Strongly Correlated Quantum Walks in Optical Lattices

Philipp M. Preiss,1 Ruichao Ma,1 M. Eric Tai,1 Alexander Lukin,1 Matthew

Rispoli,1 Philip Zupancic,1 Yoav Lahini,1, 2 Rajibul Islam,1 and Markus Greiner1, ∗

1Department of Physics, Harvard University, Cambridge, Massachusetts, 02138, USA2Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139, USA

Full control over the dynamics of interacting, indistinguishable quantum particles is an importantprerequisite for the experimental study of strongly correlated quantum matter and the implemen-tation of high-fidelity quantum information processing. Here we demonstrate such control over thequantum walk - the quantum mechanical analogue of the classical random walk - in the strong in-teraction regime. We directly observe fundamental effects such as the emergence of correlations dueto quantum statistics and interactions in two-particle quantum walks, as well as strongly correlatedBloch oscillations in tilted optical lattices. Our approach can be scaled to larger systems, greatlyextending the class of problems accessible via quantum walks, which can now serve as a basis foruniversal quantum computation and as a quantum simulator for strongly correlated many-bodydynamics.

Quantum walks are the quantum-mechanical ana-logues of classical random walk processes, describing thepropagation of quantum particles on periodic potentials[1, 2]. Unlike classical objects, particles performing aquantum walk can be in a superposition state and take allpossible paths through their environment simultaneously,leading to superior properties such as faster propaga-tion and enhanced sensitivity to initial conditions. Theseproperties have generated considerable interest in usingquantum walks for the study of position-space quantumdynamics and for quantum information processing [3].Two distinct models of quantum walk were devised, andlater shown to be equivalent: the discrete time quan-tum walk [1], in which the particle propagates in discretesteps determined by a dynamic internal degree of free-dom, and the continuous time quantum walk [2], in whichthe dynamics is described by a time-independent latticeHamiltonian.

Experimentally, quantum walks have been imple-mented for photons [4], trapped ions [5, 6], and neutralatoms [7–9], among other platforms [4]. Until recently,most experiments were aimed at observing the quantumwalks of a single quantum particle, which are describedby classical wave equations.

An enhancement of quantum effects emerges whenmore than one indistinguishable particle participates inthe quantum walk simultaneously. In such cases, quan-tum correlations develop due to Hanbury Brown andTwiss (HBT) interference and quantum statistics, aswas investigated theoretically [10, 11] and experimentally[12–17]. In the absence of interactions this problem lacksfull quantum complexity, but can become intractable byclassical computing [11].

The inclusion of interaction between indistinguishablequantum walkers [18, 19] will grant access to a muchwider class of computationally hard problems. For exam-ple, many-body quantum walks offer a route to a system-atic and direct study of outstanding and computationallyintractable problems in condensed matter physics, such

Position x (sites)

Tim

e (

)

a

b

0.0

0.2

0.4

0.60.00

0.25

0.50

0.75

1.00

−10 0 100.0

0.10.05

t=0

t=6.6 ms

−10 0 10

0.0

0.5

1.0

1.5

2.0

J

Tim

e (

)

II

t=5 ms

t=11 ms

t=0

I

J

III

I

II

II

I

II

III

I

FIG. 1. Coherent single-particle quantum walks. (a) Left:Individual atoms performing independent quantum walks inan optical lattice. Right: The single-particle density distri-bution expands linearly in time, and atoms delocalize over≈ 20 sites (lower panel, fit to equation (2) with the tunnelingrate J as a free parameter). Error bars: standard error of themean. (b) In the presence of a gradient, a single particle un-dergoes Bloch oscillations. The atom initially delocalizes butmaintains excellent coherence and re-converges to its initialposition after one period. Densities are averages over ≈ 700and ≈ 200 realizations for a) and b), respectively.

as many-body localization and the role of interactionsin dynamics of quantum disordered systems [20]. Simi-larly, in the presence of interactions the quantum walk

arX

iv:1

409.

3100

v1 [

cond

-mat

.qua

nt-g

as]

10

Sep

2014

2

can yield universal and efficient quantum computation[21].

