Theoretical description of two ultracold atoms in finite 3D optical lattices using realistic interatomic interaction potentials

A theoretical approach is described for an exact numerical treatment of a pair of ultracold atoms interacting via a central potential that are trapped in a finite three-dimensional optical lattice. The coupling of center-of-mass and relative-motion coordinates is treated using an exact diagonalization (configuration-interaction) approach. The orthorhombic symmetry of an optical lattice with three different but orthogonal lattice vectors is explicitly considered as is the Fermionic or Bosonic symmetry in the case of indistinguishable particles.Comment: 19 pages, 5 figure

Técnicas analíticas en problemas multilineales

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Departamento de Análisis Matemático, leída el 16/09/2016This Ph.D. dissertation mainly focuses on three multilinear problems and itsaimistodescribe analytical and topological techniques that we found useful to tackle these problems. The first problem comes from Quantum Information theory, it is the so-called the Separability Problem, and the other two were proposed by Gurariy. Let Mk denote the set of complex matrices of order k and let Pk be the set of positive semidefinite Hermitian matrices of Mk. The aim of this problem is to find a deterministic criterion to distinguish the separable states from the entangled states. In this work we shall only deal with the bipartite finite dimensional case, therefore the states are elements in the tensor product space Mk ⊗Mm. We say that B ∈ Mk ⊗Mm is separable if B =Σin=1 Ci ⊗Di, where Ci ∈ Pk and Di ∈ Pm, for every i. If B is not separable then B is entangled. Denote by VMkV the set {V XV,X ∈ Mk}, where V ∈ Mk is an orthogonal projection. We say that a linear transformation T :VMkV →WMmW is a positive map, if T(Pk ∩VMkV )⊂ Pm ∩WMmW. We say that a non null positive map T : VMkV →VMkV is irreducible if V ′ MkV ′ ⊂ VMkV is such that T(V ′ MkV ′)⊂ V ′ MkV ′ then V ′ = V or V ′ = 0. Let us say that T : VMkV → VMkV is a completely reducible map, if it is a positive map and if there are orthogonal projections V1,...,Vs ∈ Mk such that ViVj = 0 (i ≠ j), ViV = Vi (1 ≤ i ≤ s), VMkV = V1MkV1 ⊕ ... ⊕ VsMkVs ⊕ R, R ⊥ V1MkV1 ⊕ ... ⊕ VsMkVs satisfying: T(ViMkVi)⊂ ViMkVi (1 ≤ i ≤ s), TSis irreducible (1 ≤ i ≤ s), TSR ≡ 0. Let A =Σni=1 Ai ⊗Bi ∈ Mk ⊗Mm. Define GA : Mk →Mm, as GA(X)=Σni=1 tr(AiX)Bi and FA : Mm → Mk, as FA(X)=Σin=1 tr(BiX)Ai. Our main results are the following: If A ∈ Mk ⊗Mm is positive under partial transposition (PPT) or symmetric with positive coefficients (SPC) or invariant under realignment then FA ○GA : Mk → Mk is completely reducible...Esta tesis doctoral se centra principalmente en tres problemas multilineales y su objetivo es describir las técnicas analíticas y topológicas útiles para atacar estos problemas. El primer problema tiene su origen en la Teoría de Información Cuántica, es el llamado problema de la separabilidad de los estados cuánticos, y los otros dos fueron propuestos por Vladimir I. Gurariy. Denotemos por Mk al conjunto de las matrices complejas de orden k y Pk será el conjunto de matrices Hermíticas semidefinidas positivasde Mk. El objetivo de nuestro primer problema es encontrar un criterio determinístico para distinguir los estados separables de los estados entrelazados. Aqui sólo trabajamos con el caso bipartito de dimensión finita, luego los estados son los elementos del producto tensorial Mk ⊗Mm. Decimos que B ∈ Mk ⊗Mm es separable si B =Σni=1 Ci ⊗Di, donde Ci ∈ Pk y Di ∈ Pm, para cada i. Si B no es separable entonces B está entrelazada. Sea VMkV el conjunto {V XV,X ∈ Mk}, donde V ∈ Mk es una proyección ortogonal. Se dice que una transformación lineal T : VMkV →WMmW es una aplicación positiva, si T(Pk ∩VMkV )⊂ Pm ∩WMmW. Se dice que una aplicación no nula positiva T : VMkV → VMkV es irreducible si V ′ MkV ′ ⊂ VMkV es tal que T(V ′ MkV ′)⊂ V ′ MkV ′ entonces V ′ = V o V ′ = 0. Digamos que T : VMkV → VMkV es una aplicación completamente reducible, si es positiva y si hay proyecciones ortogonales V1,...,Vs ∈ Mk tales que ViVj = 0 (i ≠ j), ViV = Vi (1 ≤ i ≤ s), VMkV = V1MkV1⊕...⊕VsMkVs⊕R, R ⊥V1MkV1⊕...⊕VsMkVs y que satisfacen: T(ViMkVi)⊂ ViMkVi (1 ≤ i ≤ s), TSViMkVi es irreducible (1 ≤ i ≤ s), TSR ≡ 0. Sea A =Σni=1 Ai ⊗Bi ∈ Mk ⊗Mm. Defina GA : Mk →Mm, GA(X)=1 tr(AiX)Bi y FA : Mm →Mk, FA(X)=Σin=1 tr(BiX)Ai. Nuestros resultados principales son los seguintes: Si A es positiva bajo transposición parcial (PPT) o simétrica con coeficientes positivos (SPC) o invariante bajo realineamiento luego FA ○GA :Mk →Mk es completamente reducible...Depto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEunpu

Electron in higher-dimensional weakly charged rotating black hole spacetimes

We demonstrate separability of the Dirac equation in weakly charged rotating black hole spacetimes in all dimensions. The electromagnetic field of the black hole is described by a test field approximation, with vector potential proportional to the primary Killing vector field. It is shown that the demonstrated separability can be intrinsically characterized by the existence of a complete set of mutually commuting first order symmetry operators generated from the principal Killing-Yano tensor. The presented results generalize the results on integrability of charged particle motion and separability of charged scalar field studied in [1].Comment: 12 pages, no figure

Domination of operators in the non-commutative setting

We consider majorization problems in the non-commutative setting. More specifically, suppose $E$ and $F$ are ordered normed spaces (not necessarily lattices), and $0 \leq T \leq S :E \to F$. If $S$ belongs to a certain ideal (for instance, the ideal of compact or Dunford-Pettis operators), does it follow that $T$ belongs to that ideal as well? We concentrate on the case when $E$ and $F$ are $C^*$-algebras, preduals of von Neumann algebras, or non-commutative function spaces. In particular, we show that, for $C^*$-algebras \A and ${\mathcal{B}}$, the following are equivalent: (1) at least one of the two conditions holds: (i) \A is scattered, (ii) ${\mathcal{B}}$ is compact; (2) if 0 \leq T \leq S : \A \to {\mathcal{B}}, and $S$ is compact, then $T$ is compact