Calculated electron dynamics in an electric field

Transcription

1 PHYSICAL REVIEW A VOLUME 56, NUMBER 1 JULY 1997 Calculated electron dynamics in an electric ield F. Robicheaux and J. Shaw Department o Physics, Auburn University, Auburn, Alabama Received 18 December 1996; revised manuscript received 14 February 1997 We present the details o a theoretical method or calculating the wave-packet dynamics o a Rydberg electron in a strong electric ield. Results or Rb are presented, and are compared to recent experiments that measure the time dependence o the electron lux. These interesting experiments provide a dierent possibility or the experimental detection o wave packets. The results o ully quantum and classical calculations are compared to each other. Several aspects o this system that are easier to study theoretically are presented. The main property that can only be obtained theoretically is the orm o the wave unction near the nucleus. We ound that the lux o electrons entering the detector accurately relects the lux o electrons leaving a sphere o radius 2000 a.u. centered on the atomic nucleus. The classiication o the autoionizing states contributing to the wave packet is also easier to obtain theoretically. S PACS numbers: i, Re, Dz I. INTRODUCTION Recent advances allow the exploration o electron dynamics in atoms in a dynamical ashion 1 20 through the creation and detection o time-dependent electron waves. These waves are oten called wave packets, although they may not be very localized in phase space. A pulsed laser ield or a pulsed electric ield creates the wave packet and determines its initial conditions. The electron wave rapidly evolves in the Coulomb and external ields. Measuring the electron dynamics poses a diicult technological problem that has been solved several dierent ways. The ocus o this paper is on the creation o wave packets in the autoionizing regime or an alkali-metal atom in a strong and static electric ield. This is one o the ew systems involving more than one degree o reedom that has been explored The time-dependent lux o electrons ejected rom autoionizing states o Rb atoms has been measured in a static electric ield using an electron streak camera 13. These striking measurements orm a point o comparison between calculations and experiment. The experiments excite Rb rom the ground state to autoionizing states just above the classical ionization threshold in the ield. These experiments provide a very interesting way to probe the wave packet near the atom. Autoionizing states naturally eject electrons, and these electrons can be detected ar rom the atom. Wave packets constructed rom autoionizing states eject electrons in a periodic ashion that relects the periodicity o the transiently bound electron. The wave-packet dynamics is determined by several parameters: the static electric ield, the laser polarization, the main requency o the laser, and the pulse duration. The lux o electrons rom the Rb can be a very complicated unction o time but the time dependence o this lux relects the time dependence o the wave packet in the autoionizing states. By varying the parameters, a good match between the calculated and experimental luxes may be obtained. However, the asymptotic lux is only a small part o the inormation that may be obtained rom the calculated wave packet. The calculated wave packet allows the precise determination o the dynamics that led to the measured lux. By linking our computational and experimental eorts, we gain a deeper insight into this dynamical system. The creation and detection o wave packets allows a somewhat more direct connection to and exhibition o classical dynamics in quantum-mechanical systems. Many o the experiments on simple quantum systems have eatures that may easily be interpreted using classical ideas. Relatively little work has been done on complex systems, especially systems involving two or more coupled degrees o reedom. Wave packets generated on heavy alkali-metal atoms in a strong static electric ield are a prototype or the dynamics o complex wave packets. The electric ield plus Coulomb potential leads to a separable Hamiltonian in parabolic coordinates; thus the motion is independent in each o the parabolic degrees o reedom. However, or the heavy alkali-metal atoms, the core electrons break the parabolic symmetry or the valence electron, causing strong scattering between the dierent types o parabolic motion. The quantum-mechanical wave unction can be described as a superposition o unctions in the dierent parabolic channels, with the core causing a coupling between the channels. The motion o electron wave packets on alkali-metal ions in constant electric ields provides an ideal setting or comparison between theory and experiment. Further, the dynamics o this system may be calculated in two dierent ways, with only a modest investment o computational resources. The two methods that are utilized complement each other, in that there are some parameters that may be calculated using either method e.g., total photoionization cross section and there are some parameters that can only be obtained using one o the methods. By comparing the results rom the two methods where possible, we can gain conidence that we have implemented them both in an accurate and bug-ree manner. The simpler o the two methods to implement constructs the wave packet using a superposition o unctions that are solutions o an inhomogeneous Schrödinger equation. These unctions depend on the energy, and are constructed by calculating the Green s unction using a basis set o unctions in spherical coordinates. The more diicult method to implement constructs wave packets by superposing homogeneous /97/561/27812/$ The American Physical Society

