Quantum Resonances and Ratchets in Free-Falling Frames

Quantum resonance (QR) is defined in the free-falling frame of the quantum kicked particle subjected to gravity. The general QR conditions are derived. They imply the rationality of the gravity parameter $\eta$, the kicking-period parameter $\tau /(2\pi)$, and the quasimomentum $\beta$. Exact results are obtained concerning wave-packet evolution for arbitrary periodic kicking potentials in the case of integer $\tau /(2\pi)$ (the main QRs). It is shown that a quantum ratchet generally arises in this case for resonant $\beta$. The noninertial nature of the free-falling frame affects the ratchet by effectively changing the kicking potential to one depending on $(\beta ,\eta)$. For a simple class of initial wave packets, it is explicitly shown that the ratchet characteristics are determined to a large extent by symmetry properties and by number-theoretical features of $\eta$.Comment: To appear in Physical Review E (Rapid Communications

Entanglement conditions and polynomial identities

We develop a rather general approach to entanglement characterization based on convexity properties and polynomial identities. This approach is applied to obtain simple and efficient entanglement conditions which work equally well in both discrete as well as continuous-variable environments. Examples of violations of our conditions are presented.Comment: 5 pages, no figure

q-Newton binomial: from Euler to Gauss

A counter-intuitive result of Gauss (formulae (1.6), (1.7) below) is made less mysterious by virtue of being generalized through the introduction of an additional parameter

Block circulant matrices with circulant blocks, weil sums and mutually unbiased bases, II. The prime power case

In our previous paper \cite{co1} we have shown that the theory of circulant matrices allows to recover the result that there exists $p+1$ Mutually Unbiased Bases in dimension $p$, $p$ being an arbitrary prime number. Two orthonormal bases $\mathcal B, \mathcal B'$ of $\mathbb C^d$ are said mutually unbiased if $\forall b\in \mathcal B, \forall b' \in \mathcal B'$ one has that $$| b\cdot b'| = \frac{1}{\sqrt d}$$ ($b\cdot b'$ hermitian scalar product in $\mathbb C^d$). In this paper we show that the theory of block-circulant matrices with circulant blocks allows to show very simply the known result that if $d=p^n$ ($p$ a prime number, $n$ any integer) there exists $d+1$ mutually Unbiased Bases in $\mathbb C^d$. Our result relies heavily on an idea of Klimov, Munoz, Romero \cite{klimuro}. As a subproduct we recover properties of quadratic Weil sums for $p\ge 3$, which generalizes the fact that in the prime case the quadratic Gauss sums properties follow from our results

Phase operators, phase states and vector phase states for SU(3) and SU(2,1)

This paper focuses on phase operators, phase states and vector phase states for the sl(3) Lie algebra. We introduce a one-parameter generalized oscillator algebra A(k,2) which provides a unified scheme for dealing with su(3) (for k < 0), su(2,1) (for k > 0) and h(4) x h(4) (for k = 0) symmetries. Finite- and infinite-dimensional representations of A(k,2) are constructed for k < 0 and k > 0 or = 0, respectively. Phase operators associated with A(k,2) are defined and temporally stable phase states (as well as vector phase states) are constructed as eigenstates of these operators. Finally, we discuss a relation between quantized phase states and a quadratic discrete Fourier transform and show how to use these states for constructing mutually unbiased bases

Roots of the derivative of the Riemann zeta function and of characteristic polynomials

We investigate the horizontal distribution of zeros of the derivative of the Riemann zeta function and compare this to the radial distribution of zeros of the derivative of the characteristic polynomial of a random unitary matrix. Both cases show a surprising bimodal distribution which has yet to be explained. We show by example that the bimodality is a general phenomenon. For the unitary matrix case we prove a conjecture of Mezzadri concerning the leading order behavior, and we show that the same follows from the random matrix conjectures for the zeros of the zeta function.Comment: 24 pages, 6 figure

Nilpotent Classical Mechanics

The formalism of nilpotent mechanics is introduced in the Lagrangian and Hamiltonian form. Systems are described using nilpotent, commuting coordinates $\eta$. Necessary geometrical notions and elements of generalized differential $\eta$-calculus are introduced. The so called $s-$geometry, in a special case when it is orthogonally related to a traceless symmetric form, shows some resemblances to the symplectic geometry. As an example of an $\eta$-system the nilpotent oscillator is introduced and its supersymmetrization considered. It is shown that the $R$-symmetry known for the Graded Superfield Oscillator (GSO) is present also here for the supersymmetric $\eta$-system. The generalized Poisson bracket for $(\eta,p)$-variables satisfies modified Leibniz rule and has nontrivial Jacobiator.Comment: 23 pages, no figures. Corrected version. 2 references adde

Partition Functions, the Bekenstein Bound and Temperature Inversion in Anti-de Sitter Space and its Conformal Boundary

We reformulate the Bekenstein bound as the requirement of positivity of the Helmholtz free energy at the minimum value of the function L=E- S/(2\pi R), where R is some measure of the size of the system. The minimum of L occurs at the temperature T=1/(2\pi R). In the case of n-dimensional anti-de Sitter spacetime, the rather poorly defined size R acquires a precise definition in terms of the AdS radius l, with R=l/(n-2). We previously found that the Bekenstein bound holds for all known black holes in AdS. However, in this paper we show that the Bekenstein bound is not generally valid for free quantum fields in AdS, even if one includes the Casimir energy. Some other aspects of thermodynamics in anti-de Sitter spacetime are briefly touched upon.Comment: Latex, 32 page

On the sums of two cubes

We solve the equation $f(x,y)^3 + g(x,y)^3 = x^3 + y^3$ for homogeneous $f, g \in \mathbb C(x,y)$, completing an investigation begun by Vi\`ete in 1591. The usual addition law for elliptic curves and composition give rise to two binary operations on the set of solutions. We show that a particular subset of the set of solutions is ring-isomorphic to $\mathbb Z[e^{2 \pi i / 3}]$.Comment: Revised version, to appear in the International Journal of Number Theor