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Programmable Interactions and Emergent Geometry in an Atomic Array
Avikar Periwal, Eric S. Cooper, Philipp Kunkel, Julian F. Wienand, Emily J. Davis, Monika Schleier-Smith
TL;DR
The paper develops dynamical and correlation-based tools for reconstructing emergent geometries from atomic-array interactions. It finds momentum-mode growth and treelike correlation structure, while identifying limits of truncated-Wigner modeling for late-time states.
Problem
Bulk reconstruction seeks to infer an emergent geometry from boundary correlations, motivated by the relation between strongly correlated sites and small bulk areas.
Method
The approach combines discrete momentum-mode dynamics, pair-creation evolution, inverse correlation matrices, and iterative coarse-graining of correlations to reconstruct geometry.
Results
Momentum modes grow according to the dispersion relation, while simulations reproduce experimentally observed block structure in correlations after two Bloch periods.
Takeaways & Limitations
Correlation-based coarse-graining provides a physically motivated route for connecting boundary correlations with an emergent bulk geometry.
Takeaways & Limitations
The truncated Wigner approximation fails to represent the exact non-Gaussian entangled states at longer evolution times.
Abstract
from arXiv · showhide
Interactions govern the flow of information and the formation of correlations in quantum systems, dictating the phases of matter found in nature and the forms of entanglement generated in the laboratory. Typical interactions decay with distance and thus produce a network of connectivity governed by geometry, e.g., by the crystalline structure of a material or the trapping sites of atoms in a quantum simulator. However, many envisioned applications in quantum simulation and computation require richer coupling graphs including nonlocal interactions, which notably feature in mappings of hard optimization problems onto frustrated spin systems and in models of information scrambling in black holes. Here, we report on the realization of programmable nonlocal interactions in an array of atomic ensembles within an optical cavity, where photons carry information between distant atomic spins. By programming the distance-dependence of interactions, we access effective geometries where the dimensionality, topology, and metric are entirely distinct from the physical arrangement of atoms. As examples, we engineer an antiferromagnetic triangular ladder, a Moebius strip with sign-changing interactions, and a treelike geometry inspired by concepts of quantum gravity. The tree graph constitutes a toy model of holographic duality, where the quantum system may be viewed as lying on the boundary of a higher-dimensional geometry that emerges from measured spin correlations. Our work opens broader prospects for simulating frustrated magnets and topological phases, investigating quantum optimization algorithms, and engineering new entangled resource states for sensing and computation.
B. Microtraps and Lattice Transfer
The experiment uses microtraps for preparation and imaging, then transfers atoms into an intracavity optical lattice optimized for cavity-mediated interactions.
- B. Microtraps and Lattice Transfer: Atoms are cooled, prepared, and imaged in a microtrap array before transfer to an intracavity optical lattice.The hybrid scheme separates trapping stages for state preparation and interaction induction.
- B. Microtraps and Lattice Transfer: The 1560 nm intracavity lattice is registered with the 780 nm interaction light to maximize atom–light coupling.
- B. Microtraps and Lattice Transfer: The array contains M × 2 optical microtraps, with the long axis aligned to the cavity axis and 60 µm spacing between traps.Each microtrap has a 6 µm waist and depth h×4 MHz during loading.
- B. Microtraps and Lattice Transfer: For imaging, atoms are transferred back from the lattice to microtraps and moved approximately 15 µm away to suppress ac Stark shifts.
C. Imaging and Spin Readout
Spin populations are measured through state-selective fluorescence, while local Raman rotations convert transverse spin components into measurable population differences.
- C. Imaging and Spin Readout: Four fluorescence images independently measure the three Zeeman populations in F = 1 and residual atoms in F = 2.Each imaging pulse uses a retro-reflected beam resonant with the F = 2 → F′ = 3 transition for 100 µs.
- C. Imaging and Spin Readout: Background subtraction uses principal component analysis of approximately 100 images recorded without atoms.
- C. Imaging and Spin Readout: Local Raman pulses implement π/2 spin rotations that map F_x,i onto the measurable population difference n+,i − n−,i.The beam is steered site-by-site with an acousto-optic deflector, and each pulse lasts 3 µs.
- C. Imaging and Spin Readout: Randomizing the order of local rotations suppresses correlation bias from shot-to-shot Larmor-frequency fluctuations.The fluctuation-induced reduction depends on the time between corresponding Raman pulses.
D. Computation of Correlations
The analysis computes normalized correlation functions from repeated measurements, choosing normalizations according to whether spatial structure or total pair creation is being characterized.
