Browsing by Author "Smit, Andre Marius"

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    Generalized time-frequency representations and their application to quantum control
    (Stellenbosch : Stellenbosch University, 2015-12) Smit, Andre Marius; Uys, Hermann; Rohwer, Erich G.; Stellenbosch University. Faculty of Science. Dept. of Physics.
    ENGLISH ABSTRACT: We present here a new generalized Time-Frequency-Represenation (TFR) for use in describing ultra-fast laser pulses. The TFR is developed in terms of Fourier- Hermite-Gauss (FHG) polynomials on a von Neumann TFR lattice, by expanding the lattice in a third dimension, thus forming a lattice cube. The temporal and spectral coefficients of the Hermite-Gauss (HG) clearly are inherently functionally Fourier transform invariant. The enhanced numerical complexity of the FHG TFR is greatly reduced by exploiting the translation property of Hermite polynomials by means of Pascal matrices. Although the new FHG TFR represents an over complete basis set, it can be reduced by a subset selection to a complete basis. This method and the accompanying overlap integral is then developed and the complex orthogonality and similarity of the temporal and spectral overlap integral matrices is then analytically proved. Numerically the Pascal matrices are unstable. The new two correlation FHG TFR lattice cubes present an improvement on the traditional TFRs in that they have the advantage that it is unnecessary to compute the undesirable inverse overlap matrix to reconstruct the signal, temporally and spectrally, i.e. they contain all the information necessary to reconstruct the signal. It is then proven that it is a digital requirement to double the original proposed bandwidths of the signal inputs, here and also for the von Neumann TFR. The Hermite-Gauss polynomials correspond to the number states {nk} of the Glauber-Sudarshan coherent states in Fock space. A classical correspondence analogy between Glauber-Sudarshan coherent states in Fock space and the temporal FHG TFR is then considered under certain conditions and thus allows for a comparison of the amplitudes between the two decompositions for each kkk = !k/c, culminating in an optimization procedure to determine a “classical” coherent state correlation TFR. Application simulation results of quantum coherent control of IR ultra-short laser pulse interaction with octahedral molecules utilizing an optimal genetic algorithm are presented. A representative shaped laser pulse is used throughout to compare various TFRs.