Coherent wave packet

As an example of this formulation we consider pulsed photoassociation of a coherent wave packet of cold Na atoms [345], The colliding atoms are described by an (energetically narrow) normalized Gaussian packet of J = 0 radial waves ... 

The first question, which is raised by this simple analogy, concerns the very possibility of exciting slow mode wave packets in a hydrogen bond at all. Taking a different perspective it touches the very issue of interpretation of the notoriously complex IR spectra [9]. In the condensed phase much of this complexity is hidden under bands broadened by the solvent interaction. Hence it was only recently that coherent wave packet motion of a 100 cm i hydrogen bond mode could be observed after OH-stretch excitation, although in a system which has only a single minimum potential [10]. Meanwhile coherent low-frequency dynamics has also been observed in a double minimum system (acetic acid dimer) [11]. With this... 

In the case of intramolecular double proton transfer a wavepacket motion is found which depends, via the excess energy, on the branching ratio between concerted double and single proton transfer. It demonstrates that the coherent wave-packet dynamics in ESIPT molecules is driven by the ESIPT itself and is specific for the reaction path. 

With the advent of femtosecond lasers, it became possible to observe in real time the actual motion of nuclei and to study the elementary mechanisms pictured by Bodenstein in his description of gas-phase reactions. In all branches of femto-chemistry, this study of elementarity is basic and is due to the inherent resolution achieved in femtochemical studies. Since the velocity of atoms in reactions is 1 km/sec, with 10 fs resolution the distance scale reached is 0,1 A, the atomic scale of motion. As discussed below, this ability to create such localized, coherent wave packets with the atomic scale of distance resolution was part of the development of quantum mechanics as a theoretical construct, but was not an experimental reality until the development of the required time resolution of motion in atoms, molecules, and reactions. 

In view of the foregoing discussion, one might ask what is a typical time evolution of the wave packet for the isolated molecule, what are typical tune scales and, if initial conditions are such that an entire energy shell participates, does the wave packet resulting from the coherent dynamics look like a microcanonical... 

Electron Nuclear Dynamics (48) departs from a variational form where the state vector is both explicitly and implicitly time-dependent. A coherent state formulation for electron and nuclear motion is given and the relevant parameters are determined as functions of time from the Euler equations that define the stationary point of the functional. Yngve and his group have currently implemented the method for a determinantal electronic wave function and products of wave packets for the nuclei in the limit of zero width, a "classical" limit. Results are coming forth protons on methane (49), diatoms in laser fields (50), protons on water (51), and charge transfer (52) between oxygen and protons. 

The ability to create and observe coherent dynamics in heterostructures offers the intriguing possibility to control the dynamics of the charge carriers. Recent experiments have shown that control in such systems is indeed possible. For example, phase-locked laser pulses can be used to coherently amplify or suppress THz radiation in a coupled quantum well [5]. The direction of a photocurrent can be controlled by exciting a structure with a laser field and its second harmonic, and then varying the phase difference between the two fields [8,9]. Phase-locked pulses tuned to excitonic resonances allow population control and coherent destruction of heavy hole wave packets [10]. Complex filters can be designed to enhance specific characteristics of the THz emission [11,12]. These experiments are impressive demonstrations of the ability to control the microscopic and macroscopic dynamics of solid-state systems. 

As the number of eigenstates available for coherent coupling increases, the dynamical behavior of the system becomes considerably more complex, and issues such as Coulomb interactions become more important. For example, over how many wells can the wave packet survive, if the holes remain locked in place If the holes become mobile, how will that affect the wave packet and, correspondingly, its controllability The contribution of excitons to the experimental signal must also be included [34], as well as the effects of the superposition of hole states created during the excitation process. These questions are currently under active investigation. 

Coherent optical phonons can couple with localized excitations such as excitons and defect centers. For example, strong exciton-phonon coupling was demonstrated for lead phtalocyanine (PbPc) [79] and Cul [80] as an intense enhancement of the coherent phonon amplitude at the excitonic resonances. In alkali halides [81-83], nuclear wave-packets localized near F centers were observed as periodic modulations of the luminescence spectra. 

Propagation in a medium of a coherent optical wave packet whose longitudinal and transverse sizes are both of a few wavelength and whose field amplitude can induce relativistic motion of electrons is a novel challenging topic to be investigated in the general field of the so-called relativistic optics [11]. Theory and simulation have been applied to this problem for a few decades. A number of experiments have been performed since ultrashort intense laser pulses became available in many laboratories. 

Describing complex wave-packet motion on the two coupled potential energy surfaces, this quantity is also of interest since it can be monitored in femtosecond pump-probe experiments [163]. In fact, it has been shown in Ref. 126 employing again the quasi-classical approximation (104) that the time-and frequency-resolved stimulated emission spectrum is nicely reproduced by the PO calculation. Hence vibronic POs may provide a clear and physically appealing interpretation of femtosecond experiments reflecting coherent electron transfer. We note that POs have also been used in semiclassical trace formulas to calculate spectral response functions [3]. 

In Eq. (109), y denotes an n-dimensional diagonal matrix, with element y being the width parameter for the coherent state of the y th dimension. The coordinate space representation of an n-dimensional coherent state is the product of n onedimensional minimum uncertainty wave packets... 

The matter wave function is formed as a coherent superposition of states or a state ensemble, a wave packet. As the phase relationships change the wave packet moves, and spreads, not necessarily in only one direction the localized launch configuration disperses or propagates with the wave packet. The initially localized wave packets evolve like single-molecule trajectories. 

The photochemical excitation delivered by a narrowly defined pump laser pulse achieves three indispensable things it sets time = 0, energizes the reactant molecules, and localizes them in space. It induces molecular coherence as excitation of each of the individual molecules involved leads to a coherent superposition of separate wave packets, a highly locahzed, geometrically well-defined and moving packet—analogous to a classical system, one that can be described using classical concepts of atomic positions and momentum. 

This experimental work on the dissociation of excited Nal clearly demonstrated behavior one could describe with the vocabulary and concepts of classical motions.The incoherent ensemble of molecules just before photoexcitation with a femtosecond laser pump pulse was transformed through the excitation into a coherent superposition of states, a wave packet that evolved as though it represented a single vibrationally activated molecule. 

Excitations of molecules with femtosecond laser pulses lead to excited-state matter wave packets coherently, launching them with such well-defined spatial resolution and coherence in nuclear motions that they evolve like single-molecule trajectories. Both electronically excited and vibrationally excited ground-state species may be studied. The structural change versus time profile of a reaction turns out to be compatible with classical modes of thinking. 

As illustrated earlier in the text (Figure 10.5), molecules released from the centrifuge generate an oscillatory Raman signal, characteristic of the coherent rotation with well-defined relative phase relation between the quantum states inside a rotational wave packet. Time-resolved coherent Raman response from a wave packet centered at A = 69 in oxygen is plotted at the bottom of Figure 10.9a. Knowing the wave packet composition from the state-resolved detection discussed above. 

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