Multiparticle Interactions for Ultracold Atoms in Optical Tweezers: Cyclic Ring-Exchange Terms

Multiparticle interactions for ultracold atoms in optical tweezers: Cyclic ring-exchange terms

Annabelle Bohrdt,1, 2, ∗ Ahmed Omran,3, ∗ Eugene Demler,3 Snir Gazit,4 and Fabian Grusdt5, 2, 1, † 1Department of Physics and Institute for Advanced Study, Technical University of Munich, 85748 Garching, Germany 2Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 M¨nchen,Germany 3Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 4Racah Institute of Physics and The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University, Jerusalem 91904, Israel 5Department of Physics and Arnold Sommerfeld Center for Theoretical Physics (ASC), Ludwig-Maximilians-UniversitätMünchen,Theresienstr. 37, MünchenD-80333, Germany (Dated: October 2, 2019) Dominant multi-particle interactions can give rise to exotic physical phases with anyonic excita- tions and phase transitions without local order parameters. In spin systems with a global SU(N) symmetry, cyclic ring-exchange couplings constitute the first higher-order interaction in this class. In this letter we propose a protocol how SU(N) invariant multi-body interactions can be implemented in optical tweezer arrays. We utilize the flexibility to re-arrange the tweezer configuration on time scales short compared to the typical lifetimes, in combination with strong non-local Rydberg interactions. As a specific example we demonstrate how a chiral cyclic ring-exchange Hamiltonian can be implemented in a two-leg ladder geometry. We study its phase diagram using DMRG simulations and identify phases with dominant vector chirality, a ferromagnet, and an emergent spin-1 Haldane phase. We also discuss how the proposed protocol can be utilized to implement the strongly frustrated J − Q model, a candidate for hosting a deconfined quantum critical point.

Introduction.– Ultracold atoms in optical lattices have strong frustration can be realized whose ground states are become a versatile platform for performing analogue strongly correlated quantum liquids. quantum simulations, with widely tunable interactions In condensed matter systems multi-spin interactions [1] and the ability to control the single-particle band of this type emerge from higher-order virtual processes structure [2–8]. Using atoms with permanent electric or [42], leading to corrections to the pairwise Heisenberg magnetic dipole moments [9] or in Rydberg states [10] couplings of SU(2) spins in a half-ﬁlled Hubbard model. allows to study systems with long-range dipole-dipole or These cyclic ring-exchange terms play a role in frustrated van-der Waals interactions, which can mimic the long- quantum magnets like solid 3He [43] and possibly also range Coulomb repulsion between electrons in a solid. for the phase diagram of high-Tc cuprate superconductors These ingredients can be combined to study exotic phe- nomena in strongly correlated many-body systems, re- lated for example to quantum magnetism [11–17] or the fractional quantum Hall eﬀect [18–20]. Leveraging the capabilities of ultracold atoms, such experiments promise new insights for example to directly measure topological invariants [21–25] or image the quantum mechanical wavefunction with single-site resolution [26–31]. In this letter, we go beyond the two-body interactions realized so far and propose a general protocol to implement highly symmetric multiparticle interactions with ultracold atoms in optical tweezer arrays. Multiparti- cle interactions can lead to exotic ground states with in- trinsic topological order [32, 33], with applications for FIG. 1. Proposed setup: SU(2)-invariant chiral cyclic ring- quantum computation [34, 35], and they are an impor- exchange interactions can be realized by combining state-

arXiv:1910.00023v1 [cond-mat.quant-gas] 30 Sep 2019 tant ingredient for realizing lattice gauge theories [36–39] dependent lattices generated by optical tweezer arrays and central to the quantum simulation of high-energy phe- strong Rydberg interactions with a central Rydberg-dressed nomena or deconﬁned quantum criticality [40, 41]. If control qubit (C). The auxiliary states |τ = 1i|σi with σ =↑, ↓ these higher-order couplings possess additional symme- (orange) of the atoms on the sites of the plaquette are subject tries, e.g. SU(N) invariance in spin systems, models with to a state-dependent tweezer potential which allows to per- mute them coherently around the center. Our protocol makes use of stroboscopic π pulses between the physical states τ = 0 (green) and the auxiliary states τ = 1, which only take place ∗ These authors contributed equally. collectively on all sites and conditioned on the absence of a † Corresponding author email: [email protected] Rydberg excitation in the control atom. 2