2 56 CALCULATED ELECTRON DYNAMICS IN AN ELECTRIC FIELD 279 unctions calculated within the ormalism developed by Harmin 31 using the local rame transormation suggested by Fano 32. In this method, the parabolic symmetry o the Hamiltonian outside o the core region is exploited in the construction o the energy-dependent wave unction. The core couples together the dierent parabolic channels, which causes the scattering o the electron rom one channel to another. We have completed some preliminary calculations o the classical dynamics or this system, as well as the exact wavepacket calculations. The connection between some aspects o the classical and quantum dynamics should be more easily made using wave packets. In general, we have ound this to be true. For the systems that we present in this paper, the connection is sometimes diicult to make, because the wave packets are constructed rom relatively ew states. Certainly, the connection o our packets to periodic orbits cannot be made cleanly. The inal part o this paper involves the presentation o calculations o electron wave packets or Rb in an electric ield. These calculations ocus on explaining the experimental results o Lankhuijzen and Noordam 13. To this end we calculate the requency-dependent cross section, we calculate the asymptotic lux or one o the experimental geometries, we show several o the wave packets, and we show some o the classical trajectories that seem to have some correspondence with the wave packet. Calculations or another o the experimental geometries has already been presented 33. Atomic units are used throughout this paper unless speciically noted. II. LASER PULSE EXCITATION OF WAVE PACKETS In this paper, we are concerned with the dynamics o wave packets that have been created by shining a weak pulsed laser on an initial state. The wave packet is the solution o the inhomogeneous Schrödinger equation i t H r,tftˆ r cos 0 t expie I t I r, 1 where H is the static Hamiltonian incorporating the atomic Hamiltonian and the potential rom the static electric ield, I (r) is the initial state, E I is the initial state energy, 0 is the main requency o the laser pulse, ˆ is the polarization, and F(t) is the relatively slowly varying laser envelope. The unction (r,t) may be obtained in several dierent ways. Two methods based on time-dependent perturbation theory may be used. Both methods use a Fourier decomposition o the laser envelope F Fte it dt. The quantity F( 0 ) is proportional to the amplitude or inding a photon o requency in the laser pulse. The unction F() is strongly peaked around 0, which gives a strong peaking o the photon amplitude near 0. We are interested in calculating wave packets o continuum states. This makes the construction o wave packets 2 slightly more diicult than the usual case where the packets are constructed rom bound states. It is now necessary to superpose an ininite number o states. The wave packet can be constructed by superimposing the solutions o the inhomogeneous Schrödinger equation EH E r ˆ r I r, where the superscript indicates that E is composed o outgoing waves in every open channel. By simple substitution into Eq. 1, it is possible to show that r,t 4 1 E r e iet FEE I de 4 where the rotating-wave approximation has been used i.e., the expi(e I )t term in Eq. 1 is neglected. The wave packet may also be constructed by superimposing the solutions o the homogeneous Schrödinger equation EH E r 0. For energies in the continuum, there are, in general, several open channels at energy E. The number o linearly independent solutions, N, equals the number o open channels. There are ininitely many dierent ways o constructing the linearly independent solutions, so we will choose the linearly independent solutions in a orm that will most easily allow the construction o the outgoing lux. These unctions, E, are constructed so that asymptotically they only have outgoing waves in channel. These unctions have the general orthogonality property E E O EE, 6 where oten the overlap matrix O the overlap matrix is not proportional to unctions or the outgoing waves in parabolic coordinates, which orces us to use the more general ormalism. In terms o these unctions, the wave packet may be constructed or times ater the laser pulse has gone to zero as r,t i D 2 E E e iet FEE I de, 7 where the dipole matrix elements or these unctions have the general orm D E 3 5 O 1 E ˆ r I. 8 The two methods presented above give equivalent results. But there are advantages to using dierent methods in dierent regions o space, as will be described below. An important point o contact between the dierent methods is the total photoionization cross section. The total cross section may be obtained as E D E 2 Im I ˆ r E. 9

3 280 F. ROBICHEAUX AND J. SHAW 56 This relationship may be exploited as a test o the implementation o the dierent methods. It is important to note that nothing in this section depends on the particular problem that is being examined. The only requirement is that H be time independent, and that the excitation be perturbative. We note that several dynamical situations may be examined immediately using existing computer programs and Eqs. 4 and 7. In particular, wave packets excited in constant electric and magnetic ields or only in magnetic ields may be explored. These problems may have deep interest, since it would be possible to excite wave packets or parameter ranges that would lead to chaos in classical mechanics. III. CALCULATING E One method or calculating the wave packet is to superimpose the inhomogeneous unctions that are solutions o Eq. 3. The E (r) are the solutions with outgoing waves in every open channel. Enorcing this large distance boundary condition in an elegant and accurate manner is very diicult. However, a brute orce approach borrowed rom codes involving the direct solution o the time-dependent Schrödinger equation will give an accurate without the need or an explicit solution o the boundary conditions. The inhomogeneous equation was solved by expressing E (r) as a superposition o radial unctions times spherical harmonics. The radial unctions were the solution o the radial Rb Hamiltonian with a model potential near the origin to give the correct quantum deects or the atom. The radial unctions were orthonormal since they were all generated rom the same potential, and they were all orced to zero at r2800 a.u. We used 89 radial unctions or each l, and a maximum l o 59. Convergence was tested by increasing the number o radial unctions by ten, keeping everything else ixed, or by increasing the maximum l by ten and keeping everything else ixed. Neither increase changed the results by more than 0.5%. The inhomogeneous unction was calculated by direct solution or x in the Axb linear problem, taking ull advantage o the block-tridiagonal nature o H. The algorithm was based on the ormal solution o a tridiagonal linear equation, except that the elements o the tridiagonal linear system were themselves matrices. This allowed the