28 August 1995

PHYSICS LETTERS A

ELSEVIER Physics Letters A 204 (1995) 359-372

Magnetic field stabilization of Rydberg, Gaussian wave packets in a circularly polarized microwave field

David Farrelly ‘, Ernestine Lee a, T. Uzer b ’ Depurtment of Chemistry and Biochemistry, Utah State University, Logan, UT 84322-0300, USA

h School c>f Physics, Georgia Institute of Technology, Atlanta, GA 30332-0430, USA

Received 15 March 1995; revised manuscript received 5 June 1995; accepted for publication 30 June 1995 Communicated by A.R. Bishop

Abstract

The dynamics of a hydrogen atom subjected simultaneously to a circularly polarized microwave field and a magnetic field perpendicular to the plane of polarization are investigated. Stability analysis of the flow and numerical simulations reveal that the magnetic field can have the effect of stabilizing Gaussian wavepackets prepared at a global equilibrium point - a maximum - against dispersion and spreading.

1. Introduction

Attempts to form nondispersive electronic wave packets in Rydberg atoms [ 11 (monograph [ I] provides an excellent overview of many experimental and theoretical aspects of the field) generally emphasize state

preparation, possibly with the use of external fields to manipulate the energy levels of the atom [ 2-41: ideally,

to assemble a wave packet without dispersion a quantum system with constant or almost constant energy level spacings should be used. Thus, in a static electric field the energy levels of the hydrogen atom are given by E ,,k z -I /2n2 + $Fnk, where n is the principal quantum number and k the parabolic quantum number. The

spacing between subsequent energy levels is AE = 3Fn, and, since the spacing is independent of k, then,

in principle, there will be minimal dispersion of a wave packet prepared as a superposition of states within a single n-manifold. For the harmonic oscillator itself the problem is obviously straightforward because the

energy level spacings are constant and, therefore, truly nondispersive coherent states are possible. In fact, these states have found numerous applications in studies of the radiation field, lasers, squeezed states and statistical physics [ 5-71 ’ . Naturally, the question arises as to whether it might be possible to use combinations of

static and time dependent fields to prepare locally harmonic regimes in atoms, thereby allowing the creation of

almost completely nondispersive coherent atomic states (CATS). There have, of course, been many attempts to prepare atomic analogs of coherent states (e.g., elliptic quantum states [ 81 and coherent states of the hydrogen

’ Ref. [ 61 use the Kustaanheimo-Stiefel transformation [ 71 to obtain coherent states for the hydrogen atom. Johnson points out that the fictitious time so introduced in evaluating the sum over classical paths in the path integral is none other than the eccentric anomaly of the corresponding Kepler orbit.

0375-9601/95/$09.50 8 1995 Elsevier Science B.V. All rights reserved SSDI 0375-9601(95)00505-6

360 D. Farrelly et al. /Physics Letters A 204 (1995) 359-372

atom Lb1 1, but this Letter is concerned with the cons~uction of atomic states that are actually identical to the coherent states of the harmonic oscillator - these are the CATS.

Similar states can, of course, be anticipated in the vibrational dynamics of molecules if the potential energy surface has one or more minima which are locally harmonic - this is the basis of the concept of a normal mode. In Rydberg atoms, however, the potential is essentially Coulombic, which prevents the direct preparation of genuinely coherent states. TO this end there have been some attempts to create “outer potentials” in Rydberg atoms in crossed electric and magnetic (E x B) fields, but in most cases these suggestions have been blighted by the existence of a velocity-dependent paramagnetic term in the Hamiltonian [9]. One approach has been to neglect the paramagnetic term altogether in the one-particle Hamiltonian in the symmetric gauge, This is clearly incorrect because the procedure is gauge-dependent and does not reproduce the E x B-drift of the electron (one exception is the one-dimensional model of an electron bound at a liquid helium surface for which there is a real outer potential in crossed electric and magnetic fields f93 in the direction ~~ndicul~ to the He surface). For highly excited atomic systems there exists another possibility considered by Gorkov and Dzyaloshinski [ IO] : The two-body atomic problem can be “pseudo-separated” and the Coulomb potential is centered far from the origin at which point there is, instead, an oscillator potential. The paramagnetic term can be neglected if the wave-function is concentrated in the oscillator potential and does not overlap the Coulomb potential from which it is displaced by a targe distance. This situation corresponds to two oppositely charged particles performing E x B-drift without mutual interaction. Thus, the potential model applies for “ionized” atoms in crossed fields for which it is natural that the relative motion takes place in an oscillatory potential, the extension of which is determined by the heavier particle. Incidentally, these considerations are apropos to two oppositely charged particles in a pure magnetic field when the particles each execute cyclotron motions, provided that the centers of the cyclotron motions are displaced from each other by more than the cyclotron radius of the heavier particle. This model fails, however, when the particles are strongly interacting, since in this case the paramagnetic term is no longer conserved, i.e., the model is only appropriate for ionized atoms in crossed fields. Finally it is possible to create a true outer potential in the positronium atom in crossed fields when the paramagnetic term vanishes.

