Femtosecond time-resolved photoelectron spectroscopy with high frequency probe pulses

10 January 2002

Chemical Physics Letters 351 (2002) 275–280 www.elsevier.com/locate/cplett

Femtosecond time-resolved photoelectron spectroscopy with high frequency probe pulses M. Erdmann, Z. Shen, V. Engel

*

Institut f€ur Physikalische Chemie, Universit€at W€urzburg, Am Hubland, D-97074 W€urzburg, Germany Received 19 October 2001; in final form 8 November 2001

Abstract The interaction of an intense 2 eV femtosecond pulse with the Na2 molecule prepares vibrational wavepackets in several electronic states. Subsequent ionization with a time-delayed probe pulse produces photoelectron kinetic energy distributions which depend on the pump–probe delay and reflect the vibrational dynamics in the various electronic states. We compare the resulting spectra with others obtained by direct one-photon ionization and show that in the latter case the contributions from different electronic states can be separated. Ó 2002 Elsevier Science B.V. All rights reserved.

1. Introduction Time-resolved spectroscopy using femtosecond pulses is able to shed light on intra- and intermolecular dynamical processes [1–4]. In a pump– probe experiment a first ultrashort pulse prepares a non-stationary state of the system under investigation. The time-delayed interaction with a second pulse then yields the desired dynamical information. A signal, detected as a function of the pump–probe delay, can consist of e.g., fluorescence, transient absorption or non-linear coherent emission. Yet another possibility is to measure ion yields or photoelectron spectra. In the latter case, the observable is the number of ejected electrons

*

Corresponding author. Fax: +49-931-888-6362. E-mail address: [email protected] (V. Engel).

detected as a function of their kinetic energy and the delay-time. The idea that these time-resolved photoelectron spectra reflect changes of the nuclear probability density goes back to the theoretical work of Seel and Domcke [5,6]. Various femtosecond experiments employing this idea have been performed up to date [7–17]. The connection between changes of the nuclear probability density and those observed in the electron spectra is most directly demonstrated in diatomic molecules. In the latter case, the potential energy depends on a single bond distance and it is possible to establish a one-to-one correspondence [18]. This was illustrated theoretically [19,20] as well as experimentally [8,21,22], using the sodium dimer as a prototype system. In the experiments of Baumert and co-workers, 2 eV (620 nm) pump and probe pulses were employed. As is illustrated in Fig. 1, such pulses are able to induce resonant transitions between three electronic states (0), (1),

0009-2614/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 1 ) 0 1 3 7 4 - 4

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M. Erdmann et al. / Chemical Physics Letters 351 (2002) 275–280

(a)

are separated by about 2 eV. As it will be illustrated in Section 3, this allows for a separate mapping of the vibrational dynamics in the various states. The Letter is organized as follows: Section 2 contains a description of the theoretical model to calculate the photoelectron spectra. The numerical results and conclusion are given in Section 3.

(b)

E

80000

ω2

V [1/cm]

60000

(I) 40000

(I) (2)

(2)

(1)

(1)

2. Theory and model

20000

(0)

(0) 0

2

4

6

8 –10

R [10

4

6

8

m]

Fig. 1. Photoionization of the sodium dimer originating in different neutral ionic states ((0), (1), (2)), yielding photoelectrons of energy E and ionic molecules in the ground electronic state (I). Panel (a): a one-color experiment, where x1 ¼ x2 ¼ 2 eV is illustrated. Here, it is not possible to identify electrons produced by a one-, two- or three-photon process. Panel (b): photoionization with a high frequency probe pulse. The ionization processes starting from the neutral electronic states produce photoelectrons with different kinetic energies E as is suggested by the varying length of the dashed arrows. 1 þ 1 (2) (X1 Rþ g ; A Ru ; 2 Pg ) of the neutral molecule. As a consequence, vibrational wavepackets are prepared in the pump process, moving with different characteristic periods. The detected photoelectron spectra show clear features of the 1 wavepacket motion in the A1 Rþ u and 2 Pg states. The electrons originate from two ionization pathways to the ionic state (I): a one-photon pump/ two-photon probe transition and a two-photon pump/one-photon probe transition. Whereas the origin of the temporal variations of the photoelectron spectrum is clear from the excitation scheme (see Fig. 1), it is, in general, difficult to separate the various contributions. The purpose of the present Letter is to demonstrate that, by employing probe pulses with high energy photons, the vibrational dynamics in different electronic states can be well characterized. This idea is illustrated in the right-hand panel of Fig. 1 for a 2 eV pump and 6 eV (206 nm) probe pulse. As is indicated in the figure, photoionization from the three neutral electronic states yield photoelectron spectra which

