## Visualizing edge states in atomic systems

Visualizing edge states in atomic systems Simulating the solid state using ultracold atoms is an appealing research approach. In solids, however, the charged electrons are susceptible to an external magnetic field, which curves their trajectories and makes them skip along the edge of the sample. To observe this phenomenon with cold atoms requires an artificial magnetic field to have a similar effect on the neutral atoms (see the Perspective by Celi and Tarruell). Stuhl *et al.* obtained skipping orbits with bosonic atoms using a lattice that consisted of an array of atoms in one direction and three internal atomic spin states in the other. In a complementary experiment, Mancini *et al.* observed similar physics with fermionic atoms.

## Abstract

Chiral edge states are a hallmark of quantum Hall physics. In electronic systems, they appear as a macroscopic consequence of the cyclotron orbits induced by a magnetic field, which are naturally truncated at the physical boundary of the sample. Here we report on the experimental realization of chiral edge states in a ribbon geometry with an ultracold gas of neutral fermions subjected to an artificial gauge field. By imaging individual sites along a synthetic dimension, encoded in the nuclear spin of the atoms, we detect the existence of the edge states and observe the edge-cyclotron orbits induced during quench dynamics. The realization of fermionic chiral edge states opens the door for edge state interferometry and the study of non-Abelian anyons in atomic systems.

Ultracold atoms in optical lattices represent an ideal system for studying the physics of condensed matter problems in a fully tunable, controllable environment (*1*, *2*). One of the notable achievements in recent years has been the realization of synthetic background gauge fields, akin to magnetic fields in electronic systems. By exploiting light-matter interaction, it is possible to imprint a Peierls phase onto the atomic wave function. This phase is analogous to the Aharanov-Bohm phase experienced by a charged particle in a magnetic field (*3*–*5*). These gauge fields, first synthesized in Bose-Einstein condensates (*6*), have recently allowed for the realization of the Harper-Hofstadter Hamiltonian in ultracold bosonic two-dimensional (2D) lattice gases (*7*, *8*), whereas, following a complementary route based on accurate engineering of the single-particle Hamiltonian, Chern insulators have been also realized (*9*) in systems that lack a net magnetic field (*10*). These works are paving the way toward the investigation of different forms of bulk topological matter in atomic systems (*5*, *11*). In the present work, we are instead interested in the edge properties of fermionic systems under the effects of a synthetic gauge field. Fermionic edge states are a fundamental feature of 2D topological states of matter, such as quantum Hall and chiral spin liquids (*12*, *13*). Moreover, they are robust against changing the geometry of the system and can be observed even on Hall ribbons (*14*). In addition, fermionic edge states offer appealing prospects in quantum science, such as the realization of robust quantum information buses (*15*), and they are ideal starting points for the realization of non-Abelian anyons akin to Majorana fermions (*16*, *17*).

