Abstract
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, photonlike, dispersion relation for rubidium atoms is realized with a bichromatic optical lattice potential. We rely on a Raman πpulse 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 subde Broglie wavelength imaging applications.
Introduction
Veselago lensing is a concept based on negative refraction^{1,2,3}, where a spatially diverging pencil of rays is focused to a spatially converging one during the transition from a medium with positive refractive index to a negative index material, as depicted in Fig. 1. Pendry^{4} has shown that the spatial resolution of a negative index lens is not limited by the usual diffraction limit, because it restores the evanescent waves. Using a thin silver film, a spatial resolution below the diffraction limit has indeed been experimentally observed^{5}, and other experiments have demonstrated negative refractive index optics with optical metamaterials^{6}. Veselago lensing has also been proposed for electrons in graphene material^{7} and cold atoms in dark state media^{8}, but so far no experimental realization has been reported. The refractive action of a Veselago lens differs clearly from usual lenses, for the latter see also earlier work on the lensing of atomic de Broglie waves using light potentials with quadratic dispersion^{9,10}. The essential prerequisite to transfer such novel light optics effects towards matter waves is a relativistic, that is, photonlike, dispersion relation, an issue that has been achieved in some recent experiments with cold atoms in optical lattices, including the observation of Klein tunnelling, the moving and creation of Dirac points, and Zitterbewegung^{11,12,13}, see also Cheianov et al.^{7}, SoltanPanahi et al.^{14}, Lamata et al.^{15} and Gerritsma et al.^{16}
Here we demonstrate Veselago lensing for matter waves. Our experiment uses ultracold rubidium atoms in a bichromatic optical lattice with a linear, that is, relativistic, dispersion relation. A Raman πpulse transfers the atomic de Broglie waves between different branches of the dispersion relation, and the relativistic lensing occurs by a backwards propagation of atomic wavepackets on an energetically mirrored branch of the dispersion relation. We study negative refraction and Veselago lensing both in a onedimensional geometry and perform a raytracing simulation of a twodimensional Veselago lens.
Results
Experimental background
To tailor the atomic dispersion relation to resemble that of light in positive and negative index materials respectively, we modify the free atomic dispersion by adding a suitable periodic potential induced by laser fields. They imprint a bichromatic optical lattice potential of the form V(x)=(V_{1}/2)cos(2κx)+(V_{2}/2)cos(4κx+ϕ), where V_{1} denotes the potential depth of a usual standing wave lattice potential of spatial periodicity λ/2 and V_{2} that of a higher lattice harmonic of spatial periodicity λ/4, generated by the dispersion of multiphoton Raman transitions^{17}. Here, λ denotes the laser wavelength with wavevector κ=2π/λ, and ϕ the relative phase between lattice harmonics. For a relative phase ϕ=180° and a suitable choice of the amplitude of lattice harmonics, the splitting between the first and the second excited Bloch band vanishes and the corresponding dispersion relation near the crossing becomes linear, that is, relativistic, see the solid line in Fig. 2. This is understood from the destructive interference of the contributions from first order Bragg scattering of the higher periodicity lattice and secondorder Bragg scattering of the standing wave lattice. In general, if we set the zero of the energy scale to the position of the crossing, the energy of atomic eigenstates near the crossing is , where ħk_{x} denotes the atomic quasimomentum along the lattice axis, and the splitting between band is in the limit k_{x}<<κ and ΔE<<E_{r}, with E_{r}=(ħκ)^{2}/(2m) as the recoil energy. Further, c_{eff}=2ħκ/m is an effective light speed, which for rubidium atoms and a 783 nm lattice laser wavelength is 1.1 cm s^{−1}, ten orders of magnitude smaller than the speed of light in vacuum. The dynamics of atoms in the bichromatic lattice near the crossing is well described by a onedimensional Dirac equation, which allows us to study relativistic wave equation predictions with ultracold atoms^{18}. By tuning of the splitting ΔE, moreover different projections of a twodimensional Dirac cone can be realized.
