Generalized Hydrodynamics on an Atom Chip

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) and B. Bertini et al., Transport in Out-of-Equilibrium XXZ Chains: Exact Profiles of Charges and Currents, Phys. Rev. Lett. 117, 207201 (2016), is established experimentally, by monitoring the time evolution of the in situ density profile of a single 1D cloud of ${}^{87}\mathrm{Rb}$ 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.

Figure 1

(i) In situ density profile after longitudinal expansion from a harmonic trap of a 1D cloud of $N=4600\pm 100$ ${}^{87}\mathrm{Rb}$ atoms; the smooth curve is the theoretical prediction of GHD and the noisy one is the experimental data. (ii) Initial profile obtained from the Yang-Yang equation of state (YY), Gross-Pitaevskii, ideal Bose gas, and classical field [40], with the same temperature and chemical potential as for YY. (iii) Evolution from the YY initial profile with GHD and conventional hydrodynamics.

Figure 2

(a) The atom chip setup with the four wires (blue) creating the longitudinal potential and the three wires (red) creating the strong transverse confinement. (b) Position of our three data sets in the thermal equilibrium phase diagram of the Lieb-Liniger gas with $\gamma =mg/({\hslash }^{2}n)$ and $\theta =2{\hslash }^2{k}_{\mathrm{B}}T/(m{g}^2)$ [51]. At the center of the cloud $\gamma$ is of order $10^{-2}$, but it increases in the wings as the density decreases; we display the segments $[{\gamma }_{\min },{\gamma }_{\min }/10]$ corresponding to a local density $n(x)$ not smaller than a tenth of the maximal density in the cloud. The asymptotic regimes of the Lieb-Liniger gas, separated by smooth crossovers, are shown in colors. Our data sets lie at the crossover between the quasicondensate and the ideal Bose gas regimes.

Figure 3

(i) Longitudinal expansion of a cloud of $N=6300\pm 200$ atoms initially trapped in a double-well potential, compared with GHD. (ii) Even though the initial state is the same for GHD and CHD, both theories clearly differ at later times. CHD wrongly predicts the formation of two large density waves. The error bar shown at the center at $t=40\mathrm{ms}$ corresponds to a 68% confidence interval, and is representative for all data sets.

Figure 4

Quench from double-well to harmonic potential, compared to the GHD prediction, with an atomic cloud that contains $N=3500\pm 140$ atoms initially. The main features of the experimental data are well reproduced by GHD. One experimental effect, not modeled in GHD, that appears to be particularly important, are the three-body losses: after 180 ms, the number of atoms drops by approximately 15%.