Deterministic walks over a random set of $N$ points in one and two dimensions ( $d=1,2\mathrm{}$) are considered. Points (“cities”) are randomly scattered in ${\mathbf{R}}^d$ following a uniform distribution. A walker (“tourist”), at each time step, goes to the nearest neighbor city that has not been visited in the past $\tau$ steps. Each initial city leads to a different trajectory composed of a transient part and a final $p$-cycle attractor. Transient times (for $d=1,2\mathrm{}$) follow an exponential law with a $\tau$-dependent decay time but the density of $p$ cycles can be approximately described by $D\left(p\right)\propto p^{-\alpha \left(\tau \right)}$. For $\tau \gg 1$ and $\tau /N\ll 1$, the exponent is independent of $\tau$. Some analytical results are given for the $d=1\mathrm{}$ case.

A recent proof of Bell’s theorem without inequalities [A. Cabello, Phys. Rev. Lett. 86, 1911 (2001)] is formulated as a Greenberger-Horne-Zeilinger–type proof involving just two observers. On one hand, this new approach allows us to derive an experimentally testable Bell inequality which is violated by quantum mechanics. On the other hand, it leads to a new state-independent proof of the Kochen-Specker theorem and provides a wider perspective on the relations between the major proofs of no hidden variables.

The effective diffusion coefficient for the overdamped Brownian motion in a tilted periodic potential is calculated in closed analytical form. Universality classes and scaling properties for weak thermal noise are identified near the threshold tilt where deterministic running solutions set in. In this regime the diffusion may be greatly enhanced, as compared to free thermal diffusion with, for a realistic experimental setup, an enhancement of up to 14 orders of magnitude.

We consider a tapping dynamics, analogous to that in experiments on granular media, on spin glasses and ferromagnets on random thin graphs. Between taps, zero temperature single spin flip dynamics takes the system to a metastable state. Tapping corresponds to flipping simultaneously any spin with probability $p$. This dynamics leads to a stationary regime with a steady state energy $E\left(p\right)$. We analytically solve this dynamics for the one-dimensional ferromagnet and $\pm J$ spin glass. Numerical simulations for spin glasses and ferromagnets of higher connectivity are carried out; in particular, we find a novel first order transition for the ferromagnetic systems.

We introduce and analyze numerically a nonequilibrium model with a conserved dynamics which is a realization of the contact process in an ensemble of constant particle number. The model possesses just one process in which particles jump around landing only on empty sites next to an existing particle. Particles are not allowed to land on a vacant site surrounded by empty sites. In contrast with the ordinary contact process, the present model does not have an absorbing state. In spite of lacking an absorbing state, the model displays properties that, in the thermodynamic limit, are identical to those of the ordinary contact process.

We investigate dark-bright vector solitary wave solutions to the coupled nonlinear Schrödinger equations which describe an inhomogeneous two-species Bose-Einstein condensate. While these structures are well known in nonlinear fiber optics, we show that spatial inhomogeneity strongly affects their motion, stability, and interaction, and that current technology suffices for their creation and control in ultracold trapped gases. The effect of controllably different interparticle scattering lengths is examined, and stability against three-dimensional deformations is considered.

We discuss a way of creating, manipulating, and detecting anyons in rotating Bose-Einstein condensates consisting of a small number of atoms. By achieving a quasidegeneracy in the atomic motional states we drive the system into a $\frac{1}{2}$-Laughlin state for fractional quantum Hall bosons. Localized $\frac{1}{2}$-quasiholes can be created by focusing lasers at the desired positions. We show how to manipulate these quasiholes in order to probe directly their $\frac{1}{2}$-statistics.

Using a novel approach, we investigate the shape of the average spectrum and the spectral fluctuations of the $k$-body embedded unitary ensemble in the limit of a large matrix dimension. We identify the transition point between a semicircle and a Gaussian shape. The transition also affects the spectral fluctuations which deviate from the Wigner-Dyson form and become Poissonian in the limit $k\ll m\ll l$. Here $m$ is the number of fermions and $l$ the number of degenerate single-particle states.

We have produced an interacting quantum degenerate Fermi gas of atoms composed of two spin states of magnetically trapped ${}^{40}\mathrm{K}$. The relative Fermi energies are adjusted by controlling the population in each spin state. Thermodynamic measurements reveal a resulting imbalance in the mean energy per particle between the two species, which is a factor of 1.4 at our lowest temperature. This imbalance of energy comes from a suppression of collisions between atoms in the gas due to the Pauli exclusion principle. Through measurements of the thermal relaxation rate we have directly observed this Pauli blocking as a factor of 2 reduction in the effective collision cross section in the quantum degenerate regime.

We present a quantitative analysis of the experimental accessibility of the Tonks-Girardeau gas in present-day experiments with cigar-trapped alkalis. For this purpose we derive, using a Bethe ansatz generated local equation of state, a set of hydrostatic equations describing one-dimensional, $\delta$-interacting Bose gases trapped in a harmonic potential. The resulting solutions cover the entire range of atomic densities.

