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Conical Intersections and Avoided Crossings of Electronic Energy LevelsGamble, Stephanie Nicole 14 January 2021 (has links)
We study the unique phenomena which occur in certain systems characterized by the crossing or avoided crossing of two electronic eigenvalues. First, an example problem will be investigated for a given Hamiltonian resulting in a codimension 1 crossing by implementing results by Hagedorn from 1994. Then we perturb the Hamiltonian to study the system for the corresponding avoided crossing by implementing results by Hagedorn and Joye from 1998. The results from these demonstrate the behavior which occurs at a codimension 1 crossing and avoided crossing and illustrates the differences. These solutions may also be used in further studies with Herman-Kluk propagation and more.
Secondly, we study codimension 2 crossings by considering a more general type of wave packet. We focus on the case of Schrödinger equation but our methods are general enough to be adapted to other systems with the geometric conditions therein. The motivation comes from the construction of surface hopping algorithms giving an approximation of the solution of a system of Schrödinger equations coupled by a potential admitting a conical intersection, in the spirit of Herman-Kluk approximation (in close relation with frozen/thawed approximations). Our main Theorem gives explicit transition formulas for the profiles when passing through a conical crossing point, including precise computation of the transformation of the phase and its proof is based on a normal form approach. / Doctor of Philosophy / We study energies of molecular systems in which special circumstances occur. In particular, when these energies intersect, or come close to intersecting. These phenomena give rise to unique physics which allows special reactions to occur and are thus of interest to study. We study one example of a more specific type of energy level crossing and avoided crossing, and then consider another type of crossing in a more general setting. We find solutions for these systems to draw our results from.
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A new class of coherent states and it's properties.Mohamed, Abdlgader January 2011 (has links)
The study of coherent states (CS) for a quantum mechanical system has
received a lot of attention. The definition, applications, generalizations of
such states have been the subject of work by researchers. A common starting
point of all these approaches is the observation of properties of the original
CS for the harmonic oscillator. It is well-known that they are described
equivalently as (a) eigenstates of the usual annihilation operator, (b) from
a displacement operator acting on a fundamental state and (c) as minimum
uncertainty states. What we observe in the different generalizations proposed
is that the preceding definitions are no longer equivalent and only some of
the properties of the harmonic oscillator CS are preserved.
In this thesis we propose to study a new class of coherent states and
its properties. We note that in one example our CS coincide with the ones
proposed by Glauber where a set of three requirements for such states has
been imposed. The set of our generalized coherent states remains invariant
under the corresponding time evolution and this property is called temporal
stability. Secondly, there is no state which is orthogonal to all coherent states (the coherent states form a total set). The third property is that we
get all coherent states by acting on one of these states [¿fiducial vector¿] with
operators. They are highly non-classical states, in the sense that in general,
their Bargmann functions have zeros which are related to negative regions of
their Wigner functions. Examples of these coherent states with Bargmann
function that involve the Gamma and also the Riemann ¿ functions are represented.
The zeros of these Bargmann functions and the paths of the zeros
during time evolution are also studied. / Libyan Cultural Affairs
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Gazeau-Klauder coherent states in one-mode systems with periodic potentialKonstadopoulou, Anastasia, Chountasis, S., Hollingworth, J.M., Vourdas, Apostolos, Backhouse, N.B. January 2001 (has links)
No / Gazeau-Klauder coherent states for a one-mode system with sinusoidal potential, are introduced. Their quantum statistical properties and their uncertainties are studied. The effect of dissipation on these states is estimated. The evolution of the ordinary (Glauber) coherent states in this system, is also studied. It is shown that these states evolve into superpositions of many macroscopically distinguishable states (`multi-Schrödinger cats').
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The groupoid of bifractional transformationsAgyo, Sanfo D., Lei, Ci, Vourdas, Apostolos 05 1900 (has links)
Yes / Bifractional transformations which lead to quantities that interpolate between other known quantities are considered. They do not form a group, and groupoids are used to describe their mathematical structure. Bifractional coherent states and bifractional Wigner functions are also defined. The properties of the bifractional coherent states are studied. The bifractional Wigner functions are used in generalizations of the Moyal star formalism. A generalized Berezin formalism in this context is also studied.
