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# Direct imaging of topological edge states in cold-atom systems

Edited by Allan H. MacDonald, University of Texas at Austin, Austin, TX, and approved March 7, 2013 (received for review January 4, 2013)

## Abstract

Detecting topological order in cold-atom experiments is an ongoing challenge, the resolution of which offers novel perspectives on topological matter. In material systems, unambiguous signatures of topological order exist for topological insulators and quantum Hall devices. In quantum Hall systems, the quantized conductivity and the associated robust propagating edge modes—guaranteed by the existence of nontrivial topological invariants—have been observed through transport and spectroscopy measurements. Here, we show that optical-lattice-based experiments can be tailored to directly visualize the propagation of topological edge modes. Our method is rooted in the unique capability for initially shaping the atomic gas and imaging its time evolution after suddenly removing the shaping potentials. Our scheme, applicable to an assortment of atomic topological phases, provides a method for imaging the dynamics of topological edge modes, directly revealing their angular velocity and spin structure.

The integer quantum Hall (QH) effect revolutionized our understanding of quantum matter, revealing the existence of exotic phases not described by the standard theory of phase transitions (1, 2). In this phenomenon, the Hall conductivity is quantized, —where *e* is the electron charge, *h* is Planck’s constant and *ν* is an integer—whenever the Fermi energy resides in an energy gap. The integers *ν* are related to topological invariants—Chern numbers—that are associated with the bulk energy bands (1, 3, 4). Their topological origin guarantees that the Chern numbers are constant as long as the bulk gaps remain open, explaining the signature plateaus in the Hall resistivity, present when external parameters, such as magnetic fields, are varied. Moreover, a holographic principle stipulates that a topologically ordered bulk gap, with topological invariant *ν*, necessarily hosts *ν* propagating modes localized on the sample’s edge (5). These topological edge states are chiral—their motion has a well-defined orientation—inhibiting scattering processes in the presence of disorder.

In condensed matter physics, direct observations of edge states remain relatively rare. A first signature was obtained from magnetoplasmons created by pulsed voltages (6). Other evidence arises from edge transport in engineered Aharonov–Bohm interferometers with QH systems (7, 8). By contrast, the “routinely used” spectroscopic reconstruction of mid-gap states (1) is consistent with the expected topological band structure but does not prove their chiral nature.

Cold atoms trapped in optical lattices and subjected to synthetic gauge fields (9, 10) are an ideal platform for realizing topological insulating phases. Making topology manifest in experiments, however, is a fundamental challenge. In this context, transport experiments are conceivable (11) but technically demanding. Existing proposals for measuring topological invariants (12⇓⇓⇓–16) have experimental drawbacks and can only be applied to very specific configurations. Likewise, detecting topological edge states (17⇓⇓⇓⇓–22), for example through light-scattering methods (23), requires complicated manipulations to separate the small edge-state signal from the bulk background (20, 24).

Here, we introduce a simple method to directly visualize the propagation of topological edge modes, by studying the time evolution of an atomic QH system after suddenly releasing constraining walls (Fig. 1). We show that the movement of the chiral edge states, encircling the initially vacant regions, is directly visible in the atomic density. This reveals the edge states' angular velocity and provides an unambiguous signature of chiral edge modes in the atomic system (Fig. 1). Our method is straightforward and insensitive to experimental imperfections: It only relies on a large initial occupancy of edge states. Crucially, our method requires that the edge states contribution to the density remains spatially separated from the bulk, which can be realized by populating a dispersionless bulk band with a nonzero Chern number. We present several detection techniques, applicable to both flat and dispersive bands, that demonstrate the universal applicability to atomic systems with propagating edge modes.

We consider a two-dimensional optical lattice filled with noninteracting fermions, subjected to a uniform synthetic magnetic flux (25, 26), and confined by a circular potential, . In the experiment, can be made nearly arbitrarily sharp (27, 28); this configuration is of particular interest for our scheme, as demonstrated below. The resulting system realizes the Hofstadter model (29) with second-quantized Hamiltonian describes the creation of a fermion at lattice site , where are integers; *J* is the tunneling amplitude; and we take the lattice period *a* as our unit of length. This model has a topological band structure (1, 4): When , the bulk energy spectrum splits into *q* subbands (29), each associated with a nonzero Chern number (4). This guarantees the existence of robust edge states in the bulk energy gaps (5). These edge states are chiral in the sense that they propagate along the Fermi radius (i.e., the edge of the atomic cloud delimited by the confining potential ) with a definite orientation of propagation. It is convenient to represent such nontrivial spectra by diagonalizing the Hamiltonian Eq. **1** on a cylindrical geometry (5) (Fig. 2). This picture shows a clear separation of the bulk and edge states dispersions that survive in the experimental circular geometry produced by (20, 24, 30). In the following, we specifically study the configurations and and set the Fermi energy inside the lowest bulk energy gap (Fig. 2). In both these situations, the lowest energy band is associated with the Chern number , which guarantees the occupancy of a single edge mode with same chirality . These two configurations differ significantly in that the occupied bulk band is nearly flat (dispersionless) in the case , while it is dispersive for (Fig. 2).

