Probing the Excitations of a Lieb-Liniger Gas from Weak to Strong Coupling

We probe the excitation spectrum of an ultracold one-dimensional Bose gas of cesium atoms with a repulsive contact interaction that we tune from the weakly to the strongly interacting regime via a magnetic Feshbach resonance. The dynamical structure factor, experimentally obtained using Bragg spectroscopy, is compared to integrability-based calculations valid at arbitrary interactions and finite temperatures. Our results unequivocally underlie the fact that holelike excitations, which have no counterpart in higher dimensions, actively shape the dynamical response of the gas.

Figure 1

Sketch of the experimental setup. (a) A pair of retro-reflected laser beams creates an ensemble of $\approx 4000$ independent one-dimensional Bose gases. (b) The excitation spectrum is probed by illuminating the gas with a pair of Bragg laser beams. (c)–(d) Zero-temperature dynamical structure factor $S(k,\omega )$ (value shown in gray scale) for a moderately ($\gamma =3.3$) (c) and a strongly ($\gamma =45$) (d) interacting homogeneous gas. The solid (dashed) line in (d) shows the dispersion of the Lieb-I (Lieb-II) mode. Insets indicate averaging over an ensemble of trapped systems. The thin lines show fixed momentum cuts at representative densities. The corresponding values for $k/k_F$ are indicated as vertical lines in the $k-\omega$ plane. The thick line shows the averaged response $S(\omega )$ in a local density approximation.

Figure 2

Bragg-excitation spectra for different values of the 1D interaction strength $\gamma$. (a)–(e) Transferred energy $\sim ⟨p^2⟩$ (normalized to unit area) as a function of the Bragg detuning $\omega$. The scattering length is set to 15$a_0$ (a), $173a_0$ (b), $399a_0$ (c), $592a_0$ (d), and $819a_0$ (e), giving an average $\gamma [k_F/\mu {\mathrm{m}}^{-1}]$ of 0.12(5) [9(1)], 3.3(1) [4.4(2)], 11.0(4) [3.7(1)], 21.7(7) [3.5(1)], and 45(1) [3.3(1)], respectively. Solid lines show fits to the data using a Gaussian multiplied by $\omega$. The vertical dashed lines indicate the position of the Lieb-I (L1) and Lieb-II (L2) mode calculated with the averaged values for $\gamma$ [23]. The dashed line in (e) shows the calculated response for ensemble-averaged trapped TG gases. (f) Central excitation energy ${\omega }_c$ extracted from the Gaussian fit model as a function of $a_s$ (circles). Solid (dashed) lines denote the calculated position of the Lieb-I or -II modes (Bogoliubov mode). The dotted lines indicate the energy of particle and hole excitations in the TG limit. (g) ${\omega }_{\mathrm{c}}$ extracted from the data shown in (a) (triangle), (b) (square), (c) (diamond), and (e) (inverted triangle) plotted in the dimensionless energy-momentum plane. Solid (dashed) lines denote the Lieb-I (Lieb-II) mode. The shaded area shows the continuum of excitations in the TG limit. Vertical lines in (f) and (g) give the fitted full width at half maximum (FWHM).

Figure 3

Comparison of the Bragg-excitation spectra with theoretical predictions at finite temperature. Symbols depict the data presented in Fig. 2 for $\gamma =3.3(1)$ (a), $\gamma =11.0(4)$ (b), $\gamma =21.7(7)$ (c), and $\gamma =45(1)$ (d). The vertical dashed lines indicate the position of the Lieb-I and Lieb-II modes calculated with the averaged values for $\gamma$. The solid lines show the ensemble-averaged response derived from the finite temperature dynamical structure factor. (e)–(h) Reduced ${\chi }^2$ analysis of the corresponding experimental data in the top row with theoretical predictions for different temperatures.

Figure 4

Momentum distribution $n(p)$ for resonant and off-resonant Bragg excitation measured at weak and strong interactions. (a)–(c) Integrated line density after 50 ms time of flight when exciting the 1D systems at $\gamma =0.12(5)$ (a), $\gamma =3.3(1)$ (b), and $\gamma =45(1)$ (c) for red-detuned (left), resonant (middle), and blue-detuned (right) excitation with respect to the Bragg resonance. Data points show the average of four measurements. The frequencies given denote the Bragg detuning $\omega /(2\pi )$. The red line shows a fit to the central part of the line density around $p=0$, using a linear combination of a Lorentzian and Gaussian profile to extract the width $w$. The red shaded area denotes the fraction of atoms asymmetrically scattered to higher momentum states. (d)–(f) Imparted mean momentum $⟨p⟩$ extracted from $n(p)$ (squares) and relative change in the fitted width $\delta w$ of the central peak around $p=0$ (triangles) as a function of the detuning $\omega /(2\pi )$ for $\gamma =0.12(5)$ (d), $\gamma =3.3(1)$ (e), and $\gamma =45(1)$ (f). Solid lines are interpolating functions to guide the eye.