www.sciencemag.org/content/349/6255/1510/suppl/DC1

Supplementary Materials for

Observation of chiral edge states with neutral fermions in synthetic Hall

ribbons

M. Mancini, G. Pagano, G. Cappellini, L. Livi, M. Rider, J. Catani, C. Sias, P. Zoller, M.

Inguscio, M. Dalmonte, L. Fallani*

*Corresponding author. E-mail: fallani@lens.unifi.it

Published 25 September 2015, Science 349, 1510 (2015)

DOI: 10.1126/science.aaa8736

This PDF file includes:

Supplementary Text

Figs. S1 to S4

Full Reference List

Supplementary Text

Experimental Setup

The starting point for our experiment is a 173Yb spin-polarized degenerate Fermi gas withNat ' 1.6× 104 atoms at a temperature T ' 0.2TF (where TF is the Fermi temperature). Quantumdegeneracy is achieved by forced evaporation of a (m = −5/2) + (m = +5/2) nuclear spin mixturein an optical dipole trap with mean geometric frequency ω/2π ' 80 Hz. After evaporation, theatoms in the m = +5/2 state are removed by a resonant laser pulse and a spin-polarized Fermi gasin the m = −5/2 state is left. The atoms are then confined in a three-dimensional cubic opticallattice with periodicity d = λL/2 = 380 nm. The three lattice depths are set to V0x = 6.5ERand V0y = V0z = 30ER (where ER = h2/2Mλ2L is the recoil energy, h is the Planck’s constantand M is the atomic mass). Along y and z the tunneling rates (ty,z/2π ' 1 Hz) are negligible onthe timescale of the experiment, leading to the realization of an array of ≈ 1000 independent 1Dfermionic chains characterized by a longitudinal harmonic confinement with frequency ωx/2π ' 55Hz. The dynamics is allowed only in the shallow lattice along the x direction where t/2π ∼ 90 Hz.The lattice occupation is less than 1 atom per site on the central chain, ensuring that the atoms arein a metallic state in the lowest energy band.

Realization of the fermionic Hall ribbons

In this section we describe the laser configuration that produces a synthetic magnetic field in theHall ribbon. The complex tunneling along the synthetic dimension m is implemented with two off-resonant λ = 556 nm laser beams with angular frequencies ω and ω + ∆ω. We choose a detuningof +1.87 GHz with respect to the narrow intercombination transition 1S0 → 3P 1(F

′ = 7/2) inorder to reduce the inelastic photon scattering rate. The different periodicity of the Raman couplingdR = λ/ [2 sin(θ/2)], with θ being the relative angle between the two beams, with respect to thelattice spacing d, gives rise to a non-zero Peierls phase ϕ = 2π(d/dR). In our setup we chooseθ = 19.5 that, considering the projection along the longitudinal axis of the fermionic chains, leadsto a flux ϕ = 0.37π (see Fig. S1). The sign of the flux can be reversed by swapping the frequenciesof the two Raman beams. The Raman beams couple up to three nuclear spin projections of the1S0 ground states m = (−5/2,−1/2,+3/2) via ∆m = ±2 processes. These states play the roleof lattice sites in the extra-dimension m, while the Rabi frequencies of the two-photon processesare equivalent to the intersite tunneling amplitudes. The atoms are subjected to a B0 = 152 Gauss(real) magnetic field along z generating a linear Zeeman splitting ∆Z = 31.6(7) kHz betweenadjacent nuclear spin components. In order to excite σ+/σ− Raman transitions (Fig. S2), the beamsfrequency difference is set to ∆ω/2π ' 2∆Z and the polarization is carefully chosen depending onthe ladder configuration (see below).

