An iterative method to generate a discrete path integral solution of the Kramers problem is presented. It is based on a straightforward derivation of the functional formalism from the underlying Langevin equations. The method is rather simple and systematic and allows us to analytically evaluate the short time propagator up to and including terms of fourth order in a time increment $\tau$. This means a significant reduction of the number of time steps $N$ that are necessary to obtain convergent results for a given net increment $t=N\tau$.

The Kramers escape problem for an overdamped Brownian particle subjected to both dichotomous and thermal noise is studied. In the limit of weak thermal noise an asymptotically exact rate formula is obtained that is valid for general potentials and arbitrary correlation times of the dichotomous noise. Our predictions are verified by accurate numerical results and their consequences with regards to nonequilibrium potentials, resonant activation, and Brownian motors are discussed.

This Letter reports the experimental observation of light confinement effects by near-field optical microscopy. Depolarization effects giving rise to light confinement close to nanoscopic objects have been unambiguously observed in near-field optical images of subwavelength dielectric pads etched on a flat glass substrate. According to the incident polarization, this phenomenon leads to reverse contrasts in the near-field optical image of the same subwavelength objects.

We introduce a new parameter, which is related to the well known traveling salesman problem (TSP), by making an analogy with a classical particle traveling around a closed trajectory of the TSP. In order to exhibit the broken symmetry associated with the rotation invariance of the TSP set of cities around any space axis, the parameter behavior, as a function of the temperature, is found along a simulated annealing process. As expected, this behavior resembles the order parameter evolution, in temperature, of some physical systems. The discussion includes a set of a few cities up to a set of 500 cities.

Operators conjugate to the Hamiltonian are constructed explicitly for the quantum harmonic oscillator by two approaches in the space spanned by the eigenstates of $q$ and the eigenstates of $p$. The operators are quantum analogs of a classical angle variable divided by the oscillator frequency. Matrix elements have been evaluated in the coherent state representation. Either conjugate operator can be used to construct an explicitly time-dependent operator invariant. It can be used as the starting point in a new perturbative procedure for constructing invariant operators for nonlinear, nonautonomous Hamiltonians.

We demonstrate how Bragg diffraction of atomic matter waves at a time-modulated thick standing light wave can be used to coherently shift the de Broglie frequency of the diffracted atoms. The coherent frequency shift is experimentally confirmed by interferometric superposition of modulated and unmodulated atoms resulting in time-dependent interference fringes. Our frequency shifter for atomic matter waves is a generalization of an acousto-optic frequency shifter for photons.

We derive a trace formula for the splitting-weighted density of states suitable for chaotic potentials with isolated symmetric wells. This formula is based on complex orbits which tunnel through classically forbidden barriers. The theory is applicable whenever the tunneling is dominated by isolated orbits, a situation which applies to chaotic systems but also to certain near-integrable ones. It is used to analyze a specific two-dimensional potential with chaotic dynamics. Mean behavior of the splittings is predicted by an orbit with imaginary action. Oscillations around this mean are obtained from a collection of related orbits whose actions have nonzero real part.

We present experiments studying the coherent motion of atoms in crystals made from on and off resonant light. The experiments confirm that inside the light-field atoms fulfilling the Bragg condition form a standing matter wave pattern. As a consequence we observed anomalous transmission of atoms through resonant light fields.

We measure the ground-state occupation and energy of a dilute Bose gas of ${}^{87}\mathrm{Rb}$ atoms as a function of temperature. The ground-state fraction shows good agreement with the predictions for an ideal Bose gas in a 3D harmonic potential. The measured transition temperature is $0.94\left(5\right)T_o$, where $T_o$ is the value for a noninteracting gas in the thermodynamic limit. We determine the energy from a model-independent analysis of the velocity distribution, after ballistic expansion, of the atom cloud. We observe a distinct change in slope of the energy-temperature curve near the transition, which indicates a sharp feature in the specific heat.

Simple models of directed bond percolation with temporal disorder are introduced and studied via series expansions and Monte Carlo simulations. Series have been derived for the percolation probability on the directed square lattice. Analysis of the series revealed that the critical exponent $\beta$ and critical point $p_c$ change continuously with the strength of the disorder. Monte Carlo simulation confirmed the continuous change of critical exponents. Estimates for the temporal correlation-length exponent ${\nu }_{\parallel }$ for weak disorder showed that ${\nu }_{\parallel }<2\mathrm{}$ in apparent violation of the Harris criterion.