Onedimensional Veselago lensing of matter waves
We have studied the onedimensional analogue of Veselago lensing with cold atoms in the optical lattice, in the presence of an additional, spatially slowly varying potential. In a single dimension, two rays are possible (atomic paths along and opposed to the x axis of Fig. 1 respectively), and Fig. 3a shows a schematic of the relevant part of the atomic dispersion relation for the case of a vanishing splitting between bands, with E(k_{x})=±ħc_{eff}k_{x}. The atoms initially propagate in the lattice with linear dispersion relation with an energyindependent velocity of ±c_{eff}, respectively, which above the crossing, that is, for E>0, in analogy to light optics can well be described by propagation in a medium with positive refractive index, say n=1. Due to the linear dispersion relation, the atoms remain at a group velocity of ±c_{eff} even in the presence of the additional (here confining) potential. Transitions to states below the crossing, that is, with E<0, are induced with a short (broadband) fourphoton Raman πpulse. This reverses the path of propagation, resulting in spatially converging atom paths. The group velocity is now opposed to the wavevector k_{x}, and we conclude that the regime with E<0 corresponds to propagation in a medium with negative refractive index, n=−1. The πpulse reverses the temporal evolution, similar as in a spinecho experiment^{19}, and after the second propagation time, the paths close in a focus.
In our experiment, a Bose–Einstein condensate of rubidium (^{87}Rb) atoms in the m_{F}=0 spin projection of the F=1 ground state hyperfine component is loaded into a weakly confining optical dipole trapping potential with a 70 Hz radial trapping frequency (see Methods). The atoms are irradiated with a short pulse performed with optical beams simultaneously coupling the two Raman transitions m_{F}=0,ħk_{x}›→m_{F}=−1,±2ħκ+ħk_{x}› respectively, after which the lattice beams are activated. This leaves the atomic cloud in the lattice with a coherent superposition of two spatially diverging atomic paths of ‘quasirelativistic’ velocities, and energies tuned slightly above the position of the crossing shown in Fig. 3a. Figure 3b shows a series of absorption images for different propagation times in the lattice. After an expansion time of 1.3 ms, a fourphoton Raman πpulse coupling the states m_{F}=−1,−2ħκ+ħk_{x}› and m_{F}=−1,2ħκ+ħk_{x}› is applied, which causes the spatially diverging atom paths to converge. The paths meet after a second propagation time and form an image of the initial atomic cloud (Due to the used coherent preparation of the two rays, one here also expects an atomic interference pattern^{20,21}, of spatial periodicity λ/4, which is below our 3.8 μm experimental imaging resolution.). Figure 3c shows the variation of the total width of the atomic distribution versus time, which again clearly shows the focusing of the spatially expanding cloud subject to the relativistic dispersion relation in the lattice by the πpulse. The observed 16 μm width of the image formed by the onedimensional Veselago lens at ≅2.3 ms propagation time equals the initial cloud size within experimental uncertainties. For comparison, Fig. 3d,e show experimental data recorded for a nonvanishing energy gap ΔE between the two Bloch bands, where the image is clearly larger than the initial cloud size. This is attributed to the atomic velocity now depending on energy, as in Eiermann et al.^{9} and Fallani et al.^{10}, which does not correspond to the case of Veselago lensing. The energy spread acquired from the spatially varying dipole potential here causes different group velocities, which results in larger aberrations. The Veselago lens, see the data shown in Fig. 3c, achieves sharp refocusing even in the presence of the additional potential, as is understood from the (in an ideal experimental situation) energy independence of the group velocity. Our experimental findings are supported by numerical simulations (see Methods).
Raytracing simulation of twodimensional relativistic lensing
We also carried out a raytracing simulation of a twodimensional Veselago lens. The twodimensional optical dispersion is for a n=±1 refractive index respectively, where the group velocity is parallel (antiparallel) to the wavevector in the positive (negative) energy branch, respectively. Our onedimensional optical lattice is used to realize different cuts along the x axis of the twodimensional Dirac cone, with ΔE=c_{eff}2ħk_{z}. The optical propagation along the z axis is here replaced by time. The corresponding transformation for a ray with angle α to the optical axis is , which ensures that rays emitted from the source under different angles α reach the Veselago lens at a fixed value of z=L, see Fig. 1. The refraction induced by a fourphoton Raman πpulse between momentum states along the lattice axis of −2ħκ+ħk_{x}› and 2ħκ+ħk_{x}› conserves the wavevector component orthogonal to the lens surface k_{x} in the reduced band structure notation, but as desired reverses the direction of group velocity ν_{g} to convert spatially diverging to converging rays, see Fig. 4a. In particular, the wave with velocity and wavevector , where , is refracted to with . With k′=k and the wavevector along the lattice (x axis) being conserved, we obtain α′=−α and recover Snell’s law for the negative refraction: sin(α)/sin(α′)=n/n′=−1. We thus expect that πpulse transfer in the lattice can indeed realize Veselago lensing of the diverging de Broglie waves. To verify that the onedimensional lattice produces the expected dispersion along a projection of the Dirac cone, we have measured the atomic group velocity along the lattice axis for different values of k_{x}. Corresponding results, see Fig. 4b, are in good agreement with the expected dependence.
