Retardation Effects on Quantum Reflection from An

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PHYSICAL REVIEW A VOLUME 58, NUMBER 5 NOVEMBER 1998

Retardation effects on quantum reﬂection from an evanescent-wave atomic mirror

R. Coˆte´ and B. Segev* Institute for Theoretical Atomic and Molecular Physics (ITAMP), Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, Massachusetts 02138

M. G. Raizen Department of Physics, The University of Texas at Austin, Austin, Texas 78712-1081 ͑Received 19 May 1998͒ We calculate the reﬂection probability for ultracold sodium atoms incident on an evanescent-wave atomic mirror. For low enough energies the reﬂection probability curves exhibit quantum effects that are sensitive to the long-range part of the potential and can therefore be used to resolve Casimir retardation effects. We also explore the accuracy of model potentials approximating the exact atom-wall effective potential. ͓S1050-2947͑98͒08811-8͔

PACS number͑s͒: 42.50.Vk, 34.20.Ϫb

I. INTRODUCTION it, the phase of the reflected laser beam. This phase shift can be used to detect the passage of a cloud of atoms without An electromagnetic mirror for neutral atoms was first sug- destroying the coherence of the reflection ͓8͔. Experiments gested by Cook and Hill in 1982 ͓1͔. Their idea was to use with a time-dependent mirror, in which the potential changes the radiation force of an evanescent-electromagnetic wave rapidly and continuously by controlled modulation of the outside a dielectric surface to repel slow atoms. Such an intensity of the laser light, were performed ͓9͔. The use of evanescent wave is formed when light undergoes total inter- evanescent-light mirrors to build a surface trap was recently nal reflection at the dielectric-vacuum interface. The induced reported ͓10͔, and the creation of an atomic funnel was also dipole force is repulsive if the light is tuned toward higher proposed ͓11͔. The penetration depth and wave vector of the frequencies than the corresponding natural dipole frequency evanescent-electromagnetic waves were studied using on of the atom, and attractive if tuned toward lower frequencies. resonance laser atomic spectroscopy ͓12͔. Finally, atom This evanescent-wave atomic mirror was realized experi- guiding using evanescent waves in small hollow optical fi- bers was demonstrated ͓13͔. mentally by Balykin et al. ͓2͔. Almost perfect reflection was In the simplest theoretical approach, the center-of-mass demonstrated by using laser light of high intensity to create motion of the atom is treated classically while the atom- the evanescent wave, and Na atoms incident on the mirror in mirror interaction is derived from a model of a two-level grazing angles so as to minimize their velocity at the direc- system in the electromagnetic field of the evanescent laser tion perpendicular to the dielectric surface. Normal incidence light. Neglecting spontaneous emission, internal transitions, reflection in the form of an atomic ‘‘trampoline,’’ where at- and the attractive van der Waals interaction between the oms falling onto the mirror surface due to gravity bounce atom and the dielectric surface, the interaction of the atom back up because of the evanescent wave, has also been dem- and the mirror is then described by an exponentially decay- onstrated ͓3,4͔. ing optical potential. Although one would like to preserve as Recent work in atom optics has emphasized, on the one simple a physical picture of the reflection as possible, for hand, the need to develop elements, like mirrors and lenses, actual experiments it may be important to give a more com- for the coherent manipulation of atomic matter waves ͓5͔. plete treatment. On the other hand, the experimental progress brought about We consider here the following questions. Is the center of the ability to cool and launch atoms with extremely low and mass motion of the atom properly described by classical me- well-defined velocities ͓6͔. This, combined with the ability to chanics or does it exhibit quantum effects? Is the reflection continuously vary the effective potential of the mirror by sensitive to the atom-dielectric attractive interaction, and changing the intensity and the frequency of the laser light, how sensitive is it to details of the potential curves? Can it be motivates both theoretical and experimental studies of fun- used to distinguish among different theoretical models, for damental and practical aspects of the evanescent-wave mir- example, resolving retardation effects? ror. Aspect and collaborators studied an analytical solution of Experimental applications of the evanescent-wave mirror the Schro¨dinger equation obtained by neglecting the attrac- already include a variety of studies. It was recently incorpo- tive atom-wall interaction, and identified both classical and rated into an atomic interferometer ͓7͔. The presence of the pure-quantum regimes ͓14͔. The same group later used the reflected atoms near the dielectric-vacuum interface changes reflection from an evanescent-wave atomic mirror in the the index of refraction outside the dielectric prism and with classical regime to measure the van der Waals force between the dielectric surface and the atom in its ground state, dem- onstrating that the attractive interaction can hardly be ne- *Present adderess: Department of Chemistry, Ben-Gurion Univer- glected ͓15͔, as was also demonstrated by Desbiolles et al. sity, Beer Sheva 84105, Israel. ͓16͔. The purpose of this paper is to give a quantum treat-

tices to launch a subrecoil sample with negligible heating, as described by Raizen, Salomon, and Niu ͓6͔. Atom-surface scattering experiments can be performed by launching the atoms in an upward ͑nearly vertical͒ parabolic trajectory, with the surface in the vertical direction. We show that the long-range ͑Casimir͒ interactions due to retardation have in this case an observable effect on the quantum reflection probabilities and that the classically for- bidden above-barrier reflection is particularly sensitive to details of the atom-surface potential. If measured, this would constitute a low-energy experimental signature of QED effects, and would supplement recent different measurements of the Casimir forces, e.g., in cavity QED ͓20͔. FIG. 1. Schematic of an atomic mirror. The blue-detuned laser The theoretical analysis is divided into two steps. First, in light produces a repulsive force, after total internal reflection. An Sec. II, we discuss the effective potential, and second, in Sec. approaching atom will feel the effective potential formed by the III, we analyze the time evolution of the particle in this po- atom-wall and light induced potentials and be reflected. Classically, tential. There are standard physical assumptions and approxi- the atom is reflected only if its incoming energy E is less than the mations involved in being able to treat the process in this barrier height Vmax , contrary to the quantum regime where it can be reflected even if EϾV . way. Note the apparent contradiction between the two steps. max The internal structure of the atom is essential for the evalu- ment of the reflection, which takes the attractive interaction ation of the potential curves in the first step. In general, an into account. effective potential for an atom or a molecule is the result of In a recent Rapid Communication, we have shown that changes to its internal structure. Optical potentials, for ex- the center-of-mass motion of cold atoms incident on the ample, are determined by the space dependence of the atom’s evanescent-wave atomic mirror with sufficiently small ve- energy levels, and therefore of the induced dipole interaction locities ͑e.g., 10.0 cm/s for sodium atoms͒ will exhibit quan- with the laser’s electric field. In the second step, however, tum dynamics ͓17͔. We predicted an S-shaped probability after a potential curve has been determined, one solves for curve, which is typical of quantum behavior, for the fraction the dynamics, i.e., for the time evolution of the particle in the of reflected atoms as a function of the logarithm of the laser given potential, treating the atom as a pointlike structureless intensity, and suggested a way to measure it by minimizing particle. The disregard of the internal structure of the atom, the variation of the intensity across the atomic mirror. We at the second step of the calculation is justified by the as- have also referred to the possibility of observing Casimir sumption that actual changes to this internal structure can be retardation effects in the reflection process. However, at that neglected, while the space dependence of the effective po- time we were not able to make definite predictions as to the tentials is the result of virtual, not actual, transitions. For magnitude of the retardation effects because an exact poten- optical potentials, for example, this amounts to neglecting tial curve including the QED effects for a ground-state so- spontaneous emission. This approximation renders our treat- dium atom and the dielectric prism was not yet calculated. ment meaningful only in the large detuning regime. Further In this paper we review the previous results in more detail discussion of this point can be found, for example, in ͓21͔.In and present new ones. In particular, the numerical method Sec. IV, we give the results of our calculations for the reflec- that we have used for the calculations is presented and re- tion probability curves with and without retardation. Finally, flection from the physical potential is compared to the reflec- discussion and conclusions are presented in Sec. V. tion from model potentials. Finally, we use recently calculated potential curves ͓18͔ to obtain precise predictions for II. ATOMIC MIRROR: THE POTENTIAL CURVE the size and nature of the QED retardation effects on the quantum reflection probability curves. A. Optical potential The experimental setting considered here and depicted in Optical potentials are one example where an effective po- Fig. 1 relates to an uniform flux of Na atoms with an average tential for cold atoms is induced by space-dependent changes velocity of a few cm/s and a velocity dispersion of a few to the internal energy of the atoms. In the same way as elec- mm/s that would be created and launched in the direction tric fields induce Stark shifts and magnetic fields induce Zee- normal to a dielectric prism. The repulsive optical potential man shifts, radiation fields have been shown to induce light of an evanescent wave of blue-detuned laser light would shifts to the atom’s level structure, thereby creating con- combine with the attractive atom-wall interaction to create trolled and adjustable effective potentials. These potentials an effective potential barrier for the atoms. The flux of re- are important in the field of atom optics as they enable the flected atoms would be measured. coherent manipulation of atoms. Such an atomic beam with subrecoil velocity spread con- A simple successful model for the interaction of a two- ditions has been demonstrated experimentally by the group level atom with the radiation of a single-mode laser and the of Phillips ͓19͔. In this work, a Bose condensate of 106 so- vacuum fluctuations was given within the dressed atom ap- dium atoms was formed. The momentum spread was reduced proach to light-induced potentials ͓21͔. By solving the opti- to 0.1 recoil ͑3 mm/s͒ by adiabatic expansion of the conden- cal Bloch equations in the electric dipole approximation, one sate. The atoms were then used in an atomic interferometer. finds that two forces act on the atom in its ground state: a This method can be combined with accelerating optical lat- radiation pressure force and a dipole force ͓21͔. The dipole PRA 58 RETARDATION EFFECTS ON QUANTUM REFLECTION... 4001

metal TABLE I. Parameters used in this paper. The values of C3 metal (n) and K4 are taken from Karchenko et al. ͓25͔, and K4 is from ͓17͔. We also give the sodium atom mass in atomic units.