We use a model of the Na2 molecule which was employed before to simulate pump–probe ionization measurements, see [23] and references therein. The potential energy curves of the relevant electronic states are displayed in Fig. 1. Thus, we in1 þ clude the neutral electronic states X1 Rþ g , A Ru , 1 2 þ 2 Pg and the ionic ground state Rg which will, in what follows, be assigned as (0), (1), (2) and (I). Here we concentrate on the vibrational motion and do not include rotations which take place on a longer time scale. In the case of a strong pump pulse a perturbative treatment of the excitation process is not valid and it is necessary to integrate the time-dependent Schr€ odinger equation for the nuclear motion in the coupled electronic states (atomic units are employed) 0 10 1 H0 W10 ðtÞ 0 w0 ðR; tÞ @ W01 ðtÞ H1 W12 ðtÞ A@ w1 ðR; tÞ A 0 W12 ðtÞ w2 ðR; tÞ H2 0 1 w0 ðR; tÞ o ¼ i @ w1 ðR; tÞ A: ð1Þ ot w2 ðR; tÞ Here the field–molecule interaction is of the form Wnm ðtÞ ¼ lnm f ðtÞ cosðx1 tÞ ð2Þ and Hn is the nuclear Hamiltonian in state ðnÞ. The transition dipole moment between two states ðnÞ and ðmÞ is denoted as lnm and the pulse is characterized by an envelope function f ðtÞ and a frequency x1 . In the probe step, the interaction with a timedelayed pulse produces molecular ions and photoelectrons. We treat the ionization step using

M. Erdmann et al. / Chemical Physics Letters 351 (2002) 275–280

time-dependent perturbation theory. In what follows we distinguish two scenarios: in the first case the probe pulse frequency x2 equals the pump pulse frequency x1 . This corresponds to the experimental conditions as used in the pump–probe experiments on Na2 [8,24,25]. The ionic state can be populated by a three-photon process from state j0i, a two-photon transition from state j1i and also by a one-photon transition from state j2i. Accordingly, the final state nuclear wavefunction corresponding to the ejection of a photoelectron with energy E can be written as the coherent sum wE ðR; tÞ ¼

3 X

ðnÞ

wE ðR; tÞ:

ð3Þ

1 Wnm ðtÞ ¼  lnm f2 ðt  sÞeix2 ðtsÞ : 2



Z

1 t00

dt000 U1 ðt00  t000 ÞW10 ðt00 Þw0 ðR; t00 Þ:

1

ð4Þ A two-photon transition from state (1) yields  2 Z t 1 ð2Þ wE ðR; tÞ ¼ dt0 UI ðt  t0 ÞWI2 ðE; t0 Þ i 1 Z t0

dt00 U2 ðt0  t00 ÞW21 ðt00 Þw1 ðR; t0 Þ: 1

ð5Þ Finally, the one-photon ionization results in the wavefunction Z 1 t ð1Þ wE ðR; tÞ ¼ dt0 UI ðt  t0 ÞWI2 ðE; t0 Þw2 ðR; t0 Þ: i 1 ð6Þ Here Uk ðtÞ is the propagator in the electronic state ðkÞ and the nuclear wavepacket in this state is denoted as wk ðR; tÞ. The coupling terms are of the form 1 ð7Þ WI2 ðE; tÞ ¼  lI2 f2 ðt  sÞeiðx2 EÞðtsÞ ; 2

ð8Þ

The probe pulse is characterized by the shape function f2 ðtÞ and frequency x2 . Furthermore, lI2 denotes the dipole moment for the neutral-ionic transition. The above given expressions for the wavefunctions depend on the photoelectron energy E. In deriving them it was assumed that the photoelectron decouples from the nuclei and core electrons [5,6]. The photoelectron spectrum may now be calculated as PE ðsÞ ¼ hwE ðt1 ÞjwE ðt1 Þi

n¼1

The third-order wavefunction initiating in the (0)-state is of the form  3 Z t 1 ð3Þ dt0 UI ðt  t0 ÞWI2 ðE; t0 Þ wE ðR; tÞ ¼ i 1 Z t0

dt00 U2 ðt0  t00 ÞW21 ðt00 Þ

277

¼

3 X

ðnÞ

ðnÞ

hwE ðt1 ÞjwE ðt1 Þi

n¼1 ð3Þ

ð2Þ

þ 2RefhwE ðt1 ÞjwE ðt1 Þi ð3Þ

ð1Þ

ð2Þ

ð1Þ

þ hwE ðt1 ÞjwE ðt1 Þi þ hwE ðt1 ÞjwE ðt1 Þig:

ð9Þ

Here the brackets indicate integration over the nuclear coordinate, t1 is a time after the probe interaction and Re denotes the real part. Note that the interference terms appearing in Eq. (9) vary with factors as cosðjx2 sÞ, ðj ¼ 1; 2Þ. Although these oscillations have been detected experimentally [26,27], we here assume that the obtained signals are averaged over an interval of at least Ds ¼ 2p=x2 , so that the interference terms are canceled. This average was performed in the earlier Na2 experiments [24,25]. As a consequence, the photoelectron spectrum consists of three terms where each term contains information about the vibrational dynamics in a different electronic state. It is important to note that the photoelectrons resulting from the three ionization pathways fall in the same range of kinetic energies (see Fig. 1a). This means, that the nuclear dynamics in the electronic ground as well as in the two excited electronic states, as reflected in the signal, cannot be separated. We now turn to the second case, where the probe frequency is substantially larger than the pump frequency (Fig. 1b). As can be taken from the figure, ionization from the three neutral electronic states to the ionic ground state, produces

M. Erdmann et al. / Chemical Physics Letters 351 (2002) 275–280

photoelectrons with very different kinetic energies. If there is no overlap between electrons obtained from different ionization pathways the total photoelectron spectrum is of the form 3 D

E X

ðnÞ ðnÞ PE ðsÞ ¼ uE ðt1 Þ uE ðt1 Þ ; ð10Þ n¼1

where the nuclear wavefunctions now corresponds to an one-photon ionization processes initiating in the different neutral states: Z 1 t ðnÞ uE ðR; tÞ ¼ dt0 UI ðt  t0 ÞWIn ðE; t0 Þwn i 1

ðR; t0 Þ:

1

0.8

0.6

0.4

(0) 0.2

(2) 0

ð11Þ

The coupling term WIn is of the form (7) but contains the transition dipole-moment lIn between the neutral electronic state ðnÞ and the ionic ground state. In our numerical calculation we solve the timedependent Schr€ odinger equation on a grid [28]. In doing so, all transition dipole-moments are set constant and the envelope functions are of Gaussian form. For a more advanced treatment of the ionization process, see the work of McKoy and co-workers [29].

3. Results A 2 eV femtosecond pump pulse is resonant with the (0)–(1) and also with the (1)–(2) transition and thus is able to induce a population transfer between the three neutral states of Na2 . This can be taken from Fig. 2 which shows the populations Pn ðtÞ ¼ hwn ðtÞjwn ðtÞi in the neutral electronic states during and after the interaction of the molecule with a pump pulse having a width of 30 fs and an intensity of 1012 W=cm2 . For this intensity, the ground state is efficiently de-populated giving rise to an essential excitation of the higher states. The oscillations in the curves are Rabi-oscillations which are less pronounced for an excitation pulse with finite length and a system with nuclear dynamics [30] than in an continuously excited N-level system. Since the pump excitation prepares non-stationary vibrational states, the photoelectron spec-

(1)

Pn(t)

278

0

100

200

300

time [fs]

Fig. 2. Population changes in the neutral electronic states ðnÞ of Na2 upon femtosecond excitation with an intense 30 fs pulse.

tra will, in general, reflect this wavepacket dynamics. In Fig. 3 we show spectra which result from 2 eV probe excitation. Here the probe pulse intensity was set to a value of 1011 W=cm2 . This yields to a comparable contribution of the ionization pathways initiating in the electronic states (1) and (2) and a negligible contribution from the electronic ground state (0). As a consequence, the photoelectron spectrum contains two oscillating distributions. They reflect the vibrational wavepacket dynamics in state (1) and (2) with vibrational periods of  310 fs (state (1)) and  360 fs

0 PE(τ)

200 400 600

0.5 0.6 0.7 0.8 0.9 E [eV]

τ [fs]

800 1

1.1 1.2 1000

Fig. 3. Time-resolved photoelectron spectra obtained from a one color pump–probe experiment x1 ¼ x2 ¼ 2 eV. Two oscillating contributions can be distinguished, reflecting the vibrational motion in the neutral states (1) and (2).