Figure 4c shows experimental results for a raytracing simulation of a twodimensional Veselago lens with cold atoms in the lattice for a diverging pencil of rays with energy E=0.104c_{eff}ħκ. To simulate a ray with propagation angle α to the optical axis, the energy gap in the lattice is set to ΔE(α)=2Ecos(α) and atoms are prepared with a quasimomentum ħk_{x}=(E/c_{eff})sin(α) in the upper excited Bloch band by a Raman pulse. During the ballistic fall of atoms in the lattice potential, no additional dipole trapping potential is applied in this measurement and one of the lattice beams is swept with a constant chirp rate to compensate for the earth’s gravitational acceleration in the atomic rest frame. The data points give the measured centres of the atomic cloud versus the scaled time τ=t·c_{eff}cos(α)/L, with L/c_{eff}=0.25 ms. The πpulse transfer, imparted at a scaled time of τ=1, reverses the atomic paths and focuses the different rays to a single image. This demonstrates negative refraction of the rays, and provides a raytracing simulation of Veselago lensing. The variation of the positions where the different rays cross is 2 μm, which is the order of our measurement accuracy and gives an upper estimate for spherical aberrations of the lens. This value is below the size of the effective de Broglie wavelength λ_{eff}≅h/Δp≅7.5 μm of the velocity selected atomic cloud, indicating that spherical aberrations are small.
Discussion
We have demonstrated negative refraction and Veselago (relativistic) lensing for ultracold atoms in an optical lattice. In the onedimensional case, we observe refocusing despite the presence of an inhomogeneous external potential. For the case of twodimensional lensing, a raytracing simulation technique was applied. In future, it will be important to study Veselago lensing in optical lattices with higherdimensional quasirelativistic dispersion^{12,14,22,23} to achieve a wavepacket refocusing in different spatial dimensions. While in the absence of a bias force, the refocusing achieved by the πpulse transfer occurs at the original position of the wavepacket, forming a temporal version of a Veselago lens, gravity or the Stern–Gerlach force from magnetic gradients can be used to translate the second focus to a spatially separate location. For the future, imaging below the diffraction limit should be examined when imaging objects, such as microscopic mechanical structures, of size smaller than the atomic de Broglie wavelength. Other prospects of πpulse refocusing in optical lattices with linear dispersion relation include atom interferometric metrology applications, which benefits from the group velocity here being independent of external potentials, as well as its use for novel probes of quantum manybody physics in optical lattices.
Methods
Experimental details and setup
Our experiment is based on ultracold rubidium atoms in a bichromatic optical lattice potential used to tailor the atomic dispersion relation. Rubidium atoms (^{87}Rb) are initially cooled to quantum degeneracy by evaporative cooling in a quasistatic optical dipole trap, formed by a tightly focused horizontally oriented optical beam derived from a CO_{2}laser operating near 10.6 μm wavelength. During the final stage of the evaporation, a magnetic field gradient is activated, which allows us to produce a spinpolarized Bose–Einstein condensate of 6 × 10^{4} atoms in the m_{F}=0 spin projection of the F=1 hyperfine ground state component^{24}. A homogeneous magnetic bias of 1.0 G (corresponding to a Δω_{z}~2π × 700 kHz splitting between adjacent Zeeman sublevels) is then applied to remove the degeneracy of magnetic sublevels. By means of Dopplersensitive Raman transitions, atoms can subsequently be transferred from the m_{F}=0,ħk_{x}› to the m_{F}=−1,−2ħκ+ħk_{x}› and m_{F}=−1,2ħκ+ħk_{x}› states by a 30 μs long optical pulse. The transitions are driven by simultaneous irradiation with two copropagating optical beams of frequencies ω±(Δω_{z}+Δω_{d}) respectively, where Δω_{d} corresponds to the Doppler shift of about 2π × 15 kHz of atoms in the corresponding momentum states, and a counterpropagating optical beam of frequency ω. By this, atoms are accelerated to ‘quasirelativistic’ velocities near ±1.1 cm s^{−1} in the lattice.