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Bi-fractional transforms in phase spaceAgyo, Sanfo David January 2016 (has links)
The displacement operator is related to the displaced parity operator through a two dimensional Fourier transform. Both operators are important operators in phase space and the trace of both with respect to the density operator gives the Wigner functions (displaced parity operator) and Weyl functions (displacement operator). The generalisation of the parity-displacement operator relationship considered here is called the bi-fractional displacement operator, O(α, β; θα, θβ). Additionally, the bi-fractional displacement operators lead to the novel concept of bi-fractional coherent states. The generalisation from Fourier transform to fractional Fourier transform can be applied to other phase space functions. The case of the Wigner-Weyl function is considered and a generalisation is given, which is called the bi-fractional Wigner functions, H(α, β; θα, θβ). Furthermore, the Q−function and P−function are also generalised to give the bi-fractional Q−functions and bi-fractional P−functions respectively. The generalisation is likewise applied to the Moyal star product and Berezin formalism for products of non-commutating operators. These are called the bi-fractional Moyal star product and bi-fractional Berezin formalism. Finally, analysis, applications and implications of these bi-fractional transforms to the Heisenberg uncertainty principle, photon statistics and future applications are discussed.
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Bi-fractional transforms in phase spaceAgyo, Sanfo D. January 2016 (has links)
The displacement operator is related to the displaced parity operator through a two dimensional
Fourier transform. Both operators are important operators in phase space
and the trace of both with respect to the density operator gives the Wigner functions
(displaced parity operator) and Weyl functions (displacement operator). The generalisation
of the parity-displacement operator relationship considered here is called
the bi-fractional displacement operator, O(α, β; θα, θβ). Additionally, the bi-fractional
displacement operators lead to the novel concept of bi-fractional coherent states.
The generalisation from Fourier transform to fractional Fourier transform can be
applied to other phase space functions. The case of the Wigner-Weyl function is considered
and a generalisation is given, which is called the bi-fractional Wigner functions,
H(α, β; θα, θβ). Furthermore, the Q−function and P−function are also generalised to
give the bi-fractional Q−functions and bi-fractional P−functions respectively. The
generalisation is likewise applied to the Moyal star product and Berezin formalism for
products of non-commutating operators. These are called the bi-fractional Moyal star
product and bi-fractional Berezin formalism.
Finally, analysis, applications and implications of these bi-fractional transforms
to the Heisenberg uncertainty principle, photon statistics and future applications are
discussed.
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Détection Optique Homodyne: application à la cryptographie quantiqueXu, Qing 28 April 2009 (has links) (PDF)
Les réseaux et systèmes de télécommunications mondiaux fondent aujourd'hui leur confidentialité sur la cryptographie classique, qui repose sur des hypothèses mathématiques fragiles. La distribution quantique de clef (QKD) est aujourd'hui la seule façon connue pour distribuer des clefs avec une sécurité inconditionnelle. Ce travail de thèse contribue à combler de manière pluridisciplinaire et polyvalente le gap entre les limites physiques fondamentales et l'implémentation expérimentale, en termes de vitesse, fiabilité et robustesse. Dans un premier temps, nous avons donc proposé une implémentation du protocole BB84 utilisant les états de phase cohérents. Le récepteur homodyne a été conçu de manière à compenser les fluctuations de phase et de polarisation dans les interféromètres, ainsi que dans le reste du canal de propagation. Ensuite, nous avons mis en place un dispositif expérimental de système QKD à la longueur d'onde 1550 nm, avec une modulation QPSK fonctionnant avec un trajet et un sens de parcours uniques, dans une fibre optique mono-mode. Les deux schémas de détection: le comptage de photons (PC) et la détection homodyne équilibrée (BHD) ont été mis en œuvre. Enfin, nous avons effectué des comparaisons théoriques et expérimentales de ces deux récepteurs. Le récepteur BHD a été élaboré avec une décision à double seuil. La mise en œuvre d'un tel processus accepte des mesures non-conclusives, et réduit l'efficacité de génération des clés, mais reste encore bien meilleur que celle des PCs à 1550 nm. Nous avons également prouvé que ce système est robust sous la plupart des attaques potentielles.
