We examine photodetachment of H− in parallel electric and magnetic fields, hν+H−→H+e− using semiclassical approximations. The fields cause the electron to return to the atom, producing recurrences that are visible as interference oscillations in the photodetachment cross section. As the energy is varied, new returning orbits are created through bifurcations. Each such new recurrence increases the complexity of the absorption spectrum, and each bifurcation causes a local failure of the semiclassical approximation. The failure is repaired by a Fresnel diffraction integral

Delos, John B.

Jaffé, Charles

Peters, Aaron D.

English

College of William & Mary: W&M Publish

W&M ScholarWorks Arts & Sciences Articles Arts and Sciences 11-21-1994 Quantum Manifestations of Bifurcations of Classical Orbits: An Exactly Solvable Mode Aaron D. Peters Charles Jaffé J. B. Delos Follow this and additional works at: https://scholarworks.wm.edu/aspubs  Part of the Physics Commons VOLUME 73, NUMBER 21 PH YSICAL REVIEW LETTERS 21 NovEMBER 1994Quantum Manifestations of Bifurcations of Classical Orbits: An Exactly Solvable ModelAaron D. Peters and Charles JaffeDepartment of Chemistry, West Virginia University, Morgantown, West Virginia 23506John B. DelosDepartment of Physics, College of William and Mary, Williamsburg, Virginia 23187(Received 23 May 1994)We examine photodetachment of H in parallel electric and magnetic fields, hv + H H + eusing semiclassical approximations. The fields cause the electron to return to the atom, producingrecurrences that are visible as interference oscillations in the photodetachment cross section. As theenergy is varied, new returning orbits are created through bifurcations. Each such new recurrenceincreases the complexity of the absorption spectrum, and each bifurcation causes a local failure of thesemiclassical approximation. The failure is repaired by a Fresnel diffraction integral.PACS numbers: 32.80.Fb, 03.65.Sq, 05.45.+b, 32.60.+iPeriodic orbits play a central role in nonlinear dy-namics —in Poincare's words, they offer "the only open-ing through which we might try to penetrate the fortress(Chaos) which has the reputation of being impregnable. "This classical statement holds a1so in quantum mechan-ics, wherein the periodic orbit theory [1] of Gutzwiller,Balian, and Bloch and Berry provides a general theoreticalframework for studying quantum manifestations of classi-cal chaos. Periodic orbits produce scars in wave func-tions and oscillations in the density of states, while closedorbits produce oscillations in absorption spectra and real-time recurrences that have been observed in many atomsand molecules.Bifurcations of periodic orbits and of closed orbits areof particular interest [2]. A bifurcation is defined as thecreation of new periodic orbits as a fixed parameter ofthe system is varied (such as the total energy or the mag-netic field strength). Bifurcations are readily observablebecause they create new recurrences in absorption spectra,and because at a bifurcation observed recurrences are es-pecially strong. In fact, semiclassical theory predicts thatthe recurrence strength diverges at every bifurcation, be-cause a bifurcation is correlated with a focus of classicalorbits. We have seen many such cases in atomic spectra.This is a deep problem for periodic orbit theory, becauseevery stable periodic orbit produces such focusing effects,which in turn lead to vanishing denominators in the peri-odic orbit sum.We consider photodetachment of an electron from anegative ion, hv + H H + e, in the presence ofstatic parallel electric and magnetic fields [3]. This sys-tem constitutes a key model which opens the door to thesolution of the above problems. This model has the fol-lowing properties: (1) The relevant parts of the modelare exactly solvable. Whereas the mathematical theoryof bifurcations is very abstract, everything in the presentsystem can be understood by elementary methods. (2) Itadmits a simple structure of closed orbits and