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Dynamics of high molecular Rydberg states in the presence of a weak dc field
29 April 1994
CHEMICAL PHYSICS LETTERS ELSEVIER
Chemical Physics Letters 22 1 ( 1994) 473-48 1
Dynamics of high molecular Rydberg states in the presence of a weak dc field Eran Rabani
a,L.Ya.
Baranov a, R.D. Levine a, U. Even b
a The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University, Jerusalem 91904, Israel b School of Chemistry, Sadder Faculty of Exact Sciences, Tel-Aviv University, Tel Aviv 69978, Israel
Received 25 January 1994; in final form 14 February 1994
Abstract It is argued that under the conditions of current experimental interest, one role of a weak electrical dc field on the dynamics of a high Rydberg molecular electron is to stretch the time axis. The stretch factor is essentially independent of the field and provides a classical analog of the quantum mechanical ‘dilution’ effect. A second role of the field is to lower the threshold for ionization thereby reducing the energetic requirements for auto-ionization. It is argued that the implications of the presence of the field are quite different for lower Rydberg states. Classical trajectory computations using action-angle variables are used to
validate the proposed interpretation and to probe the nature of the possible intramolecular decay channels.
1. Introduction
The issue addressed in this Letter, is the role of an external weak electrical dc field on the intramolecular dynamics of Rydberg states at energies just below ionization. It has been argued that such states manifest both short time decay [ l-3 1, which is possibly slower than that expected on the basis of extrapolation of observations on lower Rydberg states of the same molecule [ 3 ] and also exhibit even longer time stability [4-l 0 1. Our question is what determines the time scales for these processes and how does the magnitude of the dc field affect the conclusions. In this Letter, we use a model Hamiltonian where, in addition to the external electrical field, there are two internal perturbations to the purely Coulombic motion of the electron about the center of charge of the molecular ionic core. One is spherical and induces a precession of the elliptical orbit of the electron and the other, which is anisotropic, couples the electron’s motion to the rotation of the core. We show,
with support from classical trajectory computations,
that at the high principal quantum numbers of interest, the primary role of the weak external dc electrical field is to ‘stretch’ the time axis. The strong increase of the orbital period of the electron with its principal quantum number n is used to argue, again with support from the trajectory computations, that this stretching of the time is special to the region of high n as probed by recent experiments. The scaling with n of the boundary between the intermediate and higher n regions is shown to be n5 so that at intermediate and high n the physics can be different. The use of classical trajectories to probe the dynamics enables us to demonstrate that, for our model Hamiltonian, there are two competing decay channels. These correspond to an exit of the Rydberg electron from the detection window by a gain or by a loss of energy due to the coupling to the rotation of the core. Two such competing channels have previously [ 1,2] been proposed as responsible for the frequency dependence of the lifetime of the higher Rydberg
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states. The two channels were referred to as ‘up’ and ‘down’ processes, respectively, and we follow this terminology here. In the real molecule there can be additional couplings, e.g. to the vibrations of the core, and these can contribute to the up and the down channels. Furthermore, there can be other external perturbations besides that of a static dc field. There is considerable interest in the role of electrical fields, both intentional and so-called stray fields, on high atomic [ 1l-l 61 and molecular [ 17-271 Rydberg states. Both enhanced decay [ 17,18 ] and increased stability [ 19 ] are reported for molecular states showing that there is not a unidirectional effect. One clear role of the field is to lower the threshold for ionization #l. States of the electron orbiting around an excited ionic core, which in the absence of the field are bound, can ionize. This has been observed both for atoms other than hydrogen [ 13 ] and for molecules [ 7,271. In the latter case the loss of energy by the core can be detected by monitoring the rotational states of the ion. Another effect, discussed for both atoms [ 11,20,29] and molecules [ 17-201 is that known as ‘I mixing’. This refers to the angular momentum quantum number I of the electron not being conserved in the presence of the field. The Hamiltonian for a Coulomb potential with a dc field can, however, be exactly solved by using, so-called, parabolic coordinates [ 301. The eigenstates in these coordinates [ 291 are superpositions of the familiar stationary states of sharp angular momentum; hence the term ‘I mixing’. The optically accessed initial state has a fairly, or even very, well-defined 1value. In the discussion below we shall therefore find it more useful to think in terms of non-stationary or ‘coherent’ states [ 20,3 1 ] which have a rather localized range of I values. The potential seen by a Rydberg electron moving about an ionic molecular core is non-Coulombic [ 3235 1. Such a potential has a non-Coulombic ‘central part which can induce a precession of the BohrSommerfeld elliptical orbit [ 30 1. This will give rise to a ‘quantum defect’, as in non-hydrogenic atoms. As will be discussed below, the precession of the orbit x1 In principle, an electron in the presence of a field, however weak, can always tunnel out [ 281. However, at high n, the width of the barrier to ionization is so large that the tunneling rate is negligible [ 271. We therefore only consider ionization to be possible only at energies above the classical threshold.
can average out the effect of an external weak electrical field which is in a given direction. We shall conclude that this effect is too weak at the high n of interest but this will not be so at lower n. The other role of the non-Coulombic part of the potential is to couple the motion of the electron to the molecular degrees of freedom and thereby allow the exchange of energy and of angular momentum between the electron and the core [ 361. We have previously argued that this intramolecular transfer is responsible for both the up process in which the electron gains energy at the expense of the core and for the down process where the electron cascades down in its orbital energy. A computational study of such processes in the absence of the field [ 361 has explicitly demonstrated the expectation that the electron is only effectively coupled to the core when it is near to it. The nature of the Kepler-like orbit is that the electron rapidly transverses this region of close coupling. Over most of its orbit the electron sees only the long-range Coulomb potential #2.To examine the combined effects of the core and the external field on the motion of the Rydberg electron we present in section 2 the results of classical trajectory computations. These are for the motion of a high n electron around an anisotropic ionic core with a weak external dc field in the z direction. The two perturbations affect the electron in different ways. It is strongly coupled to the core only when it is close to it while the coupling to the field increases the further out the electron moves in the z direction #3.In the absence of the field, the electron is least perturbed near the aphelion of its orbit. For the present problem the electron is most stable when it is away from both the core and the z axis. The Hamiltonian used in the simulations introduces only the electrical anisotropy of the ionic core. The electron can thereby couple only to the rotation of the core. For large polyatomic molecules one can also provide an additional mechanism for long time stability namely the temporary storage of the electronic energy in the high density vibrational manifold of #*This
point is also made in multi-channel quantum defect (MQDT) theory [37-40). N3 Recall that for a Bohr-Sommerfeld orbit the core is at the focal point of an ellipse. The aphelion and perihelion are at nZ( 1+ e) atomic units respectively, where t= (1-1*/n*)“* is the eccentricity. When n is high and I-Cn the perihelion is at t/(1+ 1) atomic units and its velocity there is 2/ (I+ 1) in atomic units.
