The technology for producing and controlling femtosecond laser pulses has advanced greatly during the last few years. Today laser systems can deliver reproducible pulses as short as 10 fs, making it possible to follow real-time evolution on an atomic scale. Molecular vibrations, for example, occur on a 100 fs timescale and it is possible to excite localised states and interrogate their time evolution with lasers.
This renews hopes from the 1970's that lasers could steer chemical reactions into preselected final products. Conventional chemical techniques rely on random thermal processes which, frequently, are neither economic nor efficient. The femtosecond timescale is also suitable for exciting electronic states in multiple quantum well semiconductor structures; the time evolution of these states provides valuable information about the transport properties of mesoscopic devices. Commercial applications, therefore, seem certain to follow all the recent activity in femtosecond laser development.
However, many researchers are inspired by the challenge of perturbing and probing physical systems, forcing them to give up their secrets. Quantum mechanics, for example, is a spectacularly successful theory but many aspects of it have yet to be realised and verified experimentally. Most applications of quantum theory are still concerned with energy eigenstates or stationary flows of probability (i.e. scattering theory). The only experiments to reveal the real-time evolution induced by microscopic Hamiltonians - pulse spectroscopy, quantum beats and coherent transients - have been restricted to a few discrete quantum levels. In addition to the experimental difficulties, computing the time development of quantum systems has posed formidable numerical obstacles. But the availability of high-performance computers has now made it feasible to integrate the Schrödinger equation of experimentally interesting cases with a view to understanding their evolution.
Femtosecond experiments
Mode-locking and pulse compression are two of the techniques which have made the production of pulses much shorter than 100 fs possible. Recently Ti:sapphire lasers have generated pulses shorter than 10 fs - pulses containing only a few optical vibrations. Other techniques such as harmonic generation and parametric mixing have also enabled the centre frequency of the pulses to be shifted.
Many investigations have focused on a single vibrational motion in molecules consisting of two atoms (or radicals) combined in a simple "dumb-bell" structure. The single bond between them acts as a spring and supports harmonic oscillations for small amplitudes, but the bond can break (dissociate) when stretched too much. These phenomena usually occur on timescales between a few picoseconds and a few hundred femtoseconds, and successful experiments have been performed on several molecules containing iodine such as I-I, I-CH, I-CN and I-Na (see Gruebele and Zewail in Further reading).
The theoretical description of molecular quantum states is based on the Born-Oppenheimer separation of nuclear and electronic degrees of freedom. The electrons are much lighter than nuclei and can adjust rapidly to the instantaneous nuclear configuration. Therefore molecules are characterised by "potential energy surfaces" - plots of energy versus internuclear separation - and there is a separate surface for each electronic configuration. The vibration of the molecule along this surface may be considered as continuous (as we do in this article) or as quantised in vibrational energy levels. The molecule also rotates as a rigid body, but these degrees of freedom are neglected here. The potential energy surfaces are coupled by, for example, the spin-orbit and relativistic corrections to the Born-Oppenheimer approximation, but these are not relevant when the surfaces are coupled by lasers.
In 1989 Charles Shank and collaborators at the AT&T Bell Laboratories in New Jersey used 6 fs laser pulses to observe molecular vibrations with a period of 60 fs in the dye molecule Nile blue - one of the fastest molecular processes ever observed (see Fragnito et al. in Further reading). The fact that several coherent oscillations survived against a background of many coupled degrees of freedom in a complicated molecule was an unexpected and pleasant surprise. In 1991 Stuart Rice and co-workers at the James Franck Institute in Chicago observed the oscillation of a wave packet on an excited potential energy surface in iodine molecules (I2). In Rice's experiment a wave packet is excited from the ground state with a 50-70 fs laser; a second pulse, exactly one vibrational period later, excites a second wave packet which adds coherently to the first (figure 1a). This addition can be constructive or destructive depending on the relative phases of the pulses. By controlling this phase difference Rice and co-workers were able to observe the interference between successively excited wave packets (figure 1b).
1 (a) Ground and excited state potential energy surfaces (red) and electronic wave functions (green) for a diatomic molecule. A short laser pulse (black arrow) excites a wave packet, which is centred at the minimum of the ground state, to the excited state where it oscillates about the minimum with a period T. A second laser pulse T seconds after the first can excite another wave packet (blue) on top of the first wave packet. The phase of the wave functions can be controlled by the phase of the laser pulse: if both packets have the same phase, they interfere constructively (left); if the relative phase of the two pulses is changed by 180¡, they interfere destructively (right). (b) The interference is also a function of the delay between the pulses. In this case the 60 fs pulses are in phase and the wavelength is 611.12 nm. The vibrational period of this excited state in iodine is 400 fs (see Scherer et al. and Rice in Further reading)
The molecule sodium iodide (NaI) has an ionic ground state whereas the well separated atoms are stable in their neutral states. This means that the potential energy levels must cross at some internuclear separation because there is a strong repulsive force between the neutral atoms at short distances. However, internal coupling perturbs the crossing, making the system a prototype for the "avoided crossing" situation treated in quantum mechanics text books. In 1989 Ahmed Zewail and collaborators at the California Institute of Technology used 60 fs laser pulses to follow the evolution of wave packets on the excited level of NaI (period = 1.3 ps) in one of the most beautiful experimental demonstrations of time resolved molecular vibrations (see figure 2 and Zewail in Further reading).
