Topological quantum interference in a pumped Su-Schrieffer-Heeger lattice

Topological quantum interference emerges from the interplay between quantum mechanics and topology. We present evidence for two types of such interference phenomenon that can result from the quantum dynamics of initial topological states. We realize both types of topological quantum interference in a pumped non-Hermitian Su-Schrieffer-Heeger lattice that can be implemented by creation and coherent control of excitonic states of trapped neutral atoms. On quenching the system from the topological to the gapless phases and then back again, we find that interference patterns develop in the gapless phase and also after switching back to the topological phase. These patterns occur both as many-excitation interferences generated in the presence of pumping the atoms at the end sites, and as one- and two-excitation interferences seen in the absence of pumping when starting with edge excitations. Investigation of the excitation dynamics shows that these interference patterns originate from the topological nature of the initial states and are very different from quantum interferences originating from nontopological states of the lattice. Our results also reveal that unlike well-known situations where topological states are protected against local perturbations, in the non-Hermitian SSH systems resulting from driving the excited-state populations, a local dissipation at each lattice site can suppress both the topological interference and the total population of the lattice.

Figure 1

(a) Schematic diagram of the SSH lattice with pumping fields applied at both end sites. (b) Energy spectrum plotted as a function of the parameter $\alpha$ specifying the relative phase of the three laser fields generating the lattice potential. Vertical green, blue, and cyan lines denote the topologically trivial ($\alpha =0.25\pi$), gapless ($\alpha =0.5\pi$), and topological ($\alpha =0.75\pi$) phases, respectively. (c) Effective intra- and intercell hopping amplitudes $J_1$ and $J_2$ as a function of $\alpha$. The other parameter values for evaluating energy spectrum of $H_{\mathrm{SSH}}$ in Eq. (1) and hopping amplitude in Eq. (2) are $\varepsilon =1, \gamma =0, N=20, V_0=0.125$, and $\mu /k^2=0.25$.

Figure 2

(a) Disconnected entanglement entropy $S^D$ as a function of $\alpha$, for the ground state of the SSH model at half filling on a chain of $2N=40$ sites. $S^D$ is evaluated by dividing the $2N$ sites evenly into four nonoverlapping segments $A, B, D$, and $C$, each having $N/2$ sites, then evaluating $S^D=S_{AB}+S_{BC}-S_{ABC}-S_B$, where $S_i=-{\mathrm{Tr}}_i{\rho }_{i}log\left({\rho }_i\right)$ is the von Neumann bipartite entanglement entropy (with base $e$ of the logarithm). The panels on the right show the distribution of the amplitudes of the highest filled eigenstates over all sites for (b) $\alpha =0.4\pi$, (c) $\alpha =0.5\pi$, and (d) $\alpha =0.522\pi$. All parameters are the same here as in Fig. 1.

Figure 3

Time-dependent occupation number $P_{\mathrm{pop}}$ of the SSH lattice during an interference inducing quantum quench protocol under pumping at both ends ($F_{\mathrm{A}}=F_{\mathrm{B}}=0.01$, upper panels), and under pumping at one end only ($F_{\mathrm{A}}=0.01, F_{\mathrm{B}}=0$, lower panels). The initial lattice condition at $t=0$ is no excitation, i.e., $A_l\left(0\right)=B_l\left(0\right)=0$. Topological interference is observed in panels (a) and (c) with parameters ${\alpha }_{\mathrm{T}}=0.75\pi , {\alpha }_{\mathrm{G}}=0.5\pi$, and switching time interval specified by $t_a=10T_p, t_b=30T_p$. Panels (b) and (d) are zoom-in views of the patterns in the gapless phase quench period $10T_p<t<30T_p$ of panels (a) and (c), respectively. The other parameters used are $\gamma =0.0025, N=20, \varepsilon =1, V_0=0.125, \mu /k^2=0.25, {\omega }_{pA}={\omega }_{pB}={\omega }_p=1, {\varphi }_{0A}={\varphi }_{0B}={\varphi }_0=0$. Unless otherwise specified, all energy quantities are defined in units of ${\omega }_p$.

Figure 4

(a) A zoom-in view of Fig. 3 around the first switch at $t/T_p=10$. The occupation number $P_{\mathrm{pop}}$ across the SSH lattice when (b) $t/T_p=10$, (c) $t/T_p=10.25$, (d) $t/T_p=10.5$, and (e) $t/T_p=10.75$.

