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Coherent multichannel nonadiabatic dynamics and parallel excitation pathways in the blue-violet absorption band of Rb2
Chemical Physics Letters 368 (2003) 202–208 www.elsevier.com/locate/cplett
Coherent multichannel nonadiabatic dynamics and parallel excitation pathways in the blue-violet absorption band of Rb2 Niklas Gador a, Bo Zhang a, Renee Andersson a, Pia Johansson b, Tony Hansson b,* a
Department of Physics, Royal Institute of Technology, AlbaNova University Center, SE-106 91 Stockholm, Sweden b Department of Physics, AlbaNova University Center, Stockholm University, SE-106 91 Stockholm, Sweden Received 8 May 2002; in final form 8 November 2002
Abstract The blue-violet absorption band in the Rb2 molecule at 430 nm is studied by ultrafast pump–probe spectroscopy and quantum dynamical calculations. We find that two electronic states are excited simultaneously and thereby the dynamics essentially proceeds in two independent channels. One channel exhibits vibrational motion in the D0 ð3Þ1 Rþ u shelf state while the other involves the Dð3Þ1 Pu state which is heavily perturbed by spin–orbit interactions. Moreover, we 3 directly observe the build-up of the wavefunction in the ð4Þ3 Rþ u state which together with ð1Þ Du are proposed to be the major perturbers. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction The Rb2 molecule recently has attracted considerable interest as the collision of two atoms is a pivotal process in ultracold rubidium gas and Bose–Einstein condensates. Extremely cold molecules can be produced in such environments [1] by, for example, photoassociation and stimulated Raman scattering, the application of which requires detailed knowledge of the molecular elec-
*
Corresponding author. Fax: +46-8-5537-8601. E-mail address: [email protected] (T. Hansson).
tronic states. The excited electronic states in Rb2 are frequently mixed by spin–orbit coupling and provide numerous instances of weak to intermediate electronically nonadiabatic processes. Rubidium molecules thus may serve as testing ground for detailed studies of nonadiabatic wavepacket dynamics, in particular intersystem crossings, in systems of reduced dimensionality and with multiple channel interactions, which is the main motivation of the present study. As a side benefit we also obtain new spectroscopic information on the Rb2 molecule. Our present work concerns ultrafast pump– probe spectroscopy of the Ôblue-violetÕ absorption band in Rb2 . This band extending from 425 to
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 2 ) 0 1 8 4 8 - 1
N. Gador et al. / Chemical Physics Letters 368 (2003) 202–208
450 nm was analysed by Tsi-Ze and San-Tsiang [2] who extracted the still most recent molecular constants of what is conventionally labelled the D state. Breford and Engelke [3] obtained by laserinduced fluorescence an excitation spectrum exhibiting a strong progression in the D state vibrational levels as well as predissociation. Below the energy corresponding to 429.5 nm weak predissociation was found while above that the molecule is strongly predissociated. The atomic products yield in the strong predissociation regime reflects the vibrational period of the D state and thus vibration and predissociation must occur on similar timescales. Recent ultrafast pump–probe spectroscopy by some of us [4] confirmed this qualitative picture and provided quantitative data on the fast decay rate. Probing wavepacket motion in the D state, we obtained pure vibration below the energy barrier for strong predissociation, which above that limit was superimposed on a decay-like feature with a time constant of 2–5 ps. The strong predissociation threshold corresponds to the 42 D þ 52 S dissociation limit. Consequently, Breford and Engelke suggested [3] the state causing the disappearance of the D1 Pu state correlating to this limit. We [4] proposed ð1Þ3 Du to be the responsible perturber interacting with the D state by a spin–orbit coupling element of 10 cm1 . We here extend our previous ultrafast pump– probe study by applying new probe channels and quantum dynamical calculations. This brings out two hitherto unobserved features. First, the ultrafast coherent build-up of a wavefunction in a bound triplet state connected to the D state. Second, the existence of a second absorption channel in the blue-violet band. The latter potentially provides a way to create an electronically entangled molecular state.
