Quench spectroscopy for Lieb-Liniger bosons in the presence of harmonic trap

TL;DR

This work demonstrates quench spectroscopy for a Lieb-Liniger/Bose-Hubbard system in a harmonic trap, by quenching the longitudinal lattice depth and tracking post-quench momentum evolution to reconstruct the quench spectral function $S(k,\omega)$. The confined 1D gas yields a broadened spectral feature around the Mott gap $\hbar\omega\approx U$ and a sharp cutoff at $ka=\pi/2$, signatures explained by trap-induced non-degenerate nearest-neighbor excitations; a contrasting gapless spectrum appears for the superfluid phase. The experimental results align qualitatively with DMRG-TEBD simulations, and an inverse quench with larger amplitude provides the clearest spectral signal, delivering practical guidance for applying quench spectroscopy in realistic, trapped systems. Overall, the work validates quench spectroscopy as a robust tool for mapping excitation spectra in confined 1D quantum gases and informs future studies of disordered or more complex lattice models where confinement plays a role.

Abstract

Quench spectroscopy has emerged as a novel and powerful technique for probing the energy spectrum of various quantum phases for quantum systems from out-of-equilibrium dynamics. While its efficacy has been demonstrated in the homogeneous systems theoretically, most experimental setups feature a confining potential, such as a harmonic trap, which complicates the practical implementations. In this work, we experimentally probe the quench spectroscopy for one-dimensional bosons in optical lattices with the presence of a harmonic trap, and comparing our results with the density matrix renormalization group simulation. For the Mott insulator phase, although a gap is still observed, the band signal is broadened along the frequency space and cut at the half Brillouin zone, which can be explained by the nearest-neighbor tunneling excitations under harmonic confinement. Comparing with the superfluid spectrum, we can see a clear distinction between the two phases and find the inverse quench with larger amplitude yields the clearest spectrum. Our work offers pivotal insights into conducting quench spectroscopy effectively in practical systems.

Figures (14)

• Figure 1: The quench spectroscopy experiment. (a) Sketch of the experimental setup. With two strong lattice beams($y$ and $z$ directions, red arrow), we form a bunch of one-dimensional atomic tubes (red arrays in set). Along the third direction, a relatively weak lattice ($x$ direction, blue) is added, on top of the presenting harmonic trap (black dashed lines) induced by the $y$-$z$ lattice beams. (b) Phase diagram for one-dimensional Bose-Hubbard model around the Mott lobe $na=1$. Superfluid (SF, red) and Mott insulator (MI, blue) phases are found at weak and strong interaction regimes, correspondingly. (i)-(iii) represent the three regimes we focused, as we changing the lattice depth along $x$ direction $V_x$. The typical density distribution in real space are presented in the inset pictures. (c) The sequence of quench spectroscope. Below $t=0$ ms, we load the transverse lattices ($y$ and $z$ directions, red) and longitudinal lattices ($x$ direction, blue) and hold the system for $20$ ms. Correspondingly, a harmonic trap potential $V_{HT}$ is induced (black). At $t=0$ ms, we quench $V_x$. Then, all lasers are shut down at $t=20$ ms and we perform the TOF detection. (d) Demonstration of data processing. From the TOF detection, we measure the momentum distribution evolution $n(k,t)$(left). By performing Fourier transform (FT) for parameter $t$ or $k$, we obtain the QSF $S(k,\omega)$(right top) or correlation spreading $G^{(1)}(x,t)$, correspondingly. The insets show the top view.
• Figure 2: QSF measured from the experiment when quenching from different initial trap depth $V_x^i$ to a lower final depth $V_x^f$. (a). $V_x^i=26E_{\textrm{r}}$ to $V_x^f=22E_{\textrm{r}}$, (b) $V_x^i=24.5E_{\textrm{r}}$ to $V_x^f=22E_{\textrm{r}}$, (c) $V_x^i=23E_{\textrm{r}}$ to $V_x^f=22E_{\textrm{r}}$, (d) $V_x^i=6E_{\textrm{r}}$ to $V_x^f=5E_{\textrm{r}}$. (a)-(c) are in the deep MI regime and (d) is in the deep SF regime. The values in each figures result from the average over $6$ sets of experimental data, and the average relative error is around $20\%$ (details of error analysis, see Supplementary Material \ref{['SM sec.6']}). Inset figures represent the corresponding DMRG simulations. In (a)-(c), the blue and green dashed lines are guidance for the eyes to view the broadened peaks and momentum cutoff suggested by the simulation. The blue dashed curves in (d) are Bogoliubov spectral branch suggested by the simulation result. The X-axis represents $k$ rescaled by $1/a$, Y-axis is $\omega$ rescaled by $U/\hbar$. The Colorbars represent $S(k,\omega)$ rescaled by $a\hbar/U$.
• Figure 3: Amplitude of QSF when quenching from different $V_x^i$ to a higher $V_x^f=22E_{\textrm{r}}$ in the deep MI phase. (a)-(c) are cases quenching from different $V_x^i=20E_r,19E_r,18E_r$ to a fixed $V_x^f=22E_r$ respectively.(a1)-(c1) shows the original QSF with the same range of color bar, while (a2)-(c2) adjust the color bar to reveal the detailed structure of the QSF. The values in each figures result from the average over $6$ sets of experimental data, and the average relative error is around $20\%$. Inset figures represent the corresponding DMRG simulations. The blue and green dashed lines are guidance for the eyes to view the broadened peaks and momentum cutoff suggested by the simulation. The axis and Colorbars represent the same meaning as in Fig.\ref{['fig:Fig.2']}.
• Figure 4: Visibility of the energy gap. (a) $\eta$ for different experiments. The blue and purple data points are calculated at cutting $ka=0$ and $ka=\pi/5$, respectively. The light green area are the line $\eta=1\pm 0.1$ to guide the eyes. (b) cutting at $ka=0$ for SF quench. (c) cutting at $ka=0$ for MI quench from $19E_{\textrm{r}}$ to $22E_{\textrm{r}}$. The red and yellow region in (b) and (c) represent the spectral area and gap area used to calculate $\eta$. (d) Position of the energy gap. The red dash line shows the theory position $\hbar\omega/U=1$; (e) Bandwidth of the energy gap. The blue and purple data points in (d) and (e) is calculated according to integral and cut of $S(k,\omega)$, respectively.
• Figure S1: Amplitude of QSF for three simulations of different typical regimes under original and higher $n_{max}$ and $\chi_{max}$, where (a) is from $V_x=24.5Er$ to $V_x=22Er$, (b) is from $V_x=19Er$ to $V_x=22Er$ and (c) is from $V_x=6Er$ to $V_x=5Er$. And parameters $U$ and $J$ are calculated using the s-wave scattering length $a_s=100.4a_0$, where $a_0$ is Bohr radius.