NEGATIVE-INDEX MEDIA FOR MATTER WAVES

Transcription

1 NEGATIVE-INDEX MEDIA FOR MATTER WAVES F. PERALES, V. BOCVARSKI, J. BAUDON *, M. HAMAMDA, J. GRUCKER, G. DUTIER, C. MAINOS, M. BOUSTIMI and M. DUCLOY Laboratoire de Physique des Lasers, CNRS UMR 7538, Université Paris 13, Av. J.B. Cléent, 9343-Villetaneuse, France (*) Corresponding author, One reviews the recently introduced field of atter-wave eta-optics, i.e. the extension of optical negative-index edia (NIM) to ato optics. After ephasizing the differences with light eta-optics and particularly the necessary transient character of NIM s in ato optics, we present the way of generating atter-wave NIM s and their general properties: negative refraction, ato eta-lenses. Finally their specific features are reviewed: longitudinal wave pacet narrowing associated to a tie-reversal effect, transient revivals of evanescent atter waves and ato reflection echoes at a potential barrier. 1. Introduction: Matter-Wave Optics With the fast developent of atter-wave optics, any of the functions previously operated in light optics have been realised: ato diffraction and irrors, bea splitters, ato lasers, ato holography, quantu reflection, etc. [1]. Specific characters of those processes originate in the properties of the associated particle: non-zero ato ass, vacuu dispersion for the de Broglie waves (iplying longitudinal wave pacet spreading), scalar character of the atoic wave function, influence of the internal atoic degrees of freedo Along this viewpoint, novel areas in the field of ato optics are presently explored, including e.g., the devising of non-diffracting ato nano-beas via a specially designed transverse Stern-Gerlach interferoeter []. Recently, we have proposed to extend the concept of eta-optics (nown as negative index aterials, NIM-s, in light optics [3-4]) to atter waves [5]. The ain specificity 1

2 of NIM eta-edia for ato optics is their transient character, lined to the fact that for atter waves the reversal of ato group velocity is operated and can be only transient contrary to the equivalent process in light optics where the phase velocity is reversed. This basic difference is connected to the fact that group velocities are well defined in de Broglie-Schrödinger optics - as they are related to the particle s velocity - while the phase velocity varies, depending on the gauge. Indeed it appears quite delicate to reverse the phase velocity for atter waves. In ost cases, slow assive particles, e.g. atos, cannot penetrate dense atter. Nevertheless external potentials can be used to perturb the trajectories in vacuo. As shown in [5] and discussed below, cooving potentials [6], conveniently applied to ato waves, allow one to reverse the ato group velocity, with respect to the phase velocity. This is enough to ae appear features characteristic of NIM edia, especially the negative refraction phenoenon in which incident and refracted trajectories are on the sae side of the noral to the interface, i.e. to the plane beyond which the potential begins to act (part ). In a sei-classical description and because the NIM effect is transient, one has to consider wave pacets synchronised with the applied cooving potential pulse. Therefore not solely the wave-pacet centre trajectory has to be considered, but also the dynaics of the wave pacet itself, in particular the evolution of its width. As it will be shown in part 3, the NIM effect is equivalent to a tie reversal process resulting in a transient narrowing of the wave pacet. Finally the evolution of (sei) evanescent wave-pacets generated beyond a static potential barrier and experiencing a cooving potential pulse is exained in part 4.. Negative index edia (NIM) for atter waves and negative refraction Cooving potentials provide us with a rearably siple solution to devise negative-index edia for atoic waves. Such potentials have been previously used in several experients, ost of the dealing with Stern-Gerlach ato interferoetry. They have been described in detail in [6]. Their general for is: V(t, x) = s(t) cos(π x/λ) (1) where Λ is a spatial period and s(t) a real tie-dependent factor which differs fro zero within a finite interval [, τ 1 ]. Any ind of potential (agnetic, optical ) is usable. In the case of atos possessing a spin J, such as etastable argon atos Ar*( 3 P ), a tie-dependent, space-odulated agnetic field leads

