Coherent superposition state motion

From a theoretical perspective, the object that is initially created in the excited state is a coherent superposition of all the wavefunctions encompassed by the broad frequency spread of the laser. Because the laser pulse is so short in comparison with the characteristic nuclear dynamical time scales of the motion, each excited wavefunction is prepared with a definite phase relation with respect to all the others in the superposition. It is this initial coherence and its rate of dissipation which determine all spectroscopic and collisional properties of the molecule as it evolves over a femtosecond time scale. For IBr, the nascent superposition state, or wavepacket, spreads and executes either periodic vibrational motion as it oscillates between the inner and outer turning points of the bound potential, or dissociates to form separated atoms, as indicated by the trajectories shown in Figure 1.3. 

From the point of view of the study of dynamics, the laser has three enormously important characteristics. Firstly, because of its potentially great time resolution, it can act as both the effector and the detector for dynamical processes on timescales as short as 10 - s. Secondly, due to its spectral resolution and brightness, the laser can be used to prepare large amounts of a selected quantum state of a molecule so that the chemical reactivity or other dynamical properties of that state may be studied. Finally, because of its coherence as a light source the laser may be used to create in an ensemble of molecules a coherent superposition of states wherein the phase relationships of the molecular and electronic motions are specified. The dynamics of the dephasing of the molecular ensemble may subsequently be determined. 

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. 

Figure 4. Schematic of the potential energy curves of the relevant electronic states The pump pulse prepares a coherent superposition of vibrational states in the electronic A 1 EJ state at the inner turning point. Around v = 13 this state is spin-orbit coupled with the dark b 3n state, causing perturbations. A two-photon probe process transfers the wavepacket motion into the ionization continuum via the (2) llg state [7].

A Schrbdinger-cat state is taken to be a coherent superposition of classical-like motional states. In Schrodinger s original thought experiment [30], he described how we could, in principle, entangle a superposition state of an atom with a macroscopic-scale superposition of a live and dead cat. In our experiment [24], we construct an analogous state, on a snuiller scale, with a single atom. We create the state... 

As described above, a pulsed laser with a sufficiently narrow pulse width will overlap a group of vibrational/rotational eigenstates and will prepare a coherent superposition, Eq. (4.5), of some or all of these eigenstates in a molecule. This superposition state evolves in time and is thus identified as (t). Its time dependence is given by the equation of motion (Merzbacher, 1970) ... 

The principle of the pump-probe observation scheme is plotted in Fig. 1. The clusters are excited from a low vibrational level of the electronic ground state to an excited state. Due to the spectral width of the employed fs-pump pulse, a coherent superposition of several vibrational states is created, which leads to the formation of an evolving wave packet. If a bound electronic state is excited [Fig. 1(a)], this wave packet will oscillate between the inner and outer turning point of the potential energy curve or surface, reflecting the vibrational motion of the excited molecule. The temporal evolution of this... 

The article is organized as follows in Sec. 2 we outline the scheme of a pump/probe experiment which is performed with two ultrashort laser pulses. Then the excitation mechanism which prepares the system in a coherent superposition of eigenstates in the transition-state region of the respective potential surface is described (Sec. 3). This localized wave packet evolves along the reaction path, as described in Sec. 4. In Sec. 5 it is shown how the wave-packet motion can be probed. A short summary concludes this chapter. 

The results presented in this chapter show that the use of proper effective models, in combination with calculations based on the exact vibrational Hamiltonian, constitutes a promising approach to study the laser driven vibrational dynamics of polyatomic molecules. In this context, the MCTDH method is an invaluable tool as it allows to compute the laser driven dynamics of polyatomic molecules with a high accuracy. However, our models still contain simplifications that prevent a direct comparison of our results with potential experiments. First, the rotational motion of the molecule was not explicitly described in the present work. The inclusion of the rotation in the description of the dynamics of the molecule is expected to be important in several ways. First, even at low energies, the inclusion of the rotational structure would result in a more complicated system with different selection rules. In addition, the orientation of the molecule with respect to the laser field polarization would make the control less efficient because of the rotational averaging of the laser-molecule interaction and the possible existence of competing processes. On the other hand, the combination of the laser control of the molecular alignment/orientation with the vibrational control proposed in this work could allow for a more complete control of the dynamics of the molecule. A second simplification of our models concerns the initial state chosen for the simulations. We have considered a molecule in a localized coherent superposition of vibrational eigenstates but we have not studied the preparation of this state. We note here that a control scheme for the localiza-... 

One of the most interesting applications of Femtochemistry is the stroboscopic measuring of observables related to molecular motion, for instance the vibrational periods or the breaking of a bond [1], Because femtosecond laser fields are broadband, a wave packet is created by the coherent excitation of many vibrational states, which subsequently evolves in the electronic potential following mostly a classical trajectory. This behavior is to be contrasted to narrow band selective excitation, where perhaps only two (the initial and the final) states participate in the superposition, following typically a very non-classical evolution. In this case one usually is not interested in the evolution of other observables than the populations. 

The equation of motion (115) allows us to analyze conditions for population trapping in the driven A system. In the steady state (p = 0) with p / 1 and Ac = 0 the population in the upper state p33 = 0. Thus the state 3) is not populated even though it is continuously driven by the laser. In this case the population is entirely trapped in the antisymmetric superposition of the ground states. This is the CPT effect. However, for p 1 and Ac = 0, the antisymmetric state decouples from the interactions, and then the steady-state population p33 is different from zero [46]. This shows that coherent population trapping is possible... 

We have created and analyzed thermal, Fock, squeezed, coherent, Schrodinger-cat states, and other superpositions of Fock states [21,24,25] here we briefly describe the creation and measurement of coherent and Schrddinger-cat states [21,24], We note that a scheme recently proposed for producing arbitrary states of the electromagnetic field [26] should be directly applicable to the ion case for producing arbitrary states of motion. 

For the last decade or so, a new method has been developed to control chemical reactions that it is based on the wave nature of atoms and molecules. The new methodology is called quantum control , or coherent control of chemical reactions, and is based on the coherent excitation of the molecule by a laser. Generally speaking, an ultra-short laser pulse creates a wave packet whose time evolution describes the molecular evolution in the superposition of excited states. Quantum control tries to modify the superposition of such an ensemble of excited states and, therefore, influences the motion of the wave... 

SET is sometimes categorized as coherent limit in the sense we mentioned in the previous subsection the nuclear motion keeps on a single trajectory and the electronic state evolves coherently as superposition of various electronic state, which makes no sense in the asymptotic region after wavepacket bifurcation. (We will discuss what the coherence is all about later in this book.) Although being derived in rather intuitive manner, SET can also be derived from Pechukas formulation with an (often not correct) additional assumption. 

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