arXiv subjects

Oktay K. Pashaev

Publications and source records attributed to Oktay K. Pashaev.

At least 19 recordsLinked to original sources

arXiv2024

Quantum Calculus of Fibonacci Divisors and Fermion-Boson Entanglement for Infinite Hierarchy of N = 2 Supersymmetric Golden Oscillators

The quantum calculus with two bases, as powers of the Golden and the Silver ratio, relates Fibonacci divisor derivative with Binet formula of Fibonacci divisor number operator, acting in Fock space of quantum states.It provides a tool to study the hierarchy of Golden oscillators with energy spectrum in form of Fibonacci divisor numbers. We generalize this model to supersymmetric number operator and corresponding Binet formula for supersymmetric Fibonacci divisor number operator. The operator determines the Hamiltonian of hierarchy of supersymmetric Golden oscillators, acting in fermion-boson Hilbert space and belonging to N=2 supersymmetric algebra. The eigenstates of the super Fibonacci divisor number operator are double degenerate and can be characterized by a point on the super-Bloch sphere. By the supersymmetric Fibonacci divisor annihilation operator, we construct the hierarchy of supersymmetric coherent states as eigenstates of this operator. Entanglement of fermions with bosons in these states is calculated by the concurrence, represented by the Gram determinant and hierarchy of Golden exponential functions. We show that the reference states and corresponding von Neumann entropy, measuring fermion-boson entanglement, are characterized completely by the powers of the Golden ratio. The simple geometrical classification of entangled states by the Frobenius ball and meaning of the concurrence as double area of parallelogram in Hilbert space are given.

quant-ph↗

arXiv2024

Geometry and Entanglement of Super-Qubit Quantum States

We introduce the super-qubit quantum state, determined by superposition of the zero and the one super-particle states, which can be represented by points on the super-Bloch sphere. In contrast to the one qubit case, the one super-particle state is characterized by points in extended complex plain, equivalent to another super-Bloch sphere. Then, geometrically, the super-qubit quantum state is represented by two unit spheres, or the direct product of two Bloch spheres. By using the displacement operator, acting on the super-qubit state as the reference state, we construct the super-coherent states, becoming eigenstates of the super-annihilation operator, and characterized by three complex numbers, the displacement parameter and stereographic projections of two super-Bloch spheres. The states are fermion-boson entangled, and the concurrence of states is the product of two concurrences, corresponding to two Bloch spheres. We show geometrical meaning of concurrence as distance from point-state on the sphere to vertical axes - the radius of circle at horizontal plane through the point-state. Then, probabilities of collapse to the north pole state and to the south pole state are equal to half-distances from vertical coordinate of the state to corresponding points at the poles. For complimentary fermion number operator, we get the flipped super-qubit state and corresponding super-coherent state, as eigenstate of transposed super-annihilation operator. The infinite set of Fibonacci oscillating circles in complex plain, and corresponding set of quantum states with uncertainty relations as the ratio of two Fibonacci numbers, and in the limit at infinity becoming the Golden Ratio uncertainty, is derived.

quant-ph↗

arXiv2021

PQ-Calculus of Fibonacci Divisors and Method of Images in Planar Hydrodynamics

By introducing the hierarchy of Fibonacci divisors and corresponding quantum derivatives, we develop the golden calculus, hierarchy of golden binomials and related exponential functions, translation operator and infinite hierarchy of Golden analytic functions. The hierarchy of Golden periodic functions, appearing in this calculus we relate with the method of images in planar hydrodynamics for incompressible and irrotational flow in bounded domain. We show that the even hierarchy of these functions determine the flow in the annular domain, bounded by concentric circles with the ratio of radiuses in powers of the Golden ratio. As an example, complex potential and velocity field for the set of point vortices with Golden proportion of images are calculated explicitly.

math-ph↗

arXiv2021

Quantum Coin Flipping, Qubit Measurement and Generalized Fibonacci Numbers

The problem of Hadamard quantum coin measurement in $n$ trials, with arbitrary number of repeated consecutive last states is formulated in terms of Fibonacci sequences for duplicated states, Tribonacci numbers for triplicated states and $N$-Bonacci numbers for arbitrary $N$-plicated states. The probability formulas for arbitrary position of repeated states are derived in terms of Lucas and Fibonacci numbers. For generic qubit coin, the formulas are expressed by Fibonacci and more general, $N$-Bonacci polynomials in qubit probabilities. The generating function for probabilities, the Golden Ratio limit of these probabilities and Shannon entropy for corresponding states are determined. By generalized Born rule and universality of $n$-qubit measurement gate, we formulate problem in terms of generic $n$-qubit states and construct projection operators in Hilbert space, constrained on the Fibonacci tree of the states. The results are generalized to qutrit and qudit coins, described by generalized Fibonacci-$N$-Bonacci sequences.