2. Experimental Briefly, the experiments were performed in a molecular beam set-up using femtosecond pump– probe spectroscopy with fluorescence detection. The cross-correlation time of the pump and probe pulses tuned to 425–435 nm ðkpu Þ and 927 nm ðkpr Þ, respectively, was measured in situ to be 180
203
fs (FWHM) by non-resonant 1 + 1-photon laser~ –X e transition in induced fluorescence on the A acetone. This also facilitated accurate zero delaytime determination. The two linearly polarised pulses at magic angle configuration were focused by f ¼ 50 cm spherical lenses into the vacuum chamber propagating at a small angle to each other. No significant effect on the shape of the recorded transients was found varying the two pulse intensities. The mildly cooled Rb2 beam was produced by unseeded expansion of rubidium vapour at 670 K through a /50 lm nozzle. The background pressure in the vacuum chamber during operation of the beam source was lower than 106 mbar. The light pulses crossed the molecular beam approximately 4 mm downstream of the nozzle and the induced fluorescence was collected at right angle to both the molecular and the light beams. Detection was done either through a low-resolution spectrometer (H20VIS, JobinYvon) equipped with a R928 photomultiplier tube (Hamamatsu) for obtaining emission spectra, or by spectral filtering (UG11, Schott) and a 1P28 photomultiplier tube (Hamamatsu) for the purpose of measuring pump–probe transients. The latter resulted in a spectral sensitivity in the 250– 400 nm range ðkdet Þ suitable to detect Rb 82 P, 72 P– 52 S fluorescence. For each point in a delay-time scan, Dt, the fluorescence intensity was measured by gated photon counting (SR400, Stanford Research Systems) averaging over 1000 cycles. The experimental data presented here was obtained by averaging 20 delay-time scans.
3. Results The to our discussion relevant potential energy curves are displayed in Fig. 1. These are essentially all the ungerade states in the energy range except for the ð1Þ1 Du state. The studies mentioned in the Introduction only reported absorption to the D1 Pu state. Kotnik-Karuza and Vidal [6] indicate, however, that some lines may arise from R to R transitions. Our pump–probe fluorescence spectrum in Fig. 2 is consistent with this notion. The molecular fluorescence band at around 495 nm, produced by the pump pulse alone, can only arise by transition
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Fig. 1. Potential energy, U, curves of Rb2 [5]. Note, the ð3Þ1 Rþ u curve has been shifted down by 250 cm1 . Arrows indicate radiative transitions involved in the pump–probe experiment. See text for definition of the labels I and II of the two probe transitions.
from some state other than D to the electronic ground state. Analysis of the Franck–Condon 1 þ factors [7] for the tentative ð3Þ1 Rþ u ! X Rg transition reveals suitable overlap in the correct wavelength range. Thus, we assign the 495 nm band to this transition which we will see shortly is consistent also with the pump–probe results. To simplify notation and following convention, we henceforth 0 denote the ð3Þ1 Rþ u state by D .
The pump–probe fluorescence spectrum contains several atomic lines that depend on the presence of both pulses. The main probe product is Rb(82 P) while the rest of the pump–probe correlated weak atomic transitions are likely due to cascade fluorescence from this state. The transients we get monitoring 82 P, 72 P ! 52 S fluorescence intensity, see Fig. 3a, are clearly molecular in origin. As the probe pulse is tuned to the blue side of the atomic 82 P 42 D transition, the primary molecular probe state correlates to the 82 P þ 52 S dissociation limit and does not present any substantial energy barrier to dissociation from the probe point as indicated schematically in Fig. 1. Looking closer at the transients in Fig. 3a we note first that the signal is delayed with respect to the instant of excitation by 600 fs. This is substantially longer than the 445 fs for half a vibrational period of the D state [4]. Hence, the signal stems from some state other than the D state. Second, there is a phase of step-like build-up of the signal with a step interval of 1 ps. A slow oscillation, 4.7 ps, is the third characteristic feature. Note the slight but consistent shortening of this period with decreasing pump wavelength.
4. Modelling and calculations Three states were invoked so far, D1 Pu ; D0 1 Rþ u, and ð1Þ3 Du , to explain the spectra and the fast predissociation of the D state. To understand the
Fig. 2. Pump (429 nm)–probe (927 nm) fluorescence spectrum for Dt ¼ 20 ps. The molecular bands arise from the pump pulse alone while the atomic lines except for 72 D ! 52 P (probe alone) and 52 P ! 52 S correlate to the presence of both pulses.
N. Gador et al. / Chemical Physics Letters 368 (2003) 202–208
Fig. 3. Pump–probe transients for several kpu as indicated in the figure. (a) Measured traces normalised to the same maximum amplitude and arbitrarily set off for clear presentation. The signal at Dt ¼ 0 ps is in each case essentially zero. (b) Corresponding calculated transients normalised to the experimental max. amplitudes.
appearance of the step-like structure in the present signal we need at least one more. Spiegelmann et al. [8] suggested ð4Þ3 Rþ u as responsible for the slow predissociation in the lower part of the D state. Including also this state we obtain a minimal and as we will show sufficient model to describe the gross quantum dynamics ensuing excitation of the Rb2 blue-violet absorption band. However, it should be kept in mind that it is likely that at least one more state, ð3Þ3 Pu , is involved [4], although, by the results obtained here, to a lesser extent.
