Coherence of atomic systems

Two levels of an atom are said to be coherently excited if their corresponding wave functions are in phase at the excitation time. With a short laser pulse of duration Al which has a Fourier-limited spectral bandwidth Aa l/Af, two atomic levels a and b can be excited simultaneously if their energy separation AF is smaller than hl co (Fig. 2.29). The wave function of the excited atom is then a linear combination of the wave functions lfa and and the atom is said to be in a coherent superposition of the two states a) and b). 

An ensemble of atoms is coherently excited if the wave functions of the excited atoms, at a certain time t, have the same phase for all atoms. This phase relation may change with time due to differing frequencies 00 in the time-dependent part exp(iu 0 of the excited-state wave functions or because of relaxation processes, which may differ for the different atoms. This will result in a phase diffusion and a time-dependent decrease of the degree of coherence. 

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If definite phase relations exist between the wave functions of the atoms, the systen is in a coherent state. The nondiagonal elements of (2.158) describe the degree of coherence of the system and are therefore often called coherences. ... 

Despite the difficulty cited, the study of the vibrational spectrum of a liquid is useful to the extent that it is possible to separate intramolecular and inter-molecular modes of motion. It is now well established that the presence of disorder in a system can lead to localization of vibrational modes 28-34>, and that this localization is more pronounced the higher the vibrational frequency. It is also well established that there are low frequency coherent (phonon-like) excitations in a disordered material 35,36) These excitations are, however, heavily damped by virtue of the structural irregularities and the coupling between single molecule diffusive motion and collective motion of groups of atoms. 

The scheme we employ uses a Cartesian laboratory system of coordinates which avoids the spurious small kinetic and Coriolis energy terms that arise when center of mass coordinates are used. However, the overall translational and rotational degrees of freedom are still present. The unconstrained coupled dynamics of all participating electrons and atomic nuclei is considered explicitly. The particles move under the influence of the instantaneous forces derived from the Coulombic potentials of the system Hamiltonian and the time-dependent system wave function. The time-dependent variational principle is used to derive the dynamical equations for a given form of time-dependent system wave function. The choice of wave function ansatz and of sets of atomic basis functions are the limiting approximations of the method. Wave function parameters, such as molecular orbital coefficients, z,(f), average nuclear positions and momenta, and Pfe(0, etc., carry the time dependence and serve as the dynamical variables of the method. Therefore, the parameterization of the system wave function is important, and we have found that wave functions expressed as generalized coherent states are particularly useful. A minimal implementation of the method [16,17] employs a wave function of the form ... 

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. 

SLI is not specific to molecular eigenstates, but universal to the superposition of any eigenstates in a variety of quantum systems. It is thus expected as a new tool for quantum logic gates not only in MEIP but also for other systems such as atoms, ions, and quantum dots. SLI also provides a new method to manipulate WPs with fs laser pulses in general applications of coherent control. 

A continuous laser operates by continually pumping atoms or moie-cules into the excited state from which induced decay produces a continuous beam of coherent radiation. The He—Ne laser is an example of continuous system. Another mode of operation is to apply an energy pulse to the system, exciting a considerable fraction into the excited state. When all these molecules or atoms are induced to decay simultaneously, intense but exteremely short pulse of coherent radiation is emitted. The ruby laser falls in this category. 

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