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Imaging topology of Hofstadter ribbons
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New J. Phys. 21 (2019) 053021

https://doi.org/10.1088/1367-2630/ab165b

PAPER

Imaging topology of Hofstadter ribbons
OPEN ACCESS

Dina Genkina1, Lauren M Aycock1, Hsin-I Lu1,2, Mingwu Lu1, Alina M Pineiro1 and I B Spielman1
RECEIVED

1

6 November 2018
REVISED

1 March 2019
ACCEPTED FOR PUBLICATION

5 April 20192

Joint Quantum Institute, National Institute of Standards and Technology, and University of Maryland, Gaithersburg, MD 20899, UnitedStates of America
Currently Modern Electron, Bellevue, WA 98007, United States of America

E-mail: ian.spielman@nist.gov
Keywords: ultracold atoms, quantum simulation, quantum Hall effect, quantum transport

PUBLISHED

8 May 2019

Abstract
Physical systems with non-trivial topological order ﬁnd direct applications in metrology (Klitzing et al1980 Phys. Rev. Lett. 45 494–7) and promise future applications in quantum computing (Freedman
2001 Found. Comput. Math. 1 183–204; Kitaev 2003 Ann. Phys. 303 2–30). The quantum Hall effectAny further distribution ofderives from transverse conductance, quantized to unprecedented precision in accordance with thethis work must maintainattribution to thesystem’s topology (Laughlin 1981 Phys. Rev. B 23 5632–33). At magnetic ﬁelds beyond the reach of
author(s) and the title of
the work, journal citation current condensed matter experiment, around 10 4 T, this conductance remains precisely quantizedand DOI.
with values based on the topological order (Thouless et al 1982 Phys. Rev. Lett. 49 405–8). Hitherto,
quantized conductance has only been measured in extended 2D systems. Here, we experimentallystudied narrow 2D ribbons, just 3 or 5 sites wide along one direction, using ultracold neutral atoms
where such large magnetic ﬁelds can be engineered (Jaksch and Zoller 2003 New J. Phys. 5 56; Miyake
et al 2013 Phys. Rev. Lett. 111 185302; Aidelsburger et al 2013 Phys. Rev. Lett. 111 185301; Celi et al2014 Phys. Rev. Lett. 112 043001; Stuhl et al 2015 Science 349 1514; Mancini et al 2015 Science 349
1510; An et al 2017 Sci. Adv. 3). We microscopically imaged the transverse spatial motion underlying
the quantized Hall effect. Our measurements identify the topological Chern numbers with typicaluncertainty of 5%, and show that although band topology is only properly deﬁned in inﬁnite systems,
its signatures are striking even in nearly vanishingly thin systems.
Original content from thiswork may be used under
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Commons Attribution 3.0
licence.

1. Introduction
The importance of topology in physical systems is famously evidenced by the quantum Hall effect’s role as anultra-precise realization of the von Klitzing constant RK = h e 2 of resistance [1]. Although topological order isonly strictly deﬁned for inﬁnite systems, the bulk properties of macroscopic topological systems closely resemble
those of the corresponding inﬁnite system. For 2D systems in a magnetic ﬁeld B0, the topology is characterized
by an integer invariant called the Chern number. Even at laboratory ﬁelds of tens of Tesla, crystalline materials
have a small magnetic ﬂux Φ=AB0 per individual lattice plaquette (with area A) compared to the ﬂux quantumΦ0=h/e. Superlattice [2–5] and ultracold atom [6–9] systems now realize 2D lattices in a regime where themagnetic ﬂux per plaquette Φ is a signiﬁcant fraction of Φ0.
Experimental signatures of Chern numbers generally leverage one of two physical effects: in condensed
matter systems the edge-bulk correspondence allows the Chern number to be inferred from the quantized Hall
conductivity sH = C RK , and in cold-atom experiments direct probes of the underlying band structure at every
value of crystal momentum give access to the Chern number through either static [10, 11] or dynamic [12–15]
signatures. Both of these connections derive from the pioneering work of Thouless, Kohmoto, Nightingale, andden Nijs [16], in the now famous TKNN paper. Going beyond these well known techniques, the TKNN paper
showed that for rational ﬂux Φ/Φ0=P/Q (for relatively prime integers P and Q) the integer solutions s and C to
the Diophantine equation© 2019 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft

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Figure 1. Quantum Hall effect in Hofstadter ribbons. (a) 5-site wide ribbon with real tunneling coefﬁcients along es and complex
tunneling coefﬁcients along e x , creating a non-zero phase f around each plaquette. (b) After applying a force along e x for a time Δt,
atomic populations shift transversely along es , signaling the Hall effect. (c), (d) TOF absorption images giving hybrid momentum/
position density distributions n(kx, m). Prior to applying the force (c), the m=0 momentum peak is at kx=0, marked by the redcross. Then, in (d), the force directly changed qx, evidenced by the displacement Δqx of crystal momentum, and via the Hall effectshifted population along es .

1 = Qs - PC

(1)

uniquely3 determine the Chern number C of the lowest band. Following theoretical work [11, 17–20], weleveraged this TKNN equation to determine the Chern number of our system.

2. Experimental setup
We studied ultracold neutral atoms in a square lattice with a large magnetic ﬂux per plaquette. As pictured inﬁgure 1(a), our system consisted of a 2D lattice that was extremely narrow along one direction, just 3 or 5 sites
wide—out of reach of traditional condensed matter experiments, with hard wall boundary conditions: a ribbon.
Our system was qualitatively well described by the Harper–Hofstadter Hamiltonian in the Landau gauge [24, 25]
Hˆ = - å (tx e ifm∣ j , mñá j + 1, m∣ + ts∣ j , mñá j , m + 1∣) + h.c .,

(2)

m, j

where j and m label lattice sites along ex and es , with tunneling strengths tx and ts respectively. As shown inﬁgure 1(a), tunneling along ex was accompanied by a phase shift e ifm . Hopping around a single plaquette of thislattice imprints a phase f, analogous to the Aharanov–Bohm phase, emulating a magnetic ﬂux Φ/Φ0=f/2π.
We implemented this 2D lattice by combining a 1D optical lattice deﬁning sites along an extended direction ex ,with atomic spin states forming lattice sites along a narrow, synthetic [26–28, 29] direction es . The exact
Hamiltonian of the underlying atomic system differs from the Harper–Hofstadter Hamiltonian above in that ts
3

Subject to the constraint ∣C∣  ∣Q∣ 2 [16, 21]. The integer s has no bearing on our argument, but has been interpreted as the chargetransported when the periodic potential is adiabatically displaced [22, 23].

