Bloch Oscillations of a Soliton in a 1D Quantum Fluid

TL;DR

The study addresses Bloch-like oscillations in a 1D quantum fluid without a lattice by realizing a magnetic soliton in a two-component Bose gas and subjecting it to a constant force. A combination of NLSE‑based theory and a particle‑like reduction shows that the soliton carries a canonical momentum $P = \hbar n_0 ΔΦ_1$ and experiences a periodic energy landscape in linear geometry, while ring geometry introduces a backflow current and topological winding. Experimentally, BOs are observed in both linear and ring geometries, with the period $T = n_0 h/(N_2 f)$ and clear signatures in bath phase via matter‑wave interference; two-soliton configurations reveal synchronized dynamics and bath‑mediated interactions. The work provides insight into macroscopic quantum motion in 1D fluids, connects Bloch dynamics to bath phase coherence and topological currents, and suggests avenues for atomtronics and studies of dissipation and macroscopic quantum tunneling.

Abstract

The motion of a quantum system subjected to an external force often defeats our classical intuition. A celebrated example is the dynamics of a single particle in a periodic potential, which undergoes Bloch oscillations under the action of a constant force. Surprisingly, Bloch-like oscillations can also occur in one-dimensional quantum fluids without requiring the presence of a lattice. The intriguing generalization of Bloch oscillations to a weakly-bounded ensemble of interacting particles has been so far limited to the experimental study of the two-particle case, where the observed period is halved compared to the single-particle case. In this work, we observe the oscillations of the position of a mesoscopic solitonic wave packet, consisting of approximately 1000 atoms in a one-dimensional Bose gas when subjected to a constant uniform force and in the absence of a lattice potential. The oscillation period scales inversely with the atom number, thus revealing its collective nature. We demonstrate the pivotal role of the phase coherence of the quantum bath in which the wave packet moves and investigate the underlying topology of the associated superfluid currents. Our measurements highlight the periodicity of the dispersion relation of collective excitations in one-dimensional quantum systems. We anticipate that our observation of such a macroscopic quantum phenomenon will inspire further studies on the crossover between classical and quantum laws of motion, such as exploring the role of dissipation, similarly to the textbook case of macroscopic quantum tunneling in Josephson physics.

Figures (8)

