Selected for a Viewpoint in Physics week ending PRL 119, 165701 (2017) PHYSICAL REVIEW LETTERS 20 OCTOBER 2017 Quantum criticality and the Tomonaga-Luttinger liquid in one-dimensional Bose gases Bing Yang,1,3 Yang-Yang Chen,2 Yong-Guang Zheng,1,3 Hui Sun,1,3 Han-Ning Dai,1,3 † ‡ Xi-Wen Guan,2,4,* Zhen-Sheng Yuan,1,3,5,6, and Jian-Wei Pan1,3,5,6, 1Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China 2State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China 3Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 226, 69120 Heidelberg, Germany 4Department of Theoretical Physics, Research School of Physics and Engineering, Australian National University, Canberra ACT 0200, Australia 5CAS-Alibaba Quantum Computing Laboratory, Shanghai 201315, China 6CAS Centre for Excellence and Synergetic Innovation Centre in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China (Received 26 December 2016; revised manuscript received 18 May 2017; published 18 October 2017) 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. DOI: 10.1103/PhysRevLett.119.165701 Quantum many-body systems can exhibit phase transi- temperatures and chemical potentials. Based on the thermo- tions even at zero temperature [1,2]. Here, quantum fluc- dynamic relations [20–22], we derive the pressures and tuations arising from Heisenberg’s uncertainty relation drive entropy densities, which exhibit similar universal scaling the transition from one phase to another. In this regard, one- around the critical point. Moreover, we determine two dimensional (1D) quantum systems are special owing to crossover branches that distinguish the QC regime from the the significant microscopic fluctuations that induce a classical gas (CG) and the TLL through the double-peak continuous phase transition between a disordered state structure of the specific heat. To further investigate the and a Tomonaga-Luttinger liquid (TLL) [3–7].Nearthe degenerate gas, we probe the propagations of density transition point, a quantum critical (QC) regime emerges at disturbances and acquire the Luttinger parameters. Then finite temperatures and separates these two phases [1,2,8]. we characterize the phase correlation of the 1D ultracold Although the 1D low-energy physics is generally described gas through its momentum distribution. According to the by the well-established TLL theory [3], experimental bosonization-based theory [3,23], the obtained power-law investigations of the TLL and its related quantum criticality behavior in the momentum profiles confirms the existence are rare [9–11]. In this context, signatures of TLL were of the TLL. found in some 1D systems, such as organic conductors [12], The experiment starts by adiabatically loading a Bose- carbon nanotubes [13], spin ladders [10], and quantum Einstein condensate of ∼2 × 105 atoms into a single layer gases [14,15]. Among these strongly correlated systems, of a pancake-shaped trap. We then confine the atoms into ultracold atomic gases offer a great precision and tunability an array of isolated tube-shaped traps arranged in a plane by for studying quantum phase transitions [16,17] and critical superimposing another red-detuned lattice with wavelength phenomena [18,19]. However, observation of quantum λr ¼ 1534 nm into the system [see Fig. 1(a)]. Owing to the criticality and determination of the TLL boundary in 1D homogeneity of the light beams among these tubes, they ω ¼ quantum gases remains elusive. are identical to each otherp withffiffiffiffiffiffiffiffiffiffiffi trap frequencies x In this Letter, we report the observation of quantum 2π × 22.2ð1Þ Hz and ω⊥ ¼ ωyωz ¼ 2π × 7.99ð1Þ kHz. criticality and evidence of TLL in 1D ultracold Bose gases The spatial resolution of the imaging system (1.0 μm) is 87 of Rb. The atomic samples at different temperatures are slightly larger than the lattice spacing λr=2 ¼ 767 nm. prepared in well-designed 1D harmonic potentials. Using a After acquiring around 400 high-resolution images for each high-resolution microscope, we measure the density pro- experimental setting, we then obtained very precise 1D files by in situ absorption imaging. The density scaling law density profiles by averaging these images. The measured is obtained by rescaling these measurements at different temperatures [24], T ¼ 18–74 nK, and corresponding 0031-9007=17=119(16)=165701(6) 