[44, 45]. In this letter we demonstrate how such multi- cally protected edge states [57–59] at intermediate values spin interactions can be realized and independently tuned of π/4 . φ . 3π/4, a ferromagnetic phase for φ & 3π/4 in ultracold atom systems without resorting to high-order and a dominant vector chirality for φ . π/4. virtual processes. Implementation.– For simplicity we consider only a sin- A promising route to implementing multiparticle pro- gle plaquette and for concreteness we restrict ourselves to cesses is to use strong interactions between atoms in dif- Np = 4 sites, see Fig.1. Generalizations of our scheme ferent Rydberg states representing spin degrees of free- to more than one plaquette with any number Np of sites dom. This allows to build a versatile quantum simulator are straightforward, however. which can be used to realize ring-exchange interactions Each of the four sites, labeled j = 1, ..., 4, consists of a in spin systems by representing them as sums of products static optical tweezer trapping a single atom, where re- of Pauli matrices [46], or to implement local constraints cently demonstrated rearrangement methods [49–51] al- giving rise to emergent dynamical gauge fields [47, 48]. low for populating each site with high fidelity. We assume Here we follow a similar strategy but propose to com- that the atoms remain in the vibrational ground states bine strong Rydberg interactions with the capabilities to of the microtraps throughout the sequence. Every atom quickly change the spatial configuration of atoms trapped has two internal states σ =↑, ↓ which we use to implement in optical tweezer arrays [49–51]. We consider general an effective spin-1/2 system. As a specific configuration lattice models with one N-component particle per lattice we suggest to use 133Cs atoms and utilize their F = 3, site (fermionic or bosonic) and show, as an explicit exam- mF = 2, 3 hyperfine states to represent the two spins. ple, how a general class of SU(N)-invariant chiral cyclic Optical pumping with site-resolved addressing can then ring-exchange (CCR) interactions can be realized. They be employed to prepare arbitrary initial spin patterns [56] are described by a Hamiltonian (~ = 1) and study their dynamics under Eq. (1). The key ingredient for our proposed implementation ˆ X iφ ˆ −iφ ˆ† HCCR(φ) = K (e Pp + e Pp ). (1) of CCR interactions is to realize collective permutations p of the entire spin configuration in the plaquette. This can be achieved by physically rotating the tweezer array where the sum is over all plaquettes p of the underly- ˆ† ˆ around the center of the plaquette while ensuring that the ing lattice, the operator Pp (Pp) cyclically permutes the motional and spin states of the atoms are preserved and spin configuration on plaquette p in clockwise (counter- coherence is not lost. The effect of clockwise rotations clockwise) direction and φ is a tunable complex phase. of the microtraps on the spin states is described by the A generalization to finite hole doping, with zero or one operator Pˆ, particle per lattice site, is straightforward.

Non-chiral cyclic ring-exchange interactions, realized Pˆ|σ1, σ2, σ3, σ4i = |σ4, σ1, σ2, σ3i. (2) by Eq. (1) for φ = 0, are believed to play a role in the high-Tc cuprate compounds. These materials can be de- Optimized trajectories can be chosen to cancel heating scribed by the 2D Fermi-Hubbard model on a square effects from the motion [60]. These require a timescale lattice, with weak couplings between multiple layers in set by the quantum speed limit that scales as the inverse z-direction [52]. For the relevant on-site interactions energy gap of each traps trot ∼ 1/∆ε. For deep trapping U, which dominate over the nearest-neighbor tunneling potentials where ∆ε ≈ 150 kHz, rotation times of trot < t U, this model can be simplified by an expansion in 10 µs are achievable. powers of t/U. To lowest order, one obtains a t−J model In contrast to Eq. (2), the effective Hamiltonian leads [53] with nearest-neighbor spin-exchange interactions of to a superposition of permuted and non-permuted states strength J = 4t2/U. Next to leading order, cyclic ring- in every infinitesimal time step ∆t, as can be seen from −iHˆCCR∆t exchange terms on the plaquettes of the square lattice a Taylor expansion: e = 1 − iHˆCCR∆t. To cre- contribute with strength K = 20t4/U 3. By comparison ate such superposition states in our time evolution, we of first principle calculations and measurements in the assume that every atom has a second internal degree of high-temperature regime it was shown that K ≈ 0.13×J freedom labeled by τ = 0, 1. Concretely we propose to re- in La2CuO4 [54] but its effect on the phase diagram realize the new states |τ = 1i|σi in 133Cs atoms by F = 4, mains debated. In ultracold atoms, similar higher-order mF = 3, 4 hyperfine levels, where mF = 3 (mF = 4) processes have been used to realize non-chiral cyclic ring- corresponds to σ =↓ (σ =↑). These additional levels will exchange couplings [55, 56]. be used as auxiliary states, whereas the states |τ = 0i|σi We start by explaining the general scheme using the introduced before – implemented as F = 3, mF = 2, 3 example of CCR interactions. Our method is more ver- levels in 133Cs– realize the physical spin states. satile however, and we discuss how it can be adapted to One part of our protocol consists of a permutation of implement the J − Q model which has been proposed as the spins σ, but only in the manifold of auxiliary states. a candidate system realizing deconfined quantum criti- This step requires a total time trot and can be described cality [40, 41]. We also analyze the phase diagram of the by the unitary transformations CCR Hamiltonian (1) in a ladder geometry, with exactly ˆ Y ˆ Y ˆ ˆ ˆ † one SU(2) spin per lattice site. We show that the phase U+ = |1ijh1| ⊗ P + |0ijh0| ⊗ 1σ, U− = U+ (3) diagram contains a gapped Haldane phase with topologi- j j 3