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The interactions among the four states we take to be mediated solely by spin–orbit coupling. This yields two decoupled excitation channels each corresponding to one of the singlet D0 and D states. In Fig. 1 we labelled the two channels by their probe transitions as I and II, respectively. The channel I dynamics would consist in a simple vibrational motion in the D0 shelf state and from analysis of plausible difference potentials the probe transition is located close to the outer turning point. The D state, on the other hand, may couple to both 3 ð4Þ3 Rþ u and ð1Þ Du . The channel II dynamics is therefore quite complex. The probe point consistent with the observations, mainly the initial delay of the signal, is located at the outer turning point of the ð4Þ3 Rþ u state. The probe points in the two channels are thus well separated and seemingly have different upper states in the corresponding transitions. This means interference effects from possible entanglement of the two channels are improbable and the final signal may be obtained as the incoherent sum of the independent channel contributions. We choose to ignore any nuclear dynamics taking place during the periods of light pulse-molecule interaction. That is, the simulations start out with wavepackets halfwidth put in each channel that are of 0.1 A propagated in time by the split-operator method in conjunction with an analytical expansion scheme for the exponential of the potential operator [9]. The signal from a specific channel we then get from integrating the amplitude density of the wavefunction, projected onto the appropriate HundÕs case a electronic state, inside a square de and outwards, tection window; in channel I 11.5 A in channel II 7.0–7.4 A. We do not observe significant differences using (narrower) Gaussian detection windows instead. Finally, an optical potential is employed to impose absorbing boundary conditions for dissociating components of the wavefunction. Fig. 3b displays three of our calculated transients. Within the three traces all parameters are kept constant except for the total energy of the wavepacket. In the process we slightly adjusted the potential energy curves by Park et al. In channel I the D0 state was shifted down in energy by . This was 250 cm1 and outwards by 0.105 A
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necessary in order to reach above the shelf by a classical excitation from the rovibronic ground state and influenced mainly the modulation amplitude of the calculated signal from this channel. 3 In channel II the D, ð4Þ3 Rþ u , and ð1Þ Du states were and the all shifted inwards: the first two by 0.05 A . Again the first shift was remaining one by 0.13 A done to facilitate a classically allowed transition and the same shift was applied to ð4Þ3 Rþ u to preserve the relative position to the D state. The ð1Þ3 Du , finally, was shifted relative to the D state as to increase the Franck–Condon overlap between the two states, which was necessary to achieve strong enough coupling between these states assuming reasonable spin–orbit interaction strength [10] and that the 3 Du state is responsible for the observed fast predissociation of the D state [4]. Except for the last one these shifts were made essentially for computational convenience and may be indicative. They should at this stage not be considered significant, however, as the thermal distribution of the initial state has been ignored and therefore some of them may artificially compensate for this. The spin–orbit coupling matrix 1 elements for the D1 Pu ð4Þ3 Rþ u and D Pu 3 ð1Þ Du perturbations were both set to 20 cm1 . Finally, the two individual channel contributions were weighed as to reproduce faithfully at each wavelength the first main peak in the experiment which resulted in a channels I–II intensity ratio of about 1:100. A big part of this difference originates in the Franck–Condon overlaps for the excitation transitions, in accordance with the absorption spectrum being dominated by the D state [2,3]. The main remaining part probably comes from a low electronic transition moment for the detection step in channel I.
5. Discussion
Fig. 4. The separate channel contributions to the simulated signal and wavefunction probability density at Dt ¼ 0 ps (solid) and 1.65 ps (other), respectively, projected onto the respective HundÕs case a electronic states. Position and extension of probe windows are indicated by arrows and horizontal lines. Note, the individiual probability densities have been scaled for visibility.
