Semigroups with the Erdös-Turán Property

A set X in a semigroup G has the Erdös-Turán property ET if, for any basis A of X, the representation function rA is ubounded, where rA(x) counts the number of representations of x as a product two elements in A. We show that, under some conditions, operations on binary vectors whose value at each coordinate depends only on neighbouring coordinates of the factors give rise to semigroups with the ET{property. In particular countable powers of semigroups with no mutually inverse elements have the ET{property. As a consequence, for each k there is N(k) such that, for every ¯nite subset X of a group G with X \ X¡1 = f1g, the representation function of every basis of XN ½ GN, N ¸ N(k), is not bounded by k. This is in contrast with the known fact that each p{elementary group admits a basis of the whole group whose representation function is bounded by an absolute constan

On the cardinality of sumsets in torsion-free groups

Let $A, B$ be finite subsets of a torsion-free group $G$. We prove that for every positive integer $k$ there is a $c(k)$ such that if $|B|\ge c(k)$ then the inequality $|AB|\ge |A|+|B|+k$ holds unless a left translate of $A$ is contained in a cyclic subgroup. We obtain $c(k)<c_0k^{6}$ for arbitrary torsion-free groups, and $c(k)<c_0k^{3}$ for groups with the unique product property, where $c_0$ is an absolute constant. We give examples to show that $c(k)$ is at least quadratic in $k$

Phase diagram of the vortex system in layered superconductors with strong columnar pinning

We present the results of a detailed investigation of the low-temperature properties of the vortex system in strongly anisotropic layered superconductors with a random array of columnar pinning centers. Our method involves numerical minimization of a free energy functional in terms of the time-averaged local vortex density. It yields the detailed vortex density distribution for all local free-energy minima, and therefore allows the computation of any desired correlation function of the time-averaged local vortex density. Results for the phase diagram in the temperature vs. pin concentration plane at constant magnetic induction are presented. We confirm that for very low pin concentrations, the low-temperature phase is a Bragg glass, which melts into an interstitial liquid phase via two first-order steps, separated by a Bose glass phase. At higher concentrations, however, the low-temperature phase is a Bose glass, and the melting transition becomes continuous. The transition is then characterized by the onset of percolation of liquid-like regions across the sample. Inhomogeneous local melting of the Bose glass is found to occur. There is also a depinning crossover between the interstitial liquid and a completely unpinned liquid at higher temperatures. At sufficiently large pin concentrations, the depinning line merges with the Bose glass to interstitial liquid transition. Many of the features we find have been observed experimentally and in simulations. We discuss the implications of our results for future experimental and theoretical work.Comment: 15 pages including Figure

The phase diagram of vortex matter in layered superconductors with tilted columnar pinning centers

We study the vortex matter phase diagram of a layered superconductor in the presence of columnar pinning defects, {\it tilted} with respect to the normal to the layers. We use numerical minimization of the free energy written as a functional of the time averaged vortex density of the Ramakrishnan-Yussouff form, supplemented by the appropriate pinning potential. We study the case where the pin density is smaller than the areal vortex density. At lower pin concentrations, we find, for temperatures of the order of the melting temperature of the unpinned lattice, a Bose glass type phase which at lower temperatures converts, via a first order transition, to a Bragg glass, while, at higher temperatures, it crosses over to an interstitial liquid. At somewhat higher concentrations, no transition to a Bragg glass is found even at the lowest temperatures studied. While qualitatively the behavior we find is similar to that obtained using the same procedures for columnar pins normal to the layers, there are important and observable quantitative differences, which we discuss.Comment: 12 pages, including figure

Quantum Ratchets for Quantum Communication with Optical Superlattices

We propose to use a quantum ratchet to transport quantum information in a chain of atoms trapped in an optical superlattice. The quantum ratchet is created by a continuous modulation of the optical superlattice which is periodic in time and in space. Though there is zero average force acting on the atoms, we show that indeed the ratchet effect permits atoms on even and odd sites to move along opposite directions. By loading the optical lattice with two-level bosonic atoms, this scheme permits to perfectly transport a qubit or entangled state imprinted in one or more atoms to any desired position in the lattice. From the quantum computation point of view, the transport is achieved by a smooth concatenation of perfect swap gates. We analyze setups with noninteracting and interacting particles and in the latter case we use the tools of optimal control to design optimal modulations. We also discuss the feasibility of this method in current experiments.Comment: Published version, 9 pages, 5 figure

Optomechanics assisted with a qubit: From dissipative state preparation to many-body physics

We propose and analyze nonlinear optomechanical protocols that can be implemented by adding a single atom to an optomechanical cavity. In particular, we show how to engineer the environment in order to dissipatively prepare the mechanical oscillator in a superposition of Fock states with fidelity close to one. Furthermore, we discuss how a single atom in a cavity with several mechanical oscillators can be exploited to realize nonlinear many-body physics by stroboscopically driving the mechanical oscillators. We show how to prepare non-classical many-body states by either applying coherent protocols or engineering dissipation. The analysis of the protocols is carried out using a perturbation theory for degenerate Liouvillians and numerical tools. Our results apply to other systems where a qubit is coupled to a mechanical oscillator via a bosonic mode, e.g., in cavity quantum electromechanics

Master equation approach to optomechanics with arbitrary dielectrics

We present a master equation describing the interaction of light with dielectric objects of arbitrary sizes and shapes. The quantum motion of the object, the quantum nature of light, as well as scattering processes to all orders in perturbation theory are taken into account. This formalism extends the standard master equation approach to the case where interactions among different modes of the environment are considered. It yields a genuine quantum description, including a renormalization of the couplings and decoherence terms. We apply this approach to analyze cavity cooling of the center-of-mass mode of large spheres. Furthermore, we derive an expression for the steady-state phonon numbers without relying on resolved-sideband or bad-cavity approximations.Comment: 17 pages, 5 figure