Damping of phonons in one-dimensional quantum fluids

TL;DR

The study directly probes phonon fate in a weakly interacting 1D Bose gas by resonantly exciting single phonon modes and tracking their time evolution in box-trap geometries. It confirms the universal non-analytic damping predicted by Andreev’s hydrodynamics, with a damping rate $\Gamma_k$ scaling as $\Gamma_k \propto \sqrt{\frac{k_B T}{m n_{1D}}}\,k^{3/2}$ and a fitted coefficient $\alpha \approx 0.76$. At larger perturbations, the phonon dynamics crossover to nonlinear wave-breaking, captured by finite-temperature NPSE simulations and evidenced by deviations from the $k^{3/2}$ law. The results establish a quantitative benchmark for hydrodynamic relaxation and nonlinear behavior in 1D quantum fluids, while outlining pathways to explore crossover to nonlinear Luttinger-liquid physics and integrability breaking.

Abstract

Collective excitations in one-dimensional (1D) quantum fluids are expected to propagate almost without dissipation. Here we directly excite phonon modes in a weakly interacting 1D Bose gas and study their time evolution. In the linear response regime, damping is surprisingly fast and quantitatively follows the non-analytic scaling predicted by Andreev's hydrodynamic description. For stronger excitations, we observe a crossover to a highly nonlinear regime characterized by wave breaking, captured by the finite-temperature nonlinear Schrödinger evolution. Our results resolve a long-standing question on the fate of phonons in 1D Bose gases, and open new pathways to study non-linear relaxation in quantum many-body systems.

Figures (9)

• Figure 1: Density perturbation dynamics. Time evolution of three distinct density perturbations: second mode excited via shaking of the box walls (a), density wave packets emerging from ramping down the potential amplitude in $1$ ms (b), density wave packets emerging after quenching a potential with central dimple to a flat box (c). The x-axis corresponds to the longitudinal direction $z$, while the y-axis shows the evolution time. For panel (a) negative times indicates the driving time, while $t>0$ always represent free dynamics. The position of the box walls during the free evolution is shown with dashed lines. The colorbar indicates the amplitude of the perturbation. The inital temperature for the three datasets is $64\,(11)\,$, $40\,(4)$ and $69\,(5)$ nK, respectively, while the linear density is $73$, $78$ and $68$$\mu$m$^{-1}$
• Figure 2: Damping extracted from single mode dynamics.(a): Behaviour of the normalized damping rate $\tilde{\Gamma} = (m\lambda_T^2\,\Gamma)/(4\hbar)$ at different $k$. Experimental data are shown with dots, with errorbars indicating the $68\%$ confidence interval, computed via bootstrap. The dashed line is the curve obtained with fitted coefficient $\alpha = 0.76\,(0.02)$ and fitted exponent $\beta = 1.48\,(0.03)$, where the errors correspond to two standard deviations (see Supplemental material). The shaded area represents the NPSE simulation of experimental data, taking into account uncertainty on $\tilde{\Gamma}$ due to temperature variation and perturbation amplitude. Four experimental data sets with distinct $k_j$ are highlighted in different colors, and their schematic density profile is shown in the inset of panel (a). Mode $j=2,4,6,8$ are indicated with green, orange, red and purple, respectively. (b-e): Mode decomposition of the selected data sets. On the x-axis is the mode number $j$ and on y-axis is time. Negative values indicate the driving time $\tau_{dr}$; at $t=0$ the modulation is turned off, followed by the free evolution in the box for $t>0$. (f-i): Full evolution of the addressed mode at a fixed $k$: $k_2 = 2\pi/L$, $k_4 = 4\pi/L$, $k_6 = 6\pi/L$, $k_8 = 8\pi/L$, respectively. The driving time for the four data sets is $\tau_{dr} = 60, 45, 60$ and $40$ ms, respectively. Of the full driving time only the last $20$ ms are showed in the plots.
• Figure 3: Damping extracted from wave packets dynamics. Panel a): behaviour of $\tilde{\Gamma}$ for different $k$. Data points of the same color represent the lower modes excited through different protocols: in magenta are the data obtained via a fast ramp in the box trap and their mode decompostion shown in panel b); in orange are the data obtained by symmetric box walls displacement, while in green are modes excited after quenching a dimple potential to a flat box. The mode decompostion for the orange and green data set are displayed in panel c) and d), respectively. The correspondent evolution of the density perturbation in real space is show in Fig.\ref{['fig:Density_carpets']}b), Fig. \ref{['fig:wp_wave_breaking']}a) and Fig.\ref{['fig:Density_carpets']}c). The black solid line is the expected Andreev damping obtained with coefficient $\alpha = 0.76$ as obtained from the fit of the shaking measurements (Fig. \ref{['fig:main_damping_shaking']}a)). The shaded areas represent finite temperature NPSE simulations of experimental data, taking into account uncertainty on temperature and on perturbation amplitude. Panels f) to h): Schematic illustration of the three protocols; relative proportions are not to scale.
• Figure 4: Comparison between NPSE simulations and experimental data of a large amplitude wave packet. Panel a): experimental data for the evolution of a density perturbation which is about $20\%$ of the linear density $n_{1D} = 70\,\mathrm{\mu}$m$^{-1}$. Panel b): NPSE simulations at $T=65\,$nK. Panel c) and d): density perturbation profiles $\Delta n/n$ at $t=0$ (left) and $t=10$ (right). Experimental profile is plotted in green, in dark blue is the profile obtained via finite temperature NPSE simulations and in light blue is the profile obtained with the same experimental parameter, but at zero temperature. At $t=10\,$ms fast oscillations are present in the zero-temperature profile, indicating that a wave-breaking phenomena has occoured, but thermal fluctuations cover the high-frequency oscillations.
• Figure 5: Excitation of higher order phononic modes. The $4^{\mathrm{th}}$, $6^{\mathrm{th}}$ and $8^{\mathrm{th}}$ modes are addressed in panel (a), (b) and (c) respectively. Each 2D plot shows the evolution of the density perturbation profile $\delta n(z,t)$ during the last $20\,$ ms of shaking (negative times) and for the full subsequent evolution. The lifetime of the mode noticibly decreases for higher order modes. The dashed black lines represent the position of the box walls which is used in the computation of the Fourier decomposition (see Fig. \ref{['fig:main_damping_shaking']}(b-e))