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1、arXiv:0811.2479v2 quant-ph 27 Jan 2009 APS/123-QED Negative-index media for matter-wave optics J. Baudon,M. Hamamda, J. Grucker, M. Boustimi,F. Perales, G. Dutier, and M. Ducloy Laboratoire de Physique des Lasers, Universit e Paris 13, 93430-Villetaneuse, France (Dated: January 27, 2009) We consider
2、 the extension of optical meta-materials to matter waves and then the down scaling of meta-optics to nanometric wavelengths. We show that the generic property of pulsed comoving magnetic fi elds allows us to fashion the wave-number dependence of the atomic phase shift. It can be used to produce a tr
3、ansient negative group velocity of an atomic wave packet, which results into a negative refraction of the matter wave. Application to slow metastable argon atoms Ar*(3P2) shows that the device is able to operate either as an effi cient beam splitter or an atomic meta-lens. PACS numbers: 03.75.-b, 03
4、.75.Be, 37.10.Gh, 42.25.-p Since the pioneering work of H. Lamb 1 and V.G. Veselagos seminal paper 2 about so-called “left- handed” or “meta” media for light optics, a number of studies have been devoted to these new media and their applications (negative refraction, reversed Doppler ef- fect, perfe
5、ct lens, van der Waals atom-surface interaction, etc.) 3, 4, 5, in various spectral domains 6, 7, 8, some of them being even extended to acoustic waves 9. Such media are essentially characterised by a negative value of the optical index, which results into opposite directions of the wave vector k an
6、d the Poynting vector R. Our goal here is to extend this concept to matter waves, and the fi rst arising question is the following: what should be the “de-Broglie optics” equivalent of those meta-materials? To the energy fl ux in electromagnetism (R vector) cor- responds the atomic probability fl ux
7、, namely the current density of probability J, or equivalently the group veloc- ity vg= |2J, where is the wave-function. There- fore, here also, one has to reverse vgwith respect to the wave vector k or the phase velocity.However, as discussed below, contrarily to what occurs in light op- tics where
8、 R remains directed outwards whereas k is di- rected towards the light source 10, for matter waves the direction of the phase velocity (k) remains unchanged, whereas vgis now directed towards the source. Obvi- ously, because of the conservation of probability, such an eff ect is necessarily a transi
9、ent eff ect. In light optics, meta-media generally consist of periodic ensembles of micro- or nano- structures embedded in an ordinary material or organized into an optical band-gap crystal 11. Therefore their counterpart in atom optics is far from being obvious because atoms at mean and a fortiori
10、low velocity (a few hundreds of m/s down to a few m/s or less) cannot penetrate dense matter. How- ever a possible way to act on atomic waves in vacuo is to use an interaction potential due to some external fi eld, for instance magnetic, electric or electromagnetic fi elds, or the van der Waals fi e
11、ld appearing outside a solid at Electronic address: jacques.baudonuniv-paris13.fr Present adress : Department of Physics, Umm Al-Qura University, Mekkah, Saudi Arabia the vicinity of its surface. Indeed when a semi-classical description of the external atomic motion is justifi ed, an inhomogeneous s
12、tatic potential V(r) is equivalent to an optical index n(r).This comes from the fact that the optical path accumulated along a ray C (i.e.a classi- cal trajectory) is given by the integral R C K(r)ds, where K(r) = k1 V (r)/E01/2is the local wave number, k and E0being respectively the wave number and
13、 the ki- netic energy of the atom in absence of potential, s is the curvilinear abscissa along the ray 12.This naturally leads us to set n(r) = 1 V (r)/E01/2. Nevertheless, in agreement with our previous remark about the transient character of the eff ect, such a type of potential cannot be a soluti
14、on for our purpose because the index n is either real and positive, or purely imaginary in classically for- bidden regions, but it is nowhere real negative. This looks like an impasse.However our choice of potential was too restrictive: a much larger class of interactions is of- fered by position- a
15、nd time-dependent potentials. Time- dependent potentials have already been widely used for devising atom optics elements in the time domain, oper- ating either on free atom waves (modulated atomic mir- rors 13), or trapped atoms (pulsed interferometers 14, dynamics of Bose-Einstein condensates 15).
