The geometrical Berry phase is key to understanding the behavior of quantum states under cyclic adiabatic evolution. When generalized to non-Hermitian systems with gain and loss, the Berry phase can become complex and should modify not only the phase but also the amplitude of the state. Here, we perform the first experimental measurements of the adiabatic non-Hermitian Berry phase, exploring a minimal two-site PT-symmetric Hamiltonian that is inspired by the Hatano-Nelson model. We realize this non-Hermitian model experimentally by mapping its dynamics to that of a pair of classical oscillators coupled by real-time measurement-based feedback. As we verify experimentally, the adiabatic non-Hermitian Berry phase is a purely geometrical effect that leads to significant amplification and damping of the amplitude also for noncyclical paths within the parameter space even when all eigenenergies are real. We further observe a non-Hermitian analog of the Aharonov-Bohm solenoid effect, observing amplification and attenuation when encircling a region of broken PT symmetry that serves as a source of imaginary flux. This experiment demonstrates the importance of geometrical effects that are unique to non-Hermitian systems and paves the way towards further studies of non-Hermitian and topological physics in synthetic metamaterials.
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Realizing topological edge states with Rydberg-atom synthetic dimensions
Abstract A discrete degree of freedom can be engineered to match the Hamiltonian of particles moving in a real-space lattice potential. Such synthetic dimensions are powerful tools for quantum simulation because of the control they offer and the ability to create configurations difficult to access in real space. Here, in an ultracold 84 Sr atom, we demonstrate a synthetic-dimension based on Rydberg levels coupled with millimeter waves. Tunneling amplitudes between synthetic lattice sites and on-site potentials are set by the millimeter-wave amplitudes and detunings respectively. Alternating weak and strong tunneling in a one-dimensional configuration realizes the single-particle Su-Schrieffer-Heeger (SSH) Hamiltonian, a paradigmatic model of topological matter. Band structure is probed through optical excitation from the ground state to Rydberg levels, revealing symmetry-protected topological edge states at zero energy. Edge-state energies are robust to perturbations of tunneling-rates that preserve chiral symmetry, but can be shifted by the introduction of on-site potentials.
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- Award ID(s):
- 1904294
- NSF-PAR ID:
- 10331079
- Date Published:
- Journal Name:
- Nature Communications
- Volume:
- 13
- Issue:
- 1
- ISSN:
- 2041-1723
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract Synthetic dimensions, wherein dynamics occurs in a set of internal states, have found great success in recent years in exploring topological effects in cold atoms and photonics. However, the phenomena thus far explored have largely been restricted to the non-interacting or weakly interacting regimes. Here, we extend the synthetic dimensions playbook to strongly interacting systems of Rydberg atoms prepared in optical tweezer arrays. We use precise control over driving microwave fields to introduce a tunable
U (1) flux in a four-site lattice of coupled Rydberg levels. We find highly coherent dynamics, in good agreement with theory. Single atoms show oscillatory dynamics controllable by the gauge field. Small arrays of interacting atoms exhibit behavior suggestive of the emergence of ergodic and arrested dynamics in the regimes of intermediate and strong interactions, respectively. These demonstrations pave the way for future explorations of strongly interacting dynamics and many-body phases in Rydberg synthetic lattices. -
INTRODUCTION Solving quantum many-body problems, such as finding ground states of quantum systems, has far-reaching consequences for physics, materials science, and chemistry. Classical computers have facilitated many profound advances in science and technology, but they often struggle to solve such problems. Scalable, fault-tolerant quantum computers will be able to solve a broad array of quantum problems but are unlikely to be available for years to come. Meanwhile, how can we best exploit our powerful classical computers to advance our understanding of complex quantum systems? Recently, classical machine learning (ML) techniques have been adapted to investigate problems in quantum many-body physics. So far, these approaches are mostly heuristic, reflecting the general paucity of rigorous theory in ML. Although they have been shown to be effective in some intermediate-size experiments, these methods are generally not backed by convincing theoretical arguments to ensure good performance. RATIONALE A central question is whether classical ML algorithms can provably outperform non-ML algorithms in challenging quantum many-body problems. We provide a concrete answer by devising and analyzing classical ML algorithms for predicting the properties of ground states of quantum systems. We prove that these ML algorithms can efficiently and accurately predict ground-state properties of gapped local Hamiltonians, after learning from data obtained by measuring other ground states in the same quantum phase of matter. Furthermore, under a widely accepted complexity-theoretic conjecture, we prove that no efficient classical algorithm that does not learn from data can achieve the same prediction guarantee. By generalizing from experimental data, ML algorithms can solve quantum many-body problems that could not be solved efficiently without access to experimental data. RESULTS We consider a family of gapped local quantum