[Paper Review] Observation of brane parity order in programmable optical lattices

David Wei, Daniel Adler+5 more|arXiv (Cornell University)|Jan 27, 2023

Cold Atom Physics and Bose-Einstein Condensates37 references4 citations

TL;DR

This study experimentally observes brane parity order in two-dimensional ultracold atomic gases within programmable optical lattices, using dynamically tunable lattices and site-blocking potentials to realize square, triangular, kagome, and Lieb lattices. The authors demonstrate that brane parity serves as a robust non-local order parameter for the superfluid-to-Mott insulator transition, with its geometry-dependent fluctuations quantitatively validated across lattice types, establishing it as a key probe of strongly correlated quantum phases in 2D systems.

ABSTRACT

The Mott-insulating phase of the two-dimensional (2d) Bose-Hubbard model is expected to be characterized by a non-local brane parity order. Parity order captures the presence of microscopic particle-hole fluctuations and entanglement, whose properties depend on the underlying lattice geometry. We realize 2d Bose-Hubbard models in dynamically tunable lattice geometries, using neutral atoms in a novel passively phase-stable tunable optical lattice in combination with programmable site-blocking potentials. We benchmark the performance of our system by single-particle quantum walks in the square, triangular, kagome and Lieb lattice. In the strongly correlated regime, we microscopically characterize the geometry dependence of the quantum fluctuations and experimentally validate the brane parity as a proxy for the non-local order parameter signaling the superfluid-to-Mott insulating phase transition.

Motivation & Objective

• To experimentally realize and probe non-local brane parity order in two-dimensional Mott insulating phases of the Bose-Hubbard model.
• To overcome limitations of fixed-lattice optical systems by developing a passively phase-stable, dynamically tunable optical lattice platform with programmable site-blocking potentials.
• To benchmark lattice quality and control fidelity using single-particle quantum walks across multiple lattice geometries (square, triangular, kagome, Lieb).
• To quantitatively measure the geometry dependence of quantum fluctuations and validate brane parity as a non-local order parameter for the superfluid-to-Mott insulator phase transition.
• To extract and converge the phase transition point across lattice types using finite-size scaling of the integer brane parity observable.

Proposed method

• Employing a passively phase-stable tunable optical lattice formed by retro-reflection of a bow-tie lattice beam (L2) and a controllable in-plane polarized beam (L1), enabling dynamic reconfiguration of lattice geometry.
• Using programmable site-blocking potentials to define unit cells and realize distinct lattice structures—square, triangular, kagome, and Lieb—within a single experimental setup.
• Performing single-particle quantum walks in each lattice geometry to benchmark system fidelity, coherence, and site-resolved control before entering the strongly correlated regime.
• Measuring the integer brane parity $ O_P $ on $ L imes L $ subregions of the atomic cloud to probe non-local quantum fluctuations and extract the phase transition point.
• Applying finite-size scaling of $ O_P $ across increasing $ L $ to estimate the critical $ (J/U)_{0} $, with linear fits to the sloped regime to locate the phase transition.
• Correcting for experimental biases via sign-flipping of $ O_P $ in odd-sized Lieb lattice subregions and accounting for inhomogeneous potentials and finite-temperature effects.

Experimental results

Research questions

• RQ1 Can brane parity order be experimentally observed in two-dimensional ultracold atomic systems with tunable lattice geometries?
• RQ2 How does the geometry of the underlying lattice (square, triangular, Lieb, kagome) influence the non-local quantum fluctuations captured by brane parity?
• RQ3 Is the brane parity a reliable non-local order parameter for the superfluid-to-Mott insulator phase transition in 2D Bose-Hubbard models?
• RQ4 To what extent do finite-size effects, inhomogeneous potentials, and thermal fluctuations affect the measurement and interpretation of brane parity?
• RQ5 Does the phase transition point extracted from brane parity scaling converge with increasing analysis size and match theoretical predictions?

Key findings

• The brane parity $ O_P $ exhibits perimeter-law scaling $ ext{log} O_P ightarrow -L $ in the Mott insulating phase, confirming its sensitivity to non-local quantum fluctuations.
• Significant sublattice-dependent variance in parity fluctuations is observed only in the Lieb lattice, with $ s_{ ext{hub}}^2 - s_{ ext{rim}}^2 $ deviating from zero, indicating geometry-specific quantum correlations.
• Finite-size scaling of $ O_P $ across $ L = 2 $ to $ 5 $ (and $ L = 2,4 $ for Lieb) shows convergence of the extracted phase transition point $ (J/U)_0 $, with values approaching those predicted by quantum Monte Carlo simulations.
• The phase transition point in the Lieb lattice is found to be $ (J/U)_0 ightarrow 0.03 $, consistent with theoretical expectations and indicating a critical scaling behavior.
• The brane parity shows a peak in variance difference near the phase transition, attributed to boundedness of the observable and suppression of fluctuations at high $ J/U $, consistent with inhomogeneous mean-field calculations.
• The method successfully realizes and benchmarks four distinct lattice geometries (square, triangular, kagome, Lieb) in a single setup, demonstrating high-fidelity, dynamically reconfigurable quantum simulation.