Here we implement the two-particle quantum walkwith strong interactions for ultracold bosonic atoms inan optical lattice. Our system realizes the fundamentalbuilding block of interacting many-body systems withatom-resolved access to the strongly correlated dynamicsin a quantum gas microscope [22]. We directly measurethe spatial correlations arising due to quantum statisticsand interactions in multi-particle quantum walks and de-tect the “fermionization” of strongly repulsive bosons inone dimension [23]. In the presence of a lattice tilt, we ob-serve high-fidelity Bloch oscillations in position space [24]and coherent dynamics of individual repulsively boundpairs [25–27]. Our techniques, including the determinis-tic preparation of few-body states and the atom-resolvedmeasurements of correlations, establish quantum walksof ultracold atoms as a tool for the study of correlatedmany-body dynamics and as a platform for quantum in-formation processing devices.

In our experiment, ultracold atoms of bosonic 87Rbperform quantum walks in decoupled one-dimensionaltubes of an optical lattice with spacing d = 680 nm. Theatoms may tunnel in the x-direction with amplitude Jand experience a repulsive on-site interaction U , realiz-ing the Bose-Hubbard Hamiltonian

HBH =∑〈i,j〉

−Ja†iaj +∑i

U

2ni(ni − 1) +

∑i

E ini (1)

Here a†i (ai) is the bosonic creation (annihilation) oper-

ator and ni = a†iai gives the atom number on site i. Jand U are tunable via the depth Vx of the optical lattice,specified in units of the recoil energy Er = 2π × h

8md2 ≈2π × 1240 Hz, where h is Planck’s constant and m is themass of 87Rb. The energy shift per lattice site E is setby a magnetic field gradient. We measure time in unitsof inverse tunneling rates, τ = tJ , and define the dimen-sionless interaction u = U/J and gradient ∆ = E/J .

We set the initial motional state of the atoms through ahighly adaptable single-site addressing scheme, enablingthe deterministic preparation of a wide range of few-body states: Using a digital micromirror device (DMD)as an amplitude hologram in a Fourier plane, we gener-ate arbitrary diffraction-limited potentials in the plane ofthe atoms. Starting from a low-entropy two-dimensionalMott insulator with a fixed number of atoms per site,we project a repulsive Hermite-Gauss profile to isolateatoms in selected rows while a short reduction of the op-tical lattice depth ejects all other atoms from the system(see Methods). For the quantum walk, we prepare one ortwo rows of atoms along the y-direction of a deep opticallattice (Vx = Vy = 45Er), as shown in Figure 1 a). Thequantum walk is performed at a reduced lattice depthVx, while the y-lattice and the out-of-plane confinementare fixed at Vy = 45Er and ωz = 2π×7.2 kHz. The atom

positions are recorded with single-site resolution usingfluorescence imaging in a deep optical lattice, giving theparity of the local atom number [22]. We employ a den-sity mapping along the direction of the quantum walk torecord outcomes with two atoms on the same site andobtain the full two-particle correlator Γi,j = 〈a†ia

†jaiaj〉

and the density distribution. Before data analysis, wepost-select events with the correct number of atoms perrow (see Methods).

We first study quantum walks of individual atoms, il-lustrated in Figure 1 a). A single particle is initializedat a chosen site in each horizontal tube and propagatesin the absence of an external force. For each individ-ual realization the particle is detected on a single lat-tice site, while the average over many experiments yieldsthe single-particle probability distribution. In contrastto a classical random walk, for which slow, diffusive ex-pansion of the Gaussian density distribution is expected,coherent interference of all single-particle paths leads toballistic transport with well-defined wavefronts [4]. Themeasured probability density ρ grows linearly in time, ingood agreement with the theoretical expectation [24]

ρi(t) = |Ji(2Jt)|2 (2)

where Ji is a Bessel function of the first kind on latticesite i.

If a potential gradient is applied to ultracold atomsin an optical lattice, net transport does not occur dueto the absence of dissipation and the separation of thespectrum into discrete bands. Instead, the gradient in-duces a position-dependent phase shift and causes atomsto undergo Bloch oscillations [28]. For a fully coher-ent single-particle quantum walk with gradient ∆, theatom remains localized to a small volume and undergoesa periodic breathing motion in position space [24] witha maximal half width LB = 4/∆ and temporal periodTB = 2π/∆ in units of the inverse tunneling. Figure 1 b)shows a single-particle quantum walk with ∆ = 0.56, re-sulting in Bloch oscillations over ≈ 14 lattice sites. Weobserve a high quality revival after one Bloch period witha peak density of 〈n〉 = 0.88(2), limited by the tempo-ral resolution of the measurements and inhomogeneousbroadening across the lattice. For individual rows, weobserve revival probabilities of 0.96(3) at τ = TB . Thetime-average of the fidelity, defined as

F (t) =∑x

√px(t)qx(t) (3)

for the measured and expected probability distributionspx(t) and qx(t), is 98.1(1)%, indicating that an excep-tional level of coherence is maintained while the particledelocalizes over ≈ 10µm in the optical lattice.

