Master Degree / Yüksek Lisans Tezleri
Permanent URI for this collectionhttps://hdl.handle.net/11147/3008
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Master Thesis Entanglement and Invariance of Qubit-Qubit, Qubit-Qutrit and Qutrit-Qutrit Quantum States(01. Izmir Institute of Technology, 2024) Kızılkaya, Betül; Pashaev, OktayMevcut tez, saf iki kübit, kübit-kütrit ve iki kütrit durumlarının dolanıklık özelliklerinin incelenmesine ayrılmıştır. Dolanıklık, esasen bileşik kuantum durumlarının klasik olmayan bir özelliğidir ve kuantum hesaplamasında ve kuantum bilgi teorisinde önemli bir rol oynar. Durumların dolanıklığını karakterize etmek için, saf bileşik durumun dolanıklığını karışık azaltılmış yoğunluklu matrisle ilişkilendiren azaltılmış yoğunluk matrisi yaklaşımını kullanırız. Azaltılmış yoğunluk matrisinin Von Neumann entropisi ve karesel eşzamanlılık olarak doğrusal entropi, dolanıklığı ölçmek için kullanılır. Üniter bir kübit ve bir kütrit kapılarını kullanarak, dönüşümler altında dolanıklığın değişmezliğini gösteririz. Bu, aynı seviyedeki dolanık durumların sürekli olarak parametrelendirilmiş kümesini oluşturmamızı sağlar. Sonuçları, verilen karışık durum için arıtmanın hesaplanması ve manyetik alanda iki kübit spin XYZ modeli için ortalama enerjinin maksimum dolaşık minimumunu bulmak için uyguluyoruz.Master Thesis Algebraic Methods and Exact Solutions of Quantum Parametric Oscillators(Izmir Institute of Technology, 2019) Çetindaş, Osman; Atılgan Büyükaşık, Şirin; Pashaev, OktayIn this thesis, we study different approaches for solving the Schrödinger equation for quantum parametric oscillators. The Wei-Norman algebraic approach, the Lewis- Riesenfeld invariant approach, the Malkin-Manko-Trifonov approach are investigated. For each approach, the wave function solutions of the Schrödinger equation, the propagator and dynamical invariants are found and their relations with each other are shown. In the Wei-Norman Algebraic approach, for constructing wave functions, explicit form of evolution operator is obtained uniquely in terms of two linearly independent classical solutions of the corresponding classical equation of motion. In Lewis-Riesenfeld approach, quadratic invariants are found in terms of the solution of Ermakov-Pinney equation and using the eigenstates of these invariants, wave function solutions are constructed. Setting initial values for Ermakov-Pinney solution, results of Wei-Norman and Lewis- Riesenfeld approaches are compared, then this solution is expressed in terms of same two linearly independent classical solutions. In Malkin-Manko-Trifonov approach, linear invariants which are symmetry operators for the Schrödinger equation, are constructed in terms of complex-valued solutions of the classical equation. Using these invariants, quadratic invariants are constructed and their eigenstates are used to find wave function solutions. Moreover, initial values for complex solutions of classical equation of motion are posed, and comparison of the three approaches is given.Master Thesis Kaleidoscope of Quantum Coherent States and Units of Quantum Information(Izmir Institute of Technology, 2018) Koçak, Aygül; Pashaev, OktayIn the present thesis, we study superposition of coherent states as the kaleidoscope of quantum coherent states, associated with regular n-polygon symmetry and the roots of unity q2n = 1. These states are generalizations of the Schrödinger cat states, corresponding to the roots of unity q2 = −1. To describe physical characteristics of kaleidoscope states, we introduce new type of mod n exponential functions as a superposition of exponential functions in the form of discrete Fourier transform. These functions are also known as generalized hyperbolic functions, satisfying ordinary differential equations with proper initial conditions. Kaleidoscope states are eigenstates of n-th order eigenvalue problem for annihilation operator and are not minimal uncertainty states. These states are described as quantum Fourier transform of Glauber coherent states. Normalization factors, uncertainty relations, average number of photons and coordinate representation for these states are found in a compact form by mod n exponential functions. The set of kaleidoscope states, as orthonormal computatitonal basis of quantum states, describes generic qudit unit of quantum information. Relations of kaleidoscope states with quantum group symmetry are discussed. The special cases of trinity and quartet states, corresponding to qutrit and ququat units of quantum information are treated in details.Master Thesis Generalized Golden-Fibonacci Calculus and Applications(Izmir Institute of Technology, 2018) Özvatan, Merve; Pashaev, OktayIn the present thesis the Golden-Fibonacci calculus is developed and several applications of this calculus are obtained. The calculus is based on the Golden derivative as a finite difference operator with Golden and Silver ratio bases, which allowed us to introduce Golden polynomials and Taylor expansion in terms of these polynomials. The Golden binomial and its expansion in terms of Fibonomial