• Figure 1: Soliton preparation and Bloch oscillations in a tube. a, Reconstructed absorption image of the initial density profiles in both $\left\vert 1 \right\rangle$ and $\left\vert 2 \right\rangle$. The colours represent the relative weight of the atomic densities $n_1$ and $n_2$ of both components. b, Integrated density profile of component $\left\vert 2 \right\rangle$ along the transverse direction of the absorption image. The solid line is a fit of the data to a $1/\cosh^2$ function, which corresponds to the expected shape of the soliton in the limit of low values of $n_2/n_1$ at the soliton position. c, Time evolution of the minority component in $\left\vert 2 \right\rangle$ for $N_2 = 1300(100)$ atoms and $n_0\approx350\,\rm µm^{-1}$ and under the action of a uniform differential force of $f=6.6(5)\times 10^{-4}\,Ma_\mathcal{G}$, showing the phenomenon of Bloch oscillations. The error bars correspond to the statistical deviation obtained from 10 repetitions of each experiment. The solid line is a sinusoidal fit to the data. Images are shown every 30 ms from the initial preparation of the wave packet. The dashed line is a guide to the eye to mark the initial position of the wave packet. d, Measured Bloch period when vayring $N_2$ and $f=\mu_B b'$, where $b'$ is the applied magnetic field gradient and $\mu_B$ the Bohr magneton. The different colors correspond to different forces applied on the soliton. The bath density $n_0$ is fixed. The solid black line is the prediction of Eq. \ref{['eq:Tsoliton']} without any free parameter.
• Figure 2: Evolution of the phase of the bath component during Bloch oscillations.a, Calculated phase time evolution corresponding to the experimental case. The line is represented by a dash when it is on the rear surface of the cylinder. At $t=0$, the phase is uniform. At later times, the phase varies on a short distance scale across the wave packet position, where the bath density is the lowest. The phases on both sides of the wave packet are approximately constant and the phase does not wind around the cylinder. The phase is uniform again at $t=T$, giving rise to periodic oscillations. The motion of the wave packet occurs on a short length scale and is thus not visible on the graph. b, Schematic of the configuration used to perform matter wave interference experiments. The left tube is used as a phase reference with all atoms in $\left\vert 1 \right\rangle$. The right tube is identical to the left one except for the presence of a localised wave packet in $\left\vert 2 \right\rangle$. c,d,e,f, Experimental absorption images of atoms in $\left\vert 1 \right\rangle$ obtained after a time-of-flight (ToF) expansion from the configuration shown in b. Images are on purpose saturated to highlight the position of the fringes and the color scale is thus qualitative. The matter-wave interference fringes reveal the relative phase between the two clouds. The black lines are the reconstructed positions of the bright fringes. The red dashed lines show the position of the soliton measured independently. A discontinuity of the fringes, corresponding to a $\pi$-phase shift of the phase of the bath is observed at the positions where the wave packet's velocity changes from positive to negative values (d,f).
• Figure 3: Oscillation of one and two solitons under a constant force in a ring geometry.a, Simulated time evolution of the bath phase. Periodic boundary conditions impose the same phase on both sides of the cylinder. Just after $t=T/2$, the phase profile winds once around the cylinder leading to the creation of a supercurrent in the bath. b, Time evolution of the polar angle $\alpha$ associated with the position of the soliton along the ring (see the sketch in the inset) for $N_2=1100(60)$, $n_0=320(10)\,\rm µm^{-1}$ and $F=6.6(5)\times 10^{-4}\,Ma_\mathcal{G}$. The soliton performs Bloch-like oscillations combined with a drift of the center of mass position due to the generation of supercurrents in the ring. The solid line is the prediction from the particle-like model (see Methods). c, Time evolution of the winding number $w$ of the bath phase. The winding number is measured using matter-wave interference fringes obtained after an expansion from the configuration sketched in d with two concentric rings. Absorption images of the fringes at three different times are shown in e,f,g. We observe concentric rings (e), an anticlockwise spiral pattern (f) and an anticlockwise double spiral pattern ( g), which we assign to $w=0$, $w=1$ and $w=2$, respectively. h Same as in b but for two solitons initially at diametrically opposed positions denoted by $\alpha_L$ and $\alpha_R$. The two solitons have similar initial atom numbers ($\approx 1500$ atoms). They perform in-phase Bloch oscillations with no clear drift of their positions. The solid lines represent a sinusoidal fit to the data with opposite amplitude. i, Winding number measured in the two-soliton case. The data are consistent with no observed winding. j, Sketch of the geometry used for matter-wave interference experiments in the two-soliton case. k, Example of measured fringes, corresponding to $w=0$. In all plots, the error bars represent the statistical errors obtained from the 4 repetitions of each experiment.
• Figure 4: Magnetic soliton at rest.a, Absorption image of the minority component wave packet in $|2\rangle$. The axis of the tube is horizontal. The color is in arbitrary units proportional to the atomic surface density. The horizontal solid line corresponds to a length of $10\,\rm µm$. b, Mean density along the tube direction ($x$). The solid line is a fit of the data to the analytical profile of the magnetic soliton. c, We vary the atom number transferred to the minority component and monitor the short time evolution of the size of the wave packet. Solid lines are fits of the function $t \mapsto \sigma_0+\gamma t^2$ to the data. d, Evolution of the expansion coefficient $\gamma$ as a function of the atom number $N_2$. The value $N_s$ corresponds to the atom number for which the wave packet is stationary. This experiment thus demonstrates the realization of a magnetic soliton at rest.
• Figure 5: Full width at half maximum and depletion of a magnetic soliton. We focus on the soliton at rest ($\tilde{v}=0$ and $\tilde{J}=0$). This graph is plotted using typical parameters of the experiment: $n_0 = 330 \, \rm µm^{-1}$ and $\xi_s = 2 \, \rm µm$.