To implement this evolution, two sets of optical tweezer arrays can be used, of which only one is rotating. We suggest to realize it by the near-magic wavelength λmagic ≈ 871.6 nm in 133Cs which strongly confines atoms in the state τ = 1 but almost does not affect atoms in τ = 0. By applying Uˆ± to superposition states with either all atoms in τ = 1 or all atoms in τ = 0, one can realize the desired superpositions of permuted and non-permuted spin configurations. Such states can be realized by collective π-pulses conditioned upon a control qubit trapped in the center of the plaquette [61], as described next. If the control atom is in the state |+ic it is transferred to a Rydberg state |ric with a resonant π-pulse and Rabi FIG. 2. Proposed protocol: The sequence in (a) is repeated frequency Ω , see Fig.1. If the control atom is in state r periodically with period T = 2(tc + trx). When tc 2π/∆c, |−ic, the laser Ωr is off-resonant and no Rydberg ex- 1/Ωc it implements a trotterized time evolution of the effec- citation is created. Next a Raman transition by lasers tive Hamiltonian Eq. (8), which realizes CCR couplings when (1) (2) ∆ Ω . The individual time steps are illustrated in (b). Ωj ,Ωj through an intermediate Rydberg state |rij c c is used to implement a π-pulse transferring the physical states |0ij to |1ij, without changing their spin state |σij. EIT ˆ ˆ ˆ ˆ ˆ In the presence of a coupling field Ω that establishes where we defined Urx,± = Usw U±ϕc ⊗ U± Usw. As will two-photon resonance to the Rydberg state with each be shown below, Hêff realizes CCR interactions with a Raman laser, electromagnetically induced transparency tunable phase φ = −ϕc and amplitude (EIT) [62] suppresses the transition |0ij ↔ |1ij. How- 2 ever, the EIT condition is lifted by the Rydberg block- 1 Ωc ade mechanism if the control atom is in the Rydberg K = − (tc∆c) (6) 2T ∆c state |ric [61], enabling the transfer. After the transfer is complete, another π-pulse by Ωr is applied to the con- provided that trol atom. This ensures that the control atom remains t 2π/∆ , Ω ∆ . (7) trapped during the protocol, even if the Rydberg excited c c c c state is not subject to a trapping potential. In summary, Now we estimate the strength |K| of the CCR interac- this part is described by the unitary transformation tions that can be achieved with the proposed setup. To satisfy Eq. (7) we assume Ωc = 0.2∆c and tc∆c = 0.4. Y Uˆ = |+i h+| ⊗ |1i h0| + h.c. ⊗ 1ˆ For a rotation time trot = 10µs and assuming tsw, tc sw c j σ t a reasonable strength of K/ = 50Hz × 2π can be j rot ~ achieved. This requires Ωc/2π 1.3kHz, which can be + |−i h−| ⊗ 1ˆ ⊗ 1ˆ . (4) c τ σ easily realized; the condition tsw 10µs can also be met, as the Rydberg π-pulses on the control atom can be The total time required to implement this switch (sw) is executed in ∼ 100 ns each and the Raman transfer be- denoted by tsw. tween the states |0ij and |1ij can be driven with coupling Finally, we need to introduce quantum dynamics be- strengths above 1 MHz. tween the states of the control atom. This can be realized Effective Hamiltonian.– Next we show that our proto- by a dressing laser Ωc driving transitions between |±ic, col realizes the Hamiltonian in Eq. (1). When 2πt at a detuning ∆ . These dynamics take place over a pe- c c 1/∆ , 1/Ω , we can write Uˆ = 1 − iHˆ t and calculate riod of time t and are described by the unitary evolution c c c c c c ˆ −iHˆctc exp[−iHeffT ] to leading order in tc. Eqs. (3)-(4) yield Uˆc = e with Hˆc = ∆c|+ich+| + Ωc |+ich−| + h.c. . During the remaining steps of the protocol, Eqs. (3)- t (4), we assume that Ω = 0 is off and the control atom Hˆ = c 2∆ |+i h+|+ c eff T c c picks up a phase ±ϕc if it is in state |+ic. This phase h i can be adjusted by the detuning ∆c and the duration iϕc † + Ωc |−ich+| 1 + e Pˆ + h.c. . (8) trx = 2tsw + trot, during which the time evolution of the ˆ ∓iϕc control is U±ϕc = |+ich+|e + |−ich−|. The complete protocol is summarized in Fig.2. It When Ωc ∆c we can eliminate the state |+ic which consists of a periodic repetition of the individual steps is only virtually excited. This further simplifies the ef- described above. At the discrete time steps nT , where fective Hamiltonian and we obtain T = 2(tc + trx), the unitary evolution is described by an −iϕc iϕc † Hêff = K 2 + e Pˆ + e Pˆ . (9) effective Hamiltonian Hêff:

