We give a physical interpretation of the entries of the reflection $K$ matrices of Baxter's eight-vertex model in terms of an Ising interaction at an open boundary. Although the model still defies an exact solution, we nevertheless obtain the exact surface free energy from a crossing-unitarity relation. The singular part of the surface energy is described by the critical exponents ${\alpha }_s=2-\pi /2\mathrm{}\mu$ and ${\alpha }_1=1-\pi /\mu$, where μ controls the strength of the four-spin interaction. These values reduce to the known Ising exponents at the decoupling point $\mu =\pi /2\mathrm{}$ and confirm the scaling relations ${\alpha }_s={\alpha }_b+\nu$ and ${\alpha }_1={\alpha }_b-1\mathrm{}$.

We have determined the frequency of the ${}_{}{}^3{}_{}{}^{}P_1^{}{}_{}{}^{}$- ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ intercombination transition of atomic ${}^{40}\mathrm{Ca}$ stored in a magneto-optical trap to be $\nu =455\mathrm{}986240493.95\mathrm{kHz}$ with an estimated standard uncertainty of $0.43\mathrm{kHz}$ $(\delta \nu /\nu <{10}^{-12\mathrm{}})$ using a phase-coherent optical frequency chain from the Cs atomic clock to the visible. This allows the realization of the SI-unit meter according to its definition by visible radiation with 25-fold reduced uncertainty compared to previous measurements.

In light of the recent experimental observation of Bose-Einstein condensation in dilute ${}^{87}\mathrm{Rb}$ gas cooled to nanokelvin-scale temperatures, we give a quantitative account of the ground-state properties of magnetically trapped Bose gases. Using simple scaling arguments, we show that at large particle number the kinetic energy is a small perturbation, and find a spatial structure of the cloud of atoms and its momentum distribution dependent in an essential way on particle interactions. We also estimate the superfluid coherence length and the critical angular velocity at which vortex lines become energetically favorable.

We apply the thermodynamic formalism to discrete random walks in inhomogeneous environments. For simple one-dimensional examples we prove the existence of first and second order phase transitions. A typical mechanism for the latter is identified as localization-delocalization transition of relevant eigenfunctions of the generalized propagator. This can be understood by a mapping to quantum mechanical tight-binding models. Within a bivariate version of the formalism we show the occurrence of anomalous diffusion in the delocalized phase.

Using a renormalization group method, we calculate 800 high-temperature coefficients of the magnetic susceptibility of the hierarchical Ising model. The conventional quantities obtained from differences of ratios of coefficients show unexpected smooth oscillations with a period growing logarithmically and can be fitted assuming corrections to the scaling laws with complex exponents.

We report on a high-intensity source of polarization-entangled photon pairs with high momentum definition. Type-II noncollinear phase matching in parametric down conversion produces true entanglement: No part of the wave function must be discarded, in contrast to previous schemes. With two-photon fringe visibilities in excess of 97%, we demonstrated a violation of Bell's inequality by over 100 standard deviations in less than 5 min. The new source allowed ready preparation of all four of the EPR-Bell states.

A simple framework for accurate solution of a general class of one-dimensional Fokker-Planck and/or Schrödinger equations is presented. The main idea is representing the propagator in the form $P(x, t|x_0)=P_0(x, t\left|x_0\right)\exp \left[W\right(x, t\left|x_0\right)]$ and expanding the exponent $W$ in a power series in a given function of $t$, where $P_0$ is the exact solution of a reference harmonic-oscillator problem. The expansion coefficients are analytically evaluated from recursive relations. This approach is shown to be a dramatic improvement over the standard Taylor series expansion for the propagator in that just a few terms of the present expansion are sufficient to attain a very accurate description in the whole time domain.

Trace formulas relate the quantum density of states to the properties of the periodic orbits of the underlying classical system. The expressions depend critically on the nature of the classical dynamics. The near-integrable regime is most importantly characterized by the destruction of rational invariant tori being replaced by chains of resonances, yet without globalized chaos being introduced. We give new, generalized expressions appropriate for resonances and apply them to a system that has served as a paradigm for the transition from regular to chaotic dynamics.

The nonlinear response of noisy bistable systems driven by a strong amplitude-periodic force is investigated by physical experiment. The new phenomenon of locking of the mean switching frequency between states of a bistable system is found. It is shown that there is an interval of noise intensities in which the mean switching frequency remains constant and coincides with the frequency of the external periodic force. The region on the parameter plane "noise intensity—amplitude of periodic excitation" which corresponds to this phenomenon is similar to the synchronization (phase locking) region (Arnold's tongue) in classical oscillatory systems.

