Breakdown of Disorder-Suppressed Floquet Heating under Two-Frequency Driving

Abstract

Periodic (Floquet) driving enables Hamiltonian engineering and nonequilibrium phases, but interacting systems eventually heat by absorbing energy from the drive. Disorder can greatly delay this process, yielding long-lived prethermal plateaus. Here we show that this protection can fail when pulse-train control introduces a second driving frequency and when the disorder fluctuates. Using a natural-abundance 13C nuclear-spin network in diamond, we observe sharp peaks in the late-time heating rate at the double- and triple-spin-flip resonance conditions predicted by bimodal Floquet interference, and track their evolution with drive frequency. A switching-noise model attributes the resonant absorption to stochastic electron-spin dynamics that intermittently tune rare nuclear clusters into multi-photon resonance. Our results reveal a resonance-activated limit for disorder-stabilized Floquet phases and suggest new routes to DC-field quantum sensing based on an abrupt breakdown of prethermalization.

Figures (10)

• Figure 1: (a) Floquet driving sequence: initial $\vartheta_y$ pulse followed by train of detuned $\vartheta_x$ pulses (flip angle $\vartheta=\omega_{1}\tau_{p}$, detuning $\delta\omega$, inter-pulse spacing $\tau_s$) with period $T$. During pulse (i) $H_{\rm drive}=\omega_{1} I_x+\delta\omega I_z$; during delay (ii) $H_{\rm drive}=\delta\omega I_z$. (b) Net rotation over one period defining $\hat{n}_{\rm eff}$ and $\omega_{\rm eff}$. (c) Schematic of positionally disordered $^{13}$C dipolar network coupled to randomly distributed electron spins (red), illustrating triple-resonance condition ($3\omega_{\rm eff}=\omega_d$) leading to triple-spin-flip. (d) Prethermal magnetization vs. $\delta\omega$: (i) definition of $M_{\rm pre}=\sqrt{\langle I_x\rangle^{2}+\langle I_y\rangle^{2}}$, where $\langle I_\mu\rangle={\rm Tr}[I_\mu\rho_{\rm pre}]$; (ii) numerics for random 10-spin cluster on diamond lattice at 1.1%, 5%, and 100% $^{13}$C occupancy, showing resonances near 2.5 kHz ($k=3$) and 4.9 kHz ($k=2$). (e) Average Pauli weight of prethermal state, with resonance-enhanced operator spreading; insets show schematic operator support far from (i) and near (ii) resonance. Simulations performed with $\tau_{p}=56$$\mu$s, $T=92$$\mu$s and $\omega_{1}=4.46$ kHz.
• Figure 2: (a) Experimental prethermal magnetization $M_{\rm pre}$ vs. detuning $\delta\omega$ obtained by integrating signal from 10--20 ms. Feature near 4.9 kHz corresponds to double-spin-flip resonance; transient effects are small due to weak dipolar network. (b) Long-time heating rate vs. $\delta\omega$ (obtained from fits to decay, see SI Sec. \ref{['sec:SI_fitting']}), showing pronounced peaks at triple- and double-spin-flip resonances. (c) Representative decays of $M_{\rm pre}(t)$ away from resonance ($\delta\omega=0$) and near double-spin-flip resonance ($\delta\omega=4.9$ kHz); dashed lines are fits used to extract heating rates (see SI Sec. \ref{['sec:SI_fitting']}). Data taken with $\tau_{p}=56$$\mu$s, $T=92$$\mu$s and $\omega_{1}=4.46$ kHz.
• Figure 3: (a) Heating rate vs. detuning near double-spin-flip resonance measured for $\tau_p=32-56$$\mu$s and fixed inter-pulse spacing of 36 $\mu$s and $\omega_{1}=4.46$ kHz; solid lines are Lorentzian fits. (b) Resonance locations extracted from fits for double- (red) and triple- (blue) spin-flip resonances; black line: theory (see SI Sec. \ref{['sec:SI_Res']}). (c) Resonant contribution to heating rate vs. $\omega_d$, compared with expected scaling $\propto\omega_d^{-2}$ (black lines). (d) Schematic of stochastic electron-state switching that modulates resonance condition and dephases higher-order correlations (top: example electron trajectory; bottom: three-spin cluster at resonance).
• Figure S1: Semiclassical Monte Carlo simulations of nuclear polarization decay rates as a function of detuning $\delta\omega$. The model reproduces the overall decrease of heating with increasing detuning. A weak reduction near $\delta\omega\approx2.3$ kHz is attributed to suppressed spin diffusion when strongly relaxing nuclei near paramagnetic impurities become dynamically decoupled from their neighbors. Sharp resonance peaks observed experimentally are not captured, as they arise from many-body double- and triple-spin-flip processes beyond first-order Floquet theory and beyond the scope of the semiclassical model.
• Figure S2: Numerically simulated heating rates are shown for a minimal model comprising three nuclei and one electron under detuned pulsed spin-locking. The results represent an average over 20 randomly generated three-spin clusters confined within a 10 nm sphere sampled from the diamond lattice. For each configuration, the heating rates are further averaged over electron positions distributed on a 10 point Lebedev grid located 40 nm from the center of the three-spin cluster. The pulse sequence parameters are chosen in agreement with experimental details: $\vartheta_{x}=\pi/2$, $\tau_{p}=56\;\mu{\rm s}$, and $T=92\;\mu{\rm s}$, $T_{1e}=50$ ms.