## Abstract

In Bohr’s original planetary model of the atom the electron moves along orbits of special geometric simplicity. While wave mechanics precludes the idea that a physical path could be ascribed to the electron, a classical or planetary atom can still be envisaged in which the electronic wavepacket neither spreads nor disperses as its center moves along the Kepler orbit, and this orbit is confined to a single plane in space. We show theoretically how an electronic wavepacket may be localized in this fashion in a similar way to ion confinement in a Penning trap. Because external fields are needed to keep the packet confined, a more fitting analogy than a planetary orbit is the motion of a charged dust grain in one of the rings of a giant planet such as Saturn.

© Optical Society of America

In order to explain the scattering of alpha particles by atoms, Rutherford^{1} considered Nagaoka’s Saturnian atom^{2} which consisted of rings of rotating electrons.^{2–4} Although wave mechanics forced the abandonment of the idea that a physical path could be ascribed to the electron, a classical or *planetary* atom can still be envisaged in which: *(i)* the electronic wavepacket neither spreads nor disperses as it moves along a Kepler orbit, *(ii)* this orbit is confined to a single plane in space. Physically, this atom is a rotating giant dipole.^{5} We show theoretically how this can be achieved using external fields similar to ion or electron confinement in the Penning trap^{6} except that the motion is associated with a genuine harmonic *minimum* in the effective potential. The dynamics resembles the motion of a charged dust grain in one of the ethereal rings of a giant planet, *e.g.*, the Gossamer Ring of Jupiter or Saturn’s F, G and E rings^{7,8}, and in this sense, the system may be considered to be a one-electron Saturnian atom.

Much recent work in quantum physics has been concerned with the dynamics of atoms in which a single electron is highly excited, *i.e.*, a Rydberg atom.^{9} A particular goal has been the creation of non-spreading electronic wavepackets or coherent states that move along Keplerian orbits in a similar fashion to an electron in Bohr’s planetary atom.^{10–12} However, the best that has been accomplished are atoms in which the spreading of the wavepacket is localized on a Kepler orbit—*i.e.*, radial but not angular confinement.^{10–12} In these studies the strategy has often been to work at very high quantum numbers for which the local energy spacings are approximately constant and use laser excitation to form a spatially localized superposition of atomic states.

The close analogy between a Rydberg atom in a circulary polarized microwave field (CP)^{13–17} and the Restricted Three-Body Problem^{18} led us^{19} and Bialynicki-Birula, Kalinski and Eberly (BKE)^{20} to discover independently that in the CP problem, stable equilibrium points exist that are analogous to the Lagrangian equilibrium points in celestial mechanics.^{18} This analogy led BKE to expect that wave packets launched from the equilibrium points (analogs of the so-called Lagrange points *L*
_{4} and *L*
_{5}) would orbit the nucleus without spreading. The Lagrange equilibrium points are stable maxima that support the Trojan asteroids of Jupiter, making the term “Trojan” wave packet appropriate for these states.^{21–23} However, the analogy between Rydberg atoms and planetary systems turns out to be fruitful but not perfect since the finite size of Planck’s constant imposes an absolute scale on the atomic problem.^{24,25} The atomic analogs of these points are stable only over a limited range of parameters, and placing a finite-size minimum uncertainty wave packet at such an equilibrium point becomes a delicate balancing act.

The announcement of the feasibility of nonstationary, nondispersive wave packets in the CP problem was greeted with a flurry activity. For example, BKE showed that a curved wave packet^{21} suffers very little, if any, of the dispersion that plagued their original wave packet because it nestles inside the effective potential of the CP field. Following the early discovery of similar Floquet states anchored to stable islands in the classical phase space of the linearly polarized microwave problem,^{26–27} Zakrzewski, Delande, and Buchleitner ^{28,29} have shown that it is possible to find eigenstates of the problem in a rotating frame that, being eigenstates, are immune to spreading. In the laboratory frame such eigenstates indeed orbit the nucleus without spreading. These states are neither wave packets nor coherent states in the sense of Schrödinger and will not mimic the harmonic oscillator coherent states; i.e. they are not minimum uncertainty wave packets since locally the equilibria in the CP problem are not harmonic.^{30} Suggestions for the experimental preparation of these states can be found in the literature.^{29}

Our approach differs in using a combination of external electric and magnetic fields to produce a harmonic minimum directly in the effective potential.^{30–34} Associated with this minimum are the usual non-dispersive coherent states of the harmonic oscillator.^{11}

In the dipole approximation and atomic units (ħ = *m*_{e}
= *e* = 1) the energy for a hydrogen atom subjected to a circularly polarized microwave field and a magnetic field perpendicular to the plane of polarization is

where the terms are as follows; the kinetic energy, the Coulomb potential, the paramagnetic energy, the diamagnetic energy and the interaction with the radiation field. The magnetic field is taken to lie along the positive *z*–direction and, *ω*_{c}
= *eB*/*m*_{e}*c* is the cyclotron frequency, *ω*_{f}
is the microwave field frequency and *F* its strength. In a synodic frame rotating with the field frequency *ω*_{f}
the Hamiltonian becomes ^{30–34}

where *K* is the Jacobi constant^{18} and the coordinates are now interpreted as being in the rotating frame.