3 for s(t) to the expression s(t) = g µ B M B f(t), where g is the Landé factor, μ B the Bohr agneton, M the agnetic nuber, B the axiu agnitude of the field and f(t) a noralized tie factor. As shown in [5], in the sei classical regie, the effect of the cooving field is siply to add to the atoic phase at a specific value of the oentu, Φ (, t, x), a tie-dependent phase-shift: ϕ(, t) = 1 t h dt s(t ) cos [π h t /(Λ)] () where is the ato ass. This real phase-shift taes a liiting value at t infinite, naely, assuing a perfect synchronisation of the wave pacet with the field pulse: ϕ 1 h () = h Re[H ] (3) Λ where H(ν) is the frequency spectru of s(t). The total perturbed atoic phase is then: h Φ(, t, x) = x - t + ϕ(, t) (4) The stationary-phase approxiation readily gives the otion ξ(t) of the wavepacet centre: ht Φ] = = ξ ( t) O + ϕ] (5) where is the central value in the wave-pacet oentu distribution. The last ter in (5) represents a shift of the trajectory induced by the cooving pulse along the x axis. Note that the sign of this shift depends on the sign of the agnetic nuber M, aing the device usable as a bea splitter. The coponent of the group velocity along x is: v gx = h t ϕ] (6) Clearly, by a proper choice of ϕ (i.e. of s(t)), v gx can be ade transiently negative. Indeed it is a transient effect since, for t > τ 1, ϕ reaches its liiting value and v gx recovers its initial value h /. When two spatial coordinates, e.g. x and z, are involved, the group velocity v gz is a constant and the z- coordinate of the wave-pacet centre is ζ(t) = v gz t. In figure 1, trajectories of 3

4 .6 x () S z () Figure 1. Trajectory of the wave pacet centre (coordinates ξ(t), ζ(t) along x- and z-axes) in the cooving agnetic field propagating in the ± x directions, in the region z. The entrance plane is perpendicular to the z-axis. Rays start fro a point-lie source S located at c fro the entrance plane. These rays are in plane x, z. They initially ae with the z axis different incident angles ranging fro to.1 rad. All rays exhibit a negative refraction and finally eerge, for z > 4, parallel to their initial direction. Calculations are ade for Ar*( 3 P, M=) etastable atos, the velocity of which is in odulus v = /s. The axiu agnitude of the agnetic field (see text) is 4 Gauss. Distances are in eters. Note the different scales for x and z axes. wave-pacet centres starting fro a point-lie source S (x =, z = - c) are shown in the plane x, z. They initially ae with the z axis different incident angles ranging fro to.1 rad. The interfaces between free space and potential zone are defined by ties t = and t = τ 1 at which s(t) is switched on and off respectively, i.e. by definite values of z, and v gz τ 1. Therefore the interfaces are two planes perpendicular to z axis: the potential zone is equivalent to a parallel plate whose noral is the z axis. It is seen that all rays exhibit a negative refraction and finally eerge parallel to their initial direction. Calculations are ade for argon etastable atos Ar*( 3 P, M=), the velocity of which is, in odulus, v = /s. The paraeters used here are, for the agnetic potential, Λ = 5, B = 4 Gauss, f(t) = ε (t + ε) - e -t/τ for t τ 1, f(t) = elsewhere, with ε =7.4 s, τ =.37 s, τ 1 = 1. s. Actually, as would do a plate of 1D-eta aterial in light optics, this parallel plate acts as a cylindrical eta-lens. It is easily transfored into a spherical eta-lens by using a D-cooving potential pulse such as: s(t)[cos (π x/λ) + cos (π y/λ)] [5]. In real experiental situations, e.g. in a cooled theral or nozzle bea, an atoic velocity distribution is present. The angular velocity spreading is related to the stigatis of the syste. For an angular aperture of.8 rad, the axial stigatis (along the z axis) is liited to 3.3. The dispersion of the velocity odulus gives rise to a chroatic effect. For a wide variation (± 1 %) of v, 4

5 this effect is axiu (± 4 ) at θ = and cancels at θ.9 rad. The theral dispersion we assue here is in fact uch larger than those actually accessible in standard nozzle bea experients ( δθ <.1 rad, δv /v 1- %). Consequently both negative-refraction and eta-lens effects are expected to be rather easily observable. When a series of subsequent cooving pulses is applied, a re-focussing effect is obtained. For instance, with three identical pulses (apart fro their relative heights which are 1, 1.5, 1), starting at ties, 1. s,.4 s, trajectories issued fro S within the incident angular range [-.1,.1 rad], are focussed within a spot whose diaeter is less than 1 μ (see fig. ).,1,1,8,6,4 Distance in,, -, -,4 -,6 -,8 -,1 -,1 -,,,,4,6,8,1,1,14,16 Distance in Figure. Effect of three subsequent pulses s(t) s(t 1s) + s(t s) of cooving potential on the trajectories of the wave pacet centre at incidence angles θ ranging fro -16 rad to 16 rad. A refocusing is seen, with a final transverse astigatis saller than 1 µ. The vertical axis is x, the horizontal axis is z. 3. Tie reversal and wave-pacet narrowing So far we have used the stationary-phase approxiation (eq. (5)) to deterine the wave-pacet centre otion. This approxiation can be used as well to investigate the tie evolution of the wave-pacet width. Indeed it is based upon an expansion, up to second order, of Φ over in the vicinity of : ] Φ(, t, x) Φ(, t, x) +. (- ) + 1 [ Φ (- ) + (7) When it is incorporated in the integral expression of the wave function - assuing a Gaussian oentu distribution, naely: Φ Ψ(t,x) = N d Exp[ ( ) /(δ )] Exp[i ] (8) 5

