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arXiv:2210.05515 · Physical Review A

Topological charge pumping with subwavelength Raman lattices

D. Burba, Mantas Račiūnas, Ian B. Spielman, Gediminas Juzeliūnas

Published Feb 15, 2023• 4 citations
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Abstract

Recent experiments demonstrated deeply subwavelength lattices using atoms with $N$ internal states Raman coupled with lasers of wavelength $\ensuremath{\lambda}$. The resulting unit cell was $\ensuremath{\lambda}/2N$ in extent, an $N$-fold reduction compared to the usual $\ensuremath{\lambda}/2$ periodicity of an optical lattice. For resonant Raman coupling, this lattice consists of $N$ independent sinusoidal potentials (with period $\ensuremath{\lambda}/2$) displaced by $\ensuremath{\lambda}/2N$ from each other. We show that detuning from Raman resonance induces tunneling between these potentials. Temporally 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 topological charge pump that counterintuitively 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.

DOI
10.1103/physreva.107.023309

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Recent experiments demonstrated deeply subwavelength lattices using atoms with NNN internal states Raman coupled with lasers of wavelength \ensuremathλ\ensuremath{\lambda}\ensuremathλ. The resulting unit cell was \ensuremathλ/2N\ensuremath{\lambda}/2N\ensuremathλ/2N in extent, an NNN-fold reduction compared to the usual \ensuremathλ/2\ensuremath{\lambda}/2\ensuremathλ/2 periodicity of an optical lattice. For resonant Raman coupling, this lattice consists of NNN independent sinusoidal potentials (with period \ensuremathλ/2\ensuremath{\lambda}/2\ensuremathλ/2) displaced by \ensuremathλ/2N\ensuremath{\lambda}/2N\ensuremathλ/2N from each other. We show that detuning from Raman resonance induces tunneling between these potentials. Temporally modulating the detuning couples the sss and ppp bands of the potentials, creating a pair of coupled subwavelength Rice-Mele chains. This operates as a topological charge pump that counterintuitively 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.
We present a general framework for engineering two-dimensional subwavelength topological optical lattices using spatially dependent atomic dark states in a Λ -type configuration of the atom-light coupling. By properly designing the spatial profiles of the laser fields inducing coupling between the atomic internal states, we show how to generate subwavelength Kronig-Penney-like geometric scalar potential accompanied by narrow and strong patches of the synthetic magnetic field localized in the same areas as the scalar potential. These sharply peaked magnetic fluxes are compensated by a smooth background magnetic field of opposite sign, resulting in zero net flux per unit cell while still enabling topologically nontrivial band structures. Specifically, for sufficiently narrow peaks, their influence is minimum, and the behavior of the system in a remaining smooth background magnetic field resembles the Landau problem, allowing for the formation of nearly flat energy bands with unit Chern numbers. Numerical analysis confirms the existence of ideal Chern bands and the robustness of the topological phases against nonadiabatic effects and losses. This makes the scheme well suited for simulating quantum Hall systems and fractional Chern insulators in ultracold atomic gases, offering a platform for exploring strongly correlated topological phases with high tunability.
Recent experiments with ultracold quantum gases have successfully realized integer-quantized topological charge pumping in optical lattices. Motivated by this progress, we study the effects of static disorder on topological Thouless charge pumping. We focus on the half-filled Rice-Mele model of free spinless fermions and consider random diagonal disorder. We use both static and time-dependent simulations to characterize the charge pump. The set of measures that we compute in the instantaneous basis include the polarization, the entanglement spectrum, and the space-integrated local Chern marker. As a first main result, we conclude that the space-integrated local Chern marker is best suited for a quantitative determination of topological transitions in a disordered system. In the time-dependent simulations we use the time-integrated current to obtain the pumped charge in slowly periodically driven systems. As a second main result, we observe and characterize a disorder-driven breakdown of the quantized charge pump. There is an excellent agreement between the static and the time-dependent ways of computing the pumped charge. The topological transition occurs well in the regime where all states are localized on the given system sizes. This observation is consistent with previous studies and the topological transition is therefore not tied to a delocalization-localization transition of Hamiltonian eigenstates. For individual disorder realizations, the breakdown of the quantized pumping occurs for parameters where the spectral bulk gap inherited from the band gap of the clean system closes, leading to a globally gapless spectrum. As a third main result and with respect to the analysis of finite-size systems, we show that the disorder average of the bulk gap severely overestimates the stability of quantized pumping. A much better estimate is the typical value of the distribution of energy gaps, also called mode of the distribution. We discuss our results in the context of recent quantum-gas experiments that realized charge pumps.
We investigate topological charge pumping in a system of interacting bosons in the tight-binding limit, described by the Rice-Mele model. An appropriate topological invariant for the many-body case is the change of polarization per pump cycle, which we compute for various interaction strengths from infinite-size matrix-product-state simulations. We verify that the charge pumping remains quantized as long as the pump cycle avoids the superfluid phase. In the limit of hardcore bosons, the quantized pumped charge can be understood from single-particle properties such as the integrated Berry curvature constructed from Bloch states, while this picture breaks down at finite interaction strengths. These two properties---robust quantized charge transport in an interacting system of bosons and the breakdown of a single-particle invariant---could both be measured with ultracold quantum gases extending a previous experiment [Lohse et al., Nat. Phys. 12, 350 (2016)]. Furthermore, we investigate the entanglement spectrum of the Rice-Mele model and argue that the quantized charge pumping is encoded in a winding of the spectral flow in the entanglement spectrum over a pump cycle.
Experiments with bosonic atoms in optical superlattices allow for the interesting possibility to study the adiabatic quantized pumping of bosonic atoms in the presence of interactions. We investigate this exotic phenomenon for bound bosonic pairs in the paradigmatic Su-Schrieffer-Heeger model where the ground state exhibits topological properties due to dimerized hoppings. We show that at unit filling there exist crossovers and phase transitions to bond-order phases of paired bosons known as the pair-bond-order phase for attractive interactions. The pair-bond-order phase is found to exhibit effective topological properties such as the presence of polarized paired edge states. This is further analyzed by studying the emergence and breakdown of the Thouless charge pumping of bound bosonic pairs by a parametric extension to the famous Rice-Mele model. Finally, we discuss how the pumping of paired bosons or different regimes of breakdown of charge pumping can be probed by state-of-the-art experiments, for example, with repulsively bound bosons.
We realized a quantum geometric "charge" pump for a Bose-Einstein condensate (BEC) in the lowest Bloch band of a novel bipartite magnetic lattice. Topological charge pumps in filled bands yield quantized pumping set by the global-topological-properties of the bands. In contrast, our geometric charge pump for a BEC occupying just a single crystal momentum state exhibits nonquantized charge pumping set by local-geometrical-properties of the band structure. Like topological charge pumps, for each pump cycle we observed an overall displacement (here, not quantized) and a temporal modulation of the atomic wave packet's position in each unit cell, i.e., the polarization.
We propose a scheme for realizing lattice potentials of subwavelength spacing for ultracold atoms. It is based on spin-dependent optical lattices with a time-periodic modulation. We show that the atomic motion is well described by the combined action of an effective, time-independent lattice of small spacing, together with a micromotion associated with the time modulation. A numerical simulation shows that an atomic gas can be adiabatically loaded into the effective lattice ground state, for time scales comparable to the ones required for adiabatic loading of standard optical lattices. We generalize our scheme to a two-dimensional geometry, leading to Bloch bands with nonzero Chern numbers. The realization of lattices of subwavelength spacing allows for the enhancement of energy scales, which could facilitate the achievement of strongly correlated (topological) states.