How to construct spin chains with perfect state transfer

It is shown how to systematically construct the $XX$ quantum spin chains with nearest-neighbor interactions that allow perfect state transfer (PST). Sets of orthogonal polynomials (OPs) are in correspondence with such systems. The key observation is that for any admissible one-excitation energy spectrum, the weight function of the associated OPs is uniquely prescribed. This entails the complete characterization of these PST models with the mirror symmetry property arising as a corollary. A simple and efficient algorithm to obtain the corresponding Hamiltonians is presented. A new model connected to a special case of the symmetric $q$-Racah polynomials is offered. It is also explained how additional models with PST can be derived from a parent system by removing energy levels from the one-excitation spectrum of the latter. This is achieved through Christoffel transformations and is also completely constructive in regards to the Hamiltonians.Comment: 7 page

The Geodesic Deviation Method and Extreme Mass-Ratio Systems: Theoretical methods and application to the calculation of gravitational waves

Brand, J.F.J. van den [Promotor]Holten, J.W. van [Promotor

New connection formulae for some q-orthogonal polynomials in q-Askey scheme

New nonlinear connection formulae of the q-orthogonal polynomials, such continuous q-Laguerre, continuous big q-Hermite, q-Meixner-Pollaczek and q-Gegenbauer polynomials, in terms of their respective classical analogues are obtained using a special realization of the q-exponential function as infinite multiplicative series of ordinary exponential function

Quantum state transfer in spin chains with q-deformed interaction terms

We study the time evolution of a single spin excitation state in certain linear spin chains, as a model for quantum communication. Some years ago it was discovered that when the spin chain data (the nearest neighbour interaction strengths and the magnetic field strengths) are related to the Jacobi matrix entries of Krawtchouk polynomials or dual Hahn polynomials, so-called perfect state transfer takes place. The extension of these ideas to other types of discrete orthogonal polynomials did not lead to new models with perfect state transfer, but did allow more insight in the general computation of the correlation function. In the present paper, we extend the study to discrete orthogonal polynomials of q-hypergeometric type. A remarkable result is a new analytic model where perfect state transfer is achieved: this is when the spin chain data are related to the Jacobi matrix of q-Krawtchouk polynomials. The other cases studied here (affine q-Krawtchouk polynomials, quantum q-Krawtchouk polynomials, dual q-Krawtchouk polynomials, q-Hahn polynomials, dual q-Hahn polynomials and q-Racah polynomials) do not give rise to models with perfect state transfer. However, the computation of the correlation function itself is quite interesting, leading to advanced q-series manipulations

Interpolation of SUSY quantum mechanics

Interpolation of two adjacent Hamiltonians in SUSY quantum mechanics $H_s=(1-s)A^{\dagger}A + sAA^{\dagger}$, $0\le s\le 1$ is discussed together with related operators. For a wide variety of shape-invariant degree one quantum mechanics and their `discrete' counterparts, the interpolation Hamiltonian is also shape-invariant, that is it takes the same form as the original Hamiltonian with shifted coupling constant(s).Comment: 18 page

A study on the fourth q-Painlev\'e equation

A q-difference analogue of the fourth Painlev\'e equation is proposed. Its symmetry structure and some particular solutions are investigated.Comment: 18 page

Epicycles and Poincar\'{e} Resonances in General Relativity

The method of geodesic deviations provides analytic approximations to geodesics in arbitrary background space-times. As such the method is a useful tool in many practical situations. In this note we point out some subtleties in the application of the method related to secular motions, in first as well as in higher order. In particular we work out the general second-order contribution to bound orbits in Schwarzschild space-time and show that it provides very good analytical results all the way up to the innermost stable circular orbit.Comment: 24 pages, 4 figure

Coherent States and a Path Integral for the Relativistic Linear Singular Oscillator

The SU(1,1) coherent states for a relativistic model of the linear singular oscillator are considered. The corresponding partition function is evaluated. The path integral for the transition amplitude between SU(1,1) coherent states is given. Classical equations of the motion in the generalized curved phase space are obtained. It is shown that the use of quasiclassical Bohr-Sommerfeld quantization rule yields the exact expression for the energy spectrum.Comment: 14 pages, 2 figures, Uses RevTeX4 styl

More on the q-oscillator algebra and q-orthogonal polynomials

Properties of certain $q$-orthogonal polynomials are connected to the $q$-oscillator algebra. The Wall and $q$-Laguerre polynomials are shown to arise as matrix elements of $q$-exponentials of the generators in a representation of this algebra. A realization is presented where the continuous $q$-Hermite polynomials form a basis of the representation space. Various identities are interpreted within this model. In particular, the connection formula between the continuous big $q$-Hermite polynomials and the continuous $q$-Hermite polynomials is thus obtained, and two generating functions for these last polynomials are algebraically derived

A relativistic model of the NN-dimensional singular oscillator

Exactly solvable $N$-dimensional model of the quantum isotropic singular oscillator in the relativistic configurational $\vec r_N$-space is proposed. It is shown that through the simple substitutions the finite-difference equation for the $N$-dimensional singular oscillator can be reduced to the similar finite-difference equation for the relativistic isotropic three-dimensional singular oscillator. We have found the radial wavefunctions and energy spectrum of the problem and constructed a dynamical symmetry algebra.Comment: 8 pages, accepted for publication in J. Phys.