Quantum reflection and Liouville transformations from wells to walls
Gabriel Dufour, Romain Guérout, Astrid Lambrecht, Serge Reynaud
QQuantum reflection and Liouville transformations from wells to walls
G. Dufour, ∗ R. Gu´erout, A. Lambrecht, and S. Reynaud Laboratoire Kastler Brossel, UPMC-Sorbonne Universit´es, CNRS,ENS-PSL Research University, Coll`ege de France, Campus Jussieu, F-75252 Paris, France. (Dated: October 14, 2018)Liouville transformations map in a rigorous manner one Schr¨odinger equation into another, witha changed scattering potential. They are used here to transform quantum reflection of an atom onan attractive well into reflection of the atom on a repulsive wall . While the scattering propertiesare preserved, the corresponding semiclassical descriptions are completely different. A quantitativeevaluation of quantum reflection probabilities is deduced from this method.
Quantum reflection of atoms from the van der Waalsattraction to a surface has been studied theoreticallysince the early days of quantum mechanics [1, 2]. Thoughthe classical motion would be increasingly acceleratedtowards the surface, the quantum matter waves are re-flected back with a probability that approaches unity atlow energies, because the potential varies more and morerapidly close to the surface. Experiments have seen quan-tum reflection for He and H atoms on liquid helium films[3–5] and for ultracold atoms or molecules on solid sur-faces [6–12]. Meanwhile various fundamental aspects andapplications have been analyzed in a number of theoret-ical papers [13–23].Paradoxical phenomena appear in the study of quan-tum reflection from the Casimir-Polder (CP) interactionwith a surface. The potential is attractive, with charac-teristic inverse power laws at both ends of the physicaldomain z ∈ ]0 , ∞ [ delimited by the material surface lo-cated at z = 0 : V ( z ) (cid:39) − C /z at the cliff-side , closeto the surface and V ( z ) (cid:39) − C /z at the far-end , awayfrom it. Strikingly, the probability of reflection increaseswhen the energy E of the incident atom is decreased,and increases as well when the absolute magnitude ofthe potential is decreased. For example, the probabilityof quantum reflection is larger for atoms falling onto sil-ica bulk than onto metallic or silicon bulks [24] and iseven larger for nanoporous silica [25].In the present letter, we use Liouville transformationsto study quantum reflection (QR). In quantum mechan-ics, a Liouville transformation maps in a rigorous mannerone Schr¨odinger equation into another, with a changedscattering potential. In a semiclassical picture however,the problem can be transformed from QR of an atom onan attractive well into a problem of reflection on a repul-sive wall . Remarkably, scattering properties are invariantunder the Liouville transformation and the paradoxicalfeatures of the initial QR problem become intuitive pre-dictions of the better defined problem of reflection on therepulsive wall. We will also obtain a quantitative evalu-ation of QR probabilities in this way.We consider a cold atom of mass m incident with anenergy E > V ( z ) in the half-line z ∈ ]0 , ∞ [. For plane material surfaces, the motion or- thogonal to the plane (along the z − direction) is decou-pled from the transverse motions and described by a 1DSchr¨odinger equation: Ψ (cid:48)(cid:48) ( z ) + F ( z ) Ψ ( z ) = 0 , F ( z ) ≡ m ( E − V ( z )) (cid:126) . (1)Throughout the letter, primes denote differentiation withrespect to the argument of the function.In the semiclassical WKB approximation, the function F ( z ) is seen as the square of the de Broglie wave-vector k dB associated with the classical momentum p ≡ (cid:126) k dB .As the CP potential is attractive and the incident energypositive, F is positive, so that a classical particle under-goes an increasing acceleration towards the surface. Fora quantum particle in contrast, QR occurs when the vari-ation of k dB becomes significant on a length scale of theorder of the de Broglie wavelength: λ dB ≡ λ dB π ≡ k dB = 1 √ F . (2)The Schr¨odinger equation (1) can be solved in full gen-erality by writing its solution as a linear combinationof