Fractals on a benchtop: Observing fractal dimension in a resistor network

Our first experience of dimension typically comes in the intuitive Euclidean sense: a line is one dimensional, a plane is two-dimensional, and a volume is three-dimensional. However, following the work of Mandelbrot \cite{mandelbrot}, systems with a fractional dimension, ``fractals'', now play an important role in science. The novelty of encountering fractional dimension, and the intrinsic beauty of many fractals, have a strong appeal to students and provide a powerful teaching tool. I describe here a low-cost and convenient experimental method for observing fractal dimension, by measuring the power-law scaling of the resistance of a fractal network of resistors. The experiments are quick to perform, and the students enjoy both the construction of the network and the collaboration required to create the largest networks. Learning outcomes include analysis of resistor networks beyond the elementary series and parallel combinations, scaling laws, and an introduction to fractional dimension

Erratum: Location of crossings in the Floquet spectrum of a driven two-level system (vol B 67, art no 165301, 2003)

©2004 The American Physical Society.Depto. de Física de MaterialesFac. de Ciencias FísicasTRUEpu

Perturbative analysis of coherent quantum ratchets in cold atom systems

We present a perturbative study of the response of cold atoms in an optical lattice to a weak time- and space-asymmetric periodic driving signal. In the noninteracting limit, and for a finite set of resonant frequencies, we show how a coherent, long lasting ratchet current results from the interference between first and second order processes. In those cases, a suitable three-level model can account for the entire dynamics, yielding surprisingly good agreement with numerically exact results for weak and moderately strong driving.Comment: 8 pages, 6 figure

Directed transport in driven optical lattices by gauge generation.

We examine the dynamics of ultracold atoms held in optical-lattice potentials. By controlling the switching of a periodic driving potential we show how a phase-induced renormalization of the intersite tunneling can be used to produce directed motion and control wave-packet spreading. We further show how this generation of a synthetic gauge potential can be used to split and recombine wave packets, providing an attractive route to implementing quantum computing tasks

Sublattice dynamics and quantum state transfer of doublons in two-dimensional lattices

We analyze the dynamics of two strongly interacting fermions moving in two-dimensional lattices under the action of a periodic electric field, both with and without a magnetic flux. Due to the interaction, these particles bind together forming a doublon. We derive an effective Hamiltonian that allows us to understand the interplay between the interaction and the driving, revealing surprising effects that constrain the movement of the doublons. We show that it is possible to confine doublons to just the edges of the lattice and to a particular sublattice if different sites in the unit cell have different coordination numbers. Contrary to what happens in one-dimensional systems, here we observe the coexistence of both topological and Shockley-like edge states when the system is in a nontrivial phase

Topology and Interactions in the photonic Creutz and Creutz-Hubbard ladders

The latest advances in the field of photonics have enabled the simulation of an increasing number of quantum models in photonic systems, turning them into an important tool for realizing exotic quantum phenomena. In this paper, different ways in which these systems can be used to study the interplay between flat band dynamics, topology, and interactions in a well-known quasi-1 D topological insulator-the Creutz ladder-are suggested. First, a simple experimental protocol is proposed to observe the Aharonov-Bohm localization in the noninteracting system, and the different experimental setups that might be used for this are reviewed. The inclusion of a repulsive Hubbard-type interaction term, which can give rise to repulsively bound pairs termed doublons, is then considered. The dynamics of these quasiparticles are studied for different points of the phase diagram, including a regime in which pairs are localized and particles are free to move. Finally, a scheme for the photonic implementation of a two-particle bosonic Creutz-Hubbard model is presented

Dynamical instability in kicked Bose-Einstein condensates

Bose-Einstein condensates subject to short pulses (kicks) from standing waves of light represent a nonlinear analog of the well-known chaos paradigm, the quantum kicked rotor. Previous studies of the onset of dynamical instability (i.e., exponential proliferation of noncondensate particles) suggested that the transition to instability might be associated with a transition to chaos. Here we conclude instead that instability is due to resonant driving of Bogoliubov modes. We investigate the Bogoliubov spectrum for both the quantum kicked rotor (QKR) and a variant, the double kicked rotor (QKR-2). We present an analytical model, valid in the limit of weak impulses which correctly gives the scaling properties of the resonances and yields good agreement with mean-field numerics

Effective Josephson dynamics in resonantly driven Bose-Einstein condensates

We show that the orbital Josephson effect appears in a wide range of driven atomic Bose-Einstein condensed systems, including quantum ratchets, double wells and box potentials. We use three separate numerical methods: Gross-Pitaevskii equation, exact diagonalization of the few-mode problem, and the Multi-Configurational Time-Dependent Hartree for Bosons algorithm. We establish the limits of mean-field and few-mode descriptions, demonstrating that they represent the full many-body dynamics to high accuracy in the weak driving limit. Among other quantum measures, we compute the instantaneous particle current and the occupation of natural orbitals. We explore four separate dynamical regimes, the Rabi limit, chaos, the critical point, and self-trapping; a favorable comparison is found even in the regimes of dynamical instabilities or macroscopic quantum self-trapping. Finally, we present an extension of the (t,t')-formalism to general time-periodic equations of motion, which permits a systematic description of the long-time dynamics of resonantly driven many-body systems, including those relevant to the orbital Josephson effect.Comment: 14 pages, 9 figure

Spin Filtering and Entanglement Swapping through Coherent Evolution of a Single Quantum Dot

We exploit the non-dissipative dynamics of a pair of electrons in a large square quantum dot to perform singlet-triplet spin measurement through a single charge detection and show how this may be used for entanglement swapping and teleportation. The method is also used to generate the AKLT ground state, a further resource for quantum computation. We justify, and derive analytic results for, an effective charge-spin Hamiltonian which is valid over a wide range of parameters and agrees well with exact numerical results of a realistic effective-mass model. Our analysis also indicates that the method is robust to choice of dot-size and initialization errors, as well as decoherence introduced by the hyperfine interaction.Comment: 5 pages, 3 figure