Quantum Energy Flow in Atomic Ions Moving in Magnetic Fields

VOLUME 84, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 28 FEBRUARY 2000

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Quantum Energy Flow in Atomic Ions Moving in Magnetic Fields

V. S. Melezhik1,* and P. Schmelcher2

1Physique Nucléaire Théorique et Physique Mathématique, CP 229, Université Libre de Bruxelles, B 1050 Brussels, Belgium2Theoretische Chemie, Physikalisch-Chemisches Institut, Universität Heidelberg, INF 229, D-69120 Heidelberg, Germany

(Received 12 March 1999)

Using a combination of semiclassical and recently developed wave packet propagation techniques wefind the quantum self-ionization process of highly excited ions moving in magnetic fields which has itsorigin in the energy transfer from the center of mass to the electronic motion. It obeys a time scaleorders of magnitude larger than the corresponding classical process. Importantly a quantum coherencephenomenon leading to the intermittent behavior of the ionization signal is found and analyzed. Universalproperties of the ionization process are established.

PACS numbers: 32.60.+ i

Rydberg atoms in strong external fields represent an ex-citing and very active research area both experimentallyas well as theoretically. During the 1980s the main focuswas on the hydrogen atom [1] assuming an infinite nu-clear mass which reduces the dimensionality of the system.However, in general, the atom possesses a nonvanishingcenter-of-mass (CM) motion in the magnetic field, givingrise to a variety of two-body phenomena [2]. Turning tocharged two-body systems like, for example, the He1 ionthe residual coupling of the CM and electronic motion isrepresented by an oscillating electric field term yieldingfive relevant degrees of freedom. One of the most strikingeffects caused by the two-body character of the ion is therecently found classical self-ionization process [3] whichoccurs due to energy transfer from the CM to the elec-tronic motion. Since it is well known that quantization canseverely change the effects observed in classical dynam-ics (see Ref. [4], and references therein) we develop in thepresent work a quantum approach to the moving ion in amagnetic field.

We are interested in the regime of high level density,i.e., high-lying excitations, for both the collective as wellas the electronic motion which depend on a number ofparameters (field strength, total energy consisting of theinitial CM, and internal energies). The ab initio descrip-tion of the quantum dynamics in the above regime goeseven beyond modern computational possibilities and wethus seek a semiclassical approach that is capable of de-scribing the essential physics of the problem. The totalpseudomomentum [5] is a conserved quantity associatedwith the CM motion. In spite of the fact that its compo-nents perpendicular to the magnetic field are not indepen-dent, i.e., do not commute, it can be used to find a suitabletransformation of the Hamiltonian to a particularly simpleand physically appealing form [6,7]. For the He1 ion itreads H � H1 1 H2 1 H3, where

H1 �1

2M

µP 2

Q2

B 3 R∂2

, (1)

H2 � aeM

∑B 3

µP 2

Q2

B 3 R∂∏

r , (2)

870 0031-9007�00�84(9)�1870(4)$15.00

H3 �1

2m

µp 2

e2

B 3 r 1Q2

m2

M2 B 3 r∂2

11

2M0

∑p 1

µe2

2Q

2MmM

�M 1 M0�∂B 3 r

∏2

22e2

r. (3)

Here, m, M0, and M are the electron, and the nuclear andtotal mass, respectively. a � �M0 1 2m��M and Q is thenet charge of the ion. B is the magnetic field vector whichis assumed to point along the z axis. �R, P� and �r, p�are the canonical pairs for the CM and internal motion, re-spectively. The CM motion parallel to the magnetic fieldseparates exactly and undergoes a free translational mo-tion.H1 and H3 depend exclusively on the CM and elec-

tronic degrees of freedom, respectively. H1 describesthe free motion of a CM pseudoparticle with charge Qand mass M. H3 stands for the electronic motion inthe presence of paramagnetic, diamagnetic, as well asCoulomb interactions which, in analogy to the hydrogenatom [1], exhibits a variety of classical and quantum prop-erties with changing parameters, i.e., energy and/or fieldstrength. H2 contains the coupling between the CM andelectronic motion of the ion and represents a Stark termwith a rapidly oscillating electric field 1�M�B 3 �P 2

Q�2B 3 R�� determined by the dynamics of the ion. Thiscoupling term is responsible for the effects and phenomenadiscussed in the present investigation.

