Mechanisms of localization in a finite harmonically confined optical superlattice

TL;DR

This work addresses localization in finite optical superlattices under harmonic confinement, revealing three distinct mechanisms: quasi-classical localization at high trap strength, topological edge-state localization in the topological configuration at low to intermediate trap strengths, and a novel intermediate regime where intra-band avoided crossings create an effective four-level system localized at central sites. The authors combine exact diagonalization with a tight-binding mapping to an extended SSH/eSSH model, and they elucidate the role of Zak phase $\gamma_{Zak}$ in defining topology, along with the impact of boundary extensions and trap center alignment. A key finding is the universality of the intermediate four-level localization across OS parameters ($V_{high}$, $u$, $M$), including cases without boundary extension, and the ability to observe this dynamics via the time evolution of a localized Wannier excitation. The results offer experimentally accessible signatures and transport protocols, with potential extensions to controlled state transfer and many-body settings, aided by the interplay between lattice topology and external confinement.

Abstract

We investigate the impact of harmonic confinement in a finite optical superlattice and reveal the different mechanisms that can lead to the emergence of localized states. The optical superlattice, with odd or even number of unit cells, can exhibit either a trivial or a non-trivial underlying topology, characterized by the corresponding Zak phase. We focus on a distinct localization mechanism in the intermediate harmonic trapping frequency regime. Specifically, the four lowest-lying eigenstates in this regime form an effective four-level system in the topologically non-trivial configuration. Larger trapping frequency values drive the system into a harmonic trap dominated regime, featuring classical pairing and localization of all states of the lower band, as in a usual optical lattice. For the lower trapping frequency regime, the fate of topological edge states is discussed. Our results are based on exact diagonalization and on a tight-binding approximation that maps the continuous to a discrete system. We address several aspects relevant to the experimental implementation of optical superlattices and provide a brief illustration of the dynamics, highlighting direct ways to observe and distinguish between the different localization mechanisms.

Figures (23)

• Figure 1: Schematics of the extended optical lattice and superlattice potentials in the left column (a,c,e), and the same potentials with the addition of the harmonic confinement in the right column (b,d,f). Specifically, (a,b) the extended optical lattice, (c,d) and (e,f) the extended optical superlattice in the trivial and topological configuration, respectively. The solid lines represent the potentials, and the dashed lines show the corresponding Wannier functions. The vertical dashed lines highlight the starting point of the extension of the potential area. The dotted lines show the harmonic oscillator function. The right y-axis shows in olive color (medium gray in grayscale) the squared norm of the corresponding Wannier functions for each setup.
• Figure 2: Schematic of the discrete model that is produced by the TB approximation of a continuous optical lattice or superlattice system with $\mathcal{N}=8$ minima (sites). Each site corresponds to a Wannier function, and the connecting lines show the hopping amplitudes expressed by the $J_i$'s and $J_{t_i}$'s parameters. The upward arrows denote the on-site potential $\mu_i$ at site $i$. The widths of the lines and the heights of the arrows respectively reflect the mirror symmetry of the lattice about its central site.
• Figure 3: Profiles of the (a) on-site, (b) on-site relative to the central site, (c) nearest neighbor hopping and (d) next-to-nearest neighbor hopping amplitudes of the TB approximation for an optical superlattice system with the addition of a harmonic trap. The extended optical superlattice system has $M=4$ ($\mathcal{N}= 8$) cells (minima/sites), $V_{high} = 5.0E_r$, $u = 0.6$, $x_0 = L$, $d=0.36\pi/k_r$, $\omega = 0.25E_r/\hbar$. All amplitudes are in units of $E_r$.
• Figure 4: (a-c) Energy spectrum versus the trapping frequency $\omega$ of an optical lattice potential with the addition of a harmonic trap. The optical lattice has $\mathcal{N}=8$ minima (sites) and $V_{0}=5E_r$. (a) Energy spectrum from low to high $\omega$ regime, (b) and (c) the low and mid regimes, respectively. In (a) and (c) the vertical dashed lines indicate the values of $\omega$ where crossing with the upper band occurs, and in (b) the approximate value of $\omega$ where the effect of the harmonic trap starts to become apparent in the system. (d) Spatial profile of the densities of the first twelve (12) eigenstates of the system for low values of $\omega$. The first eight (8) are from the first band, and the next four (4) correspond to the first states of the second band.
• Figure 5: The on-site and hopping amplitudes versus the trapping frequency $\omega$ of the TB approximation for an optical lattice system with the addition of a harmonic trap. The optical superlattice system has $\mathcal{N}=8$ minima (sites) and $V_0 = 5.0E_r$ and has no extension. The index $i$ identifies each on-site and hopping term in the Hamiltonian, according to the definition in Eq. \ref{['TB_eq_Terms']}. All amplitudes are in units of $E_r$.