Experimental studies of dynamics in gas-phase diatomic molecules. From lifetime-

measurements of BaF to femtosecond pump-probe Spectroscopy of Rb2

Niklas Gador

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Licentiate theses

Department of atomic and Molecular Physics Royal Institute of Technology SCFAB Stockholm, June 2002

TRITA-FYS 2002:11

ISSN 0280-316X ISRN KTH/FYS/2002:11—SE

ISBN 91-7283-305-X

Niklas Gador, Department of Atomic and Molecular Physics, Royal Institute of Technology, Stockholm Centre for Physics Astronomy and Biotechnology SE-106 91 Stockholm, Sweden Niklas Gador 2002 TRITA-FYS 2002:11 ISSN 0280-316X ISRN KTH/FYS/2002:11—SE ISBN 91-7283-305-X

A kind of magic…

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Acknowledgements My colleges: Bo Zhang Pia Johansson Tony Hansson Renee Andersson My supervisor: Tony Hansson My Boss: Lars- Erik Berg Everybody at Fysik 1,KTH and at Fysikum, Sthlm. Univ. Thanks for your help and team-work ! My friends not included in the above: take care! My family: Judith Gador Marton Gador Robin, Alex, farmor Rita Gador, Marie, Mikaela, Hanna, Elin Anka och Lennart Thank you always for your support. I appreciate it Lots and Lots !

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My Licentiate theses is based on these three papers: Paper 1: Lifetime measurements of the A 2/1

2 Π state of BaF using laser spectroscopy L.-E. Berg, N. Gador, D. Husain, H. Ludwigs, P. Royen Chem. Phys. Lett. 287 (1998) Paper 2: Time-resolved optical double resonance spectroscopy of the G +Σ2 state of BaCl H. Ludwigs, N. Gador, L-E. Berg, P. Royen, L. Vikor Chem. Phys. Lett. 288 (1998) Paper 3: Coherent multichannel nonadiabatic dynamics and parallel excitation pathways in the blue- violet absorption band of Rb2 N. Gador, B. Zhang, R. Andersson, P. Johansson, T. Hansson Chem. Phys. Lett. submitted, may 2002

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Foreword Starting my research career in the spring of 1997, I did my diploma work at the Atomic and molecular physics department at KTH. The subject was lifetime measurements of BaF and BaCl, and, as a collaboration with Mats Larsson’s group, Molecular physics, at Fysikum, Sthlm Univ, I performed the experimental work at Fysikum together with Peder Royen , Henrik Ludwigs and Juliana Vikor. This work led to the first two papers of this thesis. These lifetime measurements, in the ns timescale, was my first contact with intra-molecular dynamics. Even though I quickly moved into shorter timescales, down to femtoseconds, lifetimes of molecular and atomic levels frequently pops up and play an important role when setting up, doing, and analysing a wave-packet experiment. The ‘red line’ through my work, and hopefully through this thesis, has been to study dynamics in diatomic molecules, from ns to fs, in the molecules BaF, BaCl and Rb2. In this thesis, most attention will be given to the femtosecond studies of Rb2, while the lifetime- measurements on BaF and BaCl is briefly presented in chapter 2. In the fall of 1997, taking up my PhD studies at KTH, I switched to the Rb2 molecule, which is more suitable for our wave-packet studies taken up later on. Since then I have worked on the Rb2 molecule, which, even though it’s a small molecule produced in gas- phase, shows a complicated (but interesting!) potential curve structure with several curve crossings. Studying wavepacket motions and transfers between potential curves has motivated me since then. One of my responsibility areas of our femtosecond work has been the development and maintenance of the computer controlling of the experiments. A short summary is given in chapter 3.1.1. During 1998 and 1999, I worked on a new design of a crossed molecular beam apparatus, consisting of an effusive Rb2 beam and a supersonic Ar beam. The purpose of this machine was to produce a cold, and collision- free, Rb2 beam, planned to be used in the femtosecond lab. Sadly, I failed to prove the cooling effect of the molecules in the machine. Also, the design of the effusive Rb2 oven turned out to give too low molecule density at the laser interaction region, and too unstable performance, to be used for the femtosecond work. After some successful work with another new-built effusive beam source, designed for maximum signal intensity, I’m really excited if cooling can be achieved in our upcoming design of a third Rb2 beam source. In chapter 3.1.2, I describe the two molecular beam source’s used so far. In chapter 3.2, theoretical background of pump- probe femtosecond spectroscopy is presented, followed by wavepacket dynamics results and discussion in chapter 3.3. During the years, my main responsibility areas has been : the lab- computer, lab- electronics and the beam- machine. Step by step, the Rb2 molecule reveals its dynamical behaviour, and I’m looking forward with great enthusiasm to discover more details of (hopefully) cold Rb2.

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Contents Foreword IX 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Lifetime measurements of BaF 2/1A Π and BaCl +Σ2G . . . . . . . 5

2.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Possible systematic errors . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3. Rb2 quantum dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.1.1 Computers and electronics . . . . . . . . . . . . . . . . . . . 9 3.1.2 Beam machine . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 General Molecular dynamics theory . . . . . . . . . . . . . . . . . . . 18 3.2.1 Two level atomic beats . . . . . . . . . . . . . . . . . . . . 18 3.2.2 Wave packet dynamics . . . . . . . . . . . . . . . . . . . . 20

3.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3.1 Atomic beats, Rb 6P Spin orbit doublet . . . . . . . . . . . . 29 3.3.2 Time resolved fluorescence from laser- excited

Rb2 (3) u1 Σ shelf state . . . . . . . . . . . . . . . . . . . . . 30

3.3.3 Wave packet motion originating from the

Rb2 D u1Π state and from the (3) u

1 Σ shelf state . . . . . . . 31 4. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Papers

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1. Introduction Working in the field of Physical Chemistry, the goal is to gain more knowledge about the dynamics of molecules, e.g. the process of a chemical reaction, AB+C → A+BC or the energy flows within a molecule. These phenomena occur in Polyatomic molecules and mostly, in nature, in the liquid phase. Most attention is also directed toward this field nowadays, especially since the non- linear technique of pump-probe spectroscopy provide more information than traditional high resolution linear spectroscopy does. This is because of the heterogeneous line broadenings in liquids. However, in order to make physical models, predictions and gain insight of a complicated molecule in a complicated surrounding, one has to break down the system into smaller pieces. Our strategy is to start with a diatomic molecule, Rb2, in the gas- phase (effusive beam), characterizing and study it with pump- probe spectroscopy, and thereafter add another atom to the Rb2, for instance one Ar atom by crossing the Rb2 beam with an Ar beam. Thus we can study how a perturber change the dynamics of the Rb2 molecule. And further on, by adding atom after atom, making a small cluster in the gas- phase, we want to trace down the transition from gas- phase to the liquid- phase. In our case, we focus on the dynamical behaviour of the molecules. In principle, working on a diatomic molecule in collision- free effusive beam, the time domain and frequency domain are connected via a Fourier transform. So , again in principle, knowing all energy levels’ positions and shapes, one could calculate all potential curves, eigenstates, coupling matrix elements and the dynamics of a wave packet ( localized superposition of eigenstates). In reality, experimental conditions such as noise, frequency resolution and range limits, make it advantageous to work directly in the time domain, at least if you are looking for wavepacket dynamics. But, certainly, high- resolution spectroscopy would be a great aid to attack the complicated Rb2 potential curve system from another angle. As mentioned already, the Rb2 molecule has a quite complicated potential curve structure with several curves crossing and coupling via different mechanisms. So far, we have used Spin- orbit, L-uncoupling, and ion-ion couplings, in different cases. This nonadiabatic dynamics, when wave packets jump between adiabatic potential curves as they move, is actually closely connected with a chemical reaction, which also proceed via at least one curve crossing. This type of study is presented in paper 3. From a more physical point of view, our wave packet studies on Rb2 serves as a direct experimental test of the validity of theoretical models, e.g. the split operator fourier transform method. Even if the full quantum mechanical treatment is well accepted, there are always needs for different semi- classical models of different degrees of ‘classicalness’. This saves computer effort and usually brings out the physical understandings in a clearer way. Wave packets have an inherent semiclassical behaviour, where the expectation values often follow classical trajectories. Hopefully, our experimental results will not only test existing theories, but also induce new theoretical developments.