Parameter Value

⑀ 3.258 n 1.805 Ϫ4 kL 5.645ϫ10 a.u. ␪ 45° ␬ 4.4771ϫ10Ϫ4 a.u. metal C3 1.889 a.u. (n) C3 1.0017342 a.u. metal K4 2676.71 a.u. (n) K4 1081.03 a.u.

m 41907.782 a.u. FIG. 2. Atom-wall potential Vatom wall for a conducting and a dielectric wall ͑with nϭ1.805). Also shown are the asymptotic 4 3 3 forms K4 /z and the Lennard-Jones forms C3 /z . We plot V(z)z 3 force can be described by the effective potential as a function of z so that C3 /z becomes a straight line. Numerical ប␦ ⍀2/2 values of C3 and K4 are given in Table I. For both metallic and V ϭ ln 1ϩ , ͑1͒ dielectric walls, retardation effects reduce the strength of the poten- dipole 2 2 2 ͫ ␦ ϩ⌫ /4ͬ tial, and the potentials take their asymptotic form near z ជ ជ ϳ5000a0 . where ⍀ is the Rabi frequency, and ␦ϵ␻LϪkL•pϪ␻0 is the detuning from the atomic resonance of frequency ␻ and 0 If we take into account retardation effects, the Casimir- ជ natural linewidth ⌫ of an atom of momentum p interacting Polder potential is obtained where the ﬁnite propagation time with photons of momentum kជ L and frequency ␻L . For large between the dipole and its image results in a different detuning the dipole force dominates ͓21͔ and the potential asymptotic power-law behavior ͓23͔, reduces to lim V ͑z͒ϰz Ϫ4. ͑4͒ large ␦ 2 CP ប⍀ z ϱ V , ͑2͒ → dipole → 4␦ The complete QED treatment gives with ⍀2ϰId2 where I is the laser intensity and d is the atomic dipole. The dipole potential is therefore proportional ␣3 ϱ ϱ to the intensity of the laser and inversely proportional to V ͑z͒ϭϪ d␰␰3d͑i␰͒ dpeϪ2␰zp␣␨͑p,⑀͒, QED 2␲͵ ͵ detuning. The dipole force can be attractive or repulsive, 0 1 ͑5͒ according to the sign of the detuning. For red detuning ␻L Ͻ␻0 , ␦Ͻ0, and the atom is attracted by high intensity, in where accordance with the classical intuition for a dipole in a ﬁeld. However, for blue detuning, ␻LϾ␻0 , ␦Ͼ0, and the atom is ͱ⑀Ϫ1ϩp2Ϫp ͱ⑀Ϫ1ϩp2Ϫ⑀p repelled by high intensity. This repulsion is being used in the ␨͑p,⑀͒ϵ ϩ͑1Ϫ2p2͒ , 2 2 evanescent-wave atomic mirror. ͱ⑀Ϫ1ϩp ϩp ͱ⑀Ϫ1ϩp ϩ⑀p ͑6͒

B. Atom-dielectric interaction and d(i␰) is the dynamic dipole polarizability function ͓24͔. with and without retardation effects Using semiempirical results for the dynamic dipole polar- The simplest model for the interaction of a ground-state izability, the numerical integration of Eq. ͑5͒ was performed atom and a wall of dielectric constant ⑀ considers the inter- to obtain accurate QED potential curves ͓25͔. We present in action between a dipole dជ and its mirror image and gives the Fig. 2 the result of a recent calculation ͓18͔ of VQED(z) for Lennard-Jones potential, sodium and a dielectric prism with the index of refraction 2 2 ͑n͒ considered here, namely, nϭ1.805 ͓26͔. The deviation from ⑀Ϫ1 ͗d͉͉͘ϩ2͗dЌ͘ C3 the Lennard-Jones potential of Eq. 3 as well as the Casimir- V ϭϪ zϪ3ϭϪ , ͑3͒ ͑ ͒ LJ ⑀ϩ1 ͩ 64␲⑀ ͪ 3 Polder asymptotic behavior of Eq. ͑4͒ are clearly demon- ͩ ͪ 0 z 2 2 strated. where ͗d͉͉͘ and ͗dЌ͘ are the expectation values of the squared dipole parallel and perpendicular to the surface ͓22͔, C. Effective potential of the evanescent-wave mirror ⑀ϭn2 and n is the index of refraction of the dielectric. This expression for the potential is approximately valid for con- An evanescent-wave mirror for cold atoms is created (n) stant ⑀ and small z. C3 is related to the constant C3 of a when a blue-detuned laser beam (␦Ͼ0) with wave number (n) 2 2 pure metallic wall by C3 ϭC3(n Ϫ1)/(n ϩ1). The nu- kជ L undergoes total internal reﬂection inside a dielectric merical values for Na atoms used in this paper are given in prism. The optical dipole potential Voptical for the atom is Table I. proportional to the intensity of the laser beam, which drops 4002 R. COˆ TE´ , B. SEGEV, AND M. G. RAIZEN PRA 58

TABLE II. Parameters for the potential curves of Fig. 3. For a given value of the optical potential intensity C0 , there are two potential curves, Vexact with retardation and Vno ret without retardation. For each graph we give the height of the potential Vmax as well 1 2 as the corresponding velocity vtop ,(Vmaxϭ 2 mvtop), the location of the maximum of the barrier zmax , and the curvature ␣ at the top of the barrier. For Vno ret of Figs. 3͑e͒ and 3͑f͒, there is no barrier, hence no Vmax , vtop , zmax , and ␣.

Figure C0 Type Vmax vtop zmax ␣ Ϫ10 Ϫ11 Ϫ16 (10 (10 ͑cm/s͒͑units of a0)(10 a.u.͒ a.u.͒ a.u.͒

3͑a͒ 9.877 ret. 13.454 17.53 1629.5 Ϫ3.7754 no ret. 4.378 10.00 2243.4 Ϫ1.0531

3͑b͒ 7.463 ret. 8.207 13.69 1793.0 Ϫ2.1277 no ret. 1.576 6.00 2614.9 Ϫ0.4077 FIG. 3. The effective potential with retardation Vexact ͑solid line͒ and without retardation Vno ret ͑dashed line͒ as a function of the 3͑c͒ 6.023 ret. 5.483 11.19 1935.2 Ϫ1.3649 distance z for various values of the laser intensity and hence of C0 . no ret. 0.394 3.00 3022.2 Ϫ0.1542 The parameters of these potentials are given in Tables I and II. As the value of C decreases from ͑a͒ to ͑f͒, the barrier diminishes and 0 3͑d͒ 5.375 ret. 4.378 10.00 2017.5 Ϫ1.0541 is pushed out. Notice that, for ͑e͒ and ͑f͒, there is no barrier for the no ret. 0.013 0.54 3334.6 Ϫ0.0739 case without retardation Vno ret . exponentially with the distance z outside the surface of the 3͑e͒ 3.391 ret. 1.576 6.00 2416.9 Ϫ0.3447 dielectric prism. The optical potential is therefore no ret.

