
T. Vovk, H. Pichler Quantum trajectory entanglement in various unravelings of Markovian dynamics,
Phys. Rev. A 110 12207 (20240708),
http://dx.doi.org/10.1103/PhysRevA.110.012207 doi:10.1103/PhysRevA.110.012207 (ID: 721269)
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The cost of classical simulations of quantum manybody dynamics is often determined by the amount of entanglement in the system. In this paper, we study entanglement in stochastic quantum trajectory approaches that solve master equations describing open quantum system dynamics. First, we introduce and compare adaptive trajectory unravelings of master equations. Specifically, building on [Phys. Rev. Lett. 128, 243601 (2022)], we study several greedy algorithms that generate trajectories with a low average entanglement entropy. Second, we consider various conventional unravelings of a onedimensional open random Brownian circuit and locate the transition points from area to volumelawentangled trajectories. Third, we compare various trajectory unravelings using matrix product states with a direct integration of the master equation using matrix product operators. We provide concrete examples of dynamics, for which the simulation cost of stochastic trajectories is exponentially smaller than the one of matrix product operators.

Z. J. Li, G. Giudici, H. Pichler Variational manifolds for ground states and scarred dynamics of blockadeconstrained spin models on two and three dimensional lattices,
Phys. Rev. Research 6 23146 (20240509),
http://dx.doi.org/10.1103/PhysRevResearch.6.023146 doi:10.1103/PhysRevResearch.6.023146 (ID: 721240)
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We introduce a variational manifold of simple tensor network states for the study of a family of constrained models that describe spin1/2 systems as realized by Rydberg atom arrays. Our manifold permits analytical calculation via perturbative expansion of one and twopoint functions in arbitrary spatial dimensions and allows for efficient computation of the matrix elements required for variational energy minimization and variational time evolution in up to three dimensions. We apply this framework to the PXP model on the hypercubic lattice in 1D, 2D, and 3D, and show that, in each case, it exhibits quantum phase transitions breaking the sublattice symmetry in equilibrium, and hosts quantum many body scars out of equilibrium. We demonstrate that our variational ansatz qualitatively captures all these phenomena and predicts key quantities with an accuracy that increases with the dimensionality of the lattice, and conclude that our method can be interpreted as a generalization of meanfield theory to constrained spin models.

I. Cong, N. Maskara, M. C. Tran, H. Pichler, G. Semeghini, S. F. Yelin, S. Choi, M. Lukin Enhancing Detection of Topological Order by Local Error Correctio,
Nat. Commun. 15 1527 (20240220),
http://dx.doi.org/10.1038/s41467024455846 doi:10.1038/s41467024455846 (ID: 720882)
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The exploration of topologicallyordered states of matter is a longstanding goal at the interface of several subfields of the physical sciences. Such states feature intriguing physical properties such as longrange entanglement, emergent gauge fields and nonlocal correlations, and can aid in realization of scalable faulttolerant quantum computation. However, these same features also make creation, detection, and characterization of topologicallyordered states particularly challenging. Motivated by recent experimental demonstrations, we introduce a new approach for quantifying topological states  locally errorcorrected decoration (LED)  by combining methods of error correction with ideas of renormalizationgroup flow. Our approach allows for efficient and robust identification of topological order, and is applicable in the presence of incoherent noise sources, making it particularly suitable for realistic experiments. We demonstrate the power of LED using numerical simulations of the toric code under a variety of perturbations, and we subsequently apply it to an experimental realization of a quantum spin liquid using a Rydbergatom quantum simulator. Extensions to the characterization of other exotic states of matter are discussed.

R. Ott, T. Zache, N. Maskara, M. Lukin, P. Zoller, H. Pichler Probing topological entanglement on large scales,
(20240822),
arXiv:2408.12645 arXiv:2408.12645 (ID: 721272)
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Topologically ordered quantum matter exhibits intriguing longrange patterns of entanglement, which reveal themselves in subsystem entropies. However, measuring such entropies, which can be used to certify topological order, on large partitions is challenging and becomes practically unfeasible for large systems. We propose a protocol based on local adiabatic deformations of the Hamiltonian which extracts the universal features of longrange topological entanglement from measurements on small subsystems of finite size, trading an exponential number of measurements against a polynomialtime evolution. Our protocol is general and readily applicable to various quantum simulation architectures. We apply our method to various stringnet models representing both abelian and nonabelian topologically ordered phases, and illustrate its application to neutral atom tweezer arrays with numerical simulations.

C. Fromonteil, R. Tricarico, F. Cesa, H. Pichler HamiltonJacobiBellman equations for Rydbergblockade processes,
(20240220),
arXiv:2402.12956 arXiv:2402.12956 (ID: 721246)
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We discuss timeoptimal control problems for two setups involving globally driven Rydberg atoms in the blockade limit by deriving the associated HamiltonJacobiBellman equations. From these equations, we extract the globally optimal trajectories and the corresponding controls for several target processes of the atomic system, using a generalized method of characteristics. We apply this method to retrieve known results for CZ and Cphase gates, and to find new optimal pulses for all elementary processes involved in the universal quantum computation scheme introduced in [Physical Review Letters 131, 170601 (2023)].

Z. Zeng, G. Giudici, H. Pichler Quantum dimer models with Rydberg gadgets,
(20240216),
arXiv:2402.10651 arXiv:2402.10651 (ID: 721245)
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The Rydberg blockade mechanism is an important ingredient in quantum simulators based on neutral atom arrays. It enables the emergence of a rich variety of quantum phases of matter, such as topological spin liquids. The typically isotropic nature of the blockade effect, however, restricts the range of natively accessible models and quantum states. In this work, we propose a method to systematically overcome this limitation, by developing gadgets, i.e., specific arrangements of atoms, that transform the underlying Rydberg blockade into more general constraints. We apply this technique to realize dimer models on square and triangular geometries. In these setups, we study the role of the quantum fluctuations induced by a coherent drive of the atoms and find signatures of U(1) and Z2 quantum spin liquid states in the respective ground states. Finally, we show that these states can be dynamically prepared with high fidelity, paving the way for the quantum simulation of a broader class of constrained models and topological matter in experiments with Rydberg atom arrays.