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In the search for a deeper understanding of diatomic molecular physics, dynamical studies using lasers is an alternative to high resolution spectroscopy. In this thesis, the molecular dynamics behaviour in a perturbed region, e.g. curve- crossing, is the central theme. Starting with the dynamics of a single state, lifetime measurements were done on BaF and BaCl. The discussions of the results, although molecular specific, give a flavour on what information can be extracted from the lifetime. Of special interest for the Rb2 curvecrossing dynamics, is the complementary general technique of lifetime measurements to detect perturbations. The choice of Rb2 as a molecule to study wavepacket curve-crossing dynamics was based on the article of Breford and Engelke [1], in which The Rb2 D uΠ1 state is shown to be predissociated into the proposed 4D + 5S atomic state (via spin orbit coupling to the dissociative u∆3 state). This has been verified in our experiments. Breford’s reported LIF spectrum of the D state in Rb2, shows a vibrational structure even in the perturbed region, which suggests that the coupling strength, say 1 to 100 cm-1, is suitable for dynamics in a time scale of a few ps. While Bo Zhang performed our first pump- probe experiments on the Rb2 D state using a heat pipe oven, the continued work, presented in paper 3, has been carried out in a collision free effusive beam. The main advantage with a beam is the collision free environment, which simplify the analysis ( better controlled probing) . An illustration of the collision free environment is given in chapter 3.1.2. Besides predissociating out to the atomic 4D state, we have experimentally and theoretically found that the molecular D u

1Π state population is coupled over to a

u3 Σ state, crossing at the bottom of the D state potential curve. Moreover, simultaneously with the D state dynamics, a smaller wavepacket is excited on a ‘shelf state’, (3) u

1 Σ , making classical oscillations just above the shelf, having a very long oscillation period of 4.8 ps. This excitation pathway is a clear example of an advantage of femtosecond spectroscopy over traditional high resolution spectroscopy. Breford and Engelke show an X Σ1 - D Π1 absorption spectrum, which certainly contain irregular peak structure, but do not attempt to clear it out. D. Kotnik et al. [2] suggest a Σ - Σ transition from the absence of magnetic rotation spectra. Even though the D u

1Π state and the u1 Σ shelf state are embedded spectroscopically, they show

very different dynamics, revealing the two excitation channels. This is reported in paper 3. Bridging the single state dynamics, lifetimes, and the multiple state wavepacket dynamics, a two-state quantum beat experiment has been done on the Rb 6P atomic spin-orbit doublet, using pump- probe spectroscopy. The purpose of this experiment is twofold:

1. Minimizing the quantum beat oscillations, the ‘magic angle’ between pump and probe polarization directions can be experimentally found which is used in the Rb2 D – Shelfstate experiment.

2. The agreement between experiment and theoretical analysis conform the reliability of our experimental conditions.

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In the analysis of our experimental results, we have mainly used ab- initio potential curves from Spiegelmann et al. [3], and also the resent published ab- initio potential curves by Su Jin Park et al. [4]. For the higher lying states, above the D- state, the accuracy of these calculations decrease as one approaches the Rydberg states. This is because of the increase in number of overlapping states. Therefore, for the experiments included in paper 3, the probe’s upper state, within the molecule is virtually unknown. We assume that they are Rydberg- like, dissociating into atomic states. Atomic Rb levels are taken from tables [5]. For the groundstate of Rb2, experimental data by C. Amiot [6] was used. Paper 3 in this thesis is the first publication of our femtosecond studies in the effusive beam source. Prior to these experiments, Bo Zhang et al. [7] performed pump-probe experiments on the Rb2 D state in a heat pipe oven. Even though the molecules do not have time to collide in the timescale of femtoseconds, they do collide during the fluorescence detection, which is in the nano- or millisecond time regime. Our next step is to make a new attempt in cooling the Rb2 molecules. This has the advantage of simplifying the wavepacket analysis, by narrowing down the energy spread of ground state Rb2 molecules. Ideally, if all molecules are sitting in the lowest vibrational and rotational state before laser interaction, and neglecting translational temperature, they would all behave as one molecule, when coherently excited. This is precisely what we are searching for; how one molecule behave ( note that the superposition of states, wave packet, is done in every single molecule).

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2. Lifetime measurements of BaF 2/1 A Π and BaCl Σ2G • Measuring lifetime on an unperturbed state, information on the upper and lower wavefunctions can be obtained, from the electronic transition moment. • A lifetime measurement is also a means of studying a perturbed region, e.g. curvecrossing, since the lifetime can change depending on the character of the perturbing state. High resolution laser excitation enable single level lifetimes to be measured. • The electronic dipole moment, eR ’s r- dependence is reflected in lifetime measurements. • Monitoring eR ’s energy dependence give information on the electronic structure change with excitation energy. 2.1 Experimental setup: BaF 1/2A Π : A single mode Ti: Sapphire ringlaser ( Coherent autoscan 899),pumped by Ar ion laser, was used to excite a single rotational level of the v’ = 0 respectively v’ = 1. Excitation is in the near infrared, around 830 nm. Fluorescence detection to the groundstate is made via a high-resolution Jobin- Yvon monochromator, capable of resolving individual rotational lines. This eliminate cascading effects. The laserbeam was opto- acoustically chopped into 150 ns pulses and the lifetime was obtained with time to amplitude conversion technique. BaCl +Σ2G : In order to reach the high- lying G state, optical double resonance excitation was used. The Ar ion laser 514 nm line is resonantly exciting the C state, and the shopped Ti: Sapphire laser pump the C to G transition, at around 850 nm. Detection is made via an Jobin- Yvon H20 monochromator, detecting G → X fluorescence. Both BaF and BaCl was produced in a heat pipe oven. See fig 1 in paper 1 for a schematic layout of the experimental setup. The reason for using photon as start and pulse generator as stop, is to eliminate the problem of ‘dead- time’ of the electronics. It is useful when photon rate is much less then laser pulse rate. 2.2 Possible systematic errors: Pile- up: If the photon rate is too high, more than one detected photon per laser pulse (0.5MHz), the short delay time photons will be favoured over long delay time

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photons, giving a shorter effective lifetime. This is avoided by lowering the temperature in the oven, to have less density of molecules. Cascading effects: This problem is eliminated by the high resolution laser’s ability to excite a single rovib state. Also, in the case of the BaF measurement, the high resolution monochromator detect a single rovib transition, in the A→ X fluorescence. Photon capture: If the molecular density is high, fluorescence photons may be reabsorbed on the way to the detector, and subsequently sent out at a later time, increasing the effective lifetime. This problem is reduced if using a molecular beam instead. Complementary to the wavepacket experiments in the Rb2 beam, lifetime measurement was done on the Rb2 (3) u

1 Σ state, shown in chapter 3.3.2. Collisional quenching: The population in the excited state is transferred to nearby states by molecular collisions. This reduce the lifetime. To adjust for this, a preassure dependent lifetime plot was made, with following extrapolation to zero pressure. The inverse lifetime is assumed to depend lineary on preassure. This problem is completely eliminated in a molecular beam, where preassure is = 0 (collision free). The drawback with the beam, is the difficulty to reach the required temperatures of over 1000 K for BaF and BaCl. Flight out of view: If excited molecules escape the fluorescence observation region before they decay, a shortening of lifetime occur. Keeping the entrance slit large on the monochromator, the observation region is large and this error is eliminated. 2.3 Results and discussion: The measured lifetimes was 56 ± 1 ns for the BaF A state and 41 ± 1 ns for BaCl G state. Neglecting decay to lower lying excited states, and setting the Frank Condon factor to 1 for ∆v = 0 ( excited state has similar spectroscopic data as the groundstate for both BaF and BaCl), the electronic transition moment, Re, between the excited state and groundstate is obtained from the formula,

231eRKντ =− ( 2.3.1 )

where τ is the zero pressure lifetime, K a numerical constant, ν is the transition frequency. The electronic transition moment, Re, is given by:

∫ ΨΨ= eeeee dR τµ ''' ( 2.3.2 ) where eµ is the electronic dipole moment. The value of 2

eR was calculated to 6 (au)2 for BaF 1/2A Π state and 0.3 (au)2 for BaCl +Σ2G state.

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BaF 1/2A Π : Comparison with the other Barium monohalides, like BaCl and BaBr, BaF show an abrupt increase of the eR value. This difference is not surprising since BaF differ from the other Ba- halides in having a shorter equilibrium nuclear distance value, and following change of electronic structure. A series of lifetime measurements were done on different J’s for v = 0 and v = 1, to investigate if there is any lifetime change. If a state is perturbed by a nearby, or crossing state, lifetime can abruptly change for nearby states, depending on which states are perturbed. This is because the perturbed state take character of the perturbing state, which may have very different electronic dipole moment. In the case of dissosiative curve-crossing, the lifetime is shortened due to the extra escape channel. For the BaF 1/2A Π state, all measurements gave the same lifetime, implying that this state is perturbation- free. In general, the lifetime dependence on perturbation are often much more sensitive than the energy eigenvalue shift. Using a high resolution laser, perturbations on single states can be detected in the change of lifetime. In general, eR is dependent on internuclear distance, r, due to the dependence of eΨ on r. For an unperturbed state, eΨ , vary slowly with r within a small r range, but for a perturbed system, e g curve crossing, eΨ , may depend strongly on r, and even change symmetry for different r values. In our lifetime experiments, the vibrational wavefunction is not localized in r, giving an average eR value. On the contrary, for wavepacket studies, the total vibrational wavefunction, the wavepacket, is localized in r, and will take new r values as time proceed and the wave packet move. The probed signal will thus also depend on eR ’s r dependence. In our lifetime measurements on BaF and BaCl the eR ’s r- dependence could not be studied, due to the limited r range of the excited v = 0 and v = 1 levels. Because of the very low Frank Condon factors for ∆v ≠ 0, together with the drop of thermally excited groundstate molecules for v > 1, only v’ = 0, 1 can be sufficiently populated in the excited state. Illustrated in fig 2.3.1 is a case when lifetime measurements can be used to study the r- dependence of eR . If the two potential curves have different equilibrium r distance values, laser excitation will populate a higher vibrational level in the excited state. Fig 2.3.1 Transition 1 occur at a different r value than transition 2. The lifetime of the excited state depends on the strength ( including eR (r) ) of both transitions.