Our scheme (*i*) demonstrates the existence of propagating modes that are localized close to the Fermi radius and (*ii*) identifies their chirality and angular velocity . To achieve this goal, we consider a geometry that constrains the QH system within two regions of the trap, as sketched in Fig. 1, resembling a bat in flight. This initial “bat” geometry is shaped by a pair of sharp potential walls defined by , creating holes in the density distribution (*SI Appendix*). In the bat geometry, we set the Fermi energy within the lowest bulk gap and suddenly remove at time . We then study the dynamics of the atomic density with all other parameters unchanged. The bat shape is optimized for visualizing the time-evolving chiral edge states in the density for (Fig. 1*B*). In the following, we discuss how this “wall-removal” strategy can be exploited to reveal the edge states properties, as they progressively encircle the initially empty regions in a chiral manner.

## Results

### Time-Evolving Density for Dispersive Systems.

Fig. 3*A* shows the time-evolving density distribution for . This example highlights the importance of the bulk band structure, because it demonstrates the drawbacks encountered when preparing a system with dispersive bulk bands (Fig. 2*A*). The time evolution in Fig. 3*A* illustrates two main effects: (*i*) the progressive encircling of the holes by particles at the system’s radial-edge (with localization length ) and (*ii*) the undesired and rapid filling of the holes by bulk states (Fig. 1*B*). Once , the atomic cloud’s initial bat shape has become cyclonic, already indicating the presence of chiral edge states. To provide further insight, we separately calculated the contribution stemming from the initially populated edge states, (*Materials and Methods*). In the corresponding Fig. 3*B*, we observe that the edge states, which propagated along the edges delimited by the bat potential at , become localized along the circular edge at , and that they follow a “chiral” motion. These edge states remain localized on the edge for very long times and only slightly disperse into the bulk of the system, as can be anticipated from the small wavefunction overlap between edge and bulk states. Fig. 3*A* emphasizes the problematic (nonchiral) filling of the holes by the many dispersive bulk states, which strongly inhibits the detection of the edge states in experiments. The speed at which this filling occurs is to be compared with the circular motion of the edge states, which can be estimated from the group velocities associated with the bulk and edge states (Fig. 2*A*). Additional interference takes place within the system, leading to small but visible patterns in the density. Fig. 3*B* shows that these patterns mainly originate from interferences between bulk states.

### Flat Topological Bulk Bands.

For and , the dispersionless (flat) bulk band represented in Fig. 2*B* is totally filled, and most of the edge states lying in the first bulk gap are populated. The corresponding time-evolving density , depicted in Fig. 4*A*, is radically different than for (Fig. 3*A*). For , the edge states encircle the initially forbidden regions in a chiral manner, largely unperturbed by the now motionless bulk, making them directly visible in in situ images of the cloud. The dispersionless nature of the bulk states is further illustrated in Fig. 4*C*, which shows the evolution of . The initial bulk states are immobile for times . In Fig. 4*B*, the evolution of the initially populated edge states shows an interesting behavior: Some edge states with energies close to the flat bulk band are almost dispersionless and remain localized on the edges delimited by the bat potential. In contrast, the many edge states at higher energies are dispersive, encircling the holes in a clear and chiral manner. The instructive dynamics of the density , which is due to the clear separation of the edge and bulk states during the evolution, can also be understood by studying the population of the single-particle eigenstates (*SI Appendix*). Moreover, our method is highly robust against perturbations in the density (or equivalently in the Fermi energy, ), because it only relies on the occupation of dispersionless bulk states and sufficiently many edge states. We verified that the edge-state signal is unambiguous when a high energy (dispersive) band is initially filled (*SI Appendix*). Thanks to the topological quasi-flat band configuration, the edge states propagation can be visualized on long time scales, without being affected by the bulk dispersion. For and a typical system size , we find , which would require a realistic time to observe the -rotation undergone by the edge states. We verified that our results are stable when the spacious ellipsoidal walls are replaced by small perturbative potentials (*SI Appendix*). Finally, the edge/bulk ratio can be further improved by initially confining the entire atomic cloud to a small region located in the vicinity of the circular edge (*SI Appendix*).

