Control of Dipolar Dynamics by Geometrical Programming

TL;DR

The paper addresses motional dephasing and limited entanglement in dipolar molecular tweezer arrays by introducing geometry as a programmable control knob. It develops static geometries to suppress thermally induced dephasing and dynamic geometry protocols, including a geometry-echo sequence, to cancel motion-induced phases during entanglement. Through classical expansions, quantum-matrix analyses, and DTWA simulations, it shows that dynamic rearrangement during evolution can yield robust multiparticle entanglement and enhanced spin squeezing, approaching all-to-all scaling with appropriate sequencing. Realistic parameters for CaF, NaCs, RbCs, and KRb suggest feasible rearrangement speeds and dipolar strengths at $r=2~\mu\mathrm{m}$, making geometry-controlled dipolar dynamics a practical route for programmable quantum simulation and metrology with molecular systems.

Abstract

We propose and theoretically analyze methods for quantum many-body control through geometric reshaping of molecular tweezer arrays. Dynamic rearrangement during entanglement is readily available due to the extended coherence times of molecular rotational qubits. We show how motional dephasing can be suppressed and enhanced spin squeezing can be achieved in an actively rearranged short-range XY model. We also analyze in detail a specific static geometry that significantly suppresses decoherence. These general methods as applied to programmable quantum systems offer robust control modalities that are well suited to molecules.

Figures (8)

• Figure 1: Angular dependence of dipolar interactions in the presence of thermal motion. (a) Molecules trapped in optical tweezers (red) have thermally broadened spatial distributions (blue). The dipole alignment is set by the external magnetic field $\mathbf{B}$, forming an angle $\theta$ with the interparticle displacement vector $\vec{r}$ in the $x$-$y$ plane. (b) Distribution of dipolar couplings $J$ for various field angles $\theta$, showing minimal width at the magic angle, where leading-order sensitivity to axial motion vanishes. (c) Quality factor $Q = \tau |J|$ (normalized to its value at $0^\circ$) versus $\theta$ for different averaged axial thermal occupations $\bar{n}_z$. (d) Quality factor $Q$ versus $\bar{n}_z$ for different $\theta$. Smaller thermal extent leads to stronger suppression of interaction noise and a sharper optimum near the magic angle. We assume a thermal distribution over motional states $\{n_x, n_y, n_z\}$, corresponding to the same temperature along all three directions. Simulations are performed for CaF molecules trapped in a tweezer array with axial (radial) trap frequencies of 20 kHz (100 kHz) and an interparticle spacing of $r = 2\,\mu\mathrm{m}$. See the supplementary material SM for simulation details.
• Figure 2: Dynamical control of dipolar Hamiltonians via geometrical programming. Time-dependent control over both molecular positions (black dashed arrows) and dipole orientations via the external field $\hat{\mathbf{B}}(t)$ (purple arrows) enables reconfiguration of the dipolar interaction graph, at timescales much faster than both dipolar interaction and coherence times. This capability allows dynamic tuning of the anisotropic dipolar couplings, $J_{ij}(t) \propto \left(1 - 3\cos^2\theta_{ij}(t)\right)/r_{ij}^3(t)$ (blue bonds), during coherent many-body evolution.
• Figure 3: Suppression of leading-order thermal dephasing via geometry echo in a square array. (a) Schematic of a four-step "bang-bang" protocol that cyclically reconfigures the geometry of a square array by rearranging half of the molecules (marked with dashed circles) with relative displacement $\vec{r}_i$. The displacement vectors $\vec{r}_0, \vec{r}_1, \vec{r}_2, \vec{r}_3$ satisfy symmetry constraints (as illustrated) to preserve the spatial symmetry of the square array over a full cycle. Dipole orientations are rotated at each step via control of the external magnetic field $\vec{B}$. (b) Effective dipolar interaction $J_\mathrm{eff}$ (averaged over a full echo cycle) as a function of relative displacement $\vec{r}_0=(x_0, y_0)$ used in the first step. The dashed black contour indicates configurations for which the sum of geometric sensitivities, as defined in Eq. \ref{['eq:axial_sensitivity']}, from all four nearest neighbors vanishes, thereby canceling leading-order dephasing. Selecting a representative point $\vec{r}_0$ on this contour and applying the protocol in (a) enables robust decoupling of thermal motion from nearest-neighbor dipolar interactions in square arrays while maintaining a non-zero effective interaction strength $J_{\mathrm{eff}}$.
• Figure 4: Enhanced spin squeezing with dynamic rearrangement (a) Illustration of a rearrangement protocol. (b) Squeezing parameter $\xi^2$ calculated from DTWA plotted vs. time for $N = 20$ spins. Colors indicate an increasing number of repetitions of the move shown in (a). (c) Optimal squeezing for different system sizes and "move" steps. The dashed black line shows the expected scaling for all-to-all interactions. The colored points show DTWA simulations, with darker blue circles show increasing number of moves.
• Figure S1: Angular dependence of the sensitivity coefficients $A_{i,n}$, which quantify how thermal motion along the $i$ axis contributes to fluctuations in the dipolar coupling strength $J$. Notably, $A_{z,2}$ vanishes near $\theta \approx 63.4^\circ$, indicating suppressed sensitivity to motion along $\hat{z}$ at that angle. This is distinct from $\theta \approx 54.7^\circ$, where the dipolar interaction $J_{dd} \propto 1 - 3\cos^2 \theta$ vanishes, indicated by the dashed vertical lines.