Topological Switch for Non-Hermitian Skin Effect in Cold-Atom Systems with Loss

We propose a realistic cold-atom quantum setting where topological localization induces nonreciprocal pumping. This is an intriguing non-Hermitian phenomenon that illustrates how topology, when assisted with atom loss, can act as a “switch” for the non-Hermitian skin effect (NHSE), rather than as a passive property that is modified by the NHSE. In particular, we present a lattice-shaking scenario to realize a two-dimensional cold-atom platform, where nonreciprocity is switched on only in the presence of both atom loss and topological localization due to time-reversal symmetry breaking. The resultant nonreciprocal pumping is manifested by asymmetric dynamical evolution, detectable by atomic populations along the system edges. Our setup may trigger possible applications in nonreciprocal atomtronics, where loss and topological mechanisms conspire to control atomic transport. Its quantum nature will also facilitate future studies on the interplay between non-Hermiticity and many-body physics.

Figure 1

(a) 2D optical lattice depicted by Eq. (1); dark blue color indicates minima of the potential, generating the $a$, $b$ sublattices. (b) Effective 1D potential experienced by each sublattice along the $x$ direction, with $|\uparrow ⟩$ and $|\downarrow ⟩$ representing the $p_x$ and $s$ orbitals. Atoms in the $|\downarrow ⟩$ state are resonantly excited to a third state $|e⟩$, leading to loss. (c) The corresponding lattice structure of (a); pink arrows show the direction of nonreciprocity of each sublattice. (d) Two-leg system for each sublattice [white and gray rows in Fig. 1], with yellow and green sites representing up and down pseudospins. (e) Mechanism for topology-induced nonreciprocal corner modes. The reciprocity is broken when topological boundary modes localize on the $a$ ($b$) sublattices along the lower (upper) edge [white (gray) bars], such that nonreciprocal pumping toward the left (right) dominates.

Figure 2

$x\text{−}\mathrm{PBC}/y\text{−}\mathrm{OBC}$ spectrum of $h_{2\mathrm{D}}$ without atom loss. Almost degenerate chiral edge modes (red curves) reflect the Chern topology, with their presence (or absence) at each $k_x$ depending on whether the nearly quantized Berry phases ${\gamma }_1^y$ and ${\gamma }_4^y$ are close to $\pi$ or zero. Effective parameters are $\phi =\pi /2$ and $\{t_{+,x},t_{-,x},t_{+,y},t_{-,y},t_{+,y}^{\prime },t_{-,y}^{\prime },{\mathrm{\Delta }}_{-},t_d,t_v\}=\{0.38,0.43,-0.11,-0.41,0.44,0.14,2.07,-0.09,0.16\}$ $h\times \mathrm{kHz}$, corresponding to the experimentally realistic parameters $V_{xy}=3E_r=\frac{3}{2}{V}_y$ and ${\omega }_2=2{\omega }_1=10\times 2\pi \mathrm{kHz}$, with $E_r=({\pi }^2{\hslash }^2/2m{d}^2)=h\times 4.1\mathrm{kHz}$.

Figure 3

(a) Imaginary and real parts of the $x\text{−}\mathrm{PBC}/y\text{−}\mathrm{OBC}$ spectrum for nonzero loss $g=0.2$ $h\times \mathrm{kHz}$, color indicating different bands. Other parameters follow Fig. 2, except for larger frequencies ${\omega }_2=2{\omega }_1=12\times 2\pi \mathrm{kHz}$. (b) The corresponding summed eigenmode distribution ${\rho }_{\mathrm{sum}}(x,y)$, with corner mode densities $>\mathcal{O}(1)$, much greater than those of ordinary topological corner modes. (c) Scaling of ${\rho }_{\mathrm{sum}}(x,y)$ with system length $N_x$ at various positions indicated in (b), with the NHSE giving rise to linear, i.e., extensive, scaling at the corner $(N_x,1)$ (blue) and to a smaller extent along an edge $(N_x,N_y/2+1)$ (yellow), but not deep in the bulk at $(N_x/2,N_y/2+1)$ (purple) and the other edge at ($N_x/2$,1) (red). (d) The $x$ direction IPR for different sublattices $y$ as the intersublattice couplings $(t_{\sigma ,y},t_{\sigma ,y}^{\prime })\to c(t_{\sigma ,y},t_{\sigma ,y}^{\prime })$ are suppressed, computed for $N_x=40$. (e) Phase diagram showing anticlockwise (clockwise) topology-induced nonreciprocal pumping phases over a large range of optical lattice parameters $\phi$ and $V_{xy}/V_y$. The average Berry phase ${\overline{\gamma }}^y$ is nonzero in the topological phases, where the ${\mathrm{IPR}}_{\uparrow }^x$ for pseudo-spin-up component also peaks. The shaking frequency is set as ${\omega }_2=2{\omega }_1={\epsilon }_{s{p}_x}/\hslash$.

Figure 4

(a)–(d) Population distribution snapshots of the states ${\mathrm{\Psi }}_t^{L/R}$ after $t_0=2\mathrm{ms}$ of evolution with $g=0.5h\times \mathrm{kHz}$, as experimental signature of topology-induced corner accumulation. The initial state is prepared at (a),(c) $(\sigma ,x_0,y_0)=(\uparrow ,1,N_y)$ and (b),(d) $(\uparrow ,N_x,N_y)$, as indicated by the dashed lines. Pink arrows in (a),(b) indicate the direction of corner accumulation. For comparison, gray shadows indicate the same states evolving without dissipation. Other parameters are $\phi =\pi /2$, $V_y=2E_r$, and (a),(b) $V_{xy}=3E_r$, ${\omega }_1=6\mathrm{kHz}$; (c),(d) $V_{xy}=4.5E_r$, ${\omega }_1=9\mathrm{kHz}$. In (a), ${\mathrm{\Psi }}^R$ evolves almost identically as the no-loss case, hence covering the gray shadow. (e) Evolution of total population $n^{L/R}$ for cases (a)–(d) with different initial states and topology. Case (a), i.e., ${\mathrm{\Psi }}^R$ with ${\overline{\gamma }}^y>0$ (blue circles), decays least, manifesting topology-induced nonreciprocal pumping.