Experimental Observation of Single- and Multisite Matter-Wave Solitons in an Optical Accordion Lattice

TL;DR

The paper reports the experimental observation of both single-site and multisite lattice solitons formed by attractive Bose-Einstein condensates in an optical accordion lattice whose spacing $d_L$ can be tuned. A Gaussian variational framework predicts energy minima $M_{SS}$ and $M_{MS}$ separated by barriers $B_{SS}$ and $B_{MS}$, and these predictions are tested and refined by full 3D-GPE simulations with a quintic loss term accounting for three-body losses; the experimental results show stable solitons across varying $V_0$, $d_L$, and $g$, with lattice-spacing-dependent transitions and collapse dynamics captured by the models. The combination of quench dynamics, variable-spacing lattices, and loss-enabled evolution provides quantitative agreement with simulations and qualitative alignment with variational predictions, revealing rich nonlinear transport and stability behavior in lattice-confined matter waves. This work advances understanding of nonlinear wave dynamics in structured media and lays groundwork for soliton-based applications in interferometry, precision sensing, and controlled quantum transport.

Abstract

We report the experimental observation of discrete bright matter-wave solitons with attractive interaction in an optical lattice. Using an accordion lattice with adjustable spacing, we prepare a Bose-Einstein condensate of cesium atoms across a defined number of lattice sites. By quenching the interaction strength and the trapping potential, we generate both single-site and multisite solitons. Our results reveal the existence and characteristics of these solitons across a range of lattice depths and spacings. We identify stable regions of the solitons based on interaction strength and lattice properties, and compare these findings with theoretical predictions. The experimental results qualitatively agree with a Gaussian variational model and match quantitatively with numerical simulations of the three-dimensional Gross-Pitaevskii equation extended with a quintic term to account for the loss of atoms. Our results provide insights into the quench dynamics and collapse mechanisms, paving the way for further studies of transport and dynamical properties of matter-wave solitons in lattices.

Figures (10)

• Figure 1: Experimental setup and stability diagrams. (a) Energy $E(\eta)$ for a Gaussian wave packet with $V_0=1.1\,E_r$, $a_s=-6.2\,a_0$, $d_\text{L}=2\,\upmu$m. Single-site (SS) and multi-site (MS) solitons are stable at minima $M_{SS}$ and $M_{MS}$ with barriers $B_{SS}$ and $B_{MS}$. (b) Sketch of experimental setup. (c) Stable regions of SS and MS solitons for varying parameters $g$ and $V_0$, with $N=1800$, $\omega_{\perp}=2\pi\times30$ Hz, $d_\text{L}=3.2\,\upmu$m. No solitons exist in dark blue regions. (d) Stable regions for varying $d_\text{L}$, same parameters as (c) with constant $V_0=1.3\,E_r$ set at $d_\text{L}=3.2\,\upmu$m.
• Figure 2: Stability of single-site solitons. (a) Measured density distribution after a quench of $a_s$ and $t=100$ ms hold time with $d_\text{L}=3.2(2)\,\upmu$m, $V_0=1.3(1)\,E_r$, $\omega_\perp = 2\pi\times 40(1)$ Hz, $N\approx1800$. White lines mark atoms in the central site. (b) Measured relative central-site atom number $N_c/N_\text{tot}$ vs. $a_s$ and $V_0$ with same parameters as (a). (c) Energy $E_{SS}$ of the barrier $B_{SS}$; (i-iii) indicate regions of varying stability in (b,c). See SuppMat for the definition of $E_{SS}$ for $a_s>0\,a_0$. (d) Density distribution for varying $d_\text{L}$ after 100 ms with $a=-6.4\,a_0$, $N\approx1800$, constant $V_0=1.3(1)\,E_r$ set at $d_\text{L}=3.2(2)\upmu$m. (e) Calculated $E(\eta)$ for (d) with $d_\text{L}=3.5\,\upmu$m (dotted line) and $d_\text{L}=2.0\,\upmu$m (solid line). Measured data is averaged over typically seven repetitions.
• Figure 3: Stability of a multi-site wave packet. (a,b) Time evolution of a wave packet after a quench of $a_s$, averaged over ten repetitions with $V_0=1.3\,E_r$, $d_\text{L}=2.6\,\upmu$m, $\omega_\perp=2\pi\times 25\,$Hz, $\omega_z=2\pi\times25$ Hz, $N\approx2900$. (a) The wave packet disperses for a quench to $a_s=+2.0\,a_0$ and (b) mostly retains its overall shape for $-5.7\,a_0$. Site occupation numbers for both data sets are provided in Ref. SuppMat. (c) Density profiles of the wave packet immediately after the quench (gray), and after $250$ ms for $+2.0\,a_0$ (red) and $-5.7\,a_0$ (blue). (d) Atom number for data in (b), error bars denote the standard deviation. A comparison of the data in (c,d) with a 3D-GPE simulation is provided in Ref. SuppMat.
• Figure 4: Collapse of a multi-site wave packet. (a) Width $w_m$ of the wave packet at $t=150\,$ms after quenching to different values of $a_s$, with $V_0=1.4\,E_r$, $d_\text{L}=2.6\,\upmu$m, $\omega_\perp=2\pi\times 30\,$Hz, $N\approx1700$. The gray patch shows the variation in $w_m$, calculated using the 3D-GPE, resulting from uncertainties in the three-body loss coefficient $L_3$ and $N$. The line is an average of the calculations SuppMat. (Inset, left to right) Typical images of the density profiles after collapse ($a_s=-17\,a_0$), shrinking towards the central site ($a_s=-10\,a_0$) and expanding wave packet ($a_s=0\,a_0$). (b) Stability regions calculated using Eq. (\ref{['Eq:energy']}) with an existing minimum $M_{MS}$ (brown) and without (blue), with breathing oscillations for $E(\eta)<E_\infty$ and $E_{MS}$ (yellow), and with stable multi-site solitons for $E(\eta_0)\approx E_{MS}$ (black). (c),(d) The calculated time evolution of the density distribution and relative atom number show collapse followed by expansion for $a_s=-9.5\,a_0$, and (e),(f) dispersion for $a_s = -1.7\,a_0$. Calculations use $L_3=5\times 10^{-39}\,$m$^6$s$^{-1}$ and other parameters as in (a).
• Figure S1: Measurement of the atom number per lattice site. Density profile of a wave packet after a hold time of $250$ ms, with $V_0=1.3\,E_r$, $d_\text{L}=2.6\,\upmu$m, $\omega_\perp=2\pi\times 25\,$Hz, $N\approx2200$, $a_s=+2.0\,a_0$. Patches denote regions used for the calculation of occupation numbers, $N_j$, at the respective sites.