Bound States of Guided Matter Waves : An Atom and a Charged Wire

We argue that it is possible to bind a neutral atom in stable orbits around a wire charged by a time-varying sinusoidal voltage. Both classical and quantum-mechanical theories for this system are discussed, and a unified approach to the Kapitza picture of effective potentials associated with high-frequency fields is presented. It appears that cavities and waveguides for neutral-atomic-matter waves may be fashioned from these considerations

course of our discussion we comment on the connections with the work of Cook, Shankland, and Wells [6], Com- bescure  [7], and Brown [g], aimed mainly at understand- ing ion motion in a Paul trap [9], which is governed by the Mathieu equation.
The static 1/r potential is known mostly to physicists for the role it plays in accounting for the effects of con- served angular momentum in the radial equation of motion for a particle moving in a central field.This effective potential is always repulsive and leads to re- duced binding for increasing angular momentum when combined with an attractive force that supports bound states.The attractive 1/r potential has been discussed classically [10] and quantum mechanically [11 -13] and has been of interest mostly for the peculiarities of the motions it engenders rather than for its importance in practical problems.In the following we point out that it is exact1y these peculiarities that must and can be over- come to enable the realization of new stable states of matter based on the interaction of polarizable atoms with a charged wire.First we review the basic features of the motion.
The attractive 1/r potential stands on the border be- tween highly singular and regular potentials for which the centrifugal barrier can keep a particle from passing through the origin.Only motions with angular momen- tum below a critical value can be bound and all of these do in fact pass through the origin.
Consider now a neutral atom and a line of uniform charge density as might be contained on an extremely thin wire.The atom is polarized by the electric field from the wire and is consequently attracted to the wire by the electric-field gradient.This results in a 1/r interaction potential when the induced dipole moment is linearly dependent on electric-field strength.
The fate of an atom bound in this way is an unavoid- able collision with the wire, which most likely results in absorption of the atom to the surface of the wire or in in- elastic reflection.Either of these would severely limit the lifetime of bound states where the atom moves in stable orbits around the wire.A main motivation of our studies is to show how long- lifetime, stable motions can be realized.It is one of our central themes that this can be accomplished by adding a high-frequency component to the static interaction po- tential.The strategy by which we demonstrate the efficacy of this procedure is remarkably similar to one that can be used for introducing the notion of a centrifu- gal barrier or effective potential in central force prob- lems.The latter involves a canonical transformation from Cartesian to polar coordinates, which has the favor- able consequence of making the angular coordinate ig- norable in the transformed Hamiltonian.
The radial motion is, however, influenced by an effective potential 45 6468 1992 The American Physical Society whose magnitude depends on the conserved angular momentum.Similarly, we find that under certain circumstances, a high-frequency potential-energy function can be canonically transformed away, leaving a time- independent effective potential in its place.The transformed Hamiltonian yields insight into the particle motion, and, in our case, it demonstrates how bound motions of the type we desire can be obtained.
A major achievement of this strategy is that it can be extended from the classical to the quantum regime, which is important for the systems of atomic waveguides and cavities we envision.In this way our approach represents an extension of the analysis presented by Ka- pitza [14]in his study of the inverted pendulum.BOUND STATES: CLASSiCAL DESCRIPTION Consider a neutral atom of mass M and polarizability a, situated a distance r from the center of a wire with static charge per unit length q and radius ro.A grounded cylinder of radius r, ))ro surrounds the wire to give the system a well-defined capacitance.The atom experiences a radial force of attraction towards the center of the wire, derivable from the potential-energy function 2M r 2Mr (2) with radial momentum P"=Mdr/dt and conserved angu- lar momentum L =Mr de/dt along the wire.Bound states have energy of transverse motion E & 0, which can occur only when L (4Maq .Integration of the corn- plete equations of motion gives A free translational motion along the wire is obtained, and we shall focus on the separate problem of transverse-motion confinement.
The Hamiltonian for the transverse motion of the system is given by An elementary discussion of the consequences on motion described by such a Hamiltonian is given by Landau and   Lifshitz [15] under the condition that to is suSciently high.They show that Newton's equations of motion may then be integrated approximately, yielding a solution for r that consists of a fast component W'costot/Mtv super- imposed on a slow motion that is governed by the static interaction potential and an effective potential, which we shall refer to as the Kapitza potential.Unfor- tunately, the aforementioned derivation gives neither a completely clear picture of what the expansion parameter and correction terms to the Kapitza description are, nor a clear path to the role this potential might play in quan- tum descriptions of the motion.We therefore divert tem- porarily from our problem of an atom and charged wire to fully develop the Kapitza picture in a classical descrip- tion.
with apogee b determined from v(b)+L /(2Mb2)=E, and in which X and Px are canonically conjugate variables.