Recently, interest has resurfaced in this problem and it has been suggested that it may be possible to create locally harmonic regimes in Rydberg atoms using time dependent fields [ 111, possibly in combination with magnetic fields [ 121: e.g., BiaIyni~ki-Birula et al. [ 1 I] have proposed that in a circularly polarized (CP) microwave field [ 13-181, by finetuning of the parameters, stable equilibrium points can be created that are analogous to gravitational equilibrium points in celestial mechanics. Wave packets launched from these points will be expected to orbit the nucleus without spreading. In reality, the equilibrium points reported in Ref. [ 11 I correspond to maxima in the effective potential. Stable maxima are not common in atomic systems, but a compelling example of this phenomenon in celestial mechanics is provided by Jupiter’s Trojan asteroids whose existence was originally predicted by Lagrange in 1772 1201. Like their celestial counte~~s (h and Ls) whose stability can be attributed to the interplay of Coriolis and gravitational forces, the atomic Lagrangian equilibrium points are stable over only a quite limited range of parameters. Recently we have shown that the addition of a magnetic field perpendicular to the plane of polarization of the CP field can be used to eliminate

the paramagnetic term and produce a minimum - an outer potential - in the system { 121. In this Letter we

demonstrate, using classical mechanics, that a magnetic (B) field in combination with a CP microwave field can be used to stabilize coherent states associated with the m~imum in the effective potential for the system

studied in Ref. [ II]. The Letter is organized as follows; in Section 2 the Hamiltonian for the diamagnetic hydrogen atom in

a CP field - denoted DCP - is introduced together with the concept of a zero-velocity surface, which is a method, adapted from celestial mechanics, to visualize the dynamics in problems that feature velocity dependent (Coriolis) forces. A stability analysis of the ~uilibrium points in the DCP problem is presented in Section 3, where it is demonstrated that the B field can stabilize the equilibrium points and extend the volume of phase space that is locally harmonic around the equilibrium point. Section 4 is given over to a series Of classical

D. Farrelly et al. /Physics Letters A 204 (1995) 3.59-372 361

X

Fig. 1. Isometric view of the ZVS in scaled units as defined in E?q (21) with F = 0.6, wC = 0, a+ = I. The saddle Ls and the maximum

L,,, are shown.

simulations; in particular, we propagate swarms of traj~tories with initial conditions chosen to simulate a CAT. Conclusions are in Section 5.

2. HamiItonian and zero-velocity surface

The Lagrangian for a hydrogen atom (in atomic units and assuming an infinite nuclear mass) subjected sjmultaneousiy to a CP microwave field (intensity F and frequency wf) and a static magnetic field perpendicular to the plane of polarization of the CP field is

362 D. Farrelty er nt. /Physics L.&em A 204 (199.7) 359372

where wc is the cyclotron frequency and the choice of sign is determined by the direction of the magnetic field. At this point we specialize to a planar mode1 since, by virtue of Cauchy’s uniqueness theorem, a particle starting out in the plane of polarization, with an initial velocity contained in that plane, will never leave the

plane [ 191. The time dependence in Eq. ( I ) may be eliminated by going to a frame that rotates at the constant angular velocity wr, which finally produces the Hamiltonian

