Breathing and Fission of Magnetic Multi-Solitons

Abstract

We report the deterministic experimental realization and controlled fission of magnetic multi-soliton states in a uniform quasi-one-dimensional immiscible two-component Bose gas. We explore the Manakov regime, where the spin dynamics is well described by the easy-axis Landau-Lifshitz equation (LLE). The gauge equivalence between the easy-axis LLE and the attractive nonlinear Schrödinger equation (NLSE) enables the direct construction of magnetic multi-solitons from the well-known NLSE solutions. We observe the two- and three- soliton states, which exhibit robust breathing in quantitative agreement with integrable theory. By introducing a weak, localized perturbation, we controllably break integrability and induce the splitting of a two-soliton into its fundamental constituents. This process reveals the composite structure of multi-soliton states and realizes an experimental analog of the inverse scattering transform.

Figures (7)

• Figure 1: Analogy between an easy-axis ferromagnetic spin chain and a two-component Bose gas.(a) Schematic representation of a one-dimensional easy-axis ferromagnetic spin chain, governed by the LLE in the continuous limit. The magnetization has a constant modulus. It points along $+\boldsymbol{e}^{(z)}$ at infinity, while it bends toward the opposite direction in the middle, representing a magnetic soliton. (b) Experimental realization of an effective spin degree of freedom using two hyperfine states of the $F=1$ ground manifold of $^{87}$Rb. The system is characterized by intra- and inter-species interaction constants $g_{i,j}$. Owing to the states symmetry, $g_{1,1} = g_{2,2} \equiv g$. This immiscible two-component Bose gas maps onto an easy-axis ferromagnet, with magnetic solitons encoded in the local spin imbalance and relative phase between the components.
• Figure 2: Realization of an arbitrary spin mixture.(a) Schematic of the relevant levels and transfer process. Two co-propagating Raman beams transfer part of the cloud to the intermediate $\ket{F=2, m_F=0}$ state in a spatially-resolved way. A global micro-wave $\pi$-pulse is then applied to drive the atoms to $\ket{2}$. (b) Schematic densities in each spin component at the end of the preparation sequence. Starting from a homogeneous gas, atoms are selectively transferred from $\ket{1}$ to $\ket{2}$, such that the total density remains uniform. (c) Spin-selective absorption imaging of either component. Top: bath component $\ket{1}$. Bottom: minority component $\ket{2}$. (d) One-dimensional density obtained by integrating the images along the $y$ direction. The solid line is a fit to the data using the profile of a magnetic soliton at rest, given by Eq. \ref{['eq:mag_soliton']}.
• Figure 3: Breathing behavior of two-solitons.(a) Example of a breathing two-soliton for $\kappa=0.23(1)$. Shown are the time evolution of the soliton width (top left) and depletion (top right). Error bars correspond to the 1-$\sigma$ statistical uncertainty obtained from typically 15 repetitions of the experiment. The width $\ell(t)$ is obtained from a fit of the density at every time of the evolution with a function $\propto 1/\cosh^2(x/\ell)$. The solid line in the top panel corresponds to a damped sinusoidal fit, from which we extract the frequency. The decay of the oscillation amplitude is attributed to the presence of a small residual potential. The bottom panel shows the integrated atomic densities over the length $L$ of the segment, at times corresponding to $t \approx 0, T/2, T$, where $T$ denotes the breathing period. In the last panel, the initial profile (dashed line) is overlaid to emphasize the periodic breathing dynamics. (b) Same as in (a), but for a two-soliton in the strongly depleted regime with $\kappa=0.30(1)$. (c) Evolution of the atom number in the two-soliton with $\kappa$, obtained by adding the atom numbers of the two constituent one-solitons given in \ref{['eq:mag_soliton_N']}. The three dashed lines correspond to the three values $\kappa_i$ investigated experimentally and to the associated atom numbers. The soliton mass $N_{\mathrm at}$ diverges at $\kappa=\kappa_c(2)=1/3$. The dash-dotted line indicates the low-depletion NLSE limit. (d) Measured breathing frequency for two-solitons with different inverse widths $\kappa$. The solid black line indicates the frequency of the exact two-soliton as $\kappa$ is varied continuously. (e) Breathing frequency in the vicinity of the exact two-soliton solution. For each $\kappa_i$, the wavepacket amplitude is varied around its expected value within a small range. The colored lines represent the predicted frequencies with no adjustable parameters; they are dashed when the predicted quantity of radiated atoms is below 4 %, solid when it is below 1 %, and omitted otherwise. The filled symbols correspond to the data points shown in (d). For each investigated value $\kappa_i$, they are determined by selecting the point whose atom number $N_{\mathrm{at}}(\kappa_i)$ is closest to the predicted value indicated in (c) by the horizontal dashed lines.
• Figure 4: Observation of a magnetic three-soliton.(a) Absorption images of a magnetic three-soliton, whose initial profile is obtained from Eq. \ref{['eq:diff_eq_theta']} using $u(x)=3\kappa/\cosh(\kappa x)$ with $\kappa=0.19\pm0.01$. (b-d) One-dimensional density profiles at different evolution times. A local minimum forms around half a period of the oscillation ($t \approx60\,$ms). A pronounced asymmetry is observed, originating from the relative spatial shifts of the constituent solitons induced by the gauge transformation. In (d), the initial profile (dashed line) is overlaid to emphasize the periodic breathing dynamics.
• Figure 5: Fission protocol of a two-soliton. Numerical simulation of the time evolution of the eigenvalues for a two-soliton state of NLSE with initial wavepacket $u(x)=2/\cosh(x)$, in the presence of the localized perturbation $V_\mathrm{ext}(x)$. The parameters are $\sigma=0.1$, $\epsilon=0.02$, and $x_0=0.25$. (a-b) Illustration of the initial and final densities, where the two solitons are separated. The external potential is shown as a filled black line and magnified by a factor $100$ for clarity. (c) Time evolution of the complex IST spectrum. The initial discrete eigenvalues, $\lambda_1=\mathrm{i}/2$ and $\lambda_2=3\mathrm{i}/2$, are marked with a cross, and their time-evolution under the action of the perturbation is color-coded. The imaginary parts remain approximately constant. (d) Early-time evolution of the real parts of the two eigenvalues. Time is rescaled in units of the two-soliton breathing period. Solid lines correspond to the numerical prediction, solving Eq. \ref{['eq:IST_eigenL']}. Dashed lines show the result of perturbation theory [Eq. \ref{['eq:perturb_lambda']} at $t=0$].