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O grupo unitário simplético: propriedades gerais e estados coerentes / The simplectic unitary group: general properties and choerent statesRamos, Alexandre Ferreira 21 September 2004 (has links)
Neste trabalho fizemos uma revisão geral e encontramos resultados novos sobre a simetria unitária simplética. Obtivemos uma fórmula simples para a exponencial da álgebra de Lie simplética complexa em quatro dimensões, sp(4, C). A partir da decomposição de Gauss do referido grupo, impusemos a unitariedade para obtermos expressões analíticas para esta decomposição. Ao impormos a condição unitária ao grupo simplético, formamos o grupo unitário simplético e obtivemos as regras de multiplicação deste grupo, as quais estão implementadas simbólicamente tendo em mente aplicações futuras. Como uma consequência encontramos uma representação da álgebra de Lie em termos de operadores diferenciais. Uma segunda e mais importante conseqüência foi a obtenção da métrica de Haar deste grupo, a qual é fundamental no estudo dos estados coerentes. Um rápido estudo da quebra de simetria entre a cadeia canônica e a cadeia de termos Majorana é apresentada no apêndice tendo em vista futuras aplicações ao estudo algébrico do código genético. Os estados coerentes do grupo Usp(4) foram calculados para uma representação arbitrária e a supercompleteza foi demonstrada devido a métrica de Haar, isto completa o programa iniciado por Novaes em sua tese de PhD. Os valores médios dos geradores da álgebra de Lie foram obtidos tendo em mente a aplicação a um hamiltoniano algébrico. Por fim, obtivemos a forma simplética numa representação arbitrária, preparando o campo para aplicações aos sistemas dinâmicos. / In this work we take a general revision and take new results on the unitary symplectic symmetry. We have obtained a simple form for the exponential of the complex symplectic Lie algebra on four dimensions, sp(4, C). With the Gauss decomposition for this group, we impose the unitarity to obtain analytical expressions for that Gauss decomposition. Imposing the analytical expressions to the Gauss decomposition for the complex symplectic algebra, we have been obtained explicit multiplication formulas for the unitarian group and iinplemented symbolically have in mind further application. As a consequence a representation of the Lie algebra in terms of differential operators have been obtained. The Haar measure that plays a fundamental role in the study of coherent states is calculated in an arbitrary representation. An early study envolving the symmetry breaking of canonical Sp(4) tree by Majorana operators is presented in the appendix in the spirit of algebraic approach to genetic code. The coherent states of USp (4) have been calculated for an arbitrary representation and the overcompletness is demonstred thanks to the Haar measure, the program initiate by Novaes in his PhD thesis is now fully completed. The mean values of the Lie algebra generators in a coherent state base are calculated having in mind application to algebraic hamiltonian. Finally we obtained the symplectic form in a arbitrary representation have also been calculate preparing the field for applications to dynamical systems.
