Dynamic Signatures of Non-Hermitian Skin Effect and Topology in Ultracold Atoms

The non-Hermitian skin effect (NHSE), the accumulation of eigen–wave functions at boundaries of open systems, underlies a variety of exotic properties that defy conventional wisdom. While the NHSE and its intriguing impact on band topology and dynamics have been observed in classical or photonic systems, their demonstration in a quantum gas system remains elusive. Here we report the experimental realization of a dissipative Aharonov-Bohm chain—non-Hermitian topological model with NHSE—in the momentum space of a two-component Bose-Einstein condensate. We identify signatures of the NHSE in the condensate dynamics, and perform Bragg spectroscopy to resolve topological edge states against a background of localized bulk states. Our Letter sets the stage for further investigation on the interplay of many-body statistics and interactions with the NHSE, and is a significant step forward in the quantum control and simulation of non-Hermitian physics.

Figure 1

Topological Aharonov-Bohm chain with NHSE. (a) Schematic illustration of the Aharonov-Bohm chain. Each unit cell includes three sites ($a$, $b$, and $c$), forming a closed triangular loop. Site $c$ is further coupled to a reservoir that introduces dissipation to the system. While hopping rates $J_{p,t,r,s}$ are induced by Raman or Bragg processes, their phases contribute to a synthetic magnetic flux $\varphi$ through each ring. (b) The momentum lattice in (a) is generated by multifrequency Raman and Bragg processes that couple discrete momentum and hyperfine states (both with $m_F=0$) in the $F=1$ and $F=2$ manifolds of ${}^{87}\mathrm{Rb}$ atoms. Within each unit cell, sites $(a,c)$ and $(b,c)$ are coupled through resonant Raman processes, with a momentum difference $2\hslash k$; while sites $(a,b)$ are coupled through a four-photon resonant Bragg process, with a momentum difference of $4\hslash k$. Here $k$ is the wave vector of a 795 nm laser. Subsequently, in the $n$th unit cell ($n\in \mathbb{Z}$), sites $a$ and $b$ represent the momentum states $|p=3n⟩$ and $|p=3n+2⟩$ (in units of $2\hslash k$), respectively, of the hyperfine ground state $|F=1,m_F=0⟩$ in the ${{}^{5}S}_{1/2}$ manifold. Site $c$ is encoded into the momentum state $|p=3n+1⟩$ of the hyperfine ground state $|F=2,m_F=0⟩$. Dissipation is induced by a near-resonant coupling of atoms on site $c$ to the ${{}^{5}P}_{3/2}$ manifold.

Figure 2

Bulk dynamics along the momentum lattice, where we show the normalized atom population for each sublattice site. (a), (b) are the non-Hermitian cases with $\gamma =h\times 1.3(2)\mathrm{kHz}$; (c), (d) are the Hermitian cases with $\gamma =0$. Atoms are initially prepared in the state $|n=0,a⟩$ in (a), (c), and $|n=0,b⟩$ in (b), (d). The color bar indicates the atom population in each sublattice site, normalized by the initial atom number and shown in the order $|n,a⟩,|n,c⟩,|n,b⟩$ (from left to right) for a given unit cell $n$. The experimental parameters are $\{J_s,J_r,J_p,J_t\}=h\times \{1.07(2),1.07(1),0.65(3),0.81(1)\}\mathrm{kHz}$, $\varphi =-\pi /2$, and $\mathrm{\Delta }=0$.

Figure 3

Signatures of the NHSE. (a)–(e) Measured growth rate $\overline{\lambda }$ with a fixed $\gamma =h\times 1.3(2)\mathrm{kHz}$, and varying synthetic flux $\varphi$. (f) $\overline{\lambda }(v)$ with $\varphi =0$ and $\gamma =0$ and an evolution time of 0.4 ms. The blue dots are the experimental data, and the error bar represents the SE of the mean. The solid lines are numerical results using the effective Hamiltonian for a lattice with $N=221$ unit cells. The red triangles are numerical simulations using the experimental parameters. For all panels, the hopping rates are $\{J_s,J_r,J_p,J_t\}=h\times \{1.13(2),1.04(1),0.62(4),0.80(2)\}\mathrm{kHz}$, and $\mathrm{\Delta }=0$.

Figure 4

Topological edge states through the Bragg spectroscopy. (a) Schematic illustration of the Bragg spectroscopy. (b), (c) show typical experimental data (dots) of the Bragg spectrum, with $J_t/J_p=0.34$ in (b) and $J_t/J_p=2.27$ in (c). The solid lines are from the double-Gaussian fit. Other experimental parameters are $\{J_s,J_r,J_p\}=h\times \{1.08(2),1.02(3),1.21(3)\}\mathrm{kHz}$, $\gamma =h\times 1.0(1)\mathrm{kHz}$, and $\varphi =-\pi /2$. An energy offset $\mathrm{\Delta }=-h\times 2.0(3)\mathrm{kHz}$ on site $|n,c⟩$ is also imposed to reduce its impact on the topological gap. (d) Real components of the eigenspectrum (grey shaded), spectroscopic measurements (blue strips), and numerical simulations (red curves and shade). The band center is shifted from zero due to the light shift of the four-photon process [41]. The red curves and shade, respectively, indicate the valley location and width of the numerically simulated spectra. The blue strips are determined by fitting the experimental data: the length of each strip corresponds to $2w$, and the blue dots indicate valley locations, calculated from ${\delta }_0$ and $\mathrm{\Delta }\delta$ (see main text for definition). The insert of (d) shows the distance-to-width ratio $r$ with increasing $J_t/J_p$, where the dashed horizontal line indicates the location of the phase transition with $r=1$. Error bars show the SE of the mean.