R. Ott, T. Zache, M. Prüfer, S. Erne, M. Tajik, H. Pichler, J. Schmiedmayer, P. Zoller Hamiltonian Learning in Quantum Field Theories,
(20240102),
arXiv:2401.01308 arXiv:2401.01308 (ID: 721175)
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We discuss Hamiltonian learning in quantum field theories as a protocol for systematically extracting the operator content and coupling constants of effective field theory Hamiltonians from experimental data. Learning the Hamiltonian for varying spatial measurement resolutions gives access to field theories at different energy scales, and allows to learn a flow of Hamiltonians reminiscent of the renormalization group. Our method, which we demonstrate in both theoretical studies and available data from a quantum gas experiment, promises new ways of addressing the emergence of quantum field theories in quantum simulation experiments.

F. Cesa, H. Pichler Universal Quantum Computation in Globally Driven Rydberg Atom Arrays,
Phys. Rev. Lett. 131 170601 (20231024),
http://dx.doi.org/10.1103/PhysRevLett.131.170601 doi:10.1103/PhysRevLett.131.170601 (ID: 721145)
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We develop a model for quantum computation with Rydberg atom arrays, which only relies on global driving, without the need of local addressing of the qubits: any circuit is executed by a sequence of global, resonant laser pulses on a static atomic arrangement. We present two constructions: for the first, the circuit is imprinted in the trap positions of the atoms and executed by the pulses; for the second, the atom arrangement is circuitindependent, and the algorithm is entirely encoded in the global driving sequence. Our results show in particular that a quadratic overhead in atom number is sufficient to eliminate the need for local control to realize a universal quantum processor. We give explicit protocols for all steps of an arbitrary quantum computation, and discuss strategies for error suppression specific to our model. Our scheme is based on dualspecies processors with atoms subjected to Rydberg blockade constraints, but it might be transposed to other setups as well.

D. GonzálezCuadra, D. Bluvstein, M. Kalinowski, C. R. Kaubrügger, N. Maskara, P. Naldesi, T. Zache, A. Kaufman, M. Lukin, H. Pichler, B. Vermersch, J. Ye, P. Zoller Fermionic quantum processing with programmable neutral atom arrays,
PNAS 120 e2304294120 (20230822),
http://dx.doi.org/10.1073/pnas.2304294120 doi:10.1073/pnas.2304294120 (ID: 721066)
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Simulating the properties of manybody fermionic systems is an outstanding computational challenge relevant to material science, quantum chemistry, and particle physics. Although qubitbased quantum computers can potentially tackle this problem more efficiently than classical devices, encoding nonlocal fermionic statistics introduces an overhead in the required resources, limiting their applicability on nearterm architectures. In this work, we present a fermionic quantum processor, where fermionic models are locally encoded in a fermionic register and simulated in a hardwareefficient manner using fermionic gates. We consider in particular fermionic atoms in programmable tweezer arrays and develop different protocols to implement nonlocal tunneling gates, guaranteeing Fermi statistics at the hardware level. We use this gate set, together with Rydbergmediated interaction gates, to find efficient circuit decompositions for digital and variational quantum simulation algorithms, illustrated here for molecular energy estimation. Finally, we consider a combined fermionqubit architecture, where both the motional and internal degrees of freedom of the atoms are harnessed to efficiently implement quantum phase estimation, as well as to simulate lattice gauge theory dynamics.

C. Fromonteil, D. Bluvstein, H. Pichler Protocols for Rydberg entangling gates featuring robustness against quasistatic errors,
PRX Quantum 4 (20230605),
http://dx.doi.org/10.1103/PRXQuantum.4.020335 doi:10.1103/PRXQuantum.4.020335 (ID: 720959)
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We introduce a novel family of protocols for entangling gates for neutral atom qubits based on the Rydberg blockade mechanism. These protocols realize controlledphase gates through a series of global laser pulses that are on resonance with the Rydberg excitation frequency. We analyze these protocols with respect to their robustness against calibration errors of the Rabi frequency or shottoshot laser intensity fluctuations, and show that they display robustness in various fidelity measures. In addition, we discuss adaptations of these protocols in order to make them robust to atomicmotioninduced Doppler shifts as well.

M.T. Nguyen, J.G. Liu, J. Wurtz, M. Lukin, S.T. Wang, Hannes Pichler Quantum optimization with arbitrary connectivity using Rydberg atom arrays,
PRX Quantum 4 010316 (20230214),
http://dx.doi.org/10.1103/PRXQuantum.4.010316 doi:10.1103/PRXQuantum.4.010316 (ID: 720883)
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Programmable quantum systems based on Rydberg atom arrays have recently been used for hardwareefficient tests of quantum optimization algorithms [Ebadi et al., Science, 376, 1209 (2022)] with hundreds of qubits. In particular, the maximum independent set problem on the socalled unitdisk graphs, was shown to be efficiently encodable in such a quantum system. Here, we extend the classes of problems that can be efficiently encoded in Rydberg arrays by constructing explicit mappings from the original computation problems to maximum weighted independent set problems on unitdisk graphs, with at most a quadratic overhead in the number of qubits. We analyze several examples, including: maximum weighted independent set on graphs with arbitrary connectivity, quadratic unconstrained binary optimization problems with arbitrary or restricted connectivity, and integer factorization. Numerical simulations on small system sizes indicate that the adiabatic time scale for solving the mapped problems is strongly correlated with that of the original problems. Our work provides a blueprint for using Rydberg atom arrays to solve a wide range of combinatorial optimization problems with arbitrary connectivity, beyond the restrictions imposed by the hardware geometry.