Figure 5

A zoom-in view of the interference pattern in the topological phase quench period $30T_p<t\le 40T_p$ of Fig. 3. The plot shows the occupation number $P_{\mathrm{pop}}$ of the SSH lattice with quantum quench protocol under pumping at both ends ($F_{\mathrm{A}}=F_{\mathrm{B}}=0.01$, upper panels), with initial lattice condition at $t=0$ given by $A_l\left(0\right)=B_l\left(0\right)=0$ (no excitation). Other parameters are the same as in Fig. 3, i.e., ${\alpha }_{\mathrm{T}}=0.75\pi , {\alpha }_{\mathrm{G}}=0.5\pi , t_a=10T_p, t_b=30T_p, \gamma =0.0025, N=20, \varepsilon =1, V_0=0.125, \mu /k^2=0.25, {\omega }_{pA}={\omega }_{pB}={\omega }_p=1, {\varphi }_{0A}={\varphi }_{0B}={\varphi }_0=0$.

Figure 6

Time-dependent occupation number $P_{\mathrm{pop}}$ of the SSH lattice in the topological phase (${\alpha }_{\mathrm{T}}=0.75\pi$, left panel) and in the gapless phase (${\alpha }_{\mathrm{G}}=0.5\pi$, right panel) under pumping at both ends ($F_{\mathrm{A}}=F_{\mathrm{B}}=0.01$, upper panels) and under pumping at one end only ($F_{\mathrm{A}}=0.01, F_{\mathrm{B}}=0$, lower panels). No quench is applied here. The initial condition here is no excitation at time $t=0$, i.e., $A_l\left(0\right)=B_l\left(0\right)=0$ (same initial condition as in Fig. 3). All other parameters are the same as in Fig. 3.

Figure 7

Occupation number $P_{\mathrm{pop}}$ at (a), (c) sublattice A with odd sites and (b), (d) sublattice B with even sites during the time period $10T_p–30T_p$. Panels (a) and (b) correspond to Fig. 3 with the quantum quench scheme starting from the topological phase, while panels (c) and (d) correspond to Fig. 6 in the gapless phase without quench.

Figure 8

The superposition of highest filled and lowest unfilled eigenstates of the SSH model in the gapless phase obtained by setting $\alpha$ to ${\alpha }_{\mathrm{G}}=0.5\pi$: (a) $\left(\right|{\psi }_{\mathrm{G}}^{\left(20\right)}\rangle -|{\psi }_{\mathrm{G}}^{\left(21\right)}\rangle )/\sqrt{2}$ and (b) $\left(\right|{\psi }_{\mathrm{G}}^{\left(20\right)}\rangle +|{\psi }_{\mathrm{G}}^{\left(21\right)}\rangle )/\sqrt{2}$, with corresponding eigenenergy in Fig. 1. The parameters are the same as in Fig. 1.

Figure 9

Time-dependent occupation number $P_{\mathrm{pop}}$ in the case of two excitations (top panels) and of one excitation (bottom panels) inserted in the nonpumped SSH lattice at $t=0$. (a), (b) The two excitations initially occupy both ends $[A_1\left(0\right)=B_{20}\left(0\right)=1]$. (c), (d) One excitation initially occupies the first site [$A_1\left(0\right)=1, B_{20}\left(0\right)=0$]. Topological interference is observed in panel (a) with ${\alpha }_{\mathrm{T}}=0.75\pi , {\alpha }_{\mathrm{G}}=0.5\pi$, and the switching time $J{t}_a/2\pi =10, J{t}_b/2\pi =30$. Panels (b) and (d) show zoom-in views of the patterns during $10<Jt/2\pi <30$ in panels (a) and (c), respectively. Here the pumping fields are switched off ($F_A=F_B=0$), $\gamma =0.0025$, and all other parameters are the same as those in Fig. 3. All energy quantities are defined in units of $J(={\omega }_p)$ (e.g., $J_1=J_2=0.5J$ at ${\alpha }_{\mathrm{G}}=0.5\pi$) if not otherwise specified.