M. Erdmann et al. / Chemical Physics Letters 351 (2002) 275–280

(state (2)), respectively. This identification is straightforward within the numerical simulation but is, in general, not unique. Let us now turn to the second excitation scheme where a probe pulse frequency much larger than that of the pump pulse is employed. In doing so, we set x2 ¼ 3x1 leading to the spectrum as defined in Eq. (10). Fig. 4 displays the calculated spectra. Here, three contributions are found which are well separated on the energy scale. They oscillate with different frequencies which are characteristic for the vibrational motion in the three neutral electronic states as employed in our model. The spectrum reflecting the vibrational motion in state (2) is identical to the respective contribution as contained in Fig. 3, with the only difference that it is shifted on the absolute energy scale. This is not true for the spectrum originating in state (1). There, a two-photon resonant transition to the final ionic state is only possible at times when the wavepacket w1 ðRÞ is located close to its inner classical turning point, so that no photoelectrons are obtained at other times (see Fig. 3). On the other hand, in the direct photoionization process no Franck–Condon window exists which restricts the times when an effective ionization can take place. The energy variation of the three components of the spectrum as shown in Fig. 4 depends on the steepness of the difference potentials DIn ðRÞ ¼ VI ðRÞ  Vn ðRÞ, where VI is the potential curve of the ionic ground state and Vn ðRÞ is the potential of a neutral state. Since in the n ¼ 2 case, this quantity is almost constant, only smaller variations are found in the spectrum (middle panel in Fig. 4). On the other hand, DI0 ðRÞ is comparably steep so that a large amplitude motion is observed in the lower panel of Fig. 4. Fig. 4 clearly demonstrates that it is indeed possible to separate the motion of the components of the total nuclear wavefunction in using high energy probe photons. If the second harmonic of the pump pulse carrier frequency is used for the probe pulse, the contribution from the electronic ground state would be altered. Since the ð2Þ ð0Þ transition is forbidden ionization could only proceed either non-resonantly of via another electronic state of appropriate symmetry [31].

279

0 PE(τ)

200 400 τ [fs]

600 4.7 4.75 4.8 4.85 800 4.9 4.95 5 5.05 1000 E [eV] 5.1

0 PE(τ)

200 400 τ [fs]

600 2.7 2.75 2.8 2.85 800 2.9 2.95 3 3.05 1000 E [eV] 3.1

0 PE(τ)

200 400

600 0.8 0.85 0.9 0.95 800 1 1.05 1.1 1.15 1000 E [eV] 1.2

τ [fs]

Fig. 4. Time-resolved photoelectron spectra obtained from 620 nm pump/206 nm probe femtosecond ionization. The spectra shown in the different panels correspond to direct ionization from state (0) (lower panel), (1) (middle panel) and (2) (upper panel). Note that the spectra clearly separate on the energy scale.

It has to be kept in mind that in our second scenario (high frequency probe) excited electronic states of an ion could be populated. This, in fact, is likely to happen in the Na2 molecule where several

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M. Erdmann et al. / Chemical Physics Letters 351 (2002) 275–280

excited ionic states could be accessed [32,33]. As a consequence overlapping bands will appear in the spectra which are due to excitation from different initial neutral to different final ionic states. Then, it is necessary to vary the probe pulse frequency in order to identify the various contributions. In conclusion, we have demonstrated that the vibrational dynamics taking place in different electronic states of a neutral molecule can be mapped using pump–probe photoelectron spectroscopy. In order to separate the contributions to the signal, an ionization pulse with a high frequency can be employed. The latter allows for a direct photoionization so that resonant transitions via an intermediate state which might obscure the underlying dynamics are excluded. Pulses with high photon energies are available now [34] and effort is directed into the development of X-ray femtosecond pulses [35] which will allow for the mapping of molecular processes involving the excitation of core electrons.

Acknowledgements This work was funded by the Deutsche Forschungsgemeinschaft within the Graduiertenkolleg ‘Electron Density’ and the SFB 347, C-5. We gratefully acknowledge support by the Fonds der Chemischen Industrie.

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