Topological charge pumping with subwavelength Raman lattices

TL;DR

The paper addresses topological charge pumping in deeply subwavelength lattices formed from $N$ Raman-coupled internal states. It develops a Hamiltonian that yields a reduced unit cell $a=a_0/N=\lambda/(2N)$, and shows that time-periodic detuning couples the $s$- and $p$-bands to realize two coupled Rice–Mele chains. Through an analytic Floquet-like treatment and a tight-binding // rotating-wave framework, it identifies novel pumping regimes where per-cycle displacements can be $0$, $a$, or $2a$, governed by Chern numbers and Zak phases; finite-size edge states corroborate the bulk topology via bulk-edge correspondence. The work provides a tunable platform for robust geometric pumping in subwavelength lattices, with prospects for longer-range detuning and higher-dimensional generalizations.

Abstract

Recent experiments demonstrated deeply subwavelength lattices using atoms with $N$ internal states Raman-coupled with lasers of wavelength $λ$. The resulting unit cell was $λ/2N$ in extent, an $N$-fold reduction compared to the usual $λ/2$ periodicity of an optical lattice. For resonant Raman coupling, this lattice consists of $N$ independent sinusoidal potentials (with period $λ/2$) displaced by $λ/2N$ from each other. We show that detuning from Raman resonance induces tunneling between these potentials. Periodically modulating the detuning couples the $s$- and $p$-bands of the potentials, creating a pair of coupled subwavelength Rice--Mele chains. This operates as a novel topological charge pump that counter-intuitively can give half the displacement per pump cycle of each individual Rice--Mele chain separately. We analytically describe this behavior in terms of infinite-system Chern numbers, and numerically identify the associated finite-system edge states.

Figures (8)

• Figure 1: Lattice concept. (a) Experimental geometry with a single frequency Raman beam traveling along $\mathbf{e}_x$ and $N$ Raman laser beams sharing the same spatial mode traveling along $-\mathbf{e}_x$. The level diagram for cyclic coupling is depicted on the right. (b) Dressed state energies for $N=3$ and $\Omega_0 = \Omega_1 = \Omega_2 = 1E_{\mathrm R}$. The dashed curves are computed for zero detuning, whereas the solid ones are calculated for a detuning described by Eq. \ref{['eq:delta_j-oscil-detun']} with $l=1$ and $\delta=0.5E_{\mathrm R}$. All curves are colored according to ternary plot on the right, marking the occupation probabilities in the three dressed states (not the bare internal atomic states) obtained by diagonalizing Eq. \ref{['eq:H_0']}. (c) Resonant driving gives nearest neighbor coupling $J_{\pm1}$ between the $s$- and $p$-bands. Coupling within bands is induced by a static detuning with matrix elements $J_{0s}$ and $J_{0p}$. (d) The same lattice unraveled into coupled RM chains.
• Figure 2: Two band model. (a) and (b) Wannier functions for the $s$-band and $p$-band respectively, computed for $\Omega=1E_{\mathrm R}$, $\delta=0E_{\mathrm R}$, and $N=3$. The colors correspond to Wannier states for each of the three dressed states. (c) and (d) Natural tunneling $J^{(\alpha)}$ dependence on Rabi frequency $\Omega$ for the $s$- and $p$- bands. The solid and dashed curves plot the nearest neighbor (NN) and next nearest neighbor tunneling (NNN). The black lines denote the threshold for the applicability of the tight binding approximation: for $\alpha=s$ this threshold is $\Omega>0.4E_{\mathrm R}$, and for $\alpha=p$ it is $\Omega>1.0E_{\mathrm R}$.
• Figure 3: Modulation induced tunneling ratio $J_{0p}/J_{0s}$ calculated exactly, plotted as a function of $\Omega$.
• Figure 4: Contours of integration. Due to the periodicity of the vector potential the boundary integration vanishes. Thus the integral is fully defined by the small integrals around the excluded singular points in ${\bf A}$ given by Eq. \ref{['eq:k,t-zero-points']}.
• Figure 5: Adiabatic pumping in the $\epsilon$ scheme. (a) The three circles show the cases when: both critical points $(\pm 2J_0, 0)$ are encircled (solid), one of them is encircled (dashed) or neither is encircled (dotted). (b) Zak phase $\gamma_\mathrm{Zak}$ dependence on time $t$ for the three aforementioned trajectories.