16、In this let- ter, we show that a novel class of recently introduced po- tentials - i.e. comoving potentials 16 - provides us with a remarkably simple solution to devise negative-index me- dia for atomic waves. Comoving potentials have been previously used in several experiments, most of them dealing
17、 with Stern-Gerlach atom interferometry 16. They have been described in detail in 17. Only the way to produce them and their main characteristics will be recalled here. Two identical planar systems of currents, periodic in space (period ) and symmetric with respect to x-axis (fi g.1), produce in the
18、 vicinity of this axis a transverse fi eld, e.g. parallel to y, periodic in x with the same period. Actually only the lowest spatial frequency will have a signifi cant eff ect. Then the fi eld can be assumed to be proportional to, for instance, cos(2x/). Now if the circuit is supplied with an A.C. i
19、ntensity of frequency , then the result- ing fi eld is proportional to cos(2(tt0)cos(2x/) = 1 2cos(2(tt0)x/)+cos(2(tt0)+x/), where t0 is some reference time. Only the fi rst term propagates 2 x ? y FIG. 1: Scheme of the device generating a transverse mag- netic fi eld “comoving” along the x axis.It
20、consists of two identical planar periodic circuits (spatial period ). When the circuits are supplied with an A.C. current of frequency , they produce along x a fi eld that propagates at the velocity u = . When a frequency spectrum H( ) is used, a pulsed fi eld is generated (see text). in the same di
21、rection as the atoms (z 0), at velocity u = . The period being fi xed, the velocity u can be varied via .In particular it can be tuned on the atomic velocity, giving rise to an especially important phase-shift on the atomic wave contrarily to the second counter-propagating term which can be ignored.
22、 When a continuous spectrum H() is used instead of a single fre- quency, the resulting (pulsed) interaction potential takes the general factorized form: V (t,x) = s(t) cos ? 2 x ? (1) The time-dependent factor is real and can be expressed as: s(t) = g BMBf(t). Here g is the atomic Land e fac- tor, B
23、the Bohr magneton, M the magnetic quantum number and B the maximum of the fi eld magnitude, f(t) being a reduced dimensionless signal the maximum am- plitude of which is 1.Since s(t) is real, its frequency spectrum H() obeys the relation: H() = H()(2) where means complex conjugate. The motion along
24、the x axis of a wave packet submitted to the potential V (t,x) is governed by the time-dependent Schr odinger equation: i t = 2 2m 2 x + V (t,x) (3) where m is the atomic mass. Taking the Fourier trans- form of eq. (3) for variables x,k (k being the wave num- ber) one gets: i tC(t,k) = 2k2 2m C + W
25、C(4) where C and W are the spatial Fourier transforms of and V , and the convolution product. Setting C(t,k) = exp(ik 2 2mt)(t,k), one readily gets: i t(t,k) = s(t) e “ ik2 2m t ” ? (t,k )ei (k)2 2m t + (t,k + )ei (k+)2 2m t ? (5) -0.02-0.010.010.020.030.04 0.001 0.002 0.003 0.004 0.005 0.006 X (m)
26、S Z (m) FIG. 2: Trajectory of the wave packet centre (coordinates (t), (t) along x- and z-axes) in the comoving magnetic fi eld (dashed line) propagating in the x direction in the region z 0.The entrance plane is perpendicular to the z-axis. Rays start from a point-like source S located at 2 cm from
27、 the entrance plane. These rays are in plane x, z. They ini- tially make with the z axis diff erent incident angles ranging from 0 to 0.12 rad. All rays exhibit a negative refraction and fi nally emerge, for z 24 mm, parallel to their initial direc- tion. Calculations are made for Ar*(3P2, M=2) meta
28、stable atoms, the velocity of which is v0= 20 m/s. The maximum magnitude of the magnetic fi eld (see text) is 400 Gauss. Dis- tances are in meters. Note the diff erence in scale for x and z axis. where = 2/ ( k). A simplifi cation arises be- cause the k-derivatives of are largely dominated by those
29、of the exponential factor exp(ik 2 2mt) as soon as the wave packet has moved over a distance large com- pared to its own width 17. Under such conditions: i t(t,k) s(t)cos ? 2 k mt ? (t,k)(6) Then: (t,k) (0,k) expi(k,t)(7) where the phase shift is given by : (t,k) = 1 Z t 0 dts(t)cos ? 2 k mt ? (8) T
30、his real phase-shift takes a limiting value at t infi nite, namely, assuming a perfect synchronisation of the wave packet with the fi eld pulse (eqs. 1 and 2): (k) = 1H( k m) (9) This can be seen as a genericity property of comoving fi elds, in the sense that, in principle, any k-dependence of the p
31、hase-shift can be fashioned, using a convenient H() spectrum. As suggested previously in 17, it could be used to balance the natural spreading of a wave packet. The factor s(t), or its frequency spectrum H(), being 3 given, one may derive the evolution of the wave packet in the corresponding potenti
32、al. In particular, the semi- classical motion of the wave packet center is derived from the stationary phase condition: kkx k2 2m t + (k,t) = x k m t + k(k,t) = 0 (10) Then the abscissa of the center as a function of time is simply: (t) = k0 m t k(k,t)|k0(11) k0being the central momentum value.The q
33、uantity (t) = k(k,t)|k0represents the spatial shift induced by the potential. It is given by: (t) = 2 m Z t 0 dtts(t)sin ? 2 k mt ? (12) At large values of t, when the potential pulse is over, this shift naturally tends to a defi nite limiting value, namely: = 1 mH (k0 m) (13) where His the derivati
34、ve of H. The group velocity along x is readily derived from (11) and (12): vgx(t,k0) = k0 m tk(k,t)|k0 = k0 m 2 mt s(t)sin ? 2 k0 mt ? (14) The quantity 0= k0/(m) can be called the “reso- nance” frequency in the sense that the velocity u(0) = 0 of this spectral component of the comoving fi eld co- i
35、ncides with the atomic group velocity.For t 0) is sub- jected to a negative refraction, another signature of a left-handed or “meta” medium, whereas for M 0 the trajectory undergoes an ordinary refraction, with an ef- fective index smaller than 1.In the present case, the medium is anisotropic. It is
36、 uni-axial and optically ac- tive. Fig.2 shows rays issued from a point-like source S located at 2 cm from the interface, experiencing a neg- ative refraction (M = +2) in the xz plane, for diff erent incidence angles ranging from 0 to 0.12 rad. Numerical calculations have been carried out in the cas
37、e of a nozzle beam of metastable argon atoms Ar*(3P2), slowed down 0.020.040.060.080.1 0.01 0.02 0.03 0.04 0.05 ?Z (m) ?rad? FIG. 3: Position of the image S, where the support of the emerging ray crosses the z axis. The distance Z = SSis shown (in meters) as a function of the incident angle (in rad)
38、. At v0= 20 m/s, the stigmatism is realized with an accuracy better than 90% for incident angles lower than 0.04 rad. The chromatic dispersion is also shown (lower curve : v0= 18 m/s, mid curve : v0= 20 m/s, upper curve : v0= 22 m/s). Note the absence of chromatism at an incidence angle of 0.09 rad.