calculation o E to be roughly times aster than when working with the ull Hamiltonian matrix o the same size. This speed was crucial, because E needed to be calculated at several hundred energy points or each set o experimental parameters. The description to this point leaves out the important description o how to obtain the correct asymptotic boundary conditions. Remember, E (r) must have outgoing waves in all o the open channels with incoming waves in none o the open channels. The correct boundary conditions are automatically incorporated i we add to H an unphysical imaginary potential. The imaginary potential decreases the norm o the wave unction, where the imaginary potential is nonzero. The requirement on the absorbing potential is that it absorbs the vast majority o the lux moving to large r and does not absorb lux at an r that is too small, thus aecting the autoionizing dynamics. Two possible problems with the TABLE I. Parameters or the Rb model potential. l l (1) l (2) l (3) absorbing potential must be avoided. First, the absorbing potential should not turn on so quickly in r that it relects electrons back into the region o small r. Second, the absorbing potential should not be so weak that the electron can travel all o the way to r2800 a.u. and relect back into the small-r region. Both these restrictions can be satisied or our wave packets, because we are working in a very narrow energy range. The only drawback to using this method or constructing wave packets is that it cannot be used to compare with the experimental results o Lankhuijzen and Noordam 13 without irst comparing to some other computational technique. In these experiments, the time dependent lux o electrons ejected rom alkali atoms in static E ields was measured in a time-dependent manner using an electron streak camera. Unortunately, the electrons ejected rom Rb may not exactly maintain their spatial relationship as they travel rom the atom to the detector. This dispersion may prevent a quantitative comparison to their experiments using the superimposition o E (r) that are constructed rom a inite basis. A comparison o the lux obtained rom this method with a computational technique that calculates the lux a macroscopic distance rom the atom would also allow us to answer the question: How aithully does the time-dependent lux entering the detector relect the dynamics near the atom? An interesting aspect o this method involves its lexibility. Any type o alkali-metal atom in a static ield can be described with this method. There does not appear to be any reason why we could not make wave packets on any o the heavily explored ield atom combinations: magnetic ields and parallel E and B ields, or example. The zero-ield Rb radial unctions were generated using a model potential o the orm V l r Z l r d l l 1 r 2r 4 1expr/r c 3 2 2r 2, 10 where d is the dipole polarizability o Rb 34, r c 1.0, and Z l (r)136exp( (1) l r) (2) l rexp( (3) l r), with the l given in Table I. This potential gives quantum deects with errors less than or l 3. The radial unctions were generated on a grid o radial points on a square root mesh in r i.e., equally spaced points in s where rs 2 ). The derivatives in the one electron Hamiltonian were approximated by a ive-point or ourth-order inite dierence. The orbitals were generated using a relaxation technique by repeated multiplication o a trial R nl by E nl H l 1. Usually, only two or three multiplies were needed to achieve convergence to double precision roundo accuracy.

4 56 CALCULATED ELECTRON DYNAMICS IN AN ELECTRIC FIELD 281 IV. CALCULATING HOMOGENEOUS PARAMETERS The dynamical system o an alkali-metal atom in a static electric ield was theoretically described by Harmin 31 based on a method suggested by Fano 32. This method is ideally suited to describe the detection o electron lux ar rom the atom, since the dipole matrix elements o Eq. 7 and the asymptotic orm o the unctions are generated. The cross section may be obtained rom the dipole matrix elements, orming a point o contact with the method o Sec. III. We recast the derivation o Re. 31 into a orm more amenable to a calculation o the asymptotic time-dependent lux. The basic idea o this method is that the wave unction near the core is well described by standard radial Coulomb unction shited in phase by times the quantum deect times the appropriate spherical harmonic. Outside o the core region the electron moves in a pure Coulomb potential plus static electric ield. This type o Hamiltonian can be separated into parabolic coordinates. The solutions in spherical coordinates are connected to the solutions in parabolic coordinates at a distance larger than the core radius, but at a distance small enough that the electric ield has not distorted the electron motion; this requirement can always be satisied or the small experimental electric ields. The parabolic coordinates are connected to spherical coordinates through rz, rz, arctany/x, 11 where 0 and 0. In terms o these coordinates, the wave unction may be written as a superimposition o separable unctions Em,, 1 2 Em Em e im, Em,, 1 2 Emg Em e im, where and g are the regular and irregular solutions o d 2 y d 2 E F 2 and are solutions o solution o d 2 d 2 E 2 2 F m y0, 13 8 m , 14 with m the azimuthal quantum number and the separation constant. The dierential equations were written in this orm in order to draw attention to generic properties o these unctions. The terms in the square brackets play the role o a squared wave number or equivalently a squared momentum, and the terms in parentheses play the role o a potential. As goes to ininity the potential goes to positive ininity, which means there is only bounded motion in this direction. There is an ininite number o bound states, and as the number o nodes in this direction increases or a ixed E), monotonically increases. As goes to ininity the potential goes to minus ininity, which means that at every energy there is a continuum wave solution. However, or energies E2(1)F, the electron must tunnel through a potential barrier to go rom regions near the core to large distances. Since the direction or escape is in the direction, we will think o motion in this direction as longitudinal motion, and motion in the direction as transverse motion. As the amount o energy in transverse motion increases i.e., as n increases there is less energy available