2 VopticalϭC0exp͑Ϫ2␬z͒exp͑Ϫ␳ ͒, ͑7͒ 3͑f͒ 2.119 ret. 0.394 3.00 3000.4 Ϫ0.0828 no ret. with

2 2 ␬ϭkLͱn sin ␪Ϫ1, ͑8͒ at a distance of roughly 2000a0 , i.e., far out from the prism 2 itself. This enables us to treat the system without taking into where C0ϰId /␦, the refractive index of the prism is n, and account many surface phenomena, e.g., the sticking of the 2 the incident angle is ␪. The exp(Ϫ␳ ) factor accounts for the atoms to the surface, the creation of phonons, etc. Gaussian profile of the focused laser beam. We have shown Since one is probing distances in the neighborhood of in Ref. ͓17͔ that this Gaussian profile is of considerable im- 2000a0 and more, one could expect to be in a regime where portance for analyzing experimental results due to the aver- retardation effects could be noticeable. Indeed, when the dis- aging over a cloud of incoming atoms and that minimizing tance between the atom and the wall becomes of the order of this averaging effect is essential for observing quantum be- the transition wavelength ␭/2␲, the time of flight of the vir- havior. In the following, we assume a constant intensity and tual photons becomes large enough to cause retardation ef- 2 set ␳ ϭ0 for simplicity. fects. For sodium, ␭/2␲ϭ1772a0 , and a barrier located at The exponentially decaying optical potential and the at- 2000a0 should probe retardation corrections. In fact, one no- tractive atom-wall interaction add up to generate an effective tices from Fig. 2 that retardation corrections for the dielectric potential barrier. It is given by one of the following two considered in this paper become important near that distance. formulas, which, respectively, neglect retardation effects or To illustrate this, we also plotted the effective potentials with take them into account, and without retardation corrections in Fig. 3: the height, its location and the curvature of the potential at the top of the ͑n͒ barrier vary strongly when one compares the two sets of C3 V ͑z͒ϭC exp͑Ϫ2␬z͒Ϫ , ͑9͒ effective potentials. For small enough values of C0 , the bar- no ret 0 3 z rier disappear ͓see Vno ret in Figs. 3͑e͒ and 3͑f͔͒, but quantum reflection still occurs, as we discuss below. The numerical Vexact͑z͒ϭC0exp͑Ϫ2␬z͒ϩVQED͑z͒, ͑10͒ parameters corresponding to Vexact and Vno ret of Fig. 3 are given in Table II. where VQED is depicted in Fig. 2. The various numerical values used here are listed in Table I. The height of the III. REFLECTION FROM POTENTIAL BARRIERS barrier Vmax vary as a function of the intensity of the laser beam, as do its location and shape. This is illustrated in Fig. A. Numerical calculation of the reflection probability 3. One can see that as the intensity is reduced, the barrier is In this section, we elaborate on the specific numerical decreasing in height and is moving out to larger distances. computations involved in the evaluation of the reflection 1 2 For a barrier with Vmaxϭ 2 mvtop corresponding to sodium probability for an arbitrary potential. As described in Coˆte´ atoms with vtopϭ10.0 cm/s, the top of the barrier is located et al. ͓27͔, our procedure is based on matching a superposi- PRA 58 RETARDATION EFFECTS ON QUANTUM REFLECTION... 4003 tion of incoming and reflected Wentzel-Kramers-Brillouin ͑WKB͒ waves to an exact or accurate approximate solution of the Schro¨dinger equation bridging the region where the WKB approximation is inaccurate. The potential curves of Eqs. ͑9͒ and ͑10͒ and Figs. 2 and 3 are defined only for zϾ0. In a complete treatment the potential should be properly modified at zϭ0 to account for the presence of the interface with the dielectric layer. We assume perfect sticking at the dielectric surface and neglect any reflection from the region closer to the interface than some finite small z, determined numerically as explained below. Under this assumption, the singularity at zϭ0isofno physical importance and we solve for the reflection from the potential barriers of Eqs. ͑9͒ and ͑10͒ in the same way we would have solved for the reflection from a bounded finite range potential. We apply a numerical calculation in which we propagate a wave function and an auxiliary function, defined in Eq. FIG. 4. The dimensionless badlands ␦␭͑dashed lines͒ for the ͑20͒ below, from z ϩϱ inward. Assuming perfect sticking exact potentials Vexact of Fig. 3 ͑solid line͒, for an incoming atom → Ϫ11 at the surface, we prove that this auxiliary function con- with Eϭ5.2975ϫ10 a.u. ͑or vϭ11.0 cm/s͒. The potential Vexact verges to the reflection probability as zϭ0 is approached. In and energy E ͑the horizontal dashed line͒ are on a scale of 10Ϫ10 the proof, we assume that the potential that vanishes for z a.u., and the badlands on a unit scale. ͑a͒, ͑b͒, and ͑c͒ correspond to tunneling through the barrier ͑the classical turning points are indi- ϩϱ, takes a constant negative value ϪV0 for z Ϫϱ. → → cated by the intersection of E with Vexact) and ͑d͒, ͑e͒, and ͑f͒ We then show that the value of V0 does not influence Eq. ͑20͒, which is all we need for the calculation itself. represent the above-barrier cases. ͓The legend shown in ͑f͒ applies In order to obtain Eq. ͑20͒ which converges to the reflec- for all plots.͔ tion probabilities of a particle of mass m and kinetic energy Eϭប2k2/2m, incident on the barrier from the right, we for- we illustrate these ‘‘badlands’’ for the exact potentials shown mally consider the reciprocal problem of the particles inci- in Fig. 3: a more complete discussion is given in the next section. dent from z Ϫϱ with kinetic energy ប2kЈ2/2m where kЈ is → 2 2 By requiring Eq. ͑11͒ to asymptotically (z Ϫϱ) match defined by EϩV0ϭkЈ ប /2m. According to the reciprocity → relation, the reflection probability is the same whether the the exact wave function exp(ikЈz)ϩRexp(ϪikЈz)wefind atom is incident on a barrier from the left or the right. ͑A proof is given in Appendix A for the configuration of a step RWKBϭRexp͓i␩͑zm͔͒, ͑14͒ function.͒ Coming from z Ϫϱ, the exact wave function can be i.e., that RWKB and the conventionally defined reflection am- approximated by the→ WKB ansatz, plitude R differ by a phase ␩(zm) that accounts for the fact that the matching point zm is not at zϭ0, and that the local 1 i z momentum p may differ from its asymptotic value បkЈ at ␺WKB͑E,z͒ϭ exp ͵ p͑E,zЈ͒dzЈ finite values of z: ͱp͑E,z͒ͫ ͩ ប zm ͪ z i z 1 ␩͑zm͒ϭ2 lim p͑E,zЈ͒dzЈϪkЈz . ͑15͒ ϩRWKBexp Ϫ p͑E,zЈ͒dzЈ , ប ͵ ͵ z Ϫϱͩ zm ͪ ͩ ប zm ͪͬ → ͑11͒ Knowing a wave function ␺(z) ͑exact or accurate ap- proximation͒ corresponding to a purely outgoing wave at z where zm is an arbitrary matching point, and we are assum- ϩϱ, we can determine the reflection amplitude by matching that in the entire region zрz , the essential condition for → m ing at zm the logarithmic derivatives of the WKB wave func- applicability of the WKB approximation is valid. Namely, tion and ␺(z) ͓Eq. ͑11͔͒, that the de Broglie wavelength ␭ϭ2␲ប/p, with

i pЈ͑zm͒ p͑E,z͒ϭͱ2m͓EϪV͑z͔͒, ͑12͒ ␺Ј͑zm͒/␺͑zm͒Ϫ p͑zm͒ϩ ប 2p͑zm͒ RWKBϭϪ . ͑16͒ i pЈ͑zm͒ varies sufficiently slowly, ␺Ј͑zm͒/␺͑zm͒ϩ p͑zm͒ϩ ប 2p͑zm͒ 1 d␭ m dV The usual reflection amplitude R can be obtained from R ␦␭ϵ ϭប 3 Ӷ1. ͑13͒ WKB 2␲ͯdzͯ ͯp dzͯ through the phase correction ͑14͒, and the reflection probability is simply In the remainder of the paper, we refer to the regions where the condition ͑13͒ is not fulfilled as ‘‘badlands.’’ In Fig. 4, ϭ͉R͉2ϭ͉R ͉2. ͑17͒ R WKB 4004 R. COˆ TE´ , B. SEGEV, AND M. G. RAIZEN PRA 58

FIG. 5. In ͑a͒, we show Vexact corresponding to Fig. 3͑d͒͑with vtopϭ10.0 cm/s͒ and the badlands ␦␭ for a sodium atom incident on 2 the prism with a velocity vϭ10.1 cm/s. The real (␺C) and imagi- FIG. 6. The dimensionless auxiliary function (E,z)ϭ͉R(z)͉ R 2 nary (␺ ) parts of the wave function ͑in arb. units͒ are plotted in as it approaches the reflection probability (E)ϭ͉R͉ at small z, S R ͑b͒, as well as the dimensionless auxiliary function (E,z) for three incoming velocities: 20.0 cm/s in ͑a͒, 10.1 cm/s in ͑b͒, and 2 R 2 ϭ͉R(z)͉ as it approaches the reflection probability (E)ϭ͉R͉ at 3.0 cm/s in ͑c͒, for the exact potential Vexact of Fig. 3͑d͒͑solid line͒ R small z. Convergence is reached after the wave functions have and the hybrid potential Vhyb of Fig. 8͑c͒ below ͑dahsed line͒. Both passed through the badlands. potentials have a height corresponding to vtopϭ10.0 cm/s. ͑d͒, ͑e͒, and ͑f͒ show enlargements of the convergence region ͑at small z) 2 Ϫ5 Equation ͑16͒ is used to derive the results and predictions for Vexact :in͑d͒,͉R(z)͉ oscillates between 3ϫ10 and Ϫ5 presented below from an accurate numerical solution of the 9ϫ10 ,in͑e͒between 0.4932 and 0.4992, and in ͑f͒ between stationary Schro¨dinger equation obtained by propagating two 0.999 730 and 0.999 742. Finally, the values of (E) for the exact Ϫ5 R Ϫ6 independent solutions from large z inward toward zϭ0: and hybrid potentials, are: in ͑a͒ 6.91ϫ10 and 3.99ϫ10 ,in͑b͒ ␺ (z) and ␺ (z). They are, respectively, defined by the 0.4948 and 0.4192, and in ͑c͒ 0.999 738 9 and 0.997 813 4, respec- C S tively. boundary conditions at z ϱ, →