C. Miles, R. Samajdar, S. Ebadi, T. T. Wang, H. Pichler, S. Sachdev, M. Lukin, M. Greiner, K. Q. Weinberger, E. Kim Machine learning discovery of new phases in programmable quantum simulator snapshots,
Phys. Rev. Research 5 13026 (20230119),
http://dx.doi.org/10.1103/PhysRevResearch.5.013026 doi:10.1103/PhysRevResearch.5.013026 (ID: 720731)
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Machine learning has recently emerged as a promising approach for studying complex phenomena characterized by rich datasets. In particular, datacentric approaches lend to the possibility of automatically discovering structures in experimental datasets that manual inspection may miss. Here, we introduce an interpretable unsupervisedsupervised hybrid machine learning approach, the hybridcorrelation convolutional neural network (HybridCCNN), and apply it to experimental data generated using a programmable quantum simulator based on Rydberg atom arrays. Specifically, we apply HybridCCNN to analyze new quantum phases on square lattices with programmable interactions. The initial unsupervised dimensionality reduction and clustering stage first reveals five distinct quantum phase regions. In a second supervised stage, we refine these phase boundaries and characterize each phase by training fully interpretable CCNNs and extracting the relevant correlations for each phase. The characteristic spatial weightings and snippets of correlations specifically recognized in each phase capture quantum fluctuations in the striated phase and identify two previously undetected phases, the rhombic and boundaryordered phases. These observations demonstrate that a combination of programmable quantum simulators with machine learning can be used as a powerful tool for detailed exploration of correlated quantum states of matter.

J. Choi, A. L. Shaw, I. S. Madjarov, X. Xie, J. P. Covey, J. S. Cotler, D. K. Mark, H. Y. Huang, A. Kale, H. Pichler, F. G. Brandão, S. Choi, M. Endres Preparing random states and benchmarking with manybody quantum chaos,
Nature 613 473 (20230118),
http://dx.doi.org/10.1038/s41586022054421 doi:10.1038/s41586022054421 (ID: 720643)
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Producing quantum states at random has become increasingly important in modern quantum science, with applications being both theoretical and practical. In particular, ensembles of such randomly distributed, but pure, quantum states underlie our understanding of complexity in quantum circuits1 and black holes2, and have been used for benchmarking quantum devices3,4 in tests of quantum advantage5,6. However, creating random ensembles has necessitated a high degree of spatiotemporal control7,8,9,10,11,12 placing such studies out of reach for a wide class of quantum systems. Here we solve this problem by predicting and experimentally observing the emergence of random state ensembles naturally under timeindependent Hamiltonian dynamics, which we use to implement an efficient, widely applicable benchmarking protocol. The observed random ensembles emerge from projective measurements and are intimately linked to universal correlations built up between subsystems of a larger quantum system, offering new insights into quantum thermalization13. Predicated on this discovery, we develop a fidelity estimation scheme, which we demonstrate for a Rydberg quantum simulator with up to 25 atoms using fewer than 104 experimental samples. This method has broad applicability, as we demonstrate for Hamiltonian parameter estimation, targetstate generation benchmarking, and comparison of analogue and digital quantum devices. Our work has implications for understanding randomness in quantum dynamics14 and enables applications of this concept in a much wider context4,5,

G. Giudici, F. M. Surace, H. Pichler Unraveling PXP ManyBody Scars through Floquet Dynamics,
(20231226),
arXiv:2312.16288 arXiv:2312.16288 (ID: 721171)
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Quantum scars are special eigenstates of manybody systems that evade thermalization. They were first discovered in the PXP model, a wellknown effective description of Rydberg atom arrays. Despite significant theoretical efforts, the fundamental origin of PXP scars remains elusive. By investigating the discretized dynamics of the PXP model as a function of the Trotter step τ, we uncover a remarkable correspondence between the zero and twoparticle eigenstates of the integrable FloquetPXP cellular automaton at τ=π/2 and the PXP manybody scars of the timecontinuous limit. Specifically, we demonstrate that PXP scars are adiabatically connected to the eigenstates of the τ=π/2 Floquet operator. Building on this result, we propose a protocol for achieving highfidelity preparation of PXP scars in Rydberg atom experiments.

P. Grochowski, H. Pichler, C. A. Regal, O. RomeroIsart Quantum control of continuous systems via nonharmonic potential modulation,
(20231128),
arXiv:2311.16819 arXiv:2311.16819 (ID: 721151)
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We present a theoretical proposal for preparing and manipulating a state of a single continuousvariable degree of freedom confined to a nonharmonic potential. By utilizing optimally controlled modulation of the potential's position and depth, we demonstrate the generation of nonGaussian states, including Fock, GottesmanKitaevPreskill, multileggedcat, and cubicphase states, as well as the implementation of arbitrary unitaries within a selected twolevel subspace. Additionally, we propose protocols for singleshot orthogonal state discrimination and algorithmic cooling and analyze the robustness of this control scheme against noise. Since all the presented protocols rely solely on the precise modulation of the effective nonharmonic potential landscape, they are relevant to several experiments with continuousvariable systems, including the motion of a single particle in an optical tweezer or lattice, or current in circuit quantum electrodynamics.

K. Vodenkova, H. Pichler Continuous Coherent Quantum Feedback with Time Delays: Tensor Network Solution,
(20231113),
arXiv:2311.07302 arXiv:2311.07302 (ID: 721144)
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In this paper we develop a novel method to solve problems involving quantum optical systems coupled to coherent quantum feedback loops featuring time delays. Our method is based on exact mappings of such nonMarkovian problems to equivalent Markovian driven dissipative quantum manybody problems. In this work we show that the resulting Markovian quantum manybody problems can be solved (numerically) exactly and efficiently using tensor network methods for a series of paradigmatic examples, consisting of driven quantum systems coupled to waveguides at several distant points. In particular, we show that our method allows solving problems in so far inaccessible regimes, including problems with arbitrary long time delays and arbitrary numbers of excitations in the delay lines. We obtain solutions for the full realtime dynamics as well as the steady state in all these regimes. Finally, motivated by our results, we develop a novel meanfield approach, which allows us to find the solution semianalytically and identify parameter regimes where this approximation is in excellent agreement with our exact tensor network results.

