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arXiv:0709.1899 · Physical Review Letters

Yang-Yang Thermodynamics on an Atom Chip

Aaldert H. van Amerongen, J.J.P. van Es, Philipp Wicke, Karen V. Kheruntsyan, N. J. van Druten

Published Mar 3, 2008• 267 citations
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Abstract

We investigate the behavior of a weakly interacting nearly one-dimensional trapped Bose gas at finite temperature. We perform in situ measurements of spatial density profiles and show that they are very well described by a model based on exact solutions obtained using the Yang-Yang thermodynamic formalism, in a regime where other, approximate theoretical approaches fail. We use Bose-gas focusing [I. Shvarchuck, Phys. Rev. Lett. 89, 270404 (2002)] to probe the axial momentum distribution of the gas and find good agreement with the in situ results.

DOI
10.1103/physrevlett.100.090402

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We investigate the behavior of a weakly interacting nearly one-dimensional trapped Bose gas at finite temperature. We perform in situ measurements of spatial density profiles and show that they are very well described by a model based on exact solutions obtained using the Yang-Yang thermodynamic formalism, in a regime where other, approximate theoretical approaches fail. We use Bose-gas focusing [I. Shvarchuck, Phys. Rev. Lett. 89, 270404 (2002)] to probe the axial momentum distribution of the gas and find good agreement with the in situ results.
We describe experiments investigating the (coherence) properties of a finite-temperature one-dimensional (1D) Bose gas with repulsive interactions. The confining magnetic field is generated with a micro-electronic circuit. This microtrap for atoms or `atom chip' is particularly suited to generate a tight waveguide for atoms close to the chip surface. In contrast to the usual case of Bose-Einstein condensation in 3D, in 1D the system is not characterized by long-range phase coherence. Further interest in the 1D Bose gas stems from the fact that it forms the textbook example for the many-body quantum-mechanical systems that can be exactly solved using the Bethe Ansatz. Moreover, using a method developed by Yang and Yang, exact expressions for the thermodynamics of this system can be given. We summarize the theoretical background that is relevant to describe the (nearly) 1D gas in our experiments. The design and construction of our microtrap is described in some detail. We discuss the technique of Bose-gas focusing, in particular how it applies to our 1D system, and demonstrate that it can be used to probe the momentum distribution of the 1D gas. Finally, we perform a direct comparison of the measured in situ density distribution to the predictions of a model based on the exact Yang-Yang thermodynamics and find very good agreement.
The emergence of a special type of fluidlike behavior at large scales in one-dimensional (1D) quantum integrable systems, theoretically predicted in O. A. Castro-Alvaredo et al., Emergent Hydrodynamics in Integrable Quantum Systems Out of Equilibrium, Phys. Rev. X 6, 041065 (2016)10.1103/PhysRevX.6.041065 and B. Bertini et al., Transport in Out-of-Equilibrium XXZ Chains: Exact Profiles of Charges and Currents, Phys. Rev. Lett. 117, 207201 (2016)10.1103/PhysRevLett.117.207201, is established experimentally, by monitoring the time evolution of the in situ density profile of a single 1D cloud of ^{87}Rb atoms trapped on an atom chip after a quench of the longitudinal trapping potential. The theory can be viewed as a dynamical extension of the thermodynamics of Yang and Yang, and applies to the whole range of repulsive interaction strength and temperature of the gas. The measurements, performed on weakly interacting atomic clouds that lie at the crossover between the quasicondensate and the ideal Bose gas regimes, are in very good agreement with the theory. This contrasts with the previously existing "conventional" hydrodynamic approach-that relies on the assumption of local thermal equilibrium-which is unable to reproduce the experimental data.
We investigate theoretically the behavior of a one-dimensional interacting Bose-Fermi mixture with equal masses and equal repulsive interactions between atoms at finite temperature in the scheme of thermodynamic Bethe ansatz. Combining the Yang-Yang thermodynamic formalism with local-density approximation in a harmonic trap, we calculate the density distribution of bosons and fermions numerically by treating the radially and axially excited states as discrete and continuous ones, respectively. Our result from exactly solvable solutions may be used as a touchstone for one-dimensional interacting Bose-Fermi mixture for experimental data fitting where mean-field theoretical approaches fail.
We describe the use of the exact Yang-Yang solutions for the one-dimensional Bose gas to enable accurate kinetic-energy thermometry based on the root-mean-square width of an experimentally measured momentum distribution. Furthermore, we use the stochastic projected Gross-Pitaevskii theory to provide a quantitative description of the full momentum distribution measurements of Van Amerongen et al. [Phys. Rev. Lett. 100, 090402 (2008)]. We find the fitted temperatures from the stochastic projected Gross-Pitaevskii approach are in excellent agreement with those determined by Yang-Yang kinetic-energy thermometry.
We measure the zero-temperature equation of state of a homogeneous Bose gas of (7)Li atoms by analyzing the in situ density distributions of trapped samples. For increasing repulsive interactions our data show a clear departure from mean-field theory and provide a quantitative test of the many-body corrections first predicted in 1957 by Lee, Huang, and Yang [Phys. Rev. 106, 1135 (1957).]. We further probe the dynamic response of the Bose gas to a varying interaction strength and compare it to simple theoretical models. We deduce a lower bound for the value of the universal constant ξ > 0.44(8) that would characterize the universal Bose gas at the unitary limit.