The Raman Rabi frequencies and the light shifts are determined by considering the weightedsum of the matrix elements over the hyperfine structure F ′ = (7/2, 5/2, 3/2) of the 3P1 electronicexcited state:

1

ξα =∑q

∑F ′

|Ωqα1(F

′)|2 + |Ωqα2(F

′)|2

2∆F ′, (S.1)

Ωα =∑F ′

Ω+α1(F

′)Ω−α2(F′)

2∆F ′, (S.2)

where q = (+,−, 0) is the photon polarization index, ∆F ′ is the single-photon detuning withrespect to the F ′ state of the 3P1 manifold and Ωq

αi is the single-photon Rabi frequency related tothe i = 1, 2 Raman beam and q-polarization component. Here α is the synthetic lattice site index(see also Fig. 1 of the main text) and Ωα indicates the coupling between α and α+ 1. With thesedefinitions and considering also the overall harmonic confinement along x, the total Hamiltonian ofthe system can be written as:

H =N∑j=1

3∑α=1

[−t(c†j,αcj+1,α + h.c.)

]+

N∑j=1

2∑α=1

Ωα

2(eiϕjc†j,αcj,α+1 + h.c.)

+N∑j=1

3∑α=1

ξαnj,α +Wx

N2

N∑j=1

3∑α=1

nj,α

(j − N + 1

2

)2

.

(S.3)

In order to take into account the modification of the atomic density caused by the interparticlerepulsion, we consider a slightly lower effective trap strength, Wx ' 0.025N2t (corresponding to atrap frequency ∼ 40 Hz along x). Actually, the results depend only weakly on the particular trapfrequency used in the calculations.

In order to calibrate the parameters of the system, we set the optical lattice beam along x atV0x = 30ER so that the coherent dynamics occurs only along the synthetic dimension (t Ωα),where we observe coherent spin oscillations up to 10 periods. Under these conditions the kineticenergy and harmonic confinement can be neglected and the Hamiltonian can be written as a 3× 3matrix in a frame rotating at frequency 2∆Z :

H/~ =

ξ1 − δ Ω1/2 0

Ω1/2 ξ2 Ω2/2

0 Ω2/2 ξ3 + δ

, (S.4)

where δ is the detuning with respect to the two-photon resonance and Ω2 = 1.41Ω1. By tuning thepolarization of the two Raman beams we can choose whether to implement the 2- or the 3-leg Hallribbon.

2-leg ladder. In this case, we choose ε = (ε+ + ε−)/√

2, i.e. linear polarization in the planexy, for both Raman beams. By setting δ = ξ1 − ξ2, we let the first two states to be resonantlycoupled and exploit the light shift to isolate the third one, obtaining diagonal matrix elements(ξ1 − δ, ξ2, ξ3 + δ) = (0, 0, 2.64)Ω1 in the rotating frame, after the subtraction of a constant energy

2

shift (which is irrelevant for all the measurable properties of the system). Under this condition thepopulation of the third state is at most a few percent, and can thus be neglected.

3-leg ladder. In order to resonantly couple all three states, we use an out-of-plane linearpolarization ε = (ε+ + ε− + ε0)/

√3. In this case the light shifts are approximately the same for

all the states (apart from corrections induced by the Zeeman shift of the excited state 3P1) and bysetting δ = ξ1 − ξ2 we obtain diagonal matrix elements (ξ1 − δ, ξ2, ξ3 + δ) = (0, 0, 0.16)Ω1, afterthe subtraction of a constant energy shift. Under this condition the three states are all substantiallypopulated.

Adiabatic loading procedure

In order to adiabatically load the equilibrium state of the Hall ribbon we start with a spin-polarized m = −5/2 Fermi gas and load it into the optical lattice (Fig. S3) with a 150 msexponential ramp to the final values V0x = 6.5ER and V0y = V0z = 30ER. During the latticeloading, the dipole trap is switched off with a 40 ms exponential ramp to decompress the atomiccloud and reduce the overall harmonic confinement to ωx/2π ' 55 Hz. After 5 ms, we switch onthe Raman beams with an initial detuning δin ∼ −25Ω1 and perform an exponential frequencysweep of the form:

δ(t) = δin + (δf − δin)

(1− e−t/τ

1− e−Tadiab/τ

), (S.5)

where δf = ξ1−ξ2. The ramp duration Tadiab ranges from 20 to 80 ms depending on the experimentalconfiguration, with τ ranging from 5 to 25 ms accordingly. The adiabaticity of the whole process isverified experimentally by reversing the whole procedure to recover a spin-polarized Fermi gas.