6 where N is a noralisation constant, - expansion (7) leads to a readily calculable Gaussian integral. Without external potential, one gets: Ψ (t,x) N π ( δ ih + t) 1/ Exp 1 ( δ δ + i x) ih + t (9) The width of the free wave pacet along x axis at a given tie is given by: σ(t) = ih t + δ 1/ (1) This is the standard wave-pacet spreading, lined to vacuu dispersion for atter waves (we assue a iniu width, σ = δ -1, at t = ). When the cooving potential pulse is present, the expansion (7) reains valid provided that the ter ϕ] is added to the second -derivative of Φ. Consequently, when the potential pulse is present, the width is given by: σ 1 (t) = ih t + δ i ϕ] 1/ (11) It naturally involves the concavity of ϕ as a function of (cf. [6]). The tie evolution of Δ = σ(t) σ and σ 1 (t) σ are shown in figure 3 (left). The calculation uses the sae set of paraeters as the previous ones. It is seen that the width is transiently reduced by the potential pulse and then recovers its natural evolution, exactly as if the tie had been transiently reversed. This interpretation is confired by an approxiate calculation (reaining nevertheless close to reality), in which the arguent of the cosine in the expression () of ϕ is assued to be uch saller than one. Under such condition, one readily gets: ϕ(, t) ϕ 1 (t) + ϕ (t) (1) 1 with ϕ 1 (t) = dt' s(t') h t h π ; ϕ (t) = dt' t' Λ t s( t') (13) This leads to ϕ = ϕ (t) and finally to: 6

7 σ 1 (t) ih t eff (t) + δ 1/ (14) where the effective tie is given by: t eff (t) = t 4 ϕ ( t) = t π Λ dt' t' s( t' ) h t (15) This effective tie is shown in figure 3 (right) as a function of t. A tie reversal is clearly seen. It could be even lengthened by using several subsequent pulses. The validity of the previous approxiation is warranted by the fact that σ[t eff (t)] is very close to σ 1 (t). A question that iediately arises is: how to observe this phenoenon? For what concerns the narrowing of the wave pacet, the answer is far fro being obvious. Indeed our previous assuption that the iniu of the width (σ = δ -1 ) occurs at t =, is unrealistic. The ties at which σ(t) = σ are actually distributed over a range depending both on the source and on the chopping process. In any case, because of the additional contribution coing fro the statistical distribution of the atoic velocities, this range is such that the contribution of the widths to the tie-of-flight (t-o-f) spectru is at ost equal to the t-o-f resolution, typically a few percent. Even if the wave-pacet Δ (s) t eff (s) t (s) b t (s) Figure 3. (Left) Difference Δ = σ(t) σ as a function of tie; σ(t) is the wave-pacet width at tie t and σ = σ() = 1 μ is the iniu width. Blue curve (left): natural width evolution; red curve (right): evolution when a cooving potential pulse is present. A narrowing is seen, the iniu width being recovered at t = s. (Right) Effective tie t eff (see text) as a function of real tie. A reversal is seen in the interval < t < 1. s, corresponding to the pulse duration. 7

