Phd Degree / Doktora

Permanent URI for this collectionhttps://hdl.handle.net/11147/2869

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Author: search.filters.author.Pashaev, OktayLanguage: search.filters.language.en

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  • Doctoral Thesis
    Supersymmetric Coherent States and Superqubit Units of Quantum Information
    (01. Izmir Institute of Technology, 2024) Koçak Özvarol, Aygül; Pashaev, Oktay
    Bu tezde, hem fermiyonik hem de bozonik bileşenleri içeren, maksimum dolanık Bell tabanlı sÜper-eş uyumlu durumlar kümesini inceliyoruz. Aragone ve Zypmann tarafından tanıtılan süpersimetrik yok edici operatörü genişleterek, Bell iki-kübit kuantum durumlarıyla ilişkili dört farklı süpersimetrik eş uyumlu durum geliştiriyoruz. Bu Bell süper-kübit durumları, yer değiştirme operatörü kullanılarak inşa edilen Bell tabanlı süpersimetrik eş uyumlu durumların temelini oluşturur. Bu durumlar, süper-Bloch küresi üzerinde noktalar olarak temsil edilen ayrık bozonik eş uyumlu durumlarla birleştirildiğinde, ortaya çıkan yapıyı Bell tabanlı süper-eş uyumlu durumlar olarak adlandırılır.. Bozonik ve fermiyonik bileşenler arasındaki dolanıklık, süper-kübit referans durumu üzerinde etkili olan bir bozonik yer değiştirme operatörü aracılığıyla analiz edilir. Bu dolanık süper-eş uyumlu durumlar için belirsizlik ilişkileri concurrence $C$ ile ifade edilir. Belirsizlik ile concurrence arasındaki monoton ilişki, dolanıklığın belirsizlik ilişkileri üzerindeki etkisini göstermektedir. Daha sonra, konum ve momentum belirsizliklerinde kuadratür sıkışması gözlemliyoruz. Ayrıca, belirsizlik ilişkileri iki Fibonacci sayısının oranı ile karakterize edilen sonsuz bir süper-eş uyumlu durum dizisi tanımlıyoruz. Önceki sonuçları genelleştirmek amacıyla, tek bir süper-parçacık durumunun karmaşık bir parametre $\zeta$ ile tanımlandığı genel bir süper-kübit kuantum durumu tanıtıyoruz. Bu tanımlama, iki birim küre ile karakterize edilen PK-süper-kübit kuantum durumlarına yol açmaktadır. Bu durumlar, PK-süpersimetrik eş uyumlu durumlar olarak adlandırdığımız yapıların temelini oluşturur ve bu durumların dolanıklık özelliklerini inceliyoruz. Son olarak, pq-deforme süper-eş uyumlu durumları ve özel bir durum olarak q-deforme süper-eş uyumlu durumları ele alıyoruz.
  • Doctoral Thesis
    Quantum Calculus of Classical Heat-burgers' Hierarchy and Quantum Coherent States
    (Izmir Institute of Technology, 2017) Nalcı Tümer, Şengül; Pashaev, Oktay
    The purpose of this thesis is an application of quantum calculus to classical Heat- Burgers’ hierarchy and quantum coherent states. First we construct random walk on q-lattice, corresponding q-heat equation and exact solutions in terms of new family of q-exponential functions. Then we introduce a new type of q-diffusive heat equation and q-viscous Burgers’ equation, their polynomial solutions as generalized Kampe-de Feriet polynomials, corresponding dynamical symmetry and description in terms of Bell polynomials. Shock soliton solutions with fusion and fission of shocks are found and studied for different values of q. The q-semiclassical expansion of these equations in terms of Bernoulli polynomials is derived as corrections in power of ln q. A new class of complex valued function of complex argument as q-analytic functions in terms of q-analytic binomials is introduced and shown that these binomials are generalized analytic functions. As an application, we construct a new type of quantum states as q-analytic coherent states and corresponding q-analytic Fock- Bargmann representation. Then, we extend the concept of q-analytic function for two complex arguments, called double q-analytic functions, which has q-Hermite binomial expansion. As hyperbolic extension, we describe the q-analogue of traveling waves and find the D’Alembert solution of q-wave equation. By introducing q-translation operators we obtain q-binomials, q-analytic and q-anti analytic functions, q-travelling waves and non-commutative binomials. New type of quantum states as Hermite coherent states and Kampe-de Feriet coherent states are studied by generalization of the known Mehler formula. We introduce Golden quantum calculus, and as an application we study Golden quantum oscillator and its angular momentum representations.
  • Doctoral Thesis
    Entanglemend and Topological Soliton Structures in Heisenberg Spin Models
    (Izmir Institute of Technology, 2010) Gürkan, Zeynep Nilhan; Pashaev, Oktay
    Quantum entanglement and topological soliton characteristics of spin models are studied. By identifying spin states with qubits as a unit of quantum information, quantum information characteristic as entanglement is considered in terms of concurrence. Eigenvalues, eigenstates, density matrix and concurrence of two qubit Hamiltonian of XY Z, pure DM, Ising, XY , XX, XXX and XXZ models with Dzialoshinskii- Moriya DM interaction are constructed. For time evolution of two qubit states, periodic and quasiperiodic evolution of entanglement are found. Entangled two qubit states with exchange interaction depending on distance J(R) between spins and influence of this distance on entanglement of the system are considered. Different exchange interactions in the form of Calogero- Moser type I, II, III and Herring-Flicker potential which applicable to interaction of Hydrogen molecule are used. For geometric quantum computations, the geometric (Berry) phase in a two qubit XX model under the DM interaction in an applied magnetic field is calculated. Classical topological spin model in continuum media under holomorphic reduction is studied and static N soliton and soliton lattice configurations are constructed. The holomorphic time dependent Schrödinger equation for description of evolution in Ishimori model is derived. The influence of harmonic potential and bound state of solitons are studied. Relation of integrable soliton dynamics with multi particle problem of Calogero-Moser type is established and N soliton and N soliton lattice motion are found. Special reduction of Abelian Chern-Simons theory to complex Burgers. hierarchy, the Galilean group, dynamical symmetry and Negative Burgers. hierarchy are found.
  • Doctoral Thesis
    Enriched Finite Elements Method for Convevtion-Diffusion Problems
    (Izmir Institute of Technology, 2012) Şendur, Ali; Pashaev, Oktay
    In this thesis, we consider stabilization techniques for linear convection-diffusionreaction (CDR) problems. The survey begins with two stabilization techniques: streamline upwind Petrov-Galerkin method (SUPG) and Residual-free bubbles method (RFB). We briefly recall the general ideas behind them, trying to underline their potentials and limitations. Next, we propose a stabilization technique for one-dimensional CDR problems based on the RFB method and particularly designed to treat the most interesting case of small diffusion. We replace the RFB functions by their cheap, yet efficient approximations which retain the same qualitative behavior. The approximate bubbles are computed on a suitable sub-grid, the choice of whose nodes are critical and determined by minimizing the residual of a local problem. The resulting numerical method has similar stability features with the RFB method for the whole range of problem parameters. We also note that the location of the sub-grid nodes suggested by the strategy herein coincides with the one described by Brezzi and his coworkers. Next, the approach in one-dimensional case is extended to two-dimensional CDR problems. Based on the numerical experiences gained with this work, the pseudo RFBs retain the stability features of RFBs for the whole range of problem parameters. Finally, a numerical scheme for one-dimensional time-dependent CDR problem is studied. A numerical approximation with the Crank-Nicolson operator for time and a recent method suggested by Neslitürk and his coworkers for the space discretization is constructed. Numerical results confirm the good performance of the method.