- D. Computation of Correlations: Correlation functions C_pm, C_xx, and C_b are computed from at least 50 measurements using specified observables A and B.
- D. Computation of Correlations: Covariance and variance are defined by Cov(A, B) = ⟨AB⟩ − ⟨A⟩⟨B⟩ and Var(A) = Cov(A, A).
- D. Computation of Correlations: Shot-to-shot variance normalization preserves relevant spatial information while remaining insensitive to the total amount of pair creation.
- D. Computation of Correlations: For pair-creation dynamics, covariance matrices measured in the x-basis are normalized by each site’s atom population rather than variance.This normalization makes the extracted correlation sensitive to the total amount of pair creation and its growth over time.
E. Interaction Parameters
A magnetic-field gradient and drive-field modulation control which ensembles interact, while optical parameters set the coupling strength and interaction signs.
- E. Interaction Parameters: The magnetic-field gradient is parameterized by the adjacent-site Zeeman-splitting difference ω_B and is superposed on a bias field B_0.The bias field produces both linear and quadratic Zeeman shifts.
- E. Interaction Parameters: The experiment operates with ω_z/M > ω_B > q, using magnetic fields between 2 and 4 G and a typical gradient ω_B = 2π × 1.52(1) kHz/site.
- E. Interaction Parameters: The instantaneous spin-exchange coupling is controlled by drive detuning and intracavity photon number, with a typical collective strength 2n J̃ = 2π × 3 kHz.The drive is detuned by δ_c = −2π × 4 MHz to −2π × 7 MHz from cavity resonance.
- E. Interaction Parameters: Drive-intensity modulation produces distance-dependent couplings J(r), while phases φ_r ∈ {0, π} set interaction signs.
- E. Interaction Parameters: Periodic boundary conditions are implemented by pulsing the drive at Mω_B, with each pulse lasting 0.3τ_B/M = 11 µs.
F. Cavity Parameters
The cavity parameters characterize atom–cavity coupling, optical linewidths, detuning, and spatial variation in dispersive coupling.
- F. Cavity Parameters: The near-concentric cavity has a 5 cm length and an 18 µm waist at 780 nm.
- F. Cavity Parameters: The cavity parameters are 2g = 2π × 2.6 MHz, κ = 2π × 250(20) kHz, and η = 4.5.
- F. Cavity Parameters: Thermal motion reduces the average dispersive coupling to Ω = 2π × 13 Hz, while displacement can reduce coupling by up to 20%.Each ensemble lies within 520 µm of the cavity center, or 0.4zR.
G. Interaction Hamiltonian
The effective interaction Hamiltonian is obtained by transforming the cavity-mediated spin system into a rotating frame and time-averaging under a weak-coupling condition.
- G. Interaction Hamiltonian: The derivation starts from the lab-frame Hamiltonian Hlab after adiabatically eliminating the cavity mode.
- G. Interaction Hamiltonian: A rotating-frame transformation uses local magnetic fields hl = ωBl and leaves the quadratic Zeeman shift defined by Hq.
- G. Interaction Hamiltonian: When nJ, q ≪ ωB, the effective Hamiltonian is given to first order by the time average of the transformed Hamiltonian.
- G. Interaction Hamiltonian: The distance dependence J(r) of the effective couplings is the Fourier transform of the drive waveform.
H. Momentum-Space Dynamics
The gradient organizes spin-wave dynamics into momentum modes that are periodically coupled through the cavity, with mode growth governed by the programmed dispersion relation.
- H. Momentum-Space Dynamics: In the lab frame, the magnetic-field gradient drives Bloch oscillations at frequency ωB, while the rotating-frame cavity-coupled mode follows k = ωt.
- H. Momentum-Space Dynamics: The finite system has M orthogonal momentum modes, and a pulsed drive produces discrete momentum-space couplings while decoupling the modes.
- H. Momentum-Space Dynamics: Each momentum mode alternates between cavity-mediated spin-spin interactions and evolution under the quadratic Zeeman shift.
- H. Momentum-Space Dynamics: For |χk| > ωB, a mode grows per Bloch period by λ ≈ |2χkτB| sin(qτB), and this growth appears in the structure factor.
- H. Momentum-Space Dynamics: The pulsed-drive derivation assumes periodic boundary conditions, whereas continuous driving produces open boundaries and small deviations from the model.