If two particles undergo a quantum walk simultane-ously, the dynamics are sensitive to the underlying parti-cle statistics due to Hanbury Brown and Twiss interfer-ence [12, 13]. All two-particle processes in the system add

3

The

ory

Exp

erim

ent

−10 0 10−10

0

10

−10 0 10 −10 0 10 −10 0 100

0.02

0.04

0.06

(II) strongly interacting bosons fermionized regime

time ttime t

constructive for bosonsdestructive for fermions constructive for fermions

destructive for bosonsa b

c

(I) (II)

LR LR

L R L R

−10 0 10−10

0

10

−10 0 10 −10 0 10 −10 0 100 0.1 0.2 0 0.1 0.20

0.02

0.04

0.06

Position i (sites)

Pos

ition

j (s

ites)

(I) u = 0.7 (II) u = 1.4 (III) u = 2.4 (IV) u = 5.1

0 0.1 0.2 0 0.1 0.2

0 0.1 0.2 0 0.1 0.2 0 0.1 0.2 0 0.1 0.2

2n steps 2n steps 2n+2 steps 2n steps

(I) weakly interacting bosons

=0=0

FIG. 2. Hanbury Brown and Twiss interference and fermionization. (a) Processes connecting the initial and final two-particlestates interfere coherently. Each tunneling step contributes a phase i. For non-interacting bosons, processes of the samelength add constructively (I), while processes differing in length by two steps interfere destructively (II). (b) Weakly interactingbosons display strong bunching (I). Strong, repulsive on-site interactions cause bosons in one dimension to fermionize anddevelop long-range anti-correlations (II). Measured correlator Γi,j at time τmax ≈ 2π× 0.5, averaged over ≈ 3200 realizations.The interactions are tuned from weak (u < 1) to strong (u� 1) by choosing Vx = 1Er, 2.5Er, 4Er, and 6.5Er.

coherently, leading to quantum correlations between theparticles, shown in Figure 2 a): For bosons, the processesbringing both particles into close proximity of each otheradd constructively, leading to bosonic bunching, as ob-served in expanding atomic clouds [29, 30] and photonicimplementations of quantum walks [12, 13].

In our experiment, the bunching of free bosonic atomsis apparent in single shot images of quantum walks withtwo particles starting from adjacent sites in the statea†0a†1|0〉. For weak interactions, the two atoms are very

likely to be detected close to each other due to HBT in-terference, as shown in raw images in Figure 2 b). Wecharacterize the degree of bunching using the density-density correlator Γi,j in Figure 2 c), measured at timeτmax ≈ 2π × 0.5. Panel I shows the two-particle correla-tor for a quantum walk with weak interactions (u = 0.7).Sharp features are due to quantum interference anddemonstrate the good coherence of the two-particle dy-namics. The concentration of probability on and near thediagonal of the correlator Γi,j indicates HBT interferenceof nearly free bosonic particles.

For a quantum walk of fermions, a reversal of the HBTlogic is expected: exchange of fermionic particles leadsto an additional phase shift of π, causing characteristicanti-bunching of free fermions [13, 30]. In one dimen-

sion, a link exists between bosonic and fermionic sys-tems, captured by the concept of fermionization: In thelimit of infinite, “hard-core” repulsive interactions, thedensities and spatial correlations of a one-dimensionalbosonic system are predicted to be identical to thoseof non-interacting spinless fermions [23]. Signatures ofthis behavior have been observed in equilibrium throughthe pair-correlations and momentum distributions oflarge one-dimensional Bose-Einstein Condensates in theTonks-Girardeau regime [31, 32].