counter-propagating WKB waves with z − dependentcoefficients and matching it to the appropriate bound-ary conditions at both ends of the physical domain [13].Matter-waves can be reflected back from the cliff-side sothat the complete problem depends on the details of thephysics of the surface. In this letter, we focus our atten-tion on the one-way problem where the CP potential iscrossed only once and, therefore, do not discuss this sur-face physics problem any longer. The numerical solutionof (1) leads to reflection and transmission amplitudes de-pending on the incident energy E or, equivalently, of theparameter κ ≡ √ mE/ (cid:126) which is also the asymptoticvalue of de Broglie wavevector in the far-end.In spite of its effectiveness, the numerical solution ofthe QR problem leaves open questions. First, the scatter-ing problem, where matter waves are reflected or trans-mitted on the CP potential, is not well defined with thepotential diverging at the cliff-side. Second, an intuitiveunderstanding of the dependence of QR probability onthe parameters is missing. The Liouville transformationsconsidered in the following will give clear answers to thesequestions. a r X i v : . [ qu a n t - ph ] D ec The Schr¨odinger equation (1) is an example of a Sturm-Liouville equation in canonical form [26], which can besubmitted to transformations introduced by Liouville [27]and often named after him (see the historical notes atthe end of ch.6 in [28]). We stress at this point thatwe use these transformations to relate exactly equivalentscattering problems, with no approximation (see a similarapproach to the study of differential equations in [29]).Liouville transformations are gauge transformationsconsisting in a change of coordinate z → ˜ z , with ˜ z ( z )a smooth monotonously increasing function, and an as-sociated rescaling of the wave-function:˜ Ψ (˜ z ) = (cid:112) ˜ z (cid:48) ( z ) Ψ ( z ) . (3)Equation (1) for Ψ is transformed under (3) into an equiv-alent equation for ˜ Ψ with [28]:˜ F (˜ z ) = F ( z ) − { ˜ z, z } ˜ z (cid:48) ( z ) = z (cid:48) (˜ z ) F ( z ) + { z, ˜ z } . (4)The curly braces denote the Schwarzian derivative of thecoordinate transformation: { ˜ z, z } = ˜ z (cid:48)(cid:48)(cid:48) ( z )˜ z (cid:48) ( z ) −
32 ˜ z (cid:48)(cid:48) ( z ) ˜ z (cid:48) ( z ) . (5)These transformations form a group, with the compo-sition of z → ˜ z and ˜ z → ˆ z being a transformation z → ˆ z .The compatibility of relations obeyed by ( Ψ, F ), ( ˜
Ψ , ˜ F )and ( ˆ Ψ , ˆ F ) is ensured by Cayley’s identity: { ˆ z, z } = (˜ z (cid:48) ( z )) { ˆ z, ˜ z } + { ˜ z, z } . (6)The inverse transformation, used for the second equalityin (4), is obtained by applying (6) to the case ˆ z = z .The group of transformations preserves the Wronskianof two solutions Ψ , Ψ of the Schr¨odinger equation, whichis a constant independent of z and skew symmetric in theexchange of the two solutions: W ( Ψ , Ψ ) = Ψ ( z ) Ψ (cid:48) ( z ) − Ψ (cid:48) ( z ) Ψ ( z ) . (7)In particular, when Ψ solves (1), its complex conjugate Ψ ∗ solves it as well. As the probability density currentis proportional to the Wronskian W ( Ψ ∗ , Ψ ), it is invari-ant under the transformation. The reflection and trans-mission amplitudes r and t are also preserved, as theycan be written in terms of Wronskians of solutions whichmatch incoming and outgoing WKB waves [30]. Theycan be calculated equivalently after any Liouville trans-formation, with ˜ r = r and ˜ t = t . These transformations,which do not necessarily simplify the resolution of (1),have to be considered as gauge transformations relatingequivalent scattering problems to one another.These quantum-mechanically equivalent scatteringproblems may correspond to extremely different classi-cal descriptions. We now write a specific Liouville gaugewhich maps the initial problem of QR on an attractivewell into an intuitively different problem of reflection on a repulsive wall. This choice brings clear answers to thequestions discussed above, and it will allow us to un-cover scaling relations between the QR probabilities andthe parameters of the problem.This specific Liouville gauge is written in terms of theWKB phase