We have observed Bose-Einstein condensation of sodium atoms. The atoms were trapped in a novel trap that employed both magnetic and optical forces. Evaporative cooling increased the phase-space density by 6 orders of magnitude within seven seconds. Condensates contained up to 5×${10}^5$ atoms at densities exceeding ${10}^{14}$ ${\mathrm{cm}}^{-3\mathrm{}}$. The striking signature of Bose condensation was the sudden appearance of a bimodal velocity distribution below the critical temperature of ∼2μK. The distribution consisted of an isotropic thermal distribution and an elliptical core attributed to the expansion of a dense condensate.

We have scattered single photons from interfering de Broglie waves in an atom interferometer and observed contrast loss and revivals as the separation of the interfering paths at the point of scattering is increased. Additionally, we have demonstrated that the lost coherence can be recovered by observing only atoms that are correlated with photons emitted into a limited angular range.

We use the theory of continuous measurement to analyze the effects of decoherence on a realistic model of a quantum computer based on cavity QED. We show how decoherence affects the computation, and methods to prevent it.

We show that the use of the recently proposed thermostatistics based on the generalized entropic form $S_q\equiv \frac{k(1-\Sigma i^{}{p}_i^q)}{\mathrm{}(q-1\mathrm{})\mathrm{}}$ (where $q\in R$, with $q=1\mathrm{}$ corresponding to the Boltzmann-Gibbs-Shannon entropy $-k\Sigma i^{}{p}_i \ln p_i$), together with the Lévy-Gnedenko generalization of the central limit theorem, provide a basic step towards the understanding of why Lévy distributions are ubiquitous in nature. A consistent experimental verification is proposed.

A new generally applicable method to solve differential equations, based on neural networks, is proposed. Straightforward to implement, finite differences and coordinate transformations are not used. The neural network provides a flexible and compact base for representing the solution, found through the global minimization of an error functional. As a proof of principle, a two-dimensional ideal magnetohydrodynamic plasma equilibrium is solved. Since no particular topology is assumed, the technique is especially promising for the three-dimensional plasma equilibrium problem.

Accuracy of the electron's atomic mass has been improved tenfold by comparing cyclotron frequencies of electrons and single ${\mathrm{C}}^{6+}$ ions alternately confined to the same uniform magnetic field in a Penning trap. Cyclotron resonances are observed via frequency shifts in the particle's continuously monitored harmonic motion parallel to the field. Field instability and relativity cause, respectively, the leading statistical and systematic errors. Combining the electron's atomic mass $M_e=0.000\mathrm{}5485799111\left(12\right)\mathrm{}$ u with the proton's yields the mass ratio $\frac{m_p}{m_e}=1836.152\mathrm{}6665\left(40\right)\mathrm{}$.

We describe the representation of arbitrary density operators in terms of expectation values of simple projection operators. Two representations are presented which yield nonrecursive schemes for experimentally determining the density operator of any quantum system. We suggest a possible experimental implementation in quantum optics.

Polarized neutron interferometry has been used to measure the amplitudes for the exchange of up to five photons between the neutron and an oscillating magnetic field. The Rabi spin flip configuration and a configuration in which spin flip is suppressed were investigated, because they raise different questions on the conservation of angular momentum. Because of the linearity of the time-dependent interference signal in the exchange amplitudes, photon exchange probabilities below 1% could be resolved.

Models of random walks are considered in which walkers are born at one site and die at all other sites. Steady-state distributions of walkers exhibit dimensionally dependent critical behavior as a function of the birth rate. Exact analytical results for a hyperspherical lattice yield a second-order phase transition with a nontrivial critical exponent for all positive dimensions $D\ne 2$, 4. Numerical studies of hypercubic and fractal lattices indicate that these exact results are universal. This work elucidates the adsorption transition of polymers at curved interfaces.

A drift motion of kinks in the ${\varphi }^4$ model, induced by an oscillating external field, is investigated and the expression for nonlinear mobility coefficients is obtained. The general theory is applied to the concrete system described in the framework of the ${\varphi }^4$ model, namely, to the drift of a small-angle domain wall in an easy-axis ferromagnet near the reorientation phase transition.

We present various experiments demonstrating the mutual exclusivity of observing interference and which-path information, as demanded by Bohr's complementarity principle. Using photon pairs created in parametric down-conversion, no which-path measurements need to be performed on the interfering photon itself. Instead the other photon can be used to introduce distinguishability, which consequently destroys interference. However, a suitable measurement erases this distinguishability and interference can be recovered.