The calculated signal is broken down into its two constituents in Fig. 4. It is clear that the main oscillation stems from channel I and is due to a wavepacket essentially undergoing a simple vibrational motion in the D0 state. This state is one of the in alkali molecules frequently encountered shelf states arising from perturbations involving
the ion-pair potential [5] which explains the extraordinarily long vibrational period. There is a barrier at the entrance of the shelf that reflects part of the wavepacket causing interference structures to appear well before the first round-trip has been completed. Note that the calculation reproduces
N. Gador et al. / Chemical Physics Letters 368 (2003) 202–208
well the slight shift in period of the main peaks with excitation wavelength. Channel II then contributes the step-like structure and decay of the D state population. The signal is obtained from the outer turning point of the ð4Þ3 Rþ u state and reflects the dynamics of the small part of the initial wavepacket that undergoes intersystem crossing. What is not directly seen in the experiment is the predissociated fraction of the wavefunction propagating along the ð1Þ3 Du potential curve. We see that the 3 Du channel not only causes the decay of the D state [4] but also suppresses otherwise larger peaks subsequent to the main peak in channel II. As a further figure of merit, in addition to the position of the steps this signal contribution also reflects the steps becoming more pronounced at shorter wavelengths. In general the calculations reproduce satisfactorily the characteristic features of the measurements. They underestimate, however, the constant background at longer delay times. The main reason for this we believe is our neglect of vibrational and rotational excitation in the electronic ground state. Turning to the magnitude of the spin–orbit coupling matrix elements applied in the calculations we note that they depend critically on the accuracy of the available potential energy curves and that they at this time should be considered as order of magnitude estimates only. Using the curves by Spiegelmann et al. [8], for instance, we [4] previously estimated from the observed decay rate of the D state the D1 Pu ð1Þ3u coupling to 10 cm1 , in good agreement with calculations [10]. The presently used curves by Park et al. [5], however, do not supply sufficient Franck–Condon overlap between the two states to sustain this mechanism. Hence the triplet state was shifted to smaller internuclear separation as to keep the spin–orbit coupling strength within reasonable limits, less than 20 cm1 . As stated above, it is an open issue whether this is a significant shift until more detailed calculations are available and, consequently, we can at present only provide qualitative arguments regarding the interstate spin–orbit couplings and their magnitudes. We ignore in the calculations some interstate couplings of which the most important but indirect
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ones involving the ð3Þ3 Pu state were briefly discussed above and in [4]. In addition to this the D1 Pu and D0 1 Rþ u states cross and may couple through the L-uncoupling mechanism [11]. We find, however, that the possible range of coupling strength for this interaction, less than 4 cm1 for a typical rotational quantum number of 100, is not large enough to account for any of the main observed features in the quantum dynamical calculations. Finally, we do not display in Fig. 1 the ð1Þ1 Du state but in a detailed treatment it should be included as well. A consequence of our results is that the main population of the D0 state must come from direct excitation from the electronic ground state. This implies we can identify D0 ð3Þ1 Rþ u as the upper state in the R–R lines observed in [6]. Moreover, the D and D0 states are excited simultaneously. Our two channels thus may be entangled which would add one degree of freedom to explore in quantum computation by wavepackets like those in [12]. We can now say that this situation of overlapping 1 ð3Þ1 Rþ u and ð3Þ Pu states prevails in all the heavier homonuclear alkali molecules [13–15]. In conclusion, we measured and simulated the molecular quantum dynamics ensuing excitation of the Rb2 blue-violet absorption band. Two electronic states, Dð3Þ1 Pu and D0 ð3Þ1 Rþ u , are excited simultaneously and the dynamics are described by the wavefunction propagating in two independent channels. The D0 channel exhibits essentially vibrational motion in a shelf state while the D state is heavily perturbed by spin–orbit interaction. We now directly observed the build-up of the wavefunction in the ð4Þ3 Rþ u state which together with the previously invoked [4] ð1Þ3 Du is responsible for the decay of the D state population. In a forthcoming Letter we plan to include details on the probe transition and transient anisotropy as well as more detailed calculations and comparison to results in [4].
Acknowledgements We thank the authors of [5] for kindly providing us with their original data files. This work was supported by the Swedish Research Council (VR).
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References [1] K. Burnett, P.S. Julienne, P.D. Lett, E. Tiesinga, C.J. Williams, Nature 416 (2002) 225. [2] N. Tsi-Ze, T. San-Tsiang, Phys. Rev. 52 (1937) 91. [3] E.J. Breford, F. Engelke, Chem. Phys. Lett. 75 (1980) 132. [4] B. Zhang, L.-E. Berg, T. Hansson, Chem. Phys. Lett. 325 (2000) 577, erratum 339 (2001) 438. [5] S.J. Park, S.W. Suh, Y.S. Lee, G.-H. Jeung, J. Mol. Spectrosc. 207 (2001) 129. [6] D. Kotnik-Karuza, C.R. Vidal, Chem. Phys. 40 (1979) 25. [7] R.J. LeRoy, LEVEL 7.4: A computer program for solving the radial Schr€ odinger equation for bound and quasibound levels, University of Waterloo Chemical Physics Report CP-642R3 , 2001.