M. Cain, S. Chattopadhyay, J. Liu, R. Samajdar, H. Pichler, M. Lukin Quantum speedup for combinatorial optimization with flat energy landscapes,
(20230622),
arXiv:2306.13123 arXiv:2306.13123 (ID: 721087)
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Designing quantum algorithms with a speedup over their classical analogs is a central challenge in quantum information science. Motivated by recent experimental observations of a superlinear quantum speedup in solving the Maximum Independent Set problem on certain unitdisk graph instances [Ebadi et al., Science 376, 6598 (2022)], we develop a theoretical framework to analyze the relative performance of the optimized quantum adiabatic algorithm and a broad class of classical Markov chain Monte Carlo algorithms. We outline conditions for the quantum adiabatic algorithm to achieve a quadratic speedup on hard problem instances featuring flat lowenergy landscapes and provide example instances with either a quantum speedup or slowdown. We then introduce an additional local Hamiltonian with no sign problem to the optimized adiabatic algorithm to achieve a quadratic speedup over a wide class of classical simulated annealing, parallel tempering, and quantum Monte Carlo algorithms in solving these hard problem instances. Finally, we use this framework to analyze the experimental observations.

Giacomo Giudice, F. M. Surace, Hannes Pichler, G. Giudici Trimer states with Z3 topological order in Rydberg atom arrays,
Phys. Rev. B 106 195155 (20221128),
http://dx.doi.org/10.1103/PhysRevB.106.195155 doi:10.1103/PhysRevB.106.195155 (ID: 720842)
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Trimers are defined as two adjacent edges on a graph. We study the quantum states obtained as equalweight superpositions of all trimer coverings of a lattice, with the constraint of having a trimer on each vertex: the socalled trimer resonatingvalencebond (tRVB) states. Exploiting their tensor network representation, we show that these states can host Z3 topological order or can be gapless liquids with U(1)×U(1) local symmetry. We prove that this continuous symmetry emerges whenever the lattice can be tripartite such that each trimer covers all the three sublattices. In the gapped case, we demonstrate the stability of topological order against dilution of maximal trimer coverings, which is relevant for realistic models where the density of trimers can fluctuate. Furthermore, we clarify the connection between gapped tRVB states and Z3 lattice gauge theories by smoothly connecting the former to the Z3 toric code, and discuss the nonlocal excitations on top of tRVB states. Finally, we analyze via exact diagonalization the zerotemperature phase diagram of a diluted trimer model on the square lattice and demonstrate that the ground state exhibits topological properties in a narrow region in parameter space. We show that a similar model can be implemented in Rydberg atom arrays exploiting the blockade effect. We investigate dynamical preparation schemes in this setup and provide a viable route for probing experimentally Z3 quantum spin liquids.

K Slagle, Yue Liu, David Aasen, Hannes Pichler, R. S. Mong, Xie Chen, Manuel Endres, Jason Alicea Quantum spin liquids bootstrapped from Ising criticality in Rydberg arrays,
Phys. Rev. B 106 115122 (20220914),
http://dx.doi.org/10.1103/PhysRevB.106.115122 doi:10.1103/PhysRevB.106.115122 (ID: 720829)
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Arrays of Rydberg atoms constitute a highly tunable, strongly interacting venue for the pursuit of exotic states of matter. We develop a new strategy for accessing a family of fractionalized phases known as quantum spin liquids in twodimensional Rydberg arrays. We specifically use effective field theory methods to study arrays assembled from Rydberg chains tuned to an Ising phase transition that famously hosts emergent fermions propagating within each chain. This highly entangled starting point allows us to naturally access spin liquids familiar from Kitaev's honeycomb model, albeit from an entirely different framework. In particular, we argue that finiterange repulsive Rydberg interactions, which frustrate nearby symmetrybreaking orders, can enable coherent propagation of emergent fermions between the chains in which they were born. Delocalization of emergent fermions across the full twodimensional Rydberg array yields a gapless Z2 spin liquid with a single massless Dirac cone. Here, the Rydberg occupation numbers exhibit universal powerlaw correlations that provide a straightforward experimental diagnostic of this phase. We further show that explicitly breaking symmetries perturbs the gapless spin liquid into gapped, topologically ordered descendants: Breaking lattice symmetries generates toriccode topological order, whereas introducing chirality generates nonAbelian Ising topological order. In the toriccode phase, we analytically construct microscopic incarnations of nonAbelian defects, which can be created and transported by dynamically controlling the atom positions in the array. Our work suggests that appropriately tuned Rydberg arrays provide a coldatoms counterpart of solidstate 'Kitaev materials' and, more generally, spotlights a new angle for pursuing experimental platforms for Abelian and nonAbelian fractionalization.

G. Giudici, M. Lukin, H. Pichler Dynamical preparation of quantum spin liquids in Rydberg atom arrays,
Phys. Rev. Lett. 129 90401 (20220826),
http://dx.doi.org/10.1103/PhysRevLett.129.090401 doi:10.1103/PhysRevLett.129.090401 (ID: 720742)
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We theoretically analyze recent experiments [G. Semeghini et al., Science 374, 1242 (2021)] demonstrating the onset of a topological spin liquid using a programmable quantum simulator based on Rydberg atom arrays. In the experiment, robust signatures of topological order emerge in outofequilibrium states that are prepared using a quasiadiabatic state preparation protocol. We show theoretically that the state preparation protocol can be optimized to target the fixed point of the topological phase  the resonating valence bond (RVB) state of hard dimers  in a time that scales linearly with the number of atoms. Moreover, we provide a twoparameter variational manifold of tensor network (TN) states that accurately describe the manybody dynamics of the preparation process. Using this approach we analyze the nature of the nonequilibrium state, establishing the emergence of topological order.