␺C͑z͒ cos͑kz͒, ͑18͒ ␦␭ is not negligible͒ located around 3000a : the wave func- → 0 tions ␺C and ␺S start to be affected and (E,z) begins to ␺ ͑z͒ sin͑kz͒. ͑19͒ R S → grow from its initial zero value ͓see Fig. 5͑b͔͒. Its behavior follows the shape of the badlands. One notices that although An auxiliary reflection probability function is then defined the wave functions oscillate rapidly because of the singular- by ity at the origin, (E,z) is well behaved right after the bad- R 1 i 2 lands are passed. In Figs. 6͑a͒–6͑c͒, we also present the nu- lnp͑z͒Ϫ p͑z͒ merical calculation of (E,z) for energies high above and ץ ln␺͑z͒ϩ ץ z 2 z ប R ͑E,z͒ϭ , ͑20͒ way below the same barrier and the corresponding badlands R 1 i are plotted in Figs. 7͑a͒–7͑c͒. ͑These figures contain also ͒.zlnp͑z͒ϩ p͑z͒ͯ results for an hybrid potential discussed in a next sectionץ zln␺͑z͒ϩץ ͯ 2 ប The behavior of (E,z) as it converges to (E) is general: where we obtain the sameR pattern for all three cases.R Going inward, (E,z) grows first at the approach of the outer badland ͓if it R ␺͑z͒ϵ␺C͑z͒ϩi␺S͑z͒. ͑21͒ is sizable as in ͑b͒ and ͑c͒ but not ͑a͔͒ and overshoots its asymptotic value before decreasing until the inner badland Note that Eq. ͑20͒ does not depend on V0 or kЈ, and that the starts. It then grows again and reaches another maximum singularity at z 0 is avoided. The auxiliary function before correcting its value by decreasing too much, to finally (E,z) approaches→ the actual reflection probability as z ap- R stabilize its value to (E). In Figs. 6͑d͒–6͑f͒, we demon- proaches zero, strate the convergenceR more explicitly by enlarging the re- ϭ ͑E͒ϭ lim ͑E,z͒. ͑22͒ gion near z 0: in all three cases, the behavior is similar and R z 0R convergence is reached at zϳ100a0 . → In Fig. 5, for example, we present the solution (E,z) for an incoming sodium atom with velocity vϭ10.1R cm/s inci- B. Quantum versus classical reflection dent on a potential barrier Vexact ͑calculated with retardation͒, A classical particle with mass m and velocity v ͑momen- 1 2 of height Vmaxϭ 2 mvtop with vtopϭ10.0 cm/s, and the corre- tum បkϭmv) incident on a one-dimensional potential bar- sponding ‘‘badlands.’’ As the atom approaches the barrier rier V(z) can be either transmitted or reflected. The classical from ϩϱ, it reaches the border of the outer badland ͑where reflection probability is a Heavyside function (E) Rclassic PRA 58 RETARDATION EFFECTS ON QUANTUM REFLECTION... 4005

4͑d͒–4͑f͔͒, we observe a larger effect of the inner badland than the outer one, and conclude that most of the above- barrier reflection is caused by the inner portion of the effective potential, contrary to normal intuition. For underbarrier energies the badlands are centered around the turning point, their height extending to infinity. For above barrier energies there are no turning points, but two badlands can still be recognized. In this respect the concept of the badlands includes the concept of turning points but is more general. Note that close enough to the dielectric- vacuum interface there is no badland and WKB applies, which is the reason why (E,z) converges to (E). R R C. Model potentials While numerical calculation is the only way to obtain accurate predictions for a realistic effective potential, it is often helpful to consider simplified models that can be FIG. 7. Comparison of the dimensionless badlands ␦␭ between solved exactly. In this section three models are presented for the exact ͑solid line͒ and the hybrid ͑dashed line͒ potentials shown the potential barrier of the evanescent wave mirror, and in Fig. 8͑c͒, for different incoming velocities ͑or energies͒: ͑a͒ 20.0 closed form formulas for the probabilities are obtained from the analytic solutions to the corresponding equations. After cm/s ͑much higher than Vmax corresponding to vtopϭ10.0 cm/s͒, ͑b͒ 10.1 cm/s ͑just above vtop), and ͑c͒ 3.0 cm/s ͑much lower than studying different model potentials, we found that a better vtop). agreement with the exact results was obtained when the barrier’s height and curvature were chosen to fit the exact physi-

1 2 cal potential. Keeping these parameters fixed, we chose three ϵ⌰(VmaxϪE), where Eϭ 2 mv is the energy of the atom potentials with very different asymptotic behaviors: and Vmaxϭmax͓V(z)͔ is the height of the potential barrier. Ϫ2 In quantum mechanics the reflection probability (E)isa Vcosh͑z͒ϭVmaxcosh ͓͑zϪzmax͒/␨͔, ͑23͒ smooth function of the energy: the classical step functionR is 2 replaced by a quantum S-shaped curve, with a finite over- Vosc͑z͒ϭVmax͕1Ϫ͓͑zϪzmax͒/␨͔ ͖, ͑24͒ barrier reflection probability as well as a finite underbarrier transmission ͑or tunneling͒ probability. Vcosh͑z͒, zϾzmax Traditionally, one compares the de Broglie wavelength of Vhyb͑z͒ϭ ͑25͒ ͭ Vosc͑z͒, zϽzmax . the particle, ␭dBϭ2␲ប/mv, with some scale in the potential, say, the width of the potential barrier, to determine if sub- All these potentials have the same height Vmax and curvature stantial quantum behavior should be expected. According to ␣ at the top of the barrier as the exact physical potential, our analysis, while the underbarrier transmission is mostly 2 2 Ϫ2 z ͉zϭz ϭϪ2Vmax␨ . ͑26͒ץ/V ץsensitive to the width and height of the barrier, the overbar- ␣ϵ rier reflection is determined by regions of the potential where max the semiclassical treatment fails. The concept of the ‘‘bad- Both parameters Vmax and ␣ depend on C0 , i.e., on the in- lands’’ was introduced to quantify these regions ͓27͔. In Fig. tensity of the laser, and are sensitive to retardation effects, as 4, we illustrate ␦␭ for a sodium atom with vϭ11.0 cm/s depicted in Fig. 3 and given in Table II. Solving for Vosc going through the exact atom-wall potential, i.e., with retar- gives the leading order in a semiclassical approximation, as dation corrections. Figs. 4͑a͒–4͑c͒ show the case of tunnel- it amounts to replacing the potential by the closest inverted ing, and as one can expect, the regions where the system harmonic oscillator ͓28͔. cannot be described by WKB are located near the classical As proven in Appendix B, the reflection probabilities turning points, i.e., when V(z)ϭE or alternatively p(z)ϭ0. from these potentials are, respectively, given by As the height of the barrier is progressively decreased ͑by lowering the intensity of the laser͒, the turning points are cos2͑␲u͒ getting closer to each other: the total region spent by bad- ͑E͒ϭ , ͑27͒ Rcosh 2 2 lands is diminished, hence a larger tunneling probability. No- cos ͑␲u͒ϩsinh ͑␲k␨͒ tice that the inner badland is steeper than the outer one be- 1 cause the slope of the potential is larger near that turning ͑E͒ϭ , ͑28͒ point; for the same reason, the outer badland is more ex- Rosc 1ϩexp͑Ϫ2␲a͒ tended. When the barrier is lowered further as in Figs. 4͑d͒– 4͑f͒, there are no turning points anymore, but the badlands l͑1ϪF͒Ϫi͑1ϩF͒ 2 ͑E͒ϭ , ͑29͒ persist with the same ‘‘topography,’’ namely, a steeper inner Rhyb ͯl͑1ϪF͒ϩi͑1ϩF͒ͯ badland and a more extended outer one. From Figs. 6͑d͒– 6͑f͒, one notices that (E,z) takes on its value after passing where through the regions whereR the condition ͑13͒ is not satisfied. For above barrier reflection ͓see Figs. 6͑a͒ and 6͑b͒ or Figs. lϭͱ1ϩexp͑2␲a͒Ϫexp͑␲a͒, ͑30͒ 4006 R. COˆ TE´ , B. SEGEV, AND M. G. RAIZEN PRA 58

