice, in the presence of an additional, on an energetically mirrored branch of the dispersion relation. spatially slowly varying potential. In a single dimension, two rays We study negative refraction and Veselago lensing both in a one- are possible (atomic paths along and opposed to the x axis of dimensional geometry and perform a ray-tracing simulation of a Fig. 1 respectively), and Fig. 3a shows a schematic of the relevant two-dimensional Veselago lens. part of the atomic dispersion relation for the case of a vanishing splitting between bands, with E(kx) ¼ ±hc eff|kx|. The atoms initially propagate in the lattice with linear dispersion relation Results with an energy-independent velocity of ±ceff, respectively, which Experimental background. To tailor the atomic dispersion above the crossing, that is, for E40, in analogy to light optics can relation to resemble that of light in positive and negative index well be described by propagation in a medium with positive materials respectively, we modify the free atomic dispersion by refractive index, say n ¼ 1. Due to the linear dispersion relation, adding a suitable periodic potential induced by laser fields. They imprint a bichromatic optical lattice potential of the form V(x) ¼ (V1/2)cos(2kx) þ (V2/2)cos(4kx þ j), where V1 denotes Four-photon the potential depth of a usual standing wave lattice potential of -pulse 4 transfer spatial periodicity l/2 and V2 that of a higher lattice harmonic 2 ) r E E 0 ΔE n=+1 n=–1 Energy ( –2 –4 –1 –0.5 0 0.5 1 – Quasi-momentum (hkx) Figure 2 | Atomic dispersion in lattice. The figure shows the dispersion relation of rubidium atoms in the bichromatic optical lattice for V1 ¼ 2.1Er, V2 ¼ 0.28Er and a relative phase between lattice harmonics of j ¼ 180° x (solid) and j ¼ 0° (dashed). In the former case, the splitting between the first (red solid line) and the second (blue solid line) excited Bloch band z vanishes and the dispersion near the crossing becomes linear, that is, Figure 1 | Schematic of a Veselago lens in the x-z plane. A diverging relativistic. The zero of the energy scale is here set to the position of the pencil of rays is focused upon entry into a medium with refractive index crossing. Transitions between the branch of the dispersion relation with n ¼À1 by negative refraction. The negative refraction can be described E40, corresponding to the case with collinear wavevector and group by Snell’s law when noting that the refractive index at the surface velocity, and the regime with Eo0, for which wavevector and group velocity changes sign from positive to negative. are antiparallel, can be induced by four-photon Raman p-pulses. 2 NATURE COMMUNICATIONS | 5:3327 | DOI: 10.1038/ncomms4327 | www.nature.com/naturecommunications & 2014 Macmillan Publishers Limited. All rights reserved. NATURE COMMUNICATIONS | DOI: 10.1038/ncomms4327 ARTICLE the atoms remain at a group velocity of ±ceff even in the pre- optical dipole trapping potential with a 70 Hz radial trapping sence of the additional (here confining) potential. Transitions to frequency (see Methods). The atoms are irradiated with a short states below the crossing, that is, with Eo0, are induced with a pulse performed with optical beams simultaneously coupling S- short (broadband) four-photon Raman p-pulse. This reverses the the two Raman transitions |mF ¼ 0,hk x |mF ¼À1,±2hk þ path of propagation, resulting in spatially converging atom paths. hk xS respectively, after which the lattice beams are activated. This The group velocity is now opposed to the wavevector kx, and we leaves the atomic cloud in the lattice with a coherent super- conclude that the regime with Eo0 corresponds to propagation position of two spatially diverging atomic paths of ‘quasi- in a medium with negative refractive index, n ¼À1. The p-pulse relativistic’ velocities, and energies tuned slightly above the reverses the temporal evolution, similar as in a spin-echo position of the crossing shown in Fig. 3a. Figure 3b shows a series experiment19, and after the second propagation time, the paths of absorption images for different propagation times in the lattice. close in a focus. After an expansion time of 1.3 ms, a four-photon Raman p-pulse S In our experiment, a Bose–Einstein condensate of rubidium coupling the states |mF ¼À1, À 2hk þ hk x and |mF ¼À1,2 87 ( Rb) atoms in the mF ¼ 0 spin projection of the F ¼ 1 ground hk þ hk xS is applied, which causes the spatially diverging atom state hyperfine component is loaded into a weakly confining paths to converge.
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