N=K=#+p;) - l/r-- (wf’f~Oc)(Xp~-yp,)+Fx+$w~(x2+v2), (2)

where K is the Jacobi constant. In this Letter we specialize to the plus sign in Eq. (2), for which the effective

potential possesses both a saddle point and a m~imum. The presence of a velocity dependent paramagnetic

term in the Hamiltonian prevents the separation of H into a positive definite quadratic form in momenta and

a potential energy term. Nevertheless, an effective potential (in the sense of celestial mechanics, i.e., a zero-

velocity surface (ZVS) ) can be constructed and provides an excellent starting point for studying the dynamics [ 201. For the DCP setup the ZVS is

V=H- ;(i2+j2) =-l/r-f-Fx-~wf(o,+wf)(X*+y*). (3)

Fig. I is an isometric portrayal of the ZVS and clearly shows the existence of two global Lagrangian equilibrium

points - a saddle (L,) and a maximum (L,). It is the latter which provides the analogy with the Lagrange

equilibrium points LJ and Ls in the restricted three-body problem. Upon the addition of a magnetic field the

con~guration of ~uilibrium points is unchanged as compared to the pure CP case (i.e., o, = 0) provided

that the Larmor frequency and the helicity of the wave are chosen such that the plus sign emerges in Eq. (2) (for later use we define w = or + iw,). However, the sta~~l~~ of these points may be affected by the relative

magnitudes of F, wf and w,. We pause to remark that the analysis of Ref. [ 1 l] is based on the launching of wave packets centered on

L, in the pure CP limit. These packets will be good approximations to the ideal nondispersive CAT only if the size of the locally harmonic regime surrounding L, is large compared to the wavelength of the electron.

Of course, in the classical limit, a trajectory started precisely at L, will remain at that point, but dressing

such a periodic orbit (a circular orbit in the laboratory frame) with a Gaussian wave packet opens up the

possibility of spreading due to nonlinearities and chaos. We have argued elsewhere, and will do so again here,

that atoms whose diameters exceed N 1 cm will need to be prepared to insure against dispersion in the pure

CP limit. In the following sections we justify this assertion and, further, show how the magnetic field can lead

to stabilization of L,.

3. Stability analysis

Weierstrass [21] pointed out that stable motion is possible only at a potential minimum if the Hamiltonian

can be separated into a sum of a positive definite “kinetic” term depending quadratically on momenta and

a “potential” part depending exclusively on coordinates. While this is the most common situation in atomic physics, the problem in hand does not meet this criterion because of the presence of the paramagnetic term, and, therefore, the stability of equilibrium points must be calculated explicitly (e.g., by computing the eigenvalues of the infinitesimally symplectic mapping governing the flow). To this end we expand the ZVS around the equilibrium point L, to give the following HamiItonian function that describes librations around Lm,

The equilibrium points lie along the n-axis and are given as the solutions of the equation

FfI/x:-~f(~f+~c)x~=O, (5)

D. Farrelly et al. / Physics Letters A 204 f1995) 359-372

where x+ corresponds to the maximum and x- to the saddle point in the ZVS, and at L,,

363

n=$(+Z/x:). b= -&w;+ l/x:).

The Hamiltonian (4) is identical to the cranked anisotropic oscillator model that has been used extensively in nuclear physics to generate basis vectors for self-consistent calculations to model collective rotations [ 22-251.

More recently, this problem has also been addressed (based on the Bogoliubov-Tyablikov transformation [ 261) in molecular physics to simplify rotational-vibrational Hamiltonians [ 271.

