Cooling ultracold bosons in optical lattices by spectral transform

It is shown theoretically how to directly obtain the energy distribution of a weakly interacting gas of bosons confined in an optical lattice in the tight-binding limit. This is accomplished by adding a linear potential to a suitably prepared lattice, and allowing the gas to evolve under the influence of the total potential. After a prescribed time, a spectral transform is effected where each (highly non-local) energy state is transformed into a distinct site of the lattice, thus allowing the energy distribution to be (non-destructively) imaged in real space. Evolving for twice the time returns the atoms to their initial state. The results suggest efficient methods to both measure the temperature in situ, as well as to cool atoms within the lattice: after applying the spectral transform one simply needs to remove atoms from all but a few lattice sites. Using exact numerical calculations, the effects of interactions and errors in the application of the lattice are examined.Comment: 13+e pages, two embedded figures, revte

The discretised harmonic oscillator: Mathieu functions and a new class of generalised Hermite polynomials

We present a general, asymptotical solution for the discretised harmonic oscillator. The corresponding Schr\"odinger equation is canonically conjugate to the Mathieu differential equation, the Schr\"odinger equation of the quantum pendulum. Thus, in addition to giving an explicit solution for the Hamiltonian of an isolated Josephon junction or a superconducting single-electron transistor (SSET), we obtain an asymptotical representation of Mathieu functions. We solve the discretised harmonic oscillator by transforming the infinite-dimensional matrix-eigenvalue problem into an infinite set of algebraic equations which are later shown to be satisfied by the obtained solution. The proposed ansatz defines a new class of generalised Hermite polynomials which are explicit functions of the coupling parameter and tend to ordinary Hermite polynomials in the limit of vanishing coupling constant. The polynomials become orthogonal as parts of the eigenvectors of a Hermitian matrix and, consequently, the exponential part of the solution can not be excluded. We have conjectured the general structure of the solution, both with respect to the quantum number and the order of the expansion. An explicit proof is given for the three leading orders of the asymptotical solution and we sketch a proof for the asymptotical convergence of eigenvectors with respect to norm. From a more practical point of view, we can estimate the required effort for improving the known solution and the accuracy of the eigenvectors. The applied method can be generalised in order to accommodate several variables.Comment: 18 pages, ReVTeX, the final version with rather general expression

Stochastic Duality and Orthogonal Polynomials

For a series of Markov processes we prove stochastic duality relations with duality functions given by orthogonal polynomials. This means that expectations with respect to the original process (which evolves the variable of the orthogonal polynomial) can be studied via expectations with respect to the dual process (which evolves the index of the polynomial). The set of processes include interacting particle systems, such as the exclusion process, the inclusion process and independent random walkers, as well as interacting diffusions and redistribution models of Kipnis–Marchioro–Presutti type. Duality functions are given in terms of classical orthogonal polynomials, both of discrete and continuous variable, and the measure in the orthogonality relation coincides with the process stationary measure