Engineering micromotion in Floquet prethermalization via space-time symmetries

TL;DR

The work develops a comprehensive framework for Floquet prethermalization under strong resonant driving by incorporating dynamical space-time symmetries and mapping them to the static symmetry group of the prethermal Hamiltonian via an interaction-picture, van Vleck approach. It shows that the prethermal Hamiltonian D_n inherits the extended symmetry G^{int}_{st} and that micromotion can be engineered and detected through time-resolved local observables within the prethermal window, with a concrete dual-driving spin-ladder model illustrating robust relations between observables at mT and mT+T/2. The results provide a principled route to design and diagnose nontrivial micromotion and symmetry-protected prethermal phenomena, potentially enabling space-time crystalline orders and quasiperiodic Floquet phases in many-body systems. The findings have experimental relevance for platforms such as superconducting qubits and cold atoms, where dynamical symmetries and prethermal dynamics can be controlled and probed with high fidelity.

Abstract

We present a systematic framework for Floquet prethermalization under strong resonant driving, emphasizing the pivotal role of dynamical space-time symmetries. Our approach demonstrates how dynamical space-time symmetries map onto the projective static symmetry group of the prethermal Hamiltonian governing the prethermal regime. We introduce techniques for detecting dynamical symmetries through the time evolution of local observables, facilitating a detailed analysis of micromotion within each period and surpassing the limitations of conventional stroboscopic Floquet prethermal dynamics. To implement this framework, we present a prethermal protocol that preserves order-two dynamical symmetry in a spin-ladder model, confirming the predicted relationships between the expectation values of local observables at distinct temporal points in the Floquet cycle, linked by this symmetry.

Figures (5)

• Figure 1: (a) Schematic illustrating the Floquet thermalization of a two-leg spin-$1/2$ ladder model with a Hamiltonian preserving the order-$2$ dynamical symmetry $\hat{g}_M$ in Eq. \ref{['eq:Definition of symmetries']}. The upper left and right legs are colored in orange and green, and the lower left and right legs are colored in blue and red. After relaxation at $t_{rel}$, the system enters a prethermal regime governed approximately by the static prethermal Hamiltonian in Eq. \ref{['eq:Rotation to the static Floquet Hamiltonian in the interaction picture']} until $t_{*}$ , with $t_0$ marking a reference time within this regime. Beyond $t_{*}$, heating drives the system to an infinite temperature state at $t_{th}$. (b) Time evolution of $\langle \hat{O}_{\rm odd}(t)\rangle$ with $e^{i\alpha_M}=-1$ at $t=mT$ and $t=mT+T/2$ in the prethermal regime verifies Eq. \ref{['eq:final expression comparing two values']}, detecting $\hat{g}_M$. Shaded regions denote the prethermal regime, with bipartite entanglement and energy density of the prethermal Hamiltonian consistent across a range of frequencies. Model parameters in Eq. \ref{['eq:H_0 and V']}: $\tau=0.25$, $J_1=\Omega$, $J^{\prime}=J/(0.5-\tau)=4$, $\lambda^{LR}_a=\lambda^{LR}_b=0.5$, $g_x=g_y=0.45225$, $g_z=0.7$, and $g_{zz}=1.3$ (see the SM SM for details).
• Figure 2: (a) Time evolution of $\hat{O}^{s}_{\rm odd}(mT)=\hat{O}_{\rm odd}(mT)+\hat{O}_{\rm odd}(mT+T/2)$ stabilizes at zero (red line) in the prethermal regime, confirming the presence of $\hat{g}_M$. (b) In the symmetry-broken case, $\hat{O}^{s}_{\rm odd}(mT)$ stabilizes at a non-zero value, indicating symmetry breaking. The red line marks the center of stable oscillations in the prethermal regime. (c) and (d) Energy density and bipartite entanglement entropy (insets) highlight the prethermal regime in shaded regions. The parameters used match Fig. \ref{['fig:micromotion in prethermalization']}, except for the symmetry-preserving $(\lambda^{LR}_a,\lambda^{LR}_b)=(1,1)$ and symmetry-broken $(\lambda^{LR}_a,\lambda^{LR}_b)=(0.8,1.2)$ cases.
• Figure S1: Schematic of one-dimensional spin-$1/2$ ladder of length $L$. $S$ spins reside in the upper chain and $\sigma$ spins in the lower chain. Along each chain, the spins interact via nearest neighbor couplings with strength $J_1\sim \Omega$, as described in $\hat{H}_{0,a}$ of Eq. \ref{['eq:SI four-step driven model']}. The symmetry operation $\hat{g}_M$ represents a mirror reflection about the centerline of the ladder. Here, the notation $*=L/2-1$ denotes the left sites of the upper and lower chains adjacent to the center bond of the ladder.
• Figure S2: After relaxation, the system enters a prethermal regime governed approximately by the effective prethermal Hamiltonian $\hat{D}_0$ in Eq. \ref{['eq:SI D_0']} until $t_{0}=n_0T$ ($n_0=O(\tilde{\Omega})$), with $t_0$ marking a reference temporal point within this regime. (a) Time evolution of $\langle \hat{O}_{\rm odd,2}(t)\rangle$ at $t=mT$ and $t=mT+T/2$ within the prethermal regime verifies Eq. \ref{['eq:SI expectation values of O at mT+T/2 (b)']} and detects $\hat{g}_M$. The shaded regions indicate the prethermal regime, with bipartite entanglement and energy density consistent across a range of frequencies ($J/\Omega=0.026,0.028$). (b) Time evolution of $\hat{O}^{s}_{\rm odd,2}(mT)=\hat{O}_{\rm odd,2}(mT)+\hat{O}_{\rm odd,2}(mT+T/2)$ stabilizes at zero (red horizontal line) in the prethermal regime, confirming the presence of $\hat{g}_M$. The normalized quantity $\hat{O}^{s;\rm norm}_{\rm odd,2}(mT)=\hat{O}^{s}_{\rm odd,2}(mT)/||\hat{O}_{\rm odd,2}(mT)||$, shown in the inset, distinguishes symmetry-preserving prethermal dynamics from thermalizing states approaching infinite temperature. The parameters used match Fig. 1 in the main text.
• Figure S3: In the symmetry-broken cases (a) $(\lambda^{LR}_a,\lambda^{LR}_b)=(0.8,1.2)$ and (b) $(\lambda^{LR}_a,\lambda^{LR}_b)=(0.5,1)$, $\hat{O}^{s}_{\rm odd}(mT)$ stabilizes at a non-zero value, signaling symmetry breaking. The red horizontal line denotes the center of stable oscillations within the prethermal regime. With increased symmetry breaking, the offset from zero becomes more pronounced. (c) and (d) Energy density and bipartite entanglement entropy (insets) indicate the prethermal regime, highlighted by shaded regions. The parameters used are consistent with Fig. 1, except for the symmetry-breaking values of $(\lambda^{LR}_a, \lambda^{LR}_b)$.