To simplify the equations of motion, we want to extract the presumed small, high-frequency component from both the position and momentum by a canonical transfor- A plot of a characteristic trajectory spiraling into the ori- gin is indicated in Fig. 1.Clearly, maintaining atoms in this system is impossible.
Now imagine adding a time-varying term to the Hamil- tonian in Eq. (2) so that it takes the form W'( Y) Px =Pzsincot .CO (10) The displacements in position and momentum on the right-hand sides represent the motion of a free particle exposed to the time-varying potential alone.
The transformations in Eqs. ( 9) and ( 10) are not strictly canonical [16],and therefore the motions of Y and Pr are not governed by a Hamiltonian function.Equations ( 9) and ( 10) are, however, the lowest-order terms for a whole class of transformations that are canonical, where the ex- pansion parameter e( Y) is given by ( )

W(Y)
Mco which vanishes in the limit of high frequencies.From these we select one that, though not the simplest possible, corresponds to a particularly simple unitary transforma- tion in the quantum-mechanical description.The next section contains a discussion of the quantum-mechanical case.Remarkably, the detailed canonical transformation for the classical case can be obtained in closed form, but it has little transparent physical content beyond agreeing with Eqs. ( 9) and (10) to lowest order.We therefore relegate this canonical transformation and its generating function to Appendix A as Eqs.(A4), (A5), and (A2).
Our results are greatly simplified by restricting applica- Pz HK, = +V(Z)+V~, (Z), (13) and, after solving the equations of motion in this representation, the total motion in X space is found by the in- verse transformation: Z~Y~X.
We now apply and test the above formalism on the atom-wire problem.To obtain guidance for how small the expansion parameter e must be for the Kapitza pic- ture to hold, we compare with computer calculations where trajectories are obtained from the original Hamil- tonian (8).For the charge on the wire, we shall assume a purely sinusoidal time variation at frequency co/2 and amplitude Q, which gives 2 1coscor q = (14) Thereby, the atom-wire Hamiltonian assumes the form H. "=2M+z2 2M ~g +z6M. 2   neglecting the terms multiplied by e(Z) which must then be small compared with unity in a sense to be discussed below.
We anticipate that the Hamiltonian Hz generates some motions for which the expansion parameter is small for all their accessible regions of phase space.As a result, the correction terms may be neglected or treated as a perturbation.Under such conditions we identify the Hamil- tonian with the Kapitza Hamiltonian +E(Y) 'coscot M + V~, ( Y)( coscot -cos3cot) (12) which gives rise to a new conserved quantity EK, ~=HK, ~.The total potential in HK, ~has a minimum U;n at Z;n, and if we convert the coordinate Z and en- ergy EK, into reduced variables g and 5, we may write (16) as The time-independent part of this Hamiltonian contains the static Kapitza potential in addition to the original V( Y) interaction.The canonical transformation has had the desired eFect of reducing the time-varying part of the potential by e(Y) [note that VK, /W-e(Y)].In addi- tion, a Pr term of first order in e( Y) has appeared.This can be interpreted as a spatially dependent mass correc- tion.The amplitudes of the time-varying parts of the potential, though greatly reduced as compared to the origi- nal Hamiltonian (8), still have a spatial modulation of the Kapitza type.Since the ultimate goal is to consider the time-dependent terms as a perturbation, a second canoni- cal transformation is applied to introduce another reduc- tion of these time-varying terms.The generating func- tion and the transformation to the final coordinate Z, and momentum Pz are given in Appendix A.. The new Ham- iltonian Hz is also given there to first order in e(Z).All lowest-order terms of Hz are indeed time independent.