T. Vovk, Hannes Pichler EntanglementOptimal Trajectories of ManyBody Quantum Markov Processes,
Phys. Rev. Lett. 128 243601 (20220613),
http://dx.doi.org/10.1103/PhysRevLett.128.243601 doi:10.1103/PhysRevLett.128.243601 (ID: 720844)
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We develop a novel approach aimed at solving the equations of motion of open quantum manybody systems. It is based on a combination of generalized wave function trajectories and matrix product states. We introduce an adaptive quantum stochastic propagator, which minimizes the expected entanglement in the manybody quantum state, thus minimizing the computational cost of the matrix product state representation of each trajectory. We illustrate this approach on the example of a onedimensional open Brownian circuit. We show that this model displays an entanglement phase transition between area and volume law when changing between different propagators and that our method autonomously finds an efficiently representable area law unraveling.

S. Ebadi, Alexander Keesling, Madelyn Cain, Tout T. Wang, H. Levine, D. Bluvstein, Giulia Semeghini, Ahmed Omran, Junqiu Liu, R. Samajdar, XiuZhe Luo, B. Latz, J. Gao, B. Dayan, Edward Fahri, S. Sachdev, Nathan Gemelke, Lop Zhou, Soonwon Choi, Hannes Pi Quantum Optimization of Maximum Independent Set using Rydberg Atom Arrays,
Science 376 1215 (20220505),
http://dx.doi.org/10.1126/science.abo6587 doi:10.1126/science.abo6587 (ID: 720815)
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Realizing quantum speedup for practically relevant, computationally hard problems is a central challenge in quantum information science. Using Rydberg atom arrays with up to 289 qubits in two spatial dimensions, we experimentally investigate quantum algorithms for solving the Maximum Independent Set problem. We use a hardwareefficient encoding associated with Rydberg blockade, realize closedloop optimization to test several variational algorithms, and subsequently apply them to systematically explore a class of graphs with programmable connectivity. We find the problem hardness is controlled by the solution degeneracy and number of local minima, and experimentally benchmark the quantum algorithm's performance against classical simulated annealing. On the hardest graphs, we observe a superlinear quantum speedup in finding exact solutions in the deep circuit regime and analyze its origins.

D. Bluvstein, H. Levine, G. Semeghini, T. T. Wang, S. Ebadi, M. Kalinowski, A. Keesling, N. Maskara, H. Pichler, M. Greiner, V. Vuletic, M. Lukin A quantum processor based on coherent transport of entangled atom arrays,
Nature 604 456 (20220420),
http://dx.doi.org/10.1038/s41586022045926 doi:10.1038/s41586022045926 (ID: 720723)
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The ability to engineer parallel, programmable operations between desired qubits within a quantum processor is central for building scalable quantum information systems. In most stateoftheart approaches, qubits interact locally, constrained by the connectivity associated with their fixed spatial layout. Here, we demonstrate a quantum processor with dynamic, nonlocal connectivity, in which entangled qubits are coherently transported in a highly parallel manner across two spatial dimensions, in between layers of single and twoqubit operations. Our approach makes use of neutral atom arrays trapped and transported by optical tweezers; hyperfine states are used for robust quantum information storage, and excitation into Rydberg states is used for entanglement generation. We use this architecture to realize programmable generation of entangled graph states such as cluster states and a 7qubit Steane code state. Furthermore, we shuttle entangled ancilla arrays to realize a surface code with 19 qubits and a toric code state on a torus with 24 qubits. Finally, we use this architecture to realize a hybrid analogdigital evolution and employ it for measuring entanglement entropy in quantum simulations, experimentally observing nonmonotonic entanglement dynamics associated with quantum manybody scars. Realizing a longstanding goal, these results pave the way toward scalable quantum processing and enable new applications ranging from simulation to metrology.

B. Windt, H. Pichler Squeezing Quantum ManyBody Scars,
Phys. Rev. Lett. 128 90606 (20220304),
http://dx.doi.org/10.1103/PhysRevLett.128.090606 doi:10.1103/PhysRevLett.128.090606 (ID: 720724)
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We develop an analytical approach for the description of quantum manybody scars in PXP models. We show that the scarred dynamics in the PXP model on a complete bipartite graph can be interpreted as a onedimensional chiral scattering problem, and solve this problem analytically. The insights from this analysis allow us to predict that dynamical signatures of scars in PXP models can be enhanced by spin squeezing the initial states. We show numerically that this stabilization mechanism applies not only to the complete bipartite graph but also to one and twodimensional lattices, which are relevant for Rydberg atom array experiments. Moreover, our findings provide a physical motivation for Hamiltonian deformations reminiscent of those known to produce perfect scars.

K. Slagle, D. Aasen, H. Pichler, R. S. Mong, P. Fendley, X. Chen, M. Endres, J. Alicea Microscopic characterization of Ising conformal field theory in Rydberg chains,
Phys. Rev. B 104 235109 (20211206),
http://dx.doi.org/10.1103/PhysRevB.104.235109 doi:10.1103/PhysRevB.104.235109 (ID: 720682)

G. Semeghini, H. Levine, A. Keesling, S. Ebadi, T. T. Wang, D. Bluvstein, R. Verresen, H. Pichler, M. Kalinowski, R. Samajdar, A. Omran, S. Sachdev, A. Vishwanath, M. Greiner, V. Vuletic, M. Lukin Probing Topological Spin Liquids on a Programmable Quantum Simulator,
Science 374 1247 (20211202),
http://dx.doi.org/10.1126/science.abi8794 doi:10.1126/science.abi8794 (ID: 720644)