The essential elements of our semiclassical approach arethe following. Since we consider the case of a rapidlymoving ion in a magnetic field a classical treatment ofthe CM motion coupled to the quantized electronic de-grees of freedom seems appropriate: the CM is propagatedwith an effective Hamiltonian containing the correspond-ing expectation values with respect to the electronic quan-tum states. The latter obey a time-dependent Schrödingerequation which involves the classical CM trajectory. Boththe electronic and CM motion have to be integrated si-multaneously. The key idea of this semiclassical approachgoes back to Refs. [8,9] where it has been applied to the

© 2000 The American Physical Society


dynamics of molecular processes. Our resulting time evo-lution equations read therefore as follows:

ddt

P�t� � 2≠

≠RHcl��R�t�, P�t��

ddt

R�t� �≠

≠PHcl��R�t�, P�t��

ih̄≠

≠tc�r, t� � Hq��R�t�, P�t�, r��c�r, t�

(4)

with the effective Hamiltonian

Hcl�R, P� � H1 1 �c�r, t�jH2 1 H3jc�r, t��

Hq��R�t�, P�t�, r, p�� H3�r, p� 1 H2��R�t�, P�t�, r�� .(5)

This scheme represents a balanced treatment of the cou-pled classical and quantum degrees of freedom of the ionand takes account of the energy flow among them. Itpossesses the important property of conserving the totalenergy which is particularly important for the correct de-scription of the energy transfer processes occurring in oursystem. Since the typical energies associated with the fastheavy CM degrees of freedom are many orders of mag-nitude larger than the corresponding elementary quantum(h̄QB�M) we expect the above scheme to yield reliableresults.

Our approach to the solution of the time-dependentSchrödinger equation, which yields the dynamics of an ini-tially defined wave packet c�r, t�, is based on a recentlydeveloped nonperturbative hybrid method [10–12]. It usesa global basis on a subspace grid for the angular variables(u, f) and a variable-step finite-difference approximationfor the radial variable. The angular grid is obtained fromthe nodes of the corresponding Gaussian quadrature withrespect to u and f, which is in the spirit of the discretevariable techniques yielding a diagonal representation forany local interaction [10]. As a consequence one remainswith the Schrödinger-type time-dependent radial equationscoupled only through nondiagonal matrix elements of thekinetic energy operator. This vector equation is propagatedusing a splitting-up method [13], which permits a simplediagonalization procedure for the remaining nondiagonalpart [11,12]. Our scheme is unconditionally stable, savesunitarity, and has the same order of accuracy as the conven-tional Crank-Nickolson algorithm, i.e., �O�Dt2�, whereDt is the time step size. In order to avoid reflections of thewave packet from the right edge of the radial grid we intro-duce absorbing boundary conditions. The extension of theradial grid is chosen (see below) such that it exceeds thecenter of the radial distribution of the initial wave packetby more than 1 order of magnitude. The typical frequen-cies associated with the motion of the Rydberg electronand the CM motion are different by several orders of mag-nitude (see Ref. [3]). To investigate the quantum energytransfer mechanisms requires therefore the integration ofthe above equations of motion for a typical time which is

a multiple of the time scale of the heavy particle (CM).This corresponds to many thousand cycles of the Rydbergelectron. Such a detailed investigation would have beenimpossible without the use of the above-described combi-nation of highly efficient techniques.