E. Rabani et al. /Chemical Physics Letters 221(1994) 473-481
states [ 41-43 1. An important issue, not addressed in this study, is whether the very high n electrons which are accessed by the initial optical excitation can couple directly to the vibrations #4or whether the doorway to the vibrations is a reduction in n due to rotational coupling to the anisotropic core. The discussion leading to the concept of the ‘stretching’ of the time scale of the intramolecular processes due to the presence of the stray field is presented in section 3. Specifically, what we show is that the frequency of close encounters of the electron with the core is de facto reduced by the factor Jdln, where d is the effective range for the electron-core anisotropic interaction in atomic units. The result that this factor is independent of the magnitude of the field and, in particular, of the magnitude and conservation of the projection quantum number ml, is subject to limitations which are spelled out in section 3. However, already at this point one needs to emphasize that the result is only valid in the range of (stray or intentional) weak dc fields as currently used in ZEKE-type [ 45 ] experiments (say from 0.1 to 1 V/ cm). Quite different conclusions would obtain at lower n values or at weaker fields. (At stronger external fields, the states of high n of interest will ionize. ) It is also argued in section 3 that the stretching effect is a classical analog of the so-called ‘dilution’ effect of radiationless transitions theory [ 45-48 1. This effect is derived here from a kinetic point of view but we also comment on it from a stationary point of view so as to emphasize the role of the initial preparation [ 45,481. We further derive results for more extensive dilution made possible when the orientation quantum number ml is not conserved. The non-conservation of ml is important to us also as a mechanism for keeping the electron away from the core, recall that I> mb and thereby confers an additional stability on the orbit. The extreme dilution limit that we obtain is when the precession, which results in the rotation of the axes of the elliptical orbit, is faster than the orbital angular motion. Finally, we note that extrapolating the width of lower Rydberg states to high values of n results in rates which are more than two orders of magnitude faster than the directly observed M That
states of low n can couple to the vibrations is evident in the phenomena of vibrational auto-ionization. See refs. [ 32-
35,441.
475
fast decay [ 3 1. It remains to be seen if the stretching of the time scale noted above is an important consideration for the complete explanation of this observed lengthening or whether coupling to the vibrational modes of the polyatomic core and/or the role of other external perturbations is also important. In section 4 we examine the kinetics of the up and the down processes. Specifically it will be argued that the results are consistent with these two processes representing competing decay channels of a common population.
2. Trajectory computations The potential field for the electron (discussed in more detail in ref. [ 361) has three parts. A Coulombit 1/r part with its origin at the electric center of the core, a term p( 1 +I.?) /r2 representing spherical and dipolar anisotropic deviations from the Coulomb potential and a coupling F*r to an external dc field of magnitude Fin the direction of the z axis. The Hamiltonian of the core is that of a rigid rotor of rotational constant B. The results shown below are for Bc0.15 cm-’ but higher values were also used. As far as we can ascertain the only ‘special’ values of B are those that allow for the possibility of matching between the unperturbed orbital period of the eleo tron (r= 1.52~ lo-l6 n3 S) and the period (2Bj)-' of rotation of the core when its rotational angular momentum is j [ 361. This is the so-called ‘stroboscopic effect’ [ 49 ] and does not occur for the high values of n of interest where the orbital motion of the electron is the slowest in the system. We have also verified this by examining the power spectra of the two motions [ 36 1. Trajectories were computed in action-angle variables [ 301 for the three-dimensional motion of both the electron and the core. Ensembles of initial conditions were generated by randomly sampling over the angle variables at fixed values of the action variables. The dynamics are classical but to compare with experiment we measure the value of the action variables in units of Plan&s constant. This quasi-classical correspondence cannot replace an MQDT parametrization or, better yet, a full ab initio approach [ 501. On the other hand, since n - 1 measures the number of nodes in the radial wavefunction it follows that in
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our limit there are many de Broglie wavelengths per orbit so that the classical limit is realistic. If anything, the use of classical dynamics is probably more a limitation for the rotational motion of the core whose states are more widely spaced than those of the Rydberg electron. In the absence of either the non-Coulombic part of the potential due to the core or the external field, the action variables are constant in time while the angle variables change at a constant frequency. This is unlike the ordinary coordinates and momenta of the electron which, over a typical orbital at a high n, change by many orders of magnitude. In the present study, trajectories were integrated up to 4 ILSor until the orbital motion excited from the ‘detection window’ (most trajectories satisfied the latter criterion). The ‘detection window’ is defined with reference to the experimental method of detection which is to apply a delayed electric field which ionizes Rydberg states down to below the initial value of n. We take the lower limit of the detection window to be at n = 90 which corresponds to a delayed field of about #’ 10 V/cm. The electron is deemed to have ionized and therefore to exit from the detection window when its n exceeds #6 1000. Fig. 1 shows the time evolution of the principal action n and of the orbital angular momentum 1 of the electron versus time for two trajectories which exit from the detection window in about 300 ns. One of the exits is up and the other down. There are two notable observations: firstly, the almost periodic variation in I between a minimal value of about zero #’ and a maximal value which depends on ml but can be as high as n - 1. The other feature of the trajectory results is that n changes only when 1 is in its region of low values, I< &. a5 The results shown below for the high Rydberg state are for an initial n of 150. We did, however, check that our qualitative conclusions are unaffected by increasing the delayed field by a factor of two. As will become obvious, in the absence of coupling to the vibrational modes, increasing the magnitude of the delayed ionizing field will slow the down process. M In practice, once n exceeds the threshold for ionization which is at n of about 300 for a field of 0.1 V/cm, n rises so rapidly that the exact upper cutoff is immaterial. ” The value of I has to exceed the value of M, which in the absence of the anisotropy of the core is conserved. This observation is significant in view of the discussion of stretching of time in section 3 below.
time (us) Fig. 1. Temporal evolution of typical single trajectories, one exiting by a ‘up’ route (upper panel) and the other by a ‘down’ process for a field of 0.1 V/cm. Shown are n (-) and I (- - -), the principal action and the orbital angular momentum of the Rydberg electron in units of Planck’s constant versus time in us. Computed for a core with a rotational constant of 0.15 cm-’ and an initial rotational angular momentum j= 8.5. The anisotropy of the core is dipolar with a moment of 0.3 au. The initial n is 150. Note how changes in n are fully coordinated with a low value of I and the periodic evolution of I. (The period w= 3nF of the 1 motion is modulated by the change in n with higher n resulting in more frequent passages close to the ionic core.) Note how prompt is the escape of the electron once its n increases enough so that its energy is above the threshold for ionization. It is the lowering of this threshold by the field that accounts for the up process being more probable at higher fields. It is, however, the core that provided the energy needed to increase the value of n.