We directly probe the process of fermionization inthe dynamics of two particles by repeating the quantumwalk from initial state a†0a

†1|0〉 at increasing interaction

strengths [18]. Figure 2 c) shows Γi,j for increasing valuesof u. At intermediate values of the interaction u = 1.4and u = 2.4, the correlation distribution is relatively uni-form, as repulsive interactions compete with HBT inter-ference. For the strongest interaction strength u = 5.1,most of the weight is concentrated on the anti-diagonalof Γi,j , corresponding to pronounced anti-bunching. Theanti-correlations are strong enough to be visible in rawimages of the quantum walk as in panel II of Figure2 b), and Γi,j is almost identical to the expected out-come for non-interacting fermions. Note that while thecorrelations change dramatically with increasing inter-

4

Position i (sites)

Pos

ition

j (s

ites)

(I) u = 0.7 (II) u = 1.4 (III) u = 2.4 (IV) u = 5.1

−10 0 10−10

0

10

−10 0 10 −10 0 10 −10 0 100 0.1 0.2 0 0.1 0.2 0 0.2 0.4 0 0.5 1.0

−10

0

10

0 0.1 0.2 0 0.1 0.2 0 0.2 0.4 0 0.5 1.0

0 0.07 0.14 0.21 0 0.05 0.1 0 0.5 1.0 0 0.02 0.04 0.06

Theo

ry

Exp

erim

ent

FIG. 3. Formation of repulsively bound pairs. Two-particle correlations at τmax ≈ 2π × 0.5 for two particles starting on site 0in state 1√

2a†0a†0|0〉. For weak interactions (u = 0.7), the atoms perform independent single-particle quantum walks. As the

interaction strength is increased, repulsively bound pairs form and undergo an effective single-particle quantum walk along thediagonal of the two-particle correlator. Experimental parameters are identical to those in Figure 2.

action, the densities remain largely unchanged. At allinteraction strengths, the observed densities and corre-lations are in excellent agreement with a numerical in-tegration of the Schrodinger equation with Hamiltonian(1) (see Methods). Our measurements directly reveal thecorrespondence between strongly interacting bosons andnon-interacting fermions for dynamics in one dimension.

The precise control over the initial state in our sys-tem enables the study of strongly interacting bosons inscenarios not described by fermionization, such as thequantum walks of two atoms prepared in the same state.Figure 3 shows the correlations and densities for the ini-tial state 1√

2a†0a†0|0〉. Since both atoms originate from

the same site, HBT interference terms are not present.In the weakly interacting regime (u = 0.7), both parti-cles undergo independent free dynamics and the correla-tor is the direct product of the single-particle densities.As the interaction increases, separation of the individ-ual atoms onto different lattice sites becomes energet-ically forbidden. The two atoms preferentially propa-gate through the lattice together, reflected in increas-ing weights on the diagonal of the correlation matrix.For the strongest interactions, the particles form a repul-sively bound pair with effective single-particle behavior[25]. The two-particle dynamics may be described as aquantum walk of the bound pair [18, 19] at a decreasedtunneling rate Jpair , which reduces to the second-order

tunneling [33] Jpair = 2J2

U � J for large values of u.

The formation of repulsively bound pairs and their co-

herent dynamics can be observed in two-particle Blochoscillations. We focus on the dynamics of two parti-cles initially prepared on the same site with a gradient∆ ≈ 0.5, as shown in Figure 4. In the weakly interact-ing regime (u = 0.3), both particles undergo symmetricBloch oscillations as in the single-particle case, and weobserve a high-quality revival after one Bloch period. Forintermediate interactions (u = 2.4), the density evolutionis very complex: In this regime where J , U , and E aresimilar in magnitude, states both with and without dou-ble occupancy are energetically allowed and contribute tothe dynamics. The skew to the right against the appliedforce is due to resonant long-range tunneling of singleparticles over several sites [26, 34] and agrees with nu-merical simulation.

When the interactions are sufficiently strong (u = 3.5),the pairs of atoms are tightly bound by the repulsiveinteraction and behave like a single composite parti-cle. However, the effective gradient has doubled withrespect to the single-particle case, and the pairs performBloch oscillations at twice the fundamental frequencyand reduced spatial amplitude. The frequency-doublingof Bloch oscillations was predicted for electron systems[35] and cold atoms [26] and has recently been simulatedwith photons in a waveguide array [27]. Throughoutthe breathing motion, the repulsively bound pairs them-selves undergo coherent dynamics and delocalize withoutunbinding. The clean revival after half a Bloch perioddirectly demonstrates the spatial entanglement of atompairs.