We assume that the He1 ion is accelerated up to somevalue ECM of the kinetic energy of the CM motion andits electron is being excited to some Rydberg state nlmin field-free space. Thereafter it enters the magneticfield. In the following we choose ECM � 100 a.u.,n � 25, l � m � 0, and a strong laboratory fieldB � 1024 a.u. �23.5 T �. The initial CM velocity isyCM � 0.1656 a.u. and oriented along the x axis. Weremark that taking the above values for nlm, B for the He1

ion with the assumption of an infinitely heavy nucleusthe corresponding classical phase space is dominated bychaotic trajectories. Figures 1(a) and 1(b) illustrate results

FIG. 1. (a) The intersection C�r, z � 0, t� �Rjcj2 df

along the r axis. (b) The intersection C�r � 0, z, t� along thez axis. (Atomic units are used.)

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for the propagation of the wave packet with increasingtime. More precisely we show the intersection of theintegrated quantity C�r, z, t� �

Rjcj2 df along the

cylindrical r axis for z � 0 (Fig. 1a) and its intersectionalong the z axis for r � 0 (Fig. 1b). Figure 1a demon-strates that the motion of the wave packet is confinedby the diamagnetic interaction with respect to the r

direction, i.e., the direction perpendicular to the magneticfield. For any propagation time its value drops by severalorders of magnitude at some outer value rc for the r

coordinate. As we shall see below (see also Fig. 3) thevariation of rc is accompanied by a corresponding changein the internal/CM energies thereby demonstrating theflow of energy between the CM and electronic degrees offreedom. Figure 1b demonstrates that there is for certaintime intervals (see below) almost no decay of the wavepacket for large distances of the z coordinate. Thereforewe encounter a significant flux of probability parallel tothe external field. Reaching the boundary of our radialgrid rm � 20 000 a.u. it is absorbed and considered to rep-resent the ionized state. Having established the existenceof an ionizing probability flux parallel to the magneticfield we immediately realize from Fig. 1b that this flux isby no means constant in time but varies strongly. To seethis more explicitly and also to gain an idea of the overalldecay of the wave packet we show in Fig. 2 the decay ofthe norm of the wave packet for a time which roughlycorresponds to one cyclotron period 2pM��QB� of thefree CM motion in the field. Figure 2 shows apart from anoverall monotonous decay of the norm, which is due to thequantum self-ionization process, an amazing new feature:the norm exhibits an alternating sequence of plateausand phases of strong decay. The widths of the plateausslightly increase with increasing time. This intermittentbehavior of the ionization signal from the moving ions is

FIG. 2. The norm of the electronic wave packet as a functionof time (in units of 108 atomic units).

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a pure quantum phenomenon, i.e., does not occur in thecorresponding classical ionization rates [3]. Furthermorethe calculation of the classical ionization rates for thesame parameters (field strength, energies) yields a typicalionization time which is by 2 orders of magnitude smallerthan the one obtained by the quantum calculation. Theionization process is therefore significantly slowed downthrough the quantization of the system which is in thespirit of the quantum localization processes shown to existin a variety of different physical systems (see Ref. [4],and references therein). The observed slowing down ofthe ionization process represents one important differenceof the classical and quantum behavior of the moving ionwhich occurs in spite of the fact that we are dealing witha highly excited system.

The obvious question of the origin of the intermittentoccurrence of the plateaus and ionization bursts arises now.At this point it is helpful to consider the behavior of the CMenergy as a function of time which is illustrated in Fig. 3.Starting with ECM � 100 a.u. at t � 0 we observe a fastdrop of it for short times yielding a minimum of ECM atapproximately t1 � 4 3 107 a.u. Thereafter it raises andreaches a maximum at approximately t2 � 1.2 3 108 a.u.after which it drops again; i.e., it shows an overall oscil-lating behavior. The “valleys” of ECM coincide with theplateaus of the norm decay whereas the regions with higherCM velocities correspond to phases of a strong normdecay of the wave packet. The increase of the widths ofthe plateaus in the norm decay (see Fig. 2) matches thecorresponding decrease of the frequency of the oscillationsof the CM energy. Since the total energy is conservedthis clearly shows that the ionization bursts correspondto phases of relatively low internal energy (although cer-tainly above the ionization threshold) whereas the phasesof higher internal energy go along with the plateaus of

