Goce Chadzitaskos

Feynman's path integral and mutually unbiased bases

Our previous work on quantum mechanics in Hilbert spaces of finite dimension N is applied to elucidate the deep meaning of Feynmans path integral pointed out by G Svetlichny. He speculated that the secret of the Feynman path integral may lie in the property of mutual unbiasedness of temporally proximal bases. We confirm the corresponding property of the short-time propagator by using a specially devised N × N approximation of quantum mechanics in applied to our finite-dimensional analogue of a free quantum particle.

Quantization on and coherent states over

Finite-dimensional quantum mechanics (quantum mechanics on finite discrete space - the cyclic group of order M) is developed further: in analogy with the usual harmonic oscillator coherent states, an overcomplete family of coherent states over the phase space is constructed and their properties are determined.

Systems with Intensity-Dependent Conversion Integrable by Finite Orthogonal Polynomials

We present exact solutions of a class of the nonlinear models which describe the parametric conversion of photons. Hamiltonians of these models are related to the classes of finite orthogonal polynomials. The spectra and exact expressions for eigenvectors of these Hamiltonians are obtained.

Coherent states on the circle

We present a possible construction of coherent states on the unit circle as configuration space. In our approach the phase space is the product ×S1. Because of the duality of canonical coordinates and momenta, i.e. the angular variable and the integers, this formulation can also be interpreted as coherent states over an infinite periodic chain. For the construction we use the analogy with our quantization over a finite periodic chain where the phase space was M ×M. Properties of the coherent states constructed in this way are studied and the coherent states are shown to satisfy the resolution of unity.

Transformation design and nonlinear Hamiltonians

We study a class of nonlinear Hamiltonians, with applications in quantum optics. The interaction terms of these Hamiltonians are generated by taking a linear combination of powers of a simple ‘beam splitter’ Hamiltonian. The entanglement properties of the eigenstates are studied. Finally, we show how to use this class of Hamiltonians to perform special tasks such as conditional state swapping, which can be used to generate optical cat states and to sort photons.

Telescopic system with a rotating objective element

The angular resolution is the ability of a telescope to render detail: the higher the resolution the finer is the detail. It is, together with the aperture, the most important characteristic of telescopes. We propose a new construction of telescopes with improved ratio of angular resolution and area of the primary optical element (mirror or lense). For this purpose we use the rotation of the primary optical element with one dominating dimension. The length of the dominating dimension of the primary optical element determines the angular resolution. During the rotation a sequence of images is stored in a computer and the images of observed objects can be reconstructed using a relatively simple software. The angular resolution is determined by the maximal length of the primary optical element of the system. This construction of telescopic systems allows to construct telescopes of high resolution with lower weight and fraction of usual costs.

Para-Grassmann Star Product Calculation

In this Letter, we construct the star product for polynomials over the para-Grassmann variable and we present as an example a para-Grassmann version of the model of q–quantum mechanics. Moreover, using the structural relations of the q–deformed algebra generated by annihilation and creation operators, we decompose the Jacobi matrix in the product of three matrices: the diagonal matrix, the upper and lower diagonal matrices, mutually adjoint.

A Common Assessment Space for Different Sensor Structures

The study of the evolution process of our visual system indicates the existence of variational spatial arrangement; from densely hexagonal in the fovea to a sparse circular structure in the peripheral retina. Today’s sensor spatial arrangement is inspired by our visual system. However, we have not come further than rigid rectangular and, on a minor scale, hexagonal sensor arrangements. Even in this situation, there is a need for directly assessing differences between the rectangular and hexagonal sensor arrangements, i.e., without the conversion of one arrangement to another. In this paper, we propose a method to create a common space for addressing any spatial arrangements and assessing the differences among them, e.g., between the rectangular and hexagonal. Such a space is created by implementing a continuous extension of discrete Weyl Group orbit function transform which extends a discrete arrangement to a continuous one. The implementation of the space is demonstrated by comparing two types of generated hexagonal images from each rectangular image with two different methods of the half-pixel shifting method and virtual hexagonal method. In the experiment, a group of ten texture images were generated with variational curviness content using ten different Perlin noise patterns, adding to an initial 2D Gaussian distribution pattern image. Then, the common space was obtained from each of the discrete images to assess the differences between the original rectangular image and its corresponding hexagonal image. The results show that the space facilitates a usage friendly tool to address an arrangement and assess the changes between different spatial arrangements by which, in the experiment, the hexagonal images show richer intensity variation, nonlinear behavior, and larger dynamic range in comparison to the rectangular images.

C-orbit Function and Image Filtering

We present the first attempt to use the C-orbit functions in image processing. For the image processing we perform a Fourier-like transform of the image. Then we define a convolution on C-orbit functions and we apply the simplest spatial linear filters on several examples. Finally we compare the results with filtering via an ordinary Fourier transformation.

Three boson interaction process: spectra and coherent states

Abstract We use the methods of constructions of and deformed coherent states in order to construct the coherent states for down conversion processes. The down conversion process can be understood as a quasi-exactly solvable model of quantum mechanics. After the reduction of the Hamiltonian, we use the Turbiner polynomials approach, and the eigenvalues of the Hamiltonian for low number of photons are calculated and the approximation formula is found out. After the discussion on the time evolution and the entanglement, the coherent states are constructed as the eigenstates of the reduced annihilation operator.