quant-ph↗

arXiv2020

Bernoulli-Fibonacci Polynomials

By using definition of Golden derivative, corresponding Golden exponential function and Fibonomial coefficients, we introduce generating functions for Bernoulli-Fibonacci polynomials and related numbers. Properties of these polynomials and numbers are studied in parallel with usual Bernoulli counterparts. Fibonacci numbers and Golden ratio are intrinsically involved in formulas obtained.

math.CO↗

arXiv2020

Quantum Calculus of Fibonacci Divisors and Infinite Hierarchy of Bosonic-Fermionic Golden Quantum Oscillators

Starting from divisibility problem for Fibonacci numbers we introduce Fibonacci divisors, related hierarchy of Golden derivatives in powers of the Golden Ratio and develop corresponding quantum calculus. By this calculus, the infinite hierarchy of Golden quantum oscillators with integer spectrum determined by Fibonacci divisors, the hierarchy of Golden coherent states and related Fock-Bargman representations in space of complex analytic functions are derived. It is shown that Fibonacci divisors with even and odd $k$ describe Golden deformed bosonic and fermionic quantum oscillators, correspondingly. By the set of translation operators we find the hierarchy of Golden binomials and related Golden analytic functions, conjugate to Fibonacci number $F_k$. In the limit k -> 0, Golden analytic functions reduce to classical holomorphic functions and quantum calculus of Fibonacci divisors to the usual one. Several applications of the calculus to quantum deformation of bosonic and fermionic oscillator algebras, R-matrices, hydrodynamic images and quantum computations are discussed.

math-ph↗

arXiv2018

Special functions with mod n symmetry and kaleidoscope of quantum coherent states

The set of mod $n$ functions associated with primitive roots of unity and discrete Fourier transform is introduced. These functions naturally appear in description of superposition of coherent states related with regular polygon, which we call kaleidoscope of quantum coherent states. Displacement operators for kaleidoscope states are obtained by mod $n$ exponential functions with operator argument and non-commutative addition formulas. Normalization constants, average number of photons, Heinsenberg uncertainty relations and coordinate representation of wave functions with mod n symmetry are expressed in a compact form by these functions.

quant-ph↗

arXiv2017

Generalized Fibonacci Sequences and Binet-Fibonacci Curves

We have studied several generalizations of Fibonacci sequences as the sequences with arbitrary initial values, the addition of two and more Fibonacci subsequences and Fibonacci polynomials with arbitrary bases. For Fibonacci numbers with congruent indices we derived general formula in terms of generalized Fibonacci polynomials and Lucas numbers. By extending Binet formula to arbitrary real numbers we constructed Binet-Fibonacci curve in complex plane. For positive numbers the curve is oscillating with exponentially vanishing amplitude, while for negative numbers it becomes Binet Fibonacci spiral. Comparison with the nautilus curve shows quite similar behavior. Areas under the curve and curvature characteristics are calculated, as well as the asymptotic of relative characteristics. Asymptotically for n going to infinity the region become, infinitely wide with infinitesimally small height so that the area has finite value $A_{\infty}=\frac{2}{5 π} ln(φ)$.

math.HO↗

arXiv2017

Relativistic DNLS and Kaup-Newell Hierarchy

By the recursion operator of the Kaup-Newell hierarchy we construct the relativistic derivative NLS (RDNLS) equation and the corresponding Lax pair. In the nonrelativistic limit $c \rightarrow \infty$ it reduces to DNLS equation and preserves integrability at any order of relativistic corrections. The compact explicit representation of the linear problem for this equation becomes possible due to notions of the $q$-calculus with two bases, one of which is the recursion operator, and another one is the spectral parameter.

nlin.SI↗

arXiv2017

q-Viscous Burgers' Equation: Dynamical Symmetry, Shock Solitons and q-Semiclassical Expansion

