Two-Dimensional Momentum State Lattices

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Two-Dimensional Momentum State Lattices

Shraddha Agrawal, Sai Naga Manoj Paladugu, and Bryce Gadway

PRX Quantum 5, 010310 – Published 25 January 2024

Abstract

Building on the development of momentum state lattices (MSLs) over the past decade, we introduce a simple extension of this technique to higher dimensions. Based on the selective addressing of unique Bragg resonances in matter-wave systems, MSLs have enabled the realization of tight-binding models with tunable disorder, gauge fields, non-Hermiticity, and other features. Here, we examine and outline an experimental approach to building scalable and tunable tight-binding models in two dimensions describing the laser-driven dynamics of atoms in momentum space. Using numerical simulations, we highlight some of the simplest models and types of phenomena this system is well suited to address, including flat-band models with kinetic frustration and flux lattices supporting topological boundary states. Finally, we discuss many of the direct extensions to this model, including the introduction of disorder and non-Hermiticity, which will enable the exploration of new transport and localization phenomena in higher dimensions.

Received 1 August 2023
Accepted 18 December 2023

DOI:https://doi.org/10.1103/PRXQuantum.5.010310

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Atom optics Bose-Einstein condensates Synthetic gauge fields Topological materials

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Shraddha Agrawal , Sai Naga Manoj Paladugu , and Bryce Gadway ^*

Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-3080, USA

^*bgadway@illinois.edu

Popular Summary

Quantum simulation is the approach of studying a less accessible, difficult-to-realize system with another more controllable system. Over the years, quantum simulation has led to many insights into many-body physics, topology, and quantum dynamics. Platforms consisting of ultracold atoms have played an essential role in this field, where they have been used to realize various phenomena in condensed-matter physics, such as Mott insulator-superfluid transition, BEC-BCS crossover, quantum magnetism, and many more. An important direction in the quantum simulation of various topological systems is that of synthetic dimensions.

The idea behind synthetic dimensions is the reinterpretation of degrees of freedom, such as the hyperfine or electronic levels of atoms, as lattice sites. Tunneling between these different lattice “sites” is controlled by tuning the coupling between these sites. We present a proposal for the realization of two-dimensional momentum state lattices, where the momentum states of atoms are the lattice sites coupled via Bragg transitions. We outline the realization of various two-dimensional lattices, such as triangular, kagome, Lieb, decorated honeycomb, Hofstadter, and honeycomb. We further numerically investigate the kinetic frustration in the kagome lattice by looking at the frozen atomic dynamics. We study not only the dynamics of these lattices but also demonstrate the ability to probe the eigenspectrum of these lattices. To this end, we perform a spectroscopic study on the Lieb lattice and produce the famous double Hofstadter butterfly spectrum. In addition to the bulk spectrum of the Lieb lattice, we study the topological edge states by numerically realizing the Hofstadter model with a flux boundary.

We demonstrate through a wealth of example studies the versatility of synthetic lattices based on a laser-coupled atomic momentum states platform for exploring a range of lattice transport phenomena. Our work sets the stage for exciting studies on novel topological, disordered, and frustrated systems. These capabilities, combined with interactions, have the potential to explore band engineering, nonlinear phenomena, and many more novel lattice transport studies. Specific examples include chiral solitons in the Hofstadter model, flat-band dynamics controlled with interactions, and realizing fragile topological insulators.

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Vol. 5, Iss. 1 — January - March 2024

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Images

Figure 1
Two-dimensional momentum state lattices. (a) The layout of laser beams in real space, aligned in a plane at $120^{\circ}$ relative to one another. (b) The frequency spectra of the beams 1, 2, and 3 as laid out in panel (a). Cross-interference between spectral tones in beams 1 and 2 leads to Bragg transitions along the $A$ direction denoted by the solid horizontal arrows, interference between beams 2 and 3 leads to transitions along the $B$ direction denoted by the dashed arrows, and interference between beams 1 and 3 leads to transitions along the $C$ direction denoted by the dotted arrows. (c) The resulting set of momentum orders (black dots) connected via Bragg transitions from an initial zero momentum state (blue circle) is partially depicted, filling out an effective triangular lattice.
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Figure 2
Spectral engineering of nontriangular lattices. (a)–(f) Illustration of several example lattices that can be engineered by leaving off some resonant tones of the full triangular lattice. The appropriate removal of Bragg driving terms results in (a) honeycomb, (b) square or Hofstadter, (c) dice, (d) kagomé, (e) Lieb, and (f) decorated honeycomb lattices. Lattices (c) through (f) host flat bands. In the square lattice case of (b), specific arrangements of the hopping phases can result in a uniform flux piercing every lattice plaquette, transforming the square lattice into the Hofstadter lattice.
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Figure 3
Frozen dynamics in the Lieb lattice. (a) Band structure of the Lieb lattice tight-binding model. (b) Starting from the same diamond density pattern (equal population amplitudes at the depicted $A$ and $C$ sublattice sites of a given plaquette), selective projection onto either the flat band or dispersive bands can be achieved by controlling the relative phase structure of the wave function at the various site positions. For an equal phase at the populated sites ( $ϕ_{A} = ϕ_{C}$ ), dispersing bands are populated, whereas an alternating phase structure ( $ϕ_{A} = ϕ_{C} + π$ ) projects solely on to the flat band. (c) Numerical simulations of the dynamics for a hopping rate of $J / 2 π = 202.7$ Hz, starting from the four-site equal phase (orange) and alternating phase (blue) configurations. The dynamics of the standard deviation (in units of the unit-valued lattice site spacing) reveal the difference between frozen flat-band dynamics and population spreading for dispersing band projection.
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Figure 4
Loss spectroscopy of flat and topological energy bands. (a) Tunneling microscopy of the Lieb lattice, monitoring population loss from a weakly coupled probe site (blue, a spectator site) into the synthetic lattice. (b),(c) Probe loss spectra as a function of probe bias $E$ , resulting from the resonant weak tunneling of probe atoms into the available lattice eigenstates, shown for lattice flux values of $2 π / 3$ (top) and $π / 2$ (bottom) per lattice plaquette. The dip features at $E = 0$ relate to the enhanced density of states in the flat band. Features at higher and lower energy than the $E = 0$ dip relate to the topological Hofstadter bands [31]. $2 q$ loss features can be observed for flux values of $2 π / q$ , relating to the emergent Hofstadter sub-bands. (d) The celebrated Hofstadter butterfly spectrum was obtained from performing numerical simulations such as in (b) for various flux values through the lattice.
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Figure 5
Topological boundary states in Hofstadter lattice. (a) A sketch showing the “knight-move” flux boundary implemented in Hofstadter lattice. (b) Density plots of the population distribution after $t_{a} = 4.5$ in units of $ℏ /$ J following initialization at the flux boundary represented in (a). The top image relates to the case of $ϕ = 2 π / 3$ , whereas the bottom plot relates to $ϕ = 0$ (and hence no sharp flux boundary). (c) The anisotropy, $σ_{n} / σ_{m}$ , of the density distribution following dynamics is plotted as a function of flux through the lattice for various evolution times of 1.5, 3, 4.5, and 6 in units $ℏ /$ J. The spread is uniform for $ϕ = 0, π$ while being anisotropic for other flux values.
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