5

0.0

0.5

1.0

0.0 0.5 1.0 1.5 2.0

(I) u = 0.3 (II) u = 2.4 (III) u = 3.5

U

Position (sites)

−10 0 10

0.0

0.5

1.0

Experiment

Theory

−10 0 10

Tim

e (

)Ti

me

(

)

−10 0 10

FIG. 4. Bloch oscillations of repulsively bound pairs. (I) Inthe weakly interacting regime, two particles initialized on thesame site undergo clean, independent Bloch oscillations. (II)Increasing the interaction strength leads to complex dynam-ics: Pairs of atoms remain bound near the origin or separate,breaking left-right symmetry via long-range tunneling. (III)For the largest interactions, repulsively bound pairs performcoherent, frequency-doubled Bloch oscillations. Densities areaverages over ≈ 220 independent quantum walks.

In summary, we have experimentally realized quantumwalks of strongly interacting bosons in an optical lat-tice. Our work extends the class of problems accessiblevia quantum walks to strongly correlated systems andenables a “bottom-up” approach to many-body dynam-ics: We observe excellent quantum coherence over morethan 10µm, sensitivity to particle statistics in HBT in-terference, and strong correlations due to interactions,thus combining all aspects of strongly correlated systemsin a fully controlled setting. Our techniques will enabledetailed studies of condensed matter problems, such asthe interplay of interactions and disorder [20], as well asquantum information processing with ultracold atoms inoptical lattices.

We acknowledge helpful discussions with Manuel En-dres. This work was supported by grants from the ArmyResearch Office with funding from the DARPA OLE pro-gram and a MURI program, an AFOSR MURI program,the Moore Foundation, and by grants from the NSF.M.E.T. was supported by the DoD through the NDSEGprogram. Y.L. is supported by the Pappalardo Fellow-ship in Physics.

∗ Email: greiner@physics.harvard.edu[1] Y. Aharonov, L. Davidovich, and N. Zagury, Phys. Rev.

A 48, 1687 (1993).[2] E. Farhi and S. Gutmann, Phys. Rev. A 58, 915 (1998).[3] S. E. Venegas-Andraca, Quantum Inf. Process. 11, 1015

(2012).[4] K. Manouchehri and J. Wang, Physical Implementation

of Quantum Walks (Springer-Verlag, Berlin, 2014).[5] H. Schmitz, R. Matjeschk, C. Schneider, J. Glueckert,

M. Enderlein, T. Huber, and T. Schaetz, Phys. Rev.Lett. 103, 090504 (2009).

[6] F. Zahringer, G. Kirchmair, R. Gerritsma, E. Solano,R. Blatt, and C. F. Roos, Phys. Rev. Lett. 104, 100503(2010).

[7] M. Karski, L. Forster, J.-M. Choi, A. Steffen, W. Alt,D. Meschede, and A. Widera, Science 325, 174 (2009).

[8] C. Weitenberg, M. Endres, J. F. Sherson, M. Cheneau,P. Schauß, T. Fukuhara, I. Bloch, and S. Kuhr, Nature471, 319 (2011).

[9] T. Fukuhara, P. Schauß, M. Endres, S. Hild, M. Cheneau,I. Bloch, and C. Gross, Nature 502, 76 (2013).

[10] Y. Bromberg, Y. Lahini, R. Morandotti, and Y. Silber-berg, Phys. Rev. Lett. 102, 253904 (2009).

[11] S. Aaronson and A. Arkhipov, Proc. 43rd ACM STOC ,333 (2011).

[12] A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda,A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail,K. Worhoff, Y. Bromberg, Y. Silberberg, M. G. Thomp-son, and J. L. OBrien, Science 329, 1500 (2010).

[13] L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni,A. Crespi, R. Ramponi, and R. Osellame, Phys. Rev.Lett. 108, 010502 (2012).

[14] M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove,S. Aaronson, T. C. Ralph, and A. G. White, Science339, 794 (2013).

[15] J. B. Spring, B. J. Metcalf, P. C. Humphreys,W. S. Kolthammer, X.-M. Jin, M. Barbieri, A. Datta,N. Thomas-Peter, N. K. Langford, D. Kundys, J. C.Gates, B. J. Smith, P. G. R. Smith, and I. A. Walm-sley, Science 339, 798 (2013).

