Observation of Braid-Protected Unpaired Exceptional Points

Abstract

Spectral degeneracies (dubbed nodal points in momentum space) play fundamental roles in understanding exotic properties of light and matter. In lattice systems, unpaired band-structure degeneracies are subject to well-established no-go (doubling) theorems that universally apply to both closed Hermitian systems and open non-Hermitian systems. However, the non-Abelian braid topology of non-Hermitian multi-band systems provides a loophole to these constraints. Here we successfully leverage this loophole in a non-Hermitian three-band system, implementing an unpaired third-order exceptional point (EP3), which manifests as a non-Abelian monopole. We explicitly demonstrate the intricate braiding topology and non-Abelian, path-dependent, fusion rules underlying the unpaired EP3. The experiment uses a new design of single-photon interferometry, enabling eigenstate and spectral resolutions for multi-band systems with widely tunable parameters. Thus, the union of state-of-the-art experiments, fundamental theory, and everyday concepts such as braids pave the way toward the highly exotic non-Abelian topology unique to non-Hermitian settings.

Figures (10)

• Figure 1: Experimental setup. Photon pairs (trigger and signal) are generated via type-II spontaneous parametric down-conversion through a periodically poled potassium titanyl phosphate (PPKTP) crystal. In the state space spanned by horizontally polarised light in the upper spatial mode $\ket{H_U}$, the lower spatial mode $\ket{H_L}$, and the vertically polarised lower spatial mode $\ket{V_L}$, we prepare the signal photon in a completely mixed qutrit state through two unbalanced interferometer (UBI) setups, consisting of four polarizing beam splitters (PBSs), three half-wave plates (HWPs), and a beam displacer (BD). We purify this mixed state into one of the eigenstates of the non-Hermitian Hamiltonian $H$ by way of a non-unitary operation $U_1$, realized by six BDs, HWPs and quarter-wave plates (QWPs). To measure the associated eigenenergy, the photon beams is split by a non-polarizing beam splitter (NPBS): transmitted photons evolve under $U_2$, acquiring a complex phase relative to the reflected path, corresponding to the eigenenergies of $H$. We measure these eigenenergies interferometrically using avalanche photodiodes (APDs) by detecting coincidences between the signal and trigger photons.
• Figure 2: Observation of the Non-Abelian unpaired EP3. (a) Theoretical results for the real and imaginary parts of the spectrum of $H_M^{\delta=1+2\sqrt{2}}$. The red dot marks the non-Abelian unpaired EP3. The solid lines depict the Fermi (i-Fermi) arcs of the real (imaginary) spectral components. The dashed lines are their projections onto the $k_x$--$k_y$ plane. The path $\gamma$, with base point at $k_x=k_y=-\pi$, encircles the unpaired EP3. (b) The measured eigenenergies along the (i-)Fermi arcs are shown in (a), illustrating the existence of the unpaired EP3 and its dispersion. (c) The measured braid along $\gamma$ corresponds to the braid word $B_\gamma=\sigma_1\sigma_2\sigma_1^{-1}\sigma_2^{-1}$. (d) The loop following the Brillouin zone (BZ) boundary is homotopy equivalent to $\gamma$ and thus carries the same braid. (e) Measured discriminant [see Eq. \ref{['eq:discriminant']}] for the four segments of the loop shown in (c). Experimental data are represented by dots with error bars indicating statistical uncertainties from Poissonian photon-number fluctuations; theoretical results are denoted by continuous lines.
• Figure 3: Path-dependent fusion. (a) Theoretical spectra (real parts) over the BZ, with EP2s marked as red dots. Top: $H_M^{\delta=1}=H_{s=0}$. Bottom: $H_{s=3/4}$. (b) Panel 1, Loop along the BZ boundary based at $(k_x,k_y) = (-\pi,-\pi)$ (top) and eigenenergy braid of $H_M^{\delta=1}$ along this loop (bottom). Experimental data are presented as points with error bars indicating statistical uncertainties from Poissonian photon-number fluctuations; theoretical predictions are shown as solid lines. Panel 2: as panel 1. The two EP2s carry braid invariants $B_{\gamma_1}=\sigma_2^{-1}$ and $B_{\gamma_2} = \sigma_1$. Panel 3: as panel 1. Shifting the base point of loop $\gamma_1$ by $\Delta k_x = 2\pi$ changes the braid by a non-Abelian conjugate action, resulting in $B_{\gamma_1^x} = B_x^{-1} B_{\gamma_1 } B_x = \sigma_1^{-1}\sigma_2^{-1}\sigma_1 = \sigma_2\sigma_1^{-1}\sigma_2^{-1}$. Panel 4: as panel 1. Subsequently shifting the base point by $\Delta k_y = 2\pi$ leads to $B_{\gamma_1}^{xy}=B_y^{-1} B_{\gamma_1}^x B_y = \sigma_1^{-1}$. (c) Perturbation along $H_s$ moves the EP2s across the boundary, fusing them after they have followed a non-trivial loop around the torus.
• Figure 4: Topological transition between different band-gapped phases via creation and annihilation of a pair of EP2s. eigenenergies of $H_t$ for (a) $t=1/3$ [the $(1,\sigma_2)$-phase]), (b) $t=2/3$ (EP2 pair creation), (c) $t=4/5$ (EP2 encircling the BZ), (d) $t=6/7$ (EP2 pair annihilation), (e) $t=7/8$ [the $(1,1)$-phase], respectively. Continuous lines and surfaces correspond to theoretical predictions, while measured results are represented by dots; error bars indicate the statistical uncertainty, obtained by assuming Poisson statistics in the photon-number fluctuations. Top: Real parts of eigenenergies over the BZ, with Fermi arcs (solid lines) and their projections (dashed). Bottom: Eigenenergy braids along the BZ boundary, counterclockwise from $(-\pi,-\pi)$.
• Figure S1: Observation of the unpaired third-order exceptional point (EP3) braid along an irregular loop $\tilde{\gamma}$. (a) Theoretical results for the real and imaginary parts of the spectrum of $H_M^{\delta=1+2\sqrt{2}}$. The red dot marks the non-Abelian unpaired EP3. (b) Top: Parametric irregular loop $\tilde{\gamma}$ based at ($k_x$, $k_y$) = ($-\pi, -\pi$). Bottom: Eigenvalue braid of $H_M^{\delta=1+2\sqrt{2}}$ along this loop. Theoretical predictions are shown as solid lines, while experimental data are plotted as points. Error bars indicate statistical uncertainties, assuming Poisson statistics for photon-number fluctuations.