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Physical Review A

Scaling of the density profiles of cold atoms near the quantum critical point in two- and three-dimensional optical lattices

T. A. Zaleski, T. K. Kopeć

Published Apr 24, 2012• 2 citations
Publisher

Abstract

We study the critical behavior near the quantum critical point of strongly interacting bosons placed in an optical 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 analytically the density profiles as a function of hopping and reduced chemical potential. Our approach allows us to explicitly compute the scaling form of the density ${n}_{s}$ for systems confined in two-dimensional square and three-dimensional cubic lattices. We find a good convergence of ${n}_{s}$ with the universal scaling function in a wide temperature range that is accessible in the current experiments on the density profiles in cold bosonic systems.

DOI
10.1103/physreva.85.043622

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We investigate the excitations of strongly interacting bosons confined in an optical lattice. Using the combined Bogoliubov method and the quantum rotor approach, we map the Hamiltonian of strongly interacting bosons onto U(1) phase action. This allows us to calculate the density of states for various geometries of optical lattices and its dependence on the presence of the superfluid phase. We observe that the superfluidity is manifested by a sharp coherence peak resulting from long-range order, while the excitation spectrum of the Mott insulator contains a gap.
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 interplay of temperature and trap effects in particle systems at their quantum critical regime, such as cold bosonic atoms in optical lattices at the transitions between Mott-insulator and superfluid phases. The theoretical framework is provided by the one-dimensional Bose-Hubbard model in the presence of an external trapping potential, and the trap-size scaling theory describing the large trap-size behavior at a quantum critical point. We present numerical results for the low-temperature behavior of the particle density and the density-density correlation function at the Mott transitions, and within the gapless superfluid phase, in the hard-core limit.
Critical behavior developed near a quantum phase transition, interesting in its own right, offers exciting opportunities to explore the universality of strongly correlated systems near the ground state. Cold atoms in optical lattices, in particular, represent a paradigmatic system, for which the quantum phase transition between the superfluid and Mott insulator states can be externally induced by tuning the microscopic parameters. In this paper, we describe our approach to study quantum criticality of cesium atoms in a two-dimensional (2D) lattice based on in situ density measurements. Our research agenda involves testing critical scaling of thermodynamic observables and extracting transport properties in the quantum critical regime. We present and discuss experimental progress on both fronts. In particular, the thermodynamic measurement suggests that the equation of state near the critical point follows the predicted scaling law at low temperatures.
We develop a strong-coupling ($t\ensuremath{\ll}U$) expansion technique for calculating the density profile for bosonic atoms trapped in an optical lattice with an overall harmonic trap at finite temperature and finite on-site interaction in the presence of superfluid regions. Our results match well with quantum Monte Carlo simulations at finite temperature. We also show that the superfluid order parameter never vanishes in the trap due to the proximity effect. Our calculations for the scaled density in the vacuum-to-superfluid transition agree well with the experimental data for appropriate temperatures. We present calculations for the entropy per particle as a function of temperature which can be used to calibrate the temperature in experiments. We also discuss issues connected with the demonstration of universal quantum critical scaling in the experiments.
We study quantum fluctuation-driven first-order phase transitions of a two-species bosonic system in a three-dimensional optical lattice. Using effective potential method we find that the superfluid-Mott insulator phase transition of one type of bosons can be changed from second order to first order by the quantum fluctuations of the other type of bosons. The study of the scaling behaviors near the quantum critical point shows that the first-order phase transition has a different universality from the second-order one. We also discuss the observation of this phenomenon in the realistic cold-atom experiments based on the in situ density measurements.