FIG. 9. Comparison of the three models and the exact reflection probability (E)ϭ͉R͉2 for sodium atoms incident on a barrier of FIG. 8. The three model potentials: ͑a͒ Vcosh , ͑b͒ Vosc , and ͑c͒ R 1 2 Ϫ11 height Vmaxϭ 2 mv with vtopϭ10.0 cm/s. ͑a͒ shows the curves as Vhyb . Each of them is matched to Vexact with Vmaxϭ4.378ϫ10 top a function of v, and ͑b͒, ͑c͒, and ͑d͒ enlarge the low-, medium-, and a.u. ͑or vtopϭ10.0 cm/s͒ located at zmaxϭ2017.53 a.u. with a curvature of ␣ϭϪ1.054 08ϫ10Ϫ16 a.u. ͑see also Table II͒. high-velocity regimes, respectively. Overall, Vhyb is a better approximation to Vexact , Vosc being the best for vϳvtop and Vcosh at Ϫ1/2 velocities far from vtop . Fϭ͑␨v0͒ ergy of the atom is much higher, or much lower, than the 1 ia 3 u i␨k 3 u i␨k ⌫ ϩ ⌫ ϩ Ϫ ⌫ Ϫ Ϫ barrier height, respectively. In both cases, the badlands of the ͯ ͩ 4 2 ͪͯ ͩ4 2 2 ͪ ͩ4 2 2 ͪ model and exact potentials are extremely different, and so ϫ 3 ia 1 u i␨k 1 u i␨k are the probabilities. In the second example (vϭ10.1 cm/s͒, ⌫ ϩ ⌫ ϩ Ϫ ⌫ Ϫ Ϫ where EϳV near the top of the barrier, the agreement 4 2 4 2 2 4 2 2 max ͯ ͩ ͪͯ ͩ ͪ ͩ ͪ between the badlands is better, and a fairly good agreement ͑31͒ for (E) is obtained. WeR computed (E) for the exact and three model poten- with the notation tials of Fig. 8 asR a function of the velocity of the incoming atom. We illustrate the results in Figs. 9͑a͒–9͑d͒:in͑a͒,we kϵͱ2mE/ប, ͑32͒ observe that the four curves are very close to each other, implying that the model potentials adjusted to the parameters v 2mV / , 33 0ϵͱ max ប ͑ ͒ of Vexact give good results. In ͑b͒–͑d͒, we enlarge three different portions of the same graph, corresponding to small 2 uϵͱ1/4Ϫ͑v0␨͒ , ͑34͒ velocities where tunneling is present, velocities for which the atom energy is near the top of the barrier (vtopϭ10.0 cm/s͒, m and larger velocities where above-barrier reflection is domi- aϵ ͑V ϪE͒. ͑35͒ ͱϪ␣ max nant. The hybrid potential gives an overall better agreement to the exact curve than the two other model potentials. At The model potentials are shown in Figs. 8͑a͒–8͑c͒ for a smaller v ͓see ͑b͔͒, Vcosh is the best model and Vosc the worst Ϫ11 barrier height Vmaxϭ4.378ϫ10 a.u. ͑corresponding to a up to vϳ6.75 cm/s, at which point the three models are sodium atom with velocity vtopϭ10.0 cm/s͒, with ␣ϭ equivalent. In the range of velocities near the top of the Ϫ16 Ϫ1.05408ϫ10 a.u. From Fig. 8, one expects Vhyb to be barrier ͓in ͑c͔͒, we have the reverse, namely that Vosc is the the best approximation to the exact potential Vexact . In order closest to the exact curve and Vcosh the farthest. This remains to analyze the results from Vexact and Vhyb , we compare in true up to vϳ12.75 cm/s, at which point the Vcosh is once Figs. 7͑a͒–7͑c͒ their badlands for three different velocities of again the best model and Vosc the worst one ͓see ͑d͔͒. For all the incoming sodium atoms: 20.0 cm/s, 10.1 cm/s, and 3.0 values of v, the model potentials give curves below the exact cm/s. Since Vhyb is a combination of the two symmetric po- one, and the Vhyb is always between Vcosh and Vosc . This can tentials Vcosh and Vosc , their badlands are given by the sym- be understood by looking at the potentials themselves ͑see metric image of the Vhyb badlands about zϭzmax . The aux- Fig. 8͒ or at the badlands ͑see Fig. 7͒. iliary functions (E,z) for V and V at the same three Overall, if one is interested at the reflection probability for R exact hyb velocities are shown in Figs. 6͑a͒–6͑c͒, as they converge to a given potential barrier, one can get good approximate re- the reflection probabilities. (E) for the exact and hybrid sults by using one of the three models depending on which R Ϫ5 potentials, was found to be: in ͑a͒, ϭ6.91ϫ10 and energy regime is considered. The first step to such an ap- 3.99ϫ10Ϫ6,in͑b͒ ϭ0.4948 and 0.4192,R and in ͑c͒ proximation is to fit the two parameters, namely, the height ϭ0.999 738 9 and 0.997R 813 4, respectively. In the first, andR of the barrier and the curvature at its top, for each case con- last, example with vϭ20.0 cm/s, and vϭ3.0 cm/s, the en- sidered. PRA 58 RETARDATION EFFECTS ON QUANTUM REFLECTION... 4007

FIG. 10. Comparison of the reflection probability (E)ϭ͉R͉2 FIG. 11. In ͑a͒, we compare V ͑solid line͒ and V ͑dashed R exact no ret for the exact Vexact and Lennard-Jones Vno ret potentials as a function line͒ for values of C0 corresponding to Fig. 10͑a͒, i.e., a barrier with of the incident velocity v.In͑a͒, both have the same height Vmax vtopϭ10.0 cm/s. The potential without retardation is shifted by ͑or vtopϭ10.0 cm/s͒, and in ͑b͒, both have the same C0 ͑or laser 225.9a0 to the left to overlap with Vexact ͑see Table II͒. Vexact has a intensity͒ which produces different barrier heights (vtopϭ17.53 longer tail and a faster decrease at smaller distances than Vno ret . cm/s for Vexact and 10.0 cm/s for Vno ret). In figure ͑b͒, we also We also show the energies corresponding to three velocities, shifted the exact curves by 7.53 cm/sϭ9.877ϫ10Ϫ10 a.u. to super- namely vϭ12.0, 10.0, and 8.0 cm/s. In ͑b͒, ͑c͒, and ͑d͒, we plot the pose it with the Lennard-Jones curves: although still small, the badlands for the same three velocities: here again, the curves with- shape difference is larger than in ͑a͒. out retardation have been shifted by 225.9a0 to explicitly show any difference for the case with retardation. IV. RESOLVING RETARDATION EFFECTS long-range form of Vexact has a more extended tail than We now turn our attention to the detection of retardation V . We illustrate these observations in Fig. 11͑a͒, where effects in above-barrier reflection and quantum tunneling. no ret V has been shifted by zno retϪzexact to overlap with Retardation affects the reflection probability curves in two no ret max max V . Indeed, one observes that V ͑dashed line͒ is different ways: by shifting their reflection threshold and by exact no ret slightly larger than V ͑solid line͒ at smaller distances, changing their shape. Using the technique described above, exact and vice versa at larger distances. On the same graph, we we evaluate the reflection probability (E) for sodium at- show three different kinetic energies corresponding to ve- oms approaching a dielectric infinite wallR with nϭ1.805 ͑or locities of 12.0 cm/s ͑above barrier͒, 10.0 cm/s ͑or v ), and ⑀ϭ3.258). There are two distinct ways in which an actual top 8.0 cm/s ͑below barrier͒. The corresponding badlands are experiment can be performed. One can keep the intensity of plotted in Figs. 11͑b͒,11͑c͒, and 11͑d͒, respectively. As in the laser constant and change the energy of the incoming a , the curves for V have been shifted by zno retϪzexact in atoms, or, alternatively, keep a constant energy and vary the ͑ ͒ no ret max max intensity of the laser. We show below that in both cases order to better visualize the differences in ␦␭. In Fig. 11͑b͒, retardation shifts the threshold for reflection. However, the we notice that ␦␭ reaches slightly higher values for Vexact , hence the slightly larger (E) ͓see Fig. 10͑a͔͒. Similarly, for change in the shape of the curves is mainly observable as one R varies the intensity of the laser and scans different regions of vϽvtop , the total range spent with ␦␭ϳ1 is more extended the attractive potential. for Vexact than for Vno ret ͓see Fig. 11͑c͔͒, leading to less tunneling or more reflection, as noted in Fig. 10͑a͒. In the second situation illustrated in Fig. 10͑b͒ we con- A. Fixed potential sider a case where the intensity of the laser is fixed, the two We compute and compare the reflection probability (E) curves have C ϭ9.877ϫ10Ϫ10 a.u. corresponding to the po- R 0 for two effective potentials, Vexact ͑with retardation͒, and tentials of Fig. 3͑a͒. We notice that the threshold is different Vno ret with the Lennard-Jones atom-wall interaction ͑without for the two curves, reflecting the difference in the height of retardation͒. We first compare (E) for barriers of the same the barriers ͑corresponding to vtopϭ17.53 cm/s and 10.0 Ϫ11 R height (Vmaxϭ4.378ϫ10 a.u. or, equivalently, vtop cm/s for Vexact and Vno ret , respectively͒. We shifted the ϭ10.0 cm/s͒ and two different values of C0 . The potentials curve with retardation by 7.53 cm/s to overlap with the curve are depicted as the dashed line in Fig. 3͑a͒͑without retarda- without retardation, in order to detect any effect in the shape tion͒ and solid line in Fig. 3͑d͒͑with retardation͒. C0 of (E). Once again, the difference is small: the curve for Ϫ10 R ϭ9.877ϫ10 a.u. for the case without retardation and Vexact is slightly less steep than for Vno ret . By shifting the Ϫ10 C0ϭ5.375ϫ10 a.u. for the case with retardation ͑see curve (E) for Vexact , we actually compare two (E) Table II͒. The calculated reflection probabilities are given in curvesR with different energy scales, because of the nonlinearR Fig. 10͑a͒. Both (E) curves are basically identical: no sig- relationship between v and E. This amplifies the differences nificant retardationR effects can be detected. The curve with observed in the shapes of (E). retardation lies slightly above the curve without retardation From the two situationsR illustrated in Fig. 10, one con- since the potential with retardation is less attractive: the cludes that by varying the velocity of the incoming atoms, 4008 R. COˆ TE´ , B. SEGEV, AND M. G. RAIZEN PRA 58