coefficients is derived. We proved that Golden binomials coincide with Carlitz’ characteristic polynomials. By Golden Fibonacci exponential functions and related entire functions, the Golden-heat and the Golden-wave equations are introduced and solved. By introducing higher order Golden Fibonacci derivatives, related with powers of golden ratio, we develop the higher order Golden Fibonacci calculus. The higher order Fibonacci numbers, higher Golden periodic functions and higher Fibonomials appear as ingredients of this calculus. By using Golden Fibonacci exponential function, we introduce the generating function for new type of polynomials, the Bernoulli-Fibonacci polynomials and study their properties. As a geometrical application, the Apollonious type gaskets are described in terms of Fibonacci, Lucas and generalized Fibonacci numbers. Some mod 5 congruencies associated with Fibonacci and Lucas numbers are obtained.Master Thesis Apollonius Representation and Complex Geometry of Entangled Qubit States(Izmir Institute of Technology, 2018) Parlakgörür, Tuğçe; Pashaev, OktayIn present thesis, a representation of one qubit state by points in complex plane is proposed, such that the computational basis corresponds to two fixed points at a finite distance in the plane. These points represent common symmetric states for the set of quantum states on Apollonius circles. It is shown that, the Shannon entropy of one qubit state depends on ratio of probabilities and is a constant along Apollonius circles. For two qubit state and for three qubit state in Apollonius representation, the concurrence for entanglement and the Cayley hyperdeterminant for tritanglement correspondingly, are constant along Apollonius circles. Similar results are obtained also for n- tangle hyperdeterminant with even number of qubit states. It turns out that, for arbitrary multiple qubit state in Apollonius representation, fidelity between symmetric qubit states is also constant along Apollonius circles. According to these, the Apollonius circles are interpreted as integral curves for entanglement characteristics. For generic two qubit state in Apollonius representation, we formulated the reflection principle relating concurrence of the state, with fidelity between symmetric states. The Möbius transformations, corresponding to universal quantum gates are derived and Apollonius representation for multi-qubit states is generated by circuits of quantum gates. The bipolar and the Cassini representations for qubit states are introduced, and their relations with qubit coherent states are established. We proposed the differential geometry for qubit states in Apollonius representation, defined by the metric on a surface in conformal coordinates, as square of the concurrence. The surfaces of the concurrence, as surfaces of revolution in Euclidean and Minkowski (Pseudo-Euclidean) spaces are constructed. It is shown that, curves on these surfaces with constant Gaussian curvature becomes Cassini curves. The hydrodynamic interpretation of integral curves for concurrence as a flow in the plane is given and the spin operators in multiqubit |PP...P states are discussed.Master Thesis Resonance Solitons and Direct Methods in Soliton Theory(Izmir Institute of Technology, 2009) Duruk, Selin; Pashaev, Oktay; Pashaev, OktayThe Long-Short Wave interaction equations with adding quantum potential term and the Davey-Stewartson equation with addition of both, the quantum potential and the Hamiltonian terms are studied. These equations are reduced to different cases according to the choice of the quantum potential strength. For over critical case reductions to the non-linear diffusion-antidiffusion systems are derived. By the Hirota Direct Method one dissipaton solution of the system is derived. Two and three dissipaton (soliton) solutions are constructed explicitly. For special choice of the parameters they show the resonance character of interaction by fusion and fission of solitons.Master Thesis Nonlinear Euler Poisson Darboux Equations Exactly Solvable in Multidimensions(Izmir Institute of Technology, 2008) Ateş, Barış; Pashaev, OktayThe method of spherical means is the well known and elegant method of solving initial value problems for multidimensional PDE. By this method the problem reduced to the 1+1 dimensional one, which can be solved easily. But this method is restricted by only linear PDE and can not be applied to the nonlinear PDE. In the present thesis we study properties of the spherical means and nonlinear PDE for them. First we briefly review the main definitions and applications of the spherical means for the linear heat and the wave equations. Then we study operator representation for the spherical means, especially in two and three dimensional spaces. We find that the spherical means in complex space are determined by modified exponential function. We study properties of these functions and several applications to the heat equation with variable diffusion coefficient.Then nonlinear wave equations in the form of the Liouville equation, the Sine-Gordon equation and the hyperbolic