φ ≡ ´ z k dB ( y )d y associated with the classi-cal action integral S ≡ (cid:126) φ . We fix the freedom associatedwith the arbitrariness of the phase reference by enforcing φ ( z ) → κz at z → ∞ . We then choose the coordinate z for which we get quantities identified by boldfacing: z ≡ φ √ κ(cid:96) , F ( z ) ≡ E − V ( z ) , (8) E = κ(cid:96) , V ( z ) = − κ(cid:96) (cid:113) λ ( z ) (cid:16)(cid:112) λ dB ( z ) (cid:17) (cid:48)(cid:48) . We have defined the length scale (cid:96) ≡ √ mC / (cid:126) associ-ated with the far-end tail of the CP potential. Its in-troduction in (8) has been done for reasons which willbecome clear soon, and it leads to a dimensionless en-ergy E and a dimensionless potential V .For the CP potential, the quantity V vanishes at bothends of the physical domain z ∈ ]0 , ∞ [, that is also atboth ends of the transformed domain z ∈ ] − ∞ , ∞ [,so that the problem now corresponds to a well-definedscattering problem with no interaction in the asymptoticinput and output states. In striking contrast with theoriginal QR problem, the transformed problem can haveclassical turning points where F = 0 or E = V , thoughit corresponds to the same scattering amplitudes.This important point is illustrated by the drawings onFig.1, which shows the constants E and the functions V ( z ) for different scattering problems. In all cases, theoriginal potential V is calculated for the CP interactionbetween an hydrogen atom and a silica bulk [24], whereasthe incident energies E are respectively equal to 0.001,0.1 and 10 neV. With E always positive and V ( z ) oftenpositive, a logarithmic scale is used along the verticalaxis, which makes some details more apparent.The most striking feature of these plots is the ap-pearance of classical turning points for the not too highenergies considered here, so that QR on an attractivewell is now intuitively understood as reflection on a wall.Other clearly visible properties are that E scales like √ E whereas V ( z ) has nearly identical peak shapes for dif-ferent energies. The fact that the QR probability goesto unity when E → E coming onto a wall with a peak V .In fact, the potentials V calculated for different ener-gies tend to build up a universal function at large enoughvalues of z , and this universal function has a symmetri-cal shape. These two facts can be explained by lookingat the particular model V ( z ) = − C /z , which is repre-sentative of the CP interaction in the far-end. For thissimple model, V ( z ) is given by parametric relations (with − − − − z − − − − − V , E FIG. 1. [Colors online] The plots represent the constants E (horizontal lines) and the functions V ( z ) (curves) calculatedfor different scattering problems, corresponding to the sameCP potential V ( z ) between an hydrogen atom and a silicabulk and energies E equal to 0.001, 0.1 and 10 neV (respec-tively blue, green and red from the lowest to the highest valueof E , or from the lowest to the highest value of V in theleft-hand part of the plot). The dashed (black) curve is theuniversal function V ( z ) calculated for a pure C model. e u ≡ z/z and z = (cid:112) (cid:96)/κ ): V = 58 cosh (2 u ) , (9) z = z + ˆ u (cid:112) v )d v , z = 1 √ π Γ (cid:0) (cid:1) . This function, drawn as the dashed curve on Fig.1,reaches its peak value at z = z , which lies furtherand further away from the surface when the energy de-creases. This also explains why the functions plotted onFig.1 for the full CP potential tend to align on this uni-versal form when the energy decreases. The deviationsappearing on the figure correspond to values of z near thecliff-side, for which the C model is indeed a poor rep-resentation as the potential behaves as − C /z . In theparametric definition (9), V is even and z − z odd inthe parity u → − u . It follows that the universal function V ( z ) is symmetrical with respect to z .We come now to the discussion of the dependence ofQR on the absolute magnitude of the CP potential. Todo so we consider hydrogen falling onto nanoporous sil-ica, which has a weaker CP interaction when its poros-ity increases [25]. Fig.2 shows the constants E and thefunctions V ( z ) for an energy E = 0 .