1 2

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The steady state wavefunctions have maximum values at the turning points. The fluorescence have two channels with large Frank Condon factors, at the inner and outer turning point. If eR depend noticeably on r, only Frank Condon factors and the ν3 factor cannot account for different lifetimes of different excited vibrational levels, but also eR ’s r- dependence must be included. In the case of our Rb2 femtosecond measurements, the wavepacket move back and forth over an r range of 10 Å ( the shelf state), and a perturbed region is deliberately chosen, hence eR ‘s r- dependence has to be considered. Ligand Field theory: For both BaF and BaCl, the molecular bond can be considered ionic, with one remaining valence electron on the metal ion. Thus, the molecule resemble an atomic metal ion, perturbed by the negatively charged halide. A direct consequence is that the low- lying molecular states can be attributed atomic characters like s,p,d etc. Lifetimes on different states therefore reflect atomic selectionrules, like ∆L = 1. The groundstate, X 2Σ, has s character. BaCl +Σ2G : While BaF was studied in the infrared region, BaCl was excited to a rydberg state,

+Σ2G , close to the ionization limit, with subsequent decay to the groundstate (fluorescence λ = 300 nm). Because of the fixed Ar ion laser wavelength, together with low FC factors for ∆v ≠ 0, only a single vibrational level in the G state could be excited. According to eq 2.3.1, ν3 increase rapidly with energy, which suggests that Rydberg states ( one highly excited electron, giving the molecule atomic character ) have very short lifetime. Experimentally, however, the lifetime increase with energy as n3 for rydberg states. The reason is the decreasing eR value. In the BaCl G state case, eR = 0.3. The rydberg electron wavefunction is located further and further out from the nucleus as n increase, making less and less overlap with the groundstate wavefunction, resulting in a low eR value. In the case of the Rb2 pump probe measurements, the probe final state is a Rydberg state, which thus can be expected to have a long lifetime, which determine the width of the time gate for fluorescence detection. Pure Precession : The term Pure Precession describe a situation when a Π and Σ state are identical p- orbital like states, only differing in the L projection on the internuclear axis. The BaCl G Σ and G Π Rydberg states could be a candidate of such a pure precession complex. If it truly is a pure precession case, both p like states should have the same

eR value to the s like groundstate. Our measured lifetime of the G Σ state is roughly the same as the G Π lifetime (similar spectral line broadening), fulfilling this ‘pure precession’ condition. However, a spectral analasys by H. Ludwigs and P. Royen [8], showed that the Lambda doubling constants p and q, did not follow the formula for ‘pure precession’ [9]. The conclusion is that the BaCl G complex is not a ‘pure precession’ and that the lifetimes are the same by coincidence.

3. Rb2 quantum dynamics 3.1 Experimental setup: The ‘gas phase’ setup in the fs- lab is sketched in fig 3.1. F Tpfwf C PR Te 3 Tc

r

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rimental setup.

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Fig 3.1.1 The electronic setup as used in the Rb2 pump- probe experiments. Instruments for characterising the 120 fs laser pulses: Grating spectrometer with CCD camera: The spectral range is 200 – 1100 nm.

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Fig 3.1.2 spectral distribution of the pump laser pulse. The full curve is a gaussian fit. FWHM = 2.4 nm. The temporal distribution of the frequencies of the pulse, the chirp, has not been measured. If the timescale for the dynamics in the molecule is sufficiently much larger then the temporal pulse width, about 120 fs, the chirp should play a minor role. However, in our case, dynamic timescales of down to a few hundred fs occur, thus an interferometric autocorrelation instrument to measure the chirp would be useful. The time profile of the laser pulse has to be measured in a somewhat indirect way, since no electronics has fast enough response time. If a reflex of the laser pulse is

Colour/ interf. filter And / or H20 Jobin Yvon monochromator

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detected by a photodiode and sent to an oscilloscope, the pulse seen is a few ns wide due to the ‘slow’ response time of the electronics. Four different techniques have been used: Crosscorrelation : This is the time profile of the combined pump and probe pulses. That is, both pump and probe beams are overlapped at a medium having a nonlinear response, in our case a two- photon absorption process. Two different medias was used: In a photodiode: The diode is most sensitive for the sum frequency of the pump and probe pulse, therefore an electric pulse is produced when pump and probe overlap, both in space and in time. The better the overlap, the larger amplitude of the electric pulse. Scanning the probe pulse, by moving the delay mirror, the overlap time profile is recorded. The time resolution of the delay stage is 3.3 fs, corresponding to a mirror position shift of 0.5 micrometer. In acetone gas: Instead of using a photodiode outside the vacuum chamber, the chamber is filled with acetone gas and crosscorrelation is made in situ. The acetone molecule’s first excited state fits energetically with the sum of one pump and one probe photon, see appendix. Being a large molecule, acetone does not possess the selection rules of a diatomic molecule like Rb2. The two-photon absorption is monitored by the subsequent fluorescence when decaying back to the ground state. Since the laser beam geometries are exactly the same as in the Rb2 experiment, this crosscorrelation technique proved to be an ideal ‘time zero’ calibration of the Rb2 experiments. We simply exchange the Rb2 molecule with the acetone molecule. Unfortunately, when making a larger change of pump and / or probe frequency, one falls off resonance with the first excited energy band of acetone. Another gas, or another method must then be used.

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Fig 3.1.3 Crosscorrelation in acetone gas. Pump = 429nm, probe = 927nm. The full curve is a gaussian fit. FWHM = 180 fs. Only when pump and probe pulse overlap in time and position, can one pump and one probe photon together excite the acetone molecule.

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Background free autocorrelation: An instrument, similar to a Michelson interferometer, is used to measure autocorrelation of one laser beam. The idea is to measure the time profile of the femtosecond pulse by splitting it up into two, and then to scan (in time) one of them across the other in a nonlinear medium, at a small angle. Second harmonic generation is produced when the two pulses overlap in time. Conserving the momentum, the resultant second harmonic beam will propagate inbetween the two fundamental pulses, giving a background- free signal in the detection photodiode. The photodiode signal is fed to a boxcar integrator, which is connected to the AD card in the computer. The lab- computer hardware and software: Hardware: The lab- computer talks and listen to the instruments via a GPIB card and an AD/DA card. The GPIB communication is in general faster then serial communication. These two cards are not directly compatible with the Labview software. Some useful functions of the SR 400 photon counter are: • two signal inputs with separate discriminators. This lets you perform simultaneous counting with count subtraction or count summation as options. Combining the two inputs, you can work with upper and lower discriminators. • gated counting. Having a laser pulse frequency of 1000 Hz, and a fluorescence signal lasting one microsecond following the pulse, gated counting suppress background counts, simply by only counting pulses during the gate, say one micro second. The gate time position is synchronized with the laser pulses via a trigger pulse from a photodiode, monitoring a laser reflection. The gate width and time position can easily be changed, with the minimum gate width of 5ns, setting the time resolution of time dependent fluorescence measurements. • Any number of count-periods ( = number of laser pulses ), from 1 to more than 10.000 can be added, and the sum of, say 1000 laser pulse shots, is sent via GPIB to the computer. The fluctuation of signal level in the measurements is quite high, around 20 % of the signal level, putting large demands on averaging efficiently and during long times, hours in most cases. Software: Fig 3.1.4 Software layout.

Labview Experiment

program

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SR 400 ton counte

13

DLL library subroutines: As just mentioned, the GPIB card and AD converter card are not directly compatible with the Labview program. Therefore, a DLL library with C code subroutines was written as a mid- step to communicate with the instruments. A DLL file (Dynamic Link Library) needs only be compiled once, and can then be used from non- C code programs. Writing in C code has the advantage of being at least ten times faster then Labview code. The AD card, for instance, easily handles 1000 Hz conversion rates ( the frequency of laser pulses). In our present setup, the averaging is done in the SR 400 photon counter directly, putting less requirement on computer speed. Labview- experiment program: Labview is a high- level graphical language, designed to communicate with lab instruments. Programming is done purely graphically, with symbols and execution lines. Via Call library functions, the program communicate with the instruments. Basically the program scan the delay mirror, and collects the SR 400 photon counts. 3.1.2 Beam machine Comparison Beam machine / heat pipe oven : • In a beam, effusive or supersonic, there is a collision free environment. All collision processes are absent, thereby simplifying the analysis greatly. Even in a femtosecond pump- probe experiment, the fluorescence detected has a time span of, say, 1 µs, giving plenty of time for collision population transfer. • The molecular density between the laser interaction spot and the fluorescence capture lens is low, thereby decreasing the probability of self absorption. • Using a supersonic design, cooling of the molecule is achieved. Also in an effusive molecular beam, a slight cooling in the beam hole is present, where collisions in the hole transfer heat energy to kinetic energy. Vibrationally ( and rotationally) cool molecules also simplify the analysis by having less vibrational states populated in the groundstate. If all molecules start out from the vibrational groundstate, less energy spread of the wave packet detected is achieved. • A heat pipe oven can reach higher temperatures, which might be necessary for materials with low vapour pressure at temperatures below 500 degrees C, which is the upper limit for our beam machine. Adding buffergas, a much higher molecular density can be obtained compared to a beam machine. This can be compensated to some extent in the beam by an efficient optical detection setup. Collisions is the main drawback of a heat pipe oven. The molecular beam assembly is mounted inside a vacuum chamber pumped by a diffusion pump, with capacity 1500 litre / s. The effusive beam has a negligible gas load, while the pulsed supersonic beam has an appreciable gas load. Vacuum pressure is below 10-6 mbar, using only the effusive beam, and around 10-4 mbar using both effusive and supersonic beams. The mean free path at 10-6 mbar is 30 m. At an oven- temperature of 700 K, the Rb2 density is 1014 molecules / mm3 inside the oven, giving a mean free path of 3 µm, indicating that we do have some collisions within the 50 µm oven- hole. The detection optics is designed for maximum capture angle of the fluorescence. A 5 cm focal distance lens capture about π solid angle.

