The Classical Atom: Stabilizationof Electronic Trojan Wavepackets

David Farrelly1Þ, Ernestine A. Lee

2Þ, and T. Uzer3Þ

1Þ Department of Chemistry, Utah State University, Logan, Utah 84322, USA2Þ Incyte Genomics, 3160 Porter Drive, Palo Alto, California 94304, USA3Þ Center for Nonlinear Sciences and School of Physics, Georgia Institute of Technology,Atlanta, Georgia 30332-0430, USA

Abstract

This article demonstrates that coherent states in Rydberg atoms can be produced and stabilized bycombining a circularly polarized microwave field with a static, perpendicular magnetic field. Theseelectronic wavepackets owe their stability to atomic analogs of the Lagrange equilibria which confineJupiter’s Trojan asteroids. While these ‘‘Trojan” wavepackets may slowly decay due to tunneling, amore significant source of dispersion will arise if the tails of the wavepacket penetrate appreciably intothe non-linear or chaotic parts of phase space. In the laboratory frame, if these dispersive factors canbe minimized –– and this may be accomplished using magnetic fields –– the electronic wavepacket willtravel along a circular Kepler orbit while remaining localized radially and angularly for a finite –– butlarge –– number of Kepler periods. In this sense, the system is a classical Bohr atom.

1. Introduction

The gravitational and Coulombic potentials are functionally identical and a one-electronatom is, therefore, governed by the same Hamiltonian as is the Kepler problem. The classi-cal equations of motion for the Kepler problem are easy to solve: in the Solar System, ifonly the Earth and the Sun were considered, then the Earth and Sun would endlessly orbittheir center of mass, never varying from an elliptical orbit. The presence of the Moondestroys the integrability of the system and the resulting motions become extremely compli-cated. When the number of bodies is increased beyond three –– by including the otherplanets –– the problem becomes even more of a nightmare. In fact, despite an enormousexpenditure of effort, no case beyond the two-body problem has been solved completely.This complexity has had its felicitous side, however, in that many of the important develop-ments in applied mathematics and physics, from the publication of Sir Isaac Newton’s Prin-cipia in 1687 to Max Born’s Mechanics of the Atom in 1925 [1], and down to today, havebeen fueled by progress made in studies of the n-body problem [2]. In particular, the ‘‘OldQuantum Mechanics” as described in Born’s book includes many concepts and methodolo-gies that were carried over intact from celestial mechanics. Although celestial mechanicsand atomic physics temporarily parted ways after the discovery of wave mechanics thesynergism between the two fields left a lasting impression: the notion of a ‘‘classical” or‘‘Bohr” atom in which a localized electron revolves around the nucleus in a Kepler orbitcontinues to persist not only among members of the general public but also among atomicphysicists. Indeed, the goal of making a classical atom accounts for a notable fraction ofthe vast body of contemporary experimental and theoretical research on Rydberg atoms[3, 4]. However, the three-body problem itself, arguably the raison d’etre of celestial me-chanics for several centuries, has received relatively less attention from atomic physicistsbecause it does not have a direct quantum counterpart. Three quantum particles cannot

Fortschr. Phys. 50 (2002) 5––7, 636 –– 641

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mutually attract one another and at the same time interact through a purely Coulombicforce law. Nevertheless, in the last five years it has been demonstrated by several researchgroups that quantum analogs of a particular limit of the three-body problem, the restrictedthree-body problem [5], not only exist but contain dramatically new physics. Based onideas that can be traced directly back to Schrodinger, a definition of a classical atom that isconsistent with quantum mechanics can be stated as follows;

1. The wavepacket representing the electron neither spreads nor disperses as its centermoves along a Kepler orbit about the nucleus.

2. The spatial extent of the packet is small compared to the radius of the orbit.3. The classical orbit along which the center of the wavepacket moves is confined to a

single plane in space.

