Pre-Floquet states facilitating coherent subharmonic response of periodically driven many-body systems

TL;DR

This work addresses how long-lived subharmonic responses can persist in periodically driven many-body quantum systems. It develops a semiclassical framework that combines Floquet theory with EBK-type quantization by introducing pre-Floquet states, which are tied to almost-integrable mean-field resonance islands and connected to true Floquet states through dynamical tunneling. The driven Bose-Hubbard dimer serves as a concrete model, showing robust $1:3$ subharmonic clocking and illustrating how higher-order subharmonics may be engineered by accessing additional resonance islands, with the subharmonic behavior controlled by the effective Planck constant $\hbar_{\rm eff} \sim 2/N$. The results illuminate the quantum-classical correspondence in driven many-body systems, suggesting a route to controllable subharmonic dynamics in Floquet condensates and outlining experimental pathways and fundamental limitations due to chaos and finite particle number.

Abstract

We demonstrate longtime coherent subharmonic motion of a many-boson system subjected to an external time-periodic driving force. The underlying mechanism is exemplified numerically through analysis of a periodically driven Bose-Hubbard dimer, and clarified conceptually by semiclassical requantization of invariant tubes pertaining to the system's mean-field description. In this way, one arrives at pre-Floquet states that relate to the actual many-body Floquet states in a manner similar to the relation of site-localized Wannier states to lattice-extended Bloch states in solid-state physics. It is argued that even high-order subharmonic response can be systematically engineered, and be observed experimentally, with weakly interacting Floquet condensates comprising a sufficiently large number of particles.

Figures (9)

• Figure 1: Return probability $P_r(t) = \left| \langle \psi(0) | \psi(t) \right|^2$ for a state $|\psi(t)\rangle$ of the periodically driven Bose-Hubbard dimer (\ref{['eq:DBH']}) with $N = 2000$ particles. Here, the time $t$ is scaled with respect to the cycle duration $T = 2\pi/\omega$, revealing $1:3$ subharmonic clocking. Dimensionless system parameters are $N\kappa/\Omega = 0.92$, $\mu/\Omega = 0.4$, and $\omega/\Omega = 1.9$.
• Figure 2: Poincaré map generated at time $t = 0$ by the peri-odically driven pendulum (\ref{['eq:HMF']}), providing the mean-field description of the many-body system (\ref{['eq:DBH']}), with parameters $\alpha = 0.92$, $\mu/\Omega = 0.40$, and $\omega/\Omega = 1.90$.
• Figure 3: Color-coded Husimi projections (\ref{['eq:HPN']}) of eight Floquet states $|\psi\rangle = |u(0)\rangle$ for $N = 10\,000$ particles, intersected at time $t=0$, onto the surface of section shown in Fig. \ref{['F_2']}, here depicted in black and white. Observe that each of these states is localized on its respective closed contour $\gamma_1$ selected by the upper of the conditions (\ref{['eq:SCS']}). Their semiclassical quantum numbers are $n = 0$, $109$, $193$, $275$, $767$, $971$, $1414$, and $1672$ (inner to outer).
• Figure 4: Phase-space geometry pertaining to a hypothetical $1:2$ resonance (schematically). The central $T$-periodic mean-field tube provides a proper approximate $N$-particle Floquet state upon semiclassical "requantization." The two $2T$-periodic tubes winding around it yield two $2T$-periodic pre-Floquet states. Taking even and odd superpositions of these, thus accounting for dynamical tunneling between them, gives two approximate Floquet states.
• Figure 5: (a) Projection of a tube obtained by following an invariant contour surrounding the central elliptic fixed point of the main regular island depicted in Fig. \ref{['F_2']} in time. Such tubes are $T$-periodic, providing $T$-periodic Floquet states upon semiclassical quantization. (b) Projection of a tube generated by following a contour surrounding the central elliptic fixed point of the lowest secondary island observed in Fig. \ref{['F_2']} in time. Such tubes are $3T$-periodic, and therefore provide $3T$-periodic pre-Floquet states that effectuate the $1:3$ subharmonic clocking recognized in Fig. \ref{['F_1']}.