Two dimensional sub-wavelength topological dark state lattices

TL;DR

The paper presents a general approach to engineer 2D sub-wavelength topological optical lattices using position-dependent dark states in a Λ-system. By designing spatially varying Rabi frequencies, it creates Kronig-Penney-like scalar potentials that host co-localized sharp synthetic magnetic flux tubes, arranged so the total flux per unit cell vanishes, yet enabling nontrivial band topology. When the flux tubes are narrow, the system behaves like particles in a nearly uniform background field, producing nearly flat Chern bands with unit Chern numbers; this regime is shown to be robust to non-adiabatic effects and losses. The authors verify the existence of ideal Chern bands through quantum geometric tensor criteria, highlighting the potential for simulating quantum Hall and fractional Chern insulator physics in ultracold atomic gases with high tunability. This work thus offers a tunable, experimentally feasible platform for exploring strongly correlated topological phases in 2D sub-wavelength lattices.

Abstract

We present a general framework for engineering two-dimensional (2D) sub-wavelength 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 sub-wavelength 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 non-adiabatic effects and losses. This makes the scheme well-suited for simulating quantum Hall systems and fractional Chern insulators in ultracold atomic gases, offering a new platform for exploring strongly correlated topological phases with high tunability.

Figures (8)

• Figure 1: (a) The Lambda scheme of the atom-light coupling. Two laser fields characterized by Rabi frequencies $\Omega_{1}$ and $\Omega_{2}$ couple resonantly (or nearly resonantly with a single photon detuning $\Delta$) two atomic ground states $\left|1\right\rangle$ and $\left|2\right\rangle$ to a common excited state $\left|e\right\rangle$. (b) The position dependence of the modulus of Rabi frequencies $\Omega_{1}$ and $\Omega_{2}$ given by by Eqs. \ref{['eq:Omega_1-specific']}-\ref{['eq:Omega_2-specific']} for $\epsilon=1$ and $\nu=0.95$. Arrows indicate the direction of the phase winding of $\Omega_{1}$ around its zero points.
• Figure 2: Spatial dependence of the magnetic field $B_{z}$ for $\epsilon=1$ and $\nu=\left\{ 1,0.9,0.5,0\right\}$ when $\Delta=2000E_{{\rm R}}$, $\Gamma=1000E_{{\rm R}}$ and $\Omega_{0}=2000E_{{\rm R}}$. Note that the magnetic field around the narrow peaks are far beyond the range of their values shown. Specifically, the largest $\left|B_{z}\right|$ are equal approximately to $\left\{ 1,400,16,4\right\} \hbar k^{2}$ for $\nu=\left\{ 1,0.9,0.5,0\right\}$, respectively. The geometric scalar potential $\phi$ behaves very similar to the absolute value of $B_{z}$.
• Figure 3: Dependence of the standard deviation of magnetic field $B_{z}$ on $\epsilon$ and $\nu$ (upper plot), as well as on $\nu$ for $\epsilon=1$ (lower plot). Other parameters are $\Delta=2000E_{{\rm R}}$, $\Gamma=1000E_{{\rm R}}$, $\Omega_{0}=2000E_{{\rm R}}$. For $\nu\ne1$ we have eliminated narrow peaks of the magnetic field by cutting the magnetic field exceeding the maximum of the absolute value of $B_{z}$ corresponding to $\epsilon=1$ and $\nu=0$. The standard deviation of the scalar potential $\phi$ is characterized by nearly identical plots.
• Figure 4: Spatial dependence of the magnetic field $B_{z}$ for $\nu=0$ and $\epsilon=\left\{ 0.1,0.3\right\}$ when $\Delta=2000E_{{\rm R}}$, $\Gamma=1000E_{{\rm R}}$ and $\Omega_{0}=2000E_{{\rm R}}$. Note that the magnetic field around the narrow peaks are far beyond the range of their values shown. Specifically, the largest $\left|B_{z}\right|$ are equal approximately to $\left\{ 400,44.4444\right\} \hbar k^{2}$ for $\epsilon=\left\{ 0.1,0.3\right\}$. The geometric scalar potential $\phi$ looks very similar to the absolute value of $B_{z}$.
• Figure 5: Dependence of eigenenergies $E$ on the amplitude $\Omega_{0}$ of the Rabi frequency for the four lowest Bloch bands belonging to the dark state manifold for $\epsilon=1$ and $\nu=\left\{ 1.2,1.0,0.9,0.8\right\}$, with $\Delta=2000E_{{\rm R}}$ and $\Gamma=1000E_{{\rm R}}$ The calculations have been done using the full Hamiltonian (\ref{['eq:H_full']}) with the atom-light coupling operator $\hat{V}$ involving all three atomic internal states. Colors correspond to the excited state population of the eigenstates $\left|\langle e|\psi\rangle\right|^{2}$ due to non-adiabatic effects.