165701-1 © 2017 American Physical Society week ending PRL 119, 165701 (2017) PHYSICAL REVIEW LETTERS 20 OCTOBER 2017 (a) 20 context, the particle density in QC obeys a universal scaling x Gravity Exp. Data z ðμ Þ¼ ðd=zÞþ1−ð1=νzÞF½ðμ − μ Þ ð1=νzÞ -1 15 Y-Y Eq. law, as n ;T T c =T , where y ¼ 1 Fð Þ (µm ) 10 the dimensionality is d and x is the scaling n T=27.6(5) nK function [5]. lattice light 5 Such a universal scaling law is extracted from the 0 10 20 30 40 50 density profiles at temperatures ranging from 17.9(4) nK pancake light x (µm) to 74.4(7) nK. As shown in Fig. 1(b), we identify the (b) (c) critical point using that the scaled density becomes temper- 1.2 1.5 Exp. Y-Y Eq. μ 17.9(4) nK ature independent at c, i.e., the density profiles at different z z 27.6(5) nK temperatures intersect at the critical point. The critical 0.8 1.0 38.2(4) nK +1-1/ +1-1/ 52.6(4) nK z z ν 1/ 1/ z µc=0 64.9(7) nK exponents and are determined by the overlapping ~ ~ T T / / 74.4(7) nK ~ ~ n n 0.4 0.5 feature of the rescaled density profiles [24]. The rescaled ν ¼ 0 56þ0.07 measurements fall into a single curve with . −0.08 0 0 ¼ 2 3þ0.6 -50 -25 0 25 50 -2 -1 0 1 2 and z . −0.3 [Fig. 1(c)], confirming the emergence of ~ 1/ z µ (nK) µ~/T the quantum critical scaling. Here the uncertainties corre- spond to a 95% confidence level. The critical exponents FIG. 1. Experimental setup and density scaling law. (a) The 1D agree with the predictions from the YY equation, ν ¼ 0.5 system consists of an array of tubes created by a blue-detuned ¼ 2 “pancake” lattice and a red-detuned retro-reflected lattice. The and z [21,22]. The above properties of densities at density profiles are measured by in situ absorption imaging. various temperatures and chemical potentials reveal the The inset shows an average line density in comparison with the nature of scaling invariance. prediction of the Yang-Yang (YY) equation. (b) The rescaled The thermodynamics of the 1D system at equilibrium densities at different temperatures intersect at the critical point are described by the EOS. We can derive the local μ ¼ 0 ~ ¼ μ~ ¼ μ ðℏ2 2 2 Þ ~ ¼ ðμ Þ¼ c . Here n n=c, = c = m , and T kBT= Rpressure EOS from the atomic density via p ;T ðℏ2 2 2 Þ ¼ −2 μ 0 0 c = m , with c =a1D. The symbols denote the exper- −∞ nðμ ;TÞdμ [20] by introducing a proper cutoff in imental data; solid curves stand for the theoretical predictions. the CG regime, as shown in Fig. 2(a). For the lowest (c) At different temperatures, the rescaled densities against T ¼ 17 9ð4Þ ~ 1=νz temperature experimentally probed . nK, the μ~=T collapse into a single curve around μc. population below μc is negligible and therefore the pressure approaches that of the zero temperature result. Whereas at higher temperatures, the pressure curves split clearly in the chemical potentials in the trap center, μ0 ¼ 67–93 nK, QC regime and bunch up again at large chemical potentials. satisfy the 1D conditions of kBT, μ0 ≪ ℏω⊥. The minimal entropy per particle of 0.055(1) k at T ¼ 17.9ð4Þ nK and From the pressure EOS, one can obtain other thermody- B namic properties. For example, the entropy density can be μ0 ¼ 67.1ð1Þ nK indicates that the 1D gas is strongly ðμ Þ¼½∂ ðμ Þ ∂ degenerate. The dimensionless interaction parameter γ ≈ deduced as S ;T p ;T = T μ [20]. Figure 2(b) 2 ω ðℏ Þ¼0 04 ≪ 1 shows the entropy densities extracted from experimental ma1D ⊥= n0 . suggests that the central region of the system is in the weakly interacting regime, data and the theoretical curves. Peaks arise in the entropy density curves and become flatter at higher temperatures, where m is the mass of the atom, a1D is the 1D effective scattering length, and n0 is the line density at the center of revealing enhanced disorder in the QC regime. Moreover, the 1D tube. Under such experimental conditions, the 1D both the pressure and entropy densities [Figs. 2(c) and 2(d)] Bose gas can be described by the Lieb-Liniger model [28]. have similar universal scaling laws with the same critical Within the local density approximation (LDA), the mea- exponents as those in the density scaling function [24]. In the scenario of quantum criticality, determining the sured densities agree well with the theoretical predictions à from the YY exact grand canonical theory [21] [see crossover temperatures T in quantum gases poses theo- Fig.