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arXiv:0912.4493 · New Journal of Physics

Thermalization in a quasi-one-dimensional ultracold bosonic gas

I. E. Mazets, Jörg Schmiedmayer

Published May 28, 2010• 74 citations
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Abstract

We study the collisional processes that can lead to thermalization in one-dimensional (1D) systems. For two-body collisions, excitations of transverse modes are the prerequisite for energy exchange and thermalization. At very low temperatures, excitations of transverse modes are exponentially suppressed, thermalization by two-body collisions stops and the system should become integrable. In quantum mechanics, virtual excitations of higher radial modes are possible. These virtually excited radial modes give rise to effective three-body velocity-changing collisions, which lead to thermalization. We show that these three-body elastic interactions are suppressed by pairwise quantum correlations when approaching the strongly correlated regime. If the relative momentum k is small compared with the two-body coupling constant c , the three-particle scattering state is suppressed by a factor of ( k / c ) 12 , which is proportional to γ −12 , that is, to the square of the three-body correlation function at zero distance in the limit of the Lieb–Liniger parameter γ≫1. This demonstrates that in 1D quantum systems, it is not the freeze-out of two-body collisions but the strong quantum correlations that ensure absence of thermalization on experimentally relevant time scales.

DOI
10.1088/1367-2630/12/5/055023

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We study the collisional processes that can lead to thermalization in one-dimensional (1D) systems. For two-body collisions, excitations of transverse modes are the prerequisite for energy exchange and thermalization. At very low temperatures, excitations of transverse modes are exponentially suppressed, thermalization by two-body collisions stops and the system should become integrable. In quantum mechanics, virtual excitations of higher radial modes are possible. These virtually excited radial modes give rise to effective three-body velocity-changing collisions, which lead to thermalization. We show that these three-body elastic interactions are suppressed by pairwise quantum correlations when approaching the strongly correlated regime. If the relative momentum k is small compared with the two-body coupling constant c , the three-particle scattering state is suppressed by a factor of ( k / c ) 12 , which is proportional to γ −12 , that is, to the square of the three-body correlation function at zero distance in the limit of the Lieb–Liniger parameter γ≫1. This demonstrates that in 1D quantum systems, it is not the freeze-out of two-body collisions but the strong quantum correlations that ensure absence of thermalization on experimentally relevant time scales.
We show that thermalization and the breakdown of integrability in the one-dimensional Lieb-Liniger model caused by local three-body elastic interactions is suppressed by pairwise quantum correlations when approaching the strongly correlated regime. If the relative momentum kkk is small compared to the two-body coupling constant ccc the three-particle scattering state is suppressed by a factor of (k/c)12{(k/c)}^{12}(k/c)12. This demonstrates that in one-dimensional quantum systems it is not the freeze out of two body collisions but the strong quantum correlations which ensures integrability.
We study ultra-cold bosons out of equilibrium in a one-dimensional (1D) setting and probe the breaking of integrability and the resulting relaxation at the onset of the crossover from one to three dimensions. In a quantum Newton's cradle type experiment, we excite the atoms to oscillate and collide in an array of 1D tubes and observe the evolution for up to 4.8 seconds (400 oscillations) with minimal heating and loss. By investigating the dynamics of the longitudinal momentum distribution function and the transverse excitation, we observe and quantify a two-stage relaxation process. In the initial stage single-body dephasing reduces the 1D densities, thus rapidly drives the 1D gas out of the quantum degenerate regime. The momentum distribution function asymptotically approaches the distribution of quasimomenta (rapidities), which are conserved in an integrable system. In the subsequent long time evolution, the 1D gas slowly relaxes towards thermal equilibrium through the collisions with transversely excited atoms. Moreover, we tune the dynamics in the dimensional crossover by initializing the evolution with different imprinted longitudinal momenta (energies). The dynamical evolution towards the relaxed state is quantitatively described by a semiclassical molecular dynamics simulation.
We consider decay of a quasiparticle in a nearly-one-dimensional quasicondensate of trapped atoms, where virtual excitations of transverse modes break down one-dimensionality and integrability, giving rise to effective three-body elastic collisions. We calculate the matrix element for the process that involves one incoming quasiparticle and three outgoing quasiparticles. Scattering that involves low-frequency modes with high thermal population results in a diffusive dynamics of a bunch of quasiparticles created in the system.
Ultracold Bose gases in one-dimensional optical lattices constitute an important benchmark problem in the study of strongly interacting many-body quantum phases. Here we present a combined experimental and theoretical study of their phase-coherence properties over a wide range of lattice depths. Experimentally, we extract the single-particle correlation function directly from the measured momentum distribution. Theoretically, we perform tensor-network simulations of the Bose-Hubbard model that incorporate all relevant experimental parameters. For deep lattices well within the Mott insulator regime, the experimental results are in good agreement with the expected zero-temperature behavior, with only small temperature-dependent corrections. As the lattice depth is reduced, finite-temperature effects become increasingly important. We find that the experimental data are quantitatively described by an effective temperature extracted from the tensor-network simulations, and that this effective temperature decreases markedly with increasing lattice depth. Rather than indicating actual cooling, we interpret this behavior as evidence of inhibition of thermalization caused by the formation of Mott domains that suppress heat transport. Counterintuitively, the inhibition of thermalization favors the preparation of an effectively low-entropy quantum gas in the trap center for large lattice depths.
Transport coefficients are typically divergent for quantum integrable systems in one dimension, such as a Bose gas with a two-body contact interaction. However, when a one-dimensional system is realized by confining bosons into a tight matter waveguide, an effective three-body interaction inevitably arises as leading perturbation to break the integrability. This fact motivates us to study the thermal conductivity of a Bose gas in one dimension with both two-body and three-body interactions. In particular, we evaluate the Kubo formula exactly to the lowest order in perturbation by summing up all contributions that are naively higher orders in perturbation but become comparable in the zero-frequency limit due to the pinch singularity. Consequently, a self-consistent equation for a vertex function is derived, showing that the thermal conductivity in quasi-one-dimension is dominated by the three-body interaction rather than the two-body interaction. Furthermore, the resulting thermal conductivity in the weak-coupling limit proves to be identical to that computed based on the quantum Boltzmann equation and its temperature dependence is numerically determined.