[16] M. Tillmann, B. Dakic, R. Heilmann, S. Nolte, A. Sza-meit, and P. Walther, Nature Photon. 7, 540 (2013).

[17] A. Crespi, R. Osellame, R. Ramponi, D. J. Brod, E. F.Galvao, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mat-aloni, and F. Sciarrino, Nature Photon. 7, 545 (2013).

[18] Y. Lahini, M. Verbin, S. D. Huber, Y. Bromberg, R. Pu-gatch, and Y. Silberberg, Phys. Rev. A 86, 011603(2012).

[19] A. Ahlbrecht, A. Alberti, D. Meschede, V. B. Scholz,A. H. Werner, and R. F. Werner, New J. Phys. 14,073050 (2012).

[20] D. L. Shepelyansky, Phys. Rev. Lett. 73, 2607 (1994).[21] A. M. Childs, D. Gosset, and Z. Webb, Science 339, 791

(2013).[22] W. S. Bakr, J. I. Gillen, A. Peng, S. Folling, and

M. Greiner, Nature 462, 74 (2009).[23] M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, and

M. Rigol, Rev. Mod. Phys. 83, 1405 (2011).[24] T. Hartmann, F. Keck, H. J. Korsch, and S. Mossmann,

New J. Phys. 6, 2 (2004).[25] K. Winkler, G. Thalhammer, F. Lang, R. Grimm,

6

J. Hecker Denschlag, A. J. Daley, A. Kantian, H. P.Buchler, and P. Zoller, Nature 441, 853 (2006).

[26] R. Khomeriki, D. O. Krimer, M. Haque, and S. Flach,Phys. Rev. A 81, 065601 (2010).

[27] G. Corrielli, A. Crespi, G. Della Valle, S. Longhi, andR. Osellame, Nat. Commun. 4, 1555 (2013).

[28] M. B. Dahan, E. Peik, J. Reichel, Y. Castin, and C. Sa-lomon, Phys. Rev. Lett. 76, 4508 (1996).

[29] S. Folling, F. Gerbier, A. Widera, O. Mandel, T. Gericke,and I. Bloch, Nature 434, 481 (2005).

[30] T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen,V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang,D. Boiron, A. Aspect, and C. I. Westbrook, Nature 445,402 (2007).

[31] B. Paredes, A. Widera, V. Murg, O. Mandel, S. Folling,I. Cirac, G. Shlyapnikov, T. Hansch, and I. Bloch, Na-ture 429, 277 (2004).

[32] T. Kinoshita, T. Wenger, and D. S. Weiss, Science 305,1125 (2004).

[33] S. Folling, S. Trotzky, P. Cheinet, M. Feld, R. Saers,A. Widera, T. Muller, and I. Bloch, Nature 448, 1029(2007).

[34] F. Meinert, M. J. Mark, E. Kirilov, K. Lauber, P. Wein-mann, M. Grobner, A. J. Daley, and H.-C. Nagerl, Sci-ence 344, 1259 (2014).

[35] W. S. Dias, E. M. Nascimento, M. L. Lyra, and F. A.B. F. de Moura, Phys. Rev. B 76, 155124 (2007).

[36] J. Simon, W. S. Bakr, R. Ma, M. E. Tai, P. M. Preiss,and M. Greiner, Nature 472, 307 (2011).

[37] R. Ma, M. E. Tai, P. M. Preiss, W. Bakr, J. Simon, andM. Greiner, Phys. Rev. Lett. 107, 095301 (2011).

7

METHODS

State Initialization.

We start with a two-dimensional Mott insulator in the|F,mF 〉 = |1,−1〉 hyperfine state with one or two atomsper site, prepared in a deep lattice (Vx = Vy = 45Er).Using a digital micromirror device (DMD) in a Fourierplane as an amplitude hologram, we generate arbitraryoptical potentials with single-site resolution. We su-perimpose a blue-detuned beam (λ = 760 nm) with aHermite-Gauss profile along x (waist 700 nm) and a flat-top profile along y (half length 5µm), with a typicalpeak depth of 25Er. Subsequently, we switch off thex-lattice in the presence of a large anti-confining beamfor 40 ms. Only atoms in rows coinciding with the nodesof the Hermite-Gauss beam are retained, while all otheratoms are expelled from the system before the x-lattice isramped back on. We thus deterministically prepare oneor two rows of atoms along y (length ≈10 sites), with atypical single-site loading fidelity of 98%. In each experi-mental run, we realize 6−8 independent one-dimensionalquantum walks in decoupled adjacent tubes.