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is non-uniform due to Clebsch–Gordan coefﬁcients and there is a small m2 dependent potential term due to thequadratic Zeeman shift (see appendix D).
This system exhibits a Hall effect, where a longitudinal force FP—analogous to the electric force eE inelectronic systems—drives a transverse ‘Hall’ current density j^ = sH E for non-zero Φ/Φ0. A longitudinal
force Fx would drive a change in the dimensionless crystal momentum  Dqx G and a transverse displacement
Δm, giving a dimensionless Hall conductivity NGDm Dqx = sH RK = s˜ H , where G is the reciprocal lattice
constant and N is the number of carriers per plaquette (see appendix C). Starting with Bose-condensed atoms in
the lattice’s ground state (with transverse density shown in ﬁgure 1(a)) we applied a force along ex and obtainedΔm from site resolved density distributions [11] along es (ﬁgure 1(b)). Leveraging the TKNN equation(equation (1)), we further show that the force required to move the atoms a single lattice site signals the inﬁnite
system’s Chern number.
Our quantum Hall ribbons were created with optically trapped 87Rb Bose–Einstein condensates (BECs) ineither the F=1 or 2 ground state hyperﬁne manifold, creating 3 or 5 site-wide ribbons from the 2F+1 statesavailable in either manifold. We ﬁrst loaded BECs into a 1D optical lattice along ex formed by a retro-reﬂected
λL=1064 nm laser beam. This created a lattice with period a=λL/2 and depth 4.4(1) EL, giving tunnelingstrength tx = 0.154 (4) EL . Here, EL =  2kL2 2mRb is the single photon recoil energy; kL = 2p lL is the
single photon recoil momentum; and mRb is the atomic mass. We induced tunneling along es with either aspatially uniform rf magnetic ﬁeld or two-photon Raman transitions. The tunneling strength was ts = 1.97 (8) tx
for Raman coupling in the F=2 manifold, ts = 0.97 (8) tx for Raman coupling in the F=1 manifold, andts = 7.4 (5) tx for rf coupling in both manifolds. The rf-induced tunneling imparted at most only a spatially
uniform tunneling phase, giving f/2π=0. In contrast the Raman coupling, formed by a pair of counter
propagating laser beams with wavelength λR=790 nm, imparted a phase factor exp (-2ikR x ). Here,kR = 2p lR is the Raman recoil momentum, giving f 2p » 4 3. We then applied a force by shifting thecenter of the conﬁning potential along ex , effectively applying a linear potential. Using time-of-ﬂight techniques[27], we measured hybrid momentum/position density distributions n(kx, m), a function of momentum along
ex and position along es , as seen in ﬁgures 1(c), (d).

3. Hall conductivity measurement
We measured the Hall conductivity beginning with a BEC at qx(t=0)=0 in the lowest band with transverse
modal position m̄0 = 04. Figure 2(a) shows the band structure of a system similar to ours, but extended along
the es direction, with 41 sites. The energy is plotted as a function of crystal momentum along ex , with color
indicating the expectation value of position along es , calculated by diagonalizing the full Hamiltonian with zero
quadratic shift and uniform Clebsch–Gordan coefﬁcients (see appendix D). Figure 2(b) shows the band
structure of our experimental system, calculated from the full Hamiltonian for our experimental parameters (seeappendix D). Note that the lowest, 3rd, and 5th bands of our system are akin to the three bulk bands of theextended system, while the 2nd and 4th bands of our system resemble the edge modes of the extended system.
We applied a force Fx = 0.106 (5) EL lL for varying times Δt, directly changing the longitudinal crystal
momentum from 0 to a ﬁnal qx and giving a transverse Hall displacement from 0 to a ﬁnal m̄ . Figure 2(c) shows acollection of hybrid density distributions, where each column depicts n(kx, m) for a speciﬁc ﬁnal qx, labeled by
the overall horizontal axis. For each column, the change in crystal momentum is marked by the horizontal
displacement of the diffraction orders relative to their location in the central qx=0 column. The transverse
displacement is visible in the overall shift in density along m as a function of qx, i.e. between columns.
Figure 3(left) quantiﬁes this Hall effect by plotting the modal position m̄ as a function of qx for Φ/Φ0=0,−4/3, and 4/3. The data is represented by gray dots, with uncertainty bars reﬂecting the propagated standard
uncertainty from averaging six identical runs. For zero ﬂux Φ/Φ0=0 (ﬁgure 3(a)), m̄ was independent of qx; incontrast, for non-zero ﬂux Φ/Φ0≈±4/3 (ﬁgures 3(b), (c)), m̄ depends linearly on qx with non-zero slope.
These linear dependencies evoke our earlier discussion of the Hall conductance s̃H , in which we anticipatedslopes equal to the Chern number. Linear ﬁts to the data give s˜ H = 0.01 (1), 0.87(3), and −0.85(3) for zero,
negative and positive ﬂux respectively, showing the expected qualitative behavior. The expected slopes, given by
the Chern number, σH=0, ±1 are indicated by black dashed lines in ﬁgure 3(left).
The red curves in ﬁgure 3(top) show the expected behavior for our 5-site wide system for adiabatic changes
in qx as calculated from exact diagonalization of the full Hamiltonian (see appendix D), always within the lowest
band (ﬁgure 2(a)), i.e. Bloch oscillations. This theory predicts a nearly linear slope for small qx sharply returning
to m̄ = 0 at the edges of the Brillouin zone. A linear ﬁt to this theory produces s˜ H » 0 , 0.6, and −0.6 for zero,
negative and positive ﬂux respectively, far from the Chern number. This discrepancy is resolved by recalling that4

We deﬁne the modal position m̄ as the center of a Gaussian ﬁt to the population distribution along es .