### Revealing Topological Edge States in Dispersive Systems.

We now describe two methods for isolating the edge-states contribution from that of the bulk states, useful for systems with dispersive bulk bands. The first method consists in performing two successive experiments, using the same apparatus and parameters, but with opposite fluxes and . The difference between the two images lacks the nonchiral contribution of the bulk states and is simply given by (Fig. 5). For our bat geometry, when the edge states have undergone a rotation of . This determines the time when the signal disappears, giving the angular velocity of the edge states. This situation is illustrated in Fig. 5 for the “dispersive” case, where we find , in good agreement with the angular velocity of the populated edge states (20, 24). We verified that slight differences in the filling [e.g., ], for example due to finite temperature effects between the two successive experiments, or variations in the flux (e.g., and ), do not significantly affect the signal , highlighting the robustness of this method against experimental imperfections.

The second method aims to efficiently reduce the bulk dispersion by suddenly lowering the potential walls at , instead of removing them completely. This operation can be achieved in such a way that only the edge states with sufficiently high energies are allowed to propagate, while leaving the bulk states away from the holes. This “edge-filter method” can be realized by setting the Fermi energy within the first bulk gap and then suddenly lowering the potential to the value at , where *W* is the width of the lowest bulk band. The great efficiency of this method is presented in *SI Appendix* for the case .

### Robustness of the Edge States Against Disorder.

We now investigate the robustness of the edge states motion in the presence of disorder. This perturbation, which plays a fundamental role in solid-state systems, can be engineered in optical-lattice setups, for instance using speckle potentials (31). We study the effects of disorder by considering a random potential , with energies uniformly distributed within the range . The results are presented in Fig. 6 for the case . We find that the chiral edge states signal remains robust for disorder strengths , where is the size of the bulk gap. Interestingly, we can still distinguish a cyclonic cloud—a signature of the edge states chirality—for values up to . The immunity of the chiral edge states against disorder, a hallmark of the QH effect, could thus be demonstrated using our cold-atom setup.

### Gaussian Walls and Smooth Circular Confinements.

In the absence of walls , the edge states lying in the first bulk gap are radially localized, with a radius determined by their energy and the circular confinement. Writing the circular confinement as , we find that an edge state with energy is characterized by a localization radiuswhere is the minimum of the bulk band. This result is illustrated in Fig. 7, for , , and , where the wavefunctions amplitudes are plotted as a function of the *x* coordinate, and their corresponding energies . For an infinitely abrupt trap (27), , all the edge states are located at the constant Fermi radius . Therefore, the edge states contribution to the density yields a clear circular signal, with localization length of the order of the lattice spacing *a*. In contrast, for finite *γ*, the populated edge states are localized on different radii , leading to a broadening of the edge-state signal . For the situation illustrated in Fig. 7, this broadening is of the order for and for . Let us stress another crucial aspect of these smoothly confined QH systems, which is the fact that the angular velocity of the chiral edge states, as well as the number of available edge states within a bulk gap, highly depend on the potential’s smoothness *γ*: The angular velocity is maximized for highly abrupt confinements , whereas the number is larger for smooth potentials (19, 20, 24, 30). We numerically evaluated the angular velocity of the edge states (20, 24) for and , and we found for , for , and for (and we note that the angular velocity ). Scaling to a more realistic radius , we find that the edge states, which are populated below , undergo a rotation of after a time

indicating that it is highly desirable to design a sharp circular confining trap (27), to clearly observe the edge states rotating motion during reasonable experimental times .

We now investigate the density evolution for smooth confining traps and initial Gaussian walls . First of all, we note that the presence of Gaussian walls does not destroy the edge states lying within the first bulk gap. In this bat geometry, the edge states are localized on the outer edge delimited by , but also on the smooth boundary delimited by the Gaussian potentials. Therefore, when , the edge states behave as in Fig. 7*A* in the vicinity of the outer circular edge , whereas they behave similarly as in Fig. 7*C* in the vicinity of the Gaussian walls. We point out that, in our scheme, it is the behavior of the edge states near the outer circular edge that plays an important role. Indeed, as shown in Fig. 8*A*, replacing the infinitely abrupt walls by Gaussian potentials does not qualitatively affect the evolution of the density presented in Fig. 4*A*.

