Designing an ideal optical quadratic potential for quantum control in optical lattices

Motional squeezed states in conventional single-frequency optical lattices rapidly lose coherence due to inherent anharmonic dephasing. We investigate a dual-frequency optical lattice, parameterized as $V(x)=-\frac{V_0}{1+\rho}[\cos^2(kx)+\rho\cos^2(nkx)]$, where $V_0$ is the lattice depth, $k$ is the laser spatial frequency, $\rho$ is the tunable potential-shape parameter, and $n$ is the spatial-harmonic order to effectively suppress this degradation and significantly extend state lifetimes compared to standard lattice potentials. Through numerical simulations of quantum quench protocols, we benchmark dual-frequency configurations for $n=2$ and $n=3$ against their conventional baselines, revealing clear and robust gains in squeezing preservation. We find that a double-quench (DQ) sequence further maximizes these advantages, with optimal lattice tunings shifting away from simple quartic-cancellation conditions due to higher-order anharmonicities probed by extended wavepackets. Ultimately, our analysis highlights the $n=2$ dual-frequency DQ protocol as a highly practical, resilient blueprint for high-fidelity continuous-variable quantum control of atoms.