ˆ n Up to the energy shift 2K this realizes CCR interactions −inT Heff ˆ n ˆ ˆ ˆ ˆ e = (UT ) = Urx,+UcUrx,−Uc , (5) in an isolated plaquette. The result can be extended to 4

the supplement (SM) we provide an explicitly derivation of a spin-1 model with a gapped Haldane ground state [58, 63] for φ = π/2 and K0 K. For small values of φ, the ground state of the CCR 0 .2 Hamiltonian is a dominant vector chirality phase, as discussed in Ref. [64]. This phase is characterized by corre- 0 .0 lations of the form Sˆx,y × Sˆx0,y0 in a staggered arrange- 0 .2 ment around each plaquette. We ﬁnd that the staggered

0 .1 correlation between diﬀerent rungs, measured from the center L/2 of the chain, 0 .0 D E 0 .2 x ˆ ˆ ˆ ˆ (−1) SL/2,1 × SL/2,2 · SL/2+x,1 × SL/2+x,2 , 0 .1 (10) 0 .0 decays slowly as a function of the distance x and retains signiﬁcant non-zero values over the considered system 2 0 sizes, see Fig.3 (d). The transition between the dominant vector chirality and Haldane phases is a symmetry-

0 protected topological (SPT) phase transition. 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 Using the global SU(2) symmetry, the staggered vec- D ˆx ˆy ˆx ˆy tor chirality becomes 6 SL/2,1SL/2,2 SL/2+x,1SL/2+x,2 FIG. 3. Phase diagram of the CCR Hamiltonian on a ladder, E ˆy ˆx x obtained from DMRG in a system with 64 sites: different ob- −SL/2+x,1SL/2+x,2 (−1) . Measuring it requires ac- servables are evaluated in the ground state of the Hamiltonian cess to two four-point functions of the form hSˆµSˆν SˆλSˆρi (1) to characterize the phases. Upon varying φ, three different i j k l phases can be identified: A topological Haldane phase featur- which can be detected by making use of local addressing ing a vanishing gap in the entanglement spectrum (a) and techniques, see e.g. [65]. To detect the Haldane phase z edge states with a non-zero local magnetization for Stot = 1 experimentally, we propose to study weakly magnetized (b); A symmetry-broken phase around φ = π with long-range systems and image the topological edge states. Alterna- ferromagnetic correlations (c); And a symmetric phase for tively, one could work in the plaquette basis (see SM) and small φ, where the staggered vector chirality remains non- measure the Haldane string order parameter. An inter- vanishing over long distances (d). esting future extension would be to use machine learning techniques to retrieve non-local order parameters from a series of quantum projective measurements. multiple plaquettes by implementing the trotterized time Summary and Outlook.– In summary, we propose a ˆ step UT interchangeably on inequivalent plaquettes. general method for implementing multi-body interactions Two-leg ladder with CCR.– Now we discuss the physics in ultracold atom experiments using optical tweezer ar- of the CCR Hamiltonian in a two-leg ladder. We vary the rays. The approach is particularly useful in the pres- phase φ in the Hamiltonian (1) with K = 1 and calculate ence of additional, e.g. global SU(N) spin, symmetries. the ground state phase diagram using the density-matrix Specifically, we consider a four-body cyclic ring exchange renormalization group (DMRG). For φ = π, the ground term, which can be realized with a combination of multi- state is characterized by ferromagnetic correlations, see qubit gates based on Rydberg states and movable optical Fig.3 (c). It can be readily seen that the