In order to assess the validity of our adiabatic loading procedure we also conduct a numericalstudy. We define a time-dependent Hamiltonian where the detuning plays the role of a time-dependent chemical potential. Then, using the same exponential detuning ramp used in theexperimental setup, we simulate the evolution of the system starting from a spin-polarized Fermigas in a one-dimensional optical lattice. We therefore compare the lattice momentum distributioncalculated at the end of the evolution with the one obtained from the ground state of the Hamiltonianwith δ = δf (see Fig. S4). The comparison between the two distributions indicates that the ramp ineq. (S.5) creates only a small number of excitations in the preparation of the Hall ribbon.

Spin-selective imaging

After the ground state loading, we suddenly switch off the Raman coupling, therefore freezingthe population along the synthetic direction. Then we switch off the real lattice with a tmap = 1.35ms exponential ramp, which is slower than the timescale associated to the lattice band-gap. Thisprocedure allows us to adiabatically map the lattice momentum distribution along the chains ontothe atomic velocity distribution (29, 30), which is measured by absorption imaging after 23 ms ofballistic expansion.

In order to perform a single-site imaging along the synthetic direction, we use a sequence ofspin-selective laser pulses (“blast” pulses), in resonance with different components of the narrowintercombination transition 1S0 → 3P1(F

′ = 7/2), to remove atoms in all the spin states but one.

3

The sequence is carried out during the first 2.5 ms of ballistic expansion. At this time the (real)magnetic field is Bblast = 15 Gauss (see Fig. S3), leading to a Zeeman shift ∆Z ∼ 50(Γ/2π)between adjacent spin components in the 3P1 manifold (where Γ = 2π × 180 kHz is the naturallinewidth of the transition). This separation allows us to use two opportunely detuned σ+ and σ−

beams to remove the unwanted spin population, without causing any heating to the selected spin stateleft in the expanding cloud. After ballistic expansion, absorption imaging is performed on the dipole-allowed 1S0 → 1P1 (F ′ = 7/2) transition at 399 nm (with natural linewidth Γ′ = 2π × 28.9 MHz)with a resonant pulse along z.

Image analysis

Lattice momentum distribution and J vs Ω/t. The lattice momentum unbalance in a given leg ismeasured by taking the average of at least 30 images for each spin state with the procedure shownin Fig. S3. In order to remove residual gradients or fringes due to imperfections in the imagingsetup, also background images are acquired, averaged and subtracted from those with atoms. Theresulting images (examples are shown in the upper rows of Figs. 2a and 3a) are then integratedalong the y direction in order to obtain the n(k) curves for the different legs. The n(k) curves foreach spin state are normalized in such a way as to have

∫n(k)dk = 1. Here, and in the figures

of the main text, k is expressed in units of the lattice wavenumber kL = π/d, so that k = ±1

correspond to the boundaries of the first Brillouin zone. The quantity J =∫ 1

0h(k)dk for a given

spin state is calculated by analyzing the average image, while the errors are the standard deviationof the statistical distribution obtained after performing bootstrapping on the experimental data.

Edge-cyclotron orbits. In the quench dynamics experiments shown in Fig. 4, in order to studythe evolution along real space, we determine the average position 〈x〉 of the atomic cloud assumingthe validity of the semiclassical equation of motion along the real lattice (31). In this framework,considering the lowest band dispersion as E(k) = 2t [1− cos(kd)], the velocity of the k-componentof the Fermi sea is

vk =1

~∂E(k)

∂k=

2td

~sin(kd). (S.6)

Then, knowing the tunneling t, we can measure the average velocity of the whole cloud at a giventime τ from the experimental lattice momentum distribution n(k, τ):

〈v(τ)〉 =2td

~

∫n(k, τ) sin(kd)dk. (S.7)

Finally, to obtain the average position at a given time τ , we integrate the average velocity over time:

〈x(τ)〉 =

∫ τ

0

〈v(τ ′)〉dτ ′. (S.8)

In order to estimate the errors bars of the edge-cyclotron orbits in Fig. 4d of the main text, weperform a bootstrap analysis over a set of different 〈x(τ)〉 reconstructed from a random samplingof the experimental images.