8 width contribution was doinant, it is seen that the recovering of the iniu width occurs around t = s later. To observe directly this narrowing in a t-o-f easureent would iply a tie of flight of a few s at ost, i.e. a flight path of a few. The use of a series of pulses is able to increase the value of t and then the available flight path, but such an experient reains delicate to carry out. A uch easier way to observe the tie reversal phenoenon is to easure the retardation effect induced by a potential pulse or a series of potential pulses. Figure 4 shows t-o-f spectra at a distance of 9 c fro the source, without any pulse and with 1 subsequent identical potential pulses, shifted by 1.51 s with respect to each other. A retardation effect of about 7.5 s is predicted, without apparent ato pacet broadening. The relative t-o-f dispersion (4 %) is assued to coe exclusively fro the width dispersion. It is the ost optiistic situation for our purpose, but even with an equal additional contribution of the statistical distribution of velocities, the effect would reain observable Figure 4. Tie of flight (t-o-f) spectra produced by an enseble of wave pacets, the iniu width of which occurs within a range such that the relative t-o-f resolution is 3%. Left hand side pea: no pulse; right hand side pea: 1 identical subsequent potential pulses (see text). A delay of about 7.5 s is induced. This retardation of the atoic pacets occurs without any pacet broadening. Even if a suppleentary width of 3% due to the velocity spreading was added, the effect should reain observable. 4. Revivals of atter evanescent waves and ato echoes Sei-evanescent wave pacets are readily generated beyond a repulsive potential barrier. In the case of atos possessing a spin J, such a barrier can be created using a static agnetic field B in the half-space x >. The height of the barrier is V = g μ B M B = h a /(). We are dealing now with three wave functions, naely ψ i (t, x), the incident wave propagating fro x = - to x =, ψ r (t, x), the reflected wave propagating bacwards fro x = to x = -, and the transitted wave ψ (t, x) in the region x >. For a specific value of the wave nuber, i.e. a specific value E of the total incident energy, the three wave functions have the sae tie-dependence, in exp [i h t /()]. In the absence of cooving potential pulse, the conservation of the probability flux leads to the usual reflection and transission factors (in aplitude): 8

9 R() = ( - iκ)/( + iκ); T() = /( + iκ), with κ = a. As expected, since the evanescent wave does not carry any probability flux, R = 1 when < a. Siilar expressions of R and T are obtained for a (propagation within the barrier), by replacing iκ by = a In a first approxiation, these expressions of R and T also hold if the cooving potential is applied at tie t = +, i.e. at a sall but finite distance fro the barrier ( < x << 1/κ). Then, in spite that, since it does not propagates - the evanescent wave never really leaves the barrier edge -, we shall assue that the effect of the cooving potential pulse V(t, x) can be treated within the only half-space x >. Under this condition, the tie dependent Schrödinger equation is [Ψ(t, x) is the wave function and V(t,x) is given by eq.1] : Let us set: h h tψ = x Ψ + [V + V(t, x)] Ψ (17) i Ψ(t, x) = g(t, x) Ψ (t, x) (18) where Ψ (t, x) is the solution in absence of cooving potential. We shall assue that the perturbation factor g evolves, as a function of x, uch ore slowly than Ψ. Under such conditions: x g << κ x g (19) (the validity of this assuption will be exained a posteriori) and one finally gets [7]: i e (t,x) g = t sup e ϕ 1 π ih κ where ϕ e = h dt' s(t' ) cos (u t' ) () Λ The upper bound of the integral is t sup = Min [t, τ 1 ], and u = t + i x /( hκ). Finally, the phase shift ϕ e depends on 3 independent variables, (via κ), t and x (via u). It is worth noting that ϕ e becoes identical to the phase shift ϕ introduced previously (eq.) provided that t x /( h) << t + x /( h). This eans that, for a purely propagating wave pacet, all relevant values of t and x are close to those related to the wave pacet centre. Such an 9