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Figure 2. Band structure in an extended system and our 5-site wide ribbon. (a) Band structure of an extended system, with 41 sites
along es , computed for a 4.4EL deep 1D lattice (λL=1064 nm), 0.5EL Raman coupling strength (λR=790 nm), and quadratic
Zeeman shift ò=0EL, and excluding anisotropy due to Clebsch–Gordan coefﬁcients (see appendix D). The color indicates the
expectation value of position along the synthetic direction ámñ. (b) Band structure computed using full Hamiltonian for our
experimental parameters of 4.4EL lattice depth, 0.5EL Raman coupling strength, and quadratic Zeeman shift  = 0.02EL , giving
Φ/Φ0≈4/3, tx=0.154(4)EL, ts = 1.97 (8) tx (see appendix D). The black dot indicates the initial loading parameters. (c) TOF
absorption images n(kx, m) for varying longitudinal crystal momenta qx.

Bloch oscillations require adiabatic motion. As the ribbon width grows, the band gaps at the edge of the Brillouinzone close (see ﬁgure 2(a)), making the Bloch oscillation model inapplicable. The departure of the data from theadiabatic theory at the edges of the Brillouin zone indicates a partial break down of adiabaticity was present inour data. To conﬁrm this, we performed time-dependent Schroedinger equation (TDSE) calculations for our
experimental parameters, including the magnitude of the force applied. This is displayed by the blue curves inﬁgure 3. Note that the TDSE curves (blue) lie between the lowest band theory (red) and the large system limit(black dashed lines) at the edges of the Brillouin zone, conﬁrming a partial breakdown of adiabaticity (seeappendix C for further detail).
One might suspect that limiting the domain of the linear ﬁt such that band edge effects are excluded would
still provide a good measure of the Chern number. However, as shown in ﬁgure 3(bottom), the slope of the best
ﬁt line for non-trivial topologies, and thus the measured conductivity, depends highly on the selected domain forboth the theoretical (red) , the experimental (black) and numerical (blue) data, and the appropriate choice ofrange is ambiguous. We conclude that for an extremely narrow system such as ours, a conductivity
measurement is insufﬁcient for determining the Chern number [20].
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Figure 3. Hall displacement. Top: modal position m̄ is plotted as a function of qx for the 5-site ribbon with ﬂux (a) Φ/Φ0=0, (b)Φ/Φ0≈−4/3, (c) Φ/Φ0≈4/3. Gray circles depict the measurements; black dashed lines are the prediction of our simple s̃H , red
curves are the expectation from the lowest band theory of our thin ribbon, and blue curves are the result of TDSE calculations for our
experimental parameters including force. Bottom: extracted conductivity from the slope of a line of best ﬁt to the data (gray circles),lowest band theory (red lines), and TDSE calculations (blue lines) as a function of maximum ∣q x ∣ included in the ﬁt range, for each ﬂux
value. As discussed in appendix B, the Φ/Φ0=0 data was compensated to account for non-adiabaticity in the loading procedure.