Direct examination of the trajectory results verifies what is intuitively expected, namely, that n changes only when the electron can get close to the core. Since the distance of closest approach is nZ( 1 -dm), n can change only when lcn. This is unlike the field-free case [ 36 ] where, for low values of i, n can change once per revolution. Not every close encounter with the core necessarily results in a change in the value of n. However, since changes in n occur only during the brief sojourn near the core, the rate of both up and down processes must be proportional to the frequency of passage through the perihelion which varies as n -3. At higher 1 values, the electron does not enter the region of close coupling to the core. We defer further considerations of this point to section 3 and other computational results to section 4.
E. Rabani et al. /Chemical PhysicsLetters221(1994) 473-481
3. The time stretching
In the absence of an external dc field the orbital angular momentum 1 of the electron is conserved apart from brief passages near the perihelion when I and j are coupled by the electrical anisotropy of the potential of the core. This anisotropy has a finite effective range, say d, which is quite short compared to the scale of the Bohr-Sommerfeld orbit at high n, d << n 2. Therefore, over most of the orbit the value of 1 is conserved. Computational illustrations of this conservation can be found in ref. [ 361. On the other hand, the coupling to an external field will be manifested over the entire orbit. In a weak electrical dc field, 1will undergo a periodic, harmonic motion (a so-called ‘first-order Stark effect’ [ 301). In the semiclassical limit the frequency w of the motion of I equals the Stark splitting w = 3nF, where F is the field inatomicunits(1auoftield=5.14x109V/cm).For the typical stray field of 0.1 V/cm and n = 150 the orbital angular momentum completes one revolution in about 17 ns. In the same time the electron completes about 34 revolutions around the core. Elsewhere, we shall discuss the modifications of these conclusions made necessary by the second-order Stark effect. One such modification is that the motion of I is anharmonic. The amplitude of the 1motion does depend on the value of ml but for the purpose of the discussion we take it to be n so that the value of 1 as a function of time is n sin( wt+ $), where the phase is determined by the initial value of 1. It is only low values of 1, say 1~ l,, that allow the electron to closely approach the core. Since the distance of closest approach at low 1is 4l( l+ 1) we can approximate loby Jd, where d is the effective range of the anisotropy of the core. d can be estimated from the results for the field-free case or from analytical results for the matrix elements of the anisotropy [ 5 11. The interval in time during which 1 is below 1, is, provided 1, c n, l,/ wn = Jdl 3n2F. The frequency of returns of the electron to the core is n -’ in atomic units, hence the number of close encounters of the electron and the core is Jd/3n5F per oscillation of 1 or wJd/3n5F=Jdln4 per unit time. Since the electron executes ne3 passes through its perihelion per unit time, the frequency of close encounters of the electron with the core is reduced, compared to the field-free case, by &/n=l,Jn.
417
That the periodic motion of 1scales as expected has been verified in our trajectory computations. What can be clearly seen in Fig. 1 is how the period becomes longer if n has undergone a large change due to a close coupling to the core. It should be emphasized that the Hamiltonian used does include a term which induces a quantum defect, namely a precession of the axis of the ellipse. Below we argue that for the high n of interest the quantum defect cannot prevent the oscillation in 1but that this will no longer be the case at lower n values, cf. Figs. 2 and 3 below. The essential point is that at the high n of interest the value of 1is oscillatory and that it periodically comes down, and thereby allows the electron to couple to the core.
8 2
0
6
-0.5
4 f,
21
1,
5
0
f,
\
10 15 time (ns)
1 20-l
Fig. 2. The time-evolution of I (-) and of the angle cq (which specifies the orientation of the major axis of the elliptical orbit) in the intermediate regime, n = 70 and all other conditions as in Fig. 1. Note how the precession prevents the full oscillation in 1. At lower n values, as shown in Fig. 3,Z is hardly changing.
0
2
4
6
8
10
time (ns) Fig. 3. The time-dependence of I and n, n = 50. The value of I is hardly changing. Shown for comparison (- - -) is the timedependence of 1 when the anisotropy of the core is turned off. The I mixing is present but slow as compared to the orbital period (0+&j< 1).
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Elsewhere, we shall argue that quanta1 interference effects do not change this conclusion. The result that a weak dc field stretches the time axis by n/lo has been derived within a number of assumptions. While it is numerically a stretch since da n 2 or I, < I < n, it is physically a stretch because it is only valid for such weak fields and high n values that the period n3 of the orbital motion is far shorter than the period 0-l of the 2 motion. This implies 3n4F< 1 (equality occurs at F= 3.4 V/cm for n = 150). The classical onset of ionization is at about n4F=0.13 [28] and is saturated by n4F=0.383 so that fields at which the two periods are equal with already lead to prompt ionization of the electron. The upper end of the scale is thus not a limitation. There is, however, a lower limit of validity. This is for such weak fields that the precession of the orbit, due to the non-Coulombic part of the potential, is faster than the change in I due to the external field. The frequency of the precession is 6/n3, where delta is the quantum defect (modulus one). The time interval during which 1x1, is 3n2F/lo. The lower limit of I mixing is thus at 3n5F<&,. For our Hamiltonian 6~0.3 for s states (ref. [ 301, section 26) and decreases with 1. It follows that the field at the lower limit for time stretching is l/n as large as at the upper end. At the high n of interest this leads to fields which are comparable to the Lorentz field [ 521 due to the motion in the earth’s magnetic field. On the other hand, for a given field F, the lower limit places a steep bound on the values of n which are affected by stray fields. For a stray field of 0.1 V/cm and an s state excitation, the minimal value of n for any stretch effect is n= 111 and is significantly lower for higher values of 1.Another important limitation of our derivation is that the value of mI was assumed to be constant and quite low compared to lo. When this is not the case, the scaling of the frequency of close encounters can be shown to scale more rapidly with n and can reach n -5 when ml is fully not conserved. Elsewhere, we shall discuss the mechanism for reaching higher values of ml. Fig. 2 shows the competition between the precession and the field-induced oscillation of 1in the intermediate region where the two time scales are comparable. This corresponds to n of about 70 for our Hamiltonian (this result depends on the initial value of I). At lower values of n, the precession precludes
the changing of I due to the field as shown for n = 50 in Fig. 3. An extreme limit is when the frequency of precession is so fast that it greatly exceeds the orbital frequency. In such a limit the electron does not succeed in getting away from the core #8. In order to make an analogy between our present considerations and the quantum mechanical concept of a ‘dilution effect’ [45,48] it is necessary to also discuss the role of the initial preparation. To do so consider, for the moment, the simpler problem of an electron in the presence of both a Coulomb and an external field. This problem admits a separation using parabolic action-angle variables R9[ 301. Each one of our trajectories has, initially, a sharp value for these action variables. The oscillation in 1is precisely a result of the constancy of the parabolic action variables. This oscillation leads, as discussed above, to the time stretching of every trajectory. Now consider an ensemble of initial conditions where most members of the ensemble have similar initial values for the parabolic action variables. The ensemble we just discussed is stationary and is a classical analog of an initial preparation of one or a few eigenstates in the terminology of radiationless transitions theory [ 4548 1. In this ensemble the probability of low I values is low. The dilution effect is thereby manifested in a stationary context (see also ref. [ 241). On the other hand, one can, in principle, conceive of a different initial ensemble with a sharp value of I and, therefore, a distribution in the parabolic action variables. In this second ensemble initially all the trajectories allow strong coupling of the electron to the core. The energy of the different trajectories in this second ensemble spans the entire Stark manifold which, to firstorder, has a width of 3n2F. For the high n values of interest, this width (0.29 cm-’ for n = 150 and F= 0.1 V/cm) is large enough that excitation of an entire Stark manifold requires special conditions. #* Every attempt to depart towards the aphelion is foiled by the faster rotation of the direction vector pointing along the major axis. Elsewhere, we shall derive this and other limits from an analysis of the equations of motion of I and of the direction, socalled, Runge-Lenz, vector. a~ We have computed the dynamics in terms of these variables and will present the results elsewhere.