Sinh-Gordon equations in odd space dimensions are introduced. By some combinations of functions we show that models are reducible to the 1+1 dimensional one on the half line.The Backlund transformations and exact particular solutions in the form of progressive waves are constructed. Then the initial value problem for the nonlinear Burgers equation and the Liouville equations are solved. Application of our solutions to spherical symmetric multidimensional problems is discussed.Master Thesis Integrable Vortex Dynamics and Complex Burgers' Equation(Izmir Institute of Technology, 2005) Gürkan, Zeynep Nilhan; Pashaev, OktayIntegrable dynamical models of the point magnetic vortex interactions in the plane are studied. Reformulating the Euler equations for vorticity in the Helmholtz form, the Hamiltonian and Lax representations are found. Reduction of these equations for the point vortices to the Kirchho equations, and non-integrability of the system of N 4 hydrodynamical vortices are discussed.As an integrable model of planar motion with given vorticity for the stationary and its solutions are given. For non-stationary planar vortex diffusion and exactly solvable Initial Value Problem for the one dimensional Burgers equation are solved.By the complexied Cole-Hopf transformation, the complex Burgers equation with integrable N vortex dynamics is introduced and linearization of this equation in terms of the complex Schr odinger equation is found.This allows us to construct N vortex congurations in terms of the complex Hermite polynomials, the vortex chain lattices and study their mutual dynamics. Mapping of our vortex problem to N-particle problem, the complexied Calogero-Moser system, showing its integrability and Hamiltonian structure is given. As an applicaton of the general results, we consider the problem of magnetic vortices in a magnetic fluid model. The holomorphic reduction of topological magnetic system to the linear complex Schrodinger equation, allows us to apply all results on integrable vortex dynamics in the complex Burgers equation to the magnetic vortex evolution, including magnetic vortex lattices and the bound states of vortices.Master Thesis Damping Oscillatory Models in General Theory of Relativity(Izmir Institute of Technology, 2007) Tığrak Ulaş, Esra; Pashaev, OktayIn my thesis we have studied the universe models as dynamical systems which can be represented by harmonic oscillators. For example, a harmonic oscillatior equation is constructed by the transformation of the Riccati differential equations for the anisotropic and homogeneous metric. The solution of the Friedman equations with the state equation satisfies both bosonic expansion and fermionic contraction in Friedman Robertson Walker universe with different curvatures is studied as a conservative system with the harmonic oscillator equations. Apart from the oscillator representations mentioned above (constructed from the universe models), we showed that the linearization of the Einstein field equations produces harmonic oscillator equation with constant frequency and the linearization of the metric on the de-Sitter background produces damped harmonic oscillator system. In addition to these, we have constructed the doublet and the Caldirola type oscillator equations with time dependent damping and frequency terms in the light of the Sturm Liouville form. The Lagrangian and Hamiltonian functions are calculated for all particular cases of the Sturm Liouville form. Finally, we have shown that zeros of the oscillator equations constructed from the particular cases can be transformed into pole singularities of the Riccati equations.Master Thesis Classical and Quantum Euler Equation(Izmir Institute of Technology, 2007) Eti, Neslihan; Pashaev, OktayIn the present thesis we give generalization of analytical mechanics to describe dynamical systems with dissipation. The Lagrangian function in this case is determined by nonstationary pseudo-Riemannian metric for the kinetic energy, and by general quadratic form, nondiagonal in the generalized coordinates and velocities. Skew symmetric nondiagonal terms in our approach play the role of dissipation coefficients. As an application we study in details the classical damped harmonic oscillator. We show that two known formulations of this oscillator, the Bateman dual and the Caldirola Kanai formulations are particular realizations of our general approach. The Hamiltonian formulation and quantization of the model in both representations are given. Moreover Ostrogradsky generalization of Lagrangian and Hamiltonian formalism for description of systems with higher order derivatives and its application to the constant coefficient equations of an arbitrary order are considered. We construct related with the last one the Euler differential equation of an arbitrary order and its Lagrangian and Hamiltonian structure. Quantum Euler systems are introduced and solved for the stationary Schrodinger picture. Nonstationary nonlinear quantum models corresponding to arbitrary Euler Hamiltonian are solved exactly in the Heisenberg picture.