01 neV, and thepotentials calculated for an hydrogen atom falling ontonanoporous silica with porosities η equal to 0%, 50% and90%. These potentials correspond to different far-endtails, with values of C , and therefore (cid:96) , smaller andsmaller when the porosity is increased. As on Fig.1, the transformed potentials V have nearly identical peakshapes, which tend to align on the universal curve cal-culated for a pure C potential and shown as the dashedcurve. In contrast, the transformed energies E = κ(cid:96) aredecreasing when (cid:96) is decreased, which immediately ex-plains why the QR probability increases [25]. − − − − z − − − − − V , E FIG. 2. [Colors online] The plots represent the constants E (horizontal lines) and the functions V ( z ) (curves) calculatedfor different scattering problems, corresponding to the energy E = 0 .
01 neV and the CP potentials V ( z ) between an hy-drogen atom and nanoporous silica with porosities η equal to0%, 50% and 90% (respectively blue, green and red from thehighest to the lowest value of E , or from the lowest to thehighest value of V in the left-hand part of the plot). Thedashed (black) curve is the same as on Fig.1. We finally discuss the values obtained for QR proba-bilities, by comparing the exact results for the full CPpotential with those obtained for the C model. To thisaim, we first recall the low energy behavior of the QRprobability: R ( κ ) ≡ | r ( κ ) | (cid:39) − κb , κ → , (10)where b is the opposite of the imaginary part of the scat-tering length [24]. For a pure C model, b is known tobe equal to (cid:96) [21], but this is not the case for the fullCP potential. Table I gives (cid:96) and b for nanoporous silicawith different porosities η ( η = 0% for silica bulk). η [%] 0 30 50 70 90 (cid:96) [ a ] 321.3 282.1 244.7 192.8 111.8 b [ a ] 272.7 227.8 187.5 134.0 57.0TABLE I. Values of (cid:96) and b calculated for different porosities,measured in atomic units a (cid:39)
53 pm.
We have reported on Fig.3 the calculated QR prob-abilities R as a function of the dimensionless parame-ter κb for the scattering problems discussed above. Thefull blue curve represents the values calculated for silicabulks in [24], while the circles correspond to the scat-tering problems of Fig.2 with the same color code. Thedashed black curve corresponds to the universal function R ( κb ) obtained for the pure C model, with b ≡ (cid:96) inthis case. A table of values of this function is availableas supplemental material. The exact results on Fig.3are hardly distinguishable from this universal function,except at large values of κb where QR probabilities aresmall anyway. − − − κb . . . . . . R FIG. 3. [Colors online] Quantum reflection probability R shown as a function of the dimensionless parameter κb . Thefull blue curve represents the values calculated for silica bulksin [24], while the crosses correspond to the scattering prob-lems of Fig.2 with the same color codes. The dashed (black)curve is the universal function R for a pure C model. In this letter, the problem of QR of an atom on a po-tential well has been mapped into an equivalent problemof reflection on a wall through a Liouville transformationof the Schrdinger equation. This exact transformationrelates quantum scattering processes which correspondto different semiclassical pictures. It produces a newand clear interpretation of the main features of quan-tum reflection which were counterintuitive in the initialproblem. It also allows quantitative evaluation of QRprobabilities which can be obtained from the universalfunction corresponding to the pure C model. Acknowledgements -
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