A key point is that this configuration of fields allows the coefficient of the paramagnetic term in Eq. (2) to be varied or eliminated. By mixing coordinates and momenta the paramagnetic term prevents the normal separation of the Hamiltonian into potential and (positive definite) kinetic parts: nevertheless, a potential energy may still be defined if *ω*_{f}
= *ω*_{c}
/2—thereby eliminating the paramagnetic term—and a typical section through the resulting surface is shown in Fig. 1 for experimental parameters that are consistent with those that are currently (if not routinely) achievable. Note particularly the existence of a saddle point and an outer *harmonic* minimum in the potential. The ground state (vacuum state) of the harmonic oscillator, being a coherent state, is our wave packet. In the laboratory frame the equilibrium at the minimum corresponds to a circular orbit in the plane and localization of the electron in this well produces a giant atomic dipole rotating at the microwave frequency in the *x* - *y* plane.^{5} For snapshots of its progress around its circular orbit see Fig.3 of Ref. 34.

In Fig. 2 contours of the probability density of the vacuum coherent state are superimposed on level curves of the potential. It is significant that if the particle is initially confined to the plane *z* = 0, with no component of velocity in the *z*–direction, then it is guaranteed to remain in that plane (below the zero-field ionization limit the motion in the *z*–direction is essentially harmonic around the minimum and uncouples form the planar motion). Thereby the system can be made to meet the criteria of a planetary atom—an electronic wave packet prepared in the well will move in a non-dispersive fashion along circular Keplerian orbits that lie in a plane perpendicular to the magnetic field axis.

Prior to examining the dynamics it is convenient to scale coordinates and momenta;

Dropping the primes and assuming planar motion yields the Hamiltonian

where 𝐾 = *K*/${\omega}_{c}^{2/3}$, Ω = *ω*_{f}
/*ω*_{c}
and *ϵ* = *F*/${\omega}_{c}^{4/3}$. This scaling shows that the dynamics depends only on the three parameters, 𝐾, Ω, and ϵ. Figure 3 is a Poincaré surface of section for a number of trajectories and Ω = 1/2. The set of elliptical, foliated Kolmogorov-Arnol’d-Moser (KAM) curves is a clear signature of stable (*i.e.* non-dispersive) harmonic motion in the well. Quantum mechanically, such a state is destabilized by two decay mechanism: tunneling to the core through the potential barrier in Fig. 1 and radiative decay. Semiclassical estimates of the lifetime due to these mechanisms show them to be extremely long, which is not surprising in view of the classical nature of these states. Rigurous quantal calculation for the spontaneous decay of Trojan wave packets^{35,36} are in harmony with semiclassical estimates.

In the case that Ω ≠ 1/2 it is not possible to define a potential energy surface— simply ignoring the paramagnetic term is incorrect and results in a gauge dependent potential. However, by constructing a zero-velocity surface,^{18,37}

one can show that for Ω < 1 it is *still* possible to produce an outer well at large distances from the nucleus.^{32–34} In this case the motion may be chaotic as shown in Fig. 4, but, provided tunneling is unimportant, the electron will be confined in the well by the curves of zero velocity for all values of *K* below the saddle point. Extensive numerical simulations and stability analysis^{33} show that in the three-dimensional system, below the zero-field ionization limit, the electron may be confined similarly. Remarkably, even in these cases, when the electronic motion is mostly chaotic, the regular part of the phase space is, for relevant experimental parameters, still large enough (measured in ħ) to support states (see Fig. 2 of Ref. 32). The stability analysis for such states appears in Ref. 33.

We reiterate that with a circularly polarized microwave field alone, a stable equilibrium point can also be created^{18} which corresponds to a *maximum* in the zero velocity surface.^{30} This point is similar to the equilibria *L*
_{4} and *L*
_{5} which are themselves *maxima* in the restricted three body problem associated, *e.g.*, with Jupiter’s Trojan asteroids.^{33} Like *L*
_{4} and *L*
_{5}, the atomic Lagrangian points are linearly stable over only a narrow range of parameters, although by fine tuning of parameters it may be possible to assemble non-dispersive wave packets localized around these maxima,^{20} possibly by adding a magnetic field to enhance the stability of the maximum.^{30–34} Experimental preparation of a coherent state in the well would proceed similarly but, by producing a *minimum*, we have avoided delicate stability issues that are associated with equilibrium points that are maxima in an effective potential.^{38}

The difference between our strategy and that at the maximum might be likened to the difference between trying to balance an egg on the back of a spoon (achievable, albeit with some difficulty, by vibrating the spoon) as compared to balancing it in the hollow of the spoon. The nondispersive nature of our wave packet is confirmed both by the wave packet propagation calculation in Ref. 34, as well as the extremely long lifetimes against spontaneous decay.^{35,36}

In conclusion we mention the similarity of this system to an integrable Hamiltonian model used by Schaffer and Burns^{8} to study the dynamics of a charged dust grain in an ethereal ring of a giant planet. They consider the planet’s gravitational field and assume a rotating magnetic dipole to model the planetary magnetosphere (effects due to planetary obliquity, drag, radiation pressure, dipole tilt, etc., are neglected). Inclusion of the effect of solar radiation pressure^{39} in this model gives a non-integrable Hamiltonian similar to Eq.(1) except that the magnetic field in Eq.(1) is homogeneous. Nevertheless, outer stable equilibrium points can also exist and in both systems stable motion is confined preferentially to a plane lying perpendicular to a privileged direction established by a rotation axis.

## Acknowledgments

Partial support of this work by the American Chemical Society (Petroleum Research Fund) and the US National Science Foundation is gratefully acknowledged. We also thank D. Vrinceanu for technical assistance.

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