2 (a) Sodium iodide (Nal) has an ionic bound state but the neutral atoms, when widely separated, have a lower energy than the ions. Therefore the ionic and neutral potential energy surfaces (red) must cross over as a function of internuclear separation. However, when intramolecular coupling between the levels is included, the crossing is avoided and the excited level is a shallow bound state with a period ~ 1.3 ps. A wave packet excited from the ground state by a laser (black line) oscillates with this period and its probability gets repeatedly redistributed between the two levels as it passes the level crossing. The part emerging on the flat (covalent) potential surface can be observed as free Na and I atoms if we probe the system with a laser resonant with a transition in atomic Na or I (the top curve in (b) shows the periodic accumulation of free Na). If we probe with a laser (dotted line) resonant with surface I (or II) near the avoided crossing and another electronic state of the molecule, we can see the regular appearance and disappearance of the mixed state at the crossing (lower curve in (b))
In general, experiments are determined by the availability of laser sources with the required frequency and pulse length. These experiments have been performed at modest laser intensities, requiring pulse energies ranging from a few nJ to a few microJ. The next generation of laser sources should open up entirely new areas of time-resolved physics (see Knox in Further reading). In the meantime, theorists are beginning to explore new areas of the laser-molecule interactions in an attempt to elucidate those situations which display the most interesting and challenging physical behaviour.
Molecular basics
As we have seen, the dynamics of a molecule can be described by a series of effective (vibrational) potential energy surfaces. Each electronic energy level has its own surface and transitions between these can be induced by lasers when the energy difference between two surfaces (for a given internuclear separation) is exactly equal to the energy of one laser photon. In what follows we consider a one-dimensional quantum mechanical world where we can study the time-dependent Schrödinger equation. Each surface has a simple wave equation with its own potential function, although the situation is complicated by the coupling between the different electronic surfaces (see box for details). The coupling can come from either steady-state or pulsed laser excitation and its strength is determined by the laser intensity. But, despite the apparent simplicity of this problem, its numerical solution is far from trivial, and it can be insuructive to compare these results with those from well established quantum mechanical models.
Molecular potential surfaces often have a steep inner wall, where the molecular constituents experience strong repulsion, with the potential becoming flat for large atomic separations (figure 3a). The motion of a wave packet on a vibrationally excited surface is shown in figure 3b. The wave packet is a pictorial representation of the quantum mechanical probability distribuuon of the system at a given time. In figure 3a the packet is originally localised on the left slope of the potential; as it picks up speed it increases in width according to the rules of quantum mechanics, and is rather spread out at the right-hand slope. It then reverses direction and returns to its original position where it breaks up into oscillations (caused by interference between the incident and reflected parts of the packet) before reassembling itself and starting a second period. By its second encounter with the right-hand side of the potential, the packet has miraculously contracted to a slightly broadened version of its original shape. This is a manifestation of quantum mechanical coherence. In contrast to diffusive spreading, quantum mechanical spreading is a reversible process, and any loss of shape can be regained under suitable conditions. After several oscillations, however, the initially localised wave packet will have extended over all of the available range of the potential. This derives from the very nature of quantum evolution on anharmonic potentials.
3 (a) Wave packets (yellow) on a typical molecular potential energy surface (red). The wave packet is the modulus squared of the wave function. (b) The wave packet changes shape due to diffraction and spreading as it moves in the potential
There are, of course, many recent theoretical treatments of molecules in strong laser fields, but wave packet calculations have been used only in connection with experiments such as those described earlier.
Molecular dynamics
In Helsinki we have calculated the behaviour of a wave packet as it undergoes a series of laser-induced events and encounters. In figure 4 the wave packet is initially at rest in the ground state potential (1) where the local environment is almost a harmonic oscillator. The wave packet has a simple Gaussian shape and its width is determined by the width of the potential. When the laser is turned on (A), the wave packet is excited to level 2 where it is now vibrationally excited and it slides down the slope - picking up speed and spreading as predicted by quantum theory. At B a second laser introduces a level crossing between level 2 and a third potential energy surface (level 3). This coupling transfers part of the wave packet to level 3 and the split packet continues its vibrational evolution under the influence of two different potentials. However, there is a quantum coherence between the pieces, and this can be made visible when the wave packets are remixed at a second level crossing (C).