their associ-(2)q&(t) = + tpo ~2where 80, pp are the polar and azimuthal angles definingthe initial outward motion.ated recurrences, and it possesses an orderly sequence ofbifurcations. (3) At each bifurcation a certain geometricalstructure —a cylindrically focused cusp —passes throughthe origin. This causes the semiclassical approximationto fail. (4) The failure is repaired by a simple diffrac-tion function, a Fresnel integral. The integral provides auniform approximation which is always finite and whichbehaves correctly in all limiting cases. (5) The focusedcusp is sufficiently similar to the structures found in exci-tation of neutral atoms that the present model has helpedus to derive appropriate formulas for those more diffi-cult cases. (6) Finally, the model accurately represents asystem on which experimental measurements can test thepredictions.Following conventional ideas, we describe electron de-tachment by saying that the active electron is initiallyloosely bound to the hydrogen atom by a short-range,spherically symmetric potential. The Hmniltonian gov-erning this electron is [4]t',0= p + —' + ' + —mcop2m( ~ p2) 2 8+ p, + eFoz —eVb(r).1 2 (1)If the active electron absorbs a photon, it leaves theatom traveling radially outward in any direction, but witha fixed initial speed. Qutside the short range of thebinding potential Vb(r), the electron feels only the staticparallel electric and magnetic fields. Accordingly, thez-motion is uniform acceleration, and the motion in p andq is a circular motion at the cyclotron frequency co,z(t) = t(2E/m)'1 cos80 —t eFO/2m,p(t) = (2/co, ) (2E/m)'1 sin80 [si ncot 2/(,0031-9007/94/73(21)/2825(4)$06. 00 1994 The American Physical Society 2825VOLUME 73, NUMBER 21 PHYSICAL REVIEW LETTERS 21 NovEMBER 1994Every classical orbit that subsequently returns to theatom produces a recurrence that may be visible either inreal time or as oscillations in the photoabsorption crosssection. What are the closed orbits? We find them easilyfrom Eqs. (2). For the z-motion, the time required to gofrom z = 0 up against the electric force and then returnto z = 0 is t,'„=2(2mE)' 2cosOO/eFO. The motion inp(t) is a sinusoidal oscillation and represents cyclotronmotion in the magnetic field. The electron returns top = 0 at each cyclotron period t~„=2nm. /cu, . A closedorbit occurs whenever the electron goes out with energyand direction of motion such that t,'„=t~„.It follows that one returning orbit always exists: itlies on the positive z axis, and at very low energiesthis is the only orbit that can exist. The return timefor p motion is the fixed cyclotron time, independent ofthe energy and independent of the radius of the motion.However, the return time for z motion cannot exceedt~[(E) = 2(2mE)'t /eFo, which at small E may be muchless than the cyclotron time. Therefore, for energies suchthat t~~(E) & 2m. /~„the only possible returning orbit lieson the positive z axis.If we increase the electron's energy, the return time ofthe parallel orbit also increases until, at the first "bifur-cation energy," it is exactly equal to one cyclotron time.At this point, a new returning orbit is created. Increas-ing the energy further, the return time of the parallel orbitcontinues to increase. However, by "aiming" the electronat a different angle, we put less energy into the z motionand more into the p motion. Above the first bifurcationenergy there always exists an initial direction that dividesthe energy between the two modes in such a way that thez return time equals the p return time 2m. /cu, so the orbitgoes up and down in z while simultaneously executing asingle circle in (x, Y).%e may say that the new orbit is created out ofthe parallel orbit, and it moves away as the energy isincreased. This phenomenon is what we define as abifurcation of an orbit closed at the originAs we increase the energy further, a second bifurcationenergy occurs, at which the return time of the parallel or-bit is exactly twice the cyclotron period. At this point an-other new returning orbit is created, which undergoes twocyclotron circles in (x, Y) while simultaneously moving upand down in z. As we continue to increase the energy, asequence of bifurcations occurs, with the nth bifurcationoccurring at E„=n~m2(eFO) /2m '„where t~~(E) is n cy-clotron times. Resulting closed orbits are drawn in Fig. 1.