[8] F. Spiegelmann, D. Pavolini, J.-P. Daudey, J. Phys. B: At. Mol. Opt. Phys. 22 (1988) 2465. [9] G. Peoux, M. Monerville, J.-P. Flament, J. Phys. B: At. Mol. Opt. Phys. 29 (1996) 6031. [10] D. Edvardsson, Ph.D. Thesis, Uppsala University, Uppsala, Sweden, 2000. [11] H. Nakamura, in: R.E. Wyatt, J.Z.H. Zhang (Eds.), Dynamics of Molecules and Chemical Reactions, Marcel Dekker, New York, 1996, p. 473. [12] J. Vala, Z. Amitay, B. Zhang, S.R. Leone, R. Kosloff, Available from, 2001. [13] C. Amiot, W. Demtr€ oder, C.R. Vidal, J. Chem. Phys. 88 (1988) 5265. [14] C. Amiot, J. Chem. Phys . 89 (1988) 3993. [15] W. Jastrzeßbski, P. Kowalczyk, Chem. Phys. Lett. 206 (1993) 69.
Coherent multichannel nonadiabatic dynamics and parallel excitation pathways in the blue-violet absorption band of Rb2 Niklas Gador a, Bo Zhang a, Renee Andersson a, Pia Johansson b, Tony Hansson b,* a
Department of Physics, Royal Institute of Technology, AlbaNova University Center, SE-106 91 Stockholm, Sweden b Department of Physics, AlbaNova University Center, Stockholm University, SE-106 91 Stockholm, Sweden Received 8 May 2002; in final form 8 November 2002
Abstract The blue-violet absorption band in the Rb2 molecule at 430 nm is studied by ultrafast pump–probe spectroscopy and quantum dynamical calculations. We find that two electronic states are excited simultaneously and thereby the dynamics essentially proceeds in two independent channels. One channel exhibits vibrational motion in the D0 ð3Þ1 Rþ u shelf state while the other involves the Dð3Þ1 Pu state which is heavily perturbed by spin–orbit interactions. Moreover, we 3 directly observe the build-up of the wavefunction in the ð4Þ3 Rþ u state which together with ð1Þ Du are proposed to be the major perturbers. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction The Rb2 molecule recently has attracted considerable interest as the collision of two atoms is a pivotal process in ultracold rubidium gas and Bose–Einstein condensates. Extremely cold molecules can be produced in such environments [1] by, for example, photoassociation and stimulated Raman scattering, the application of which requires detailed knowledge of the molecular elec-
*
Corresponding author. Fax: +46-8-5537-8601. E-mail address: [email protected] (T. Hansson).
tronic states. The excited electronic states in Rb2 are frequently mixed by spin–orbit coupling and provide numerous instances of weak to intermediate electronically nonadiabatic processes. Rubidium molecules thus may serve as testing ground for detailed studies of nonadiabatic wavepacket dynamics, in particular intersystem crossings, in systems of reduced dimensionality and with multiple channel interactions, which is the main motivation of the present study. As a side benefit we also obtain new spectroscopic information on the Rb2 molecule. Our present work concerns ultrafast pump– probe spectroscopy of the Ôblue-violetÕ absorption band in Rb2 . This band extending from 425 to
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 2 ) 0 1 8 4 8 - 1
N. Gador et al. / Chemical Physics Letters 368 (2003) 202–208
450 nm was analysed by Tsi-Ze and San-Tsiang [2] who extracted the still most recent molecular constants of what is conventionally labelled the D state. Breford and Engelke [3] obtained by laserinduced fluorescence an excitation spectrum exhibiting a strong progression in the D state vibrational levels as well as predissociation. Below the energy corresponding to 429.5 nm weak predissociation was found while above that the molecule is strongly predissociated. The atomic products yield in the strong predissociation regime reflects the vibrational period of the D state and thus vibration and predissociation must occur on similar timescales. Recent ultrafast pump–probe spectroscopy by some of us [4] confirmed this qualitative picture and provided quantitative data on the fast decay rate. Probing wavepacket motion in the D state, we obtained pure vibration below the energy barrier for strong predissociation, which above that limit was superimposed on a decay-like feature with a time constant of 2–5 ps. The strong predissociation threshold corresponds to the 42 D þ 52 S dissociation limit. Consequently, Breford and Engelke suggested [3] the state causing the disappearance of the D1 Pu state correlating to this limit. We [4] proposed ð1Þ3 Du to be the responsible perturber interacting with the D state by a spin–orbit coupling element of 10 cm1 . We here extend our previous ultrafast pump– probe study by applying new probe channels and quantum dynamical calculations. This brings out two hitherto unobserved features. First, the ultrafast coherent build-up of a wavefunction in a bound triplet state connected to the D state. Second, the existence of a second absorption channel in the blue-violet band. The latter potentially provides a way to create an electronically entangled molecular state.