The close analogy between a Rydberg atom in a circularly polarized microwave field andthe restricted three-body problem extends to the existence of atomic stable equilibrium pointsthat are directly analogous to the Lagrangian equilibrium points in celestial mechanics. Thisled Bialynicki-Birula, et al. [6] to expect that wavepackets launched from the atomic equili-brium point analogous to the stable Lagrange points L4 and L5 would orbit the nucleus with-out spreading. The term ‘‘Trojan” wavepacket is appropriate for such states. While the ana-logy between Rydberg atoms and planetary systems turned out to be useful, the finite size ofPlanck’s constant imposes an absolute scale on the atomic problem which soon leads to dis-persion. The atomic analogs of these points are stable only over a limited range of parameters,and placing a finite-size minimum uncertainty wavepacket at such an equilibrium point mayrequire unrealistically large principal quantum numbers to guarantee stability. In other wordslinear stability is not enough to guarantee the actual stability of a particular wavepacket.

The present authors invented a different method in which a homogeneous magnetic fieldis used in addition to the circularly polarized microwave field to continuously stabilize thepacket as it moves around the nucleus; in essence, the states are robust coherent states ofan atom dressed by both microwave and magnetic fields. Examination of all of these sys-tems in a rotating frame reveals the key role played by Coriolis forces.

2. Trojan Wavepackets

The problem of a hydrogen atom in a circularly or elliptically polarized microwave fieldhas received considerable attention, mainly from the standpoint of how the polarization ofthe field affects ionization [3]. In the dipole approximation and atomic units (�h ¼ me

¼ e ¼ 1) the Hamiltonian for a hydrogen atom subjected to a circularly polarized micro-wave field and a magnetic field perpendicular to the plane of polarization is

H ¼ ~pp2

2� 1

r� wc

2ðxpy � ypxÞ þ

w2c

8ðx2 þ y2Þ þ Fðx coswf t þ y sinwf tÞ : ð1Þ

The magnetic field is taken to lie along the positive z-direction, wc ¼ eB=mec is the cyclo-tron frequency, while wf is the microwave field frequency and F is its strength. In a framerotating with the field frequency wf the Hamiltonian becomes

K ¼ ~pp2

2� 1

r� wf þ

wc

2

� �ðxpy � ypxÞ þ Fx þ w2

c

8ðx2 þ y2Þ ; ð2Þ

where K is called the Jacobi constant in analogy with the restricted three-body problem [2]and the coordinates are now interpreted as being in the rotating frame.

Fortschr. Phys. 50 (2002) 5––7 637

The equilibria may be found by examination of curves of zero velocity used by Hill inhis studies of the motions of the Moon, see Ref. [2]. These curves are obtained most easilyby the simple short cut: re-write the Hamiltonian in terms of the velocities using Hamilton’sequations of motion and then, since the Hamiltonian is a quadratic form in the velocities,setting them to zero which gives;

� ¼ � 1

rþ Fx � wf ðwf þ wcÞ

2ðx2 þ y2Þ : ð3Þ

The equilibrium points of this zero-velocity surface (ZVS) are found to lie along the xaxis and provides the analogy with the Lagrange equilibrium points L4 and L5 in the re-stricted-three body problem. This configuration of equilibria occurs whether or not a mag-netic field is added. Bialynicki-Birula et al. [6] originally studied the case in which only acircularly polarized microwave field was used.In scaled coordinates the Hamiltonian becomes

K ¼ ~pp2

2� 1

r� ðx py � y pxÞ þ Ex þ w2

s

8ðx2 þ y2Þ ð4Þ

which shows that the classical dynamics depends only on the three parameters, K, ws, andE. Stability of the equilibrium can now be expressed in terms of the two parameters ws andE. For wc 6¼ 0 the curve separating stable from unstable motion consists of an upper branchðE>Þ and a lower branch ðE<Þ given by

E>< ¼4� 5w2

s � 2� 1

2w2

s

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4� 9w2

s

p243w

23s ð2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4� 9w2

s

pÞ23

; ð5Þ

where the upper (lower) sign is taken throughout for E> ðE<Þ. These two functions becomeequal to each other exactly at ws ¼ 2=3 which is the rightmost point in Fig. 4. It wassomething of a surprise to learn that at this point the planar limit of the problem reduces tothe rigorously integrable case discovered by Rakovic and Chu [7]

