Two-dimensional non-Hermitian skin effect in an ultracold Fermi gas

Abstract

The concept of non-Hermiticity has expanded the understanding of band topology leading to the emergence of counter-intuitive phenomena. One example is the non-Hermitian skin effect (NHSE), which involves the concentration of eigenstates at the boundary. However, despite the potential insights that can be gained from high-dimensional non-Hermitian quantum systems in areas like curved space, high-order topological phases, and black holes, the realization of this effect in high dimensions remains unexplored. Here, we create a two-dimensional (2D) non-Hermitian topological band for ultracold fermions in spin-orbit-coupled optical lattices with tunable dissipation, which exhibits the NHSE. We first experimentally demonstrate pronounced nonzero spectral winding numbers in the complex energy plane with non-zero dissipation, which establishes the existence of 2D skin effect. Further, we observe the real-space dynamical signature of NHSE in real space by monitoring the center of mass motion of atoms. Finally, we also demonstrate that a pair of exceptional points (EPs) are created in the momentum space, connected by an open-ended bulk Fermi arc, in contrast to closed loops found in Hermitian systems. The associated EPs emerge and shift with increasing dissipation, leading to the formation of the Fermi arc. Our work sets the stage for further investigation into simulating non-Hermitian physics in high dimensions and paves the way for understanding the interplay of quantum statistics with NHSE.

Figures (20)

• Figure 1: Hermitian and non-Hermitian system in an optical lattice with spin-orbit couplinga, Our optical Raman lattice system, which consists of a 1D optical lattice along the y direction and another Raman beam along the direction with angle $76^o$ tilted from the lattice incident beam, generating a periodic Raman potential. A loss beam with $\sigma^-$ polarization is further applied along -z direction to realize the non-Hermiticity. b, Band structure after the quantum quench where the band gap can be extracted by the time evolution of spin polarization and the equilibrium spin polarization which can be obtained by adiabatically loading the atoms into the lowest energy band in the Hermitian optical Raman lattice. c, Experimental measurement and numerical simulation of spin polarization of equilibrium state as a function of two photon detuning $\delta$. d, Band structure of the lowest two bands at different two-photon detuning values in the Hermitian optical Raman lattice. With increasing dissipation, atoms accumulate at the boundary of the system showing the non-trivial winding of eigenenergy in the complex energy plane. Furthermore, the energy band closes the gap at a specific quasi-momentum.
• Figure 1: a, Schematic energy level diagram with relevant transitions. Both the lattice and the Raman beams are blue-detuned from $^1S_0(F=\frac{5}{2})\rightarrow{^3}P_1(F'=\frac{7}{2})$ intercombination transition and induce the Raman transition between the $|{\uparrow}\rangle=|{m_F=5/2}\rangle$ and $|{\downarrow}\rangle=|{m_F=3/2}\rangle$ hyperfine states of the $^1S_0$ ground manifold. b, Our optical Raman lattice system, which consists of a 1D optical lattice along the y direction and another Raman beam along the direction with angle $76^o$ tilted from the lattice incident beam, generating a periodic Raman potential. A loss beam with $\sigma^-$ polarization is further applied along -z direction to realize the non-Hermiticity. c, Absorption image of $m_F$ = 5/2 atoms without quenching the two photon detuning ($t_f$=0ms) after 15ms time-of-flight. d, When the two photon detuning is quenched from $\delta_i$ to $\delta_f$, two $m_F$ = 3/2 atom clouds are coupled out from $m_F$ = 5/2 clouds after holding $t_f$=0.15ms, which shows the resonant spin flipping on particular quasi-momentum subspace.
• Figure 2: Momentum-dependent Rabi oscillation to resolve the band gap closing.a, Rabi oscillation at different quasi-momenta $\textbf{q}=(k_x,q_y)$ (points in b). Lines show the fitted curve with a damped sinusoidal function. b, Band gap between the lowest two bands simulated from plane wave expansion at different loss terms from $\gamma_{\downarrow}=0E_r$ to $2.1E_r$, while the lattice potential and Raman coupling strength are fixed to $V_{\uparrow(\downarrow)}=3.2(2.2)E_r$ and $M_0=2.0E_r$. The two-photon detuning $\delta$ (or equivalently $m_z$) is set to $\delta=-0.35E_r$ to compensate the on-site energy difference induced by spin-dependent lattice potential. The dashed line marks the positions of quasi-momentum $\textbf{q}$ where the energy gap is closed. c, With dissipation, each Dirac point in the Hermitian regime separates into a pair of EPs, which are connected by a (non-Hermitian) Fermi arc and continuously move in the momentum space as dissipation increases. d,e The change of the real energy band gap and damping rate at different quasi-momenta shown by the point $\alpha$ and $\beta$ in c with increasing dissipation. The slight increase in the real band gap when the loss rate is larger than the exceptional point is a result of the higher band effect.
• Figure 2: Experimental sequence for momentum resolved Rabi spectroscopy (a) and spin polarization texture measurement in equilibrium (b).
• Figure 3: Observation of non-Hermitian EPs and Fermi arc.a The band gap $\Re{(\Delta E)}$ (left) and damping rate $\Im{(\Delta E)}$ (right) extracted from the Rabi oscillation at two different quasi-momentum marked in the left energy band gap. Before and after the EPs, the fitting functions are different. The solid line is obtained from the simulation of plane wave expansion. The shaded region indicates the uncertainty of the band gap and damping rate associated with the uncertainty of atom loss (green color) and the uncertainty of quasi-momentum (brown color). The inset shows the fitting result with same damped oscillation fitting function before and after the exceptional point. The vertical and horizontal error bars represent the fitting errors and the experimental uncertainty related to the calibration, respectively.b The normalized $R^2$ value, obtained by fitting the experimental data with exponential decay (ED) curves, divided by the $R^2$ value obtained from fitting with a damped oscillation (DO) curve, is presented. The corresponding quasi-momentum values, $(k_x,q_y)=(-0.5,-0.9)k_0$ (top) and $(k_x,q_y)=(0.0,-0.3)k_0$ (bottom), are marked in the energy band gap shown in a, respectively. This ratio reveals that above the exceptional point (EP), the fitting using an exponential decay function performs equally well as the fitting with a damped oscillation, indicating that the spin oscillation follows a monotonic exponential behavior, while below the EP, the exponential decay function is disfavored. The horizontal error bars represent the experimental uncertainty related to the calibration.c The spin oscillation for same quasi-momentum and different dissipation strengths and marked by the arrows in a bottom. When $\gamma_\downarrow$ is smaller than the EP, the spin polarization oscillates with the frequency dependent, while above EP, the PT-symmetry broken phase shows a monotonic spin polarization. The right column shows the corresponding simulation from plane wave expansion with finite temperature considering momentum uncertainty (solid line) and without considering the momentum uncertainty (dashed line). d The spin oscillation for two different quasi-momentum and similar dissipation strength $\gamma_\downarrow\sim0.8E_r$ (top) and $1.5E_r$ (bottom). When $\gamma_{\downarrow}=1.5E_r$, the left EP moves across $k_x=-0.5k_0$, which means the quasi momentum $(k_x,q_y)=(-0.5,-0.9)k_0$ is on Fermi arc and shows a monotonic spin polarization.

Theorems & Definitions (1)

• proof