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479 F = 0.5 V/cm
4. The decay kinetics
1
400
This section briefly reports on the time-evolution for an ensemble of trajectories where the lowering of the threshold for ionization by the field is clearly evident. More detailed results will be presented in a full report which is in preparation for publication. Fig. 4 shows decay curves at several values of the field for an initial n of 150. The rate of short time decay increases strongly with increasing field. Not evident in this plot where the time is in the microsecond scale that within the first few orbital periods, and before 1 had the time to start increasing, a few trajectories exit from the detection window. Also not clearly evident are the few long-living trajectories. Elsewhere, we shall argue in detail that the long-living trajectories are those where the electron reaches a region in phase space where it is comparatively resilient to both the effect of the field and of the core. However, all trajectories which have enough energy to ionize eventually do so. One can classify the trajectories according to whether they escape up or down. Fig. 5 shows the separate decay curves for the two classes of trajectories. The separate fits to an exponential decay yields essentially identical rate constants as shown in the inset. This is the expected result if there is a common population which can competitively decay into one of two alternative channels. The rate constants for up and down decay determine the branching fraction into
300 2 Z
-I
200
100 J
0
0
0.5
1 time (ps)
1.5
2
Fig. 5. The number of trajectories which remain in the detection window up to the time t, classified according to their mode of exit for a field of 0.5 V/cm. As in Fig. 4, the fit shown is to exponential decay and the resulting decay rate (given in the inset) is essentially identical for both channels. The branching ratio is 2.8 1 versus 2.68 and 2.1 at F=0.3 and 0.1 V/cm, respectively. The increase of the branching ratio for the up process as the ionization threshold is lowered is analogous to the role of the energy gap to ionization discussed in ref. [ 11.
either channel and the overall decay rate is the sum of the two. In this fashion, one can provide a separate determination of the two rate constants and examine their dependence on both the field and on the initial value of n. Results of this type have already been reported for the field-free case [ 361 and will be provided in the full account of this work.
5. Concluding remarks
0 0.5
1 1.5
2 2.5
3
3.5
4
time @s) Fig. 4. Decay of an ensemble of high Rydberg states versus time in ps. Shown is the fraction of trajectories which remain in the experimental detection window for an integration time of 4 ps for three different values of the field in V/cm. Other conditions as in Fig. 1. The fit is to an exponential decay (in ps-‘) plus a constant (as our integration time does not suffice to establish the decay constant of the very few long-living states).
The typical stray or intended dc fields present during the time-evolution of high Rydberg states are such that their role can be summed up as inducing a stretch of the time axis. The boundary region between the high Rydberg states and the lower n states for which this field effect is averaged out, due to the precession related to the quantum defect, is rather narrow (and occurs at about n = 70 for our Hamiltonian which has a quantum defect of 0.3 for s states). A second important field effect is the lowering of the energetic threshold for ionization. In order to decay by an up process the total energy of the state need no longer be positive but only to exceed the classical onset for ionization.
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Acknowledgement
We thank Professor R. Bersohn, Professor J. Jortner and Professor E.W. Schlag, and Dr. K. Muller-Dethlefs and Dr. H. Selzle for discussions and Professor J. Jortner for his comments on the manuscript. ER is a Clore foundation scholar. This work was supported by the Volkswagenwerk Stiftung and by the German-Israel foundation for research ( GIF) . The experimental work at Tel-Aviv University is supported by the German-Israel James Franck binational program. The Fritz Haber research center is supported by MINERVA.
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[16]H.RotkeandK.H.Welge,Phys.Rev.A33 (1986) 301. [ 171 J. Chevaleyre, C. Bordas, M. Broyer and P. Labastie, Phys. Rev. Letters 57 (1986) 3027. [ 18 ] C. Bordas, M. Broyer, J. Chevaleyre and P. Labastie, J. Phys. (Paris) 48 (1987) 647. [ 191 C. Bordas, P.F. Brevet, M. Broyer, J. Chevaleyre, P. Labastie and J.P. Petto, Phys. Rev. Letters 60 ( 1988) 9 17. [20] L.D. Noordam, A. ten Wolde, H.G. Muller, A. Lagendijk and H.B. van Linden van den Heuvell, J. Phys. B 21 ( 1988) L533; A. ten Wolde, L.D. Noordam, A. Lagendijk and H.B. van Linden van den Heuvell, Phys. Rev. A 40 (1989) 485; L.D. Noordam, A. ten Wolde, A. Lagendijk and H.B. van Linden van den Heuvell, Phys. Rev. A 40 ( 1989) 6999. [21] J.L. Dehmer, P.M. Dehmer, S.T. Pratt, F.S. Tomkins and M.A. G’Halloran, J. Chem. Phys. 90 (1989) 6243. [22] S.T. Pratt, J. Chem. Phys. 98 (1993) 9241. [23] F. Merkt, H.H. Fielding and T.P. Softley, Chem. Phys. Letters 202 (1993) 153. [24] W.A. Chupka, J. Chem. Phys. 98 (1993) 4520. [25] H. Park and R.N. Zare, J. Chem. Phys. 99 (1993) 6537. [26] G.I. Nemeth, H.L. Selzle and E.W. S&lag, Chem. Phys. Letters 214 (1993) 141. [27] G.P. Bryant, Y. Jiang, M. Martin and E.R. Grant, J. Phys. Chem. 96 (1992) 6875. [ 281 L.Ya. Baranov, R. Kris, R.D. Levine and U. Even, J. Chem. Phys. 100 (1994) 186. [29] D. Park, Z. Physik 159 (1960) 155. [ 301 M. Born, Mechanics of the atom (Blackie, London, 195 1) . [ 3 1 ] M. Nauenberg, Phys. Rev. A 40 ( 1989) 1133; J.-C. Gay, D. Delande and A. Bommier, Phys. Rev. A 39 (1989) 6587. [ 321 R.S. Berry, J. Chem. Phys. 45 (1966) 1228. [ 331 A. Russek, M.R. Patterson and R.L. Becker, Phys. Rev. 167 (1968) 17. [ 341 R.S. Berry and S.E. Nielsen, Phys. Rev. A 1 (1970) 383. [35]B.Ritchie,Phys.Rev.A6 (1972) 1761. [ 361 E. Rabani, R.D. Levine and U. Even, J. Chem. Phys., submitted for publication. [37] U. Fano, J. Opt. Sot. Am. 65 (1975) 979. [ 381 C.H. Greene and Ch. Jungen, Advan. Atom Mol. Phys. 2 1 (1985) 51. [39] M.S. Child and Ch. Jungen, J. Chem. Phys. 93 ( 1990) 7756. [ 401 K. Milller-Dethlefs, J. Chem. Phys. 95 ( 1991) 482 1. [ 4 1 ] E.W. S&lag, H.J. Grotemeyer and R.D. Levine, Chem. Phys. Letters 190 (1992) 521. [42] E.W. Schlag and R.D. Levine, J. Phys. Chem. 96 (1992) 10668. [43] J. Jortner, private communication. [ 441 R.L. Whetten, G.S. Ezra and E.R. Grant, Ann. Rev. Phys. Chem. 36 (1985) 277; M. Ito, T. EbataandN. Mikami, Ann. Rev. Phys. Chem. 39 (1988) 123. [45] M. Bixon and J. Jortner, Chem. Phys. 48 (1968) 715. [ 461 E.W. Schlag, S. Schneider and S.F. Fischer, Ann. Rev. Phys. Chem. 22 (1971) 465.