We propose new type of $q$-diffusive heat equation with nonsymmetric $q$-extension of the diffusion term. Written in relative gradient variables this system appears as the $q$- viscous Burgers' equation. Exact solutions of this equation in polynomial form as generalized Kampe de Feriet polynomials, corresponding dynamical symmetry and description in terms of Bell polynomials are derived. We found the generating function for these polynomials by application of dynamical symmetry and the Zassenhaus formula. We have constructed and analyzed shock solitons and their interactions with different $q$. We obtain modification of the soliton relative speeds depending on value of $q$.For $q< 1$ the soliton speed becomes bounded from above and as a result in addition to usual Burgers soliton process of fusion, we found a new phenomena, when soliton with higher amplitude but smaller velocity is fissing to two solitons. q-Semiclassical expansion of these equations are found in terms of Bernoulli polynomials in power of $\ln q$.

nlin.SI↗

arXiv2017

Nonlinear Cauchy-Riemann Equations and Liouville Equation For Conformal Metrics

We introduce the Nonlinear Cauchy-Riemann equations as Bäcklund transformations for several nonlinear and linear partial differential equations. From these equations we treat in details the Laplace and the Liouville equations by deriving general solution for the nonlinear Liouville equation. By Möbius transformation we relate solutions for the Poincare model of hyperbolic geometry, the Klein model in half-plane and the pseudo-sphere. Conformal form of the constant curvature metrics in these geometries, stereographic projections and special solutions are discussed. Then we introduce the hyperbolic analog of the Riemann sphere, which we call the Riemann pseudosphere. We identify point at infinity on this pseudosphere and show that it can be used in complex analysis as an alternative to usual Riemann sphere to extend the complex plane. Interpretation of symmetric and antipodal points on both, the Riemann sphere and the Riemann pseudo-sphere, are given. By Möbius transformation and homogenous coordinates, the most general solution of Liouville equation as discussed by Crowdy is derived.

nlin.SI↗

arXiv2017

Kaleidoscope of Quantum Coherent States

The Schrödinger cat states, constructed from Glauber coherent states and applied for description of qubits, are generalized to the kaleidoscope of coherent states related with regular n-polygon symmetry and the roots of unity. The cases of the trinity states and the quartet states are described in details. Normalization formula for these states requires introduction of specific combinations of exponential functions with mod 3 and mod 4 symmetry. We show that for an arbitrary $n$, these states can be generated by the Quantum Fourier transform and can provide qutrits, ququats and in general, qudit units of quantum information. Relations with quantum groups and quantum calculus are discussed.

quant-ph↗

arXiv2017

Apollonius Representation of Qubits

We introduce the qubit representation by complex numbers on the set of Apollonius circles with common symmetric points at $0$ and $1$, related with $|0\rangle$ and $|1\rangle$ states. For one qubit states we find that the Shannon entropy as a measure of randomness is a constant along Apollonius circles. For two qubit states, the concurence as a characteristic of entanglement is taking constant value for the states on the same Apollonius circle. Geometrical meaning of concurence as an area and as a distance in the Apollonius representation are found. Then we generalize our results to arbitrary $n$-qubit Apollonius states and show that the fidelity between given state and the symmetric one, as reflected in an axes, is a constant along Apollonius circles. For two qubits it coinsides with the concurence. For generic two qubit states we derived Apollonius representation by three complex parameters and show that the determinant formula for concurence is related with fidelity for symmetric states by two reflections in a vertical axis and inversion in a circle. We introduce the complex concurence and an addition formula for Apollonius states and show that for generic two qubit states its modulus satisfies the law of cosine. Finally, we show that for two qubit Apollonius state in bipolar coordinates, the complex concurence is decribed by static one soliton solution of the nonlinear Schrödinger equation.

quant-ph↗

arXiv2016

Double q-Analytic q-Hermite Binomial Formula and q-Traveling Waves

Motivated by derivation of the Dirac type delta-function for quantum states in Fock-Bargmann representation, we find q-binomial expansion in terms of q-Hermite polynomials, analytic in two complex arguments. Based on this representation, we introduce a new class of complex functions of two complex arguments, which we call the double q-analytic functions. The real version of these functions describe the q-analogue of traveling waves, which is not preserving the shape during evolution as the usual traveling wave. For corresponding q-wave equation we solve IVP in the q-D'Alembert form.

