Words of Engel type are concise in residually finite groups

Given a group-word w and a group G, the verbal subgroup w(G) is the one generated by all w-values in G. The word w is said to be concise if w(G) is finite whenever the set of w-values in G is finite. In the sixties P. Hall asked whether every word is concise but later Ivanov answered this question in the negative. On the other hand, Hall\u2019s question remains wide open in the class of residually finite groups. In the present article we show that various generalizations of the Engel word are concise in residually finite groups

Ground State Laser Cooling Beyond the Lamb-Dicke Limit

We propose a laser cooling scheme that allows to cool a single atom confined in a harmonic potential to the trap ground state $|0>$. The scheme assumes strong confinement, where the oscillation frequency in the trap is larger than the effective spontaneous decay width, but is not restricted to the Lamb-Dicke limit, i.e. the size of the trap ground state can be larger than the optical wavelength. This cooling scheme may be useful in the context of quantum computations with ions and Bose-Einstein condensation.Comment: 6 pages, 4 figures, to appear in Europhysics Letter

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

Nano-friction in cavity quantum electrodynamics

The dynamics of cold trapped ions in a high-finesse resonator results from the interplay between the long-range Coulomb repulsion and the cavity-induced interactions. The latter are due to multiple scatterings of laser photons inside the cavity and become relevant when the laser pump is sufficiently strong to overcome photon decay. We study the stationary states of ions coupled with a mode of a standing-wave cavity as a function of the cavity and laser parameters, when the typical length scales of the two self-organizing processes, Coulomb crystallization and photon-mediated interactions, are incommensurate. The dynamics are frustrated and in specific limiting cases can be cast in terms of the Frenkel-Kontorova model, which reproduces features of friction in one dimension. We numerically recover the sliding and pinned phases. For strong cavity nonlinearities, they are in general separated by bistable regions where superlubric and stick-slip dynamics coexist. The cavity, moreover, acts as a thermal reservoir and can cool the chain vibrations to temperatures controlled by the cavity parameters and by the ions phase. These features are imprinted in the radiation emitted by the cavity, which is readily measurable in state-of-art setups of cavity quantum electrodynamics.Comment: 9 pages, 7 figure

Entangled light pulses from single cold atoms

The coherent interaction between a laser-driven single trapped atom and an optical high-finesse resonator allows to produce entangled multi-photon light pulses on demand. The mechanism is based on the mechanical effect of light. The degree of entanglement can be controlled through the parameters of the laser excitation. Experimental realization of the scheme is within reach of current technology. A variation of the technique allows for controlled generation of entangled subsequent pulses, with the atomic motion serving as intermediate memory of the quantum state.Comment: 4 pages, 3 figures, revised version (new scheme for generation of subsequent pairs of entangled pulses included). Accepted for publication in Phys. Rev. Let

Mott insulator states of ultracold atoms in optical resonators

We study the low temperature physics of an ultracold atomic gas in the potential formed inside a pumped optical resonator. Here, the height of the cavity potential, and hence the quantum state of the gas, depends not only on the pump parameters, but also on the atomic density through a dynamical a.c.-Stark shift of the cavity resonance. We derive the Bose-Hubbard model in one dimension, and use the strong coupling expansion to determine the parameter regime in which the system is in the Mott-insulator state. We predict the existence of overlapping, competing Mott states, and bistable behavior in the vicinity of the shifted cavity resonance, controlled by the pump parameters. Outside these parameter regions, the state of the system is in most cases superfluid.Comment: 4 pages, 3 figures. Substantially revised version. To appear in Phys. Rev. Let

Quantum coherence and population trapping in three-photon processes

The spectroscopic properties of a single, tightly trapped atom are studied, when the electronic levels are coupled by three laser fields in an $N$-shaped configuration of levels, whereby a $\Lambda$-type level system is weakly coupled to a metastable state. We show that depending on the laser frequencies the response can be tuned from coherent population trapping at two-photon resonance to novel behaviour at three photon resonance, where the metastable state can get almost unit occupation in a wide range of parameters. For certain parameter regimes the system switches spontaneously between dissipative and coherent dynamics over long time scales