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CHEMICAL PHYSICS LETTERS ELSEVIER
Chemical Physics Letters 22 1 ( 1994) 473-48 1
Dynamics of high molecular Rydberg states in the presence of a weak dc field Eran Rabani
a,L.Ya.
Baranov a, R.D. Levine a, U. Even b
a The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University, Jerusalem 91904, Israel b School of Chemistry, Sadder Faculty of Exact Sciences, Tel-Aviv University, Tel Aviv 69978, Israel
Received 25 January 1994; in final form 14 February 1994
Abstract It is argued that under the conditions of current experimental interest, one role of a weak electrical dc field on the dynamics of a high Rydberg molecular electron is to stretch the time axis. The stretch factor is essentially independent of the field and provides a classical analog of the quantum mechanical ‘dilution’ effect. A second role of the field is to lower the threshold for ionization thereby reducing the energetic requirements for auto-ionization. It is argued that the implications of the presence of the field are quite different for lower Rydberg states. Classical trajectory computations using action-angle variables are used to
validate the proposed interpretation and to probe the nature of the possible intramolecular decay channels.
1. Introduction
The issue addressed in this Letter, is the role of an external weak electrical dc field on the intramolecular dynamics of Rydberg states at energies just below ionization. It has been argued that such states manifest both short time decay [ l-3 1, which is possibly slower than that expected on the basis of extrapolation of observations on lower Rydberg states of the same molecule [ 3 ] and also exhibit even longer time stability [4-l 0 1. Our question is what determines the time scales for these processes and how does the magnitude of the dc field affect the conclusions. In this Letter, we use a model Hamiltonian where, in addition to the external electrical field, there are two internal perturbations to the purely Coulombic motion of the electron about the center of charge of the molecular ionic core. One is spherical and induces a precession of the elliptical orbit of the electron and the other, which is anisotropic, couples the electron’s motion to the rotation of the core. We show,
with support from classical trajectory computations,
that at the high principal quantum numbers of interest, the primary role of the weak external dc electrical field is to ‘stretch’ the time axis. The strong increase of the orbital period of the electron with its principal quantum number n is used to argue, again with support from the trajectory computations, that this stretching of the time is special to the region of high n as probed by recent experiments. The scaling with n of the boundary between the intermediate and higher n regions is shown to be n5 so that at intermediate and high n the physics can be different. The use of classical trajectories to probe the dynamics enables us to demonstrate that, for our model Hamiltonian, there are two competing decay channels. These correspond to an exit of the Rydberg electron from the detection window by a gain or by a loss of energy due to the coupling to the rotation of the core. Two such competing channels have previously [ 1,2] been proposed as responsible for the frequency dependence of the lifetime of the higher Rydberg
0009-2614/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDIOOO9-2614(94)00273-S
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states. The two channels were referred to as ‘up’ and ‘down’ processes, respectively, and we follow this terminology here. In the real molecule there can be additional couplings, e.g. to the vibrations of the core, and these can contribute to the up and the down channels. Furthermore, there can be other external perturbations besides that of a static dc field. There is considerable interest in the role of electrical fields, both intentional and so-called stray fields, on high atomic [ 1l-l 61 and molecular [ 17-271 Rydberg states. Both enhanced decay [ 17,18 ] and increased stability [ 19 ] are reported for molecular states showing that there is not a unidirectional effect. One clear role of the field is to lower the threshold for ionization #l. States of the electron orbiting around an excited ionic core, which in the absence of the field are bound, can ionize. This has been observed both for atoms other than hydrogen [ 13 ] and for molecules [ 7,271. In the latter case the loss of energy by the core can be detected by monitoring the rotational states of the ion. Another effect, discussed for both atoms [ 11,20,29] and molecules [ 17-201 is that known as ‘I mixing’. This refers to the angular momentum quantum number I of the electron not being conserved in the presence of the field. The Hamiltonian for a Coulomb potential with a dc field can, however, be exactly solved by using, so-called, parabolic coordinates [ 301. The eigenstates in these coordinates [ 291 are superpositions of the familiar stationary states of sharp angular momentum; hence the term ‘I mixing’. The optically accessed initial state has a fairly, or even very, well-defined 1value. In the discussion below we shall therefore find it more useful to think in terms of non-stationary or ‘coherent’ states [ 20,3 1 ] which have a rather localized range of I values. The potential seen by a Rydberg electron moving about an ionic molecular core is non-Coulombic [ 3235 1. Such a potential has a non-Coulombic ‘central part which can induce a precession of the BohrSommerfeld elliptical orbit [ 30 1. This will give rise to a ‘quantum defect’, as in non-hydrogenic atoms. As will be discussed below, the precession of the orbit x1 In principle, an electron in the presence of a field, however weak, can always tunnel out [ 281. However, at high n, the width of the barrier to ionization is so large that the tunneling rate is negligible [ 271. We therefore only consider ionization to be possible only at energies above the classical threshold.
