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AIP conference proceedings

Expansion of Tonks-Girardeau Gases on a One-Dimensional Optical Lattice

Marcos Rigol, Alejandro Muramatsu

Published Jan 1, 2006• 0 citations
Publisher

Abstract

We study the free expansion of Tonks‐Girardeau gases on one‐dimensional lattices after removing the confining potential. We find that for low initial densities the momentum distribution of the expanding gas rapidly approaches the one of noninteracting fermions. In the opposite limit, i.e, when the initial system is a pure Mott insulator with one particle per lattice site, the expansion leads to the emergence of quasi‐condensates at finite momentum. These traveling quasi‐condensates develop quasi‐long range one‐particle correlations similar to the ones shown to be universal in the equilibrium case.

DOI
10.1063/1.2354600

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AIP conference proceedings•Jan 1, 2006•0 citations
We study the free expansion of Tonks‐Girardeau gases on one‐dimensional lattices after removing the confining potential. We find that for low initial densities the momentum distribution of the expanding gas rapidly approaches the one of noninteracting fermions. In the opposite limit, i.e, when the initial system is a pure Mott insulator with one particle per lattice site, the expansion leads to the emergence of quasi‐condensates at finite momentum. These traveling quasi‐condensates develop quasi‐long range one‐particle correlations similar to the ones shown to be universal in the equilibrium case.
We summarize in the present work exact results obtained for Tonks-Girardeau gases on one-dimensional optical lattices both for the ground state and nonequilibrium dynamics. On the theoretical side, impenetrable bosons offer the opportunity to study strongly interacting systems in one-dimensional lattices exactly, by means of the Jordan-Wigner transformation, and hence contribute to the topic of strong correlations at the center of interest in both condensed matter physics and quantum gases. This motivation is further enhanced by recent experimental realizations of such systems with ultracold atoms. After having shown their universal properties in equilibrium, we concentrate on their nonequilibrium dynamics. It will be shown that, starting from a pure Fock state, quasi-long-range correlations develop dynamically and lead to the formation of quasicondensates with a momentum determined by the underlying lattice. We expect this effect to be relevant for atom lasers with full control of the wavelength. Then, we will show that the free evolution of an initially confined Tonks-Girardeau gas leads to a momentum distribution that approaches at long times that of the equivalent fermionic system, giving rise to a bosonic gas with a Fermi edge, and hence a fermionization that can only be obtained out of equilibrium. Remarkably, although the momentum distribution function of the Tonks-Girardeau gas becomes equal to the one of the fermions, no loss in coherence is observed in the system, as reflected by a large occupation of eigenstates of the one-particle density matrix.
We review recent exact results for the free expansion of impenetrable bosons on one-dimensional lattices, after switching off a confining potential. When the system is initially in a superfluid state, far from the regime in which the Mott-insulator appears in the middle of the trap, the momentum distribution of the expanding bosons rapidly approaches the momentum distribution of non-interacting fermions. Remarkably, no loss in coherence is observed in the system as reflected by a large occupation of the lowest eigenstate of the one-particle density matrix. In the opposite limit, when the initial system is a pure Mott insulator with one particle per lattice site, the expansion leads to the emergence of quasicondensates at finite momentum. In this case, one-particle correlations like the ones shown to be universal in the equilibrium case develop in the system. We show that the out-of-equilibrium behavior of the Shannon information entropy in momentum space, and its contrast with one of non-interacting fermions, allows to differentiate the two different regimes of interest. It also helps in understanding the crossover between them.
We develop an exact approach to study the quench dynamics of hard-core bosons initially in thermal equilibrium in one-dimensional lattices. This approach is used to study the sudden expansion of thermal states after confining potentials are switched off. We find that a dynamical fermionization of the momentum distribution occurs at all temperatures. This phenomenon is studied for low initial site occupations, for which the expansion of the cloud is self-similar. In this regime, the occupation of the natural orbitals allows one to distinguish hard-core bosons from noninteracting fermions. We also study the free expansion of initial Mott insulating domains at finite temperature and show that the emergence of off-diagonal one-body correlations is suppressed gradually with increasing temperature. Surprisingly, the melting of the Mott domain is accompanied by an effective cooling of the system. We explain this phenomenon analytically using an equilibrium description based on an emergent local Hamiltonian.
Recent experiments on ultracold atomic alkali gases in a one-dimensional optical lattice have demonstrated the transition from a gas of soft-core bosons to a Tonks-Girardeau gas in the hard-core limit, where one-dimensional bosons behave like fermions in many respects. We have studied the underlying many-body physics through numerical simulations which accommodate both the soft-core and hard-core limits in one single framework. We find that the Tonks-Girardeau gas is reached only at the strongest optical lattice potentials. Results for slightly higher densities, where the gas develops a Mott-like phase already at weaker optical lattice potentials, show that these Mott-like short-range correlations do not enhance the convergence to the hard-core limit.
We analyze in detail the expansion of a 1D Bose gas after removing the axial confinement. We show that during its one-dimensional expansion the density of the Bose gas does not follow a self-similar solution. Our analysis is based on a nonlinear Schrödinger equation with variable nonlinearity whose validity is discussed for the expansion problem, by comparing with an exact Bose-Fermi mapping for the case of an initial Tonks-Girardeau gas. For this case, the gas is shown to expand self-similarly, with a different scaling law compared to the one-dimensional Thomas-Fermi condensate.
We discuss in this paper ground‐state and nonequilibrium properties of ultracold atoms in optical lattices in the strongly correlated limit. We review fermionic Mott‐insulators studied on the basis of quantum Monte Carlo simulations, where local quantum criticality is displayed in one dimension. We continue with exact results for hard‐core bosons in one dimension, showing their universal properties in equilibrium, and their nonequilibrium dynamics. Here we show that starting from a Fock state, a quasi‐condensate emerges at finite momentum during free expansion. On the other hand, the free evolution of an initially confined quasi‐condensate of hard‐core bosons leads to a bosonic gas with a Fermi edge, and hence a fermionization that can only be obtained out of equilibrium.