Floquet control of interactions and edge states in a programmable quantum simulator

Abstract

Quantum simulators based on trapped ions enable the study of spin systems and models with rich dynamical phenomena. The Su-Schrieffer-Heeger (SSH) model for fermions in one dimension is a canonical example that can support a topological insulator phase when couplings between sites are dimerized, featuring long-lived edge states. Here, we experimentally implement a spin-based variant of the SSH model using one-dimensional trapped-ion chains with tunable interaction range, realized in crystals containing up to 22 interacting spins. Using an array of individually focused laser beams, we apply site-specific, time-dependent Floquet fields to induce controlled bond dimerization. Under conditions that preserve inversion symmetry, we observe edge-state dynamics consistent with SSH-like behavior. We study the propagation and localization of spin excitations, as well as the evolution of highly excited configurations across different interaction regimes. These results demonstrate how precision Floquet engineering enables the exploration of complex spin models and dynamics, laying the groundwork for future preparation and characterization of topological and exotic phases of matter.

Figures (4)

• Figure 1: Floquet Engineering of Spin Bonds and Topological Edge-States. a-c, Bond dimerization in an $L=12$ spin crystal. Measured spin-spin bond strength $\hbar |\mathcal{J}_{ij}|$ between $i$th and $j$th spins up to next-nearest neighbors (NNN) with increasing Floquet amplitude $\bar{\eta}$. a, No modulation ($\bar{\eta}=0$). b, Moderate modulation ($\bar{\eta}=0.6$). c, Full modulation ($\bar{\eta}=1$). Black spheres for spins and lines for bonds strength, pink rectangles for local Floquet fields. d-f Evolution of a single spin excitation at the edge over time for configurations a-c respectively. The Floquet field suppresses thermalization into the bulk, leading to edge localization. $J$: average nearest-neighbor spin bond coupling absent the Floquet drive; yellow lines show excitation spreading rate (see Method). g, spin excitation at the crystal edge $\langle\hat{s}_z^{(1)}\rangle$ with varying Floquet field amplitude. h, Excitation spreading rate $v_s$ (black circles) decreases with increasing Floquet field amplitude, consistent with the modified SSH model (Eq. \ref{['eq:total_Hamiltonian']}, purple) and the numerical time-dependent ion Hamiltonian (Eq. \ref{['eq:time_dependent_H']}, green). Bars represent 1$\sigma$ binomial uncertainties. Data in d-f aligns with both numerical models in Supplementary Figure 2.
• Figure 2: Edge States in a $L=22$ Spin Crystal. Evolution of a single spin-excitation in a strongly-dimerized ($\bar{\eta}=0.8$) crystal of $L=22$ spins. a edge-excitation ($j=1$) and, b bulk-excitation ($j=11$). We repeat these experiments for all $1\leq j \leq 22$ single-spin excitations, and for each calculate the late-time-averaged spin $\bar{s}_{z,j}$ excitation by averaging the values in the orange rectangle; see Supplementary Figure 3. c Late-time-averaged $\bar{s}_{z,j}$ (black circles) highlights the protection of edge excitations over the bulk. Grey line shows the mean late-time excitation across all crystal sites; each point is calculated by averaging the measurement values in the yellow boxes, excluding the orange box, as exemplified for $j=11$ in b. Bars represent 1$\sigma$ binomial uncertainties. d-e Evolution of edge excitation ($j=1$, d) and bulk excitation ($j=11$, e) as a function of the global Floquet phase shift $\phi$. d The spin excitation at the edge survives longer at $\phi=\tfrac{3\pi}{4}$. The generalized SSH model in Ref. nevado2017topological predicts a nontrivial topological phase at this value. While we do not directly measure a topological invariant, our observations are consistent with edge-state behavior in such a phase. e The bulk excitation is minimally-affected by the Floquet phase.
• Figure 3: Signatures of Fermionic Interaction terms. Evolution of a $L=12$ spin crystal with long-range interactions, initialized in a staggered spin state. a, In the absence of Floquet fields ($\bar{\eta}=0$), the spin excitations quickly hop and thermalize. With full Floquet modulation (b, $\bar{\eta}=0.6$ and c, $\bar{\eta}=1$), thermalization is partially suppressed and the the edges become more protected. d-f Simulation of the $\bar{\eta}=0.6$ configuration (subplot b) of spins $j=1$ (d), $j=4$ (e) and $j=9$ (f). Experimental data (black circle with error bars) agree well with a full simulation of an equivalent fermionic Hamiltonian containing interaction terms (green curve). Bars represent 1$\sigma$ binomial uncertainties. For the bulk spins, the data does not agree with a long-range free-fermionic Hamiltonian (magenta curve). The dynamics of all spins are shown in Supplementary Figure 1. These results are contrasted with short range spin models which are fully explained by free-fermionic evolution. See text and Supplementary Figure 5.
• Figure 4: Domain wall dynamics. The evolution of a domain wall in a $L=12$ spin crystal, comprising of multiple spin-excitations. a-b Short range spin-spin interaction, and c-d Long-range spin-spin interaction. a,c Absent the Floquet field ($\bar{\eta}=0$), the domain wall thermalizes as excitations are free to hop. b,d Fully dimerized crystal ($\bar{\eta}=1$). b Boundary excitations between the wall are exchanged coherently, and the domain wall maintains its state in b or partially thermalizes in d. The results in a-d agree well with numerical calculation of the spin evolution, see Supplementary Figure 6.