10 approxiation is clearly no longer valid for an evanescent or a sei evanescent wave pacet. Calculations have been carried out with argon etastable atos (Ar* 3 P ) the velocity of which along the x axis is v x = 4 /s (de Broglie wavelength λ db.8 n), polarized in the M = + Zeean sublevel. The axiu agnitude of the cooving field B ax is varied fro up to 4 Gauss. The centre of the Gaussian oentu distribution ρ() is = Evanescent waves appear within a repulsive potential barrier of oentu height a = (B =17 Gauss) as soon as < a. This value of the height has been chosen in order to get a sei-evanescent regie. The standard deviation in ρ() is δ =.5. In s(t), tie paraeters are different fro those used before since they ust be adapted to typical characteristics of the wave pacets in the region x >. Their values are as follows: ε = 7.4 µs, τ =.37 µs, τ 1 =.6 µs. The spatial period is Λ = μ. The effect of the cooving pulse on the wave function in half space x > is siply obtained by incorporating the factor exp[i ϕ e (, t, x)] into the expansion of this wave function over. At this point, let us exaine the validity of approxiation (19). Actually it is rather poor for the original cooving potential in cos(πx/λ), in the sense that the ratio of eq.19, κ x g / x g, which is zero for κ =, becoes larger than 1 only when > δ. The reason is that such a potential starts abruptly at x =. The situation is greatly iproved by slightly shifting the spatial dependence, into, for instance, cos (πx/λ π/5), which is quite easy fro an experiental point of view. In such a case, the ratio is larger than 1 as soon as > δ/. In other words, only a narrow central slice of the spectru violates the approxiation. It is expected and it has been verified to be of little iportance in so far as, for the spatial dependence of the wave pacets, it involves large distances (x > /δ = 1.78 μ). In what follows, the spatial dependence in cos (πx/λ π/5) has been used. In figure 5a the squared odulus ψ of the unperturbed wave function is shown as a function of t and x, whereas figure 5b shows ψ when a cooving potential pulse of agnitude B = 1 Gauss is applied. A negative refraction effect beyond the barrier edge (x > ) is clearly seen: while the tail of ψ at large x is decreased, a revival of ψ at sall x is observed. Because of the continuity conditions at x =, this revival generates in the free space (x < ) a second reflected wave pacet propagating towards x < (atoic reflection echo). The process can be repeated by applying several subsequent pulses: repeated rebounds (and reflection echoes) are generated beyond the barrier edge, aing an atoic creeping wave, or an atoic surface wave (see figure 5c). 1

11 (a) (b) t (s) (c) x () Figure 5. (a) Profile in t and x of the squared odulus ψ of the unperturbed sei-evanescent wave function. The height (in oentu) of the potential barrier is equal to the central value of the incident oentu distribution. At x =, the tie dependence is Gaussian. For x >, one observes the decay characteristic of the evanescent part together with a propagation at a positive group velocity. (b) Sae as (a), in the presence of a single cooving potential pulse of agnitude B ax = 1 Gauss. An internal negative refraction is seen, resulting in a revival (a rebound) of the wave function. (c) Exaple of successive rebounds obtained by applying two subsequent potential pulses. 5. Conclusion Provided that their (teporal) frequency spectru is properly suited, cooving potentials are able to shift the phase of an atoic wave, with alost any adjustable dependence on the oentu. Owing to this generic property, these potentials can be used to reverse the group velocity of atter wave pacets the phase velocity reaining alost unchanged and create a eta-ediu for atter waves, close to, but nevertheless different fro, eta-aterials in 11

12 light optics. These differences coe fro the respective natures of de Broglie atter waves and light waves. A wide variety of new phenoena in ato optics results fro the action of cooving potentials on atos. Soe of the have been described here: negative refraction, eta-lens effect, transient narrowing of wave pacets and tie reversal effect, rebounds at a barrier edge of sei evanescent wave pacets and generation of atoic surface waves. Most of these phenoena are expected to be experientally observable. Soe of the can find applications, for instance in ato interferoetry. Indeed when the atoic state is a linear superposition of Zeean sublevels, the NIM ediu operates as a very efficient bea splitter (the spatial shift induced between different M- states is in the illietre range). Then, applying a second delayed pulse opposite in sign to the first one, one gets trajectory loops. Using the Stern- Gerlach interferoetric technique [], one builds a closed interferoeter of large area (several ) particularly sensitive to inertial effects. Other new phenoena in ato optics can be iagined, dealing in particular with D cooving potentials, or with the use of special series of subsequent potential pulses. Authors are ebers of the Institut Francilien de Recherche sur les Atoes Froids (IFRAF). References 1. See, for instance, P. Meystre, Ato Optics (Springer Series on Atoic, Optical and Plasa Physics, Springer, 1), and references therein. F. Perales, J. Robert, J. Baudon and M. Ducloy, Europhys. Lett., 78, 63 (7) 3. V.G. Veselago, Sov. Phys. Usp., 1, 59 (1968) 4. J. Pendry, Phys. Rev. Lett., 85, 3966 (); C. Foteinopoulou et al., Phys. Rev. Lett., 9, 1741 (3) 5. J. Baudon, M. Haada, J. Grucer, M. Boustii, F. Perales, G. Dutier and M. Ducloy, Phys. Rev. Lett. 1, 1443 (9) 6. R. Mathevet et al. Phys. Rev. A, 56, 954 (1997) 7. F. Perales et al., to be published 1