D. S. Wild, D. Sels, H. Pichler, C. Zanoci, M. Lukin Quantum sampling algorithms, phase transitions, and computational complexity,
Phys. Rev. A 104 32602 (20210902),
http://dx.doi.org/10.1103/PhysRevA.104.032602 doi:10.1103/PhysRevA.104.032602 (ID: 720684)
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Drawing independent samples from a probability distribution is an important computational problem with applications in Monte Carlo algorithms, machine learning, and statistical physics. The problem can be solved in principle on a quantum computer by preparing a quantum state that encodes the entire probability distribution followed by a projective measurement. We investigate the complexity of adiabatically preparing such quantum states for the Gibbs distributions of various classical models including the Ising chain, hardsphere models on different graphs, and a model encoding the unstructured search problem. By constructing a parent Hamiltonian, whose ground state is the desired quantum state, we relate the asymptotic scaling of the state preparation time to the nature of transitions between distinct quantum phases. These insights enable us to identify adiabatic paths that achieve a quantum speedup over classical Markov chain algorithms. In addition, we show that parent Hamiltonians for the problem of sampling from independent sets on certain graphs can be naturally realized with neutral atoms interacting via highly excited Rydberg states.

D. S. Wild, D. Sels, H. Pichler, C. Zanoci, M. Lukin Quantum Sampling Algorithms for NearTerm Devices,
Phys. Rev. Lett. 127 100504 (20210902),
http://dx.doi.org/10.1103/PhysRevLett.127.100504 doi:10.1103/PhysRevLett.127.100504 (ID: 720685)
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Efficient sampling from a classical Gibbs distribution is an important computational problem with applications ranging from statistical physics over Monte Carlo and optimization algorithms to machine learning. We introduce a family of quantum algorithms that provide unbiased samples by preparing a state encoding the entire Gibbs distribution. We show that this approach leads to a speedup over a classical Markov chain algorithm for several examples, including the Ising model and sampling from weighted independent sets of two different graphs. Our approach connects computational complexity with phase transitions, providing a physical interpretation of quantum speedup. Moreover, it opens the door to exploring potentially useful sampling algorithms on nearterm quantum devices, as the algorithm for sampling from independent sets on certain graphs can be naturally implemented using Rydberg atom arrays.

S. Ebadi, T. T. Wang, H. Levine, A. Keesling, G. Semeghini, A. Omran, D. Bluvstein, R. Samajdar, H. Pichler, W. Ho, S. Choi, S. Sachdev, M. Greiner, V. Vuletic, M. Lukin Quantum phases of matter on a 256atom programmable quantum simulator,
Nature (20210707),
http://dx.doi.org/10.1038/s41586021035824 doi:10.1038/s41586021035824 (ID: 720710)

R. Samajdar, W. Ho, H. Pichler, M. Lukin, S. Sachdev Quantum phases of Rydberg atoms on a kagome lattice,
Proc. Natl. Acad. U.S.A. 118 e2015785118 (20210126),
http://dx.doi.org/10.1073/pnas.2015785118 doi:10.1073/pnas.2015785118 (ID: 720622)
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Programmable quantum simulators based on Rydberg atom arrays have recently emerged as versatile platforms for exploring exotic manybody phases and quantum dynamics of strongly correlated systems. In this work, we theoretically investigate the quantum phases that can be realized by arranging such Rydberg atoms on a kagome lattice. Along with an extensive analysis of the states which break lattice symmetries due to classical correlations, we identify an intriguing regime that constitutes a promising candidate for hosting a phase with longrange quantum entanglement and topological order. Our results provide a route to experimentally realizing and probing highly entangled quantum matter.

J. Perczel, J. Borregaard, D. Chang, H. Pichler, S. F. Yelin, P. Zoller, M. Lukin Photonic band structure of twodimensional atomic lattices,
Phys. Rev. A 96 63801 (20171204),
http://dx.doi.org/10.1103/PhysRevA.96.063801 doi:10.1103/PhysRevA.96.063801 (ID: 719916)
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Twodimensional atomic arrays exhibit a number of intriguing quantum optical phenomena, including subradiance, nearly perfect reflection of radiation, and longlived topological edge states. Studies of emission and scattering of photons in such lattices require complete treatment of the radiation pattern from individual atoms, including longrange interactions. We describe a systematic approach to perform the calculations of collective energy shifts and decay rates in the presence of such longrange interactions for arbitrary twodimensional atomic lattices. As applications of our method, we investigate the topological properties of atomic lattices both in free space and near plasmonic surfaces.
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H. Pichler, S. Choi, P. Zoller, M. Lukin Universal photonic quantum computation via timedelayed feedback,
PNAS 114 (20171017),
http://dx.doi.org/10.1073/pnas.1711003114 doi:10.1073/pnas.1711003114 (ID: 720008)
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We propose and analyze a deterministic protocol to generate twodimensional photonic cluster states using a single quantum emitter via timedelayed quantum feedback. As a physical implementation, we consider a single atom or atomlike system coupled to a 1D waveguide with a distant mirror, where guided photons represent the qubits, while the mirror allows the implementation of feedback. We identify the class of manybody quantum states that can be produced using this approach and characterize them in terms of 2D tensor network states.

