Quantum criticality and the Tomonaga-Luttinger liquid in one-dimensional Bose gases

We experimentally investigate the quantum criticality and Tomonaga-Luttinger liquid (TLL) behavior within one-dimensional (1D) ultracold atomic gases. Based on the measured density profiles at different temperatures, the universal scaling laws of thermodynamic quantities are observed. The quantum critical regime and the relevant crossover temperatures are determined through the double-peak structure of the specific heat. In the TLL regime, we obtain the Luttinger parameter by probing sound propagation. Furthermore, a characteristic power-law behavior emerges in the measured momentum distributions of the 1D ultracold gas, confirming the existence of the TLL.

Figure 1

Experimental setup and density scaling law. (a) The 1D system consists of an array of tubes created by a blue-detuned “pancake” lattice and a red-detuned retro-reflected lattice. The density profiles are measured by in situ absorption imaging. The inset shows an average line density in comparison with the prediction of the Yang-Yang (YY) equation. (b) The rescaled densities at different temperatures intersect at the critical point ${\mu }_c=0$. Here $\tilde{n}=n/c$, $\tilde{\mu }=\mu /({\hslash }^2{c}^2/2m)$, and $\tilde{T}=k_{B}T/({\hslash }^2{c}^2/2m)$, with $c=-2/a_{1\mathrm{D}}$. The symbols denote the experimental data; solid curves stand for the theoretical predictions. (c) At different temperatures, the rescaled densities against $\tilde{\mu }/{\tilde{T}}^{1/\nu z}$ collapse into a single curve around ${\mu }_c$.

Figure 2

The dimensionless pressure $\tilde{p}=p/[{\hslash }^2{c}^3/(2m)]$ and entropy density $\tilde{S}=S/(k_{B}c)$. The solid lines and the shaded curves are the theoretical predictions from the YY equation. The thickness of the shaded area indicates errors arising from the temperature uncertainties. (a) The pressure equation of state (EOS) (symbols) at different temperatures is deduced from the density profiles. (b) Symbols denote the experimental data of entropy densities extracted from the pressure. [(c) and (d)] Using the same critical exponents $\nu$ and $z$ for the density, the rescaled pressures and entropy densities overlap and collapse into a single curve around the critical point. Here $p_r$ and $S_r$ are the regular parts of the scaling functions. The error bars denote the $\pm 1\sigma$ statistical errors.

Figure 3

The dimensionless specific heat ${\tilde{c}}_V=c_V/(k_{B}c)$ and its phase diagram. (a) Experimental specific heat (red and blue circles) at the temperatures of $T=40(1)\mathrm{nK}$ and $T=50(1)\mathrm{nK}$, respectively. The double-peak feature of specific heat marks out the regimes CG, QC, and TLL. The shaded areas indicate the theoretical predictions by taking account of the temperature uncertainty. (b) Contour plot of specific heat in the $T-\mu$ plane in which its peaks (dashed lines) separate three fluctuation regimes. The QC and the TLL are classified by different critical exponents $z$ and $\nu$. The dots denote the experimental data of the specific heat peaks that mark the two crossover temperatures $T^{*}$. Here the error bars denote the $\pm 1\sigma$ fitting errors.

Figure 4

Evidences for TLL. (a) Measurements of sound velocity. The upper inset shows the generation process of the excitation in the sample. The excitation signal is obtained by subtracting the perturbated cloud from the initial density. The lower inset shows the propagation of a negative perturbation in 1D tubes with amplitude $\eta =0.17(1)$ and a Gaussian width $w=4.0(4)\mu \mathrm{m}$. The red and blue circles are experimental sound velocities versus excitation ratios at temperature $T=40(3)\mathrm{nK}$ and $T=50(3)\mathrm{nK}$. The red and blue curves are the fitting results. (b) Momentum profiles of the 1D gases. The red circles and blue squares stand for the experimental momentum distributions for a degenerate gas at $T=40(1)\mathrm{nK}$ and a classical gas at $T=209(1)\mathrm{nK}$, respectively. All the experimental data are normalized to the zero-momentum value of the degenerate gas. The red solid curve is the theoretical prediction by considering the finite-temperature effect and the averaged Luttinger parameter. The red dashed curve is an auxiliary straight line with a slope of $-1.66$, while the blue solid curve follows a Gaussian distribution. (c) Wilson ratio $R_W$ and Luttinger parameter $K$. The red and blue circles are experimental $R_W$ at $T=40(1)\mathrm{nK}$ and $T=50(1)\mathrm{nK}$, while the shaded areas indicate the theoretical predictions. Two diamond points represent the averaged Luttinger parameter of the TLL regime. The error bars represent the $\pm 1\sigma$ statistical errors.