Outer commutator words are uniformly concise

We prove that outer commutator words are uniformly concise, i.e. if an outer commutator word w takes m different values in a group G, then the order of the verbal subgroup w(G) is bounded by a function depending only on m and not on w or G. This is obtained as a consequence of a structure theorem for the subgroup w(G), which is valid if G is soluble, and without assuming that w takes finitely many values in G. More precisely, there is an abelian series of w(G), such that every section of the series can be generated by values of w all of whose powers are also values of w in that section. For the proof of this latter result, we introduce a new representation of outer commutator words by means of binary trees, and we use the structure of the trees to set up an appropriate induction

Structural defects in ion crystals by quenching the external potential: the inhomogeneous Kibble-Zurek mechanism

The non-equilibrium dynamics of an ion chain in a highly anisotropic trap is studied when the transverse trap frequency is quenched across the value at which the chain undergoes a continuous phase transition from a linear to a zigzag structure. Within Landau theory, an equation for the order parameter, corresponding to the transverse size of the zigzag structure, is determined when the vibrational motion is damped via laser cooling. The number of structural defects produced during a linear quench of the transverse trapping frequency is predicted and verified numerically. It is shown to obey the scaling predicted by the Kibble-Zurek mechanism, when extended to take into account the spatial inhomogeneities of the ion chain in a linear Paul trap.Comment: 5 pages, 3 figure

Laser cooling with electromagnetically induced transparency: Application to trapped samples of ions or neutral atoms

A novel method of ground state laser cooling of trapped atoms utilizes the absorption profile of a three (or multi-) level system which is tailored by a quantum interference. With cooling rates comparable to conventional sideband cooling, lower final temperatures may be achieved. The method was experimentally implemented to cool a single Ca$^+$ ion to its vibrational ground state. Since a broad band of vibrational frequencies can be cooled simultaneously, the technique will be particularly useful for the cooling of larger ion strings, thereby being of great practical importance for initializing a quantum register based on trapped ions. We also discuss its application to different level schemes and for ground state cooling of neutral atoms trapped by a far detuned standing wave laser field.Comment: 9 pages, 13 figures, submitted to Appl Phys B 200

Quantum quenches of ion Coulomb crystals across structural instabilities

Quenches in an ion chain can create coherent superpositions of motional states across the linear-zigzag structural transition. The procedure has been described in [Phys. Rev. A 84, 063821 (2011)] and makes use of spin-dependent forces, so that a coherent superposition of the electronic states of one ion evolves into an entangled state between the chain's internal and external degrees of freedom. The properties of the crystalline state so generated are theoretically studied by means of Ramsey interferometry on one ion of the chain. An analytical expression for the visibility of the interferometric measurement is obtained for a chain of arbitrary number of ions and as a function of the time elapsed after the quench. Sufficiently close to the linear-zigzag instability the visibility decays very fast, but exhibits revivals at the period of oscillation of the mode that drives the structural instability. These revivals have a periodicity that is independent of the crystal size, and they signal the creation of entanglement by the quantum quench.Comment: 14 pages, 8 figures; added a paragraph in the introduction providing more background, added paragraph at the end of Sec. IV discussing experimental parameter

A Convex-Nonconvex variational method for the additive decomposition of functions on surfaces

We present a Convex-NonConvex variational approach for the additive decomposition of noisy scalar f ields defined over triangulated surfaces into piecewise constant and smooth components. The energy functional to be minimized is defined by the weighted sum of three terms, namely an L2 fidelity term for the noise component, a Tikhonov regularization term for the smooth component and a Total Variation (TV)-like non-convex term for the piecewise constant component. The last term is parametrized such that the free scalar parameter allows to tune its degree of non- convexity and, hence, to separate the piecewise constant component more effectively than by using a classical convex TV regularizer without renouncing to convexity of the total energy functional. A method is also presented for selecting the two regularization parameters. The unique solution of the proposed variational model is determined by means of an efficient ADMM-based minimization algorithm. Numerical experiments show a nearly perfect separation of the different components

Systematic Analysis of Crystalline Phases in Bosonic Lattice Models with Algebraically Decaying Density-Density Interactions

We propose a general approach to analyse diagonal ordering patterns in bosonic lattice models with algebraically decaying density-density interactions on arbitrary lattices. The key idea is a systematic search for the energetically best order on all unit cells of the lattice up to a given extent. Using resummed couplings we evaluate the energy of the ordering patterns in the thermodynamic limit using finite unit cells. We apply the proposed approach to the atomic limit of the extended Bose-Hubbard model on the triangular lattice at fillings $f=1/2$ and $f=1$. We investigate the ground-state properties of the antiferromagnetic long-range Ising model on the triangular lattice and determine a six-fold degenerate plain-stripe phase to be the ground state for finite decay exponents. We also probe the classical limit of the Fendley-Sengupta-Sachdev model describing Rydberg atom arrays. We focus on arrangements where the atoms are placed on the sites or links of the Kagome lattice. \changed{Our method provides a general framework to treat cristalline structures resulting from long-range interactions.Comment: 35 pages, 11 figure

Cooling atomic motion with quantum interference

We theoretically investigate the quantum dynamics of the center of mass of trapped atoms, whose internal degrees of freedom are driven in a $\Lambda$-shaped configuration with the lasers tuned at two-photon resonance. In the Lamb-Dicke regime, when the motional wave packet is well localized over the laser wavelenght, transient coherent population trapping occurs, cancelling transitions at the laser frequency. In this limit the motion can be efficiently cooled to the ground state of the trapping potential. We derive an equation for the center-of-mass motion by adiabatically eliminating the internal degrees of freedom. This treatment provides the theoretical background of the scheme presented in [G. Morigi {\it et al}, Phys. Rev. Lett. {\bf 85}, 4458 (2000)] and implemented in [C.F. Roos {\it et al}, Phys. Rev. Lett. {\bf 85}, 5547 (2000)]. We discuss the physical mechanisms determining the dynamics and identify new parameters regimes, where cooling is efficient. We discuss implementations of the scheme to cases where the trapping potential is not harmonic.Comment: 11 pages, 3 figure