Restricting ourselves to first order librations around L, we define the vector [28,29]

and the matrix

which allows Hamilton’s equations to be written in the form

&=A& (9)

The stability of an equilibrium is determined by the eigenvalues of the matrix A, i.e., the roots of the equation

det(AJ- A) =0, (10)

where Z is the unit matrix. Rather than solving this equation directly, it is more convenient to compute the

matrix product RAR-‘, where

A -1 1 0

(11)

0 100

and the eigenvalues are then determined as the roots of the polynomial equation

det(n-a) =O, (12)

where

B=RAR-‘=

A -1 tw 1 - A2 - aw2 A-A(-l+w)+Aw

l-w A -A- A(1 -w) - Aw 1 - A2 - bw2

1 0 -A 1+w 0 1 -1 -w -A

(13)

We then compute

det(hZ- !3) = A4 +(2+a+b) A2w2+(a-I)(b-1)~~. (14)

The motion is stable if the four eigenvalues are purely imaginary: Fig. 2 illustrates the stability regions for the motion as a function of u and b. Importantly, stable motion is possible even in the regime where the potential part of the Hamiltonian corresponds to a saddle point - the rotation stabilizes the dynamics, in analogy to the dynamics in the Paul and Penning traps [ 30-331. However, in the limit wC = 0 the stable region for the problem

364 D. Farrelly et al. / Physics Letters A 204 (1995) 359-372

4

b 3

B + 2b = 0 2

1

0

-1

a

in hand is extremely limited, being defined by the inters~tion of the line a + 2b = 0 [II ] with the stability region shaded in Fig. 2. The result is a very restricted set of values that a and b may take for stable dynamics,

Fig. 2. The regions of stability (shaded) in the parameter plane ((4, h).

i.e., the parameters must be selected so that b lies in the range $ < b < 1. It should he noted that tinear stability of an equilibrium point says nothing about the size of the region surrounding the equilibrium point for which the harmonic approximation contained in Eq. (4) is justified. Whether a wavepacket will be nondispersive or not depends upon the wavelength of the electron as comp~ed to the size of the harmonic regime. This can be estimated by comparing the vacuum state for Eq. (4) with the harmonic approximation to the ZVS. Because the Coriolis term is bilinear in coordinates and momenta it is easy to transform the Hamiltonian into a separable form for which the energy eigenvalues and eigenstates can be found analytically.

The four eigenvalues of A in the stable regime are purely imaginary of the form &i.Q+ and &iD_ where L& are positive, real numbers. After a rotation in phase space (described in detail in Refs. [22-271 - see also Ref. 1341)

x’=Ax-tBp,., y’=Ay+Bp,, p:=ps+Cy, pj, = py -t- Cx, (15)

with A - BC = 1 (to preserve the commutation relations between coordinates and momenta) H can be reduced to the following separable form (here we follow the notation of Ref. [ 24]),

(16)

D. Farrelly et ai. / Physics Lerters A 208 W9sf 359-372 365

X Fig. 3. level curves of the ZVS with h = 0.9562. Thick lines are contours (at 0.25, 05, 0.75 and 0.95) of the Gaussian probabiljty density

centered at L,, (in a.u.); (a) xu = 104: (b) XI) = 107. The axis scales in the x and v directions are equivalent. but, for cku-ity. the ranges of x and ,Y are different.

where

with wx = ,/&w and w?, = I.&O. The eigenvaiues are given by

E = signtm+~(~+ + $)fil&./ + sign(m_)(n_ + ~)filLL_j. (18)

If the ground state energy is defined as Em = U2, then the vacuum state can be expressed as follows in terms of the original variables,

366 D. Farrelly et al. / Physics Letters A 204 (19951359-372

Fig. 4. Level curves of the ZVS with with the addition of a magnetic field and WI = 1 x IO-“, wC = 5 x IO-“, F = 5 x IO-x a.~. Thick

lines arc contours (at 0.25, 0.5, 0.75 and 0.95) of the Gaussian p~bability density centered at L, = IO4 u.