4. Chern number measurement via TKNN Diophantine equation
To better identify Chern numbers, we relate the TKNN equation (equation (1)) to the physical processes present
in our system. Although the Hofstadter Hamiltonian in equation (2) is only invariant under m-translations thatare integer multiples of Q, a so-called ‘magnetic-displacement’ by Δm=1 along es accompanied by adisplacement in crystal momentum along ex by Dqx 2kR = P Q leaves equation (2) unchanged—this is themagnetic translation operator. Together, these symmetry operations give a Q-fold reduction of the Brillouinzone along es , and add a Q-fold degeneracy, as illustrated in ﬁgure 4(a) for Φ/Φ0=0, 1/3, and 2/5. Recallingthat the Brillouin zone is 2kL periodic along ex , it follows that a displacement by 2kL Q to the nearest
symmetry related state involves an integer C magnetic displacements, shown in ﬁgure 4(b) for Φ/Φ0=1/3 and
2/5, given by solutions to 2kL s - 2kR C = 2kL Q , where s counts the number of times the Brillouin zone was‘wrapped around’ during the C vertical displacements. Because this is exactly the TKNN equation (1), we
identify C as the Chern number. Both C and s directly relate to physical processes. First, each time the Brillouinzone is wrapped around implying a net change of momentum by 2kL a pair of photons must be exchanged
between the optical lattice laser beams. Second, each change of m by 1 must be accompanied by a 2kR recoil
kick imparted by the Raman lasers as they change the spin state. This physical motivation of the TKNN equation
remains broadly applicable even for our narrow lattice, providing an alternate signature of the Chern number.
Figure 5 shows the full evolution of fractional population in each m site as a function of crystal momentum qx
in the lowest band. The black circles locate the peak of the fractional population in each spin state. We identify
the locations of those peaks as the crystal momenta at which the atoms were displaced by a single lattice site
along es starting at qx=0, similar to the suggestions in [19, 20]. We associate the Chern number with the slopeof a linear ﬁt through the three peak locations. The dependence of Chern number extracted in this way on thestrength of applied force is much weaker than the Hall conductivity approach (see appendix C). For the 3-sitewide ribbon, we measured a Chern number of 0.99(4),−0.98(5) for negative and positive ﬂux respectively5, inagreement with the exact theory as calculated from the full Hamiltonian (see appendix D), which predicts ±0.97(1), with uncertainties reﬂecting ﬁt uncertainty of peak locations. For the 5-site wide ribbon, we measured 1.11(2),−0.97(4), close to the theoretical prediction of ±1.07(1).
5

Our Chern number extraction scheme fails for the rf case as the fractional populations are ﬂat and there is no peak. We therefore assign a
Chern number of 0 to ﬂat distributions.

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Figure 4. Chern number from the TKNN equation. (a) Lowest band energy (left) and expected position along es , ámñ (right) within
the Brillouin zone in an extended 2D system, where qx and qs are crystal momenta along e x and es , respectively. Top. Φ/Φ0=0.
Middle. Φ/Φ0=1/3: Brillouin zone shrinks by a factor of 3 and becomes three-fold degenerate, distance between adjacent energy
minima spaced by 2kL Q is labeled. Each minimum corresponds to a different expected position along es . Bottom. Φ/Φ0=2/5. (b)Fractional population in each spin state in the lowest band at qs=0. Top. Φ/Φ0=1/3. Bottom. Φ/Φ0=2/5. A momentum shift
along e x of 2kL Q is accompanied by an integer number of spin ﬂips C. A line connecting magnetic states separated by 2kL Q , withslope C=1 (top) and −2 (bottom), is indicated.

Figure 5. Chern number measurement. Lowest band fractional population measured as a function of crystal momentum in the e x andposition in the es . Darker color indicates higher fractional population. In the Raman-coupled cases, the points represent the ﬁtted
population maxima and the Chern number is extracted from the best ﬁt line to those points. (a) 3-site (left) and 5-site (right) systems
with positive ﬂux. (b) 3-site (left) and 5-site (right) system with zero ﬂux. (c) 3-site (left) and 5-site (right) systems with negative ﬂux.
The parameters for 3-site data were identical to those for 5-site data, see ﬁgure 3(a), except ts = 0.98 (8) tx .