Using extensive numerical analysis and exact calculations we show that the relaxation of a classical particle in 1D anharmonic potential landscapes with a leading quartic term follows a $1/t$ decay law at all temperatures, leading to a logarithmically increasing mean square displacement. For leading anharmonic terms of form $x^{2n}$ we find that the asymptotic relaxation is consistent with $1/t^{\phi }$, where $\phi =1\mathrm{}/(n-1\mathrm{})$ at all temperatures. We briefly comment on the possible implications of this result in the study of displacive structural transitions and in complex systems.

Suppose a two-dimensional dynamical system has a stable attractor that is surrounded by an unstable limit cycle. If the system is additively perturbed by white noise, the rate of escape through the limit cycle will fall off exponentially as the noise strength tends to zero. By analyzing the associated Fokker–Planck equation we show that in general the weak-noise escape rate is non-Arrhenius: it includes a factor that is periodic in the logarithm of the noise strength. The presence of this slowly oscillating factor is due to the nonequilibrium potential of the system being nondifferentiable at the limit cycle. We point out the implications for the weak-noise limit of stochastic resonance models.

Imagine being shown $N$ samples of random variables drawn independently from the same distribution. What can you say about the distribution? In general, of course, the answer is nothing, unless you have some prior notions about what to expect. From a Bayesian point of view one needs an a priori distribution on the space of possible probability distributions, which defines a scalar field theory. In one dimension, free field theory with a normalization constraint provides a tractable formulation of the problem, and we discuss generalizations to higher dimensions.

Recently, the phenomenon of array enhanced stochastic resonance was demonstrated via numerical simulations by Lindner et al. [Phys. Rev. Lett. 75, 3 (1995)]. We present experimental evidence that the signal-to-noise ratio (SNR) of the output signal of a single diode resonator can be significantly improved by coupling it diffusively into an array of up to 32 nonidentical resonators. It is shown that the maximum SNR coincides with optimal spatiotemporal synchronization of the chain. We briefly address the connection between spatiotemporal stochastic resonance and the presence of kink-antikink pairs in the array.

We reconstruct the density matrices and Wigner functions for various quantum states of motion of a harmonically bound ${}^9{\mathrm{Be}}^{+}$ ion. We apply coherent displacements of different amplitudes and phases to the input state and measure the number state populations. Using novel reconstruction schemes we independently determine both the density matrix in the number state basis and the Wigner function. These reconstructions are sensitive indicators of decoherence in the system.

A new, general, field theoretic approach to the derivation of asymptotic conservation laws is presented. In this approach asymptotic conservation laws are constructed directly from the field equations according to a universal prescription which does not rely upon the existence of Noether identities or any Lagrangian or Hamiltonian formalisms. The resulting general expressions of the conservation laws enjoy important invariance properties and synthesize all known asymptotic conservation laws, such as the Arnowitt-Deser-Misner energy in general relativity.

Conventional weak-coupling perturbation theory suffers from problems that arise from the resonant coupling of successive orders in the perturbation series. Multiple-scale perturbation theory avoids such problems by implicitly performing an infinite reordering and resummation of the conventional perturbation series. Multiple-scale analysis provides a good description of the classical anharmonic oscillator. Here, it is extended to study the Heisenberg operator equations of motion for the quantum anharmonic oscillator. The analysis yields a system of nonlinear operator differential equations, which is solved exactly. The solution provides an operator mass renormalization of the theory.

Using a generalized transfer matrix method we exactly solve a Schrödinger-like equation in a time periodic potential, with discretized Euclidean space-time. The ground state wave function propagates in space and time with an oscillating solitonlike wave packet and the wave front is wedge shaped. In a statistical mechanics framework our solution represents the partition sum of a directed polymer subjected to a potential layer with alternating (attractive and repulsive) pinning centers.

It is well known that at long wavelengths $\lambda$ an s-wave scatterer can have a scattering cross section $\sigma$ on the order of ${\lambda }^2$, much larger than its physical size, as measured by the range of its potential. Very interesting phenomena can arise when two or more identical scatterers are placed close together, well within one wavelength. We show that, for a pair of identical scatterers, an extremely narrow p-wave “proximity” resonance develops from a broader s-wave resonance of the individual scatterers. A new s-wave resonance of the pair also appears. The relation of these proximity resonances (so called because they appear when the scatterers are close together) to the Thomas and Efimov effects is discussed.

A global minimum principle is reported for inverse scattering for Schrödinger's equation without bound states. For the one-dimensional problem, the line integral of the potential on the interval $\{-\infty ,x\}$ can be found by minimizing a certain functional of the scattering amplitude and a variational wave field. In three dimensions, the minimum of a closely related functional is shown to yield $-\int d^3\mathbf{y}\nu \left(\mathbf{y}\right)/|\mathbf{x}-\mathbf{y}{|}^2$, where $\nu \left(\mathbf{x}\right)$ is the potential.