Of course, the utility of this Hamiltonian depends upon The potential-energy terms of ( 16) are plotted in Fig. 2. They consist of a 1/Z interaction (just the time- independent potential in the original Hamiltonian) and a repulsive 1/Z Kapitza term.Clearly, for appropriate initial conditions on the motion, bound states are suggest- ed, which are constrained to occupy regions away from wide range of initial conditions as long as e(Z) is less than a critical value at the inner turning point in the po- tential of Fig. 2. The motion predicted by the Kapitza   Hamiltonian is then a good approximation to the slowly varying part of the trajectory.We find that bound states of extremely long lifetime compared to that in Eq. ( 5) will be obtained when e=e(Z;")&0.28 at the distance of closest approach to the wire, Z;".This distance is obtained from Eq. ( 17) with Pz =0 and gives, when inserted in (11), the 5-dependent lower limit on L in (20a), r ()tm) 2MaQ [I -0.28f (5)j &L &2MaQ (20a) FIG. 2. Total effective potential in the Kapitza picture for a sodium atom, L =49k, ra=0. 1 pm, r& =1 cm, and an applied peak voltage of 8 V at co/2=2m.X(400 kHz).Shown also are the lowest quantized energy levels for this system.where the origin or wire in our problem.
Figures 3(a) and 3(b) show plots of a stable bound tra- jectory obtained from a numerical integration of the original dynamical equations with initial conditions chosen to correspond to a bound motion in the Kapitza picture.The separation into fast and slow motion, inherent in our canonical transformations, is clearly seen.However, as can be seen in Figs.3(c) and 3(d), even for initial condi- tions that suggest bound states of the Kapitza Hamiltoni- an, the actual motion may not be stable.The difference between these two cases is related to the value of the ex- pansion parameter e(Z).
We empirically find that stable motion is obtained for a 0.5-(20b) The upper limit (L,") for L in Eq. ( 20a) is simply determined from the requirement that the total radial effective potential energy be negative.Of course, the re- duced energy 5 must be negative for bound states, i.e. , -1 & 5 &0.Note the dependence of the left-hand side of (20a) on 5.In Fig. 4, L;" is given as the left-hand side of (20a) with 5= -1.
Figure 4 shows regions of EK, -L space where the Kapitza picture should be valid.The stability region is delimited by the obvious constraints that U;"(EK, ~(0, and by the upper and lower L values dis- cussed in connection with Eq. ( 20).The bound motion shown in Figs.FIG. 4. Regions of stable bound motion for sodium atoms, with sinusoidal voltage of 8 V peak, co/2=2m.X(400 kHz).The upper boundary connecting L;"with L", is obtained from the left-hand side of condition (20a).L,"comes from the right- hand side of (20a).Points A and B correspond to the stable and unstable trajectories in Fig. 3. WKB energy levels from Eq. (B7) are also indicated.A complementary and more global view of the system dynamics is provided by the Poincare plots shown in Fig. 5.The motion in two-dimensional radial phase space is sampled at the frequency of the ac drive.Keeping the amplitude and frequency of the sinusoidal voltage fixed, we varied the initial position and velocity for constant an- gular momentum to obtain trajectories for the range of energies EK, shown in Fig. 4 by the vertical line (at L =485) crossing the boundary of the stability region.
For the largest binding energies (corresponding to the smallest values of e), the slow motion is essentially governed by the Kapitza Hamiltonian, which gives rise to the concentric rings (cross sections of tori) in Fig. 5.As this energy is decreased, the lengthening period of the slow motion comes into resonance with an integral multi- ple of the period for the ac drive, and the nonlinearities of the problem stabilize that motion, forming the ring of nine islands in Fig. 5 (corresponding to a=0.23).This is a subtle indication of the imminent breakdown of the Ka- pitza picture.
Further decreases in the binding energy (and increases in e ) give successively more complicated overlapping res- onances (and lack of energy conservation expected of the Kapitza picture).We conjecture that the stable and bound motions in this phase space correspond to those in time-independent nonlinear problems where the Kolmogorov-Arnold-Moser theorem [17,18] has been ap- plied and has provided guidance on the existence and number of such orbits.
BOUND STATES: QUANTUM DESCRIPTION We begin immediately by considering the possibility of a Kapitza description under conditions where a quantum-mechanical description is called for.In particu- lar, we note that de Broglie wave lengths for the atoms can indeed be comparable with orbital dimensions at the low energies we are discussing.