4 A laser excites a wave packet from the ground state (1) to an excited state (2) where it slides down the surface. This happens at an internuclear separation A. A second laser couples level 2 with another excited state (3). These two potential energy surfaces differ by exactly one laser photon energy at two internuclear separations, B and C. In the figure the energy of level 3 has been shifted by one photon energy to show the levels crossing. The wave packet gets redistributed between levels 2 and 3 at B and the subsequent dynamics of each wave packet now depends on the shape of the different surfaces. At C the two wave packets are remixed. Each wave packet now displays interference determined by the phase differences between the paths - this is the molecular analogue of a Michelson interferometer
Figure 5a shows the time development of the electronic excitation process labelled A in figure 4. The wave packet is initially at rest (i.e. with no vibrational energy) in the ground state (lower half of figure). A short laser pulse excites the wave packet to an electronically excited potential energy surface. When the time duration of the exciting pulse approaches zero the wave packet is excited to level 2 with very little change in its shape. However, this is totally the wrong shape for equilibrium, as the wave packet is localised at an energy where the proper eigenstates are either very spread out or dissociating. Therefore it starts moving down the surface. For pulses of finite duration, the shape of the packet may change because it slides down the slope and gets wider during the time of the pulse. Our numerical results can be compared with a simple excitation model proposed by Rosen and Zener in 1932. The Rosen-Zener model only breaks down when the laser pulse length is longer than the characteristic vibrational times of the two electronic levels (see Suominen et al. in Further reading).
Flgure 5b shows the splitting of population between levels 2 and 3 when the wave packet encounters the level crossing at B in figure 4. Before the level crossing (indicated by the broken line) all the wave packet is in level 2 (the lower sheet), but we can see a wave packet emerging in level 3 just after the level crossing. Since the splitting of the wave packet is governed by wave mechanics, both packets "remember" their interference in the interaction region and continue with a superimposed pulsation. The coupling between the levels has been chosen such that the probability is divided roughly equally between the two levels. The ability to control quantum mechanical behaviour in this way, and hence to steer molecular reactions, could be of key importance in laser chemical applications. We have also shown that an analytic model put forward by Landau and Zener in 1932 can describe the distribution of probability between the two levels accurately if the wave packet velocity through the level crossing is known (see Garraway and Stenholm 1991 in Further reading).
5 (a) Calculated probability densities for the ground (lowersheet) and first excited states of figure 4. Initially the wave packet is at rest in the ground state and there is no population in the excited state. A laser pulse then excites some of the wave packet into the excited state and the packet begins to move and spread (upper sheet) as shown in figure 3a. Yellow represents the highest probability density; red, green and blue represent progressively lower probabilities. The colour coding is the same for a, b and c but the actual value associated with a particular colour differs.
(b) The lower sheet now represents the first excited state (2) in figure 4; the upper sheet is level 3 and the dashed red line is the level crossing at B. The wave packet moves in the potential surface of level 2 until it encounters the level crossing and part of the probability is transferred to level 3. In this case the probability is divided roughly equally between the levels and coherent quantum processes leave the wave packets in pulsating states
(c) The dashed red line on the right is the level crossing at C. The oscillations between the crossings and enhanced pulsations after C can be seen. These so-called Stückelberg oscillations are caused by interference of the wave packets which have accumulated different phases on the different potential surfaces. By changing the parameters of the molecular configuration, one can monitor the interference pattern and obtain information about the potential energy surfaces
After the level crossing, both wave packets move independently on their separate electronic potentials (as long as the surfaces are well separated). However, full quantum mechanical coherence is preserved between both parts of the wave function, and the wave packets interfere when they meet at the second level crossing (C - see figure 5c). After this second crossing, both components of the vibrational wave packet have been influenced by the accumulated phase of the other one, and the emerging wave packets display strong pulsations.
The phase of each wave packet is a "memory" of its journey along its own potential, and when the wave packets overlap the interference pattern contains information on the different potentials. The accumulated phase difference along the two paths is analogous to the phase difference caused by different optical path lengths in a Michelson interferometer. It is also related to the idea which first led Schrödinger to consider particle paths in quantum mechanics as the analogue of the ray approximation to wave-front propagation in classical optics. The interferometric information contained in the pulsations of the wave packets could be used to determine the shape of unknown potential surfaces in molecules (see Garraway and Stenholm 1992 in Further reading). The technique would be highly indirect, but so are all the traditional methods such as analysis of scattering data, fitting to thermodynamic functions, or energy-level analysis based on spectroscopic data.
