z. Phys. B. 98, 319-321 (1995) ZEITSCHRIFT FOR PHYSIK B �9 Springer-Verlag 1995

Quantum reflection Carlo Carraro 1'*, Milton W. Cole 2

1Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA 2104 Davey Laboratory, Department of Physics and Center for Materials Physics, Pennsylvania State University, University Park, PA 16802, USA (e-mail: mwc@psuvm.psu.edu., Fax: (814)865-3604)

Abstract. Recent experimental and theoretical results concerning the sticking coefficient at ultralow energy are described. The need for an accurate treatment of long range forces, including retardation, is emphasized. The system involving H atoms incident on liquid helium pro- vides the first clear evidence of quantum reflection. New results are reported for the sticking of D atoms incident on helium. The energy upper bound for the regime of quan- tum reflection for alkali atoms is found to be extremely low, but ultimately achievable.

PACS: 34.50.Dy; 82.20.Tr; 67.65.Tz

1. Introduction

In the low energy (E) regime, many physical phenomena exhibit behavior which is qualitatively distinct from that seen at higher energies. The problem of sticking provides one example. It has long been realized that quantum mechanics affects the sticking coefficient, s, in a dramatic way at low energy [1]. More recently, it has been found that long range forces play a particularly important role in determining s at low E. This situation has allowed us to demonstrate that the (usually elusive) effects of retardation on these forces are clearly manifested in recent sticking experiments [2, 3]. The present paper describes the basic ideas and results, with special emphasis provided for the case of hydrogen atoms incident on liquid helium. This focus is due to the relative ease of the calculations and the fact that only this system has been explored to date in the extreme quantum regime. The basic concepts are more general, however, and have relevance to eventual measurements of the sticking of very cold alkali atoms.

The basic physics of low E sticking is straightforward. An atom, incident at low E has a wave length 2 which is

* Present address: Department of Chemistry, University of California, Berkeley, CA 94720, USA

long compared to the characteristic distance scale of the gas-surface interaction V(r). The general theory of waves yields the expectation in this limit that the wave/particle is likely to be reflected long before the atom arrives at the attractive well. Hence s falls to zero; the specific prediction is that s is proportional to the amplitude for the impinging wave within close proximity to the surface. This leads to a dependence,

s ~ ~/E (1)

This so-called quantum reflection behavior differs dramati- cally from the classical expectation that s ought to ap- proach unity at low E for the case of a very cold surface. The latter belief is intuitively obvious because the ap- proach to the surface takes infinite time, during which the particle exerts a nonzero force on the surface. This guarantees that there will ensue some excitation of the solid, resulting in energy loss and hence s = 1.

There have been many studies of the specific case of H/He because of both an experimental relevance and the theoretical convenience of this system (a well known force law and the existence of only one bound state [4]). Inter- est in this field exponentiated in 1991 when data of Doyle et al. I-5] implied that s approaches 0.3 for ultralow energy (10 .4 K) H atoms incident on liquid helium, in marked contrast to the prediction of equation (1). This apparent discrepancy was tentatively resolved by the hypothesis that the experiment actually involved a helium film, of thickness d ~ 50 A [2, 6]. Since 2 > d, the underlying sub- strate strongly alters the long range potential experienced by the impinging atom, drastically modifying s. The pre- dictions of the most recent version of the theory [7] and its subsequent confirmation [3] are discussed below. We also describe here new theoretical predictions of quantum reflection of D atoms incident on liquid helium and dis- cuss the possibility of seeing quantum reflection with ultracold alkali atoms.

Since rather complete reports of the general theory have been published recently, we shall only sketch the basic ideas of the calculation and its relation to the recent experiments. We shall mention also some intriguing results obtained [8,9] for ultralong range potentials

320

which deviate from those found for "ordinary" tong range potentials.