Hamiltonians, where the Hamiltonian H ( x ) depends smoothly on m parameters (denoted by x ). The ML algorithm learns from a set of training data consisting of sampled values of x , each accompanied by a classical representation of the ground state of H ( x ). These training data could be obtained from either classical simulations or quantum experiments. During the prediction phase, the ML algorithm predicts a classical representation of ground states for Hamiltonians different from those in the training data; ground-state properties can then be estimated using the predicted classical representation. Specifically, our classical ML algorithm predicts expectation values of products of local observables in the ground state, with a small error when averaged over the value of x . The run time of the algorithm and the amount of training data required both scale polynomially in m and linearly in the size of the quantum system. Our proof of this result builds on recent developments in quantum information theory, computational learning theory, and condensed matter theory. Furthermore, under the widely accepted conjecture that nondeterministic polynomial-time (NP)–complete problems cannot be solved in randomized polynomial time, we prove that no polynomial-time classical algorithm that does not learn from data can match the prediction performance achieved by the ML algorithm. In a related contribution using similar proof techniques, we show that classical ML algorithms can efficiently learn how to classify quantum phases of matter. In this scenario, the training data consist of classical representations of quantum states, where each state carries a label indicating whether it belongs to phase A or phase B . The ML algorithm then predicts the phase label for quantum states that were not encountered during training. The classical ML algorithm not only classifies phases accurately, but also constructs an explicit classifying function. Numerical experiments verify that our proposed ML algorithms work well in a variety of scenarios, including Rydberg atom systems, two-dimensional random Heisenberg models, symmetry-protected topological phases, and topologically ordered phases. CONCLUSION We have rigorously established that classical ML algorithms, informed by data collected in physical experiments, can effectively address some quantum many-body problems. These rigorous results boost our hopes that classical ML trained on experimental data can solve practical problems in chemistry and materials science that would be too hard to solve using classical processing alone. Our arguments build on the concept of a succinct classical representation of quantum states derived from randomized Pauli measurements. Although some quantum devices lack the local control needed to perform such measurements, we expect that other classical representations could be exploited by classical ML with similarly powerful results. How can we make use of accessible measurement data to predict properties reliably? Answering such questions will expand the reach of near-term quantum platforms. Classical algorithms for quantum many-body problems. Classical ML algorithms learn from training data, obtained from either classical simulations or quantum experiments. 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Abstract Recently, the study of topological structures in photonics has garnered significant interest, as these systems can realize robust, nonreciprocal chiral edge states and cavity-like confined states that have applications in both linear and nonlinear devices. However, current band theoretic approaches to understanding topology in photonic systems yield fundamental limitations on the classes of structures that can be studied. Here, we develop a theoretical framework for assessing a photonic structure’s topology directly from its effective Hamiltonian and position operators, as expressed in real space, and without the need to calculate the system’s Bloch eigenstates or band structure. Using this framework, we show that nontrivial topology, and associated boundary-localized chiral resonances, can manifest in photonic crystals with broken time-reversal symmetry that lack a complete band gap, a result that may have implications for new topological laser designs. Finally, we use our operator-based framework to develop a novel class of invariants for topology stemming from a system’s crystalline symmetries, which allows for the prediction of robust localized states for creating waveguides and cavities.more » « less
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Synthetic spaces allow physicists to bypass constraints imposed by certain physical laws in experiments. Here, we show that a synthetic torus, which consists of a ring trap in the real space and internal states of ultracold atoms cyclically coupled by Laguerre-Gaussian Raman beams, could be threaded by a net effective magnetic flux through its surface—an impossible mission in the real space. Such a synthetic Hall torus gives rise to a periodic lattice in real dimensions, in which the periodicity of the density modulation of atoms fractionalizes that of the Hamiltonian. Correspondingly, the energy spectrum is featured by multiple bands grouping into clusters with nonsymmorphic-symmetry-protected band crossings in each cluster, leading to swaps of wave packets in Bloch oscillations. Our scheme allows physicists to glue two synthetic Hall tori such that localization may emerge in a quasicrystalline lattice. If the Laguerre-Gaussian Raman beams and ring traps were replaced by linear Raman beams and ordinary traps, a synthetic Hall cylinder could be realized and deliver many of the aforementioned phenomena.more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1038/s41467-022-28550-y