P~)o(x,y) = Nexp ( -+x2 - i/3$ - iyxy). (19)

The parameters cr,& y are given by

a=R(l +s)/h, p = .n( 1 - s)/JL, y = ws/fi,

with

(n - b&o* s= q&p -&)*

(205

The stability of a Gaussian wavepacket launched at I+,, depends, in part, on the quality of a locally harmonic approximation to the ZVS at L,,,. If F = 0 the ZVS is flat (i.e., not harmonic at all) transverse to the field direction in the rotating frame, but becomes increasingly harmonic with increasing F. However, for W, = 0, a transition to instability [ 111 (Brown or Trojan bifurcation 1281) occurs at L, when E, = Fc/w4i3 M 0.1 I56 which limits the range of linear dynamics. Fig. 3a shows a Gaussian wavepacket defined as in Ref. [ 111 and assuming L, is located at ~0 = lo4 a.u. (the value suggested in Ref. [ 111) with b = 0.9562 and E = 0.0444 [ 1 11. It is apparent that much of the packet spills out of the harmonic regime. Simulations described in the next section reveal that a sea of classical chaos surrounds the tiny regular island centered on L, and, thus, any leakage of the packet into the chaotic zone will further enhance spreading. If x0 2 lo7 a.u. however, most of the packet can nest quite comfortably atop the harmonic part of the maximum - see Fig. 3b. Numerical arguments to justify this contention are supplied in the following section.

The addition of a magnetic field (w, # 0) changes the situation dramatic~Ily, since it is now possible to adjust the relative sizes of the coefficients a and b in order to enlarge the stable domain. In p~ticular, it allows one to increase F beyond F,, thereby increasing the size of the harmonic regime at L,. Fig. 4 should be compared with Fig. 3a and illustrates the enhancement in the size of the harmonic region that is possible upon the addition of a static magnetic field. In fact, it can be shown that the DCP Hamiltonian is actually integrable for particular values of tic and wf [ 351.

D. Farreliy et al. / Physics Letters A 204 (1995) 359-372 361

1 7

0.8

0.6

0.4

0.2

Y” -0.2

-0.4

-0.6

-0.8

-1 0.6 0.8 1 1.2 1.4 1.6

0.4

0.3

0.2

0.1

Y” -0.1

-0.2

-0.3

-0.4

X

I , I I , I I , ,

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5

X

0.4

0.3

0.2

c.1

YO

-0.1

-0.2

-0.3

-0.4 l- 0.95 1 1.05 1.1 1.15 1.2 1.23

0.4

0.3

0.2

0.1

Y” -0.1

-0.2

-0.3

-0.4

X Fig. 5. Poincati surfaces of section ( Pp = 0) in scaled units showing the Trojan bifurcation with wC = 0. In each case the energy is the energy of the maximum: (a) c = 0.0444 (b) l = 0. I. (b) (F = 0. I 156 (the Trojan bifurcation), (d) E = 0.1 170.

4. Classical dynamics and Gaussian swarms

In examining the dynamics it is convenient to scale coordinates and momenta; f = o*/“r, p’ = GJ-‘/~P. After

dropping the primes this yields the Hamiltonian

if = K = i<px” + P;) - l/r - (xp, - ypx) -I- $&x2 + y2) -i- EX,

where K = K/w213, w, = w,/w and E = F/w 4/3 This scaling shows that the classical dynamics depends only . on the three parameters, K, w,, and E. Our strategy is to integrate swarms of classical trajectories with initial conditions chosen to simulate a CAT. All the integrations are performed in scaled coordinates: however, the

368 D. Farreliy et al. / Physics Letters A 204 (1995) 359-372

0.8

06

0.4

0.2

YO -0.2

-0.4

-0.6

-0.8

'0.7 0.8 0.9 t 1.1 1.2 1.3 1.4 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

X X

Fig. 6. Swarm of initial conditions

'0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

X plotted on the ZVS with wC = 0 and .qk = IO4 at times after (a) 0. (b) (c) 5000 Kepler periods.

ensembles of initial conditions are generated in unscaled (atomic) units which are subsequently scaled. First we computed Poincare surfaces of section (SOS) in the vicinity of L, by integrating the equations of motion in cylindrical coordinates (x = p cos (ft, y = p sin 4), computing the Fp = 0 surface of section, and plotting the x - y phase plane to provide a visual impression of the spatial extent of regular and chaotic regions at the maximum. The classical scaling property means that all SOS are equivalent independent of the value used for x0 at fixed E. In the absence of a magnetic field and with E = 0.0444 the regular regime around L, in the SOS is quite small - see Fig. 5a - and the KAM curves are surrounded by a sea of chaotic or scattering trajectories