39、 FIG. 4: (a) 3D representation of half a cone of rays issued from a point-like source and making an angle = 0.1 rad with the z axis. All rays exhibit a negative refraction and fi nally emerge parallel to their initial direction.(b) Same as (a), with a 2D comoving potential s(t) = cos(2x/) + cos(2y/)
40、 (see text). The system behaves as a spherical meta-lens. Note that this surface slightly diff ers from that generated by the curve (z) rotated around the z-axis. at a velocity v0= 20 m/s by means of a Zeeman slower 18, 19. The parameters used in these calculations are, for the magnetic potential, =
41、 5 mm, B = 400 Gauss, f(t) = 2(t + )2et/for 0 t 1, = 0 elsewhere, with = 7.4 ms, = 0.37 ms, 1= 1.2 ms. Actually the negative refraction only appears when the magnitude of B exceeds a threshold value (280 Gauss in the present case). Rays symmetric of the previous ones with respect to the z axis exper
42、ience opposite refractions due to the opposite component of the comoving fi eld. The device thus behaves as a parallel plate, or a cylindrical meta- 4 - 0.020.050.1 ZHmL - 0.001 - 0.0005 0.0004 XHmL 0.0010.003 tHsL S X (m) Z (m) t(s) FIG. 5: Eff ect of three subsequent pulses s(t)+1.5s(t1ms)+ s(t2ms
43、) of comoving potential (shown in the inset), on the trajectories of the wave packet centre at incidence angles ranging from 0 to 16 mrad. A refocusing is seen, with a fi nal transverse astigmatism lower than 25m. lens which gives an image S of the source point S. In a real experimental situation, e
44、.g. in a cooled thermal or nozzle beam, there exists an atomic velocity distribution. The angular velocity spreading is related to the stigma- tism of the system, as shown in fi g.3. For an angular aperture of 0.08 rad, the axial stigmatism is limited to 3.3 mm. The dispersion of the velocity modulu
45、s gives rise to a chromatic eff ect. This eff ect is also shown in fi g.3, for a wide variation (10%) of v0. It is maximum (4 mm) at = 0 and cancels at 0.09 rad. Such “thermal” spreadings are much larger than those experimentally ac- cessible ( 0.01 rad, |v0/v0| 1 2%). Therefore eff ects related to
46、negative group velocities should be ob- servable. Obviously ultra-cold atoms, and a fortiori, con- densates are expected to give rise to even more striking eff ects. The evolution of a conical map of rays with = 0.1 rad is shown in fi g.4a. The previous treatment can be easily generalized to a 2D po
47、tential, provided it allows a separation of variables x and y, such as a potential in cos(2x/)+cos(2y/). Under such conditions the de- vice behaves at small incidence angles as a spherical lens (see fi g.4b). Atom wave re-focusing is actually predicted when using a series of two or three comoving pu
48、lses (Fig. 5). It is also worth noting that, with atoms initially spin- polarized in a linear superposition of Zeeman sub-levels, 1D-comoving potentials could be used in atom interfer- ometry, as very effi cient beam splitters. Finally, along the similarities between meta-media in light optics and a
49、tom optics, we should underline some of their basic diff erences. They originate in the distinct characteristics of wave propagation for Maxwell waves and “de Broglie - Schr odinger” waves: contrary to elec- tromagnetic waves, de Broglie waves for massive particles undergo a longitudinal wave packet spreading due to the fundamental vacuum dispersion in matter optics (as im- posed by the dispersion relation = k2/2m). Contrar- ily to optical meta-materials, in scalar de Broglie optics, atomic meta-media with negative index are charact