Cold-Atom Elevator: From Edge-State Injection to the Preparation of Fractional Chern Insulators

Optical box traps offer new possibilities for quantum-gas experiments. Building on their exquisite spatial and temporal control, we propose to engineer system-reservoir configurations using box traps, in view of preparing and manipulating topological atomic states in optical lattices. First, we consider the injection of particles from the reservoir to the system: this scenario is shown to be particularly well suited to activating energy-selective chiral edge currents, but also to prepare fractional Chern insulating ground states. Then, we devise a practical evaporative-cooling scheme to effectively cool down atomic gases into topological ground states. Our open-system approach to optical-lattice settings provides a new path for the investigation of ultracold quantum matter, including strongly correlated and topological phases.

Figure 1

The cold-atom elevator. (a) Protocol for chiral edge-state injection: setting the reservoir energy on resonance with the system’s edge modes, particles are continuously injected into edge states in an energy-selective manner, and chiral edge currents appear in the system without populating the bulk. (b) Injection protocol for state preparation: starting from a trivial state in the reservoirs, the latter are slowly lifted so as to adiabatically inject particles into the system until an insulating state, e.g., a quantum Hall state, is formed. (c) Cooling protocol for state preparation: a proper tuning of the reservoir energy can be used to retrieve excitations (hot atoms) from the system; removing the particles from the reservoir and repeating this lift-removal process over many cycles leads to the preparation of the desired insulating (QH) state in the system.

Figure 2

Edge-state injection in the HH model. (a) Spectrum as a function of the eigenstate index $j$. The blue and red dots correspond to the Hamiltonian describing the reservoir (with ${\epsilon }_R=1$) and the system, respectively; $E_F$ is the Fermi energy. (b) Population of HH eigenstates ${\rho }_j$ as a function of their energy $E_j$, at time $t=28$. (c) Edge-mode population as a function of ${\epsilon }_R$ (compensated by $E_0$) for different initial particle number $N$ in the reservoir. Snapshots of spatial density distribution for (d) ${\epsilon }_R=1$, (e) ${\epsilon }_R=2.5$, (f) ${\epsilon }_R=4$ at times $t=7$, 21. Here, a system of size $13\times 12$, with flux $\varphi =\pi /2$ per plaquette, is coupled to a reservoir of size $7\times 12$. Except for (c), the number of particles is $N=19$. Energy and time are in units of $J$ and $\hslash /J$, respectively. The arrows in (d) and (f) are a guide to the eye for the chiral motion.

Figure 3

Preparing an FCI based on injection. (a) Spatial density distribution of the initial state at ${\epsilon }_R=-3$. (b) Density distribution of the target state at ${\epsilon }_R=-2$. (c) Bulk density and the local Streda marker as a function of ${\epsilon }_R$. The shadow indicates the FCI regime. (d) Many-body energy gap as a function of ${\epsilon }_R$. Inset: the ramping protocol for ${\epsilon }_R(t)$ used in the next panels (e)–(f). (e) Bulk density as a function of ${\epsilon }_R$ for different $\tau$ and for the instantaneous ground state. (f) Streda marker as a function of $\tau$. The dashed horizontal line indicates the ideal value $C_{\mathrm{Str}}=1/2$. Inset: linear fit of the density versus flux at $\tau =160$, yielding $C_{\mathrm{Str}}=0.51$. Here, we consider $N=12$ hard-core bosons; the system of size $4\times 4$ is coupled to two reservoirs of size $4\times 4$, with $J_R=0.15$. The error bars denote the standard error of the regression slope used to extract $C_{\mathrm{Str}}$.

Figure 4

Preparing a Chern insulator with a repeated-cleaning sequence. (a) Energy spectrum of the HH model coupled to a trivial reservoir. We partition a lattice of size $20\times 20$ into a target $12\times 12$ system (central box) and a surrounding reservoir; we set a flux $\varphi =\pi /2$ in the system only. The aim is to empty the excited band $E_1$, while keeping $E_0$ perfectly filled. Inset: simplified three-level model; $⟨E_0⟩$ and $⟨E_1⟩$ denote representative energies of the lowest two bands. (b) Population in HH eigenstates ${\rho }_j$, at various times $t=\tau n_{\mathrm{cyc}}$, for $\tau =100$ and $J_R=0.15$. Inset: HH-orbital entropy versus $n_{\mathrm{cyc}}$. (c) Uniformity factor as a function of the total cleaning time $t_{\mathrm{tot}}=\tau N_{\mathrm{cyc}}$, for different protocols (including a static evaporative scheme). The uniformity $u=1-\sqrt{{\sum }_{\ell }(n_{\ell }-\alpha {)}^2/N_{\ell }}$ is evaluated in a central disk of radius $r=4$ containing $N_{\ell }$ sites. Inset: density profile at the end of the sequence in (b). (d) The mean Streda marker ${\overline{C}}_{\mathrm{Str}}$ as a function of $N_{\mathrm{cyc}}$, while fixing the total cleaning time ($t_{\mathrm{tot}}=800$); $N_{\mathrm{cyc}}=0$ corresponds to a static reservoir. The error bars reflect the Streda marker’s standard deviation in the disk used in (c). Inset: local Streda marker along the central row at $N_{\mathrm{cyc}}=8$; the error bars reflect the standard error of the regression slope as in Fig. 3.