Generalized time-frequency representations and their application to quantum control

Smit, Andre Marius (2015-12)

Thesis (PhD)--Stellenbosch University, 2015.

Thesis

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.

AFRIKAANSE OPSOMMING: Hier bied ons aan ’n nuwe veralgemende Tyd-Frekwensie-Voorstelling (TFV) vir die gebruik in die beskrywing van ultra-vinnige laser pulse. Die TFV word ontwikkel in terme van Fourier-Hermite-Gauss (FHG) polinome op ’n von Neumann TFV diskrete rooster, deur die normale rooster uit te brei in ’n derde dimensie, om ’n kubus rooster te vorm. Die temporale en spektrale koëffisiënte van die Hermite-Gauss (HG) polinome is duidelik inherent funksioneel Fourier transform invariant. Die verhoogde numeriese kompleksiteit van die FHG TFV word aansienlik verminder deur die ontginning van die translasie eienskap van Hermitiese polinome, deur gebruik te maak van Pascal matrikse. Hoewel die nuwe FHG TFV ’n oor volledige basis voorstel, kan dit deur ’n deelversameling selektief verminder word tot ’n volledige basis. Hierdie metode en die gepaardgaande oorvleuelings integrale word dan ontwikkel en die komplekse ortogonaliteit en similartieit van die temporale en spektrale oorvleueling integrale matrikse word dan analities bewys. Numeries is die Pascal matrikse onstabiel. Die nuwe twee korrelasie FHG TFR kubus roosters bied ’n verbetering op die tradisionele TFV in dat hulle die voordeel inhou dat dit onnodig is om die ongewenste omgekeerde oorvleuelings matriks te bereken om die sein te herkonstrueer, temporaal en spektraal, d.w.s. dit bevat al die nodige inligting om die sein te herkonstrueer. Daar word dan bewys dat dit ’n digitale vereiste is om die oorspronklike voorgestelde bandwydtes van die sein insette te verdubbel, hier en ook vir die von Neumann TFV. Die Hermite-Gauss polinome stem ooreen met die aantal getal toestande {nk} van die Glauber-Sudarshan koherente toestande in die Fock ruimte. ’n Analogie tussen Glauber-Sudarshan koherente toestande in Fock ruimte en die temporale FHG TFV word dan beskou onder sekere omstandighede om gevolglik ’n vergelyking van die amplitudes tussen die twee ontbindings vir elke kkk = !k/c, wat uiteindelik kulmineer in ’n optimalisering proses om ’n “klasieke” koherente toestand korrelasie TFV te bepaal. Simulasie resultate van ’n toepassing van kwantum koherente toestand beheer van IR ultra-kort laser pulse se interaksie met oktahedriese molekules, deur gebruik te maak van ’n optimale genetiese algoritme, word dan aangebied. ’n Verteenwoordigende gevormde laser puls word deurgaans gebruik om verskeie TFVs te vergelyk.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/97845
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