Detecting Complex-Energy Braiding Topology in a Dissipative Atomic Simulator with Transformer-Based Geometric Tomography

Abstract

Machine learning (ML) is shaping our exploration of topological matter, whose existence is inherently tied to the geometry of quantum states or energy spectra. In non-Hermitian systems, distinctive spectral geometry can lead to topological braiding of complex-energy bands, yet directly probing this topology-geometry interplay remains challenging. Here, we introduce a Transformer-based ML framework to capture this interplay and experimentally demonstrate it in a dissipative cold-atom simulator. Using a Bose-Einstein condensate, we engineer tunable dissipative two-level systems whose complex eigenenergies form braids. Owing to the density-dependent dissipation, the instantaneous energy braids exhibit topologically distinct structures at short and long times. The Transformer not only accurately predicts topological invariants for diverse energy braids but also, through its self-attention mechanism, autonomously highlights band crossings as the governing underlying geometric feature. Our work paves the way for ML-guided exploration of non-Hermitian topological phases in cold atoms and beyond.

Figures (6)

• Figure 1: Illustration of topological complex-energy braiding in energy-momentum space. The gray and blue curves denote two bands, respectively, and the star labels the exceptional point. Gluing the $k = 0$ and $k = 2\pi$ planes leads to different knotted structures, such as $\mathbf{a}$ the unlink, $\mathbf{b}$ the unknot, $\mathbf{c}$ the Hopf link, and $\mathbf{d}$ the Trefoil knot, distinguished by braid degree $\nu=0,1,2,3$, respectively. In all plots, theoretical calculations are based on Eq. (\ref{['eq1']}) with $J_1 = 0.7$, $J_2 = 1.8$, $J_3 = 1$, and $\Omega = 0.5$. For parameters ($\gamma$,m), we use $\mathbf{a}$$(4.5,1)$, $\mathbf{b}$$(2, 1)$, $\mathbf{c}$$(0.5,1)$, and $\mathbf{d}$$(2,3)$.
• Figure 2: Joint learning of complex-energy braiding topology and geometry via the self-attention of the Transformer.$\mathbf{a}$ Schematics of the working principle. The complex-energy bands in the momentum space ($k\in [0,2\pi]$) are encoded as sequential input data, where the two eigenenergies $E_\pm(k)$ at each $k$-point correspond to a feature token. The Transformer processes this sequence and outputs the topological invariant $\nu$ as a functional mapping of the input energy bands. Leveraging the unique self-attention mechanism (blue shaded region) of the Transformer, the model generates attention weight matrices that, when projected into the input data, reveal its focus on distinct regions of energy bands in classifying the bands' braiding topology. The primary focus of attention locates the decisive moment-energy points underlying braiding topology. $\mathbf{b}$ Training results. Blue curves depict the prediction accuracy of Transformer on the test dataset. The green (yellow) curves show training (validation) loss. As a comparison, the prediction accuracy of CNN trained under identical conditions is shown by the brown curve.