FIG. 13. A reflection probability curve of (E)ϭ͉R͉2 as a R function of the laser intensity C0 . The areas A1 and A2 correspond to above barrier reflection and quantum tunneling, respectively, and FIG. 12. Comparison of the reflection probability (E)ϭ͉R͉2 R they are defined by the vertical line cutting the curve at ͉R͉2ϭ0.5 as a function of C0 for the exact ͑with retardation͒ and Lennard- Jones ͑no retardation͒ potentials, for three different incident veloci- located ͑by definition͒ at C0ϭC1/2 . ties: 10.0 cm/s in ͑a͒, 6.0 cm/s in ͑b͒, and 3.0 cm/s in ͑c͒. Near the classical threshold C0ϭC*,(VmaxϭE), the reflection probability ing the potentials in Fig. 3 and the corresponding numbers in increases from zero to one. These thresholds are shifted by the Table II. For each case, the location of the top of the barrier retardation effects. The shifts are larger for higher velocities. Ϫ10 Ϫ10 at C0ϭC* is moving out as the v is decreased. The barriers They are 4.502ϫ10 a.u. in ͑a͒, 4.072ϫ10 a.u. in ͑b͒, and at the two threshold cases of Fig. 12͑a͒ are depicted as the 3.904ϫ10Ϫ10 a.u. in ͑c͒. The S shape of the curve results from dashed curve of Fig. 3͑a͒͑no retardation͒ and the solid curve quantum effects and is also sensitive to retardation. As v is decreased, this S shape becomes steeper. To illustrate the variation in of Fig. 3͑d͒͑with retardation͒, for which zmaxϭ2243.4a0 and shape, with and without retardation, we shift the curves without 2017.5a0 , respectively, and the change in C0ϭC* is 4.502 Ϫ10 retardation by the differences in C*. ϫ10 a.u. ͑see Table II͒. Similarly, Fig. 11͑b͒ corresponds to the dashed curve of Fig. 3͑b͒͑no retardation͒ and the solid only a threshold change could be measured, the difference in curve of Fig. 3͑e͒͑with retardation͒ with zmaxϭ2614.9a0 and the shape of (E) being extremely small. 2416.9a , respectively, and ⌬C*ϭ4.072ϫ10Ϫ10 a.u.; and R 0 Fig. 11͑c͒ corresponds to the dashed curve of Fig. 3͑c͒͑no B. Scanning the attractive potential retardation͒ and the solid curve of Fig. 3͑f͒͑with retardation͒ by varying the laser intensity with zmaxϭ3022.2a0 and 3000.4a0 , respectively, and ⌬C* Ϫ10 A simpler and better experimental setup is to have a ϭ3.904ϫ10 a.u. So, as v is reduced, zmax is moved out, source of slow atoms incident on the prism with a given but since the retardation effects reduce the strength of the velocity, and to change the shape and extent of the barrier by attractive atom-wall potential with respect to the Lennard- varying the intensity of the laser, or, similarly, the value of Jones case, the evanescent potential has more relative C0 .AsC0 is decreased, the barrier is lowered and moved strength and therefore a smaller C* is required to get the out. By doing so, one probes different regions of the atom- same barrier height. The farther out zmax is, the weaker wall potential and at different rates. Vatom wall , and the smaller the difference between the thresh- We chose incoming atoms at three different velocities rel- olds C* with and without retardation. evant to future experiments: 10.0 cm/s, 6.0 cm/s, and 3.0 It is visible from Fig. 12 that the shape of the reflection cm/s. For each velocity, we computed the reflection prob- probability curves for potentials with and without retardation ability curves as a function of C0 for the potentials with and differs. To quantify this quantum effect, we define, as shown without retardation effects. The results are shown in Fig. 12. in Fig. 13, two regions of area A1 and A2 delimited by the For a given energy E ͑or velocity v), as one changes C , vertical line at C ϵC intersecting the curve at ϭ0.5. 0 0 1/2 R one reaches a value C0ϭC* which is the threshold for clas- This almost corresponds to the value of C0ϭC* giving a sical reflection, for which VmaxϭE. We observe a shift in the barrier of height corresponding to the atom velocity ͑here 3.0 threshold C*: for the curve with retardation, it is located at cm/s͒. In fact, in Fig. 13, lnC1/2ϭϪ22.291 a.u. which is very smaller values of C0 than for the curve without retardation. close to lnC*ϭϪ22.275 a.u. ͑see Table III͒. We define A1 We shifted (E) without retardation by the difference in C* and A2 in respect to C1/2 because the exact determination of to illustrateR the difference in shape between the two curves. C* would be experimentally more difficult. Notice here that As one can see from Figs. 12͑a͒–12͑c͒, as we reduce E ͑or classically, the reflection probability curve would be a step v), the location of the curves with retardation is slightly function located at C*, which is well approximated by C1/2 : moved to lower values of C0 , but the curves without retar- the curve for C0ϽC1/2 represents the effect of above-barrier dation are more affected. Moreover, the shift between the quantum reflection, and the curve for C0ϾC1/2 the effect of two sets of curves for a given value of v decreases slightly as quantum tunneling. Equivalently, A1 is a signature of above we lower the velocity. This can be understood from examin- barrier quantum reflection and A2 a signature of tunneling. PRA 58 RETARDATION EFFECTS ON QUANTUM REFLECTION... 4009

TABLE III. Quantum reflection and tunneling signatures. C1/2 is the value of C0 that gives ϭ0.5 while C* is the value of C0 for R Ϫ11 which VmaxϭE ͑or vtopϭv). A1 and A2 ͑in units of 10 a.u.͒ are defined in Fig. 13 and represent above-barrier reflection and quantum tunneling, respectively, for the curves of Fig. 12. Similarly, B1 and B2 denote above-barrier reflection and quantum tunneling, respectively, for the curves of Fig. 16. A1ϩA2 and B1ϩB2 are mea- sures of the total quantum signature. For all quantities, we give the ratio ret./no ret. As the velocity v is reduced, the effect of retardation is more pronounced.