We introduce the first simple mechanical system that shows fully realistic transport behavior while still being exactly solvable at the level of equilibrium statistical mechanics. The system is a Lorentz gas with fixed freely rotating circular scatterers which scatter point particles via perfectly rough collisions. Upon imposing either a temperature gradient and/or a chemical potential gradient, a stationary state is attained for which local thermal equilibrium holds. Transport in this system is normal in the sense that the transport coefficients which characterize the flow of heat and matter are finite in the thermodynamic limit. Moreover, the two flows are nontrivially coupled, satisfying Onsager’s reciprocity relations.

We explore a new general-purpose heuristic for finding high-quality solutions to hard discrete optimization problems. The method, called extremal optimization, is inspired by self-organized criticality, a concept introduced to describe emergent complexity in physical systems. Extremal optimization successively updates extremely undesirable variables of a single suboptimal solution, assigning them new, random values. Large fluctuations ensue, efficiently exploring many local optima. We use extremal optimization to elucidate the phase transition in the 3-coloring problem, and we provide independent confirmation of previously reported extrapolations for the ground-state energy of $\pm J$ spin glasses in $d=3\mathrm{}$ and $4$.

It has recently been proposed that fluctuating “pulled” fronts propagating into an unstable state should not be in the standard Kardar-Parisi-Zhang (KPZ) universality class for rough interface growth. We introduce an effective field equation for this class of problems, and show on the basis of it that noisy pulled fronts in $d+1\mathrm{}$ bulk dimensions should be in the universality class of the $\left(\right(d+1\mathrm{})+1\mathrm{})\mathrm{D}$ KPZ equation rather than of the $(d+1\mathrm{})\mathrm{D}$ KPZ equation. Our scenario ties together a number of heretofore unexplained observations in the literature, and is supported by previous numerical results.

Finding the ground state of a system with a complex energy landscape is important for many physical problems including protein folding, spin glasses, chemical clusters, and neural networks. Such problems are usually solved by heuristic search methods whose efficacy is judged by empirical performance on selected examples. We present a proof that, within the large class of algorithms that simulate a random walk on the landscape, threshold accepting is the best possible strategy. In particular, it can perform better than simulated annealing and Tsallis statistics. Our proof is the first example of a provably optimal strategy in this area.

A set of parallel replicas of a single simulation can be statistically coupled to closely approximate long trajectories. In many cases, this produces nearly linear speedup over a single simulation ( $M$ times faster with $M$ simulations), rendering previously intractable problems within reach of large computer clusters. Interestingly, by varying the coupling of the parallel simulations, it is possible in some systems to obtain greater than linear speedup. The methods are generalizable to any search algorithm with long residence times in intermediate states.

We present a proposal for protecting states against decoherence, based on the engineering of pointer states. We apply this procedure to the vibrational motion of a trapped ion, and show how to protect qubits, squeezed states, approximate phase eigenstates, and superpositions of coherent states.

The overdamped Brownian motion in a periodic potential under far from equilibrium conditions is considered. A large class of systems with an intrinsic asymmetry, called supersymmetric ratchets, is identified for which the occurrence of directed transport can be ruled out without any fine-tuning of parameters.

The frequency comb created by a femtosecond mode-locked laser and a microstructured fiber is used to phase coherently measure the frequencies of both the ${\mathrm{Hg}}^{+}$ and Ca optical standards with respect to the SI second. We find the transition frequencies to be $f_{\mathrm{Hg}}=1\mathrm{}064721609899143\left(10\right)\mathrm{Hz}$ and $f_{\mathrm{Ca}}=455\mathrm{}986240494158\left(26\right)\mathrm{Hz}$, respectively. In addition to the unprecedented precision demonstrated here, this work is the precursor to all-optical atomic clocks based on the ${\mathrm{Hg}}^{+}$ and Ca standards. Furthermore, when combined with previous measurements, we find no time variations of these atomic frequencies within the uncertainties of $|(\partial f_{\mathrm{Ca}}/\partial t){/f}_{\mathrm{Ca}}|\le 8\times {10}^{-14\mathrm{}}{\mathrm{yr}}^{-1\mathrm{}}$ and $|(\partial f_{\mathrm{Hg}}/\partial t){/f}_{\mathrm{Hg}}|\le 30\times {10}^{-14\mathrm{}}{\mathrm{yr}}^{-1\mathrm{}}$.

We demonstrate a general method to measure the quantum state of an angular momentum of arbitrary magnitude. The $(2F+1)\times (2F+1)$ density matrix is completely determined from a set of Stern-Gerlach measurements with $(4F+1)$ different orientations of the quantization axis. We implement the protocol for laser cooled Cesium atoms in the ${6S}_{1/2}(F=4\mathrm{})$ hyperfine ground state and apply it to a variety of test states prepared by optical pumping and Larmor precession. A comparison of input and measured states shows typical reconstruction fidelities $F\gtrsim 0.95$.

A trapped Bose-Einstein condensed atomic gas containing a quantized vortex is predicted to exhibit precession after a sudden rotation of the confining potential. The equations describing the motion of the condensate are derived and the effects of superfluidity explicitly pointed out. The dependence of the precession frequency on the relevant parameters of the problem is discussed. The proposed gyroscope is well suited to explore rotational effects at the level of single quanta of circulation.