H. Pichler, S. Choi, P. Zoller, M. Lukin Photonic tensor networks produced by a single quantum emitter,
PNAS 114 11362 (20171010),
http://dx.doi.org/10.1073/pnas.1711003114 doi:10.1073/pnas.1711003114 (ID: 719751)
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We propose and analyze a protocol to generate two dimensional tensor network states using a single quantum system that sequentially interacts with a 1D string of qubits. This is accomplished by using parts of the string itself as a quantum queue memory. As a physical implementation, we consider a single atom or atom like system coupled to a 1D waveguide with a distant mirror, where guided photons represent the qubits while the mirror allows the implementation of the queue memory. We identify the class of manybody quantum states that can be produced using this approach. These include universal resources for measurement based quantum computation and states associated with topologically ordered phases. We discuss an explicit protocol to deterministically create a 2D cluster state in a quantum nanophotonic experiment, that allows for a realization of a quantum computer using a single atom coupled to light.
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A. Glätzle, K. Ender, D. S. Wild, S. Choi, H. Pichler, M. Lukin, P. Zoller Quantum Spin Lenses in Atomic Arrays,
Phys. Rev. X 7 31049 (20170920),
http://dx.doi.org/10.1103/PhysRevX.7.031049 doi:10.1103/PhysRevX.7.031049 (ID: 719790)
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We propose and discuss `quantum spin lenses', where quantum states of delocalized spin excitations in an atomic medium are `focused' in space in a coherent quantum process down to (essentially) single atoms. These can be employed to create controlled interactions in a quantum lightmatter interface, where photonic qubits stored in an atomic ensemble are mapped to a quantum register represented by single atoms. We propose Hamiltonians for quantum spin lenses as inhomogeneous spin models on lattices, which can be realized with Rydberg atoms in 1D, 2D and 3D, and with strings of trapped ions. We discuss both linear and nonlinear quantum spin lenses: in a nonlinear lens, repulsive spinspin interactions lead to focusing dynamics conditional to the number of spin excitations. This allows the mapping of quantum superpositions of delocalized spin excitations to superpositions of spatial spin patterns, which can be addressed by light fields and manipulated. Finally, we propose multifocal quantum spin lenses as a way to generate and distribute entanglement between distant atoms in an atomic lattice array.
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P. Guimond, M. Pletyukhov, H. Pichler, P. Zoller Delayed Coherent Quantum Feedback from a Scattering Theory and a Matrix Product State Perspective,
Quantum Sci. Technol. 2 44012 (20170908),
http://dx.doi.org/10.1088/20589565/aa7f03 doi:10.1088/20589565/aa7f03 (ID: 719814)
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We study the scattering of photons propagating in a semiinfinite waveguide terminated by a mirror and interacting with a quantum emitter. This paradigm constitutes an example of coherent quantum feedback, where light emitted towards the mirror gets redirected back to the emitter. We derive an analytical solution for the scattering of twophoton states, which is based on an exact resummation of the perturbative expansion of the scattering matrix, in a regime where the time delay of the coherent feedback is comparable to the timescale of the quantum emitter's dynamics. We compare the results with numerical simulations based on matrix product state techniques simulating the full dynamics of the system, and extend the study to the scattering of coherent states beyond the lowpower limit.
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J. Perczel, J. Borregaard, D. Chang, H. Pichler, S. F. Yelin, P. Zoller, M. Lukin Topological Quantum Optics in TwoDimensional Atomic Arrays,
Phys. Rev. Lett. 119 23603 (20170714),
http://dx.doi.org/10.1103/PhysRevLett.119.023603 doi:10.1103/PhysRevLett.119.023603 (ID: 719767)
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We demonstrate that twodimensional atomic arrays with subwavelength spacing can be used to create topologically protected quantum optical systems where the photon propagation is robust against large imperfections while losses associated with free space emission are strongly suppressed. Breaking timereversal symmetry with a magnetic field results in gapped photonic bands with nontrivial Chern numbers. Such a system displays topologically protected bound states and unidirectional emission by individual atoms into longlived edge states. Possible experimental realizations and applications are discussed.
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B. Vermersch, P. Guimond, H. Pichler, P. Zoller Quantum State Transfer via Noisy Photonic and Phononic Waveguides,
Phys. Rev. Lett. 118 133601 (20170327),
http://dx.doi.org/10.1103/PhysRevLett.118.133601 doi:10.1103/PhysRevLett.118.133601 (ID: 719696)
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We describe a quantum state transfer protocol, where a quantum state of photons stored in a first cavity can be faithfully transferred to a second distant cavity via an infinite 1D waveguide, while being immune to arbitrary noise (e.g. thermal noise) injected into the waveguide. We extend the model and protocol to a cavity QED setup, where atomic ensembles, or single atoms representing quantum memory, are coupled to a cavity mode. We present a detailed study of sensitivity to imperfections, and develop a quantum error correction protocol to account for random losses (or additions) of photons in the waveguide. Our numerical analysis is enabled by Matrix Product State techniques to simulate the complete quantum circuit, which we generalize to include thermal input fields. Our discussion applies both to photonic and phononic quantum networks.
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P. Lodahl, S. Mahmoodian, S. Stobbe, P. Schneeweiss, J. Volz, A. Rauschenbeutel, H. Pichler, P. Zoller Chiral Quantum Optics,
Nature 541 473 (20170126),
http://dx.doi.org/10.1038/nature21037 doi:10.1038/nature21037 (ID: 719634)

M. Łącki, M. Baranov, H. Pichler, P. Zoller NanoScale `Dark State' Optical Potentials for Cold Atoms,
Phys. Rev. Lett. 117 233001 (20161130),
http://dx.doi.org/10.1103/PhysRevLett.117.233001 doi:10.1103/PhysRevLett.117.233001 (ID: 719619)
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We discuss generation of subwavelength optical barriers on the scale of tens of nanometers, as conservative optical potentials for cold atoms. These arise from nonadiabatic corrections to BornOppenheimer potentials from dressed `dark states' in atomic Λ configurations. We illustrate the concepts with a double layer potential for atoms obtained from inserting an optical subwavelength barrier into a well generated by an offresonant optical lattice, and discuss bound states of pairs of atoms interacting via magnetic dipolar interactions. The subwavelength optical barriers represent an optical `KronigPenney' potential. We present a detailed study of the bandstructure in optical `KronigPenney' potentials, including decoherence from spontaneous emission and atom loss to open `bright' channels.