14

1: st Rb2 source Crossed molecular beams: Fig 3.1.5 Crossed beam design. The Rb2 mAr atom pulses, and detected by LIF. Solid Rb metal is put into the effusive oven oven. A thermocouple is monitoring temper450 °C, the effusive beam consist of roughlybeam is crossed by a pulsed supersonic Ar aand Ar atoms, heat is transferred from the hoThe pulsed operation of the supersonic nozzdiffusion pump. The kicked down Rb2 molefluorescence, LIF. The beam assembly can bdifferent laser interaction spots. Results: •

Fig 3.1.6 Fluorescence spectrum of Rb2 C st

Ar

2

Laser

Rb

Thermocouple

Heating wire

Pulsed Supersonic

nozzle

U ( 1

03 cm

-1)

Rb, Rb

olecule

sourceature, w 1 % Rtom bet Rb2

le is thcules ae mov

ate

2

10

20

0

fluorescence

s are kicked down by supersonic

, heated by wires covering the hich range from 200 to 450 °C. At b2 and 99 % Rb atoms [10]. This am. In the collisions between Rb2 molecules to the cold Ar atoms. ere to reduce the gas load on the re detected by laser induced ed in all directions, enabling

3 4 5 6 7 8 9

3Σu

476780794

5 P1 / 2

5 P3 / 2

B 1Πu

C 1Πu

X 1Σg

r (Å)

15

Using an H20 Jobin Yvon monochromator and a plotter, fig 3.1.6 show the LIF spectrum of Rb2 C state excited by the 476 Ar ion laser line is shown. Also, a few potential curves [4] are shown. Only the Rb2 beam is used, and is lowered in order to overlap the laser beam. Note the intense 780 nm fluorescence, and the total absence of 794 nm fluorescence. This demonstrates an experimental ‘proof’ of a collision free environment in the beam. The C state is strongly predissociated by the (2) u

3 Σ state, which correlates to the upper 5P atomic state, the 5P3/2, including the Spin- orbit interaction. The recipe for adiabatic Spin- orbit correlation limits is given in the appendix. Collisions transfer population to the 5P1/2 atomic state, as seen by J.M. Brom Jr. et al. [10], who performed this experiment in a heat pipe oven . The peak at 610 nm is proposed to be fluorescence from (2) u

1 Σ to very high lying vibrational levels (and continuum) of the X g

1 Σ groundstate. The (2) u1 Σ state cross the C u

1Π state and couple via the ‘L- uncoupling matrix element’. Further, cascading B ⇒ X transition give signal around 650 nm. •

Fig 3.1.7 Time trace of a Rb2 pulse, excited by Ar Ion 476 nm and detected atomic fluorescence at 780 nm, using a 780 nm interference filter. Figure 3.1.7 shows a typical Rb2 pulse. The continuous Rb2 effusive beam is crossed by the pulsed Ar atom supersonic beam, and the kicked down Rb2 molecules are detected by LIF. The supersonic nozzle valve has a pulse width of about 1.5 ms and repetition frequency of maximum 100 hz. • Most effort was put on proving the cooling mechanism, which, looking at Fig 3.1.8, could not be conformed. A few scans pointed toward a cooling effect, but the instability of the signal made it hard to reproduce the cooled spectrums. Apart from the oven instability ( which was greatly improved in the second source), mode jumping of the dye laser added extra instability. The tactic to show the cooling of the Rb2 molecules was the following:

16

Fig 3.1.8 Absorption spectrum of the X g1 Σ ⇒ B u

1Π transition, of the kicked down Rb2 molecules by the supersonic Ar atom beam. The numbers show the vibrational progressions, v”-v’, and the wavelength axis is calibrated against ref. [11], and an H20 Jobin Yvon monochromator. The X ⇒ B (one of few unperturbed states) absorption spectrum was recorded using Ar ion laser pumped dye laser. DCM was used as lasing media. Scanning range was 650 to 680 nm. Detection with a long- pass colour filter at 695 nm. The beam assembly was repetitively moved up and down ( up with Ar beam on, down with Ar beam off), in order to compare non- kicked to kicked Rb2 molecules. If cooling is present, electronic ground state, X g

1 Σ , population in higher vibrational levels is suppressed, and peaks in the spectra originating from higher ground state levels should decrease in intensity, compared to the v” = 0 peaks. Drawbacks of the apparatus: • Stability of the Rb2 effusive beam. The 50 µm oven hole has a tendency to clog up. Since there is no separate heating for the hole, one cannot keep the hole at a higher temperature then the rest of the oven to prevent condensing in the hole. • Density of the Rb2 effusive beam. The effusive beam density distribution is proportional to cos θ, where θ is the angle from the centre direction [12]. Thus, only a small portion of the total flux, in the centre direction, can be utilized. Furthermore, the beam density decrease with the square of the distance from the hole. In this design, the distance from the hole to laser interaction point is at least 5 cm. Also, the argon beam will only kick down a portion of the effusive Rb2 beam, but this loss should be compensated to some extent by the cooling of the Rb2, putting more molecules in the ground state, from where they are excited. Another limitation is the maximum supersonic nozzle frequency of 100 Hz, thus utilizing only one tenth of the laser pulses.

17

2nd Rb2 source one effusive beam: A second, single beam, Rb2 source was designed and built. Higher molecule density, more stable beam, and the ability to change atom / molecule ratio was the goal, and which was also achieved. Fig 3.1.9 Single effusive beam oven design. T1 and T2 are two independent heating thermo- coax cables. Upper and lower temperatures are monitored by two Thermo- couples. The oven is situated inside a cooled copper house with a conical front, shaped to maximize fluorescence capture angle. The distance from oven nose to laser interaction is now only 3- 4 mm. Two, independently operated, thermo coax cables heat the front and bottom, making it possible to keep the front at a higher temperature then the bottom, to prevent condensing in the hole. The possibility of changing the front temperature, T2, while keeping the bottom temperature, T1, constant also permit a change of atomic/ molecular ratio in the beam. A constant T1 gives a constant vapour pressure. Then, by increasing T2, the increased heat at the front will dissociate molecules into atoms, thereby decreasing molecular content. This tool was used often to investigate if the signal is only due to excited atoms, or, at least partly, due to excited molecules. Running temperatures for Rb2 has been around 480 °C at the front and 430 °C at the bottom. This oven has been used in all our beam experiments done with the femtosecond laser, which are further presented in chapter 3.3. A suggested improvement of the performance, is to mount a thermally isolated cooled capture cup for the Rubidium beam on the other side of the laser interaction. Since the oven is loaded with 2g of Rb metal, which last a few days, quite a bit of Rb material is released from the oven, part of which is condensed on the fluorescence lens. A cooled capture cup would make the Rb material condense there instead.

Rb, Rb2

Fluorescence

Laser

Thermo- couples T1

T2

copper shield

18

3.2 General Molecular dynamics theory: 3.2.1 Two level atomic beats The experimental 6P Rb atomic beat signal shown in fig 3.3.3 serves as a relatively simple experiment- theory comparison case. Not shown in chapter 3.3.1 is the perpendicular pump- probe polarisation signal. The ‘ab initio’ calculations behind the simulations can be divided into two groups, one atomic- specific property part, and one geometrical part. The first group give you energy levels, wave functions, symmetries, and coupling elements between states. These data, I have picked from tables. The second group deals with angular momentum couplings, and is a general theory for any atom. It’s based on symmetry conservation, expansion coefficients, quantum numbers etc. In order to get a ‘physical picture’ of the mechanisms that give rise to the experimental signal, I will use a vector model. Second, to get a more qualitative theory to compare with experiment, we have made some density matrix time evolution calculations, convoluted with gaussian laser pulses to get a simulated time trace. Two approximations are: perfectly linear polarisations and instantaneous excitation. The pump excite coherently the 6P doublet with 421 nm from 5S groundstate, and the probe excite the 6P doublet to the final 8S state with 1895 nm. The final population in 8S is proportional to the cascading signal ( 7P to 5S ), since 8S is spherically symmetric. The situation is very similar to standard quantum beat measurements, where the probe photon take the role of the spontaneously emitted photon in quantum beat spectroscopy. The large energy spacing of 77.5 cm-1 between the 6P levels give an oscillation period of 430 fs. Further, the probe’s linear polarisation direction is varied compared to the pump’s. • Vector model At time 0, the pump excite the 6P levels, having the E field parallel with the z axis, which is taken as quantization axis. The 6P L vector will couple with the S vector via spin orbit coupling to give the J vector. The effect of the coupling can be understood in the way that L and S precess around J, and J precess around z. The L precession has a period T = 1/( E P 3/2 – E P 1/2). During the photon interaction, the photon only ‘feel’ the angular momentum L, which describe the electron cloud geometry. At an S ⇒ P transition, L is perpendicular to the E field. Fig 3.2.1 Angular momentum coupling. L ( and S) is precessing around J. L start at a horizontal direction at t = 0.