4

Theoretical description

In this section, we provide more details on the simulations presented in the paper.

J versus Ω/t. In order to quantify the chiral character of both 2- and 3-leg ladders, we definedthe quantity

J =

∫ 1

0

h(k)dk (S.9)

withh(k) = n(k)− n(−k). (S.10)

We computed this quantity for the Hamiltonian in Eq. (S.3) in the central region of the system,with a peak density of 0.8/0.9 particles per rung in the center, and an approximate size of up to15 µm (corresponding to up to L = 40 lattice rungs with density > 10−5). For the data in Figs. 2and 3 of the main text, we considered a single Hall ribbon comprising 75 sites and 20 particles,and had the light shifts as the only free parameters in the microscopic Hamiltonian for the 3-legcase (in the 2-leg case the results are not very sensitive to the light shift on the −1/2 leg, so thatwe kept the one extracted from the calibration, as described in Section S.II). In the 3-leg case, wefound that the best quantitative agreement with the experimental data is obtained by using a set oflight shifts ξα that slightly differ from the nominal ones. In particular, for the simulations displayedin Fig. 3c, we used (ξ1, ξ2, ξ3) = (0, 0.1,−0.26)Ω1. The discrepancy between these best-fittingvalues and the nominal ones can be easily ascribed to the experimental uncertainty in the Ramanbeams polarization, whose variations have a very strong effect on the light shifts.

In order to take into account thermal fluctuations and possible effects of the non-adiabatic loadingduring the lattice ramp, we simulated systems with an average of 20% thermal excitations abovethe Fermi sea (both the 2-leg and 3-leg case), and performed averages over up to 200 configurations.These results are approximately recovered by investigating a finite-temperature partition functionwith temperature T ' 0.4t. The width of the theoretical curves in Figs. 2d and 3c, representingthe effect of thermal excitations, has been evaluated as the 68.27% percentile interval centeredaround the mean value of the statistical distribution obtained with a bootstrapping analysis of 100realizations of the simulation.

In addition, we have performed simulations over the entire system (up to 2× 104 particles in∼ 1000 independent ribbons) using a local density approximation in the grand canonical ensemble.The results (not displayed) are qualitatively the same as for the simple model used here.

Chiral dynamics at the edge. For the quench dynamics illustrated in Fig. 4 of the manuscript,we solved numerically the time-evolution of systems at different densities. Indeed, it turned out thatassuming a single value for the density is usually not sufficient to capture the full time-dependentdynamics at the edge. As a matter of fact, in the different ladders of the system, each with a differentdensity, there is a very different fraction of particles participating in the edge dynamics. Whilstnot affecting drastically the behavior of J , this inhomogeneity has strong effects on the singleedge-dynamics probed by a quench. In order to take into account the inhomogeneity effect, wehave simulated systems at various system sizes up to L = 55 lattice rungs and different densities.For the data shown in Fig. 4, we considered a system of L = 35 rungs and averaged over a setof different densities (0.2, 0.4, 0.6, 0.8) (equally weighted). We notice that, when taken singularly,none of these density realizations captures the system dynamics correctly, while the average gives agood quantitative agreement with the experimental results.

5

ωω + ∆ω θ

λL

x

y

B0z

FIG. S1. Optical setup for the realization of the hybrid real+synthetic lattice. Red arrows: optical latticebeams. Green arrows: Raman beams.

6

FIG. S2. Level diagram for the ∆m = ±2 Raman transitions providing the tunneling mechanism in thesynthetic dimension of the Hall ribbon.

7

FIG. S3. Timing of the experimental procedure (see text for details).