However, as can be anticipated from the discussion above, replacing the perfectly sharp potential by smoother confinements, , has dramatic consequences on the dynamics. In Fig. 8*B*, which shows the evolution of the density for , we clearly observe the broadening of the edge-state contribution , as they progressively encircle the holes. We also note the slower motion undergone by the edge states, which have a reduced angular velocity , as seen above. An even more dramatic situation is illustrated for the case in Fig. 8*C*. These results demonstrate the robustness of the edge-state motion in the presence of smooth confining traps and Gaussian walls, but they greatly emphasize the importance of designing sharp external confinements to improve the experimental detectability of the topological edge states.

## Conclusions

In this article, we introduced a simple, yet powerful, method to image the dynamics of topological edge states in atomic systems. Our scheme shapes an atomic gas, initially prepared in a topological phase, and directly images its time evolution. By explicitly revealing the presence of propagating chiral edge modes, this method provides an unambiguous signature of topological order in the context of cold atomic gases. Importantly, we have discussed the applicability of our method under realistic experimental conditions, emphasizing the importance of using sharp confining potentials to improve the detection of the edge-states signal. The schemes introduced in this work to reduce, or even eliminate, the irrelevant contribution of dispersive bulk states can be applied to a wide family of topological atomic systems, such as the promising Haldane-like optical lattice (12, 14) and fractional QH atomic gases (32⇓–34). Finally, our method can be directly extended to visualize the propagation of topological (spin-polarized) edge states, both in 2D (18, 35) and 3D (36), by using standard spin-dependent imaging methods (28).

## Materials and Methods

The system is prepared in the ground-state of the initial Hamiltonian,

where the potential describes the walls initially present in the trap and where is given by Eq. **1**. We denote the number of available sites before and after removing the walls by and , respectively, and we define the ratio . When , the total number of sites within the trap is approximatively given by the area , while the outer circular edge contains about lattice sites. The holes in the density created by correspond to vacant sites, leading to the large ratio . For , the system initially contains sites and the number of sites delimiting the edge of the bat is sites. Thus, the spacious holes used in our calculations lead to a large edge/bulk ratio. For , and initially setting the Fermi energy in the lowest bulk gap, leads to the filling factor . After removing the walls , the filling factor is reduced to the smaller value .

The groundstate of Hamiltonian Eq. **3** is written as

where the operator creates a particle in the single-particle state , with energy located below the Fermi energy . Here represents the complete set of single-particle eigenstates and eigenvalues satisfying the stationary Schrödinger equation

We are interested in the time evolution of the spatial density after removing the walls at . The evolution of the single-particle states is then entirely governed by the Hamiltonian . It is therefore convenient to introduce the eigenstates and eigenvalues corresponding to the Hamiltonian ,

We then define as the time evolution of the initial state according to the Hamiltonian ,

The spatial density at time *t* is given by

namely, the particle density is entirely governed by the time-evolution of the initially occupied single-particle states. The time evolution of the atomic cloud, after releasing the walls at , can therefore be numerically evaluated through a direct diagonalization of the Hamiltonians and . In our study, a crucial aspect consists in identifying the regimes for which the edge states propagating around the initially forbidden regions provide a clear signal, which is not perturbed by the many bulk states. It is therefore desirable to separately evaluate the contributions of the initially populated bulk and edge states. We introduce the corresponding quantities

where the sum is restrained to the populated edge states with energies located within the bulk gap.

## Acknowledgments

We thank J. Beugnon, I. Bloch, G. Bulnes Cuetara, M. Müller, and S. Nascimbène for discussions. This work was supported by the Fonds de la Recherche Scientifique (FNRS Belgium), Agence Nationale de la Recherche (ANR) via the project AGAFON (Artificial gauge fields on neutral atoms), European Research Council (ERC) project Quantum Gauge Theories and Ultracold Atoms (QUAGATUA), the Emergences program (Ville de Paris and Université Pierre et Marie Curie), and ERC Many-Body physics in gauge fields with ultracold Yb atoms in optical lattices (ManyBo) Starting Grant. Work at Innsbruck is supported by the integrated project AQUTE, the Austrian Science Fund through Spezialforschungsbereich (SFB) F40 FOQUS, and by the Defense Advanced Research Projects Agency (DARPA) Optical Lattice Emulator (OLE) program. I.B.S. acknowledges the financial support of the National Science Foundation (NSF) through the Physics Frontier Center at Joint Quantum Institute (JQI) and the Army Research Office (ARO) with funds from both the Atomtronics Multidisciplinary University Research Initiative (MURI) and DARPA OLE Program.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: ngoldman{at}ulb.ac.be.

Author contributions: N.G., J.D., F.G., M.L., P.Z., and I.B.S. designed research; N.G. and A.D. performed research; N.G. analyzed numerical data; and N.G., J.D., M.L., P.Z., and I.B.S. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1300170110/-/DCSupplemental.

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