Thus we return to our original time-dependent Hamil- tonian, The dynamical variables are now interpreted as operators in the Heisenberg picture, which offers the closest correspondence between classical and quantummechanical descriptions.In quantum mechanics, canonical transformations are represented by unitary transfor- mations, and accordingly we anticipate that a unitary transformation that accounts for the fast motion will sim- plify the description under high-frequency conditions and again lead us to the Kapitza Hamiltonian.The Hamiltonian, which generates the equations of motion for these new operators in the form where the function T is defined in (A3).The first factor is a gauge transformation, which accounts for the proper momentum displacement, and the last factor results in the translation in position.The transformation relations for coordinate and momentum operators are obtained from FIG. 5. Poincare plots of trajectories for L =48A' along the vertical solid line of Fig. 4 with energies from U;" to just beyond the line of stability.a, -8.9 X 10 ' eV, a=0.This result agrees with the classical one in Eq. (12) except for the mass-correction term, which now appears in sym- rnetrized form due to the noncommuting behavior of Pz and Y.Note again the appearance of the Kapitza poten- tial and the expansion parameter e(Y) in this operator equation.
As in the classical case, a second transformation, to coordinate operator Z and momentum Pz, is required to eliminate the time-dependent Kapitza-like terms.We give the corresponding unitary operator and the associat- ed Hamiltonian Hz in Appendix B. Since this second transformation does not alter the mass-correction term to first order in e(Z), the Hamiltonian differs from the clas- sical result only by the symmetrization of the rnass- correction term as was found in Hz.To lowest order, we obtain the Kapitza Hamiltonian operator Pz HK, = + V(Z)+ VK,p(Z) .

Bz (27)
Here 6Hz [see (B4)] consists of the higher-order terms in E'.We have replaced the coordinate and momentum operators Z and Pz by their representatives in the coordi- nate representation.
Due to the periodicity in time of b, Hz, we may choose solutions 0'(z, t) as Bloch functions.
The classical coordinate transformations (A4) and (A8) relate eigenvalues for the X, Y, and Z operators.Hence we obtain the wave function in the X representation from the solution %(z, t) in the Z representation by using the inverse of these transformations, 1/2 0'(x, t) =0'(z(y (x)),t) Bz By By Bx (28) The Jacobians originate from the normalization of the coordinate operator eigenstates.Since the canonical transformations are periodic in time, the Bloch-function character of the solution is preserved.
We anticipate that for small values of the "expansion Let the system be in state ~4 ), and consider the probabil- ity amplitude for the observable corresponding to the coordinate operator Z to have the value z at time t.We denote it by %(z, t) From . the equations of motion (24) for the operators, we find that the amplitude 0'(z, t) must Solutions of Eq. ( 27) may then be chosen as stationary states of this opera- tor, i.e. , O'E(z, t)= exp( iEx-, "tlat')Vs(z, O),  where %E is an eigenfunction for the Kapitza Hamiltoni- an corresponding to energy EK, .Transformation back to the X representation causes a translation and a rescal- ing of the argument of the wave functions, which make explicit the high-frequency effects hidden by our transfor- mations to the Kapitza picture.
Corrections to the Kapitza approximation are obtained through perturbation theory as long as the appropriate matrix elements involving the "expansion parameter" are small.Compared to the classical case, there is, of course, the problem that the "expansion parameter" is now an operator.We are ultimately interested in the stability of states, and, for our atom-wire problem, we shall give esti- mates for the lifetime of eigenstates of the Kapitza Ham- iltonian, based upon the golden rule.
In addition to the fact that the above quantum descrip- tion corresponds to the apparently successful classical description, note that for quadratic power potentials it yields the exact solution to the problem of the Paul trap for ions in the high-frequency limit.There our results reduce to those of Combescure [7] and Brown [8], who have taken advantage of the linear (Mathieu) equations of motion for the Paul trap to find exact solutions at all fre- quencies.Our case is nonlinear, and the methods we have developed yield useful results only at high frequen- cies.A transformation corresponding to a part of ( 22) was performed by Cook, Shankland, and Wells [6].This yields a translation in the momentum operator only and results in an incomplete description of the Kapitza pic- ture.
We shall consider the lifetime of the ground state in Fig. 2 that is deeply inside the quantum regime.The en- ergy quanta of the radio-frequency driving terms in the perturbation are much larger than the depth of the effective potential in the Kapitza picture.The perturba- tion thus couples the ground state to free states.With the atom-wire potentials inserted into Eq. (B4), where we include only the cos(cot) terms, we obtain from (37) l' --, '   iA y- which is a one-dimensional time-dependent Schrodinger equation corresponding to a Hamiltonian of the form (21) which are eigenstates for the L operator with eigenvalues IR.The radial equation may be written in terms of y(r, t) = R ( r, t) v'r as (The A2/4 term above originates in the two-dimensional nature of the problem. ) By letting the operator r =X-+ Y~Z, we obtain a sta- tionary equation in the quantum-mechanical Kapitza pic- ture, which in reduced variables [see Eq. (18} with L ~A (1 -, ') j is-~= J'dC x'x, 33 This approximation can actually be used in the range of ~5~(0.9,where the error introduced for large binding is only a few percent.The eigenvalues are shown graphical- ly in Figs. 2 and 4, where they are superimposed on the classical stability region.In Appendix B, results from the numerical integration are compared to WKB energies.