FIG. 3. The CM energy as a function of time (in units of 108

atomic units).


the norm behavior, i.e., the localization of the electronicmotion. This provides the key for the understanding ofthe rich structure of the norm decay. The phase of highenergy for the electronic motion means that the magneticinteraction strongly dominates over the Coulomb inter-action. This makes the electronic motion approximatelyseparable with respect to the motion perpendicular andparallel to the magnetic field. As a consequence theenergy transfer process from the degrees of freedom per-pendicular to those parallel to the external field are veryweak; i.e., the ionization process is strongly suppressed.This corresponds to an almost integrable situation for theions dynamics. On the contrary for relatively low internalenergies the Coulomb interaction is much more relevantand mediates together with the coupling Hamiltonian H2the energy transfer from the CM to the electron motionparallel to the magnetic field. As a result we encountera flow of probability in the 1�2 z directions whichcorresponds to the ionization burst. During this periodof motion a comparatively strong dephasing of the wavepacket takes place.

The intermittent behavior of the ionization rate cantherefore be seen as a quantum manifestation for theswitching between different regimes of the internal energycorresponding to weaker or stronger Coulomb interaction.Pumping energy from the CM to the electronic motionweakens the Coulomb interaction and leads to the suppres-sion of the ionization process whereas pushing the energyback to the CM motion decreases the internal energy andenhances the Coulomb interaction. To elucidate the timescale on which this process takes place we have computedthe autocorrelation function C�t� � �c�t� jc�0�� where� � means integration over the electronic coordinates.As a result we observe a modulation and recurrence ofthe autocorrelation at a time scale t 1.6 3 108 a.u.which corresponds approximately to the recurrence ofthe plateaus for the norm decay. The correspondingpower spectrum shows a broad peak at a frequencyv 3.5 3 1028 a.u. An important feature of the quan-tum self-ionization process is the approximate stabilityof the time intervals corresponding to the plateaus of thenorm (no ionization signal) with respect to variationsof the initial CM velocity of the ion. Our investigationshows that decreasing the CM energy from 100 to 12 a.u.leads to a decrease with respect to the distances betweenthe plateaus, i.e., the difference of the norm valuesbelonging to different plateaus, roughly by a factor of2. This corresponds to a significant slowing down of theionization process. However, the widths of the plateausremain rather stable and represent therefore a universal

quantity which is approximately independent of the CMvelocity. Varying the field strength causes a change ofboth the distances between the plateaus and their widths.

The quantum self-ionization process should have impli-cations on the physics of atoms and plasmas occurring ina number of different circumstances. Apart from this itobviously suggests itself for a laboratory experiment (thelifetime of the Rydberg states exceeds the time scale ofionization by orders of magnitude) which should be veryattractive due to the expected intermittent ionization signalwhich is a process revealing the intrinsic structure and dy-namics of the system during its different phases of motion.

This work was supported by the National ScienceFoundation through a grant (P. S.) for the Institute forTheoretical Atomic and Molecular Physics at HarvardUniversity and Smithsonian Astrophysical Observatory.P. S. thanks H. D. Meyer and D. Leitner for fruitfuldiscussions. V. S. M. gratefully acknowledges the useof the computer resources of the IMEP of the AustrianAcademy of Sciences; he also thanks the PNTPM groupof the Université Libre de Bruxelles for warm hospitalityand support.

*Permanent address: Joint Institute for Nuclear Research,Dubna, Moscow Region 141980, Russian Federation.

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Quantum Energy Flow in Atomic Ions Moving in Magnetic Fields