Generalized pulse design in Floquet engineering: Application to interacting spin systems

TL;DR

This work advances Floquet engineering by relaxing rigid, weak-pulse assumptions and introducing a non-stroboscopic high-frequency expansion that accommodates arbitrary, experimentally implementable pulse shapes. By explicitly deriving an effective Hamiltonian framework with a tunable kick operator and nested commutator structure, the authors show how to tailor $H_{eff}$ from a long-range XXZ spin system toward Ising-like dynamics, while controlling errors via the small parameter $\omega^{-1}$. They demonstrate this with concrete pulse designs (cosine and square) that yield analytic conditions—via functions like $J_0$ and $\sin(z)/z$—for converting $H_{XXZ}$ into $H_{ZZ}$, and discuss feasibility domains in terms of the anisotropy ratio $s=J_z/J_\perp$. The approach broadens the accessible effective Hamiltonian landscape for quantum simulators, enabling more flexible realization of weighted graph states and related spin models in platforms such as Rydberg atoms and polar molecules.

Abstract

Floquet engineering in quantum simulation employs externally applied high-frequency pulses to dynamically design steady-state effective Hamiltonians. Such protocols can be used to enlarge the space of Hamiltonians but approximations often limit pulse profile shapes and therefore the space of available effective Hamiltonians. We consider a nonstroboscopic high-frequency expansion formalism for Floquet engineering. We generalize the pulse profiles available by rigorously keeping all necessary terms to lowest order in inverse frequency expansions used to derive the effective Hamiltonians. Our approach allows wide tunability in application of external driving fields. We apply our method to long-range interacting XXZ spin Hamiltonians. We model an example application where we derive conditions on specific pulse shapes to engineer effective Ising models from XXZ models. Our method allows the space of continuous pulse profiles, relevant to experimental control fields, to better and more accurately explore possible effective Hamiltonians available for Floquet engineering.

Figures (4)

• Figure 1: Schematic time-dependence of example pulse profiles, $\hat{V}(t)$ with cycle time $2T$. Colored bars and lines indicate pulsed application of Pauli operators, specifically application of $h_{j}^{\alpha}(t) \hat{\sigma}_{j}^{(\alpha)}$ in Eq. \ref{['eq:perturbingPotential']} such that $\hat{\sigma}^{(1)},\hat{\sigma}^{(2)},$ and $\hat{\sigma}^{(3)}$ denote the usual Pauli $x,y,$ and $z$ matrices, respectively. $T$ indicates the duration of a repeated pulse subcycle. (a) A conventional rectangular pulse profile where $\tau_w$ labels the pulse width. (b) The same as (a) but with pulses designed to replicate a continuous pulse profile with discrete pulses. (c) A general continuous pulse profile that can be accurately implemented with our approach.
• Figure 2: Example of a square wave pulse with cycle time $2T$. Top: The solid line shows $g_1(t)$ in the first subcycle: the global pulse profile for application of a $\sigma^{(1)}$ Pauli matrix. The dashed line shows the integral of the solid line, $G_1(t)$, defined by Eq. \ref{['eq:Gfunction']}. Bottom: The same as the top but for a second subcycle pulsing the $\sigma^{(2)}$ Pauli matrix.
• Figure 3: A schematic depicting the expansion procedure used to derive Eqs. \ref{['eq:lowestKick']} and \ref{['eq:effectiveEqn']}. The first row ($n=0$) in the central rectangle shows that Eq. \ref{['eq:effectiveEqn']} retains all nested commutators to lowest order in inverse frequency, $\omega^0$. The rectangle on the right shows that the lowest order kick operator must also be kept. The remaining rows ($n>0$) are truncated in our formalism. The dashed line depicts an example $\mathcal{O}(\omega^{-1})$ correction to highlight the need for higher order expansions of the kick operator, $\hat{K}^{(1)}$, when retaining $n=1$ terms.
• Figure 4: The solid lines plot the surfaces of solutions for $J_z/J_{\perp}$ obtained from Eq. \ref{['eq:IsingCondition']} plotted against the pulse strength, $v$. The pulse is assumed to be a cosine shape, Eq. \ref{['eq:cosinepulse']}. In the top, middle, and bottom panels the dashed lines plot the truncated approximations terminating at $p_{max} = 1$, $p_{max} = 8$, and $p_{max} = 16$, respectively. The vertical dashed lines indicate a pole in the truncated expansion. As expected, when $v \rightarrow 0$, lower order approximations are good fits to the solution surface. On the other hand, every truncated solution incorrectly predicts the existence of values of $v$ for $s > 0$ (i.e., $J_z$ and $J_{\perp}$ having the same sign) which are pushed further and further out to $v \rightarrow \infty$. To determine all admissible values of $v$ which effectively engineer an Ising-type interaction from a given form of $H_{XXZ}$, we must keep all terms in the expansion.