Our method used to create a variable dispersion relation for cold atoms relies on a biharmonic potential with two lattice harmonics of spatial periodicities of λ/2 and λ/4, respectively. Here, λ denotes the wavelength of the driving laser near 783 nm wavelength, which is detuned 3 nm to the red of rubidium D2line. Two counterpropagating optical waves derived from this laser are used to generate the fundamental spatial frequency of λ/2, forming a usual standing wave lattice potential. The higher spatial harmonic of periodicity λ/4 is created by virtual fourphoton Raman transitions in a threelevel scheme with two stable ground states and one electronically excited state. The atoms are irradiated by two copropagating beams of frequencies ω+Δω and ω−Δω and a counterpropagating beam of frequency ω, see Fig. 5. Here, the m_{F}=−1 and the m_{F}=0 spin projections respectively of F=1 are the used ground states of the threelevel scheme, while the 5P_{3/2} manifold serves as the electronically excited state. We typically use a frequency offset between counterpropagating beams of Δω/(2π)≅1 MHz, which is sufficiently large that twophoton standing wave processes are suppressed. The optical field exchanges momentum with atoms in units of fourphoton recoils, which is twice that of the corresponding process in a usual standing wave lattice. Correspondingly, the spatial periodicity of the induced lattice potential is a factor two lower than in a standing wave lattice, and reaches λ/4. By combining periodic potentials of λ/2 and λ/4 spatial periodicities, a variable lattice potential can be Fouriersynthesized.
The band structure of the variable lattice along with the energy splitting between the lowest two excited Bloch bands can be calculated as follows, see Salger et al.^{11}, Witthaut et al.^{18} and Ritt et al.^{25} for details. The atomic evolution in the Fouriersynthesized lattice potential is determined by the Hamiltonian
where denotes the momentum operator and the lattice potential is
Using a Bloch ansatz to solve the stationary Schrödinger equation, the obtained eigenvalue equation is
where H_{l,l’} can be written in the form
when we restrict ourselves to the ground and the first two excited Bloch bands. Assuming that k_{x}<<κ, that is, for atomic quasimomenta k_{x} near the crossing, and ΔE<<E_{r}, an adiabatic elimination of the ground band leads to a reduced 2 × 2 matrix
on the basis of the eigenstates −2ħκ+ħk_{x}› and 2ħκ+ħk_{x}›, where denotes the energy splitting between the bands. Using this effective Hamiltonian one can, with an additional basis rotation, show that the atomic dynamics near the crossing is determined by an effective Dirac equation. As shown in Witthaut et al.^{18}, this also holds in the presence of an additional (spatially slowly varying) external potential. For a finite size of the splitting, atoms will be in a coherent superposition of the eigenstates −2ħκ+ħk_{x}› and 2ħκ+ħk_{x}›, meaning that for quasimomenta near the crossing, the population in a temporal picture will oscillate between states with positive and negative ‘quasirelativistic’ velocities. An experimental determination of the group velocity in the upper band versus the atomic quasimomentum has been shown in Fig. 4b, which illustrates that the group velocity of atoms near the crossing despite being in the superposition state can well go down to zero values.
The fourphoton Raman πpulse used to transfer atoms between different branches of the dispersion relation in the lattice, between the levels m_{F}=−1,−2ħκ+ħk_{x}› and m_{F}=−1,2ħκ+ħk_{x}› respectively, uses the m_{F}=0,ħk_{x}› state as intermediate atomic ground state level. The πpulse condition can be achieved with a typically 30 μs long optical pulse performed with same optical frequencies as for the Raman beams used for initial state preparation. All required optical frequencies for the variable optical lattice, as well as the Raman beams optical fields used for state preparation and to induce the πpulse transfer, are derived acoustooptically from a single high power diode laser source. The optical beams after spatial filtering are directed in a vertical geometry (that is, along the axis of gravity) to the atoms.
