Dynamical structure factor of one-dimensional Bose gases: Experimental signatures of beyond-Luttinger-liquid physics

Interactions are known to have dramatic effects on bosonic gases in one dimension (1D). Not only does the ground state transform from a condensate like state to an effective Fermi sea, but new fundamental excitations, which do not have any higher-dimensional equivalents, are predicted to appear. In this work, we trace these elusive excitations via their effects on the dynamical structure factor of 1D strongly interacting Bose gases at low temperature. An array of 1D Bose gases is obtained by loading a ${}^{87}\mathrm{Rb}$ condensate in a two-dimensional lattice potential. The dynamical structure factor of the system is probed by energy deposition through low-momentum Bragg excitations. The experimental signals are compared to recent theoretical predictions for the dynamical structure factor of the Lieb-Liniger model at $T>0$. Our results demonstrate that the main contribution to the spectral widths stems from the dynamics of the interaction-induced excitations in the gas, which cannot be described by the Luttinger liquid theory.

Figure 1

Illustration of the effects of interactions on the excitation spectrum of a 1D gas. (a) Sketch of the excitation spectrum for a weakly interacting Bose gas. (b) Sketch of the particle-hole spectrum for 1D strongly correlated bosons. The shaded area represents the continuum of excitations bounded between the quasiparticle Lieb I mode and quasihole Lieb II mode. Gray dashed lines indicate cross sections of the spectra at fixed momentum.

Figure 2

Energy spectra calculated through thermodynamic Bethe ansatz for typical parameters of the experiment $(\rho =4.5\mu {\mathrm{m}}^{-1}$, and $\gamma =1.2)$ and different system sizes. The amplitude of the spectra is renormalized to have the same integral. The shaded areas represent the calculated uncertainty (see text). For system size $N\ge 30$ the results are close enough to the thermodynamic one.

Figure 3

Experimental data and theoretical spectra at $\mathit{T}=0$, illustrating the characteristics of different approaches (finite-temperature effects are discussed later). The graph shows the energy $\Delta E$ transferred to an array of 1D gases as a function of the Bragg excitation frequency $\omega$ for an excitation momentum $\hslash q\simeq 0.6\hslash k_F$. The 1D gases are confined in a lattice of depth $s=35$. Blue dots: experimental data. The values are averages over up to five repeated measurements and the error bars are standard deviations. The shaded area below the experimental dots is a guide for the eyes. Dashed line: energy transfer calculated according to the Luttinger liquid theory (see text). Solid thick line: Bethe ansatz solution at interaction strength $\gamma \simeq 1$ presented in this work. Inset: The same experimental data are shown along with the Bogoliubov theory for weakly interacting bosons $(\gamma \ll 1$, dotted line) and Tonks-Girardeau limit for hard-core bosons $(\gamma =\infty$, dash-dotted line). The integral of each theoretical curve is normalized to that of the experimental signal. The conclusion is thus that a correct treatment of the interactions, as performed within the Bethe ansatz, already accounts for most of the observed interaction width.

Figure 4

Distributions of ${\gamma }_{k,l}$ (left) and $q/{\left(k_F\right)}_{k,l}$ (right) in the 2D array of 1D gases as calculated from our model. The vertical dashed lines indicate the weighted mean values $\langle \gamma \rangle \simeq 1$ and $\langle q/k_F\rangle \simeq 0.6$.

Figure 5

Effect of finite temperature on the excitation spectra. The empty dots show the measured increase of energy of 1D gases confined in a 2D optical lattice of amplitude $s=45$ (a), $s=35$ (b), and $s=50$ (c), after a Bragg excitation. The error bars are standard deviations over up to five repeated measurements per frequency. The lines are the calculated response of the array of 1D systems at different temperatures. For $s=35$, the temperatures $T=25$ nK, 50 nK, and 75 nK correspond to $\tau =0.57$, 1.15, and 1.72, respectively. For $s=50,\tau =0.39$, 0.79, and 1.18. The shaded area around the theoretical curves represents their uncertainty, mainly due to the normalization of the experimental data. The insets show the mean squared residuals between data and theoretical curves at different test temperatures $T$ $({\chi }^2=1/N{\sum }_i{(\Delta E_i^e-\Delta E_i^t)}^2$, where $\Delta E_i^e$ are the experimental data, $\Delta E_i^t$ the corresponding theoretical values, and $N$ the number of data points).

Figure 6

Calculated energy spectra $\Delta E(q,\omega )$ for single 1D gases that compose the array. Atom distribution and parameters are those of the experimental spectrum of Fig. 5. (a) Calculated response of the single 1D gases of variable density ${\rho }_{k,l}$ composing the 2D array. At low densities, finite-size effects are more pronounced than at high density, leading to artificial structures of the spectrum, noticeable at $\rho =1.70\mu {\mathrm{m}}^{-1}$. (b, c) First two moments of $\Delta E(q,\omega )$ for a single 1D gas as a function of density $\rho$ and for different temperatures $T$.