Generation of Ultrahigh Anomalous Hall Conductivities via Optimally Prepared Topological Floquet States

TL;DR

The paper addresses enabling high-fidelity preparation of topological Floquet states in a 2D quantum well by shaping Floquet drive ramps with quantum optimal control. It shows monotonic ramps fail near topological gap closings, while optimally designed oscillatory ramps act as non-adiabatic topological pumps, achieving fidelities above 0.99 and generating time-averaged anomalous Hall conductivities far exceeding equilibrium expectations. The main contribution is the demonstration that the preparation path strongly controls transport observables, revealing a breakdown of the Floquet TKNN bound under non-equilibrium conditions and opening routes to ultrafast topological devices. The work suggests experimental realizations in graphene-like materials and cold-atom platforms and points to future directions including dissipation-enabled fidelity improvements and direct optimization for transport outcomes.

Abstract

Ultrafast quantum matter experiments have validated predictions from Floquet theory - notably, the dynamical modification of the electronic band structure and the light-induced anomalous Hall effect, via monotonic modulation of the driving amplitude. Here, we demonstrate how new physics is uncovered by leveraging quantum optimal control techniques to design Floquet amplitude modulation profiles. We discover a fundamentally different regime of topological transport, whereby the optimal oscillatory preparation protocol functions as a non-adiabatic topological pump: as a result, ultrahigh time-averaged anomalous Hall conductivities emerge, that reach up to around seventy times the values one would expect from the Chern number of the targeted Floquet state. The optimal protocols achieve >99% fidelity at the topological energy gap closing point - a twenty-fold improvement over standard monotonic approaches in as little as ten Floquet cycles - while unexpectedly generating the predicted ultrahigh conductivities. Our findings demonstrate that optimally prepared non-equilibrium quantum states can access transport regimes not achievable in the corresponding equilibrium system or even by applying conventional Floquet approaches, opening new avenues for ultrafast quantum technologies and topological device applications.

Figures (9)

• Figure 1: Qualitative overview of key findings in this paper. [Top] Demonstration of a topological transition induced by varying the amplitude of a Floquet drive in 2D quantum matter, as evidenced by the change in Chern number across the (quasi)energy gap closing point. [Bottom Left] Amplitude modulation from zero through the critical amplitude up to a targeted value monotonically. As one expects, monotonic passage produces diabatic errors, resulting in a poor Floquet state preparation fidelity at the topological energy gap closing point. The post-ramp time-averaged anomalous Hall conductivity is still consistent with the Floquet Chern number of the lower targeted band. [Bottom Right] The optimal oscillatory amplitude modulation results in high Floquet state preparation fidelities at the topological energy gap closing point via diabatic error cancellation. Most importantly, the oscillatory protocol functions as a non-adiabatic topological pump in 2D that results in ultrahigh post-ramp time-averaged anomalous Hall conductivities.
• Figure 2: Quasienergy band structures of the Floquet driven 2D quantum well. The Floquet photon energy is $\hbar \Omega = 4$ and each panel is the result for a different driving amplitude $V_0$. (a) $V_0 = 0$ corresponds to the limiting case of the static system, with parameters $A = -0.1$, $B = -0.1$, and $M = 0.1$. The computed Chern number of the lower band is $C=0$, indicating that the system is topologically trivial. (b) $V_0 = V_c = 0.31$ is the critical amplitude that leads to the quasienergy gap closing at a crystal momentum of $\boldsymbol k = \Gamma$. (c) $V_0 = V_T = 0.41$ is the driving amplitude of the Floquet state that we attempt to target with subsequent exact quantum dynamics. The quasienergy gap is re-opened with the characteristic appearance of band inversion. The computed Chern number of the lower band is now $C=1$, indicating that a topological transition has been induced.
• Figure 3: Fidelity as a function of crystal momentum. A linear ramp of 10 Floquet driving cycles in duration is used [Eq. \ref{['polynomial_ramp']}, with $\mathcal{P} = 1$]. As expected, the fidelity is poor at $\boldsymbol k = \Gamma$, where the topological gap closing point occurs in the quasienergy band structures from the previous figure.
• Figure 4: Fidelity improvement by parameter optimization of a simple oscillating ramp [Eq. \ref{['simple_oscillating_ramp']}]. (a) Fidelity at $\boldsymbol k = \Gamma$ as a function of the number of Floquet ramping cycles $N_R$ and the crossing number $\mathcal{C}$ that counts the number of times the amplitude is brought through the critical value $V_0 = V_c$. (b) The absolute maximum fidelity ($> 0.99$) in panel (a) is achieved for $N^*_R = 35$ and $\mathcal{C}^* = 69$. We plot the fidelity as a function of the entire $k$-space for this parameter combination, which now oscillates significantly compared to the previous figure, despite the significant improvement at $\boldsymbol k = \Gamma$.
• Figure 5: Fidelity results for optimally designed ramps. (a) Ramping profiles optimized by QOC with the requirement that the fidelity at $\boldsymbol k = \Gamma$ be improved to better than 0.99. The maximum frequency component in these solutions is considerably lower than that of the optimized simple oscillating ramp in the previous figure. (b) Corresponding fidelity map in the $k$-space for the 10 Floquet ramping cycle optimized ramp. (c) Same as panel (b), except for 20 Floquet ramping cycles. The fidelity is now a much smoother function of $\boldsymbol k$ overall.