2. Calculations

The calculations of the sticking probabili ty require a Hamiltonian; that is, we must specify the coupling be- tween the a tom and the surface, focussing on the domi- nant processes involved in sticking. In so doing, we extend and refine efforts of Mantz and Edwards. Z immerman and Berlinsky, and Goldman [10, 11]. All of these workers recognized the crucial role of the ripplons, which are quantized capillary waves on the surface of liquid helium. The sticking process involves conservation of the atom's incident energy and surface-parallel momentum (p [I ). If coq is the ripplon's energy for (two-dimensional) wave vector q, the kinematic condition describing sticking is

pZ/(2m) = E = coq + e + (p II - q)Z/(2m) (2)

where s is the atom's binding energy on the surface, and we assume that the bare mass characterizes its translation along the surface [12]. We have set Planck's constant equal to 2re. For the limiting case of virtually vanishing E, one finds that a very specific value of ripplon wave vector (qo) results (0.13 A -a for H incident on He; 0.27 A for D). As described in [7], the resulting low energy expression for the sticking coefficient in the distorted wave Born approximation is

s = [2m/(3~kz] I < 01V'qolkz > 12/[1 + 4Eqo/(3COqo)] (3)

where V'qo is the Fourier transform of the r ipplon-atom coupling and Eqo = q~/(2m). Equation 3 exhibits the ex- pected quantum reflection behavior: the matrix element involves the overlap between the wave functions of the bound state and the impinging particle, which is propor- tional to the normal component of the wave vector, kz. When the matrix element is squared and divided by the incident flux (proportional to kz), we obtain Eq. (1); the key point is that the amplitude of the state I kz > is ~ x / E in the well region.

There are several supplementary considerations to dis- cuss. One is our actual use of a modified approach, instead of (3), to calculate s. As derived by B6heim, Brenig, and Stutzki [-13], our procedure, which incorporates virtual ripplons in the scattered state, is valid if inelastic reflection is improbable, a reliable assumption at low E. A second issue is that the "quantum reflection" description is not quite appropriate when there happens to be a resonance associated with a bound state near zero energy. In such a case, there can ensue deviation from (1). This is, we believe, the situation for the initial experiments of Doyle et al. [4], and helps to explain their values s ~ 0.3. Both we and Hijmans et al. established this result as a consequence of the fact that this initial experiment involved a helium film, for which the resonance does occur, while it does not for the bulk liquid [2, 6, 7].

One of the critical features involved in low energy sticking is the role of the asymptotic interaction V(z) ~ z-". Specifically, it is important to take into ac- count retardation of the relevant forces. In our calcu-

0.1

0.01

0,1 I 10 100 1000 I I m K ]

Fig. 1. Most recent experimental measurements of the MIT group [3, circles] and Amsterdam group [16, triangles] of the sticking coefficient of H incident on cold liquid helium, as a function of incident energy (measured in units of a temperature T). Calculations are those of Hijmans et al. [6, dashed curve] and Carraro and Cole [7, full curves]; the latter curves use potentials fit to alternative values of binding energy, differing by the experimental uncertainty, 10%. Figure is taken from [3] with permission

100

10 1

10 .2

10 .3

10 -4

. . . . . . . . , ........ , ........ , ........ , ........ , ........ ,

/

. - ' + " / / /

J f

. . . . J ' ~ " n = l

/

/

1 0 5 1 0 ~ . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . , . . . . . . . . I . . . . . . .

10 `5 10 .4 10 ~ 10 2 10 "~ 10 0

Ez(K)

Fig. 2. Calculations of the sticking coefficient of D atoms incident on liquid helium and a 50 A helium film, as in [2], respectively. Shown are the capture probabilities into the ground (n = 0; solid- line = film, short-dashes =bulk) and first excited (n = 1; long dashes = film, dot-dash = bulk) states bound to the surface

lations, this means that the power n of the asymptotic interaction changes from 3 to 4, with a crossover occur- ring at about 200 A [14, 15]. Omission of this retardation effect would spuriously increase the predicted s values by about a factor of two [7].

