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arXiv:2606.03438

Populating topologically protected edge states of a Chern insulator with the cold-atom elevator scheme and measurements

Toke Marstrand Pontoppidan Lindhard, Anne E. B. Nielsen

cond-mat.quant-gascond-mat.mes-hallquant-phSubmitted Jun 2, 2026• Announced Jun 3, 2026• 0 citations
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Abstract

Two-dimensional Chern insulators support topologically protected, chiral edge currents, and these can be detected in experiments with ultracold atoms in optical lattices. It has previously been shown that one can populate a selected group of edge states of a Chern insulator by transferring particles from a reservoir. Here, we numerically investigate the effect of performing an instantaneous, projective measurement on the reservoir before the reservoir is discarded. In this way, the final state of the system is pure and described by a wavefunction. We also show that quite likely measurement outcomes can help to increase the final number or percentage of particles in the chiral edge states through postselection. Without the measurement step, the physics can be described in terms of single-particle physics. The measurement significantly complicates the description. By appropriately rewriting the analytical expressions, we show that measurement probabilities, expectation values, averages of expectation values, and purity can nevertheless be computed from the state before the measurement in a way that scales only linearly with the number of lattice sites for a fixed number of particles. This enables us to investigate a setup with, for instance, 14 particles and 198 lattice sites numerically. The approach applies generally to noninteracting, fermionic models that conserve the number of particles.

Comments
10 pages, 4 figures

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Two-dimensional Chern insulators support topologically protected, chiral edge currents, and these can be detected in experiments with ultracold atoms in optical lattices. It has previously been shown that one can populate a selected group of edge states of a Chern insulator by transferring particles from a reservoir. Here, we numerically investigate the effect of performing an instantaneous, projective measurement on the reservoir before the reservoir is discarded. In this way, the final state of the system is pure and described by a wavefunction. We also show that quite likely measurement outcomes can help to increase the final number or percentage of particles in the chiral edge states through postselection. Without the measurement step, the physics can be described in terms of single-particle physics. The measurement significantly complicates the description. By appropriately rewriting the analytical expressions, we show that measurement probabilities, expectation values, averages of expectation values, and purity can nevertheless be computed from the state before the measurement in a way that scales only linearly with the number of lattice sites for a fixed number of particles. This enables us to investigate a setup with, for instance, 14 particles and 198 lattice sites numerically. The approach applies generally to noninteracting, fermionic models that conserve the number of particles.
Optical box traps for cold atoms 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 activate 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.
Edge states of chiral topologically ordered phases are commonly described by chiral Luttinger liquids, effective theories that are exact only in the hydrodynamic limit. Motivated by recent bulk observations of fractional Chern insulators (FCIs) in two-dimensional materials and by synthetic realizations in ultracold atoms, we revisit this framework and quantify deviations from the hydrodynamic limit due to lattice effects. Using a combination of analytical arguments and numerical simulations, we disentangle universal from nonuniversal edge properties. We outline experimental probes in excitonic FCIs and in ultracold atom systems, and in particular propose time-resolved edge spectroscopy to directly access the predicted exponents and velocities.
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.
The realization and detection of topological phases with ultracold atomic gases is at the frontier of current theoretical and experimental research. Here, we identify cold atoms in optical ladders subjected to synthetic magnetic fields as readily realizable bridges between one-dimensional spin-orbit (time-reversal) topological insulators and two-dimensional quantum Hall insulators. We reveal three instances of their promising potential: (i) they realize spin-orbit coupling, with the left-right leg degree of freedom playing the role of an effective spin, (ii) their energy bands and eigenstates exactly reproduce the topological chiral edge modes of two-dimensional Chern insulators, and (iii) they can be tailored to realize a topological phase transition from a trivial to a topological insulating phase. We propose realistic schemes to observe the chiral and topological properties of ladder systems with current optical lattice-based experiments. Our findings open a door to the exploration of the physics of the edges of quantum Hall insulators and to the realization of spin-orbit coupling and topological superfluid phases with ultracold atomic gases.
We propose an experimental scheme to simulate and detect the properties of time-reversal invariant topological insulators, using cold atoms trapped in one-dimensional bichromatic optical lattices. This system is described by a one-dimensional Aubry-Andre model with an additional SU(2) gauge structure, which captures the essential properties of a two-dimensional Z2{Z}_{2}Z2​ topological insulator. We demonstrate that topologically protected edge states, with opposite spin orientations, can be pumped across the lattice by sweeping a laser phase adiabatically. This process is an elegant way to transfer topologically protected quantum states in a highly controllable environment. We discuss how density measurements could provide clear signatures of the topological phases emanating from our one-dimensional system.
We present a novel technique for detecting topological invariants ‐ Chern numbers ‐ from time-of-flight images of ultra-cold atoms. We show that the Chern numbers of integer quantum Hall states of lattice fermions leave their fingerprints as oscillations in the atoms’ momentum distribution. The number of oscillations within the lowest Brillouin zone is simply equal to the Chern number. The ubiquitous signature is strikingly enhanced for anisotropic hopping, where the momentum distribution assumes a sinusoidal form with period determined by the Chern number. Thus, an everyday observable in cold atom experiments can serve as a useful tool to characterize and visualize quantum states with non-trivial topology. Topological insulators (TI) and superfluids are many-body quantum systems with energy gaps in the bulk but topologically protected, gapless excitations on the boundary [1, 2]. Each class of topological phases is characterized by a topological invariant which can be either an integer or a binary quantity. For example, the integer quantum Hall insulators [3] are distinguished from other two-dimensional insulators with the same symmetry by the Chern number, an integer that capture the global properties of all occupied bands [4]. The Chern number coincides with the number of chiral edge modes, and the Hall conductance in units of the conductance quantum. A central problem in exploring new forms of topological matter is how to extract the topological invariants from experiments, and unambiguously determine the topological phase. Experiments with ultracold atoms has brought us opportunities to creating topological phases of matter in parameter regimes unreachable in solids. In addition to band-structure engineering using optical lattices and tuning the atom-atom interaction by Feshbach resonance, artificial magnetic fields for neutral atoms have been produced by either rotation [5] or atom-laser coupling [6, 7], making way to realizing quantum Hall states of cold atoms [8]. Moreover, non-Abelian synthetic gauge fields [7, 9] and spin-orbit coupling [10] have been demonstrated. These developments motivated theories on properties of ultracold topological matter, e.g., the fractional quantum Hall effect of interacting bosons [5, 11, 12], the quantum spin Hall effect of fermions [9], and the quantum anomalous Hall states ofp-orbital fermions [13]. While cold atom systems offer new detection techniques and great tunability, it is challenging to measure the topological invariants directly. Measurements of local density of states (of the edge states) or mass transport are marred with complexity in contrast to solid state systems. Thus, creative methods have to be invented. For example, Ref. [8] proposed to extract the quantized Hall conductance from the derivatives