25 May 2000 Ž . Optics Communications 179 2000 117–121 www.elsevier.comrlocateroptcom A bouncing wavepacket: finite-wall and resonance effects 1 Julio Gea-Banacloche Department of Physics, UniÕersity of Arkansas, FayetteÕille, AR 72701, USA Received 2 July 1999; accepted 1 September 1999 Abstract The motion of a quantum particle bouncing on a hard surface under the influence of gravity is considered, including effects arising from the repulsive barrier’s finite height and width. The possible effect of tunnelling resonances is illustrated. The analytical approach is based on the WKB approximation and on the study of the phase shift experienced by the particle’s wavefunction upon reflection at the barrier. The results of ‘exact’ numerical calculations are also shown. q 2000 Elsevier Science B.V. All rights reserved. PACS: 03.75.Be; 32.80.Pj; 32.80.Lg wx In a recent publication 1 I have considered the motion of a quantum particle bouncing on a hard w x surface under the influence of gravity 2–4 from a pedagogical viewpoint, focusing on the derivation of the classical limit and pointing out the presence in this problem, as in many other similar ones, of collapses and revivals of the expectation value of a Ž dynamical variable in this case, the particle’s posi- . tion . The present paper deals with a somewhat more realistic model, by treating the repulsive force near the surface as an exponential, rather than an infinite potential barrier, and discusses also the way in which the collapses and revivals might be affected if reso- nant tunnelling of the particle through the barrier is possible. The latter may not be the case in the simplest realizations of this system, but could happen for other very cold particles confined in other ge- Ž . ometries e.g., optical lattices . 1 This paper is affectionately dedicated to Marlan Scully on the occasion of his 60th birthday. In the interest of keeping this paper short, the wx reader is referred to 1 for many of the details. wx Ignoring the Van der Waals force 5 between the atom and the wall, and all other complications Ž . spontaneous emission, transverse motion, etc. , the potential seen by the atom can be approximated by V z s V e yz rd q mgz . 1 Ž . Ž. 0 The exponential decay of the repulsive potential holds both for evanescent wave mirrors and mag- Ž netic mirrors for the former, d is the penetration . wx depth of the evanescent wave . In 1 the repulsive Ž . barrier was taken to be infinite V ` and in- 0 Ž . finitely sharp d 0. From now on I shall assume that the time, posi- tion and energy variables have been scaled to the wx ‘gravitational scales’ introduced in 1 . That is, z Ž 2 2 . 1r3 will be measured in units of l s " r2 gm , g Ž 2 2 . 1r3 energy in units of mgl s " mg r2 , and time g Ž . Ž 2 . 1r3 in units of "r mgl s 2 "rmg . For reference, g for a Cs atom l s 0.226 mm, mgl s 3.06 = 10 y12 g g 0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. Ž . PII: S0030-4018 99 00493-9