- H. Momentum-Space Dynamics: Modes with minimum energy are maximally amplified for χk < 0, while opposite signs of χ and q allow access to low-energy interaction states through pair creation.
I. Euclidean Reconstruction
The reconstruction method infers effective coordinates and couplings from measured correlations by modeling correlation spreading and using the inverse correlation matrix.
- I. Euclidean Reconstruction: Measured correlations Cxx are used to reconstruct effective coordinates ρ and inferred couplings J′ in the engineered geometry.
- I. Euclidean Reconstruction: The Gaussian ansatz models correlation decay with distance, motivated by the structure factor and the effective geometry set by the couplings.
- I. Euclidean Reconstruction: For nearest-neighbor interactions, a cosine drive waveform produces a dispersion relation whose squared structure factor determines spatial correlations through Fourier components.
- I. Euclidean Reconstruction: After several Bloch periods, the binomial distance dependence approaches a Gaussian, analogous to a diffusion process.
- I. Euclidean Reconstruction: Multi-frequency drives produce diffusion within the effective geometry, with correlations given by multinomial-expansion terms of |χk|^2T.
- I. Euclidean Reconstruction: The inferred coupling matrix J′ = (Cxx)^−1 is motivated by the relation between amplified modes and low-energy states of the XY model.
- I. Euclidean Reconstruction: The inverse correlation matrix represents partial correlations: sites coupled directly can retain nonzero partial correlation while mediated longer-range correlations vanish.
AUTHOR INFORMATION
Avikar Periwal, Eric S. Cooper, and Philipp Kunkel contributed equally to the work.
- Avikar Periwal, Eric S. Cooper, and Philipp Kunkel contributed equally.
EXTENDED DATA
The supplementary material covers the experiment’s parameter sets, imaging sequence, analysis methods, geometry reconstruction, simulations, and effective Hamiltonian engineering.
- Experimental parameters: Extended Data Table 1 lists the magnetic offset field, Bloch oscillation frequency, quadratic Zeeman shift, and interaction time for datasets in Figs. 1–4.
- Experimental sequence and imaging: The experiment uses a sequential spin-rotation and state-sensitive fluorescence-imaging sequence after driving the cavity to induce interactions.
- Supplementary Information: The supplement derives the effective Hamiltonian in position and momentum-space representations and describes geometry reconstruction and truncated-Wigner simulations.
- Hamiltonian engineering: The implemented translationally invariant XY spin models use distance-dependent couplings J(rµν) between spins.
- Hamiltonian engineering: The drive-laser frequency spectrum determines the coupling structure, while a magnetic-field gradient introduces an energy cost proportional to the separation of flipped spins.
A. Derivation of the Effective Hamiltonian
The paper derives programmable spin-exchange dynamics from cavity-mediated interactions, magnetic-field gradients, and modulated drive waveforms. It connects the resulting dispersion and correlation dynamics to geometry reconstruction and discusses simulation validity and experimental deviations.
- Effective Hamiltonian: The optical cavity mediates all-to-all spin exchange, while detunings set the interaction strength and sign.The interaction strength depends on the cavity and Zeeman detunings; a uniform-field term can be neglected in a suitable rotating frame.
- Programmable couplings: A magnetic-field gradient makes separated spin-exchange processes off-resonant, and drive modulation at rωB selectively restores interactions at distance r.The resulting coupling J(r) is obtained from Fourier components of the modulated interaction and can be programmed subject to hermiticity.
- Early-time dynamics: The early-time approximation treats the m = 0 population as a classical pump, yielding independent momentum-mode pair-creation dynamics.For weak interactions over many Bloch periods, the trotterized evolution becomes equivalent to M independent continuous pair-creation processes.
- Early-time dynamics: In the strong-instability regime, momentum-mode growth is controlled by χk, with the most negative dispersion producing the fastest amplification.The propagator eigenvalues simplify when χk < 0 and q > 0 have opposite signs and |χk|τB sin(qτB) > 1.
- Geometry reconstruction: Metric multidimensional scaling reconstructs effective coordinates from correlation-derived distances, while iterative coarse-graining supports reconstruction of an emergent bulk geometry.The multidimensional-scaling coordinates follow from a truncated singular-value decomposition; the iterative step treats bonds as new sites and repeats the procedure on a coarse-grained correlation matrix.
- Validity and experiment: The truncated Wigner approximation captures relevant short-time dynamics but fails to represent late-time non-Gaussian entangled states exactly.It remains informative for first and second moments, while dissipation and experimental imperfections affect comparisons with measured correlations.