to surmount the barrier and escape. At a ixed energy 2(1)F will become larger as n increases, and eventually the electron will be orced to tunnel to escape. When 1, the potential in Eq. 14 is purely repulsive, and an electron would need to tunnel enormous distances to reach the core region. Channels with 1 play no role in the dynamics discussed in this paper. Following Re. 31, the unctions in Eqs. 13 and 14 are evaluated using WKB-type approximations. This is necessary because o the large number o energies that are needed in the construction o the wave packet. As a technical detail, all o the integrals were calculated numerically using Chebyshev quadrature with less than 40 quadrature points. For the sake o numerical stability we normalized the Em () and g Em () unctions, so that or small they oscillate 90 out o phase and are energy normalized. The key point o analysis is the connection between the Em and Em unctions o Eq. 12 to the unctions in spherical coordinates near the core El m El (r)y l m and El m g El (r)y l m where El and g El are the energy normalized radial Coulomb unctions o Re. 35 and the connection o the small- behavior to the unctions at asymptotically large. Note that in this section the symbols,, and mean dierent things depending on their subscripts; El m subscripts mean simple unctions in spherical coordinates, and Em subscripts mean simple unctions in parabolic coordinates. An atom that has quantum deects l has wave unctions near the nucleus El m El m El m tan( l ), when r is greater than the size o the core, r c. The connection between the unctions in spherical coordinates to those in parabolic coordinates is accomplished through the local rame transormations 31,32 El m 1/U 0 l Em, 15 El m 0 Em U l, 16 where U 0 is the transormation matrix Eqs. 17, 21, and 62 o Re. 31b. The transormation matrix is real and depends on E,l,, m, and d/dn 1, although we have only indicated the and l dependence. The U 0 matrix has a slow energy dependence compared to the U matrix that was used to calculate the cross section in Re. 31. We deine a standing wave, coupled-channel solution o the Schrödinger equation near the core to be

5 282 F. ROBICHEAUX AND J. SHAW 56 Em U 0 l El m Em l E mk, 17 where the K is the parabolic K matrix that couples the parabolic channels. The coupling arises rom the phase shits due to the core K l 0 U l U 0 l tan l. 18 Thereore, it is only the l 0 that cause the scattering between parabolic channels. The connection between the small- and large- orms o the Em unction is accomplished through a WKB-type approximation. The unctions in have the asymptotic orm given by Eqs. 44 in Re. 31b. The asymptotic orm is needed in order to obtain the lux into the detector. To this point, we have exactly ollowed Re. 31 or describing alkali-metal atoms in static ields. Now we need to slightly modiy this approach in order to cast all parameters in a orm that will simpliy the description o wave packets at asymptotically large distances rom the atom. To this end we deine new sets o parameters. I we use the deinitions R R exp i 4, S S exp i 4, 19 the wave unction that only has outgoing waves in each channel is obtained by Em 1 2i E mr S K 1 E m E m S 20 where R and S are diagonal matrices with the elements given by Eq. 19. The Eq. 49 o Re. 31a, Eq. A9 o Re. 31b, R and S Appendix A o Re. 31b have convenient deinitions in terms o WKB integrals. R and S are large when the electron must tunnel through the barrier except when the WKB integral k()d between the smallest two turning points equals an integer times or S or an integer plus 1 2 times or R. The matrix S is the nonunitary S matrix coupling the parabolic channels that is deined by the matrix equation S R * S *K R S K 1, and the asymptotic traveling wave solutions are given by 21 Em expim Emexpi k kd. 22 Using Eqs. 17 and 20, we can ind an expression or the dipole matrix element in terms o parameters that can be obtained using WKB expressions Em ˆ r I l R *KS * 1 0 U l D l m, 23 where D l m El m ˆ r I is the dipole matrix element coupling initial and inal states when there is zero ield. For the systems studied in this paper, the initial state is an s state, so we may take D l m l 1, with the only inaccuracy being the overall size o the cross section or the amount o wave unction excited. None o the time or energy dependences are aected by this choice. The overlap matrix in Eq. 6 may be obtained rom the orm o in Eq. 20 to be O 1 2 S S. 24 Simple substitution o Eq. 21 into Eq. 24 gives a orm or the inverse o this matrix that is numerically unstable when there are several strongly closed channels because R and S are very large which gives a nearly singular matrix. I we perorm several matrix manipulations and use the identity R S sin 1, we obtain the stable orm O 1 D i exp i 4 1iR 1 KW T 1WKR 2 KW T 1 W U 0 1, 25 where WR 1 (1KR 2 cot) 1, and W T is the transpose o W. V. CLOSED-ORBIT THEORY AND CLASSICAL TRAJECTORY CALCULATIONS A. Classical recurrences The wave-packet description in the previous sections can be related to the classical dynamics o the electron by constructing the semiclassical approximation to the overlap integral in Eq. 9. Let us review this briely. I a laser excites outgoing waves E near the zero ield threshold, then those waves go out ar rom the alkali core, bohr radii. Those portions o the outgoing wave that are turned around in the combined Coulomb and electric ield to return to the atomic core ollow the paths o the closed classical orbits o the system. These waves return to the source and overlap with it, contributing to the matrix element ˆ r I E in Eq. 9. The matrix element in Eq. 9 becomes a coherent sum o overlap integrals due to waves associated with distinct classical orbits labeled by (n,k), i.e., the nth return o the kth orbit. Each orbit gives a sinusoidal modulation in the oscillator strength with an energy wavelength n k h/t n k, where T n k is the return time o the closed orbit. Closed-orbit theory give ormulas or the amplitudes and phases o these modulations The sum o these modulations add up, in principle, to the peaks seen in the photoabsorption spectrum. It is worth emphasizing here that the closed orbits or an alkali-metal atom and hydrogen are the same, since the dynamics outside the core is controlled by the external ield