variational tweezers. We numerically study the ground state of the energy hHˆCCR(π)i is minimized for ferromagnetic con- cyclic ring exchange Hamiltonian and find different dom- z figurations. In the sector Stot = 0 used in our DMRG inant correlation functions as the complex phase of the simulations in Fig.3 (c), we find phase separation with ring exchange term is varied. two ferromagnetic domains of opposite magnetization. Our work paves the way for future studies of the inter- At intermediate φ we find an emergent Haldane phase, play between ring-exchange and pair-exchange terms, as with two-fold degenerate states in the entanglement spec- discussed in Ref. [66] for the non-chiral case φ = 0. In the z trum, see Fig.3 (a). For a finite Stot = 1 the expectation experimental realization proposed here, it is conceptually ˆz value hSL,1i at the edge is non-zero, see Fig.3 (b). The straightforward to introduce holes into the system, lead- spin gap ∆ES = E0,S=1 − E0,S=0, defined as the differ- ing to a finite doping. The interplay between spin and ence between the ground state energy with and without charge degrees of freedom could be further studied by finite total magnetization, is zero in this phase, since the adding direct tunneling terms, which lead to rich Hamil- additional spin can be placed in the spin-1/2 topologi- tonians in the spirit of t − J like models. The physics of cal edge states of the system without increasing the total this type of model is completely unknown and provides an energy. We corroborate this picture further by consid- exciting prospect for future theoretical and experimental ering the K − K0 model with alternating strengths K, research. The proposed protocol is versatile enough to K0 of the CCR interactions on adjacent plaquettes. In implement larger classes of models with multi-spin inter- 5 actions, such as the J −Q model [41]. In two dimensions, Union FP7 under grant agreement 291763, the Deutsche this model features a phase transition between an anti- Forschungsgemeinschaft (DFG, German Research Foun- ferromagnet and a valence-bond solid, which has been dation) under Germany’s Excellence Strategy–EXC- proposed as a candidate for a deconfined quantum criti- 2111–390814868, DFG grant No. KN1254/1-1, DFG cal point [41]. Moreover, the experimental protocol can TRR80 (Project F8). A.B. acknowledges support by be varied to study different types of problems, such as the Studienstiftung des deutschen Volkes. A.O. acknowl- discrete time-evolutions of complex models or impurity edges support by a research fellowship from the German models, which can be realized by an inclusion of the con- Research Foundation (DFG). E.D. acknowledges support trol qubits into the models. by Harvard-MIT CUA, AFOSR-MURI Quantum Phases of Matter (grant FA9550-14-1-0035), AFOSR-MURI: Acknowledgements.– The authors would like to thank Photonic Quantum Matter (award FA95501610323), T. Calarco, M. Endres, M. Greiner, A. Kaufman, M. DARPA DRINQS program (award D18AC00014). S.G. Knap and M. Lukin for fruitful discussions. A.B. and acknowledges support from the Israel Science Founda- F.G. acknowledge support from the Technical Univer- tion, Grant No. 1686/18. F.G. acknowledges support sity of Munich - Institute for Advanced Study, funded by the Gordon and Betty Moore foundation through the by the German Excellence Initiative and the European EPiQS program.