In standard quantum mechanics text books the phase of the wave function can be determined from the potential by the semi-classical Wentzel-Kramers-Brillouin (WKB) formula. We have combined this with the Landau-Zener treatment of level crossings to get an analytic model for the interferometric oscillations. This also allows us to see the effects of the phase char.tge induced within the crossings when we study the result of two consecutive crossings.This effect is not usually observable after only one crossing.
The final products of the laser-induced molecular reactions can be detected either directly, from dissociative fragments, or indirectly, by probing the different final states that levels 2 and 3 lead to. Spontaneous emission, a very sensitive detection mechanism that is sometimes carelessly called Raman scattering, can also be used. The evolving state may also be interrogated by a suitably timed laser pulse which excites the system into a short-lived state which then rapidly emits its energy at a frequency characterising the reaction product under scrutiny. The integrated spectrum of the emitted radiation may also carry interesting information about the dynamics of the time evolution (see Gruebele and Zewail in Further reading).
Outlook and limitations
We have been following the real-time evolution of laser-excited wave packets in molecules by integrating the Schrödinger equation on coupled electronic energy surfaces. In spite of many years of research into quantum mechanics, very few cases of genuine time dependence have been explored. Femtosecond lasers are now making the investigation of time-dependent quantum mechanical phenomena in molecules a reality. The calculations in figure 5 are related to recent work on Rydberg state wave packets in atoms (see the review by Alber and Zoller in Further reading). In the molecular case, however, much of the physics and the numerical work is different.
Our models are too simple to have a direct bearing on real molecules, as the computational effort required for calculations involving real configurations greatly exceeds what is currently available. However, the one-space-dimension approximation is rigorously true for diatomic molecules, so systematic investigations of generic situations may improve our understanding of quantum mechanics in the time domain. Comparison with wave packet computations also improves our knowledge of the accuracy and limitations of well established semi-classical approximations (see table).
Large deviations from semi-classical behaviour are expected to be connected with wild oscillations in the wave function, but this is where numerical methods fail. Thus we reach the almost paradoxical conclusion that we can compute the time evolution of wave functions only when they are fairly well localised and smooth. But this is just the limit where well developed semi-classical methods are adequate. Indeed, except for special situations, we have found that most of our numerical results can be described by a combination of semi-classical techniques. However, working at the boundary between classical physics and quantum physics in cases where the semi-classical description is inadequate, we can pinpoint the limitations of such methods.
The emergence of classical mechanics out of a semiclassical picture of quantum theory has been the centre of much interest recently. In particular, the semi-classical dynamics of classically chaotic systems provides valuable clues about the connection between the classical and quantum descriptions of the world. The work described here, however, does not include such complicated dynamical systems, as all classical motion is integrable in one dimension.
But the addition of internal degrees of freedom (in the form of potential energy surfaces, and coupling between these degrees of freedom) to a one-dimensional quantum system, leads to mathematically and numerically difficult problems with relevance to the interpretation of femtosecond experiments in real molecules. In this way we can begin to enhance our understanding of the intricate behaviour of time-dependent quantum mechanics.
---------------------------------------------------------- Physical situation Quantum mechanical model under investigation used for interpretation ---------------------------------------------------------- Wave packet excitation Rosen-Zener by short laser pulses Probabllity transfer at Landau-Zener level crossings Interference of accumulated phase in repeated crossings Landau-Zener and WKB ----------------------------------------------------------
and
, which obey the time-dependent
Schrödinger equation:
and 
are the
potential surfaces and the coupling, 
,
is proportional to the amplitude of the laser field and the electric dipole
matrix element between the levels. This type of Schrödinger equation is
often called a coupled channel problem.
The computational treatment of the equations above is complicated by numerical instability. In the most direct method of integrating the Schrödinger equation one replaces x, which is obviously continuous, by a discrete lattice with spacing a and uses the approximation:
This should be a good approximation for 
but it is well known
that the resulting integration procedure is unstable. A number of other methods
have been developed and we have found the so-called "split operator" method to
be the most reliable. Here one advances the wave function in time using,
alternatively, the kinetic part of the Hamiltonian and then the potential part.
This should be accurate enough for short time steps and is found to be stable.
The least trivial part of the operation is the kinetic propagation. The spatial
dependence of the wave functions has to be Fourier transformed so that the
kinetic energy becomes simply 
where k is the Fourier transform variable conjugate to x. After the kinetic
propagation one transforms back to x and repeats the process.
The slowest part of the whole procedure is the Fourier transform; even with modern fast Fourier transform routines, a powerful computer is needed to solve more than the very simplest problems in a reasonable time. To treat realistic molecular situations with many electronic levels and three spatial dimensions is totally beyond the capabilities of today's computers.