2. Experimental Briefly, the experiments were performed in a molecular beam set-up using femtosecond pump– probe spectroscopy with fluorescence detection. The cross-correlation time of the pump and probe pulses tuned to 425–435 nm ðkpu Þ and 927 nm ðkpr Þ, respectively, was measured in situ to be 180
203
fs (FWHM) by non-resonant 1 + 1-photon laser~ –X e transition in induced fluorescence on the A acetone. This also facilitated accurate zero delaytime determination. The two linearly polarised pulses at magic angle configuration were focused by f ¼ 50 cm spherical lenses into the vacuum chamber propagating at a small angle to each other. No significant effect on the shape of the recorded transients was found varying the two pulse intensities. The mildly cooled Rb2 beam was produced by unseeded expansion of rubidium vapour at 670 K through a /50 lm nozzle. The background pressure in the vacuum chamber during operation of the beam source was lower than 106 mbar. The light pulses crossed the molecular beam approximately 4 mm downstream of the nozzle and the induced fluorescence was collected at right angle to both the molecular and the light beams. Detection was done either through a low-resolution spectrometer (H20VIS, JobinYvon) equipped with a R928 photomultiplier tube (Hamamatsu) for obtaining emission spectra, or by spectral filtering (UG11, Schott) and a 1P28 photomultiplier tube (Hamamatsu) for the purpose of measuring pump–probe transients. The latter resulted in a spectral sensitivity in the 250– 400 nm range ðkdet Þ suitable to detect Rb 82 P, 72 P– 52 S fluorescence. For each point in a delay-time scan, Dt, the fluorescence intensity was measured by gated photon counting (SR400, Stanford Research Systems) averaging over 1000 cycles. The experimental data presented here was obtained by averaging 20 delay-time scans.
3. Results The to our discussion relevant potential energy curves are displayed in Fig. 1. These are essentially all the ungerade states in the energy range except for the ð1Þ1 Du state. The studies mentioned in the Introduction only reported absorption to the D1 Pu state. Kotnik-Karuza and Vidal [6] indicate, however, that some lines may arise from R to R transitions. Our pump–probe fluorescence spectrum in Fig. 2 is consistent with this notion. The molecular fluorescence band at around 495 nm, produced by the pump pulse alone, can only arise by transition
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Fig. 1. Potential energy, U, curves of Rb2 [5]. Note, the ð3Þ1 Rþ u curve has been shifted down by 250 cm1 . Arrows indicate radiative transitions involved in the pump–probe experiment. See text for definition of the labels I and II of the two probe transitions.
from some state other than D to the electronic ground state. Analysis of the Franck–Condon 1 þ factors [7] for the tentative ð3Þ1 Rþ u ! X Rg transition reveals suitable overlap in the correct wavelength range. Thus, we assign the 495 nm band to this transition which we will see shortly is consistent also with the pump–probe results. To simplify notation and following convention, we henceforth 0 denote the ð3Þ1 Rþ u state by D .
The pump–probe fluorescence spectrum contains several atomic lines that depend on the presence of both pulses. The main probe product is Rb(82 P) while the rest of the pump–probe correlated weak atomic transitions are likely due to cascade fluorescence from this state. The transients we get monitoring 82 P, 72 P ! 52 S fluorescence intensity, see Fig. 3a, are clearly molecular in origin. As the probe pulse is tuned to the blue side of the atomic 82 P 42 D transition, the primary molecular probe state correlates to the 82 P þ 52 S dissociation limit and does not present any substantial energy barrier to dissociation from the probe point as indicated schematically in Fig. 1. Looking closer at the transients in Fig. 3a we note first that the signal is delayed with respect to the instant of excitation by 600 fs. This is substantially longer than the 445 fs for half a vibrational period of the D state [4]. Hence, the signal stems from some state other than the D state. Second, there is a phase of step-like build-up of the signal with a step interval of 1 ps. A slow oscillation, 4.7 ps, is the third characteristic feature. Note the slight but consistent shortening of this period with decreasing pump wavelength.