8

FIG. S4. Simulation of the adiabatic loading of the Hall ribbon. The solid lines show the calculated latticemomentum distribution for the m = −5/2 leg in a three-leg system at different times while the ramp ofEq. (S.5) is executed. The parameters for this evolution are Tadiab = 40 ms and τ = 12.5 ms, whileΩ1/t = 3.65. The dashed line shows the lattice momentum distribution as calculated from the ground stateof the Hamiltonian with δ = δf .

9

References and Notes

1. M. Lewenstein, A. Sanpera, V. Ahufinger, Ultracold Atoms in Optical Lattices: Simulating

Quantum Many-Body Systems (Oxford Univ. Press, Oxford, 2012).

2. I. Bloch, J. Dalibard, S. Nascimbène, Quantum simulations with ultracold quantum gases. Nat.

Phys. 8, 267–276 (2012). doi:10.1038/nphys2259

3. D. Jaksch, P. Zoller, Creation of effective magnetic fields in optical lattices: The Hofstadter

butterfly for cold neutral atoms. New J. Phys. 5, 56 (2003). doi:10.1088/1367-

2630/5/1/356

4. J. Dalibard, F. Gerbier, G. Juzeliūnas, P. Öhberg, Colloquium : Artificial gauge potentials for

neutral atoms. Rev. Mod. Phys. 83, 1523–1543 (2011).

doi:10.1103/RevModPhys.83.1523

5. N. Goldman, G. Juzeliūnas, P. Öhberg, I. B. Spielman, Light-induced gauge fields for

ultracold atoms. Rep. Prog. Phys. 77, 126401 (2014). Medline doi:10.1088/0034-

4885/77/12/126401

6. Y.-J. Lin, R. L. Compton, K. Jiménez-García, J. V. Porto, I. B. Spielman, Synthetic magnetic

fields for ultracold neutral atoms. Nature 462, 628–632 (2009). Medline

doi:10.1038/nature08609

7. M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, I. Bloch, Realization of the

Hofstadter Hamiltonian with ultracold atoms in optical lattices. Phys. Rev. Lett. 111,

185301 (2013). Medline doi:10.1103/PhysRevLett.111.185301

8. H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Burton, W. Ketterle, Realizing the Harper

Hamiltonian with laser-assisted tunneling in optical lattices. Phys. Rev. Lett. 111, 185302

(2013). Medline doi:10.1103/PhysRevLett.111.185302

9. G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif, T. Esslinger,

Experimental realization of the topological Haldane model with ultracold fermions.

Nature 515, 237–240 (2014). Medline doi:10.1038/nature13915

10. F. D. M. Haldane, Model for a quantum Hall effect without Landau levels: Condensed-matter

realization of the “parity anomaly”. Phys. Rev. Lett. 61, 2015–2018 (1988). Medline

doi:10.1103/PhysRevLett.61.2015

11. L. Duca, T. Li, M. Reitter, I. Bloch, M. Schleier-Smith, U. Schneider, An Aharonov-Bohm

interferometer for determining Bloch band topology. Science 347, 288–292 (2015).

Medline doi:10.1126/science.1259052

12. X.-G. Wen, Quantum Field Theory of Many-Body Systems (Oxford Univ. Press, Oxford,

2004).

13. R. E. Prange, S. M. Girvin, The Quantum Hall Effect (Springer, New York, 1990).

14. A. Celi, P. Massignan, J. Ruseckas, N. Goldman, I. B. Spielman, G. Juzeliūnas, M.

Lewenstein, Synthetic gauge fields in synthetic dimensions. Phys. Rev. Lett. 112, 043001

(2014). Medline doi:10.1103/PhysRevLett.112.043001

15. N. Y. Yao, C. R. Laumann, A. V. Gorshkov, H. Weimer, L. Jiang, J. I. Cirac, P. Zoller, M.

D. Lukin, Topologically protected quantum state transfer in a chiral spin liquid. Nat.

Commun. 4, 1585 (2013). Medline doi:10.1038/ncomms2531

16. C. Nayak, S. H. Simon, A. Stern, M. Freedman, S. Das Sarma, Non-Abelian anyons and

topological quantum computation. Rev. Mod. Phys. 80, 1083–1159 (2008).

doi:10.1103/RevModPhys.80.1083

17. N. H. Lindner, E. Berg, G. Refael, A. Stern, Fractionalizing Majorana fermions: Non-

Abelian statistics on the edges of Abelian quantum Hall states. Phys. Rev. X 2, 041002

(2012).