P. Guimond, H. Pichler, A. Rauschenbeutel, P. Zoller Chiral quantum optics with Vlevel atoms and coherent quantum feedback,
Phys. Rev. A 94 33829 (20160916),
http://dx.doi.org/10.1103/PhysRevA.94.033829 doi:10.1103/PhysRevA.94.033829 (ID: 719590)
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We study the dissipative dynamics of an atom in a Vlevel configuration driven by lasers and coupled to a semiinfinite waveguide. The coupling to the waveguide is chiral, in that each transition interacts only with the modes propagating in a given direction, and this direction is opposite for the two transitions. The waveguide is terminated by a mirror which coherently feeds the photon stream emitted by one transition back to the atom. First, we are interested in the dynamics of the atom in the Markovian limit where the timedelay in the feedback is negligible. Specifically, we study the conditions under which the atom evolves towards a pure "dark" stationary state, where the photons emitted by both transitions interfere destructively thanks to the coherent feedback, and the overall emission vanishes. This is a singleatom analogue of the quantum dimer, where a pair of laserdriven twolevel atoms is coupled to a unidirectional waveguide and dissipates towards a pure entangled dark state. Our setup should be feasible with current stateoftheart experiments. Second, we extend our study to nonMarkovian regimes and investigate the effect of the feedback retardation on the steadystate.
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T. Ramos, B. Vermersch, P. Hauke, H. Pichler, P. Zoller NonMarkovian Dynamics in Chiral Quantum Networks with Spins and Photons,
Phys. Rev. A 93 62104 (20160602),
http://dx.doi.org/10.1103/PhysRevA.93.062104 doi:10.1103/PhysRevA.93.062104 (ID: 719497)
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We study the dynamics of chiral quantum networks consisting of nodes coupled by unidirectional or asymmetric bidirectional quantum channels. In contrast to the familiar photonic networks consisting of driven twolevel atoms exchanging photons via 1D photonic nanostructures, we propose and study a setup where interactions between the atoms are mediated by spin excitations (magnons) in 1D XXspin chains representing a spin waveguide. While Markovian quantum network theory eliminates quantum channels as structureless reservoirs in a BornMarkov approximation to obtain a master equation for the nodes, we are interested in nonMarkovian dynamics. This arises from the nonlinear character of the dispersion with bandedge effects, and from finite spin propagation velocities leading to time delays in interactions. To account for the nonMarkovian dynamics we treat the quantum degrees of freedom of the nodes and connecting channel as a composite spin system with the surrounding of the quantum network as a Markovian bath, allowing for an efficient solution with timedependent density matrix renormalization group techniques. We illustrate our approach showing nonMarkovian effects in the drivendissipative formation of quantum dimers, and we present examples for quantum information protocols involving quantum state transfer with engineered elements as basic building blocks of quantum spintronic circuits.
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H. Pichler, G. Zhu, A. Seif, P. Zoller, M. Hafezi A Measurement Protocol for the Entanglement Spectrum of Cold Atoms,
Phys. Rev. X 6 41033 (20160517),
http://dx.doi.org/10.1103/PhysRevX.6.041033 doi:10.1103/PhysRevX.6.041033 (ID: 719569)
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Entanglement plays a major role in characterizing manybody quantum systems. In particular, the entanglement spectrum holds a great promise to characterize essential physics of quantum manybody systems. While there has been a surge of theoretical works on the subject, no experimental measurement has been performed to this date, due to the lack of an implementable measurement scheme. Here, we propose a measurement protocol to access the entanglement spectrum of manybody states in experiments with cold atoms in optical lattices. Our scheme effectively performs a Ramsey spectroscopy of the entanglement Hamiltonian, and is based on the ability to produce several copies of the state under investigation together with the possibility to perform a global swap gate between two copies conditioned on the state of an auxiliary qubit. We show how the required conditional swap gate can be implemented with cold atoms, either by using Rydberg interactions or coupling the atoms to a cavity mode. We illustrate these ideas on a simple (extended) BoseHubbard model where such a measurement protocol reveals topological features of the Haldane phase.
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H. Pichler, P. Zoller Photonic Quantum Circuits with Time Delays,
Phys. Rev. Lett. 116 93601 (20160303),
http://dx.doi.org/10.1103/PhysRevLett.116.093601 doi:10.1103/PhysRevLett.116.093601 (ID: 719362)
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We study the dynamics of photonic quantum circuits consisting of nodes coupled by quantum channels. We are interested in the regime where the time delay in communication between the nodes is significant. This includes the problem of quantum feedback, where a quantum signal is fed back on a system with a time delay. We develop a matrix product state approach to solve the quantum stochastic Schrödinger equation with time delays, which accounts in an efficient way for the entanglement of nodes with the stream of emitted photons in the waveguide, and thus the nonMarkovian character of the dynamics. We illustrate this approach with two paradigmatic quantum optical examples: two coherently driven distant atoms coupled to a photonic waveguide with a time delay, and a driven atom coupled to its own output field with a time delay as an instance of a quantum feedback problem.
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M. Łącki, H. Pichler, A. Sterdyniak, A. Lyras, V. Lembessis, O. AlDossary, J. Budich, P. Zoller Quantum Hall Physics with Cold Atoms in Cylindrical Optical Lattices,
Phys. Rev. A 93 13604 (20160107),
http://dx.doi.org/10.1103/PhysRevA.93.013604 doi:10.1103/PhysRevA.93.013604 (ID: 719282)
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We propose and study various realizations of a HofstadterHubbard model on a cylinder geometry with fermionic cold atoms in optical lattices. The cylindrical optical lattice is created by counterpropagating LaguerreGauss beams, i.e. light beams carrying orbital angular momentum. By strong focusing of the light beams we create a real space optical lattice in the form of rings, which are offset in energy. A second set of LaguerreGauss beams then induces a Ramanhopping between these rings, imprinting phases corresponding to a synthetic magnetic field (artificial gauge field). In addition, by rotating the lattice potential, we achieve a slowly varying flux through the hole of the cylinder, which allows us to probe the Hall response of the system as a realization of Laughlin's thought experiment. We study how in the presence of interactions fractional quantum Hall physics could be observed in this setup.
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