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SUPPLEMENTARY INFORMATION

A. State preparation 0

In the following we present details about the prepara- -2 tion protocol and relevant experimental parameters that ultimately set the limitations on timescales of the exper- -4 iment. Trap depth (mK) (3,3) (4,4) Applying recently developed techniques of rearranging (3,2) (4,3) atoms to arbitrary patterns [49–51] allows for the initial- -6 ¸magic ization of the desired plaquette pattern with high fidelity. 865 869 873 877 Raman sideband cooling can be applied to prepare the Wavelength ¸ (nm) atoms in the motional ground state of the tweezers. To achieve a large fraction of ground state occupation in all FIG. 4. AC Stark shifts on sublevels of 133Cs vs. wavelength three directions, axial confinement of the atoms within for the rotating traps of 1 µm waist and 3 mW power. Energy the tweezers is necessary. This can be achieved using a shifts on |3, 3i and |3, 2i are smaller than the trap depth of the blue-detuned, higher-order Gaussian beam propagating static tweezers (grey-shaded area). The orange dashed line along the side of the tweezer array. Strong axial confine- marks the optimal wavelength λmagic of the rotating traps. ment also reduces the position uncertainty of the atoms, thereby suppressing fluctuations in interactions between the Rydberg states from run to run. 1 µm, an optical power of around 6 mW per trap is re- For the proposed experiments, we suggest encoding the quired. spin and pseudospin degrees of freedom in the atomic While the static trap depth is flexible, it poses strin- states of 133Cs: gent bounds on the depth of the rotating traps. On the one hand, these should be significantly deeper than the |τ = 0, σ =↑i = |F = 3, mF = 3i static traps to allow for efficient relocation of the target |0, ↓i = |3, 2i qubits in |F = 4i. On the other hand, residual light (11) |1, ↑i = |4, 4i shifts on the |F = 3i states should be small enough as to prevent the atoms in these states from being removed |1, ↓i = |4, 3i from the static traps (Fig.4). Optical pumping allows for preparation of the full pop- ulation in a single hyperfine level such as |1, ↑i, while strong magnetic field gradients or local optical address- 2. Scattering rate ing beams can be used to imprint state-dependent energy shifts and transfer the spin state on individual sites using For the trap parameters assumed earlier, the resulting global addressing with microwaves or Raman beams. off-resonant photon scattering rate of the static traps is on the order of 10 Hz, which is much slower than the desired experimental cycle time. B. Trap parameters For the rotating traps, the near-magic wavelength yields a minimal off-resonant scattering rate on the or- We propose the use of static traps at a far-detuned der of 500 Hz. Therefore it is essential to perform the wavelength to mitigate heating from off-resonant scatter- rotations sufficiently quickly to avoid destruction of the ing. 1064 nm represents a convenient and readily avail- coherent superposition of states, heating or scattering able wavelength for these traps. In addition, a different into different spin states altogether. The dependence of set of traps at a near-magic wavelength between the D1 scattering rate on the rotating trap wavelength and the 133 and D2 lines of Cs is used to transport the F = 4 atomic spin state is shown in Fig.5. hyperfine sublevels, but not F = 3. An optical trap with σ+-polarization and a wavelength of 871.6 nm can strongly shift the |F = 4i sublevels but 3. Rotation timescales not the |F = 3i ones. For trap frequencies on the order of 100 kHz, a fully adiabatic transport process would require a timescale 1. Trap depths much longer than the inverse trap frequency. The necessary number of operations to implement cyclic ring ex- We assume a static trap depth of 1 mK, where good re- change would need a very long total trap duration, at sults on ground state cooling of 133Cs is achievable. As- which the off-resonant scattering rates estimated above suming a wavelength of 1064 nm and a waist of wstat ∼ would severely limit the process fidelity. 9

700 D. The spin-1 Haldane phase (4,4) (3,3) (4,3) (3,2) To obtain a better understanding of the Haldane phase 600 ¸magic observed in our DMRG simulations, we perform a rigorous analytical analysis of a simpliﬁed model with CCR couplings of strength K, K0 on alternating plaquettes, 500 ˆ 0 X iφ ˆ −iφ ˆ† Scattering rate (Hz) HCCR(K ) = K (e Pp + e Pp ) p∈P 400 0 X iφ ˆ −iφ ˆ† 865 869 873 877 + K (e Pp + e Pp ), (15) Wavelength ¸ (nm) p∈P

