Experimental engineering of Floquet topological phases in a one-dimensional optical lattice

Abstract

Periodic driving enables realization of topological phases without static counterparts. We experimentally realize and detect a one-dimensional anomalous Floquet topological phase in an optical lattice, using multi-frequency control to manipulate the sign configuration of the gap windings $(W_0,W_π)$ associated with the $0$ and $π$ quasienergy gaps. We develop a lattice-depth modulation scheme that induces staggered nearest-neighbor $s$-$p$ orbital couplings and realize a minimal nontrivial Floquet topology under single-tone driving. Introducing a second tone, the relative phase provides a physical control knob that sets the effective coupling signs in the two gaps, such that the corresponding windings can be tuned to add or cancel. Aligned windings yield high-winding phases, whereas opposing windings cancel the net Floquet-band invariant while retaining nontrivial gap indices. We read out $(W_0,W_π)$ with a band-inversion-surface (BIS)-resolved Ramsey protocol assisted by lattice position shaking, which measures relative Floquet phases on the BISs. Controlled quenches further confirm phase-dependent band modifications even at quasimomenta far from resonance. These results establish multi-frequency control with a tunable relative phase as a quantitative route to engineering anomalous Floquet topology, and demonstrate phase-coherent coexistence of distinct drive modalities.

Figures (7)