6 56 CALCULATED ELECTRON DYNAMICS IN AN ELECTRIC FIELD 283 and a 1/rattractive potential. The only dierences are in the handling o the outgoing and returning waves near the core 37. To determine the amplitude o the modulations in the spectrum rom an orbit, you need to know the shape o the outgoing wave Y( i ) created by the laser Eq in Re. 37. This is simple i the initial state is an s state since the dipole operator o the laser can only populate p states. The laser polarization selects whether an outgoing m0 p wave or an outgoing m1 p wave is excited. For the rubidium calculations in Sec. VI, the outgoing wave at t0 isl1, m1, so a closed orbit going out at an initial angle i k,n is weighted by sin( i k,n ). Orbits going out near the electric-ield axis i 0 are suppressed, while those going out near i 90 o are emphasized. This gives the initial amplitude o the outgoing wave near the initial angle o the outgoing classical closed orbit. Starting on an initial surace 10r o 100 bohr, we use a semiclassical approximation to the Green s unction, G sc (r,r 0 )A(r,r o )e i(r,r o ), to propagate the waves along the classical paths connecting r o (r o, i k,n,0) and r(r,,) 39. The phase o this wave unction depends on the classical action along the path S k n, and the Maslov index k n, which counts the number o caustics and ocii encountered on the orbit 39,40. The semiclassical amplitude A(r,r o ) depends on the divergence rate o the neighboring trajectories. For closed orbits the amplitude A(r,r o )is related to the stability properties o the closed orbit 41. For closed orbits, we connect the semiclassical returning wave to the partial wave expansion, pw (r,), or a zeroenergy wave coming in rom ininity at a particular inal angle k,n Eq. 7.5 in Re. 37. The ratio o the semiclassical returning wave to the incoming part o an incoming zero-energy scattering wave deines a complex matching constant N n k, which multiplies pw (r,) or each closed orbit. N n k contains the amplitude and phase o the returning wave or the nth return o the kth closed orbit. An expression or this matching constant when k,n i 0 is given as Eq in Re. 37. The phase o the returning wave is set by the action and Maslov indices calculated rom the origin to the origin on an orbit. The amplitude reduces to terms involving the derivative o k,n with respect to k i evaluated on a boundary r o and is proportional to Y( k,n i ). Thereore the returning part o the wave E in the semiclassical approximation is ret,n E,k r n pw Nk k,n r,. 26 This lets us deine the recurrence integral or the nth return o the kth orbit as R n k r I ret,n E,k 4 2 Ỹ k,n n N k 27 since the overlap o r I with pw (r,) see Eq. 7.7 in Re. 37, is just 2 3/2 Ỹ( ) the Ỹ here is the unexpected conjugate 37 o Y evaluated at the inal angle o the incoming wave. R k n gives the amplitude and phase o the terms in the oscillator strength due to each closed hydrogenic orbit. We will show in Sec. V B that writing the recurrences in terms o the recurrence integral allows us to simpliy the scattering calculations. Figure 5 shows the closed classical orbits relevant to the study o Rb. We will use Eq. 27 in Sec. VI when looking at classical recurrence time versus the time o lux ejection rom the region o the core. B. Semiclassical core scattering The ormulas above partially account or the non- Coulombic ield within the core. For example the angular unctions Y() contain the phase shits l which are related to the quantum deects. However, returning waves on the kth closed orbit create core-scattered outgoing waves on all the closed orbits, including the original orbit, and this is not accounted or in Eq. 27. Core scattering in eect constitutes a new source: the core-scattered outgoing waves can be turned around by the external ields, and return to the atom and produce a whole new set o recurrences. In an electric ield, the core-scattered outgoing waves can also go out in directions that lead to ionizing trajectories or example, see Fig. 5c 33. This is the semiclassical analog to the channel coupling that leads to ionization in the quantum calculations. To obtain the quantitative description o the scattering, it helps to deine new notation. The core-scattered wave can be extracted rom the partialwave expansion pw (r,), Eqs. 7.14a and 7.14b in Re. 37, so the core-scattered wave created by the n 1 return o n the k 1 hydrogenic orbit is N 1 k1 times the core-scattered part o pw (r,). I we semiclassically propagate that part o the core-scattered wave that goes out in the direction o the k 2 closed orbit, and take the overlap ater n 2 cycles on the new closed orbit, then we obtain a combination-recurrence integral given by n 1 T n R 2 n 1 n k2 k k2 R 2 1 k 1 k2 k Y 2 k i Ỹ 1,n 1 R k 1 n 1, n 2 n 1 28 where the Y s in the denominator o R k2 cancel corresponding actors in the R s or each orbit. The angular de- k 1 n pendence o T 1 k2 k is proportional to the usual T matrix; we 1 use the deinition n T 1 i k2 k e 2i k l 1Y 1 4 l m 2 k i,0y l * m 1,n 1,0, l m 29 where m is the azimuthal quantum number appropriate to the laser polarization. The combination-recurrence integral is proportional to the recurrence integrals or both the irst and n second orbits, and T 1 k2 k constitutes an eective T matrix 1 representing the scattering by the core rom one orbit to another. By splitting the scattering expression into the recurrence integrals or each closed orbit, and a coupling term which depends only on the incoming and outgoing angles, we can simpliy the multiple-scattering expansions derived