where P denotes the set including every second plaquette FIG. 5. Off-resonant scattering rates of the rotating traps on 133 and P its complement. the sublevels of Cs for the same parameters as Fig.4. The K − K0 model (15) can be solved exactly in the limit K0/K = 0, where its ground state is a product of decoupled plaquettes P. The eigenstates of a single Therefore, we propose the implementation of opti- ˆ mized trajectories that allow for counterdiabatic trans- plaquette p ∈ P can be labeled by the total spin Sp = P4 ˆ port of atoms with no net heating on a timescale of the j=1 Sj where j = 1...4 denotes the four sites of the inverse energy gap of each trap [60]. Taking into account plaquette (labeled in anti-clockwise direction). z the ramping times of the traps, the total transfer period The two states with Sp = ±2 have an energy 2(φ) = z of a single rotation can take 100 µs, we can perform on 2K cos φ. In the sector with Sp = ±1 there exist four average ∼ 20 rotations until a single photon is scattered states corresponding to the four positions j of the mi- from the traps. nority spin. The eigenstates are plane wave superpositions of different j = 1...4 with discrete momenta pn = nπ/2 for n = 0, 1, 2, 3 and corresponding energy n C. Chiral cyclic ring-exchange on a 2 × 2 plaquette 1 (φ) = 2K cos(φ + pn). Finally there exist six states z with Sp = 0. Four of them correspond to plane-wave It is well known that the four-spin cyclic ring-exchange superpositions of domain wall configurations, including ˆ ˆ† ↑↑↓↓ and all cyclic permutations. They have discrete interaction (Pp + Pp ) can be written in terms of the spin ˆ momenta pn = nπ/2 for n = 0, 1, 2, 3 and the same operators Sj on the four sites j = 1...4 of the plaquette n energy 2 (φ) = 2K cos(φ + pn) as states in the sector (labeled in anti-clockwise direction). To relate our chiral z model in Eq. (1) to spin models of quantum magnetism, Sp = ±1. Two additional states |±i correspond to sym- and for the our numerical implementation of the model metric and anti-symmetric superpositions of Néelstates | ↑↓↑↓i ± | ↓↑↓↑i on the plaquette, with eigenenergies by DMRG, we also write it out in terms of spin operators. ± (φ) = 2K cos(φ + q±) where q+ = 0 and q− = π. As a starting point, we express Pˆp in terms of pairwise 2 spin-permutation operators, In the following we focus on the case when φ = π/2. The ground state of every plaquette is three-fold degen- 1 erate with energy (π/2) = −2K, and the states are Pˆ = Pˆ = 2 Sˆ · Sˆ + , (12) ij ji i j 4 4 1 X ˆ |↑i = e−ijπ/2Sˆ−| ↑↑↑↑i (16) for which Pij|σiσji = |σjσii. Hence 2 j j=1 Pˆp = Pˆ43Pˆ32Pˆ21. (13) 4 1 X |0i = e−ijπ/2Sˆ+Sˆ+ | ↓↓↓↓i (17) Using standard identities for spin operators, we obtain 2 j j−1 j=1 1 4 iφ ˆ −iφ ˆ† ˆ ˆ ˆ ˆ 1 X −ijπ/2 + e Pp + e Pp = cos(φ) + S1 · S2 + S2 · S3 |↓i = e Sˆ | ↓↓↓↓i (18) 4 2 j j=1 + Sˆ3 · Sˆ4 + Sˆ4 · Sˆ1 + Sˆ2 · Sˆ4 + Sˆ1 · Sˆ3 + 4(Sˆ · Sˆ )(Sˆ · Sˆ ) + 4(Sˆ · Sˆ )(Sˆ · Sˆ ) Since the single plaquette is SU(2) invariant, this triplet 1 2 3 4 1 4 2 3 of states corresponds to the sector S = 1 where the total p ˆ − 4(Sˆ1 · Sˆ3)(Sˆ2 · Sˆ4) + 2 sin(φ) Sˆ1 · (Sˆ2 × Sˆ3) spin on the plaquette is Sp = Sp(Sp + 1). The three states | ↑ip, |0ip, | ↓ip define a system of spin-1 operators Jˆ on every plaquette p ∈ P. When + Sˆ · (Sˆ × Sˆ ) + Sˆ · (Sˆ × Sˆ ) + Sˆ · (Sˆ × Sˆ ) . p 1 3 4 1 2 4 2 3 4 |K0| K, and without loss of generality K > 0, they (14) are protected by a gap ∆ ≈ K from further state and K0 10 only introduces coupling between neighboring plaquettes can be implemented. By integrating out the n-th control (n) (n) hp, qi. Making use of SU(2) invariance, we calculated the atom, with detuning ∆c Ωc , an effective Hamilto- 0 P iφ ˆ resulting matrix elements of the term K p∈P(e Pp + nian of the form −iφ ˆ† e Pp ) analytically. This leads to the following effective (n) 2 Hamiltonian, X (Ωc ) Hˆ ∝ e−iϕc Pˆ(n) + h.c. (21) eff (n) p p,n ∆c X is obtained. We envision that control atoms can be stored Hêff = 0 + λ cos(θ) Jˆp · Jˆq in a register and moved into the center of the plaquette hp,qi individually when they are needed for the protocol. 2 + sin(θ) Jˆp · Jˆq (19) As a specific example, we discuss an implementation of the J − Q model [41] on a ladder. The conventional 31 0 way to write the J − Q Hamiltonian [41] is in terms of where 0 = −2K + 72 K . The remaining two coupling ˆ constants are given by spin operators Sj on the sites j of the lattice,