Experimental sequence
For the onedimensional Veselago lensing experiment (corresponding to the data shown in Fig. 3), atoms after condensate generation are in a tightly focused dipole trap loaded into a dipole potential created with a beam near 10.6 μm wavelength and 125 μm focal diameter (roughly a factor of 5 larger than the size of the beam used for condensate preparation by evaporative cooling). This beam is irradiated in a counterpropagating direction with respect to the dipole trapping beam used for condensate production, and has an orthogonal polarization. The measured radial trapping frequency of atoms in this weak dipole trap is 70 Hz in the presence of the earth's gravitational field. The confining force is so weak that atomic interactions in the dilute atomic cloud are neglible. The velocity width of the expanded condensate atomic cloud is ≅0.7ħk/m. The onedimensional Veselago lensing experiment achieves refocusing of the two spatially diverging atomic rays in the presence of the external potential. When Fig. 3b is inspected in detail, one finds that the temporal duration required for refocusing of the wavepackets after the πpulse is some 0.2 ms shorter than the period in which the wavepackets expand. This is attributed to a remaining nonlinearity of the bands, which results in slightly different group velocities. In other words, the effective refractive indices in positive and negative energy regions do not exactly match the values ±1, respectively.
For raytracing simulation of a twodimensional Veselago lens (see Fig. 4 for corresponding experimental data) after preparation of the rubidium condensate, the dipole trapping beam is extinguished and the atoms are left in free fall in the earth’s gravitational field. The Raman beams used for the preparation and Doppler selection of a particular atomic velocity class along the lattice beams axis are activated 3 ms after release of the atoms from the dipole trapping potential, after which the atomic interaction energy of the dense condensate cloud due to the lowering of density has largely been converted to kinetic energy. The low intensity, long (1 ms duration) Raman beam pulse used here selects a narrow slice of ≅0.1ħk/m width from the initial atomic velocity distribution, while the remaining atoms are left in the (undetected) m_{F}=0 spin projection. The lattice beams are subsequently activated. The difference in frequency of both vertically upwards and downwards propagating lattice beams and the Raman beams inducing the fourphoton πpulse to reverse the temporal evolution in the course of the Veselago lensing simulation are tuned accoustooptically to yield a spatially stationary lattice field in a frame falling ballistically downwards in the earth’s gravitational field. The theory curve in the measurement of the group velocity of Fig. 4b (solid line) was derived using the analytic expression
Imaging method
At the end of the experimental sequence, the spatial distribution of atoms in the F=1, m_{F}=−1 ground state level is measured by first transferring this population to F=2, m_{F}=−1 with a 34 μs long microwave πpulse and then pulsing on a laser beam tuned to the F=2→F′=3 hyperfine component of the rubidium D2line, recording a shadow image on a CCD camera. From the observed spatial distribution, the width and the centre of mass position of the atomic cloud is calculated by fitting a Gaussian distribution to the normalized absorption imaging data.
Numerical calculations
We have carried out semiclassical numerical calculations to simulate the atomic centre of mass trajectories of the atomic path in the presence of the trapping potential and the lattice, see Fig. 3b,d. Initially, the dispersion relation in the lattice E(k_{x}) was calculated by numerically solving the Schrödinger equation for the used experimental parameters. Subsequently, the atomic centre of mass trajectory for each path was calculated by solving the classical Euler–Lagrange equations in the presence of the trapping potential and the corresponding atomic dispersion relation in the lattice.
Additional information
How to cite this article: Leder, M. et al. Veselago lensing with ultracold atoms in an optical lattice. Nat. Commun. 5:3327 doi: 10.1038/ncomms4327 (2014).
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Acknowledgements
We acknowledge discussions with L. Santos and J. Klaers. Financial support from the DFG (We174820) is acknowledged.
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M.L. and C.G. conducted the experiment and analysed the data. All authors contributed to the writing of the paper and to the interpretation of the results. M.W. planned the project.
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Leder, M., Grossert, C. & Weitz, M. Veselago lensing with ultracold atoms in an optical lattice. Nat Commun 5, 3327 (2014). https://doi.org/10.1038/ncomms4327
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DOI: https://doi.org/10.1038/ncomms4327
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