The counterpart of metamaterials in light optics for nonrelativistic matter waves governed by the Schr？dinger equation can be found by transiently reversing the group velocity using a so called comoving potential. Possible applications to wave-packet dynamics, atom interferometry, and atom deceleration are described. 1. Introduction The genuine concept of “meta” materials for electromagnetic waves originates from the now famous Veselago’s paper published in 1967 [1]. The basic idea is that, in a material with negative electric permittivity and negative magnetic permeability , Maxwell equations impose that the wave vector k and the Poynting vector S of a planar wave have opposite directions and, because of causality, the effective optical index is real negative: . The realisation of such artificial or “meta” materials, also called left-handed materials (LHM), in a wide range of wavelengths, has been—and continues to be—the subject of considerable theoretical and experimental efforts [2–4]. Compared to an ordinary material with a positive index, a metamaterial has a similar group velocity, whereas its phase velocity is reversed. This gives rise to the negative refraction phenomenon, owing to which so-called “meta” lenses are conceivable. The concept is rather easily extended to matter waves, provided that the effective mass of the particle be zero or close to zero, as it is the case for electrons in graphene, governed by a (relativistic) Dirac equation [5]. Paradoxically the situation is much more intricate with nonrelativistic particles, as atoms having a thermal velocity (a few hundreds ms？1), the dynamics of which is governed by the Schr？dinger equation. The first obstacle is the inability of atoms to penetrate dense matter: hence a “material” should be replaced by a “medium”, namely, some external potential created in vacuum. A second difficulty comes from the fact that, in this situation, the phase velocity is an ambiguous concept since it is gauge dependent and its inversion appears to be problematic, if not meaningless. Nevertheless the key property of a metamedium lies in the opposite directions of phase and group velocities, a property which will be realised in our case by simply reversing the group velocity. Obviously, given a source of atoms, this property has necessarily a transient character since the group velocity is associated to the density of the probability flux which should finally be oriented outwards from the source. As a consequence, the external potential, assumed to depend on a single spatial coordinate ( ), must be also time
References
[1]
V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of epsilon and ,” Soviet Physics Uspekhi, vol. 10, no. 4, pp. 509–514, 1968.
[2]
J. Pendry, “Negative refraction makes a perfect lens,” Physical Review Letters, vol. 85, no. 18, pp. 3966–3969, 2000.
[3]
C. Foteinopoulou, E. N. Economou, and C. M. Soukoulis, “Refraction in media with a negative refractive index,” Physical Review Letters, vol. 90, Article ID 107401, 4 pages, 2003.
[4]
A. Sambale and D.-G. Welsch, “van der Waals interaction and spontaneous decay of an excited atom in a superlens-type geometry,” Physical Review A, vol. 78, Article ID 053828, 10 pages, 2008.
[5]
V. V. Klimov, J. Baudon, and M. Ducloy, “Comparative focusing of Maxwell and Dirac fields by negative-refraction half-space,” Europhysics Letters, vol. 94, no. 2, Article ID 20006, 2011.
[6]
R. Mathevet, K. Brodsky, B. J. Lawson-Daku, C. Miniatura, J. Robert, and J. Baudon, “Atomic interferences in a comoving magnetic field,” Physical Review A, vol. 56, no. 4, pp. 2954–2958, 1997.
[7]
J. Baudon, M. Hamamda, J. Grucker et al., “Negative-index media for matter-wave optics,” Physical Review Letters, vol. 102, no. 14, Article ID 140403, 2009.
[8]
C. Miniatura, J. Robert, O. Gorceix et al., “Atomic interferences and the topological phase,” Physical Review Letters, vol. 69, no. 2, pp. 261–264, 1992.
[9]
M. Hamamda, F. Perales, G. Dutier et al., “Time reversal in matter-wave optics,” European Physical Journal D, vol. 61, no. 2, pp. 321–325, 2011.