Figure 10

(a) A zoom-in view of Fig. 9 around the first switch at $t/T_p=10$. The occupation number $P_{\mathrm{pop}}$ across the SSH lattice when (b) $Jt/2\pi =10$, (c) $Jt/2\pi =10.25$, (d) $Jt/2\pi =10.5$, and (e) $Jt/2\pi =10.75$.

Figure 11

A zoom-in view of the interference pattern in the topological phase quench period $30<Jt/2\pi \le 40$ of Fig. 9. The plot shows the occupation number $P_{\mathrm{pop}}$ in the case of two excitations initially occupying both ends, i.e., $A_1\left(0\right)=B_{20}\left(0\right)=1$ at $t=0$, with no pumping fields ($F_A=F_B=0$). Other parameters are same as in Fig. 9, i.e., ${\alpha }_{\mathrm{T}}=0.75\pi , {\alpha }_{\mathrm{G}}=0.5\pi , J{t}_a/2\pi =10, J{t}_b/2\pi =30\gamma =0.0025, N=20, \varepsilon =1, V_0=0.125$, and $\mu /k^2=0.25$.

Figure 12

Time-dependent occupation number $P_{\mathrm{pop}}$ in the case of two excitations (top panel) and one excitation (bottom panels) inserted in the nonpumped SSH lattice at $t=0$, for the topologically nontrivial phase (left panels, ${\alpha }_{\mathrm{T}}=0.75\pi$) and for the gapless phase (right panels, ${\alpha }_{\mathrm{G}}=0.5\pi$). (a), (b) The two excitations initially occupy both ends $[A_1\left(0\right)=B_{20}\left(0\right)=1]$. (c), (d) One excitation occupies the first site [$A_1\left(0\right)=1, B_{20}\left(0\right)=0$]. Here the pumping fields are switched off ($F_A=F_B=0$), $\gamma =0.0025$, and all other parameters are same as those in Fig. 9.

Figure 13

Evolution of the total number $P_{\mathrm{tot}}$ of excitations for the pumped many-excitation case [panels (a)–(c)] and nonpumped two-excitation case [panel (d)] for various values of dissipation parameter $\gamma$. (a) Interference scheme, (b) topological nontrivial phase ($\alpha =0.75\pi$), and (c) gapless phase ($\alpha =0.5\pi$) corresponding to Figs. 3, 6, and 6, respectively. In panel (d), the dotted, dash-dotted, thin-dashed, thick-dashed, and solid curves correspond to $\gamma =0$, 0.0005, 0.0025, 0.005, 0.0075, respectively, while the circle, left, and right triangles are for $\alpha =0.25\pi , 0.5\pi , 0.75\pi$, respectively. Other parameters for panel (a), panels (b) and (c), and panel (d) are the same as those in Figs. 3, 6, 9, respectively.

Figure 14

Distribution of the lowest unfilled and highest filled eigenstates for various values of $\alpha$ for the $2N=40$ finite-size SSH lattice. All parameters are the same as in Fig. 1 of the main text.

Figure 15

Dynamics of disconnected entanglement entropy $S^D\left(t\right)$ for a SSH lattice of $2N=40$ sites. The evaluation of $S^D\left(t\right)$ is performed in a similar way as in Fig. 2 of the main text, i.e., by using an even partition of the $2N$ sites into four nonoverlapping segments.

Figure 16

Time-dependent occupation number $P_{\mathrm{pop}}$ for the case of many excitations with a pumping field applied at the last site ($F_{\mathrm{A}}=0, F_{\mathrm{B}}=0.01$) of the finite SSH lattice with $2N=40$ sites. Parameters are the same as in Fig. 3 of the main text.

Figure 17

Time-dependent occupation number $P_{\mathrm{pop}}$ for the case of an excitation initially occupying the last site ($A_1\left(0\right)=0, B_{20}\left(0\right)=1$) and no pumping. Parameters are the same as in Fig. 9 of the main text.

Figure 18

Time evolution of the occupation number $P_{\mathrm{pop}}$ for many excitations pumped from (a) both ends, (b) the first site, and (c) the last site of the SSH lattice in the trivial phase regime with ${\alpha }_{\mathrm{tr}}=0.25\pi$. In this regime, large populations appear on the pumping sites, i.e., the $1\mathrm{st}$ and/or last sites and there is still a small number of excitations that hop between adjacent sites. Here we take $\gamma =0.0025$. (d) Time dependence of the total number excitations $P_{\mathrm{tot}}$ in the case of pumping from both ends [panel (a)] in the trivial phase regime. Other parameters are same as in Fig. 6 of the main text.

