This paper reviews recent experimental and theoretical progress concerning many-body phenomena in dilute, ultracold gases. It focuses on effects beyond standard weak-coupling descriptions, such as the Mott-Hubbard transition in optical lattices, strongly interacting gases in one and two dimensions, or lowest-Landau-level physics in quasi-two-dimensional gases in fast rotation. Strong correlations in fermionic gases are discussed in optical lattices or near-Feshbach resonances in the BCS-BEC crossover.

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Many-Body Physics with Ultracold Gases

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

The short-range, one-band model for electron correlations in a narrow energy band is solved exactly in the one-dimensional case. The ground-state energy, wave function, and the chemical potentials are obtained, and it is found that the ground state exhibits no conductor-insulator transition as the correlation strength is increased.

Absence of Mott Transition in an Exact Solution of the Short-Range, One-Band Model in One Dimension

Functions, types and expressions programming languages and their operational semantics compilation partial evaluation of a flow chart languages partial evaluation of a first-order functional languages the view from Olympus partial evaluation of the Lambda calculus partial evaluation of prolog aspects of Similix - a partial evaluator for a subset of scheme partial evaluation of C applications of partial evaluation termination of partial evaluation program analysis more general program transformation guide to the literature the self-applicable scheme specializer.

Partial evaluation and automatic program generation

It is a fundamental assumption of statistical mechanics that a closed system with many degrees of freedom ergodically samples all equal energy points in phase space. To understand the limits of this assumption, it is important to find and study systems that are not ergodic, and thus do not reach thermal equilibrium. A few complex systems have been proposed that are expected not to thermalize because their dynamics are integrable. Some nearly integrable systems of many particles have been studied numerically, and shown not to ergodically sample phase space. However, there has been no experimental demonstration of such a system with many degrees of freedom that does not approach thermal equilibrium. Here we report the preparation of out-of-equilibrium arrays of trapped one-dimensional (1D) Bose gases, each containing from 40 to 250 (87)Rb atoms, which do not noticeably equilibrate even after thousands of collisions. Our results are probably explainable by the well-known fact that a homogeneous 1D Bose gas with point-like collisional interactions is integrable. Until now, however, the time evolution of out-of-equilibrium 1D Bose gases has been a theoretically unsettled issue, as practical factors such as harmonic trapping and imperfectly point-like interactions may compromise integrability. The absence of damping in 1D Bose gases may lead to potential applications in force sensing and atom interferometry.

A quantum Newton's cradle

A gas of one-dimensional Bose particles interacting via a repulsive delta-function potential has been solved exactly. All the eigenfunctions can be found explicitly and the energies are given by the solutions of a transcendental equation. The problem has one nontrivial coupling constant, γ. When γ is small, Bogoliubov’s perturbation theory is seen to be valid. In this paper, we explicitly calculate the ground-state energy as a function of γ and show that it is analytic for all γ, except γ=0. In Part II, we discuss the excitation spectrum and show that it is most convenient to regard it as a double spectrum—not one as is ordinarily supposed.

Exact analysis of an interacting bose gas. i. the general solution and the ground state

Linear one step methods of a novel design are given for the numerical solution of stiff systems of ordinary differential equations. These methods permit fast integration with increments h of the independent variable adjusted to the slowly varying, dominant solutions. The integration formulas owe their efficiency to combining a reasonable order of accuracy for $h \to 0$ with a precise simulation of specific rapidly varying solutions, and with A-stability in the sense of Dahlquist.

Efficient integration methods for stiff systems of ordinary differential equations

An efficient implementation of the Crout elimination method in solving large sparse systems of linear algebraic equations of arbitrary structure is described. A computer program, GNSO, by symbolic processing, generates another program, SOLVE, which represents the optimal reduced Crout algorithm in the sense that only nonzero elements are stored and operated on. The method presented is particularly powerful when a system of fixed sparseness structure must be solved repeatedly with different numerical values. In practical examples, the execution of SOLVE was observed to be typically N times as fast as that of the full Crout algorithm, where N is the order of the system.

Symbolic Generation of an Optimal Crout Algorithm for Sparse Systems of Linear Equations

An integration method for ordinary differential equations is said to be contractive if all numerical solutions of the test equationx′=λx generated by that method are not only bounded (as required for stability) but non-increasing. We develop a theory of contractivity for methods applied to stiff and non-stiff, linear and nonlinear problems. This theory leads to the design of a collection of specific contractive Adams-type methods of different orders of accuracy which are optimal with respect to certain measures of accuracy and/or contractivity. Theoretical and numerical results indicate that some of these novel methods are more efficient for solving problems with a lack of smoothness than are the familiar backward differentiation methods. This lack of smoothness may be either inherent in the problem itself, or due to the use of strongly varying integration steps. In solving smooth problems, the efficiency of the low-order contractive methods we propose is approximately the same as that of the corresponding backward differentiation methods.

Contractive methods for stiff differential equations part I

Two of the most commonly used methods, the trapezoidal rule and the two-step backward differentiation method, both have drawbacks when applied to difficult stiff problems. The trapezoidal rule does not sufficiently damp the stiff components and the backward differentiation method is unstable for certain stable variable-coefficient problems with variable-steps. In this paper we show that there exists a one-parameter family of two-step, second-order one-leg methods which are stable for any dissipative nonlinear system and for any test problem of the form $\dot x = \lambda (t)x$, $\operatorname{Re} \lambda (t) \leq 0$, using arbitrary step sequences.

Werner Liniger

Papers

Exact analysis of an interacting bose gas. i. the general solution and the ground state

Efficient integration methods for stiff systems of ordinary differential equations

Symbolic Generation of an Optimal Crout Algorithm for Sparse Systems of Linear Equations

Contractive methods for stiff differential equations part I

Stability of Two-Step Methods for Variable Integration Steps