Figure 1 compares our theoretical predictions with the subsequent experimental data of Yu et al. for H (and older Amsterdam data [16]) incident on low temperature he- lium. One observes that the lowest energy data are bracketed by our two theoretical curves, which are de- rived using alternative estimates of the H binding energy ( ~ 1 K), differing by the experimental uncertainty of 10%. This excellent agreement has encouraged us to pro- ceed to the case of D incident on helium, another feasible experiment. Apart from the mass, one must change the potential. This may be surprising to those who realize that H and D are chemically identical. The reason is that we are dealing here with an effective potential due to the cooperative nature of the atom-surface bound state, a kind of polaron effect [10, 17]. If we were instead to let

321

the D potent ia l coincide with that used for H, the cal- culated lowest eigenvalue would be 1.8 K, quite different f rom the measured value, 2.5 K [181. Our change of the potent ia l is s imply to alter its form in the repulsive region occupied by the liquid; the difference there is

V~(z) - VH(Z) = A~/[exp(f lz) + 11

where A/~ is 12 K and fl is 0.587 1. Figure 2 shows the results of these calculat ions for s. We expect that our predict ions will soon be subjected to exper imental validation.

One of the amusing elements of the history of the sticking p rob lem is the following. The nomina l s ignature of q u a n t u m reflection has generally been deemed to be consistency with Eq. 1. The recent flurry of theoretical activity has involved a critical examina t ion of this con- clusion's underlying assumpt ions and approximat ions . Fo r example, studies by Clougher ty and K o h n and by Brenig and Russ have found tha t Eq. 1 is no~ satisfied in the case of ul t ra long range potent ia ls [8, 91. Their findings indicate tha t the low energy limit of s is finite, but is less than one, when the power of the interact ion is either n = 1 or 2; for n = 2, this unusual behavior prevails only if the coefficient of the z -2 te rm is larger than a critical value. Quali tat ively, one m a y interpret this exceptional (finite s) behavior in terms of the incoming wave 's adiabat ic adjust- men t to these very slowly varying potentials, reducing the reflection probabi l i ty at large distance.

Finally, we turn to the case of very low E alkali a toms. We have used a short range hard core interact ion and a re tarded long range interact ion of the form

V ~ - C 3 / [ z 3 ( 1 q- ~z)]

where C3 is the nonretardeod van der Waals coefficient and we have taken 1/~ = 200 A. F r o m the s tandard effective range theory of Boheim et al. [13], we can easily deduce that q u a n t u m reflection occurs at distances large com- pared to 1/~. As a consequence, the q u a n t u m reflection domain of energy is

E ~ [1/(2m)12o:/C3 ~ 2 x 1 0 - 4 / A 2 Kelvin

where A is the a tomic mass (in amu) and we have esti- ma ted C3 ~ 1 eV-A 3 [19]. This expression cor responds to the energy range E less than or of order 10-10 K for Cs a toms incident on liquid helium. While this is very low, the cri terion would involve an even lower limit ( ~ 10-12 K) in the absence of re tardat ion. Again, experi-

menta l tests await the heroic efforts of ambi t ious experi- mentalists [20].

We are grateful to Tom Greytak, Steve Chu, Dennis Clougherty, and Flavio Toigo for discussion and to the NSF for its support through grants DMR91-06237 and 90-22681 (from the Materials Research Group Program).

References

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the surface that the effective mass corrections should be small. In the calculations of [10], the 3He bound state (for which the mass correction is about 1.3) resides on average at z ~ 1.5 A above the interface, while the H and D atoms' mean positions are at z_>5A

13. B6heim, J., Brenig, W., Stutski, J.: Z. Phys. B48, 43 (1982) 14. Cheng, E., Cole, M.W.: Phys. Rev. B38, 987 (1988) 15. Margenau, H., Kestner, N.R.: Theory of Intermolecu!ar Forces,

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Phys. Rev. Lett. 57, 2387 (1986) 17. Goodman, F.O.: Surf. Sci. 214, 577 (1989) 18. Silvera, I.F., Walraven, J.T.M.: Phys. Rev. Lett. 45, 1268 (1980) 19. Vidali, G., Cole, M.W.: Surf. Sci 110, 10 (1981) 20. The most recent experiments with alkali atoms have not entered

this energy regime; see Kasevich, M., Moler, K., Riis, E., Sunder- man, E., Weiss, D., Chu, S.: In: Atomic Physics t2, Zorn, J.C., Lewis, R.R. (eds.) (American Institute of Physics, New York, 1991), p. 47. However, velocity selection techniques, yielding picokelvin temperatures, make such an experiment eventually feasible; see Kasevich, M., Riis, E., Moler, K., Kasapi, S., Chu, S.: Phys. Rev. Lett. 66, 2297 (1991)