Quantity Type vϭ10.0 cm/s vϭ6.0 cm/s vϭ3.0 cm/s ln C1/2 ret. Ϫ21.352 Ϫ21.815 Ϫ22.291 no ret. Ϫ20.739 Ϫ21.019 Ϫ21.232 ln C* ret. Ϫ21.344 Ϫ21.805 Ϫ22.275 no ret. Ϫ20.736 Ϫ21.016 Ϫ21.230

FIG. 14. Potential curves with and without retardation (Vexact A1 ret. 3.2350 2.6049 2.1485 and Vno ret) depicted to explain Fig. 12. In ͑a͒, ͑b͒, and ͑c͒, E Ϫ11 1 2 (10 a.u.͒ no ret. 4.0013 3.4540 2.9779 ϭ 2 mv with vϭ10.0 cm/s, in ͑d͒, ͑e͒, and ͑f͒, vϭ6.0 cm/s, and in ratio 0.8085 0.7542 0.7215 ͑g͒, ͑h͒, and ͑i͒, vϭ3.0 cm/s. C0 ͑or the laser intensity͒ was chosen in ͑b͒, ͑e͒, and ͑h͒ to give reﬂection probability ϭ0.5 in all these R Ϫ10 A ret. 3.5009 2.8654 2.4245 cases, namely, C0ϭC1/2 . For Vexact C1/2 is 5.333ϫ10 a.u. in 2 Ϫ10 Ϫ10 (10Ϫ11 a.u.͒ no ret. 4.2543 3.6976 3.2683 ͑b͒, 3.356ϫ10 a.u. in ͑e͒, and 2.085ϫ10 a.u. in ͑h͒; while for V C is 9.844ϫ10Ϫ10 a.u. in ͑b͒, 7.440ϫ10Ϫ10 a.u. in ͑e͒, and ratio 0.8229 0.7749 0.7418 no ret 1/2 6.013ϫ10Ϫ10 a.u. in ͑h͒. Equally spaced points in Fig. 12 on both sides of C0ϭC1/2 for these six cases are depicted in the other A1ϩA2 ret. 6.7359 5.4703 4.5729 Ϫ11 graphs, with a spacing of ⌬ϭ5.0ϫ10 a.u. Vexact is higher than (10Ϫ11 a.u.͒ no ret. 8.2556 7.1516 6.2462 Vno ret for C0ϭC1/2Ϫ⌬, and vice versa for C1/2ϩ⌬, because ratio 0.8159 0.7649 0.7321 Vatom wall is less important for Vexact , hence a change in C0 has a stronger impact. Vexact always has a more extended tail. B1 ret. 0.067 96 0.090 89 0.1289 ͑a.u.͒ no ret. 0.043 77 0.050 64 0.054 19 height and shape between Vexact and Vno ret is decreasing as ratio 1.553 1.795 2.379 well. This is related to the nonlinear dependence of the effective potentials on C0 : for a higher value of C1/2 , the B2 ret. 0.059 24 0.075 15 0.098 26 optical potential is more dominant, and a change ⌬ in its ͑a.u.͒ no ret. 0.040 31 0.045 95 0.049 86 value has a stronger effect than for a smaller value of C1/2 for ratio 1.470 1.635 1.971

B1ϩB2 ret. 0.1272 0.1660 0.2272 ͑a.u.͒ no ret. 0.084 08 0.096 59 0.010 41 ratio 1.513 1.719 2.183

Looking at the overlapping curves on Fig. 12͑c͒,wefind that (E) is larger for V than for V when C R no ret exact 0 ϽC1/2 , and vice versa when C0ϾC1/2 . In other words, the curve with retardation is steeper than the curve without, and the difference is increased as v diminishes. In order to un- derstand the reason for the shape difference in (E) with and without retardation illustrated in Fig. 12 ͑beyondR the shift in the value of C*), we plot in Fig. 14 the potential curves for the three values of incoming velocities (vϭ10.0, 6.0, and 3.0 cm/s͒, and for each velocity, we consider three values of C0 : C1/2Ϫ⌬, C1/2 , and C1/2ϩ⌬͑where ⌬ Ϫ11 ϭ5.0ϫ10 a.u.͒. The energies corresponding to the three FIG. 15. The badlands ␦␭ for the potentials of Fig. 14. For velocities are also drawn for each graph. The corresponding above-barrier reflection corresponding to ͑a͒, ͑d͒, and ͑g͒, the bad- badlands are shown in Fig. 15. As was already noted in Fig. lands for Vno ret are more prononuced than for Vexact , hence a larger 11, we observe that Vexact is more extended at large dis- reflection probability. For quantum tunneling corresponding to ͑c͒, tances, and that the variation in Vno ret is more pronounced. ͑f͒, and ͑i͒, the badlands for Vexact are more extended than for As v is reduced, the shift in zmax and the difference in both Vno ret , leading to less tunneling ͑or a larger reflection probability͒. 4010 R. COˆ TE´ , B. SEGEV, AND M. G. RAIZEN PRA 58 which the nonaffected atom-wall component is relatively more important. The increasing difference in (E) for decreasing velocities despite the fact that the barriersR actually look more alike can be understood by examining the badlands of Fig. 15. In Figs. 15͑a͒,15͑b͒, and 15͑c͒, we show ␦␭ for three values of C0 for vϭ10.0 cm/s. For C0ϽC1/2,in͑a͒, the badlands without retardation ͑dashed lines͒ are higher and more extended, hence the smaller above barrier reflection probability for Vexact .In͑b͒ (E)ϭ0.5 by construction, and we do not observe a significantR difference in the size of the region spent by ␦␭ϳ1. For C0ϾC1/2 shown in ͑c͒, we note a more extended ␦␭ for Vexact , hence a larger value for (E). As v is decreased, these differences in the badlands becomeR stronger ͑see Fig. 15͒, and so do the differences in the probability curves of Fig. 12. From the above discussion and the curves shown in Fig. 12, it is tempting to conclude that retardation effects reduce the quantum signature in the reflection probability curve. FIG. 16. Same as Fig. 12, but as a function of ln C0 . Because of But, as we mentioned before, retardation effects increase the logarithmic scaling, the smaller values of C0 are enhanced, and slightly (E) for constant effective potentials of the same the curves without retardation are steeper, in contrast to Fig. 12. height ͓seeR Figs. 10͑a͒ and 11͔. Moreover, as was noted in The quantum signature is amplified, especially the above-barrier ͓17͔ for a conducting wall, retardation effects enhance reflection that occurs at smaller values of C0 . (E). This apparent contradiction can be resolved by real- dation has a stronger effect at lower v. Moreover, from the izingR that, in fact, we are comparing two very different types ratios of B1 and B2 , we again find that above-barrier reflec- of experimental setups. In Fig. 10͑a͒ and ͓17͔, the potential is tion is more affected by retardation, although in the reverse fixed and we compute the reflection probability as a function order. That difference is also amplified by the logaritmic of the energy or velocity of the incoming atoms. This setup scale. really evaluates the effect of retardation on incoming atoms. On the other hand, the situation explored in Fig. 12 corre- V. CONCLUSION AND DISCUSSION sponds to a multiple set of experiments: for each value of C , the potential is different, and therefore comparing (E) 0 R We conclude that retardation effects could be observed in for various C0 is comparing different potentials. this system and that, in general, quantum effects may be used We have computed A1 and A2 for the various cases of for very sensitive measurements. Unlike reflection in the Fig. 12. They are listed in Table III. As described above, classical domain, quantum reflection depends on the details retardation affects the quantum signature more strongly at of the potential, both at short and large distances. In the lower v. The ratio of A1ϩA2 with and without retardation is classical regime, i.e., for reflection experiments that could be reduced, indicating a weaker quantum signature with retar- properly described by classical mechanics, reflection yields dation than without. One also notices that above-barrier re- can be used to identify thresholds. In the semiclassical reflection ͑ratio of A1) is slightly more affected than tunneling gime, the yields are also sensitive to the curvature near the ͑ratio of A2). For vϭ3.0 cm/s, the effect of retardation is of top of the barrier. Quantum reflection probabilities are deter- nearly 30%. mined by the complete potential curve. As we have demon- In Fig. 16 we plot the same results for (E) that were strated, identifying and using parameters in the quantum dy- R presented in Fig. 12, this time as a function of lnC0 , because namics regime, i.e., looking for experiments that cannot be it allows the comparison with experiments where averaging described by a semiclassical approximation, is a promising over the Gaussian profile of the laser beam is relevant ͓17͔. new way to study long-range and short-range atom-surface Because C0 takes on smaller values, the curves with retarda- interactions. tion are more affected than those without: most of the points Finally, we would like to comment on some relevant is- of the previous discussion seem to be reversed. The shift in sues that were not considered in our work. While we have threshold lnC* is larger as v is decreased, and the curves used the most accurate available calculations, to our knowl- without retardation ͑dashed lines͒ are steeper: the quantum edge, for the atom-wall interaction, and have given a com- signature appears to be stronger with retardation. Although plete quantum treatment to the center-of-mass motion of the the logarithmic scale enhances the smaller values of C0 and atoms, we still relied on the following simplifying assump- hence amplifies the signature of above-barrier reflection tions, justified by the experimental conditions that we have more than the quantum tunneling, it is still clear from Fig. 16 considered. First, we neglected surface roughness. Studies of that retardation has more influence at lower velocities. As surface roughness effects indicate that the mirror allows for previously, we can quantify the quantum signature by evalu- specular reflection if the surface is flat at the atomic scale ating similar areas as A1 and A2 , defined this time as an ͓29͔. Note that, for the mirror to be useful for atom-optics integral over lnC0 : we call them B1 and B2 , respectively. applications, it is essential that the reflection process be co- Their values are also given in Table III. Again, by comparing herent. Second, we neglected the finite response time of the the ratios of B1ϩB2 as a function of v, we notice that retar- mirror images in the dielectric, which seems reasonable since PRA 58 RETARDATION EFFECTS ON QUANTUM REFLECTION... 4011 the atoms are moving very slowly. The effect of the response R R 1 time for the rearrangement of charges in the bulk forming the . % ϩ ϭT , ͑A4͒ mirror was studied for metal ͓30͔, but to our knowledge, T T* . % % such results are not available for the rearrangement time of the dipole polarizations in dielectrics. Third, we treated the R R* atom as a two-level system, and assumed that no internal . ϩ % ϭ0. ͑A5͒ T T* transitions take place during the reflection. Observed effects % that go beyond the two-level model include state-selective % properties of the reflection ͓2,31͔, cooling during the reflec- The second equation implies that ͉R͉ϭ͉R ͉ϭ͉R ͉. Substi- tion through a spontaneous Raman transition between two tuting in the first equation, we obtain . % hyperfine levels ͓16,32͔, and, in general, an interplay between internal state transitions and the center-of-mass mo- 1 tion ͓33͔. Further, we did not take into account the depen- T ϭ ͑1Ϫ͉R͉2͒, ͑A6͒ dence of the dielectric constant on frequency, e.g., in the . T* calculations that gave the potential curves for the atom- % dielectric interaction using Eq. ͑5͒. Finally, looking only at which, in turn, implies reflection probabilities, we did not study the reflection process in the time domain and we did not look at wave packets. ͑phase of T ͒ϭ͑phase of T ͒ϭ␪. ͑A7͒ Clearly, some of these issues should be addressed in future . % research. Similarly, from the relation between R and R , we find . % ACKNOWLEDGMENTS ͑phase of R ͒ϭ␲ϩ2␪ϩ͑phase of R ͒. ͑A8͒ The authors are pleased to acknowledge helpful conver- . % sations with J. F. Babb and A. Dalgarno. The work of R.C. Finally, from the usual definition of the reflection and trans- and B.S. was supported by the National Science Foundation mission probabilities ͓34͔, i.e., through a grant for the Institute for Theoretical Atomic and Molecular Physics ͑ITAMP͒ at the Harvard-Smithsonian v R R* Center for Astrophysics. The work of M.G.R. was supported 1 2 ϵ % % ϭ͉R ͉ , ͑A9͒ by the National Science Foundation and the R.A. Welch R% v1 1 % Foundation.