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O grupo unitário simplético: propriedades gerais e estados coerentes / The simplectic unitary group: general properties and choerent statesAlexandre Ferreira Ramos 21 September 2004 (has links)
Neste trabalho fizemos uma revisão geral e encontramos resultados novos sobre a simetria unitária simplética. Obtivemos uma fórmula simples para a exponencial da álgebra de Lie simplética complexa em quatro dimensões, sp(4, C). A partir da decomposição de Gauss do referido grupo, impusemos a unitariedade para obtermos expressões analíticas para esta decomposição. Ao impormos a condição unitária ao grupo simplético, formamos o grupo unitário simplético e obtivemos as regras de multiplicação deste grupo, as quais estão implementadas simbólicamente tendo em mente aplicações futuras. Como uma consequência encontramos uma representação da álgebra de Lie em termos de operadores diferenciais. Uma segunda e mais importante conseqüência foi a obtenção da métrica de Haar deste grupo, a qual é fundamental no estudo dos estados coerentes. Um rápido estudo da quebra de simetria entre a cadeia canônica e a cadeia de termos Majorana é apresentada no apêndice tendo em vista futuras aplicações ao estudo algébrico do código genético. Os estados coerentes do grupo Usp(4) foram calculados para uma representação arbitrária e a supercompleteza foi demonstrada devido a métrica de Haar, isto completa o programa iniciado por Novaes em sua tese de PhD. Os valores médios dos geradores da álgebra de Lie foram obtidos tendo em mente a aplicação a um hamiltoniano algébrico. Por fim, obtivemos a forma simplética numa representação arbitrária, preparando o campo para aplicações aos sistemas dinâmicos. / In this work we take a general revision and take new results on the unitary symplectic symmetry. We have obtained a simple form for the exponential of the complex symplectic Lie algebra on four dimensions, sp(4, C). With the Gauss decomposition for this group, we impose the unitarity to obtain analytical expressions for that Gauss decomposition. Imposing the analytical expressions to the Gauss decomposition for the complex symplectic algebra, we have been obtained explicit multiplication formulas for the unitarian group and iinplemented symbolically have in mind further application. As a consequence a representation of the Lie algebra in terms of differential operators have been obtained. The Haar measure that plays a fundamental role in the study of coherent states is calculated in an arbitrary representation. An early study envolving the symmetry breaking of canonical Sp(4) tree by Majorana operators is presented in the appendix in the spirit of algebraic approach to genetic code. The coherent states of USp (4) have been calculated for an arbitrary representation and the overcompletness is demonstred thanks to the Haar measure, the program initiate by Novaes in his PhD thesis is now fully completed. The mean values of the Lie algebra generators in a coherent state base are calculated having in mind application to algebraic hamiltonian. Finally we obtained the symplectic form in a arbitrary representation have also been calculate preparing the field for applications to dynamical systems.
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Path integral formulation of dissipative quantum dynamicsNovikov, Alexey 06 June 2005 (has links) (PDF)
In this thesis the path integral formalism is applied to the calculation
of the dynamics of dissipative quantum systems.
The time evolution of a system of bilinearly coupled bosonic modes is
treated using the real-time path integral technique in
coherent-state representation.
This method is applied to a damped harmonic oscillator
within the Caldeira-Leggett model.
In order to get the stationary
trajectories the corresponding Lagrangian function is diagonalized and
then the path integrals are evaluated by means of the stationary-phase
method. The time evolution of the
reduced density matrix in the basis of coherent states is given in simple
analytic form for weak system-bath coupling, i.e. the so-called
rotating-wave terms can be evaluated exactly but the non-rotating-wave
terms only in a perturbative manner. The validity range of the
rotating-wave approximation is discussed from the viewpoint of spectral
equations. In addition, it is shown that systems
without initial system-bath correlations can exhibit initial jumps in the
population dynamics even for rather weak dissipation. Only with initial
correlations the classical trajectories for the system coordinate can be
recovered.
The path integral formalism in a combined phase-space and coherent-state
representation is applied to the problem of curve-crossing dynamics. The
system of interest is described by two coupled one-dimensional harmonic
potential energy surfaces interacting with a heat bath.
The mapping approach is used to rewrite the
Lagrangian function of the electronic part of the system. Using the
Feynman-Vernon influence-functional method the bath is eliminated whereas
the non-Gaussian part of the path integral is treated using the
perturbation theory in the small coordinate shift between
potential energy surfaces.
The vibrational and the population dynamics is considered in a lowest order of the perturbation.
The dynamics of a
Gaussian wave packet is analyzed along a one-dimensional reaction
coordinate.
Also the damping rate of coherence in the electronic part of the relevant system
is evaluated within the ordinary and variational perturbation theory.
The analytic expressions for the rate functions are obtained in
the low and high temperature regimes.
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