4. Modelling and calculations Three states were invoked so far, D1 Pu ; D0 1 Rþ u, and ð1Þ3 Du , to explain the spectra and the fast predissociation of the D state. To understand the
Fig. 2. Pump (429 nm)–probe (927 nm) fluorescence spectrum for Dt ¼ 20 ps. The molecular bands arise from the pump pulse alone while the atomic lines except for 72 D ! 52 P (probe alone) and 52 P ! 52 S correlate to the presence of both pulses.
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Fig. 3. Pump–probe transients for several kpu as indicated in the figure. (a) Measured traces normalised to the same maximum amplitude and arbitrarily set off for clear presentation. The signal at Dt ¼ 0 ps is in each case essentially zero. (b) Corresponding calculated transients normalised to the experimental max. amplitudes.
appearance of the step-like structure in the present signal we need at least one more. Spiegelmann et al. [8] suggested ð4Þ3 Rþ u as responsible for the slow predissociation in the lower part of the D state. Including also this state we obtain a minimal and as we will show sufficient model to describe the gross quantum dynamics ensuing excitation of the Rb2 blue-violet absorption band. However, it should be kept in mind that it is likely that at least one more state, ð3Þ3 Pu , is involved [4], although, by the results obtained here, to a lesser extent.
205
The interactions among the four states we take to be mediated solely by spin–orbit coupling. This yields two decoupled excitation channels each corresponding to one of the singlet D0 and D states. In Fig. 1 we labelled the two channels by their probe transitions as I and II, respectively. The channel I dynamics would consist in a simple vibrational motion in the D0 shelf state and from analysis of plausible difference potentials the probe transition is located close to the outer turning point. The D state, on the other hand, may couple to both 3 ð4Þ3 Rþ u and ð1Þ Du . The channel II dynamics is therefore quite complex. The probe point consistent with the observations, mainly the initial delay of the signal, is located at the outer turning point of the ð4Þ3 Rþ u state. The probe points in the two channels are thus well separated and seemingly have different upper states in the corresponding transitions. This means interference effects from possible entanglement of the two channels are improbable and the final signal may be obtained as the incoherent sum of the independent channel contributions. We choose to ignore any nuclear dynamics taking place during the periods of light pulse-molecule interaction. That is, the simulations start out with wavepackets halfwidth put in each channel that are of 0.1 A propagated in time by the split-operator method in conjunction with an analytical expansion scheme for the exponential of the potential operator [9]. The signal from a specific channel we then get from integrating the amplitude density of the wavefunction, projected onto the appropriate HundÕs case a electronic state, inside a square de and outwards, tection window; in channel I 11.5 A in channel II 7.0–7.4 A. We do not observe significant differences using (narrower) Gaussian detection windows instead. Finally, an optical potential is employed to impose absorbing boundary conditions for dissociating components of the wavefunction. Fig. 3b displays three of our calculated transients. Within the three traces all parameters are kept constant except for the total energy of the wavepacket. In the process we slightly adjusted the potential energy curves by Park et al. In channel I the D0 state was shifted down in energy by . This was 250 cm1 and outwards by 0.105 A
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necessary in order to reach above the shelf by a classical excitation from the rovibronic ground state and influenced mainly the modulation amplitude of the calculated signal from this channel. 3 In channel II the D, ð4Þ3 Rþ u , and ð1Þ Du states were and the all shifted inwards: the first two by 0.05 A . Again the first shift was remaining one by 0.13 A done to facilitate a classically allowed transition and the same shift was applied to ð4Þ3 Rþ u to preserve the relative position to the D state. The ð1Þ3 Du , finally, was shifted relative to the D state as to increase the Franck–Condon overlap between the two states, which was necessary to achieve strong enough coupling between these states assuming reasonable spin–orbit interaction strength [10] and that the 3 Du state is responsible for the observed fast predissociation of the D state [4]. Except for the last one these shifts were made essentially for computational convenience and may be indicative. They should at this stage not be considered significant, however, as the thermal distribution of the initial state has been ignored and therefore some of them may artificially compensate for this. The spin–orbit coupling matrix 1 elements for the D1 Pu ð4Þ3 Rþ u and D Pu 3 ð1Þ Du perturbations were both set to 20 cm1 . Finally, the two individual channel contributions were weighed as to reproduce faithfully at each wavelength the first main peak in the experiment which resulted in a channels I–II intensity ratio of about 1:100. A big part of this difference originates in the Franck–Condon overlaps for the excitation transitions, in accordance with the absorption spectrum being dominated by the D state [2,3]. The main remaining part probably comes from a low electronic transition moment for the detection step in channel I.