%'e need to know exactly what happens to the wholefamily of outgoing trajectories at a bifurcation. As statedearlier, each family is defined by the condition that theelectron begins at the origin (p = 0, z = 0) with fixedspeed moving in any direction. For low energies we havethe families of trajectories depicted in Fig. 2. Caustics(boundaries between allowed and forbidden regions wheretrajectories cross over each other and semiclassical ap-proximations diverge) are apparent in the figure. In the2826j=O {Pal.allel)oIt1612 ';I-2 0p (10 a.u. )FIG. 1. Some orbits closed at the nucleus for parallel electricand magnetic fields.lower part of the left-hand figure, caustics come togetherto form an upward-pointing cusp, whose tip is locatedat z = —1.8 x 103 a.u. Since q& is an ignorable coordi-nate this cusp is actually a three-dimensional structure,which we call a focused cusp, obtained by rotating thetwo-dimensional cusp through 2m. This focused cusp isdirectly connected with the bifurcations of the para11el or-bit. As we increase the energy this cusp rises toward theorigin, and there exists an energy at which the tip of thecusp precisely touches the origin. This energy is exactlythe first bifurcation energy, where a new closed orbit iscreated. On the right, we show the family of trajectoriesat an energy above the first bifurcation energy but belowthe second. The new orbit created at the first boundary en-ergy is shown along with the parallel orbit. Another cuspat z = —4.2 X 10 a.u. has appeared. As we increase theenergy to the second bifurcation energy Eb=2 this cusp willeventually touch the origin, and the second new orbit iscreated. This sequence of events continues and producesall the other bifurcations. At every bifurcation a focusedcusp touches the origin.Semiclassical methods give a simple formula forthe photodetachment cross section: o.(E) = o-0(E) +g," 0 crj,t(E), where pro(E) is the cross section in the ab-sence of fields, and o,"„(E)is the oscillatory contributionto the cross section arising from the jth closed orbit. %efind [4,5]o 3 ~h~ 'I 1~'=,'., = ——~ol2 \ 2E i sin[co, (2mE)'t /eFO]sin+0,VOLUME 73, NUMBER 21 PH YS ICAL REVIEW LETTERS 21 NovEMBER 1994I' ' ' I ' ' ' IE=2.5 cm E=5.65 cm E=15 cm. H,=2T2 - FO=100 V/cmparallel orbit0Ill. !",(2 3 -3 -2 -1 0 1 2 3-3 -2 -1 0 1 2 3p (10 a.u. )FIG. 2. Families of trajectories outgoing from the origin with fixed speed at three different energies.) !Iti. co, l! (eFO/cu, )m.& 2E ) $2mE&& (cos80)-' sin(4, —m. /4), (3b)where ill, = S,/fi— p, , n. /2 is. the action around the jthclosed orbit corrected by the Maslov index p, and wherej = 0 refers to the parallel orbit.Equations (3) say that the recurrence amplitude ofthe parallel orbit [defined as the prefactor of sin@0 inEq. (3a)] diverges at every bifurcation because of thesin in the denominator. Those divergences are visible inFig. 3, and they identify the bifurcation energies. Thisprediction is of course incorrect. To fix it we need anaccurate wave function of the cylindrically focused cusp,which can be constructed using Maslov's methods. By re-expressing the Schroedinger wave function in the mixedposition-momentum spaceA representative calculation is shown in Fig. 3. Su-perimposed on a gradually rising photodetachment crosssection there are oscillations connected with each closedorbit. Close to E = 0 is a simple pattern, then at eachof the bifurcation energies a new oscillation appears, andthe pattern grows steadily more complex as the energyincreases.'P(p„,pY, z) and using cylindrical coordinates in thatspace, the wave function and the resulting photodetach-ment cross section can easily be reduced to a Fresnelintegral [6]. Using the definitions given in Ref. [7], thecontribution to the photodetachment cross section arisingfrom the combined effects of the parallel orbit and the nthnew orbit at energies close to the nth bifurcation is [7]o = 3ao(meFO/—p cos eo)'~ Im e'P e'" duQ00.03CV Oa50.02Eb001Eb=Om --JI. . . . I. . . . I ~ ~ . I10 20 30 40I50 60 70I80Eb 4I90 100-1Energy (cm )FIG. 3. Photodetachment cross section according to semiclas-sical formula. Each bifurcation leads to a divergence followedby a more complicated oscillatory structure.2827This formula is finite at all energies and reduces toEq. (2a) plus (2b) (with j = n) when z, is sufficiently farfrom the origin.VOLUME 73, NUMBER 21 PHYSICAL REVIEW LETTERS 21 NovEMaER l9940030.0185 86 87I88I89I ~ . ~ I90 91I. . . I92 93 94 95Results are shown in Fig. 4. From the corrected for-mula, we find that (I) generally the recurrence amplitudeof the parallel orbit is much less than that of the other or-bits, but (2) near a bifurcation they become comparable,and (3) exactly at each bifurcation, the combined recur-rence amplitude of the parallel orbit and the new orbittogether is precisely half the recurrence amplitude of thenew orbit by itself.All of these consequences of the model can be tested ex-perimentally, preferably by scaled-variable spectroscopy.This research was supported by the National ScienceFoundation and the Office of Naval Research.[1] M. Gutzwiller, Chaos in Classical and QuantumMechanics (Springer-Verlag, New York, 1990); J. Main,Energy (cm )FIG. 4. Photodetachment cross section near the fourth bifur-cation using exact quantum theory (heavy line) and uniformsemiclassical approximation (light line).[2][4][5][6][7]G. Wiebusch, K. Welge, J. Shaw, and J.B. Delos, Phys.Rev. A 49, 847 (1994), and references therein.J.-M. Mao and J.B. Delos, Phys. Rev. A 45, 1746(1992); J.-M. Mao, K. A. Rapelje, S.J. Blodgett-Ford,J.B. Delos, A. Konig, and H. Rinneberg, Phys. Rev. A48, 2117 (1993); J. Gao and J.B. Delos, Phys. Rev. A49, 869 (1994). Other important studies of the behaviornear bifurcations are as follows: C. P. Malta, M. A. M.deAguiar, and A. M. Ozorio de Almeida, Phys. Rev. A47, 1625 (1993); M. Kus, F. Haake, and D. Delande,Phys. Rev. Lett. 71, 2167 (1993); M. W. Beims andG. Alber, Phys. Rev. A 48, 3123 (1993).This system was previously considered by M. L. Du,Phys. Rev. A 40, 1330 (1989) and by I. I. Fabrikant, Phys.Rev. A 43, 258 (1991), but their fully quantum treatmentdid not display any connection with classical orbits.e is the absolute value of the electron charge, cu, , is thecyclotron frequency eHo/mc, Fo and Ho are the appliedelectric and magnetic field strengths, Vb(r) is the effectiveatomic potential that binds the active electron to thehydrogen atom, and L, = xp, , —yp, is the conserved:component of the angular momentum, which we take tobe zero.A full derivation will be published elsewhere.Proof: The generator of the Lagrangian mani-fold for a two-dimensional cusp is the functionS(p„z)= $2mE —pz (z —z, ) + u p4 and the as-sociated wave function in configuration space isknown as a Pearcy function. However, for a cylin-drically focused cusp, replace p' by p„+p, andthe wave function in configuration space is 'P(xyz) =f exp{i[p„x+ p, y + S(p„p,)]/fi)dp, dp„We ne.ed toevaluate it at the origin x = 0, y = 0, and z = 0. Usecylindrical coordinates for the integration, expand thesquare root in powers of (p„' + p2), and (voila) there isthe Fresnel integral.Im means imaginary part, p = 42mE, P = pz, . —meFoz,'/4 pcos'Hofi —(po —1)vr/2, uo = —(z, /2) &&(meFO/pficos'Ho)'t2, Ho is the initial angle of the neworbit after the bifurcation, or zero before the bifurcation,z,. = z(nt, ) is the location of the cusp at the given energy.2828

Quantum Manifestations of Bifurcations of Classical Orbits: An Exactly Solvable Mode

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Quantum Manifestations of Bifurcations of Classical Orbits: An Exactly Solvable Mode

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