Dynamical Hartree-Fock-Bogoliubov Theory of Vortices in Bose-Einstein Condensates at Finite Temperature

We present a method utilizing the continuity equation for the condensate density to make predictions of the precessional frequency of single off-axis vortices and of vortex arrays in Bose-Einstein condensates at finite temperature. We also present an orthogonalized Hartree-Fock-Bogoliubov (HFB) formalism. We solve the continuity equation for the condensate density self-consistently with the orthogonalized HFB equations, and find stationary solutions in the frame rotating at this frequency. As an example of the utility of this formalism we obtain time-independent solutions for quasi-two-dimensional rotating systems in the co-rotating frame. We compare these results with time-dependent predictions where we simulate stirring of the condensate.Comment: 13 pages, 11 figures, 1 tabl

Quantum Spin Lenses in Atomic Arrays

We propose and discuss `quantum spin lenses', where quantum states of delocalized spin excitations in an atomic medium are `focused' in space in a coherent quantum process down to (essentially) single atoms. These can be employed to create controlled interactions in a quantum light-matter interface, where photonic qubits stored in an atomic ensemble are mapped to a quantum register represented by single atoms. We propose Hamiltonians for quantum spin lenses as inhomogeneous spin models on lattices, which can be realized with Rydberg atoms in 1D, 2D and 3D, and with strings of trapped ions. We discuss both linear and non-linear quantum spin lenses: in a non-linear lens, repulsive spin-spin interactions lead to focusing dynamics conditional to the number of spin excitations. This allows the mapping of quantum superpositions of delocalized spin excitations to superpositions of spatial spin patterns, which can be addressed by light fields and manipulated. Finally, we propose multifocal quantum spin lenses as a way to generate and distribute entanglement between distant atoms in an atomic lattice array.Comment: 13 pages, 9 figure

Wigner crystals in two-dimensional transition-metal dichalcogenides: Spin physics and readout

Wigner crystals are prime candidates for the realization of regular electron lattices under minimal requirements on external control and electronics. However, several technical challenges have prevented their detailed experimental investigation and applications to date. We propose an implementation of two-dimensional electron lattices for quantum simulation of Ising spin systems based on self-assembled Wigner crystals in transition-metal dichalcogenides. We show that these semiconductors allow for minimally invasive all-optical detection schemes of charge ordering and total spin. For incident light with optimally chosen beam parameters and polarization, we predict a strong dependence of the transmitted and reflected signals on the underlying lattice periodicity, thus revealing the charge order inherent in Wigner crystals. At the same time, the selection rules in transition-metal dichalcogenides provide direct access to the spin degree of freedom via Faraday rotation measurements.Comment: 15 pages, 12 figure

Discriminants, symmetrized graph monomials, and sums of squares

Motivated by the necessities of the invariant theory of binary forms J. J. Sylvester constructed in 1878 for each graph with possible multiple edges but without loops its symmetrized graph monomial which is a polynomial in the vertex labels of the original graph. In the 20-th century this construction was studied by several authors. We pose the question for which graphs this polynomial is a non-negative resp. a sum of squares. This problem is motivated by a recent conjecture of F. Sottile and E. Mukhin on discriminant of the derivative of a univariate polynomial, and an interesting example of P. and A. Lax of a graph with 4 edges whose symmetrized graph monomial is non-negative but not a sum of squares. We present detailed information about symmetrized graph monomials for graphs with four and six edges, obtained by computer calculations

Defect mediated melting and the breaking of quantum double symmetries

In this paper, we apply the method of breaking quantum double symmetries to some cases of defect mediated melting. The formalism allows for a systematic classification of possible defect condensates and the subsequent confinement and/or liberation of other degrees of freedom. We also show that the breaking of a double symmetry may well involve a (partial) restoration of an original symmetry. A detailed analysis of a number of simple but representative examples is given, where we focus on systems with global internal and external (space) symmetries. We start by rephrasing some of the well known cases involving an Abelian defect condensate, such as the Kosterlitz-Thouless transition and one-dimensional melting, in our language. Then we proceed to the non-Abelian case of a hexagonal crystal, where the hexatic phase is realized if translational defects condense in a particular rotationally invariant state. Other conceivable phases are also described in our framework.Comment: 10 pages, 4 figures, updated reference