can average out the effect of an external weak electrical field which is in a given direction. We shall conclude that this effect is too weak at the high n of interest but this will not be so at lower n. The other role of the non-Coulombic part of the potential is to couple the motion of the electron to the molecular degrees of freedom and thereby allow the exchange of energy and of angular momentum between the electron and the core [ 361. We have previously argued that this intramolecular transfer is responsible for both the up process in which the electron gains energy at the expense of the core and for the down process where the electron cascades down in its orbital energy. A computational study of such processes in the absence of the field [ 361 has explicitly demonstrated the expectation that the electron is only effectively coupled to the core when it is near to it. The nature of the Kepler-like orbit is that the electron rapidly transverses this region of close coupling. Over most of its orbit the electron sees only the long-range Coulomb potential #2.To examine the combined effects of the core and the external field on the motion of the Rydberg electron we present in section 2 the results of classical trajectory computations. These are for the motion of a high n electron around an anisotropic ionic core with a weak external dc field in the z direction. The two perturbations affect the electron in different ways. It is strongly coupled to the core only when it is close to it while the coupling to the field increases the further out the electron moves in the z direction #3.In the absence of the field, the electron is least perturbed near the aphelion of its orbit. For the present problem the electron is most stable when it is away from both the core and the z axis. The Hamiltonian used in the simulations introduces only the electrical anisotropy of the ionic core. The electron can thereby couple only to the rotation of the core. For large polyatomic molecules one can also provide an additional mechanism for long time stability namely the temporary storage of the electronic energy in the high density vibrational manifold of #*This
point is also made in multi-channel quantum defect (MQDT) theory [37-40). N3 Recall that for a Bohr-Sommerfeld orbit the core is at the focal point of an ellipse. The aphelion and perihelion are at nZ( 1+ e) atomic units respectively, where t= (1-1*/n*)“* is the eccentricity. When n is high and I-Cn the perihelion is at t/(1+ 1) atomic units and its velocity there is 2/ (I+ 1) in atomic units.
E. Rabani et al. /Chemical Physics Letters 221(1994) 473-481
states [ 41-43 1. An important issue, not addressed in this study, is whether the very high n electrons which are accessed by the initial optical excitation can couple directly to the vibrations #4or whether the doorway to the vibrations is a reduction in n due to rotational coupling to the anisotropic core. The discussion leading to the concept of the ‘stretching’ of the time scale of the intramolecular processes due to the presence of the stray field is presented in section 3. Specifically, what we show is that the frequency of close encounters of the electron with the core is de facto reduced by the factor Jdln, where d is the effective range for the electron-core anisotropic interaction in atomic units. The result that this factor is independent of the magnitude of the field and, in particular, of the magnitude and conservation of the projection quantum number ml, is subject to limitations which are spelled out in section 3. However, already at this point one needs to emphasize that the result is only valid in the range of (stray or intentional) weak dc fields as currently used in ZEKE-type [ 45 ] experiments (say from 0.1 to 1 V/ cm). Quite different conclusions would obtain at lower n values or at weaker fields. (At stronger external fields, the states of high n of interest will ionize. ) It is also argued in section 3 that the stretching effect is a classical analog of the so-called ‘dilution’ effect of radiationless transitions theory [ 45-48 1. This effect is derived here from a kinetic point of view but we also comment on it from a stationary point of view so as to emphasize the role of the initial preparation [ 45,481. We further derive results for more extensive dilution made possible when the orientation quantum number ml is not conserved. The non-conservation of ml is important to us also as a mechanism for keeping the electron away from the core, recall that I> mb and thereby confers an additional stability on the orbit. The extreme dilution limit that we obtain is when the precession, which results in the rotation of the axes of the elliptical orbit, is faster than the orbital angular motion. Finally, we note that extrapolating the width of lower Rydberg states to high values of n results in rates which are more than two orders of magnitude faster than the directly observed M That
states of low n can couple to the vibrations is evident in the phenomena of vibrational auto-ionization. See refs. [ 32-
35,441.
475
fast decay [ 3 1. It remains to be seen if the stretching of the time scale noted above is an important consideration for the complete explanation of this observed lengthening or whether coupling to the vibrational modes of the polyatomic core and/or the role of other external perturbations is also important. In section 4 we examine the kinetics of the up and the down processes. Specifically it will be argued that the results are consistent with these two processes representing competing decay channels of a common population.
2. Trajectory computations The potential field for the electron (discussed in more detail in ref. [ 361) has three parts. A Coulombit 1/r part with its origin at the electric center of the core, a term p( 1 +I.?) /r2 representing spherical and dipolar anisotropic deviations from the Coulomb potential and a coupling F*r to an external dc field of magnitude Fin the direction of the z axis. The Hamiltonian of the core is that of a rigid rotor of rotational constant B. The results shown below are for Bc0.15 cm-’ but higher values were also used. As far as we can ascertain the only ‘special’ values of B are those that allow for the possibility of matching between the unperturbed orbital period of the eleo tron (r= 1.52~ lo-l6 n3 S) and the period (2Bj)-' of rotation of the core when its rotational angular momentum is j [ 361. This is the so-called ‘stroboscopic effect’ [ 49 ] and does not occur for the high values of n of interest where the orbital motion of the electron is the slowest in the system. We have also verified this by examining the power spectra of the two motions [ 36 1. Trajectories were computed in action-angle variables [ 301 for the three-dimensional motion of both the electron and the core. Ensembles of initial conditions were generated by randomly sampling over the angle variables at fixed values of the action variables. The dynamics are classical but to compare with experiment we measure the value of the action variables in units of Plan&s constant. This quasi-classical correspondence cannot replace an MQDT parametrization or, better yet, a full ab initio approach [ 501. On the other hand, since n - 1 measures the number of nodes in the radial wavefunction it follows that in
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our limit there are many de Broglie wavelengths per orbit so that the classical limit is realistic. If anything, the use of classical dynamics is probably more a limitation for the rotational motion of the core whose states are more widely spaced than those of the Rydberg electron. In the absence of either the non-Coulombic part of the potential due to the core or the external field, the action variables are constant in time while the angle variables change at a constant frequency. This is unlike the ordinary coordinates and momenta of the electron which, over a typical orbital at a high n, change by many orders of magnitude. In the present study, trajectories were integrated up to 4 ILSor until the orbital motion excited from the ‘detection window’ (most trajectories satisfied the latter criterion). The ‘detection window’ is defined with reference to the experimental method of detection which is to apply a delayed electric field which ionizes Rydberg states down to below the initial value of n. We take the lower limit of the detection window to be at n = 90 which corresponds to a delayed field of about #’ 10 V/cm. The electron is deemed to have ionized and therefore to exit from the detection window when its n exceeds #6 1000. Fig. 1 shows the time evolution of the principal action n and of the orbital angular momentum 1 of the electron versus time for two trajectories which exit from the detection window in about 300 ns. One of the exits is up and the other down. There are two notable observations: firstly, the almost periodic variation in I between a minimal value of about zero #’ and a maximal value which depends on ml but can be as high as n - 1. The other feature of the trajectory results is that n changes only when 1 is in its region of low values, I< &. a5 The results shown below for the high Rydberg state are for an initial n of 150. We did, however, check that our qualitative conclusions are unaffected by increasing the delayed field by a factor of two. As will become obvious, in the absence of coupling to the vibrational modes, increasing the magnitude of the delayed ionizing field will slow the down process. M In practice, once n exceeds the threshold for ionization which is at n of about 300 for a field of 0.1 V/cm, n rises so rapidly that the exact upper cutoff is immaterial. ” The value of I has to exceed the value of M, which in the absence of the anisotropy of the core is conserved. This observation is significant in view of the discussion of stretching of time in section 3 below.