7 284 F. ROBICHEAUX AND J. SHAW 56 in Re. 42, and generalize to include outgoing angles that lead to ionizing trajectories in the electric ield. Equation 29 was derived by taking the outgoing angle o k the scattered wave,, to be the initial angle 2 i o a new closed orbit. Dropping this restriction gives an expression or the scattered wave which is proportional to the recurrence integral o the original closed orbit times a scattering amplitude or scattering rom a closed orbit (n 1,k 1 ) into the direction i. As we will see in this experiment with rubidium, i the recurrences described by Eq. 27 are not resolved and interere destructively, then this predicts that the lux o electrons ejected rom the atom will be small. This connects the classical recurrences to the delayed lux ejection. VI. SPECIFIC CASES IN Rb In this section, we describe the knowledge gained rom the calculated dynamics o a Rb electron wave packet in a ield o 2 kv/cm. We have chosen this case because we can compare our calculated time-dependent lux to the measurements o Lankhuijzen and Noordam 13. We are only examining cases or which the electron can classically escape to the detector, i.e., E2F. Beore presenting the speciic cases, we will irst discuss some o the generic properties o a highly excited alkali-metal atom in a strong electric ield. A. Generic properties The electric ield that we will use in this is small compared to the atomic unit o ield strength; we will be utilizing F410 7 a.u. But the states that we examine will be highly excited, and Fz will not be a perturbative interaction. Certainly, since we will be interested in energies 0E2F, the electron dynamics will be strongly inluenced by the electric ield. This is the energy range that a classical electron can escape the atom and travel to the detector i the ield is on but not i the ield is o. A measure o the nonperturbative nature o this interaction is the size o the zero-ield basis used in the calculation o the E (r); 90 radial unctions per l and 60l channels. This is very large considering we are exciting n20 states which perturbatively can only mix in l up to 20. I the atom in the electric ield is hydrogen, all o the quantum deects in Eq. 18 are zero, and thus the couplings between the parabolic channels is zero. A wave packet on hydrogen in an E ield will consist o several independent waves oscillating in the dierent parabolic channels. For energies E2F, the electron can only escape by tunneling. I E2F, there will be several channels small n )in which the electron can escape the atom by going over the top o the barrier, and there will be several channels large n )in which the electron can only escape the atom by tunneling. Qualitatively, the more energy in the coordinate this energy increases with the number o nodes, n ) the less energy is available in the direction to escape over the barrier. As n increases, the tunneling rate decreases and the lietime increases. The behavior o a wave packet on H in the range 0E2F can be described as ollows: 1 The part o the wave packet with n small enough so the electron does not need to tunnel to escape the atom will have waves starting near the nucleus that travel over the barrier directly to the detector with nearly unit probability. 2 The part o the packet with n too large will have waves starting near the nucleus that travel to the barrier. Because they do not have enough energy to go over the barrier, most o the wave is relected back to small, while a small raction o the wave tunnels through the barrier and then travels to the detector. The part o the wave relected rom the barrier travels back to small, where it is completely relected back to large in the same parabolic channel. Thus the lux o electrons rom the atom will have a peak that comes rom electrons that directly escape the atom, and several later pulses that arise rom electrons getting close to the barrier and tunneling through. The electron lux gives a measure o inding the electron near enough to the barrier to tunnel through. I the atom in the electric ield is Li, Na, K, Rb, or Cs, some o the quantum deects in Eq. 18 are nonzero, and thus there is a coupling between the parabolic channels. Remember that this arises because the core electrons break the parabolic symmetry or the valence electron. All o the discussion or H in the previous paragraph remains true except or the coupling between the parabolic channels. This small dierence qualitatively changes the wave packet dynamics by giving a third mechanism or ejecting lux rom the atom. 3 Ater scattering rom the nucleus, some o the wave packet will be in channels with small enough n to directly escape over the barrier; this occurs in the alkali-metal atoms because the core breaks the parabolic symmetry scattering some o the electron wave rom channel to channel. Thus the lux o electrons rom the atom will have a direct peak that comes rom electrons that directly escape the atom, and later pulses that arise rom electrons getting close to the barrier and tunneling through, and later pulses that arise rom electrons scattering rom the core into channels or which the electron can directly escape the atom. Except or large-m states large m being deined to be such that l 1 or l m) the main mode o ejection o electrons rom later pulses arises rom mechanism 3 i.e., the electron is more likely to scatter rom the core into classically open channels than to tunnel through the barrier in the closed channels. For later pulses, the electron lux gives a measure o inding the electron near the core in an angular momentum state small enough to scatter into open channels. B. Results A comparison between the experimental and calculated electron luxes was presented in Re. 33 or a case o linear polarization. For this case we were able to obtain excellent agreement. Several interesting eects were noted in this paper. We were also able to relate the time-dependent lux to a single orbit because only one o the orbits had a large amplitude or excitation by a laser polarized in the static ield direction. The dierence between the classical period and the measured period could be explained by trimming the autoionizing states. In this section, we will present additional results comparing calculated and experimental time-dependent luxes or a case where the laser is polarized perpendicular to the static ield. This case has several interesting eatures, which we