The deviation from unity results from Φ/Φ0−4/3≈0.01, a non-zero quadratic Zeeman shift, and ts>tx
allowing hybridization of states in the vicinity of the edge (see ﬁgure 2 in [20]). The dependence of a Chern
number inferred with this technique on both the size of the system along es and the tunneling ts is studied inﬁgure 6. Figure 6(a) shows the dependence of the Chern number on the width of the ribbon, from our
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Figure 6. Dependence of Chern number inferred via the TKNN Diophantine equation method on system size and tunneling. (a)Chern number dependence on number of sites along es . (b) Chern number dependence on coupling strength along the synthetic
direction for 3 and 5 site systems.

experimental parameters of 3 and 5 sites to an extended system of 70 sites. As seen in the ﬁgure, as the system size
grows the measured Chern number converges to the expected value of 1. These were calculated from the full
Hamiltonian (appendix D), assuming uniform tunneling along es , with tunneling tx = 0.5EL , synthetic
direction coupling  W = 0.5EL and no quadratic shift. Figure 6(b) shows the dependence of the measured
Chern number on the coupling strength along the synthetic direction ÿΩ for lattice widths relevant to ourexperiment—3 and 5 sites. We used the same Hamiltonian and parameter values listed above. In the limit ofvanishing tunneling, both the 3 and 5-site wide Chern number converge to the exact integer value of 1. This
supports the hypothesis that deviation from unity at non-zero coupling strengths is a consequence of thehybridization of states in the vicinity of the edge, which is facilitated by stronger couplings.

5. Conclusion
Our direct microscopic observations of topologically driven transverse transport demonstrate the power of
combining momentum and site-resolved position measurements. With the addition of interactions, thesesystems have been shown to display chiral currents [30], and with many-body interactions are predicted to giverise to complex phase diagrams supporting vortex lattices and charge density waves [31–33]. Realizations ofcontrolled cyclic coupling giving periodic boundary conditions [26] along es could elucidate the appearance ofedge modes as the coupling between two of the three states is smoothly tuned to zero. In addition, due to the
non-trivial topology as well as the low heating afforded by synthetic dimensional systems, a quantum Fermi gas
dressed similarly to our system would be a good candidate for realizing fractional Chern insulators [34].

Acknowledgments
This work was partially supported by the Air Force Ofﬁce of Scientiﬁc Researchs Quantum Matter MURI, NIST,
and NSF (through the Physics Frontier Center at the JQI).

Appendix A. Experimental detailWe created nearly pure 87Rb BECs in a crossed optical dipole trap [27] with frequencies(wx , wy , wz ) 2p = (27.1 (2), 58.4 (8) , 94.2 (5)) Hz . We deliberately used small, low density BECs with ≈103
atoms to limit unwanted scattering processes in regimes of dynamical instability [35]. At various times in thesequence, we used coherent rf and microwave techniques to prepare the hyperﬁne ∣F , mF ñ state of interest. The
1D optical lattice was always ramped on linearly in 300 ms. For non-zero f, we turned on the Raman beams
adiabatically in 30 ms after ramping on the lattice. For f=0 we used adiabatic rapid passage starting inmF=−F and swept the bias magnetic ﬁeld in ≈50 ms to resonance. We applied forces by spatially displacing theoptical dipole beam providing longitudinal conﬁnement (by frequency shifting an acousto-optic modulator),
7

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Figure C1. Dependence of Hall conductivity measurement on force in the 5-site strip. Left: modal position m̄ as a function of crystal
momentum for different magnitudes of applied force, obtained via numerical solutions of the time-dependent Schroedinger equation.
The smallest force (cyan) resulted in adiabatic evolution, while the strongest force (red) resulted in almost full non-adiabaticity. Right:
measured Chern number as a function of applied force, as inferred from the Hall conductivity (blue) and the TKNN Diophantine
equation method (green).

effectively adding a linear contribution to the existent harmonic potential for displacements small compared tothe beam waist.