time (us) Fig. 1. Temporal evolution of typical single trajectories, one exiting by a ‘up’ route (upper panel) and the other by a ‘down’ process for a field of 0.1 V/cm. Shown are n (-) and I (- - -), the principal action and the orbital angular momentum of the Rydberg electron in units of Planck’s constant versus time in us. Computed for a core with a rotational constant of 0.15 cm-’ and an initial rotational angular momentum j= 8.5. The anisotropy of the core is dipolar with a moment of 0.3 au. The initial n is 150. Note how changes in n are fully coordinated with a low value of I and the periodic evolution of I. (The period w= 3nF of the 1 motion is modulated by the change in n with higher n resulting in more frequent passages close to the ionic core.) Note how prompt is the escape of the electron once its n increases enough so that its energy is above the threshold for ionization. It is the lowering of this threshold by the field that accounts for the up process being more probable at higher fields. It is, however, the core that provided the energy needed to increase the value of n.
Direct examination of the trajectory results verifies what is intuitively expected, namely, that n changes only when the electron can get close to the core. Since the distance of closest approach is nZ( 1 -dm), n can change only when lcn. This is unlike the field-free case [ 36 ] where, for low values of i, n can change once per revolution. Not every close encounter with the core necessarily results in a change in the value of n. However, since changes in n occur only during the brief sojourn near the core, the rate of both up and down processes must be proportional to the frequency of passage through the perihelion which varies as n -3. At higher 1 values, the electron does not enter the region of close coupling to the core. We defer further considerations of this point to section 3 and other computational results to section 4.
E. Rabani et al. /Chemical PhysicsLetters221(1994) 473-481
3. The time stretching
In the absence of an external dc field the orbital angular momentum 1 of the electron is conserved apart from brief passages near the perihelion when I and j are coupled by the electrical anisotropy of the potential of the core. This anisotropy has a finite effective range, say d, which is quite short compared to the scale of the Bohr-Sommerfeld orbit at high n, d << n 2. Therefore, over most of the orbit the value of 1 is conserved. Computational illustrations of this conservation can be found in ref. [ 361. On the other hand, the coupling to an external field will be manifested over the entire orbit. In a weak electrical dc field, 1will undergo a periodic, harmonic motion (a so-called ‘first-order Stark effect’ [ 301). In the semiclassical limit the frequency w of the motion of I equals the Stark splitting w = 3nF, where F is the field inatomicunits(1auoftield=5.14x109V/cm).For the typical stray field of 0.1 V/cm and n = 150 the orbital angular momentum completes one revolution in about 17 ns. In the same time the electron completes about 34 revolutions around the core. Elsewhere, we shall discuss the modifications of these conclusions made necessary by the second-order Stark effect. One such modification is that the motion of I is anharmonic. The amplitude of the 1motion does depend on the value of ml but for the purpose of the discussion we take it to be n so that the value of 1 as a function of time is n sin( wt+ $), where the phase is determined by the initial value of 1. It is only low values of 1, say 1~ l,, that allow the electron to closely approach the core. Since the distance of closest approach at low 1is 4l( l+ 1) we can approximate loby Jd, where d is the effective range of the anisotropy of the core. d can be estimated from the results for the field-free case or from analytical results for the matrix elements of the anisotropy [ 5 11. The interval in time during which 1 is below 1, is, provided 1, c n, l,/ wn = Jdl 3n2F. The frequency of returns of the electron to the core is n -’ in atomic units, hence the number of close encounters of the electron and the core is Jd/3n5F per oscillation of 1 or wJd/3n5F=Jdln4 per unit time. Since the electron executes ne3 passes through its perihelion per unit time, the frequency of close encounters of the electron with the core is reduced, compared to the field-free case, by &/n=l,Jn.
417
That the periodic motion of 1scales as expected has been verified in our trajectory computations. What can be clearly seen in Fig. 1 is how the period becomes longer if n has undergone a large change due to a close coupling to the core. It should be emphasized that the Hamiltonian used does include a term which induces a quantum defect, namely a precession of the axis of the ellipse. Below we argue that for the high n of interest the quantum defect cannot prevent the oscillation in 1but that this will no longer be the case at lower n values, cf. Figs. 2 and 3 below. The essential point is that at the high n of interest the value of 1is oscillatory and that it periodically comes down, and thereby allows the electron to couple to the core.
8 2
0
6
-0.5
4 f,
21
1,
5
0
f,
\
10 15 time (ns)
1 20-l
Fig. 2. The time-evolution of I (-) and of the angle cq (which specifies the orientation of the major axis of the elliptical orbit) in the intermediate regime, n = 70 and all other conditions as in Fig. 1. Note how the precession prevents the full oscillation in 1. At lower n values, as shown in Fig. 3,Z is hardly changing.
0
2
4
6
8
10
time (ns) Fig. 3. The time-dependence of I and n, n = 50. The value of I is hardly changing. Shown for comparison (- - -) is the timedependence of 1 when the anisotropy of the core is turned off. The I mixing is present but slow as compared to the orbital period (0+&j< 1).