5. Discussion
Fig. 4. The separate channel contributions to the simulated signal and wavefunction probability density at Dt ¼ 0 ps (solid) and 1.65 ps (other), respectively, projected onto the respective HundÕs case a electronic states. Position and extension of probe windows are indicated by arrows and horizontal lines. Note, the individiual probability densities have been scaled for visibility.
The calculated signal is broken down into its two constituents in Fig. 4. It is clear that the main oscillation stems from channel I and is due to a wavepacket essentially undergoing a simple vibrational motion in the D0 state. This state is one of the in alkali molecules frequently encountered shelf states arising from perturbations involving
the ion-pair potential [5] which explains the extraordinarily long vibrational period. There is a barrier at the entrance of the shelf that reflects part of the wavepacket causing interference structures to appear well before the first round-trip has been completed. Note that the calculation reproduces
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well the slight shift in period of the main peaks with excitation wavelength. Channel II then contributes the step-like structure and decay of the D state population. The signal is obtained from the outer turning point of the ð4Þ3 Rþ u state and reflects the dynamics of the small part of the initial wavepacket that undergoes intersystem crossing. What is not directly seen in the experiment is the predissociated fraction of the wavefunction propagating along the ð1Þ3 Du potential curve. We see that the 3 Du channel not only causes the decay of the D state [4] but also suppresses otherwise larger peaks subsequent to the main peak in channel II. As a further figure of merit, in addition to the position of the steps this signal contribution also reflects the steps becoming more pronounced at shorter wavelengths. In general the calculations reproduce satisfactorily the characteristic features of the measurements. They underestimate, however, the constant background at longer delay times. The main reason for this we believe is our neglect of vibrational and rotational excitation in the electronic ground state. Turning to the magnitude of the spin–orbit coupling matrix elements applied in the calculations we note that they depend critically on the accuracy of the available potential energy curves and that they at this time should be considered as order of magnitude estimates only. Using the curves by Spiegelmann et al. [8], for instance, we [4] previously estimated from the observed decay rate of the D state the D1 Pu ð1Þ3u coupling to 10 cm1 , in good agreement with calculations [10]. The presently used curves by Park et al. [5], however, do not supply sufficient Franck–Condon overlap between the two states to sustain this mechanism. Hence the triplet state was shifted to smaller internuclear separation as to keep the spin–orbit coupling strength within reasonable limits, less than 20 cm1 . As stated above, it is an open issue whether this is a significant shift until more detailed calculations are available and, consequently, we can at present only provide qualitative arguments regarding the interstate spin–orbit couplings and their magnitudes. We ignore in the calculations some interstate couplings of which the most important but indirect
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ones involving the ð3Þ3 Pu state were briefly discussed above and in [4]. In addition to this the D1 Pu and D0 1 Rþ u states cross and may couple through the L-uncoupling mechanism [11]. We find, however, that the possible range of coupling strength for this interaction, less than 4 cm1 for a typical rotational quantum number of 100, is not large enough to account for any of the main observed features in the quantum dynamical calculations. Finally, we do not display in Fig. 1 the ð1Þ1 Du state but in a detailed treatment it should be included as well. A consequence of our results is that the main population of the D0 state must come from direct excitation from the electronic ground state. This implies we can identify D0 ð3Þ1 Rþ u as the upper state in the R–R lines observed in [6]. Moreover, the D and D0 states are excited simultaneously. Our two channels thus may be entangled which would add one degree of freedom to explore in quantum computation by wavepackets like those in [12]. We can now say that this situation of overlapping 1 ð3Þ1 Rþ u and ð3Þ Pu states prevails in all the heavier homonuclear alkali molecules [13–15]. In conclusion, we measured and simulated the molecular quantum dynamics ensuing excitation of the Rb2 blue-violet absorption band. Two electronic states, Dð3Þ1 Pu and D0 ð3Þ1 Rþ u , are excited simultaneously and the dynamics are described by the wavefunction propagating in two independent channels. The D0 channel exhibits essentially vibrational motion in a shelf state while the D state is heavily perturbed by spin–orbit interaction. We now directly observed the build-up of the wavefunction in the ð4Þ3 Rþ u state which together with the previously invoked [4] ð1Þ3 Du is responsible for the decay of the D state population. In a forthcoming Letter we plan to include details on the probe transition and transient anisotropy as well as more detailed calculations and comparison to results in [4].
Acknowledgements We thank the authors of [5] for kindly providing us with their original data files. This work was supported by the Swedish Research Council (VR).
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