z , Epump

L S

J

t = 0

19

In fig 3.2.1, the J precession around z is not shown, but must be kept in mind. Since the probe step is also P ⇒ S transition, the maximum transition probability occur when L and E is ‘the most perpendicular’. For the parallel case ( fig 3.3.3a), this occur at t = 0 and at every time t = nT; n = 1,2,3.... This explains the oscillations, phase of oscillations with respect to time zero, and signal amplitude ( at every maximum, E is perpendicular to L, regardless of J’s precession). For the perpendicular case, at times t = nT, E and L are parallel and a minimum in signal appears. Thus, the signal is 90 degrees phase shifted compared to the parallel case, and the signal intensities lower. For any intermediate pump- probe angle, the signal level is in-between these two extremes. Even though it looks like there is an angle being perpendicular to L at t = (n+0.5)T, because of the J precession, another L vector could be found not being perpendicular to E, therefore reducing the signal. At the special pump- probe angle of 54.7 degrees, the MAGIC angle, the signal actually becomes flat, time independent. In this case, as many L vectors move toward the E- field as move away from E as time goes on after the pump pulse. • Density-matrix time evolution The procedure, as described by K. Blum [13], is to divide up the pump- probe scheme in separate parts. Instead of the density operator, a tensor operator is defined called state multipole. This operator, expressed in the coupled basis, is more applicable to our measurements. It transforms easily under rotation (expressed in spherical harmonics), and it’s term’s give direct information on population, orientation and alignment. The excitation is assumed to be instantaneous, and the Spin- orbit interaction ignored at this step. The populated 6P state, is then split up when the Spin-orbit coupling is turned on, and the 6P state multipole evolves in time, governed by both zero order Hamiltonian and the spin- orbit Hamiltonian. After the delay time, the evolved multi- pole is rotation transformed to the probe’s E- field axis, at a certain angle to the pump E- field. Finally, an electric dipole transition to the final state, 8S, is carried out. Together with a convolution of pump- and probe gaussian time shapes, one obtain the signal,

M τ( ) C

1−

1

x

τ 2+

te K− x2⋅e K− t τ−( )2⋅

⋅13

19

29

cos 14.6 t x−( )⋅[ ]⋅+

3 cos θ( )( )2⋅ 1− ⋅+

⋅

⌠⌡

d

⌠⌡

⋅:=

( 3.2.1 ) where, τ : delay time in ps between pump and probe C: scaling constant K = 190, corresponding to intensity FWHM = 120 fs θ: the angle between pump and probe E- fields.

xddx

20

The expression is for a two colour experiment, λpump ≠ λprobe. Plotting M against τ, for the angles θ = 0, θ = 90, and θ = 54.7 degrees, gives : Fig 3.2.2 Simulated time traces for the 6P beating. a) θ = 0°, b) θ = 90° c) θ = 54.7° (magic angle). 3.2.2 Wave packet dynamics In this thesis, the main topic is wavepacket dynamics in the neighbourhood of a curvecrossing, where two, or more states, perturb each other. Before discussing the wavepacket and it’s behaviour, I’ll first make a brief review of molecular structure in a perturbed region. Molecular structure: The terms diabatic, adiabatic, nonadiabatic, Born- oppenheimer breakdown etc are used in different contexts, with different definitions and is thus a big source of confusion. Here, I will make a short summary of how we define these terms. The ‘true’ energy levels and eigenfunctions of a molecule is obtained by solving the time independent Schrödinger equation,

H |Ψ⟩ = E |Ψ⟩ ( 3.2.2 ) where the Hamilton operator contains all possible types of energy. This equation is impossible to solve analytically or numerically, and hence approximations are necessary. The common technique is to include the largest terms in H, calculate a numerical solution, and then to add extra terms to H, e.g. Spin- orbit operator or the L-uncoupling operator. These extra terms can both effect the energy levels ( diagonal terms) or mix the wave functions, that is transfer population between the original potentials ( off-diagonal coupling terms). By the term diabatic, we mean that we use the original potentials ( with respect to, for instance, the Spin- orbit operator), and add the extra term (e.g. Hso) as a coupling matrix element. If one instead include this extra term as well in H, and diagonalize the Hamiltonian in order to get the new potential curves and wavefunctions, we deal with

0 2 40

0.005

0.015.57610 3−

×

0

Sj

41− Xj

0 2 40

0.005

0.015.77810 3−

×

0

Sj

41− Xj

0 2 40

0.01

0.020.013

0

Sj

41− Xj

a) b) c)

τ

M

adiabatic curves. If the two original curves (diabatic) crossed, the two new adiabatic curves will not cross (avoided crossing), as long as this extra coupling term is nonzero ( which could be the case for symmetry reasons). If the electronic states are isolated, non- perturbed, the electronic wavefunction is approximately independent of the internuclear distance ( the energy potential still varies with nuclear distance), that is

r ∂Ψ∂ = 0 which is part of the Born- Oppenheimer approximation.

However, if two diabatic states cross, and are coupled, eg by Hso, the new adiabatic wavefunctions will certainly depend on r, changing in-between the two diabatic wavefunction characters, as sketched in fig 3.2.3. Fig 3.2.3 Sketch of dnear a crossing. The en

In other words, r

∂Ψ∂ i

elements, which couplthe non- crossed adiab

approximation, with th

In our simulation of ththe diabatic potential celement, V. However, equation, the straightfoadiabatic coupling termfield is often named no Wavepacket and differ • wavepacket The broadband laser eis simultaneously, all wwavetunction is a supe

exp( r)t,( =Ψ

Ψ1

21

iabatic (full lines) and adiergy split at the crossing

s nonzero, and these term

es the two adiabatic curveatic curves). This is the br

e r

∂Ψ∂ terms being large

e Rb2 D u1Π and (3) u

1 Σ urves [4] and add the couin the procedure of solvinrward way is to transforms, evolve a time- step, ann- adiabatic dynamics.

ence potential:

xcite several vibrational stavefunctions having a fix

rposition of all eigenstate

{ exp(-c (r)c t)-iE 2111 +Ψ

Ψ2

abatic (broken line) representation is twice the coupling matrix strength.

s are called nonadiabatic matrix

s ( transferring population between eak down of the Born- Oppenheimer

in the region of the diabatic crossing.

dynamics, described later on, we use pling term as an offdiagonal matrix g the time- dependent Schrödinger to the adiabatic basis, use the non-d then transform back. Thus, this

ates ( ≈ 5 vib.states) coherently, that ed phase relationship. The created

s excited.

} (r))t)E-i(E 212 Ψ ( 3.2.3 )

2V

22

If this superposition is localized in internuclear distance, r, we have a wavepacket. The wavepacket is approximately gaussian with FWHM ≈ 0.2 Å. If the potential is approximately harmonic, the wavepacket oscillate back and forth in the potential with a period T = 1/(En+1 – En). In practice, an ensemble of molecules are coherently excited, that is ‘simultaneously’ within the 120 fs laser pulse. If all molecules started out from the same state, they would all behave as if it was only one molecule. A thermal distribution in the groundstate will smear out the overall wavepacket, and a density matrix theory would be appropriate. However, a simple gaussian wavepacket is often sufficient to explain the main features of our experiments. The wavepacket possess many classical behaviours, e.g. it’s expectation position and momentum values follow classical trajectories. Exciting by itself, the wavepacket is something in- between quantum and classical physics. Whenever a classical description gives a sufficiently correct prediction or explanation, computational effort is greatly decreased, and the physics beneath the quantum mechanics can be understood more easily. • difference potential This technique is one such classical tool which immensely simplify the work load. The issue is to find out at what inter- nuclear distance a wavepacket transition between two electronic potentials take place, when excited by electro- magnetic radiation. Quantum mechanically, Frank- Condon factors gives the answer. But then, a whole set of Frank- Condon factors must be calculated. Classically, the momentum of the nucleus is conserved at an instant electronic transition or, equivalently, the kinetic energy is conserved. It follows that the point where the difference energy between the potential curves equals the laser energy, that’s where the transition occur. Another illustration of this is the following: Frank- Condon factors are the square of the vibrational wavefunction overlap integral. In order for this to be non- zero, the spatial oscillation frequency of the two wavefunctions must be the same. This frequency is determined by the d2/d2r derivative of the wavefunction. Recalling the Schrödinger equation, this derivative is precisely the Kinetic energy. When setting up a new experiment, difference potentials is frequently used to design a scheme of pump, probe and detection wavelengths. Rb2 X- D Frank Condon calculation: The Frank Condon factors were calculated between the Rb2 X g

1 Σ state [5] and the

D u1Π state [4] using the program Level 7.4 [14]. Fig 3.2.4a shows a 3-D contour plot

of F-C factors with v’’ on x- axis and v’ on y axis.