Appendix B. Rf correction
In experiments where the tunneling along es was induced by a uniform rf magnetic ﬁeld (Φ/Φ0=0), ourloading procedure had remnant non-adiabaticity that led to temporal oscillations in the fractional populations
in different m states at the 40% level. To separate the effects due to this non-adiabaticity from transverse
transport, we performed the experiment with identical preparations without applying the longitudinal force. We
then used the observed oscillations as a function of time without an applied force as a baseline, and report thedifference in fractional populations between that baseline and the cases where the force was applied.

Appendix C. Hall conductivity
The current density can be expressed as j^ = n2D v^ e , where n2D is the 2D charge carrier density, v⊥ is thetransverse velocity and e is the electron charge. Using sH E = F sH e , and choosing some increment of time Δt,
we have v^ = Dx^ Dt , and F =  Dq Dt , where qP is the crystal momentum along the direction of the force.
Re-expressing n2D in number of carriers N per plaquette, deﬁning Dx^ as transverse displacement in units oflattice periods, we obtain NGDx^ Dq = sH RK . We set N=1 as we are looking at motion of single atoms.
The dependence of the measured Hall conductivity for our experimental parameters as a fucntion of themagnitude of applied force is shown in ﬁgure C1. The modal position for every point in the Brillouin zone wascalculated by numerically solving the TDSE. These are shown in the left panel of the ﬁgure, for forces 0.165,
0.141, 0.123, 0.11 , 0.099, 0.09 , 0.082, 0.076, 0.071 and 0.066 EL lL . The right panel shows the measured Hall
conductivity from a linear ﬁt to the full range of crystal momenta as a function of applied force (blue line). For
comparison, we also show the Chern number as inferred via the TKNN Diophantine equation method as afunction of applied force (green line).

Appendix D. Full Hamiltonian
The spin dependent Hamiltonian, in the presence of Raman coupling and a 1D optical lattice can be written as
F ,¥

å

H=

m =-F , n =-¥8

H0 + HR + HL,

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Table D1. Summary of experimental parameters.
Parameter

Value

V
tx
ts

4.4(1) EL
0.154(4) EL
3-site Raman: 0.97(8) tx; 5-site Raman: 1.97(8) tx; rf:
7.4(5) tx
0.02 EL
0.00(2) EL
0.106(5) EL/λLòδFxFigure D1. Band structure of the 3-site system for our experimental parameters, with color representing the expectation value of
position along es , ámñ.

where the diagonal term
H0 = ( 2 (q x - 2mF F0 - 2n)2kL2 2mRb + hdm +  m 2)´ ∣ q x + n2kL , mñá q x + n2kL , m ∣includes the kinetic energy as well as the two-photon Raman detuning from resonance δ and the quadratic
Zeeman shift ò. Here, m represents atomic spin, n represents lattice order in the momentum basis, q is thelongitudinal crystal momentum, and mRb is the atomic mass. The second term represents the Raman couplingwith coupling strength Ω, with anisotropic tunneling arising from the spin-dependent prefactor (Clebsch–
Gordan coefﬁcient):
HR =  W F (F + 1) - m (m + 1) 2 2 ∣ q x + n2kL , mñá q x + n2kL , m + 1∣ + h.c.

Here, h.c. stands for Hermitian conjugate. The third term represents lattice coupling to higher order latticestates, with lattice depth U:
HL = U 4∣ q x + n2kL , mñá q x + (n + 1) 2kL , m ∣ + h.c.

In our experiment,  W = 0.5 (2)EL for Raman coupling and  W = 0.57 (1)EL for rf coupling, U = 4.4 (4) EL for
all preparations. In our calculations, we found that restricting −7<n<7 was sufﬁcient at our energies.
We choose the appropriate tunneling coefﬁcients tx and ts for the approximate tight-binding Hamiltonian
equation (2) as those that optimally reproduce the lowest two bands, relevant for our experiment. A summary of
our experimental parameters is presented in table D1. The band structure of our 3-site wide ribbon is presented
in ﬁgure D1, with three bands plotted as a function of crystal momentum qx colored in accordance with the
expected position along es , ámñ.

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