18. During the completion of this manuscript, we became aware of a closely related experimental

work with Raman-coupled Bose-Einstein condensates (28).

19. O. Boada, A. Celi, J. I. Latorre, M. Lewenstein, Quantum simulation of an extra dimension.

Phys. Rev. Lett. 108, 133001 (2012). Medline doi:10.1103/PhysRevLett.108.133001

20. Full details on the experimental and theoretical procedures are available as supplementary

materials on Science Online.

21. M. A. Cazalilla, A. M. Rey, Ultracold Fermi gases with emergent SU(N) symmetry. Rep.

Prog. Phys. 77, 124401 (2014). Medline doi:10.1088/0034-4885/77/12/124401

22. S. Taie, R. Yamazaki, S. Sugawa, Y. Takahashi, An SU(6) Mott insulator of an atomic Fermi

gas realized by large-spin Pomeranchuk cooling. Nat. Phys. 8, 825–830 (2012).

doi:10.1038/nphys2430

23. G. Pagano, M. Mancini, G. Cappellini, P. Lombardi, F. Schäfer, H. Hu, X.-J. Liu, J. Catani,

C. Sias, M. Inguscio, L. Fallani, A one-dimensional liquid of fermions with tunable spin.

Nat. Phys. 10, 198–201 (2014). doi:10.1038/nphys2878

24. M. Atala, M. Aidelsburger, M. Lohse, J. T. Barreiro, B. Paredes, I. Bloch, Observation of

chiral currents with ultracold atoms in bosonic ladders. Nat. Phys. 10, 588–593 (2014).

doi:10.1038/nphys2998

25. Y.-J. Lin, K. Jiménez-García, I. B. Spielman, Spin-orbit-coupled Bose-Einstein condensates.

Nature 471, 83–86 (2011). Medline doi:10.1038/nature09887

26. P. Wang, Z. Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H. Zhai, J. Zhang, Spin-orbit coupled

degenerate Fermi gases. Phys. Rev. Lett. 109, 095301 (2012). Medline

doi:10.1103/PhysRevLett.109.095301

27. L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr, M. W. Zwierlein, Spin-

injection spectroscopy of a spin-orbit coupled Fermi gas. Phys. Rev. Lett. 109, 095302

(2012). Medline doi:10.1103/PhysRevLett.109.095302

28. B. K. Stuhl, H.-I Lu, L. M. Aycock, D. Genkina, I. B. Spielman, Visualizing edge states with

an atomic Bose gas in the quantum Hall regime. Science 349, 1514–1518 (2015).

29. M. Greiner, I. Bloch, O. Mandel, T. W. Hänsch, T. Esslinger, Exploring phase coherence in a

2D lattice of Bose-Einstein condensates. Phys. Rev. Lett. 87, 160405 (2001). Medline

doi:10.1103/PhysRevLett.87.160405

30. M. Köhl, H. Moritz, T. Stöferle, K. Günter, T. Esslinger, Fermionic atoms in a three

dimensional optical lattice: Observing Fermi surfaces, dynamics, and interactions. Phys.

Rev. Lett. 94, 080403 (2005). Medline doi:10.1103/PhysRevLett.94.080403

31. For the full two-dimensional case, another term, taking into account the Berry phase in

momentum space, also affects x at the level of semiclassical trajectories [see (32)].

However, this is not applicable for the present case, in which the real-space dynamics of

the particles is strictly 1D.

32. M.-C. Chang, Q. Niu, Berry phase, hyperorbits, and the Hofstadter spectrum: Semiclassical

dynamics in magnetic Bloch bands. Phys. Rev. B 53, 7010–7023 (1996). Medline

doi:10.1103/PhysRevB.53.7010