23

Franck-Condon factors: Rb2, X->D

FCF Fig 3.2.4 a) Frank Condon factors between X and D ( lightgreen is large F-C factor). b) final D state population distribution. In order to estimate the vibrational population distribution in the D state, the Frank Condon factors are weighted by a thermal ( 700 °C ) Boltzmann groundstate population. Further, weighting with the laser spectrum, assumed gaussian with center wavelength 427 nm takes the laser frequency and bandwidth ( =3nm) into account. Finally, summation over all (v’’: 1-30) groundstate levels gives the D state vibrational population distribution, shown in fig 3.2.4b. This calculation point out the energy distribution of the created wavepacket, and the r- centroid value show at what r value the excitation proceed at ( here r = 4.2 Å ). All coherences, and all time dynamics are neglected in this calculation. The straightforward way to create the wavepacket on the D state would now be to express the wavepacket in these vibrational levels. But then, coherences play a large role, as does the electric field time properties, and a full density matrix ‘ doorway’ theory is necessary. Instead, we have assumed a starting gaussian wavepacket ( r = 4.2 Å ) and propagate it with the Split-operator Fourier transform method (see later on) as described by Garraway et al.[15] for two states, and by Péoux et al.[16] for three states. This simplify computer effort greatly. Classical trajectory in the (3) u

3 Σ state: The wavepacket’s semi- classical behaviour, within one potential curve (fig 3.2.5a), make it possible to simulate it’s time dependence by the classical Hamilton equations, as a starting point in analysing the experiments. The Hamilton equations for a ‘particle in a potential’ are:

a) b)

0 10 20 30 40 50 602.13241.10 38

0.037

0.075

0.11

0.15Final population in the D state

0.1510 0−⋅

2.1324110 38−⋅

P'sumv'

600 v'

v’

v’’

v’

Popu

latio

n

24

mp

tr =

∂∂

r V

m1

tr2

2

∂∂−=

∂∂ r(0) = 4.25

tr

∂∂ (0) = 0 ( 3.2.4 )

This system of differential equations is solved with the Runge Kutta algorithm. At t = 0, the wavepacket is created on the left hand side at r = 4.2 Å, having zero momentum. Fig 3.2.5b shows clearly that the particle spend most of it’s time on the shelf region. In fig 3.2.5c, for illustrative reason, the harmonic oscillation is obtained with r (0) = 4.35 Å.

4 6 8 10 12 14

18000

19000

20000

21000 r = 4.2 Å

U (cm

-1)

r (Å)

Fig 3.2.5 Classical trajectories in the (3) u

3 Σ state. In the experiment published in paper 3, the wavepaenergy level. Wave packet dynamics: The split operator Fourier transform method: At time zero, the wave packet has the form:

ψ0 expx 4.25 10 10−⋅−

11.97 10−

2

−

:=

where x(0) = 4.25 Å and the FWHM is 0.2 Å ( x,r,Rdistance).

a) 10 10 10−⋅ 510 10 10−⋅ 5b)

cket is created just above the shelf

( 3.2.5 )

are all used for internuclear

4 10 10−⋅

R

50000 t1

10 10 10−⋅

4 10 10−⋅

R

50000 t1

0

5

0 5t [ps]

t [ps]4

4

5

r [Å]

r [Å]

04 10 10−⋅

R

50000 t1

10 10 10−⋅

4 10 10−⋅

R

50000 t1

0

5

0 5t [ps]

t [ps]4

4

5

r [Å]

r [Å]

0

c)

25

The wavepacket propagation is governed by the time dependent Schrödinger equation:

( )( ) t)(R, R UT= t

),( i N Ψ+

Ψ∂

∂ tR! ( 3.2.6 )

where 2

22

N R m 2- T

∂∂= ! is the kinetic operator and U(R) is the potential curve.

After a time ∆t, the propagated wavefunction is:

( ) ( ) ( ) ( )001/2VT0

1/2V0 tR,tR,UUtR, Ut tR, Ψ≈∆+Ψ ( 3.2.7 )

where

∆= NT Tt i-exp U

! can be re- written as,

F m 2

k t i-exp FU2

1-T

∆= ! ( 3.2.8 )

F is the Fourier transform and F-1 the inverse Fourier transform. UV is given by:

( )

∆= 0V tR, Ut i-exp U

! ( 3.2.9 )

where U(R, t0) is the potential curve. With only one potential curve, U(R, t0) is a scalar potential function, but if there are two coupled levels, U1(R) and U2 (R), U(R,t) takes the matrix form,

( )( )

=

RUV(t)V(t)RU

t)U(R,2

1 and ( ) ( ) ( )( )tR, ,tR, tR, 21 ΨΨ=Ψ ( 3.2.10 )

V(t) is the coupling matrix element, which may be time dependent ( e.g. laser excitation). The formal way to form the exponential UV is to diagonalize U(R,t), evolve the diagonalized wave functions, and then transform back to the original wavefunctions. The final expression can be found in ref [15]. The split operator technique is based on the approximation: exp(A+B) = exp(A/2)exp(B)exp(A/2) ( 3.2.11 )

with

∆= U(R)t i-A

! and

∆= NTt i-B

!

26

which is accurate up to second order when A and B do not commute. The trick of taking the Fourier transform when calculating UT, simplify the action of TN to a multiplication by an exponential. A few snapshots of wave packet propagation in the (3) u

1 Σ shelf state is shown in fig 3.2.6. Fig 3.2.6 Wavepacket propagation in (3) u

1 Σ shelfstate. Snapshots at t = 0 ps; t = 0.5 ps; t = 3.8 ps; t = 5 ps. The x axis shows 3.5 < r < 14.0 Å and y axis shows

2Ψ on an arbitrary scale. The potential curves are inserted for illustrative purpose. The probing of the wave packet can either be implemented by adding a timedependent coupling up to a final state’s potential curve, or, in a more crude way, putting an r- window where the wavefunction is probed. The signal is then proportional to the integral of Ψ 2, over the window region. Expanding the simulation model to include three coupled states, I have followed the procedure given in ref [16] in order to handle the exponential of the 3 x 3 potential matrix. Although a Taylor expansion of the exponential, exp( kU), is always possible, this approximate technique has too poor convergence properties for being practically usable. A better method is again to diagonalize the diabatic U, propagate the

27

adiabatic U, and then transforming back to the original diabatic U. The method described in ref [16] is proved to be yet a factor of two faster. It uses a theorem, Cayley- Hamilton, in order to re-write the exponential into a second order polynomial in U. In contrast to a Taylor expansion, this polynomial is exact. The only requirement is that U is symmetric, which is fulfilled in my cases. In the three- state system { D u

1Π , (4) u3 Σ , (1) u

3 ∆ } the starting wavepacket is launched on the D state, just above the (1) u

3 ∆ dissociation limit, 4D. The split operator Fourier transform method is relying on that the r- grid is covering the whole wavefunction, which fails when the wavefunction on the u

3 ∆ state reaches the right- side limit. This results in large distortions of the wavefunction. To avoid this problem, a negative complex part is added to the u

3 ∆ potential curve on the right hand side, 12 Å < r < 20Å. According to eq. 3.2.9, the result is a negative exponential part in the time propagator, which damp, or eat up, the wavefunction gradually as it enters this zone. A compromise between reflection and non- complete damping has to be made. In my simulation, the remaining wavefunction, after the damping, give a completely neglectable distortion to the overall dynamics.

28

3.3 Results and discussion: Setting the scene for our Rb2 experiments, fig 3.3.1 shows some of the involved potential curves [4]. Shown separately in fig X are two more states, predissociating the D state and used in the simulation of the D state dynamics.

4 6 8 10 12 14 16

0

10000

20000

30000

6P

6S

4D

5P

5S

( )31Σ u

+D u

1Π

C u1Π

3Σ u

B u1Π

X g+1Σ

r [Å]

ener

gy [c

m-1]

Fig 3.3.1 Some potential curves involved in our experiments. An overview of different detection channels was obtained by using an H20 Jobin Yvon monochromator instead or together with a colour filter, in-front of the PM tube. In fig 3.3.2, the monochromator was scanned between 300 and 800 nm. In this case, laser wavelengths were set to 429 respectively 954 nm, and the delay time was kept constant at + 50 ps.

3 0 0w a v e l e n g t h [ n m ]

Fig 3.3.2 Fluorescence spectrum following the pump and probe pulse. ∆t

Pump laser 429nm

(3)1Σu - X

Phot

on c

ount

s

5P-5S

6D-5P

8S-5P

7D-5P

D-X Rb2 fluorescence

P

7 -5S

8P-5S

8 0 0

= 50 ps.

In the spectrum, we see a couple of atomic transitions, arising from cascading down from the 8P atomic state. The 7D atomic level is populated by 3 photon excitation of the 954 probe beam only. Also, molecular D state fluorescence down to the X groundstate is seen around 470 nm, in accordance with the Frank- Condon calculations shown in chapter 3.2.2. The peak next to the right of the D-X transition, at 490 nm is fluorescence from the (3) u

1 Σ shelf state down to the X g1 Σ groundstate. The area of this peak is a factor of

10 less then the area of the D-X peak at 470 nm. Although Frank- Condon factors should be included, the spectrum hints to that the X to D transition probability being much larger then X to (3) u

1 Σ shelfstate. In the simulations, I use a factor of 10 in favour of the X-D transition. 3.3.1 Atomic beats, Rb 6P Spin orbit doublet The blue pump beam is tuned to 421 nm, overlapping the two 6P spin-orbit Rb atomic levels, J= 3/2 and J= 1/2, having excitation wavelengths of 420 and 422 nm respectively. Electric dipole transition selection rule in atoms is ∆L = ± 1. The spectral laser bandwidth,FWHM, is about 3nm at 421nm. The excited wavefunction is now a coherent superposition of the two 6P spin-orbit wavefunctions. After a certain delay time, the probe further excite the 6P atoms to 8S at 1890 nm. Detection of population in 8S is done by measuring the cascade fluorescence 7P to 5S. The curve shown in fig 3.3.3a is recorded with the pump and probe both linearly polarised in the horizontal direction.

