Floquet Topological States in Shaking Optical Lattices

Abstract: In this letter we propose realistic schemes to realize topologically nontrivial Floquet states by shaking optical lattices, using both one-dimension lattice and two-dimensional honeycomb lattice as examples. The topological phase in the two-dimensional model exhibits quantum anomalous Hall effect. The transition between topological trivial and nontrivial states can be easily controlled by shaking frequency and amplitude. Our schemes have two major advantages. First, both the static Hamiltonian and the shaking scheme are sufficiently simple to implement. Secondly, it requires relatively small shaking amplitude and therefore heating can be minimized. These two advantages make our scheme much more practical.

• The paper introduces a minimal experimental scheme to induce Floquet topological states via lattice shaking in 1D and 2D geometries.
• It utilizes both numerical Floquet analysis and the Floquet-Magnus expansion to reveal topologically nontrivial phases analogous to SSH and Haldane models.
• The approach achieves robust edge-localized states with modest shaking amplitudes, significantly suppressing heating compared to alternative methods.

Floquet Topological States Induced by Shaking in Optical Lattices

Introduction

This work presents a theoretical framework and practical schemes for realizing Floquet topological states via periodic lattice shaking in cold-atom optical lattices, in both one-dimensional (1D) and two-dimensional (2D) geometries. The approach employs minimal experimental complexity and low driving amplitudes, crucially suppressing heating and facilitating experimental realization. The core achievement is the implementation of topological phases—specifically, Floquet analogues of the Su-Schrieffer-Heeger (SSH) model in 1D and the Haldane quantum anomalous Hall (QAH) model in 2D—by periodic modulation of simple, already-accessible lattice Hamiltonians.

Theoretical Methodology

The authors systematically combine exact and approximate analysis of periodically-driven quantum systems using Floquet theory. The central object is the Floquet operator $\hat{F}$, representing one period of time evolution for the driven Hamiltonian $\hat{H}(t)$. Two main strategies are deployed:

1. Direct Numerical Floquet Analysis: The eigenproblem for $\hat{F}$ is solved numerically to obtain the quasi-energy spectrum and edge-localized Floquet modes. Topologically nontrivial phases are identified via the presence of in-gap boundary modes.
2. Floquet-Magnus Expansion and Effective Hamiltonian Construction: For high-frequency drives, an effective static Hamiltonian $\hat{H}_{\text{eff}}$ is constructed using a perturbative expansion, connecting the topology of the system to established results for static models.

This dual-pronged approach enables both numerically exact and physically transparent characterization of Floquet-induced topological phenomena.

Floquet SSH Physics in One-Dimensional Shaken Lattices

A 1D optical lattice is periodically displaced by modulating the relative phase between counter-propagating laser beams, resulting in a time-dependent vector potential. In the comoving frame and tight-binding limit, the essential model reduces to two bands (associated with $s$- and $p$-orbitals), resonantly coupled by shaking near the two-phonon resonance condition $2\omega \sim \epsilon_p - \epsilon_s$.

The effective Floquet Hamiltonian takes the form $$H_{\text{eff}}(k_x) = \mathbf{B}(k_x)\cdot\boldsymbol{\sigma}$$, with $\mathbf{B}(k_x)$ winding in the $yz$-plane. This is mathematically analogous to the SSH model, with the Zak phase serving as the topological invariant. Tuning the detuning $\Delta_0$ and shaking amplitude $k_rb$ drives transitions between trivial and topologically nontrivial (Zak phase $\neq 0$) regimes. Significantly, the topological phase emerges for small $k_rb\approx 0.1$, which minimizes energy absorption and heating.

The existence of a pair of edge-localized in-gap Floquet states in the quasi-energy spectrum confirms the realization of a topologically nontrivial phase. In contrast, the topologically trivial regime lacks such edge modes. The topological phase is robust for weak shaking, in marked contrast to other schemes requiring complex engineered hoppings or larger driving amplitudes.

The role of resonance structure is emphasized: only under the two-phonon resonance does nontrivial topology arise; the single-phonon resonance regime remains topologically trivial due to the lack of band inversion or winding of $\mathbf{B}(k_x)$.

Floquet Haldane Model and Quantum Anomalous Hall States in Shaken Honeycomb Lattices

The 2D extension uses a honeycomb optical lattice configuration, periodically shaken in both $x$ and $y$ directions with a $\pi/2$ phase offset. This drive creates a circularly-polarized effective field, which explicitly breaks time-reversal symmetry. The induced vector potentials are mapped onto the underlying lattice structure using Peierls substitution and analyzed via Floquet theory.

Within the high-frequency regime ($\omega \gg$ bandwidth and mass gap), the effective Hamiltonian $$H_{\text{eff}}(\mathbf{k}) = \mathbf{B}(\mathbf{k})\cdot\boldsymbol{\sigma}$$ is constructed. The Floquet-engineered $B_z(\mathbf{k}) = M + D(\mathbf{k})$ term is crucial: $M$ is the controllable sublattice energy offset, while $D(\mathbf{k})$ arises from the drive and scales with shaking amplitude. When $D(\mathbf{k})$ exceeds $|M|$ at certain $\mathbf{k}$, $B_z$ changes sign between Dirac points, analogous to the staggered flux of the Haldane model. This results in a Chern insulating phase with chiral edge states at finite driving amplitude $k_rb \gtrsim 0.1$.

The quasi-energy spectrum displays chiral, edge-localized in-gap states in the topologically nontrivial regime. The phase diagram is delineated by $M/E_r$ and $k_rb$, with the topological region accessible for experimentally modest shaking amplitudes, ensuring minimal heating.

Experimental Feasibility and Practical Advantages

The proposed protocols stand out for their simplicity and realistic experimental requirements. Both the static Hamiltonian (pure, unmodified tight-binding with nearest-neighbor hopping) and the periodic drive (shaking via laser phase modulation) are experimentally standard and do not require complex laser-assisted tunneling, Raman couplings, or synthetic dimensions.

Because the drive amplitude needed to reach the topological regime is small, heating—a critical obstacle in Floquet-engineered systems—is strongly suppressed compared to alternative proposals. This positions the approach as highly promising for practical realization and exploration of Floquet topological phenomena in cold atomic gases.

Implications and Future Research Directions

The work establishes a minimal and experimentally accessible route to Floquet topological phases, directly connecting periodic driving to canonical models such as SSH and Haldane. The schemes pave the way toward quantum simulation of quantum anomalous Hall effects and detection of topological edge modes in ultracold atoms.

The results motivate several directions:

• Generalization to three-dimensional lattices and higher-order topological insulators via multidimensional shaking protocols
• Inclusion of interactions (e.g., using Fermi gases or bosonic mixtures), with potential for realizing Floquet-engineered topological superfluids or correlated insulators
• Exploration of driven-dissipative regimes and their connection to periodically engineered Floquet steady states
• Integration with high-resolution quantum gas microscopy for direct observation of Floquet edge modes and associated observables (e.g., Hall response)

Conclusion

Periodic shaking of optical lattices, as analyzed within this framework, enables robust and tunable realization of Floquet topological phases with minimal experimental overhead and low heating. The connection to prototypical models (SSH in 1D, Haldane in 2D) validates the use of periodic drives as a practical tool for quantum simulation of topological matter. These results substantially lower the threshold for experimental access to nontrivial Floquet topological states and set a foundation for further exploration of strongly correlated Floquet topological systems.