Here we report the observation of chiral edge states in a system of neutral fermions subjected to a synthetic magnetic field. We exploit the high level of control in our system to investigate the emergence of chirality as a function of the Hamiltonian couplings. These results have been enabled by an innovative approach, proposed in (*18*), where an internal (nuclear spin) degree of freedom of the atoms is used to encode a lattice structure lying in an “extra dimension” (*14*), providing direct access to edge physics. We synthesize a system of fermionic particles in an atomic Hall ribbon of tunable width pierced by an effective gauge field. One dimension is realized by an optical lattice, which induces a real tunneling *t* between different sites along direction (Fig. 1A). The different internal spin states are coupled by a two-photon Raman transition, which provides a coherent controllable coupling between different spin components. This can be interpreted as a complex tunneling amplitude between adjacent sites of an extra-dimensional lattice along a synthetic direction (*14*, *19*). Furthermore, the phase imprinting laid out by the Raman beams amounts to the synthesis of an effective magnetic field for effectively charged particles (*4*) with flux (in units of the magnetic flux quantum) per plaquette (*20*). The Hamiltonian describing the system is
(1)where are fermionic creation (annihilation) operators on the site (*j*,α) in the real (*j*) and synthetic (α = 1,2,3) dimensions, and . The first term describes the dynamics along , where *t* can be tuned by changing the intensity of the optical lattice beams. The dynamics along is encoded in the second term: Ω_{α} can be controlled by changing the power of the Raman beams, whereas can be tuned by changing their angle (*20*). In Eq. 1, μ_{j} describes a weak trapping potential along , whereas ξ_{α} accounts for a state-dependent light shift, providing an energy offset along . h.c. stands for Hermitian conjugate. In our experiment, we produced large synthetic magnetic fields corresponding to per plaquette. For fermionic particles, we use alkaline-earth–like ^{173}Yb atoms, initially prepared in a degenerate Fermi gas. The sites of the synthetic dimension (Fig. 1B) are encoded in a subset of spin states {*m*} out of the *I* = 5/2 nuclear spin manifold, thus providing fermionic “ladders” with up to six “legs.” These atoms show SU(*N*)-invariant interactions [SU(*N*), special unitary group of degree *N*] (*21*), inhibiting the redistribution of the atoms among the different synthetic sites by collisional processes (*22*, *23*). The effect of these interactions—which is not fundamental for explaining the observations reported in this manuscript—has been taken into account in the theoretical model as a renormalization of the trap frequency (*20*). This is possible because the maximum filling fraction is atoms per site of the real-space lattice: For larger filling fractions commensurate with the lattice, possible insulating phases can be stabilized.

The key advantage offered by the implementation in a lattice that combines real and synthetic spaces is the possibility to work with a finite-sized system with sharp and addressable edges. Specifically, we focus on elementary configurations made up of fermionic ladders with a small number of legs connected by a tunnel coupling between them. A leg is constituted by a 1D chain of atoms trapped in the sites of the real lattice in a specific spin state, whereas the ladder “rungs” are provided by the synthetic tunneling (Fig. 1A). The number of legs can be set by controlling the light shifts ξ_{α} in such a way as to choose the number of spin states that are coupled by the Raman lasers (*20*).

We first consider the case of a two-leg ladder constituted by the nuclear spin states *m* = –5/2 and *m* = –1/2. A quantum degenerate ^{173}Yb Fermi gas with 1.6 × 10^{4} atoms and an initial temperature (where *T*_{F} is the Fermi temperature) is first spin-polarized in *m* = –5/2. By slowly turning on the intensity of the optical lattice along (and of an additional strong lattice along and to suppress the dynamics along the other two real directions), we prepare a system of ladders in which all atoms occupy the *m* = –5/2 leg with less than one atom per site (i.e., in a conductive metallic state). Then, by controlling the intensity and frequency of the Raman beams (*20*), we slowly activate the tunnel coupling between the legs so as to adiabatically load the fermionic system in the lowest band of both the lattice and the Raman-dressed energy spectrum.

Despite the absence of a real bulk region, this two-leg configuration is expected to support chiral currents with atoms flowing in opposite directions along the legs (Fig. 2C), as investigated recently in bosonic systems (*24*). To observe this, we measured the relative motion of the atoms in the two legs by spin-selective imaging of the lattice momentum distribution, obtained by switching off the synthetic coupling and releasing the atoms from the lattice. In Fig. 2A (upper panel), we show two time-of-flight images corresponding to the *m* = –5/2 and *m* = –1/2 legs (Fig. 2C) for Ω_{1} = 2π × 489 Hz and *t* = 2π × 134 Hz (Ω_{1}/*t* = 3.65). Here we are interested only in direction , which reflects the distribution of the lattice momenta *k* along the legs (in units of the real-lattice wave number *k*_{L} = π/*d*, where *d* is the real-lattice spacing). The lattice momentum distribution along is a uniform square due to the presence of the strong optical lattice along the transverse (frozen) real directions (*20*). The central panel of Fig. 2A shows the lattice momentum distribution *n*(*k*) after integration of the images along and normalization according to . We observe a clear asymmetry in *n*(*k*) [similar to what was reported in experiments with spin-orbit coupling in harmonically trapped gases (*25*–*27*)], which we characterize by defining the function
(2)
which is plotted in the lower panel of Fig. 2A. The expression provides a measurement of the lattice momentum unbalance and quantifies the strength of the chiral motion of the particles along the two legs. The values *J* = +0.056(3) for *m* = –5/2 and *J* = –0.060(7) for *m* = –1/2 are approximately equal in intensity and opposite in sign, providing direct evidence for presence of chirality in the system. The small value of *J* is attributable to the fact that, in addition to states exhibiting chiral currents, fermions occupy other states at the bottom of the band, which do not display chiral features. We also performed the same experiment with a reversed direction of the synthetic magnetic field **B** (Fig. 2B), observing a change of sign in *J*, corresponding to currents circulating in the opposite direction. This behavior confirms the interpretation of our data in terms of chiral currents induced by a synthetic magnetic field in a synthetic 2D lattice.