K ¼ðp2x þ p2yÞ

2� 1

r� ðxpy � ypxÞ þ ð2=3Þ

43 x þ 1

18ðx2 þ y2Þ : ð6Þ

2.1. Coherent states

The next step is to expand the Hamiltonian at the equilibrium point using methods devel-oped in celestial mechanics [8] and nuclear physics [9]. Full details and an exhaustive setof references are provided in Refs. [10, 11].A harmonic approximation to the Hamiltonian to describe librations around the equili-

brium point shows the motion in the z (or z)-direction to be stable, harmonic, and un-coupled from the planar motion. Therefore, we ignore this degree of freedom.After a rotation in phase space

x0 ¼ Axþ Bph ; ð7Þh0 ¼ Ahþ Bpx ; ð8Þ

p0x ¼ px þ Ch ; ð9Þ

p0h ¼ ph þ Cx ð10Þ

D. Farrelly et al., The Classical Atom638

with A� BC ¼ 1 (to preserve the commutation relations between coordinates and momen-ta) H can be reduced to the following locally separable form [9]

K ¼ 1

2mxp0x

2 þ 1

2mxW

2xx

02 þ 1

2mhp0h

2 þ 1

2mhW

2hh

02 ; ð11Þ

where we will assume (without loss of generality) that wh > wx: The locally harmonicfrequencies

W2h ¼ w2 1

4ws

2 þ 1� 1

2q� 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9q2 � 8qþ 4ws

2p� �

; ð12Þ

W2x ¼ w2 1

4ws

2 þ 1� 1

2qþ 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9q2 � 8qþ 4ws

2p� �

ð13Þ

and the ‘‘masses”

mx ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9q2 � 8qþ 4w2

s

p2þ 3

2q

� �þ 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9q2 � 8qþ 4ws

2p ; ð14Þ

mh ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9q2 � 8qþ 4ws

2p

2� 3

2q

� �� 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi9q2 � 8qþ 4ws

2p ð15Þ

are here given in terms of the dimensionless quantity

q ¼ 1

w2x30: ð16Þ

The ‘‘masses” are not physical masses and can be either positive or negative; there is nobound motion if both masses are negative. In the case of motion at a maximum, the masseshave opposite signs. Based on the zero-velocity surface, this case occurs whenw2 > wx

2; wh2. When both masses are positive, stable motion at a minimum is indicated.

In order to cover the two possibilities for stable motion, we define the index

L ¼ mxmh

jmxmhjð17Þ

which is 1 for a minimum and �1 for a maximum.The energy eigenvalues are given by

E ¼ mx

jmxjnx þ

1

2

� ��h jWxj þ

mh

jmhjnh þ

1

2

� ��h jWhj : ð18Þ

The magnitude of the ground state energy is defined in terms of an average frequency Wthrough

E ¼ �h

2ðjWxj þL jWhjÞ ¼ �hW ð19Þ

Fortschr. Phys. 50 (2002) 5––7 639

with this frequency explicitly being given by

W ¼ jwj2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2� qþ 1

2w2

s þ 2Lsðq; wsÞr

; ð20Þ

where

sðq; wsÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2q� 1

4ws

2

� �1� q� 1

4w2

s

� �s: ð21Þ

2.2. The initial wavepacket

The ground-state wavefunction of the (local) electronic Hamiltonian becomes

Y000ðx; h; zÞ ¼ NwCðx; hÞ exp �wz

2z2

� �; ð22Þ

where wCðx; hÞ is the normalized ground state wavefunction of the cranked oscillator

wCðx; hÞ ¼ab

p2

� �14

exp � a

2x2 � b

2h2 � igxh

� �: ð23Þ

The parameters a; b; g are given by,

a ¼ w

3q�h

� �1þ 2q� 1

4ws

2 þLs

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2� qþ 1

2w2

s þ 2Lsðq; wsÞr

; ð24Þ

b ¼ w

3q�h

� �q� 1þ 1

4ws

2 �Ls

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2� qþ 1

2w2

s þ 2Lsðq; wsÞr

; ð25Þ

g ¼ w

3q�h

� �2þ q� 1

2ws

2 þ 2Ls

� �: ð26Þ

2.3. Phase factors

The transformation to barycentric synodical coordinates requires two shifts, one in coordi-nate and another in momentum, to reach the equilibrium point from the center of mass.These shifts introduce all-important phase factors. The quantum mechanical consequence ofthese shifts can be described by a translation operator

