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arXiv:0809.0771 · Physical Review A

Visibility of cold atomic gases in optical lattices for finite temperatures

Alexander Hoffmann, Axel Pelster

Published May 14, 2009• 31 citations
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Abstract

In nearly all experiments with ultracold atoms time-of-flight pictures are the only data available. In this paper we present an analytical strong-coupling calculation for those time-of-flight pictures of bosons in a three-dimensional optical lattice in the Mott phase. This allows us to determine the visibility, which quantifies the contrast of peaks in the time-of-flight pictures, and we suggest how to use it as a thermometer.

DOI
10.1103/physreva.79.053623

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In nearly all experiments with ultracold atoms time-of-flight pictures are the only data available. In this paper we present an analytical strong-coupling calculation for those time-of-flight pictures of bosons in a three-dimensional optical lattice in the Mott phase. This allows us to determine the visibility, which quantifies the contrast of peaks in the time-of-flight pictures, and we suggest how to use it as a thermometer.
We study the spatial correlations of strongly interacting bosons in a ground state, confined in a two-dimensional square and a three-dimensional cubic lattice. Using the combined Bogoliubov method and the quantum rotor approach, we map the Hamiltonian of strongly interacting bosons onto U(1) phase action in order to calculate the atom-atom correlations' decay along the principal axis and a diagonal of the lattice-plane direction as a function of distance. Lower tunneling rates lead to quicker decays of the correlations, whose character becomes exponential. Finally, correlation functions allow us to calculate quantities that are directly bound to experimental outcomes, namely time-of-flight absorption images and resulting visibility. Our results contain all the characteristic features present in experimental data (transition from Mott insulating blob to superfluid peaks, etc.), emphasizing the usability of the proposed approach.
We investigate the ground-state and finite-temperature properties of the Bose-Hubbard model using the standard basis operator method applied to the equation of motion for Green's functions. We show that by introducing the self-consistent approach, the theory goes significantly beyond the mean-field approximation, allowing more-precise description of quantum and thermal fluctuations. For optical lattice systems with ultracold bosons, we find relevant quantitative agreement with numerical and experimental data. In particular, the presented method reveals the importance of thermal fluctuations in time-of-flight experiments. For the most reliable fitting of the visibility analytical curve to the experimental data, the value of temperature is determined and proximity to the quantum regime is discussed. The problem of identification of the superfluid--Mott-insulator phase boundary from time-of-flight experiments is also considered.
We propose a method for measuring the temperature of strongly correlated phases of ultracold atom gases confined in spin-dependent optical lattices. In this technique, a small number of 'impurity' atoms—trapped in a state that does not experience the lattice potential—are in thermal contact with atoms bound to the lattice. The impurity serves as a thermometer for the system because its temperature can be straightforwardly measured using time-of-flight expansion velocity. This technique may be useful for resolving many open questions regarding thermalization in these isolated systems. We discuss the theory behind this method and demonstrate proof-of-principle experiments, including the first realization of a three-dimensional (3D) spin-dependent lattice in the strongly correlated regime.
In this paper, by treating the hopping parameter in the Bose-Hubbard model as a perturbation, with the help of the resummed Green's function method and cumulant expansion, the momentum distribution function of an ultracold Bose system in a triangular optical lattice is calculated analytically. By its use, the time-of-flight absorption picture is plotted and the corresponding visibility is determined. A comparison between our analytical results and the experimental data of Becker et al. [New J. Phys. 12, 065025 (2010)] exhibits a qualitative agreement.
We discuss a technique to probe the properties of an interacting cold atomic gas that can be viewed as a dynamical compressibility measurement. We apply this technique to the study of the superfluid-to-Mott-insulator quantum phase transition in one and three dimensions for a bosonic gas trapped in an optical lattice. Excitations of the system are detected by time-of-flight measurements. The experimental data for the one-dimensional case are in good agreement with the results of a time-dependent density-matrix-renormalization-group calculation.