K0 K0 ˆ X ˆ ˆ X ˆ ˆ λ cos(θ) = , λ sin(θ) = , (20) HJQ = J Si · Sj − Q Pi,j Pk,l, (22) 16 144 hi,ji hijkli i.e. λ = 0.0629K0 and θ = 0.035π. For these parameters, the effective Hamiltonian is very where hijkli denotes a sequence of corners of a plaquette. close to a Heisenberg spin-1 chain. Because of the small The second term describes projectors on singlets, second term ∝ λ sin(θ) λ cos(θ), the model interpo- lates between the exactly solvable AKLT model [67] and 1 ˆ = Sˆ · Sˆ − . (23) the simple Heisenberg spin-1 chain. Hence the ground Pi,j i j 4 state of the effective Hamiltonian (19) is gapped [57] and has spin-1/2 edge states [58] reflecting the symmetry pro- We use a representation in terms of pairwise permutation tected topological order [59, 68]. 1 operators Pî,j = 2 Sî · Sˆj + , for which We checked numerically by exact diagonalization of nu- 4 merically accessible system sizes that the system remains gapped at φ = π/2 when the ratio K0/K is continu- 2Q + J X Q + J X Hˆ = Pˆ + Pˆ ously tuned from 0 to 1. Since the system remains in- JQ 2 i,j 2 i,j version symmetric around the central bond of the ladder, hi,ji∈R hi,ji∈L and this symmetry is sufficient to protect the topologi- Q X − Pˆ Pˆ (24) cal character of the topological Haldane phase [68], this 4 i,j k,l establishes that the homogeneous ladder with K = K0 is hijkli in a non-trivial symmetry-protected phase at φ = π/2. up to an overall energy shift, which depends on the The SU(2) symmetry of the system is also sufficient to P P protect the topological Haldane phase [68]. boundary conditions. Here hi,ji∈R ( hi,ji∈L) denotes a sum over all links on the rungs (legs) of the ladder and the last term contains a sum over sequences of corners E. The J − Q model hijkli of the plaquettes. To implement Eq. (24), we propose to use one control Our proposed protocol is versatile enough to imple- atom per link hi, ji, to realize the first and second terms ment larger classes of models with multi-spin interac- ∝ Pî,j . In addition, two control atoms per plaquette are tions. In our derivation of Eq. (9) above we only used required to realize Pˆ12Pˆ34 and Pˆ14Pˆ23 respectively; here that fact that Pˆ†Pˆ = 1 and the cyclic ring-exchange op- the sites of the plaquette are labeled by integers 1, 2, 3, 4 erator Pˆ can be replaced by an arbitrary permutation P in anti-clockwise direction around the plaquette. of spins. Moreover, introducing more than one control Similar extensions can be envisioned for implementing qubit per plaquette allows to implement multiple such the J − Q model in two dimensions. This model features (n) terms Pp per plaquette p: For every control atom n a phase transition around J/Q ≈ 0.04 between an an- associated with plaquette p a coupling term tiferromagnet and a valence-bond solid, which has been proposed as a candidate for a deconfined quantum critical h (n) i (n) −iϕc ˆ(n) ∝ Ωc |−, nich+, n| 1 + e Pp + h.c. point [41].