Having found energy eigenvalues and corresponding eigenstates for the Kapitza Hamiltonian of the atom-wire problem, we can now return to the question of stability of these states.A conservative estimate of the transition rate between states y; and yf is given by the golden rule, Although it is possible to integrate this equation by nu- merical means, we have found that in the phase-space re- gions of interest here (u;")10), the Wentzel-Kramers- Brillouin (WKB) approximation yields very accurate re- sults for the reduced bound state energies in an analytic form.In the limit of weak binding (5~0) they are given as Here 5f is the energy of the final states in reduced units, and Rl,"=(2MaQ )'~i s the maximum L value given by the right-hand side of (20a).
The result is a ground-state lifetime of 3 msec.This is much shorter than the lifetime predicted by the classical description, where we found stability over seconds.The reason for this difference is clear: the ground state tun- nels into classically forbidden regions where the expan- sion parameter is large.We can approach the classical regime by choosing larger values for the voltage, frequen- cy, and angular momentum.It is then possible to obtain stability of the states for seconds or more.For these cases, the energy of the radio-frequency quanta becomes much smaller than the depth of the potential, and, ac- cordingly, scattering between Kapitza states does not necessarily lead to loss of the atom.Although the quantum-mechanical lifetime found is much smaller than the classical result, we still see enormous enhancements when compared to the static case of Fig. 1.

DISCUSSION
The bulk of the preceding has dealt with establishing and evaluating the applicability of the Kapitza picture to the atom-wire and related problems.All of our classical and quantum-mechanical intuitions about the atomic states are deduced from this description.Even where the exact equations of motion were directly integrated to study stability, the interesting parameters and initial conditions were obtained from the Kapitza picture.Note that expansions or linearization of the equations of motion around exact solutions were not necessary.In ad- dition, the mathematical formulation of the Kapitza pic- ture for power-law potentials led naturally to an expansion parameter that serves as a useful guide to the phasespace regions where the approximation is useful.
Here we shall give a physical argument explaining why W" lMco is an appropriate parameter controlling the ap- plicability of the Kapitza picture to our problem.Clear- ly, the description breaks down when the "local" fre- quency for the slow motion becomes comparable to the high frequency of the time-dependent interaction, since then energy transfers to the slow motion can be large and unpredictable.The local frequency is proportional to the square root of the curvature of the total potential in the It should be emphasized that this requirement for the local frequency is a stronger constraint than the requirernent that the overall oscillation frequency for the slow motion be small compared to the driving frequency.This global oscillation frequency for the atom-wire problem is determined mainly by the long-range, attractive 1/r potential.In the case of the Paul trap, however, the global and local oscillation frequencies coincide.We see that for the Kapitza picture to be valid, the lo- cal frequency of the slow motion should be small com- pared to the drive frequency at any point in the oscilla- tion.For our problem we then conclude that although the global frequency becomes smaller for weaker binding, the inner turning point moves closer to the origin, where the local frequency is larger.The fact that instability occurs at smaller rather than larger binding energies in Fig. 4 is thus explained.
The theoretical discussion in the preceding sections clearly shows that enormous enhancements can be achieved in the time during which a low-energy neutral atom may occupy stable, bound orbits around a wire.
Realization of such a system will most likely rely on the combined technology of laser cooling atoms to the micro- and nano-electron-volt regions and the fabrication of structures like small wires of accurate dimensions on micro-and submicrometer length scales.The plots we presented above were all for what we believe to be realis- tic experimental conditions, none of which is beyond current capabilities.We expect the observation of these states to occur shortly, followed by more-detailed spec- troscopy and experimental probes of the atomic motions.
Since our atoms are uncharged, the coupling to exter- nal electromagnetic waves at frequencies comparable to the slow atomic motions is quite small.Indeed, the radia- tive lifetime of the bound states need hardly be con- sidered.The system should surely be probed spectros- copically through signals applied directly to the wire.