FPpp Ttw

a)

0 1 2 3 4 5

0

2000

4000

Phot

on c

ount

s

∆t (ps)

ig 3.3.3 Rb atomic beat pump probe sirobe: 6P to 8S at 1890 nm ; Detection 7olarisation directions are parallel. b) Maolarisation directions.

he oscillation period is 427 fs, in agreemo 6P levels. The probe has no resonant

b)

2

gnP gi

e tr

9

al. Pump: 5S to 6P at 421nm; to 5S at 360nm. a) Pump and probe c angle, 54.7 °, between pump and probe

nt with the energy split of 78 cm-1 for the ansition from the groundstate, therefore

0 1 2 3 4 5

0

1000

2000

Phot

on c

ount

s

∆t (ps)

30

giving no signal for negative time delays. The oscillations of the signal continue with equal amplitude throughout the delay stage range ( I have scanned 500 ps from time zero), which is a second illustration of the collisionfree environment, with neglectable dephasing due to collisions . Three benefits of a more practical issue with this atomic signal are: • Finding roughly time zero. For some experiments on the Rb2 molecule, signal is only present close to time zero, that is close to when pump and probe pulses arrive at the same time at the molecular beam. Thus, it’s quite a bit of work to find the correct delay mirror position ( a few ps out of 600 ps scanning range). Using the atomic ‘step– function’, the time zero is found in a few seconds ! • Alignment. Overlapping the pump and probe beams together with the adjustment of oven position is simplified having a strong signal, which the atoms certainly provide. • Another use of the atomic beating is to find the so called magic angle. This angle, 54.7 degrees, between the pump and probe polarisation directions ( linearly polarised), make the oscillation of the atomic signal disappear. This same magic angle, also remove rotational effects in the Rb2 molecule. The wavepacket experimental curves presented in paper 3 are all done in the magic angle. Fig 3.3.3b shows the same atomic signal as in fig 3.3.3a, but with the magic angle between pump and probe. 3.3.2 Time resolved fluorescence from laser- excited Rb2 (3) u

1Σ shelfstate In the first round of experiments, only the pump laser light around 430 nm was used to excite Rb atoms (422 nm) and Rb2 molecules ( D- and u

1 Σ shelf state, 425 to 440 nm). The subsequent fluorescence time dependence was measured by scanning the delay time of a 30 ns wide gate on the photon counter. Two obvious advantages compared to the BaF lifetime measurements are: on a ns timescale, the 120 fs long pump pulse is ‘instantaneous’, and the molecules in the beam are in a collision free environment, corresponding to zero pressure. Disadvantage is the pour energy resolution of the laser light. In figure 3.3.4, the pump is set to 436 nm, reaching the molecular shelf-state just below the 4D atomic level ( see fig 3.3.1). But due to the gaussan spectral shape with FWHM of 3 nm, and the thermal ground state population ( 430 °C), part of the excited molecules may still dissociate out to the 4D atomic state ( all atomic limits considered correspond to one atom in the groundstate 5s and the other atom in the specified excited state). In fig 3.3.4a, detection is an interference filter at 488, monitoring the shelf-state molecular fluorescence, giving a lifetime of roughly 50 ns. The fluorescence may also include some D ⇒ X transition. In fig 3.3.4b, detection is an H20 Jobin Yvon monochromator tuned to 780 nm, monitoring the 5P3/2 ⇒ 5S atomic transition. The peak at x = 32 ns is scattered laser light. The excited molecules may take two routs to the 5P3/2 atomic state, direct predissociation out to the 4D atomic level with subsequent radiative decay, or via curvecrossings reaching the

dissociative u3 Σ state which correlates to 5P3/2. The 780 nm fluorescence first show a

rise during 60 ns, corresponding to the build- up of 5P3/2 atomic state population, and then a decay as the excited molecules and atoms are depleted. The fit contains two decay times, τ = 35 ns and τ = 120 ns, where the first correspond to 5P atomic lifetime, and the second to atomic 4D lifetime and/or molecular D and shelfstate decay. Inserted in the figure are the atomic 4D and 5P lifetimes [12]. For pump wavelengths closer to the atomic 5S-6P transition ( 420 and 421.7 nm), the tail of the pump spectrum will excite atomic 6P level in addition to excited molecules. These atoms will cascade to 5P, mainly via 6S, and give fluorescence at both 780 nm and 794nm. However, these excited atoms are not involved in the probe step if the probe wavelength is less then 1000 nm, which would then ionise the 6P atoms.

0 100 200 300 400

0

400

800

1200

1600

y (ph

oton

coun

ts)

x (ns)

Fig 3.3.4 Fluorescence time behaviour. Inthe time window when scattered laser pho 3.3.3 Wave packet motion originatin (3) u

1Σ shelf state. The starting point of the wavepacket dynaTwo wavepackets are launched from the g(3) u

1 Σ shelf state, and one to the D u1Π st

more intense than the shelfstate wavepacka width of 0.1 Å (= 300 cm-1 energy width Molecular Rb2 D ⇒ atomic Rb 4D predis The pump was set at 428 nm, exciting a wabove the atomic 4D level. The probe is setransition 4D ⇒ 8P. Detection is fluorescecolour filter with transmission region 270 7P ⇒ 5S transition. The long time-trace is

31

3.3.4a, the shaded region correspond to

tons may be detected, using a 30 ns gate.

g from the Rb2 D u1Π and

mics in the region of the D state is: roundstate ( re= 4.2 Å), one to the ate. The D state wavepacket is 10 times et ( F-C factors ). Both have re = 4.2 Å and ).

sociation dynamics:

avepacket on the D state, energetically just t to 954 nm, in resonance with atomic nce 8P ⇒ 5S in the Rb atom using UG 11 - 375 nm, thus also detecting the cascading shown in fig 3.3.5.

0 200 400 600

0

20

40

60

80

tabel:τ = 85 ns (4D)τ = 30 ns (5P)

y = -82*exp(-(x-32)/35)+82*exp(-(x-32)/120)

y (ph

oton

coun

ts)

x (ns)

32

0 50 100 150 200 250

0

500

1000

1500

2000

fit: y = C2-C2exp( -t / τ ) τ = 90 ps

pump 428probe 954

sign

al (

phot

on c

ount

s)

∆t (ps)

Fig 3.3.5 Pump probe trace of the excited Rb2 D state and probed atomic 4D dissociation products . The atomic 4D – 8P is not the only probe channel at 954nm, but also the (3) u

1 Σ shelfstate and (4) u

3 Σ states ( see later on) are probed to proposed rydberg states. The signal in fig 3.3.5 is therefore an overlap of molecular and atomic products. To isolate the molecular probe channel, probe wavelength was tuned off atomic resonance to 927 nm, whose dynamic signals we focused on and which is presented in the next section and in paper 3. Nevertheless, even if the short delay time ( ∆t < 10 ps ) of the signal is dominated by molecular dynamics, the atomic 4D products have a rise time of around 90 ps. One proposed state involved in the predissociation of the D state to 4D atomic level, is the (1) u

3 ∆ state, fig X, interacting via spin- orbit coupling. Probing off atomic resonance, at 927nm : Keeping the pump at 428 nm, we excite both the Rb2 D and shelfstate. We neglect coupling inbetween them, and treat them separately. Both systems will be probed (927 nm) to Rydberg states, ending up in the same 8P atomic state. Hence the signal seen in fig 3.3.6 is composed of both these two channels. The D state system is responsible for the peaks at 1 and 2.1 ps time delay, while the shelfstate contribute with large peaks at 3.5 , 8.5 ps etc with a 4.8 ps period. Fluorescence is detected from atomic 8P to atomic groundstate 5S at 335 nm. The experimental pump- probe trace shown in fig 3.3.6 is detected with a UG 11 colour filter. Pump and probe polarisation directions in magic angle.

+ 2 0 2 4 6 8 1 0 1 2

0

1 0 0

2 0 0

3 0 0si

gn

al (

ph

oto

n c

ou

nts

/ s

)

∆ t (p s )

Fig 3.3schem The tim The puconsta On theinner tThe coinvesti (3) u

1 Σ

U (c

m-1)

Fig 3.3ion- iocompl

33

.6 Pump- probe trace of the D and shelfstate systems. The time zero is

atically shown with a dashed line.

e zero is calibrated using acetone gas, see fig 3.1.3 and appendix.

mp- probe signal level out at about 200 counts /s, after 20 ps and stays at that nt level for all longer delay times.