• Figure 1: Topology from lattice-depth modulation (LDM) and the corresponding Floquet band structures for single- and two-tone driving. (a) A gauge transformation $w_p(x)\rightarrow(-1)^j w_p(x)$ renders the $p$-orbital hopping spatially uniform while keeping the $s$-$p$ coupling staggered. (b) Nearest-neighbor overlap coefficients $\nu_1^{\alpha\beta}$ for the $s$-$s$, $p$-$p$, and $p$-$s$ channels. The adjacent $s$ and $p$ Wannier overlap remains sizable up to $V_0=10\,E_r$. (c) Experimental measurements for single-tone driving at $\omega_0$ with $\delta V_0=0.875\,E_r$, and for two-tone driving at $\omega_0$ and $2\omega_0$ with $\delta V_0^{(1)}=\delta V_0^{(2)}=0.875\,E_r$, both at lattice depth $V_0=3.5\,E_r$. All data are extracted from time-of-flight (TOF) images taken after 30 ms of expansion. Each slice is averaged over five shots. (d,e) Numerical Floquet bands for single-tone driving with $\omega_0=5\,E_r/\hbar$ and $\delta V_0=0.5\,E_r$, and for two-tone driving at $\omega_0=4\,E_r/\hbar$ and $2\omega_0$ with $\delta V_0^{(1)}=0.5\,E_r$ and $\delta V_0^{(2)}=1\,E_r$. Simulations use $V_0=4.5\,E_r$ to enhance the visibility of the two pairs of edge states.
• Figure 2: Ramsey detection of the single-photon $s$-$p$ resonance induced by single-tone lattice-depth modulation (LDM) at the BIS momenta. (a) The Ramsey sequence uses a lattice-position shaking (LPS) preparation pulse with stroboscopic form $H_F^{(s)}(k,\phi_s)$ and duration $t_1$, followed by a dark evolution of duration $\tau_d$ under the static Hamiltonian $H_0$, and a LDM readout described by $H_F^{(j)}(k,\phi_0=0)$ with duration $t_2$. All pulses are set close to $\pi/2$ to maximize fringe contrast. (b) Atomic momentum space density during the sequence in experiment. (c-d) Ramsey fringes of the spin imbalance at the two BIS $k=k_R$ (yellow dots) and $k=k_L$ (blue dots) for the LPS--LDM sequence as $\tau_d$ is varied. The preparation phase choices are $\phi^{(s)}=0$ and $\pi$. A robust $\pi$ phase shift between $k_R$ and $k_L$ is observed. (e) LPS--LPS sequence showing no phase shift between the two BIS fringes. (f) In the LPS--LDM sequence, the LPS preparation phase $\phi_s$ sets the global fringe phase, while the relative $\pi$ contrast between $k_R$ and $k_L$ is preserved. The fitting function is $\tilde{n}_{f,z}(k)=\sin\!(\tfrac{2\pi}{\tilde{T}} t+\tilde{\phi})\exp(-\tilde{\lambda} t)+\tilde{c}$, where tildes denote fitted parameters. The experimental parameters are $\delta D=22.3\,\mathrm{nm}$ for the LPS with $t_1=2T$ and $\delta V_0=1.2\,E_r$ for the LDM with $t_2=4T$. The drive period is $T=2\pi/\omega_0$ with $\omega_0=2\pi\times 8\,\mathrm{kHz}$ and $V_0=3.5\,E_r$ for static lattice.
• Figure 3: Ramsey interferometry with two-tone driving $(\omega_0,2\omega_0)$. (a-b) State preparation uses a two-tone LPS pulse, which launches coherent evolution governed by the Bloch equation of motion. Readout uses a two-tone LDM pulse with matching frequency components so that both BIS pairs $(k_{j,L},k_{j,R})$ for $j=1,2$ are addressed. Here $\tau_{1,2,3}$ label data taken at $\tau_d = 20\,\mu\mathrm{s}$, $80\,\mu\mathrm{s}$, and $160\,\mu\mathrm{s}$, respectively. (c) A characteristic $\pi$ phase contrast between the left and right BIS momenta in either the $0$ gap or the $\pi$ gap indicates opposite orientations of the effective Bloch field, consistent with $h_{F,y}(-k)=-h_{F,y}(k)$ on the BIS. (d) Zero-delay Ramsey readout for two settings of the interferometric pulse phase, $\phi_0^{(2)}=0$ and $\pi$. Toggling $\phi_0^{(2)}$ produces a global $\pi$ shift of the interferometric phase and reverses the relative phase between the $j=1$ and $j=2$ BIS pairs. The experimental parameters are $\delta V_0^{(1)}=\delta V_0^{(2)}=0.98\,E_r$ for LDM. For the preparation LPS pulse, $\delta D^{(1)}=\delta D^{(2)}=22.3\,\mathrm{nm}$ and $\phi_s^{(1)}=\phi_s^{(2)}=\phi_s$, with a duration of $250\,\mu\mathrm{s}$. The modulation frequencies are $\omega_0$ and $2\omega_0$, where $\omega_0=2\pi\times 4\,{\rm kHz}$. The static lattice depth is $V_0=3.5\,E_r$.
• Figure 4: Measurement of the dynamics at the topological charge momentum $k_c$, defined by $h_{F,y}(k_c)=0$. (a) For $\phi_0^{(2)}=0$, the total winding number of the Floquet bands is $C=2$. A topological charge occurs at $k_c\approx 0.4\pi$, which suppresses the transverse drive and results in nearly frozen dynamics in (e). (b) For $\phi_0^{(2)}=\pi$, the total winding number is $C=0$. No charge lies between the BIS momentums, so $\lvert h_{F,y}(k_c)\rvert$ remains finite and damped $s$-$p$ oscillations are observed in (d). Dashed lines are guides to the eye. The color scale and the experimental LDM-pulse parameters are the same as in Fig. \ref{['fig3']}. The calculated effective Floquet Hamiltonian components $h_{F,y}(k)$ and $h_{F,z}(k)$ are obtained using the same parameters.
• Figure S1: Schematic illustration of the relevant resonant processes for single- and two-tone driving with multiple coherent channels. For two-tone driving, the single-photon resonance at the $\mathrm{BIS}_1$ is addressed by the first-order process $\delta V^{(1)}$ at frequency $\omega_0$, as well as by second-order processes involving combinations such as $\delta V^{(1)}J_1(\beta^{(2)})$ and $\delta V^{(2)}J_1(\beta^{(1)})$. The two-photon resonance at $\mathrm{BIS}_2$ is primarily driven by the first-order process $\delta V^{(2)}$ at $2\omega_0$, with additional contributions from higher-order processes involving $\delta V^{(1)}$.