The stability of chiral edge states in fermionic systems is of key importance, for example, for quantum information applications. In our system, the appearance of a chiral behavior is governed by several key parameters, including the ratio Ω_{1}/*t*, the Fermi energy *E*_{F}, and the flux . These parameters are easy to adjust, so they can be used to investigate the rise and fall of the edge currents as a function of the Hamiltonian parameters (*24*), as well as to identify which regimes exhibit stronger chiral features. By varying the tunneling rates along and , we observe a phase transition between a chiral behavior and a nonchiral regime. The lattice momentum distribution is measured as a function of Ω_{1}/*t* without affecting other relevant parameters, such as *E*_{F} and *T*. Figure 2D illustrates the measurement of as a function of Ω_{1}/*t* (circles). As expected, no chirality is observed for vanishing Ω_{1}, when the legs are decoupled. Chirality is also suppressed for large inter-edge coupling . In the latter regime, the largest energy scale in the system is the effective kinetic energy along the synthetic direction: This contribution is minimized when the fermions occupy the lowest energy state on each rung, which does not exhibit any chiral behavior. The measured values of compare well with the results of a numerical simulation that includes thermal fluctuations (shaded area in Fig. 2D) (*20*).

We next considered a three-leg ladder, which is the minimal configuration for which chiral currents at the edges can be sharply distinguished from the behavior of the bulk. The experimental procedure is analogous to that employed for the two-leg case, with the Raman parameters adjusted to extend the synthetic coupling to *m* = +3/2, with (*20*). Figure 3A shows measured *n*(*k*) and *h*(*k*) for each of the three legs for Ω_{1} = 2π × 620 Hz and *t* = 2π × 94 Hz (Ω_{1}/*t* = 6.60). We observe strong chiral currents in the upper- and lower-edge chains, showing values of *J* with opposite sign, similar to the two-leg case [*J* = +0.079(6) for *m* = –5/2 and *J* = –0.062(4) for *m* = +3/2]. In contrast, the central leg shows a much-reduced asymmetry in *n*(*k*) [*J* = 0.018(5)], signaling a suppressed net current in the bulk. This is direct evidence of the existence of chiral states propagating along the edges of the system, which leave the bulk mostly decoupled from the edges (Fig. 3C). This behavior is akin to what is expected for a fermionic system in a Harper-Hofstadter Hamiltonian. Bulk states exhibit only local circulations of current, which average to zero when all of the different states enclosed by the Fermi surface are considered. Only the edges of the system experience a nonzero current, because there the chiral nature of the states prevents this cancellation effect from occurring. In the ribbon geometry of the experiment, the bulk reduces to just a single central line. Nevertheless, the behavior discussed above is clearly present and detectable in the experimental signal. The small width of the ribbon favors the observation of edge states, given the large boundary-to-surface ratio of the system, which is reflected in a substantial population of states with edge character.