‘wrong’ side of zero, the probe pump the A u1 Σ state, and the pump probe the

urning point of b u3 Π which couple to the A state via the Spin- orbit coupling.

upling strength is around 120 cm-1, and this system is presently under gation.

shelfstate channel :

5 10 15 20 25

20000

24000

28000

32000

r (Å)

.7 Potential curves of Rb2 D and Shelf state. The dotted line illustrate the n bonding potential, responsible for the shelf shape, through a series of icated avoided crossings. The shaded region show the probe window.

34

The probe pulse is energetically in resonance with a proposed Rydberg state just at the classical outer turning point ( 11 Å < r <12 Å). The Rydberg state dissociate out to the 8P atomic level. The scheme is sketched in fig 3.3.7. The shelf state is shown together with the D state. The two states do cross, but due to symmetry reasons, they can only couple via the ‘L uncoupling element’, which is estimated to be less then 3 cm-1 at J = 100 [18]. The influence of this coupling is small and is neglected in the simulations. The wavepacket make classical oscillations back and forth, just above the shelf barrier. Simulation of the signal is shown in paper 3, fig 4. Due to the shelf shape, the signal is zero until 1.7 ps delay, and has an oscillation period of about 4.8 ps. D u

1Π state system channel : Three states are used to simulate the results. The D state is Spin- orbit coupled to both the u

3 Σ state and to the u3 ∆ state. The u

3 Σ and u3 ∆ are not coupled (no crossing).

Coupling strength is set to 20 cm-1 to both states, which is the ‘pure precession’ estimation [18]. Fig 3.3.8 D- state system Fig 3.3.8 shows the potential curves together schematic wavepackets numbered from left: 1. t = 0 fs: starting wavepacket ( on D state, re = 4.2 Å). 2. t = 450 fs: remaining D state wavepacket ( after half a D state period). 3. t = 550 fs: portion of D state wavepacket coupled over to the u

3 Σ state at the crossing at re = 5.1 Å. 4. t = 650 fs: portion of D state wavepacket coupled over to the (1) u

3 ∆ state at the crossing at re = 4.7 Å. This wavepacket dissociate into 4D3/2 + 5S atoms. Again, probing is at the right- hand side classical turningpoint ( 7 Å< r < 7.2 Å ) of the u

3 Σ state, up to a proposed triplet Rydberg state. The u3 ∆ state deplete the D

state population ( finally ending up at atomic 4D), each time the D state wavepacket

4 5 6 7 8

18500

19000

19500

20000(3) 3Π

uD 1Πu

(4) 3Σu

(3) 1Σu

(1) 3∆u

U (c

m-1)

r (Å)

35

reach the left side of the potential. Thus, during the first oscillation in the D state, the wavepacket will couple over to the u

3 Σ state twice, on the way out and then on the way in. This gives the two signal peaks, at 1 and 2.1 ps time delay in fig 3.3.6 . Thereafter the signal from u

3 Σ is exponentially decayed due to the loss of wavefunction through the u

3 ∆ route. Simulation signal is shown in paper 3, fig 4. Scaling the D state system with a factor 10 over the Shelfstate signal ( the transition probabilities are unknown), and adding the signals result in the final ab initio simulation of the wavepacket dynamics, shown in paper 3, fig 3b. Simulating the D state decay: Doing ‘computer experiments’, one is free to design any imaginable experiment, and one such is to keep the same D state system as above, but putting the probe window on the D state, close to the outer turning point. This was done to compare with the earlier experiments by Zhang et al. [7].

Fig 3.3.9 Probing the D state. For short wavelengths, the decay is around 5 ps, and for longer wavelengths the decay is around 1 ps. The reason for this effect is that the less energetic wave packet travels slower across the crossing, and couple stronger, in the sense of the Landau Zener model [15]. The oscillation is the D state vibration. The splitting is due to the probe- window being slightly inside the outer turning point. These features agrees with the results of Zhang et al. Discrepancies are the depth of the signal dips and a constant signal background. Including rotational distribution, decreasing coupling strength to u

3 ∆ , including more states and considering the two λ- doubling components are suggested improvements in the modelling, which will be investigated in the near future.

37

4. Appendix Spin- orbit adiabatic correlation limits using group theory: A set of molecular states correlate adiabatically to either the upper or lower spin- orbit splitted 5P states. Group theory is a quick way to determine which atomic level, a molecular potential curve dissociate into. The following example determine the correlation limit for the (2) u

3 Σ state, responsible for the predissociation of the C u1Π

( see chapter 3.1.2 ). All ungerade molecular states [4] correlating to 5P + 5S atomic state are shown in fig 4.1.

4 6 8 10 12 14

10000

20000

5P1/2

+ 5S5P3/2

3Πu

1Πu

3Σu

1Σu

ener

gy [c

m-1

]

r [Å]

Fig 4.1 Ungerade potential curves correlating to 5P + 5S. The Rb2 molecule belongs to the D∞ h point group. Step 1 : The molecular states are decomposed into spin- orbit symmetries: Using the direct product tables and species of spin functions tables [19], one get the following symmetries:

u3 Σ state: S=1 ⇒ { gg Π+Σ− }

−− Σ=Σ⊗Σ uug and uug Π=Σ⊗Π Likewise the other states give the symmetries :

u1Π : uΠ

u1 Σ : +Σu

u3 Π : uΠ , +Σu , −Σu , u∆

38

Step 2: Resolution of species of atoms into the D∞ h symmetry: 5S atom:

Sg : +Σg S= ½ : E1/2 g

⇒ E1/2 g ⊗ +Σg = E1/2 g 5P atom:

Pu: +Σu ; uΠ S = 1/2: E1/2 g

⇒ E1/2 g ⊗ +Σu = E1/2 u ⇒ E1/2 g ⊗ uΠ = E1/2 u ; E3/2 u

5P1/2 + 5S : E1/2 g ⊗ E1/2 u = +Σu , −Σu , uΠ 5P3/2 + 5S : E1/2 g ⊗ E1/2 u = +Σu , −Σu , uΠ E1/2 g ⊗ E3/2 u = uΠ , u∆ The correlation diagram is drawn in fig 4.2. The order of the molecular states is the order at the bound molecule, r ≈ 4.2 Å. States with the same symmetry do not cross. Fig 4.2 Correlation diagram Specifically, the u

3 Σ state correlate only to the 5P3/2 + 5S atomic state. This result agrees with experiment.

5P1/2 + 5S

5P3/2 + 5S

3Σu

1Πu

1Σu

3Πu

Πu

∆u

Σu-

Πu

Σu+

Σu+

Πu

Σu-

Time zero cross- correlation in acetone gas: This measurement is done for two purposes: calibrating the time zero, and monitoring the crosscorrelation shape between pump and probe. Fig 4.3 Two photon The two- photon abso

S ∝ ⟨ I1(t) I2(t If both pulses are assmaximum at τ = 0, anmeasured cross- corre Recalling that a distawithin 50 fs can onlyexperimental time traacetone calibration, t

A I2(t+ τ)

I1(t)

I1: 425 nm pump pulse I2: 927 nm probe pulse

X

39

absorption in Acetone

rption is proportional to the second order correlation function:

+ τ) ⟩ ( 4.1 )

umed to be gaussan, this function is a gaussan too, with d a halfwidth of 2 times each pulse’s halfwidth. The lation intensity- halfwidth is 180 fs.

nce of 10 µm equals 30 fs for the light, accuracy of time zero be achieved in situ, where the Rb2 molecules are excited. All ces in paper 3 are calibrated with acetone gas. Repeating the he error is estimated to be less then 50 fs.

41

References [1] E.J. Breford and F. Engelke, Chem.Phys.Lett. Vol 75, No1 (1980) [2] D. Kotnik-Karuza and C.R Vidal, Chem.Physics. Vol 40, p25-31 (1979) [3] F. Spiegelmann, D. Pavolini and J-P. Daudey, J.Phys.B, At.Mol.Opt.Phys, vol 22, p2465-2484 (1989) [4] Su Jin Park, Sung Won Suh, Yoon Sup Lee and Gwang-Hi Jeung, J.of Mol.Spectr. vol 207, p129-135 (2001) [5] Atomic energy levels, National Bureau of Standards [6] C. Amiot, J.Chem.Phys. vol 93, No.12, p8591-8604 (1990) [7] B. Zhang, L-E. Berg and T. Hansson, Chem.Phys.Lett. 325, p577-583 (2000) [8] H. Ludwigs and P. Royen, Chem.Phys.Lett. 223, p95-98 (1994) [9] G. Herzberg, Molecular Spectra and Molecular structure, vol I [10] J.M. Brom Jr. and H.P. Broida, J.Chem.Phys. Vol 61, No.3 p982-987 (1974) [11] C.D. Caldwell, F. Engelke and H. Hage, Chem.Phys. 54 p21-31 (1980) [12] G Scoles, Atomic and molecular beam methods, vol I [13] K. Blum, Density matrix theory and applications [14] R.J. Le Roy, Level 7.4, computer program (2001) [15] B.M. Garraway and K.A. Suominen, Rep.Prog.Phys. 58, p365-419, (1995) [16] G. Peoux, M. Monnerville and J-P Flament, J.Phys.B,At.Mol.Opt.Phys. Vol 29, p6031-6047 (1996) [17] O.S. Heavens, Rubidium atomic transition probabilities [18] H.L. Brion and R.W. Field, Perturbations in the Spectra of Diatomic Molecules [19] G. Herzberg, Molecular Spectra and Molecular structure, vol III