math.QA↗

arXiv2014

Variations on a theme of q-oscillator

We present several ideas in direction of physical interpretation of $q$- and $f$-oscillators as a nonlinear oscillators. First we show that an arbitrary one dimensional integrable system in action-angle variables can be naturally represented as a classical and quantum $f$-oscillator. As an example, the semi-relativistic oscillator as a descriptive of the Landau levels for relativistic electron in magnetic field is solved as an $f$-oscillator. By using dispersion relation for $q$-oscillator we solve the linear q-Schrödinger equation and corresponding nonlinear complex q-Burgers equation. The same dispersion allows us to construct integrable q-NLS model as a deformation of cubic NLS in terms of recursion operator of NLS hierarchy. Peculiar property of the model is to be completely integrable at any order of expansion in deformation parameter around $q=1$. As another variation on the theme, we consider hydrodynamic flow in bounded domain. For the flow bounded by two concentric circles we formulate the two circle theorem and construct solution as the q-periodic flow by non-symmetric $q$-calculus. Then we generalize this theorem to the flow in the wedge domain bounded by two arcs. This two circular-wedge theorem determines images of the flow by extension of $q$-calculus to two bases: the real one, corresponding to circular arcs and the complex one, with $q$ as a primitive root of unity. As an application, the vortex motion in annular domain as a nonlinear oscillator in the form of classical and quantum f-oscillator is studied. Extending idea of q-oscillator to two bases with the golden ratio, we describe Fibonacci numbers as a special type of $q$-numbers with matrix Binet formula. We derive the corresponding golden quantum oscillator, nonlinear coherent states and Fock-Bargman representation.

math-ph↗

arXiv2014

Bianchi I Model: An Alternative Way To Model The Presentday Universe

Although the new era of high precision cosmology of the cosmic microwave background (CMB) radiation improves our knowledge to understand the infant as well as the presentday Universe, it also leads us to question the main assumption of the exact isotropy of the CMB. There are two pieces of observational evidence that hint towards there being no exact isotropy. These are first the existence of small anisotropy deviations from isotropy of the CMB radiation and second, the presence of large angle anomalies, although the existence of these anomalies is currently a huge matter of debate. These hints are particularly important since isotropy is one of the two main postulates of the Copernican principle on which the FRW models are built. This almost isotropic CMB radiation implies that the universe is almost a FRW universe, as is proved by previous studies. Assuming the matter component forms the deviations from isotropy in the CMB density fluctuations when matter and radiation decouples, we here attempt to find possible constraints on the FRW type scale and Hubble parameter by using the Bianchi type I (BI) anisotropic model which is asymptotically equivalent to the standard FRW. To obtain constraints on such an anisotropic model, we derive average and late-time shear values that come from the anisotropy upper limits of the recent Planck data based on a model independent shear parameter of Maartens et al. (1995a,b) and from the theoretical consistency relation. These constraints lead us to obtain a BI model which becomes an almost-FRW model in time, and which is consistent with the latest observational data of the CMB.

astro-ph.CO↗

arXiv2014

Connection between Fisher Information and Wave Mechanical Interpretations of Universe

In this study, we model the dark matter and baryon matter distribution in the Cosmic Web by means of highly nonlinear Schrödinger type and reaction diffusion wave mechanical descriptions. The construction of these wave mechanical models of the structure formation is achieved by introducing the Fisher information measure and its comparison with a highly nonlinear term called the quantum potential in the wave equations. Strikingly, the comparison of the nonlinear term and the Fisher information measure provides a dynamical distinction between lack of self-organization and self-organization in the dynamical evolution of the cosmic components. Mathematically equivalent to the standard cosmic fluid equations, these approaches make it possible to follow the evolution of the matter distribution even into the highly nonlinear regime by circumventing singularities. In addition, these wave formalisms are extended to two-fluid descriptions of the coupled dark matter and baryon matter distributions in the linear regime, in the Einstein de Sitter Universe (EdS) to construct toy models of the cosmic components in this relatively simple Universe model. Based on these two different wave mechanical formalisms, here fully analytical results for the dark matter and baryon distributions are provided. Also, numerical realizations of the emerging weblike patterns are presented from the nonlinear dynamics of the baryon component corresponding to soliton-like solutions. These soliton-like solutions might represent a proper description of filamentary structures even in the linear regime.

astro-ph.CO↗

arXiv2011

Golden Quantum Oscillator and Binet-Fibonacci Calculus

The Binet-Fibonacci formula for Fibonacci numbers is treated as a q-number (and q-operator) with Golden ratio bases $q=ϕ$ and $Q=-1/ϕ$. Quantum harmonic oscillator for this Golden calculus is derived so that its spectrum is given just by Fibonacci numbers. Ratio of successive energy levels is found as the Golden sequence and for asymptotic states it appears as the Golden ratio. This why we called this oscillator as the Golden oscillator. By double Golden bosons, the Golden angular momentum and its representation in terms of Fibonacci numbers and the Golden ratio are derived.

math.QA↗