• Figure 3: Attention-based geometric tomography of complex-energy bands. Attention weights generated from the Transformer are projected onto the test data. $\mathbf{a1}$-$\mathbf{a5}$ Distribution of attention weights shown in the complex energy plane, for energy braids with distinct braiding degree $\nu$. $\mathbf{b1}$-$\mathbf{b5}$ Corresponding distribution of attention weights in the Brillouin zone (BZ). $\mathbf{c1}$-$\mathbf{c5}$ Theoretically calculated distribution of $|\partial_k\phi_k|$ in momentum space (see text). $\mathbf{d1}$-$\mathbf{d5}$ Real and imaginary parts of the complex energy bands. Dashed line denotes band crossing with extreme $|\partial_k\phi_k|$ in the real or imaginary component of energy bands. Test data are generated from Eq. (\ref{['Etest']}) with $\Omega=1$; other parameters $(\alpha, J_1,J_2,J_3,J_4,\gamma,m)$ are $\mathbf{a1}$-$\mathbf{d1}$$(0,0.1,2,1,0,1,1)$; $\mathbf{a2}$-$\mathbf{d2}$$(0,2,2,1,0.2,0.5,2)$; $\mathbf{a3}$-$\mathbf{d3}$$(0,0.1,2,1,0,0.5,2)$; $\mathbf{a4}$-$\mathbf{d4}$$(0,1.5,2,1,0,1,6)$; $\mathbf{a5}$-$\mathbf{d5}$$(1,1.5,2,1,0,1,5)$.
• Figure 4: Experimental setup with atomic BECs.$\mathbf{a}$ Level diagram. A microwave field detuned by $\Delta$ coherently couples the hyperfine states $|F=1,m_F=-1\rangle$ and $|F=2,m_F=0\rangle$ of $^{87}\textrm{Rb}$ atoms with a coupling rate $\Omega$. A resonant optical beam is used to generate dissipation (atom loss) in the $|F=2,m_F=0\rangle$ state. The BEC is initialized in $|F=1,m_F=-1\rangle$. After an evolution time $t$, the Stern-Gerlach absorption image is taken after $10$ ms time of flight. $\mathbf{b1(b2)}$ Measured $N_{2(1)}(t)/N_0$ as a function of time, where $N_0$ is initial atom number. The blue, yellow and green dots denote experimental data under various [$\Delta^{\prime}/\Omega, \Gamma'/\Omega$]: $[0.16,0.53]$ when $m=1$ (blue), $[-0.59,1.60]$ when $m=1$ (yellow), and $[1.05,1.71]$ when $m=2$ (green), respectively. Each experimental data is the average over 3 measurements. The error bars are $1\sigma$ standard deviation. Solid curves depict simulations via Hamiltonian $H_e$ in Eq. (\ref{['eq4']}) with $\Gamma'$. Dashed curves denote two-parameter fit to the data using $H_e(\Gamma_1)$ and $H_e(\Gamma_2)$; see Methods and Supplementary Sec. III. $\mathbf{c1}$-$\mathbf{c3}$ Measured energies in the short-time regime, shown in ($\textrm{Re}E$, $\textrm{Im}E$, $\theta$) space. Experimental data of eigenvalues are extracted from the fitted $H_e(\Gamma_1)$ (Methods). Solid curves denote calculated eigenvalues of $H_e(\Gamma')$; parameters $(J_1,J_2,J_3,\gamma,m)$ are $\mathbf{c1}$$(0,1/4,1/4,1,1)$; $\mathbf{c2}$$(0,1/2,1/4,2,1)$ and $\mathbf{c3}$$(1/8,1/2,1/4,2,2)$. In all measurements, $\Omega=31.4$ kHz.
• Figure 5: Experimental detection of topological and geometrical properties of energy braids in the short-time regime via a trained Transformer.$\mathbf{a1}$-$\mathbf{a3}$ Single-shot measurement of complex energy $E$ as a function of $\theta\in[0,2\pi]$. The plots correspond to Figs. \ref{['Fig4']}$\mathbf{c1}$-$\mathbf{c3}$, respectively, where experimental data denote eigenvalues of $H_e(\Gamma_1)$. Solid lines are the interpolated curve passing through experimental data (see Methods and Supplementary Sec. IV). $\mathbf{b}$ Transformer-predicted braiding degree $\nu$. For each set of single-shot measurement of complex eigenvalues, braiding degree $\nu$ is extracted using the Transformer. The average $\nu$ from three set of measurements is shown, and the error bar denotes standard deviation. $\mathbf{c1}$--$\mathbf{d3}$ Attention weights projected onto the preprocessed experimental data, shown in $\mathbf{c1}$-$\mathbf{c3}$ in complex energy plane, and $\mathbf{d1}$-$\mathbf{d3}$ as a function of $\theta\in [0,2\pi]$. In the top and bottom rows, dashed line denotes band crossings with extreme phase gradients.