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E. Rabani et al. /Chemical Physics Letters 221(1994) 473-481
Elsewhere, we shall argue that quanta1 interference effects do not change this conclusion. The result that a weak dc field stretches the time axis by n/lo has been derived within a number of assumptions. While it is numerically a stretch since da n 2 or I, < I < n, it is physically a stretch because it is only valid for such weak fields and high n values that the period n3 of the orbital motion is far shorter than the period 0-l of the 2 motion. This implies 3n4F< 1 (equality occurs at F= 3.4 V/cm for n = 150). The classical onset of ionization is at about n4F=0.13 [28] and is saturated by n4F=0.383 so that fields at which the two periods are equal with already lead to prompt ionization of the electron. The upper end of the scale is thus not a limitation. There is, however, a lower limit of validity. This is for such weak fields that the precession of the orbit, due to the non-Coulombic part of the potential, is faster than the change in I due to the external field. The frequency of the precession is 6/n3, where delta is the quantum defect (modulus one). The time interval during which 1x1, is 3n2F/lo. The lower limit of I mixing is thus at 3n5F<&,. For our Hamiltonian 6~0.3 for s states (ref. [ 301, section 26) and decreases with 1. It follows that the field at the lower limit for time stretching is l/n as large as at the upper end. At the high n of interest this leads to fields which are comparable to the Lorentz field [ 521 due to the motion in the earth’s magnetic field. On the other hand, for a given field F, the lower limit places a steep bound on the values of n which are affected by stray fields. For a stray field of 0.1 V/cm and an s state excitation, the minimal value of n for any stretch effect is n= 111 and is significantly lower for higher values of 1.Another important limitation of our derivation is that the value of mI was assumed to be constant and quite low compared to lo. When this is not the case, the scaling of the frequency of close encounters can be shown to scale more rapidly with n and can reach n -5 when ml is fully not conserved. Elsewhere, we shall discuss the mechanism for reaching higher values of ml. Fig. 2 shows the competition between the precession and the field-induced oscillation of 1in the intermediate region where the two time scales are comparable. This corresponds to n of about 70 for our Hamiltonian (this result depends on the initial value of I). At lower values of n, the precession precludes
the changing of I due to the field as shown for n = 50 in Fig. 3. An extreme limit is when the frequency of precession is so fast that it greatly exceeds the orbital frequency. In such a limit the electron does not succeed in getting away from the core #8. In order to make an analogy between our present considerations and the quantum mechanical concept of a ‘dilution effect’ [45,48] it is necessary to also discuss the role of the initial preparation. To do so consider, for the moment, the simpler problem of an electron in the presence of both a Coulomb and an external field. This problem admits a separation using parabolic action-angle variables R9[ 301. Each one of our trajectories has, initially, a sharp value for these action variables. The oscillation in 1is precisely a result of the constancy of the parabolic action variables. This oscillation leads, as discussed above, to the time stretching of every trajectory. Now consider an ensemble of initial conditions where most members of the ensemble have similar initial values for the parabolic action variables. The ensemble we just discussed is stationary and is a classical analog of an initial preparation of one or a few eigenstates in the terminology of radiationless transitions theory [ 4548 1. In this ensemble the probability of low I values is low. The dilution effect is thereby manifested in a stationary context (see also ref. [ 241). On the other hand, one can, in principle, conceive of a different initial ensemble with a sharp value of I and, therefore, a distribution in the parabolic action variables. In this second ensemble initially all the trajectories allow strong coupling of the electron to the core. The energy of the different trajectories in this second ensemble spans the entire Stark manifold which, to firstorder, has a width of 3n2F. For the high n values of interest, this width (0.29 cm-’ for n = 150 and F= 0.1 V/cm) is large enough that excitation of an entire Stark manifold requires special conditions. #* Every attempt to depart towards the aphelion is foiled by the faster rotation of the direction vector pointing along the major axis. Elsewhere, we shall derive this and other limits from an analysis of the equations of motion of I and of the direction, socalled, Runge-Lenz, vector. a~ We have computed the dynamics in terms of these variables and will present the results elsewhere.
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479 F = 0.5 V/cm
4. The decay kinetics
1
400
This section briefly reports on the time-evolution for an ensemble of trajectories where the lowering of the threshold for ionization by the field is clearly evident. More detailed results will be presented in a full report which is in preparation for publication. Fig. 4 shows decay curves at several values of the field for an initial n of 150. The rate of short time decay increases strongly with increasing field. Not evident in this plot where the time is in the microsecond scale that within the first few orbital periods, and before 1 had the time to start increasing, a few trajectories exit from the detection window. Also not clearly evident are the few long-living trajectories. Elsewhere, we shall argue in detail that the long-living trajectories are those where the electron reaches a region in phase space where it is comparatively resilient to both the effect of the field and of the core. However, all trajectories which have enough energy to ionize eventually do so. One can classify the trajectories according to whether they escape up or down. Fig. 5 shows the separate decay curves for the two classes of trajectories. The separate fits to an exponential decay yields essentially identical rate constants as shown in the inset. This is the expected result if there is a common population which can competitively decay into one of two alternative channels. The rate constants for up and down decay determine the branching fraction into
300 2 Z
-I
200
100 J
0
0
0.5
1 time (ps)
1.5
2
Fig. 5. The number of trajectories which remain in the detection window up to the time t, classified according to their mode of exit for a field of 0.5 V/cm. As in Fig. 4, the fit shown is to exponential decay and the resulting decay rate (given in the inset) is essentially identical for both channels. The branching ratio is 2.8 1 versus 2.68 and 2.1 at F=0.3 and 0.1 V/cm, respectively. The increase of the branching ratio for the up process as the ionization threshold is lowered is analogous to the role of the energy gap to ionization discussed in ref. [ 11.
either channel and the overall decay rate is the sum of the two. In this fashion, one can provide a separate determination of the two rate constants and examine their dependence on both the field and on the initial value of n. Results of this type have already been reported for the field-free case [ 361 and will be provided in the full account of this work.
5. Concluding remarks
0 0.5
1 1.5
2 2.5
3
3.5
4
time @s) Fig. 4. Decay of an ensemble of high Rydberg states versus time in ps. Shown is the fraction of trajectories which remain in the experimental detection window for an integration time of 4 ps for three different values of the field in V/cm. Other conditions as in Fig. 1. The fit is to an exponential decay (in ps-‘) plus a constant (as our integration time does not suffice to establish the decay constant of the very few long-living states).
The typical stray or intended dc fields present during the time-evolution of high Rydberg states are such that their role can be summed up as inducing a stretch of the time axis. The boundary region between the high Rydberg states and the lower n states for which this field effect is averaged out, due to the precession related to the quantum defect, is rather narrow (and occurs at about n = 70 for our Hamiltonian which has a quantum defect of 0.3 for s states). A second important field effect is the lowering of the energetic threshold for ionization. In order to decay by an up process the total energy of the state need no longer be positive but only to exceed the classical onset for ionization.
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Acknowledgement
We thank Professor R. Bersohn, Professor J. Jortner and Professor E.W. Schlag, and Dr. K. Muller-Dethlefs and Dr. H. Selzle for discussions and Professor J. Jortner for his comments on the manuscript. ER is a Clore foundation scholar. This work was supported by the Volkswagenwerk Stiftung and by the German-Israel foundation for research ( GIF) . The experimental work at Tel-Aviv University is supported by the German-Israel James Franck binational program. The Fritz Haber research center is supported by MINERVA.
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