T ðx0Þ ¼ exp ð�ix0px=�hÞ ð27Þ

and a boost-like operator

Bðwx0Þ ¼ exp ðiwx0h=�hÞ : ð28Þ

If jCi is the ket which is represented by wCðx; hÞ, then the ket jIi that we need to use asthe initial state in the barycentric coordinates is related to it by

jIi ¼ T ðx0Þ Bðwx0Þ jCi ð29Þ

D. Farrelly et al., The Classical Atom640

and therefore the wavefunctions are related by

wCðx; yÞ ¼ab

p2

� �14

exp ðinx0yÞ exp � a

2ðx � x0Þ2 �

b

2y2 � igðx � x0Þ y

� �; ð30Þ

where n ¼ w=�h.Numerical integration of the quantum motion in [11] demonstrates explicitly that these

wavepackets are stable to dispersion, and are localized on a scale small compared to thesize of the atom.

3. Conclusions

This article has demonstrated that coherent states in Rydberg atoms can be produced by acombination of circularly polarized microwave and magnetic fields. Such states neitherspread nor disperse as they execute their revolutions around the nucleus, although a Trojanwavepacket will slowly decay due to tunneling. A more significant source of dispersionwill arise if the tails of the wavepacket penetrate appreciably into the non-linear or chaoticparts of phase space. In the laboratory frame, if these dispersive factors can be minimized,the electronic wavepacket will travel along a circular Kepler orbit while remaining localizedradially and angularly for a finite (but possibly very large) number of Kepler periods. Animportant point in our study is that the stability of such a packet can be enhanced consider-ably by using a magnetic field in addition to the circularly polarized field. Indeed, theinitial wavepacket and field choices displayed remarkable localization with the addition of astatic magnetic field whereas the absence of this field can lead to rapid delocalization.

The applications of Hill’s methods and the importance of Lagrange equilibria in quantumphysics are not exhausted [2]. Applications have also been made to Rydberg molecules(important in zero electron-kinetic energy or ZEKE spectroscopy), to the stability of Bose-Einstein condensates in the time-averaged orbiting potential (TOP) trap, and to quantumdots [12]. Without doubt the legacies of Lagrange and Hill as well as other giants of celes-tial mechanics continue to influence the direction of many important areas of physics andastronomy.

References

[1] M. Born, Mechanics of the Atom (republished by F. Ungar, New York, 1960). Tranlsated by J. W.Fisher.

[2] J. M. A. Danby, Fundamentals of Celestial Mechanics, 2nd edition (Willman-Bell, Richmond,Virginia, 1992). pp. 384––385.

[3] T. Gallagher, Rydberg Atoms (Cambridge University Press, Cambridge, 1994).[4] J. P. Connerade, Highly Excited Atoms (Cambridge University Press, Cambridge, 1998).[5] V. Szebehely, Theory of Orbits: The Restricted Problem of Three Bodies (Academic, New York,

1967).[6] I. Bialynicki-Birula, M. Kalinski, and J. H. Eberly, Phys. Rev. Lett. 73 (1994) 1777.[7] M. J. Rakovic and S. I. Chu, Phys. Rev. A 50 (1994) 5077.[8] A. Deprit, Astron. J. 71 (1966) 77.[9] D. Glas, U. Mosel, and P. G. Zint, Z. Phys. A 285 (1978) 83.

[10] E. Lee, A.F. Brunello, and D. Farrelly, Phys. Rev. A 55 (1997) 2203.[11] C. Cerjan, E. Lee, D. Farrelly, and T. Uzer 55 (1997) 2222.[12] E. Lee, Theoretical Investigations of the Electronic Dynamics of Rydberg Atoms and Molecules

and Quantum Dots in the Presence of Nonconserved Paramagnetic Interactions. Doctoral Disser-tation, Utah State University, 1997 (unpublished).

Fortschr. Phys. 50 (2002) 5––7 641