8 56 CALCULATED ELECTRON DYNAMICS IN AN ELECTRIC FIELD 285 TABLE II. Classiication o autoionizing states in Fig. 1. n n n E (10 3 a.u FIG. 1. Solid line, proportional to the ininite resolution photoionization cross section as a unction o the electron s energy: F stat 2005 V/cm and m1 rom the homogeneous unction. Dotted line, same except using the inhomogeneous unction. Dashed line, proportional to F(E), the amplitude or inding a photon at each energy. will present. The experimental results were previously presented in Fig. 3 o Re. 13. All o the results in this paper are or the pulsed-laser excitation o Rb atoms in a static electric ield o a.u V/cm. The Rb atoms are initially in the 5s ground state which is unperturbed by the weak static ield. The laser excites two wave packets: one with azimuthal quantum number m1 and one with m1. These two packets do not interere with each other, since the detector averages over and contribute equivalent amounts to the measured lux in the detector. The lux into the detector is simply twice the lux rom the m1 packet. Thereore, all calculations are perormed only or m1 packets. This has the added advantage that the resulting 2 has an azimuthal symmetry, making it easier to display. Since the dipole coupling in Eq. 23 is only to l 1, the speciic value o this matrix element only provides an overall size but does not aect the dynamics. The only dynamical inormation that is necessary is the quantum deects. We have used the values , , , , and l 4 0. From these values it is clear that the main scattering between the parabolic channels occurs or l 2. The main requency o the laser pulse is such that the atom is excited to the autoionizing threshold in a kV/cm ield and the zeroield ionization threshold. In Fig. 1, we present the ininite resolution photoionization cross section versus binding energy. The solid and short dashed line is the cross section calculated using the homogeneous unction and using the inhomogeneous unctions as given in Eq. 9. It is evident rom this igure that the two methods are in very good agreement with each other. The long dashed curve is proportional to the amplitude or inding a photon at a requency to excite the atom to the energies shown on the graph. This unction was chosen to be proportional to exp(ee 0 ) 4 / 4, where E a.u. and a.u. There are basically only our states that are excited by the laser pulse. These states can be classiied in terms o parabolic quantum numbers using the semiclassical wave unctions. Beore giving the classiications, we must stress that some o these states are strongly mixed and only the dominant contribution is given. The classiication o all o the states in Fig. 1 is given in Table II. Remember that n counts the number o nodes in the direction, and n counts the number o nodes in the direction and nn n m1. When n n, the electron is localized on the up-ield side o the atom. When n n, the electron s motion is more nearly perpendicular to the ield direction; i.e., the electron is localized to z0. Note that these states belong to dierent n maniolds. The main two states that are excited are the 12,7 and 17,1 states. We should expect that the main periodicity in the timedependent lux should be 2/E, where E is the energy dierence o these two states. This E is much smaller than the E that arises rom states o the same n. Thereore, we should see peaks in the lux with a larger spacing in time than would be expected rom simple arguments based on energy splittings within an n maniold. Inspection o Fig. 1 leads us to expect that a laser pulse as sketched in Fig. 1 will produce a wave packet that has a periodicity in time given by 2/E a.u ps. We expect the timedependent lux to consist o relatively ew pulses because the autoionizing states are very broad, and hence decay quickly. However, there is some inormation that cannot be deduced. The most striking example is the relative heights o the electron pulses entering the detector. Will the electron pulse heights decrease monotonically with time, or will a more complex time dependence emerge? In Re. 13, the recurrence spectrum or this system was measured. The recurrence spectrum measures (0)(t), and may be obtained rom the Fourier transorm o the requencydependent photoionization cross section. Because the electrons can only scatter down-ield and travel to the detector i they return to the nucleus, the plausible expectation is that detecting the time-dependent electron lux with an electron streak camera will simply reproduce the recurrence spectrum which can be obtained quite simply rom Fig. 1. Lankhuijzen and Noordam 13 showed this expectation is wrong, and that interesting inormation is obtained with the streak camera that cannot be obtained in the energy domain. In Fig. 2, we present the time-dependent lux as measured in Re. 13 solid line and calculated using the homogeneous unction rom Eq. 7. This igure shows some o the expected eatures that can be obtained in the energy domain. The number and spacing o the peaks is as expected. Figure 2 also shows an interesting and unexpected eature. The heights o the peaks are very irregular with the second peak

9 286 F. ROBICHEAUX AND J. SHAW 56 FIG. 2. The relative time-dependent lux using the pulse and ield rom Fig. 1. Solid line, experiment; dashed line, calculation using the homogeneous unction. The time axes or all curves have been shited so the irst peak is at t0. being much higher than the irst; in the recurrence spectrum, the irst peak is the largest. The irst peak results rom the electrons that are excited directly down-ield. The second peak results rom electrons that are initially excited into a region o phase space that gives bounded motion. But ater 12 ps, the electron returns to the nucleus where it can be scattered into a dierent parabolic channel and travel to the detector. The third peak arises rom electrons that were not scattered down-ield during the irst scattering with the core. We have not included the calculation in Fig. 2 using the inhomogeneous unction rom Eq. 4. The two calculated luxes are very close to each other; the slight dierences could be explained as arising rom slight inaccuracies in the dierent numerical methods. This means the electron waves disperse very little in traveling rom 2000 a.u. rom the core into the detector which is a macroscopic distance rom the atom. The time dependence o the electron lux entering the detector is an extremely accurate measure o the time dependence o the electrons being ejected rom the atom. This act is vitally important or the interpretation o the measurements. The reason or this lack o dispersion is that the electron is quickly accelerated in the electric ield; the spread in the wave packet goes like zte/v, which goes to a constant at large t since v is increasing linearly with t in the electric ield. The comparison o our calculations with experiment provides us with some inormation about the dynamics. We have gained inormation about the quality o WKB approximations, about the dispersion o the electron wave, and about the lack o importance o the spin-orbit interaction we do not include the spin-orbit interaction in our calculation. There is urther inormation that may be obtained because