Figure 3C shows the values of *J* as a function of Ω_{1}/*t* for the three different legs of the ladder. The results illustrate the role of the bulk-edge coupling: Similar to the two-leg case, chirality is very weak for small coupling and increases as Ω_{1}/*t* approaches ~3. The theoretical curves show that further increasing Ω_{1}/*t* eventually leads to attenuation of the signal because of effective coupling between the edges, which smoothens the chiral features of the system. We observe a substantial agreement between experiment and theory for the range of Ω_{1}/*t* that can be explored in our experimental setup. The nonzero current in the bulk (*J* < 0.035) can be ascribed to the different couplings (Ω_{1} and Ω_{2}), as well as to a residual light shift that breaks the symmetry between the two edges (*20*).

Finally, we performed additional quench dynamics experiments that provide direct evidence of chiral transport properties along the edges. We prepared a system of lattice fermions in an initial state with zero average momentum on the lower *m* = –5/2 leg of a three-leg ladder. We then performed a quench by suddenly activating the complex tunneling in the synthetic direction. Figure 4A shows the time dependence of the average position in the synthetic direction , measured by optical Stern Gerlach detection (*23*). Figure 4B shows the time dependence of the average lattice momentum along , measured by time-of-flight imaging of the whole cloud. Figure 4D shows an experimental reconstruction of the average orbit on the ribbon surface as a plot of versus the average position in real space . The latter has been determined by evaluating the average velocity along , considering the knowledge of energy band dispersion versus lattice momentum, and then performing an integration in time (*20*). The dynamics displays a strong chiral character, demonstrated by the in-phase oscillations in Fig. 4, A and B, and the orbits in Fig. 4D. Under the effect of the synthetic magnetic field, the fermions move according to cyclotron-type dynamics, which is naturally truncated by the synthetic edge, giving rise to a skipping-type orbit, as expected for a quantum Hall system (*12*, *13*). Furthermore, the experimental data are in reasonable agreement with the theoretical predictions, represented by the thick lines in Fig. 4, A, B, and D. These dynamics are effectively damped, even in the idealized case described by theory (Fig. 4, A and B), as a result of averaging over many different fermionic trajectories, which also causes a reduction of the average orbit radius to less than one lattice site (Fig. 4D). This is markedly different from the behavior of a noninteracting Bose gas, which would occupy a single condensed wave packet undergoing undamped oscillations.

Our approach can be extended to wide ladder systems with as many as 2*I* + 1 legs, providing a setting for the investigation of both edge and bulk 2D topological matter, complementary to recent works on Chern insulators (*9*). This would allow a controlled study of the combined effect of interactions and synthetic gauge fields, crucial for the realization of fractional quantum Hall physics, potentially leading to exotic states of matter (such as chiral Mott insulator states) in ladder systems. Moreover, the flexibility offered by the present scheme allows the engineering of arbitrary lattice patterns, including disorder and constriction, in ladder systems. This opens the door for the realization of interferometers for chiral liquids, investigation of their transport properties, and the possibility of implementing interfaces between chiral edges, which, in the presence of a molecular or superconducting reservoir (*17*), can potentially host exotic non-Abelian anyons such as Majorana-like states (*16*).

## Supplementary Materials

## References and Notes

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- ↵During the completion of this manuscript, we became aware of a closely related experimental work with Raman-coupled Bose-Einstein condensates (
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- For the full two-dimensional case, another term, taking into account the Berry phase in momentum space, also affects at the level of semiclassical trajectories [see (
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**Acknowledgments:**We thank A. Celi and P. Massignan for early stimulating discussions on the synthetic dimension approach. M.D. and M.R. thank C. Laflamme and A. Sterdyniak for discussions. The experimental work in Florence, Italy, was supported by European Union (EU) grant FP7 SIQS, Ministero dell’Istruzione, dell’Universitá e della Ricerca (MIUR) grant PRIN2012 AQUASIM, and European Research Council (ERC) Advanced Grant DISQUA. The theoretical work in Innsbruck, Austria, was supported by ERC Synergy Grant UQUAM, SFB FoQuS of the Austrian Science Fund, and EU grant FP7 SIQS.