Although we have for simplicity discussed only the ap- plication of a sinusoidal voltage to the wire, we have also studied the case where an additional constant voltage is applied.The methods of this paper can be applied equal- ly well to this case, yielding similar bound-state motions.
The additional parameter adds the possibility of modify- ing these states adiabatically during an experiment.
It is quite interesting to consider the possible mecha- nisms that will limit the lifetime of the bound states.First we must admit that in spite of all our preceding theoretical and computer studies, we do not have a proof of the possibility of absolute binding for infinitely long periods of time for any classical or quantum-mechanical motions that we have discussed.It is a conjecture that the terms in the Hamiltonian beyond those of the Kapit- za type result in secular phase perturbations only.Our intuition that this is true in the classical limit is firmly based upon computer simulations following the bound system over millions of cycles.
Even if the classical trajectories could be shown to be absolutely stable, it is clear that this will not be true for the quantum-mechanical motion.Tunneling to classically forbidden regions, where the expansion parameter is large, can lead to transitions between Kapitza states and ultimate loss of binding.The effect of tunneling to these regions, or to the surface of the wire, can be probed ex- perimentally.
There is an additional quantummechanical effect that we have not discussed that may also play a role in the lifetime of stable states: in our quantum treatment the charge on the wire was taken to be a classical, time-dependent parameter.
This is, of course, an approximation.It must ultimately be con- sidered as a quantum-mechanical dynamical variable in its own right, and this can affect the stability of the atom- ic states.
We are intrigued by the fact that the atom-wire system may be investigated over a range of conditions spanning classical to quantum-mechanical regimes.The existence or interpretation of chaos in quantum systems is a topic of current interest, and we expect the experimental and theoretical study of the boundaries to motion governed by the Kapitza picture to contribute to our insight here.
However, in the final analysis, we believe that the ulti- mate utility of the new system we propose will depend directly on the lifetime of those orbits we imagine to be the most stable.Under favorable conditions, atomic con- stants and interactions between surfaces and atoms may be determined.Reference systems that depend on the values of such constants could be established.Transport of atoms in well-defined states over large distances or containment over long times would be achieved.The possibility of large numbers of atoms being contained simultaneously must also be considered.In this connec- tion it is extremely important to note that internal heat- ing from interparticle interactions will be strongly suppressed for neutral-particle confinement compared to that in ion traps.The distinction between Bose and Fermi particles must also be taken into account, and achieve- ment of a bound, condensed state in the former case is an exciting possibility.Confinement along the wire's length is then required as well, and we anticipate that nonunifor- mities in the wire or modification of external-field- forming surfaces wiB satisfy this requirement.
The general utility of the Kapitza picture in under-
mation to a new coordinate Y and momentum P~r elated toXand Px by The maximum time an orbit can have before crashing into the limit, taking r =X~Y ~Z, the motion is governed by a time-independentHamiltonian, FIG. 3. (a)Bound orbit with conditions as in Fig.1but with an applied sinusoidal voltage of 8 V peak and co/2=2m X(400 kHz).a=0.20, 5(0.(b) Stable time-dependent radial motion in (a) shown as a solid line.The dashed line shows Kapitza motion for Z. (c) Unstable orbit with @=0.32, 5 (0.(d) Unsta- ble time-dependent radial motion in (c) shown as a solid line.The dashed line shows Kapitza motion for Z.
The generating function in (A2), which led to the clas- sical is given in Appendix B. The classical rela- tions [Eqs. (A4) and (A5)] are regained apart from the symmetrization needed to insure that the momentum operator is Hermitian.
Since the angular momentum operator L involves the an- Fig.3(b), can presumably be accounted for by higher-order terms in the Hamiltonian [see (A10)].Point8 corresponds to the motion of Figs.3(c) and 3(d), with a=0.32, which is unstable and ultimately escapes to infinity.At the distance of closest approach to the wire, it is condition (19) that is violated, so the left-hand side of (20a) is not fulfilled for this trajectory.This leads to a breakdown of the Kapitza picture.For trajectories like 8 that reemerge into regions where e(Z) (0.28, motion will again be governed by the Kapitza Hamiltonian but with completely new values of 5 that are very sensitive to the original initial conditions of the problem.
Kapitza picture.The strictest requirement is obtained at the inner turning point, where the potential is approxi- mately the Kapitza potential only,