we can calculate the ull wave unction. One aspect o our understanding may be tested using the WKB wave unctions. This involves the expectation that the electron only escapes ater it scatters rom the core electrons. Equations 17 and 20 together give the transormation rom the El m unctions to the Em unctions. We can reverse the transormation to calculate the relative probability or inding the electron in a particular l wave within a FIG. 3. The relative probability or inding the electron near the nucleus in a p wave solid line or a d wave dashed line. relatively small distance e.g., 50 a.u. rom the nucleus. In Fig. 3, we plot the p-wave solid line and d-wave dashed line relative probabilities, written as A l (t) 2, as a unction o time. The p and d waves are the only m1 waves that have quantum deects substantially dierent rom 0. For the electron to scatter down-ield, it must return to the core as a p- ord-wave. At t0, there is a large p-wave component relecting the initial excitation. At t6 ps and 20 ps, the p- and d-wave probabilities have become very small, and or t13 ps and 26 ps the p- and d-wave probabilities become large. It is an interesting eature that the irst return 13 ps is dominated by d waves, and the second return 25 ps is dominated by p waves. This matches Fig. 2 in every respect. However, there is not complete agreement between the two igures because near t41 ps and 54 ps, there are large recurrences o p- and d-wave characters. Actually these recurrences extend to very long times 67 ps, 80 ps, 92 ps, etc.. This behavior must come rom the beating o the 9,11 and 7,14 states; these are very sharp, and thus have a long lietime. This does not invalidate our argument about the mechanism or electrons leaving the atom. Electrons must return to the core in low-l waves in order to scatter rom one parabolic channel to another. In particular, the electrons must return to the core in order to scatter in to classically open channels and travel to the detector. But just because the electron wave returns to the core does not mean it will scatter down ield. In Fig. 3, the relative proportion o p and d waves returning to the core is similar at 41 and 54 ps. The p and d waves do individually scatter rom the core down-ield. But the p and d waves are coherent, and the superposition o the p wave scattered down-ield and the d wave scattered down-ield destructively interere giving only little lux into the detector. I we could change the phase o the p or d wave between 41 and 54 ps, we would obtain a large lux o electrons into the detector. In Figs. 4, we present the contour plot o (,z,t) 2 or several dierent times. The amount o wave unction on the atom decreases with time so the peaks are rescaled at each t. There are a number o striking eatures o this igure. For example, we can clearly see the times when there is a large amount o lux o electrons leaving the atom. These igures

10 56 CALCULATED ELECTRON DYNAMICS IN AN ELECTRIC FIELD FIG. 4. Contour plot o r u c ( r,z,t) u 2 at times ~a! 0 ps, ~b! 6 ps, and ~c! 13 ps. show quite a complicated spatial dependence o the wave packet. As an aid in the interpretation o this packet, we perormed calculations o classical trajectories at this energy and ield strength that start at the nucleus. In Fig. 5, we present ive closed-orbit trajectories and the sequence o trajectories that directly escape the atom. A comparison o the general eatures between Figs. 4 and 5 sheds some light on the quantum processes. As in the lux going down-ield does not cover all o the classically allowed region but is curiously conined. This coninement arises rom the act that the electrons traveling down-ield must have originated rom a scattering event near the nucleus. The quantum lux only covers the region o space which a classical electron could access i it started near the core. This eect is clear in Fig. 5~c!, where the vast majority o the classical trajectories reach a maximum in r o 1000 a.u. near z52750 a.u., and then decrease in r as z decreases. Compare this with Fig. 4~c! to see the similarity. Some o the other eatures o the wave packet may be related to eatures o the closed-orbit trajectories. At t56 ps, there is a large amount o wave unction down-ield that is not traveling to the detector. The nodal structure indicates the main momentum component is in the r direction. In Fig. 5, we present some o the closed orbits at this energy. Notice that the simplest closed bold line in Fig. 5~c!# spends most o its time at negative z, and goes through the antinodal structure o the wave unctions. 287 FIG. 5. ~a! Two closed-orbit, classical trajectories that return to the nucleus ater.6 ps. ~b! Two closed-orbit, classical trajectories that return to the nucleus ater.6 ps. ~c! Bold line is a classical trajectory that returns to the nucleus ater.6 ps. The thin line trajectories are those that leave the nucleus and travel down-ield into the detector. The connection between classical orbits and the quantum wave packets is somewhat more tenuous than the case discussed in There it was clear that only one orbit was important. Here all o the orbits that seem important have return times roughly a actor o 2 smaller than the quantum period. An interpretation o this system might invoke intererence. We use Eq. ~27! to calculate the semiclassical intererence between returning orbits. Speciically, the two orbits in Fig. 5~a! have roughly the same return time,.6 ps, but return to the core out o phase with each other by 0.82p giving destructive intererence. The two orbits in Fig. 5~b! have roughly the same return time,.6 ps, but return to the core out o phase with each other by 0.84p. Upon their second return to the core the orbits are now out o phase by.2 p, giving constructive intererence. In the classical calculation, there are no closed orbits until the clustering o ive orbits near 6 ps that are shown in Fig. 5; then there are no closed orbits until a clustering o nine orbits near 12 ps. These nine orbits are the second return o the ive orbits in Fig. 5 and our new orbits returning or the irst time. This suggests that the period or this system is really near 6 ps, but an intererence eect suppresses the wave packet near the core at the irst return, and thus suppresses the scattering. An examination o the quantum energy levels in Table II supports this conjecture. I we use energy dier-