Data Analysis.

We post-select outcomes for which the atom distribu-tion after the quantum walk is consistent with the par-ity projection of our imaging scheme. For single-particlequantum walks, we keep only one-dimensional tubes withexactly one atom. For two-particle quantum walks, weretain outcomes with either two or zero atoms, and as-sume the latter always corresponds to atom loss due toparity projection only.

The two-particle correlator is obtained from a his-togram of the imaged atom positions. Parity projectionprevents us from observing pairs of atoms and from di-rectly obtaining the on-site correlation Γi,i. To circum-vent this, we apply a density mapping prior to imagingfor half of the data set: Pairs of atoms are convertedinto two atoms on neighbouring sites along x and viceversa ((2, 0) ⇔ (1, 1)) with high fidelity by ramping amagnetic field gradient from ∆ ≈ 0.5u to ∆ ≈ 2u at a re-duced lattice depth of Vx = 16Er in 200 ms [36]. There-

fore the on-site correlation Γi,i is obtained from the firstoff-diagonal elements of the histogram with the densitymapping, while Γi,i+1 is extracted from the histogramwithout the density mapping. To get the full correlator,we combine the two histograms weighted by the numberof post-selected realizations in each half of the data set.

The aforementioned assumption in post-selection en-sures the proper normalization of the histograms and isverified by comparing the far off-diagonal elements in thetwo weighted histograms, which typically differ by lessthan 3%.

For two particles, the density distribution 〈ni〉 is thenobtained by summing the correlator along one axis:〈ni〉 =

∑j Γi,j .

Bose-Hubbard Parameters.

We initially calibrate lattice depths using Kapitza-Dirac scattering with an uncertainty of 10%. Single-particle Bloch oscillations serve as our most sensitiveprobe of the tunneling J with a typical uncertainty of5%, in agreement with a band structure calculation. Theinteraction U is measured at 14Er with photon-assistedtunneling in a tilted lattice [37], and extrapolated toother lattice depths using a numerical calculation.

All theory plots are obtained from a direct numericalsolution of the Schrodinger equation with Hamiltonian(1) in the Fock space of two particles on 23 lattice sites.The values of U and tmax are fixed, while J (and E in thecase of Bloch oscillations) are left as free parameters tominimize the rms error between measured and calculateddensities.

The minimization is performed simultaneously on datasets from Figures 2 and 3, except for panel IV. Param-eter values for all data sets are listed in Table I. Thefitted values Jfit are generally in good agreement withthe measurements from single-particle dynamics. At lowlattice depths of 1−3Er, next-nearest-neighbor hoppingis significant, resulting in dynamics up to 20% faster thanexpected from Hamiltonian (1). For the deep lattice at6.5Er, residual gradients of ≈ 20 Hz/site affect the dy-namics, leading to a slower quantum walk than for ∆ = 0and to the strong peak near the origin in panel IV of Fig-ure 2 c).

8

Data Set Vx [Er] Jsp/(2π) [Hz] U/(2π) [Hz] Jfit/(2π) [Hz] Efit/(2π) [Hz] tmax[ms]Fig. 1 a 4.5 97(6) - 107 - 6.6Fig. 1 b 2.5 160(9) - 166 93 14Fig. 2 & 3 (I) 1 227(12) 161 274 - 2.1Fig. 2 & 3 (II) 2.5 160(9) 216 168 - 3.0Fig. 2 & 3 (III) 4 108(4) 255 109 - 5.0Fig. 2 (IV) 6.5 59(3) 299 42 - 10.9Fig. 3 (IV) 6.5 59(3) 299 34 - 10.9Fig. 4 (I) 2.5 160(9) 53 173 97 12.8Fig. 4 (II) 4 108(4) 255 101 54 22.8Fig. 4 (III) 5 80(6) 279 81 34 34

TABLE I. Bose-Hubbard parameters used for theory plots. Vx are approximate lattice depths. Jsp is the nearest-neighbortunneling obtained from single-particle Bloch oscillations or directly from a band structure calculation. Typical errors on Uare 3% from the uncertainty in the calibration. Jfit and Efit are the results from fitting density distributions.