[10]
M. Hamamda, M. Boustimi, F. Correia et al., “Atom slowing via dispersive optical interactions,” Physical Review A, vol. 85, no. 2, Article ID 023417, 5 pages, 2012.
[11]
S. L. Zhu, B. Wang, and L. M. Duan, “Simulation and detection of Dirac fermions with cold atoms in an optical lattice,” Physical Review Letters, vol. 98, no. 26, Article ID 260402, 4 pages, 2007.
[12]
K. L. Lee, B. Grémaud, R. Han, B. G. Englert, and C. Miniatura, “Ultracold fermions in a graphene-type optical lattice,” Physical Review A, vol. 80, no. 4, Article ID 043411, 2009.
[13]
G. Juzeliunas, J. Ruseckas, M. Lindberg, L. Santos, and P. ？hberg, “Quasirelativistic behavior of cold atoms in light fields,” Physical Review A, vol. 77, no. 1, Article ID 011802, 4 pages, 2008.
[14]
M. Hamamda, PhD thesis, University Paris 13.
[15]
F. Mezei, “Neutron spin echo: a new concept in polarized thermal neutron techniques,” Physik, vol. 255, no. 2, pp. 146–160, 1972.
[16]
M. De Kieviet, D. Dubbers, Ch. Schmidt, D. Scholz, and U. Spinola, “3He spin echo: new atomic beam technique for probing phenomena in the nev range,” Physical Review Letters, vol. 75, no. 10, pp. 1919–1922, 1995.
[17]
E. Majorana, “Atomi orientati in campo magnetico variabile,” Nuovo Cimento, vol. 9, no. 2, pp. 43–50, 1932.
[18]
G. Sulyok, H. Lemmel, and H. Rauch, “Neutrons in a time-dependent magnetic field: photon exchange and decoherence modeling,” Physical Review A, vol. 85, no. 3, Article ID 033624, 7 pages, 2012.
[19]
L. Niel and H. Rauch, “Acceleration, deceleration and monochromatization of neutrons in time dependent magnetic fields,” Zeitschrift für Physik B, vol. 74, no. 1, pp. 133–139, 1989.
[20]
H. L. Bethlem, F. M. H. Crompvoets, R. T. Jongma, S. Y. T. Van de Meerakker, and G. Meijer, “Deceleration and trapping of ammonia using time-varying electric fields,” Physical Review A, vol. 65, no. 5, Article ID 053416, 20 pages, 2002.
[21]
H. L. Bethlem, G. Berden, and G. Meijer, “Decelerating neutral dipolar molecules,” Physical Review Letters, vol. 83, no. 8, pp. 1558–1561, 1999.
[22]
E. Narevicius, A. Libson, C. G. Parthey et al., “Stopping supersonic beams with a series of pulsed electromagnetic coils: an atomic coilgun,” Physical Review Letters, vol. 100, no. 9, Article ID 093003, 2008.
[23]
Y. Yamakita, S. R. Procter, A. L. Goodgame, T. P. Softley, and F. Merkt, “Deflection and deceleration of hydrogen Rydberg molecules in inhomogeneous electric fields,” Journal of Chemical Physics, vol. 121, no. 3, p. 1419, 2004.
[24]
E. Vliegen, H. J. Worner, T. P. Softley, and F. Merkt, “Nonhydrogenic effects in the deceleration of Rydberg atoms in inhomogeneous electric fields,” Physical Review Letters, vol. 92, no. 3, Article ID 033005, 4 pages, 2004.
[25]
N. Vanhaecke, D. Comparat, and P. Pillet, “Rydberg decelerator using a travelling electric-field gradient,” Journal of Physics B, vol. 38, no. 2, p. S409, 2005.
[26]
W. D. Phillips and H. J. Metcalf, “Laser deceleration of an atomic beam,” Physical Review Letters, vol. 48, no. 9, pp. 596–599, 1982.
[27]
H. J. Metcalf and P. van der Straten, Optical Cooling and Trapping, Springer, New York, NY, USA, 2001.