0.6

Y”

-0.4

-0.6

-0.8

D. Farrelly et al. / Physics Letters A 204 (1995) 359-372 369

'0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 .0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

X X

1

0.8

0.6

0.4

0.2

YO -0.2

-0.8

Fig. 7. Swarm of initial

0.7 0.6 0.9 1 1.1 1.2 1.3 1.4

X conditions plotted on the ZVS with m, = 0 and XQ = IO’ after (a) 0, (b) 2500, fcf 5tm Kepler periods.

(ionization is possible because at energies greater than the saddle point energy, the electron can escape over

the saddle). As E is increased further the KAM curves start to break up and the motion becomes increasingly chaotic as illustrated in the sequence Figs. 5a-5d.

Figs. 6 and 7 compare the time evolution of swarms of 10000 initia1 conditions chosen to simulate a coherent state with initial conditions corresponding to xc = IO4 and IO7 respectively in the limit w, = 0. Note that, in scaled units, the ZVS is identical for the two values of xc but the spatial extent of the two swarms is quite

different, reflecting the smaller eflective value of E when xc = 10’. It is clear that the classical wavepacket for the smaller value of xa spreads and disperses considerably and it is not until the radius of the orbit (in the

370 D. Farrelly et al. /Physics Letters A 204 (1995) 359-372

l-

Y O-

-1 -

-2 -

1 I I I I I I

0.5 1 1.5 2 2.5 3

X Fig. 8. Poincark surfaces of section ( Pp = 0) in scaled units and l = 0.940905,~~ = I .42857 (corresponding to Fig. 4). For these values

IO = 2.305 and E = 0.4338.

pure CP limit) is > I cm that the packet becomes convincingly nondispersive. Even for the smallest value of

x0 = 104, however, a considerable number of trajectories remain localized in the vicinity of the equilibrium

point. This suggests that diagonalization of the local potential at the equilibrium would be one possible way

to include the nonlinearities and so produce a packet with improved stability characteristics. Equivalently one

might attempt to find the corresponding Floquet state. Clearly, a higher order Taylor expansion at the maximum will lead to a state that is a better approximation to a stationary state (in the rotating frame) but this would

not qualify as a Glauber coherent state. Production of such a coherent state may be feasible upon the addition

of a magnetic field perpendicular to the plane of polarization (i.e. wc # 0) which leads to stabilization of the

wavepacket. Fig. 8 shows the SOS for this situation. It is apparent that the size of the regular regime at Lr,, has

been increased considerably as compared to the pure CP case. Fig. 9 shows the time evolution of the swarm in the presence of

nonstationary state.

5. Conclusions

the magnetic field and the packet is obviously extremely compact, i.e., a nondispersive,

We have investigated the dynamics of a hydrogen atom subjected simultaneously to a circularly polarized

microwave field and a magnetic field perpendicular to the plane of polarization. Stability analysis and classical

simulations reveal that the magnetic field can be used to stabilize wavepackets prepared at the global equilibrium point against dispersion and spreading. In the harmonic approximation these states are identical to the coherent states of the cranked harmonic oscillator and they, therefore, qualify as genuine CATS.

Acknowledgement

We thank Dr. Georg Raithel for illuminating conversations on the problem of the hydrogen atom in crossed electric and magnetic fields. Partial support of this work by the American Chemical Society (Petroleum Research Fund) is gratefully acknowledged.

0.5

YO

-0.5

-1

D. Farrelly et al. / Physics Letters A 204 (1995) 359-372 371

-0.5

-1

0.5

-0.5

-1

-1 6 1 1.5 2 2.5 3 3.5

X Fig. 9. Swarm of initial ~onditjons piotted on the ZVS with wc. = 5 x IO-” and the other parameters as in Fig. 4 and xo = IO’ at times

after after (a) 0, (b) 2500, (c) SO00 Kepler periods.

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