Figure 19

Time evolution of the occupation number $P_{\mathrm{pop}}$ for (a) two excitations initially at both ends and one excitation initially occupying (b) the first site and (c) the last site of the SSH lattice in the trivial phase regime with ${\alpha }_{\mathrm{tr}}=0.25\pi$. An interference pattern due to nontopological states appears in panel (a). Here we set the damping parameter to $\gamma =0.0025$. All other parameters are the same as in Fig. 12 of the main text.

Figure 20

The occupation number $P_{\mathrm{pop}}$ at five specific sites (a) 1, (b) 5, (c) 10, (d) 15, and (e) 20 of Fig. 6 of the main text in the gapless phase. The parameters are same as in Fig. 6.

Figure 21

Evolution of the total number $P_{\mathrm{tot}}$ of excitations for the two-excitation case with various values of damping rate $\gamma$ under the quench scheme of the main text that generates topological interference in the region $10<Jt/2\pi <30$. The curve with $\gamma =0.0025$ corresponds to Fig. 9 under the quantum quench protocol in the main text. Other parameters are the same as Fig. 9 of the main text.

Figure 22

Time-dependent occupation number $P_{\mathrm{pop}}$ in the case of two excitations (top panels) and of one excitation (bottom panels) inserted in the SSH lattice at $t=0$. (a), (b) The two excitations initially occupy both ends $[A_1\left(0\right)=B_{20}\left(0\right)=1]$. (c), (d) One excitation initially occupies the first site [$A_1\left(0\right)=1, B_{20}\left(0\right)=0$]. Topological interference is observed in panel (a) with ${\alpha }_{\mathrm{T}}=0.75\pi , {\alpha }_{\mathrm{G}}=0.5\pi$, and the switching time $J{t}_a/2\pi =10T_p, J{t}_b/2\pi =30$. Panels (b) and (d) show zoom-in views of the patterns during $10<Jt/2\pi <30$ in panels (a) and (c), respectively. Here the pumping fields are switched off ($F_A=F_B=0$). We take $\gamma =0$ and other parameters are same as in Fig. 9 of the main text.

Figure 23

Time-dependent occupation number $P_{\mathrm{pop}}$ in the case of two excitations (top panel) and one excitation (bottom panels) inserted at $t=0$, for the topologically nontrivial phase (left panels, ${\alpha }_{\mathrm{T}}=0.75\pi$) and for the gapless phase (right panels, ${\alpha }_{\mathrm{G}}=0.5\pi$). (a), (b) The two excitations initially occupy both ends $[A_1\left(0\right)=B_{20}\left(0\right)=1]$. (c), (d) One excitation occupies the first site [$A_1\left(0\right)=1, B_{20}\left(0\right)=0$]. Here the pumping fields are switched off ($F_A=F_B=0$). We take $\gamma =0$ and all other parameters are same as in Fig. 12 of the main text.

Figure 24

Time-dependent occupation number $P_{\mathrm{pop}}$ of the SSH lattice during an alternative interference-inducing quantum quench protocol into the trivial phase ${\alpha }_{\mathrm{tr}}$. The upper panels show results under pumping at both ends ($F_{\mathrm{A}}=F_{\mathrm{B}}=0.01$), and the lower panels results under pumping at one end only ($F_{\mathrm{A}}=0.01, F_{\mathrm{B}}=0$). The initial lattice condition at $t=0$ is no excitation, i.e., $A_l\left(0\right)=B_l\left(0\right)=0$. Topological interference is observed in panels (a) and (c) with ${\alpha }_{\mathrm{T}}=0.75\pi , {\alpha }_{\mathrm{tr}}=0.25\pi$, and switching time interval specified by $t_a=10T_p, t_b=30T_p$. Panels (b) and (d) show zoom-in views of the pattern in the trivial phase quench period $10T_p<t<30T_p$ of panels (a) and (c), respectively. The other parameters used are $\gamma =0.0025, N=20, \varepsilon =1, V_0=0.125, \mu /k^2=0.25, {\omega }_{\mathrm{A}}={\omega }_{\mathrm{B}}=1, {\varphi }_{\mathrm{A}}={\varphi }_{\mathrm{B}}=\varphi =0$.