v2 R R* APPENDIX A: RECIPROCITY RELATIONS ϵ . . ϭ͉R ͉2, ͑A10͒ R. v2 1 . In this appendix, we review the reciprocity relations for the reflection and transmission amplitudes. We consider a v2 T T* k2 one-dimensional potential that goes uphill ͑left to right͒ from ϵ % % ϭ ͉T ͉2, ͑A11͒ a constant value V at z Ϫϱ to a constant value V at z T% v1 1 k1 % 1 → 2 ϩϱ͑with V1ϽV2). For a particle of mass m and kinetic energy→ E incoming from Ϫϱ, one has v1 T T* k1 ϵ . . ϭ ͉T ͉2. ͑A12͒ z Ϫϱ z ϩϱ T. v2 1 k2 . → → eik1zϩR eϪik1z T eik2z, ͑A1͒ We get % ↔ % where k1ϭͱ2m(EϪV1)/ប and k2ϭͱ2m(EϪV2)/ប. R 2 % ϭ ϭ ϭ͉R͉ , ͑A13͒ and T represent the uphill reflection and transmission am- R R% R. plitudes,% respectively. The downhill relation going from right and since ϭ1Ϫ ϭ1Ϫ͉R͉2 and ϭ1Ϫ ϭ1 to left is 2 T% R% T. R. Ϫ͉R͉ , T eϪik1x eϪik2xϩR eik2x, ͑A2͒ . ↔ . k2 k1 and R and T now represent the downhill reflection and ϭ ϭ or ͉T ͉2ϭ ͉T ͉2. ͑A14͒ transmission. amplitudes,. respectively. Dividing the first rela- T T% T. k1 % k2 . tion by T and its complex conjugate by T* , multiplying % % the first relation by R , and adding them together, one gets APPENDIX B: REFLECTION PROBABILITIES . FOR THE MODEL POTENTIALS R R 1 R R* Ϫik x ik x . % ϩ e 1 ϩ . ϩ % e 1 In this appendix we obtain the reflection probabilities for ͩ T T* ͪ ͩT T* ͪ % % % % the model potentials of Eqs. ͑23͒, ͑24͒, and ͑25͒, by exactly Ϫik x ik x solving the corresponding stationary Schro¨dinger equations. e 2 ϩR e 2 . ͑A3͒ ↔ . A general solution to the stationary Schro¨dinger equation By comparing with Eq. ͑A2͒, one must have with the potential Vcosh is given by ͓35͔ 4012 R. COˆ TE´ , B. SEGEV, AND M. G. RAIZEN PRA 58

z taking the limit z ϱ and identifying the asymptotic coeffi- ␺ ͑z͒ϭA coshuϩ1/2 → cosh ͩ ␨ͪ cients, 1 u ik 1 u ik 1 z 1 Rcosh ␨ ␨ 2 lim ␺ ϭ exp͑Ϫikz͒ϩ exp͑ikz͒. ͑B8͒ ϫ F ϩ ϩ , ϩ Ϫ , ;Ϫsinh cosh T T 2 1ͫ4 2 2 4 2 2 2 ͩ ␨ͪͬ z ϱ cosh cosh → 2 z z The reflection probability oscϵ͉Rosc͉ for the potential ϩB sinh coshuϩ1/2 R ͩ ␨ͪ ͩ ␨ͪ Vosc is obtained in a similar way by finding ␯osc and ␮osc that would satisfy 3 u ik␨ 3 u ik␨ 3 z ϫ F ϩ ϩ , ϩ Ϫ , ;Ϫsinh2 , 2 2 1 4 2 2 4 2 2 2 ␨ ͫ ͩ ͪͬ lim ␺oscϭͱ exp͓i␾͑z͔͒, ͑B9͒ z Ϫϱ ͉z͉ ͑B1͒ → x2 1 where A and B are constants, satisfying ␾͑z͒ϵ Ϫa ln xϩ arg ⌫͑1/2ϩia͒, ͑B10͒ 4 2 ␺cosh͑zϭ0͒ϭA, ͑B2͒ taking the limit z ϱ and identifying the asymptotic coefficients, → ␺coshץ ϭB/␨. ͑B3͒ ͯ zץ zϭ0 1 2 Rosc 2 lim ␺oscϭ ͱ exp͓Ϫi␾͑z͔͒ϩ ͱ exp͓i␾͑z͔͒. z ϱ Tosc z Tosc z A general solution to the stationary Schro¨dinger equation → ͑B11͒ with the potential Vosc is given by ͓36͔ Note that exp(Ϫikz) and exp(ikz) are left and right moving, ␺ ͑z͒ϭ␯W͑a,zͱ2v /␨͒ϩ␮W͑a,Ϫzͱ2v /␨͒, osc 0 0 respectively, while 2/ z exp Ϫi (z) and 2/ z exp i (z) ͑B4͒ ͱ ͉ ͉ ͓ ␾ ͔ ͱ ͉ ͉ ͓ ␾ ͔ are inward and outward moving, respectively. where ␯ and ␮ are constants, satisfying Finally, the reflection probability ϵ͉R ͉2 for the Rhyb hyb potential Vhyb is obtained by finding ␯hybϭ␯osc and ␮hyb ͉⌫͑1/4ϩia/2͉͒ ϭ␮ that would satisfy ␺ ͑zϭ0͒ϭ 2Ϫ3/4͑␯ϩ␮͒, ͑B5͒ osc osc ͱ͉⌫͑3/4ϩia/2͉͒ 2 lim ␺ ϭ exp͓i␾͑z͔͒, ͑B12͒ hyb ͱ z ͉ ͉ ␺osc ͉⌫͑3/4ϩia/͒2͉ z Ϫϱץ ͉ ϭϪͱ2v /␨ 2Ϫ1/4͑␯Ϫ␮͒. → z zϭ0 0 ͱ⌫ 1/4ϩia/2ץ ͉ ͑ ͉͒ finding then A and B from the continuity of the wave ͑B6͒ hyb hyb function and its first derivative at zϭ0, taking the limit z The reflection probability ϵ͉R ͉2 for the potential ϱ and identifying the coefficients, Rcosh cosh → Vcosh is obtained by finding a specific solution to the Schro¨- 1 Rhyb dinger equation, i.e., a choice of AϭAcosh and BϭBcosh for lim ␺hybϭ exp͑Ϫikz͒ϩ exp͑ikz͒. ͑B13͒ Eq. ͑B1͒ that would satisfy z ϱ Thyb Thyb →

lim ␺coshϭexp͑Ϫikz͒, ͑B7͒ These straightforward derivations give the reﬂection prob- z Ϫϱ abilities of Eqs. ͑27͒–͑29͒. →

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