Dipole-Dipole Interactions of Floquet States

TL;DR

The paper develops a Floquet-Markov Lindblad framework for translationally cold two-level atoms driven by a strong monochromatic field and coupled to a common electromagnetic bath, revealing a modified dipole-dipole interaction $ abla$ that emerges from Floquet sidebands. In the two-atom case, the dipole interaction $\,\mathcal{H}_{\rm dp}$ is derived with sideband-resolved couplings, and the resulting interaction energy is encoded in $\tilde{\Omega}_{ij}(\omega)$, leading to a tunable, anisotropic spin interaction when mapped onto a dressed-state basis. In the weak-driving, RWA limit, this Floquet-dipole Hamiltonian reduces to an anisotropic Heisenberg model with couplings $J_{xx}^{ij}$, $J_{yy}^{ij}$, $J_{zz}^{ij}$ and $J_{xz}^{ij}$ controlled by drive parameters via $\theta_m$ and $\tilde{\Omega}_{ij}(\omega)$, connecting Floquet engineering with open-system spin dynamics. The work clarifies the regimes of validity for the secular approximation and the OBE vs FME descriptions, highlighting how driving-induced sidebands enable new resonant reservoir couplings and thereby richer, long-range spin-swap physics relevant for quantum simulation with driven atomic ensembles, especially in Rydberg platforms.

Abstract

We formulate a Floquet-Markov Lindblad master equation for translationally cold two-level atoms driven by a strong monochromatic wave and coupled to a common electromagnetic bath. The resulting dipole-dipole interaction reproduces the anisotropic Heisenberg model.

Figures (4)

• Figure 1: Illustration of the physical setting. In the unperturbed case on the left-hand side, tracing over the degrees of freedom of the reservoir gives rise to an effective flip-flop interaction between the atoms, which transfers excitations. On the right-hand side, the presence of a strong drive (represented by the antenna) acting on the atoms modifies the atomic structure, inducing sidebands which offer new resonant transitions for the reservoir to couple. This leads to a modified dipole-dipole interaction Hamiltonian $\mathcal{H}_\mathrm{dp}$, given in Eq. \ref{['eq:6']}.
• Figure 2: Two steps on the Floquet ladder separated by $(n-m)\omega$, revealing possible resonant couplings between the atoms. The resonant interactions between the transitions ${\rm I}-{\rm I}^\prime$, ${\rm I}-{\rm II}^\prime$ and ${\rm II}-{\rm II}^\prime$, ${\rm II}-{\rm I}^\prime$ (red) correspond to the first and second line in Eq. \ref{['eq:6']}, with coupling strength (8), whereas the resonant interactions between transitions ${\rm III}-{\rm III}^\prime$ (burgundy) and ${\rm IV}-{\rm IV}^\prime$ (gray) give rise to the third line in Eq. \ref{['eq:6']}, with strength (9).
• Figure 3: (a) Two adjacent steps on the Floquet ladder, depicting the four two-atom Floquet states with corresponding two-atom quasienergies $\mu_{1}=2\mu_{+}$, $\mu_{2}=\mu_{3}=0$, $\mu_{4}=-\mu_{1}$. On the left-hand side, the levels are sufficiently separated; the distances pertaining to the inverse time scale $\tau_{\mu}^{-1}=\min(d_{1},d_{2},d_{3})$ are indicated. On the right-hand side, a quasidegeneracy leading to $\Delta\epsilon-\Delta\epsilon'\approx0$ for some $\Delta\epsilon, \,\Delta\epsilon'$ is illustrated qualitatively. (b) Illustration of regimes for which discarding of oscillating terms $\exp(i(\Delta\epsilon-\Delta \epsilon') t)$ with $\Delta\epsilon\neq \Delta\epsilon'$ is justified. Equation \ref{['eq:6']} holds everywhere except in the dark shaded regions (where $\tau_\mu$ diverges), provided that the remaining separations of time scales, as outlined in the main text, persist.
• Figure 4: (a) Qualitative illustration of regimes of validity of OBE and FME in the RWA regime (hatched lines) (b) Separation of time scales for the exemplary values $\omega_{eg}\sim 10\, GHz$, $d_{eg}\sim 1000\, ea_0$, $r_{ij}\sim 40\, µm$, $\Omega_{\rm R}\sim 0.01\omega_{eg}$, $\delta=0$, $\vec{d}_{eg} \perp\vec{r}_{12}$, and $T_{\rm R}=0$ in the main text. We have $\tau_\omega=1/\omega$, $\tau_\mu \sim 1/\Omega_{\rm gen}=\tau_{\Omega_{\rm gen}}$, and $\tau_s\sim 1/\tilde{\Omega}_{ij}(\omega_{eg})$. As remarked, the separation allows one to choose $\tau_{cg}\ll 1/\Omega_{\rm gen}$ or $\tau_{cg}\gg 1/\Omega_{\rm gen}$, giving rise to the $N$-atom OBE and FME with distinct expressions for the dipolar interaction. However, this is fully consistent, as another coarse-graining of the flip-flop interaction results in the structure presented in Eq. \ref{['eq:spin']}.