An atomic boson sampler

The work addresses whether and how boson sampling—a task believed to be classically intractable for many non-interacting identical bosons—can be realized with ultracold atoms instead of photons. The study is motivated by the challenges of photonic implementations (difficult Fock-state generation and loss scaling with circuit depth) and explores a platform where atoms can be deterministically prepared, coherently evolved, and efficiently detected. The central goal is to assemble large Fock states of identical bosons in a tunnel-coupled optical lattice, implement non-interacting unitary dynamics, and sample output configurations, while rigorously testing indistinguishability and interference signatures. Establishing such atomic boson sampling is significant for demonstrating quantum advantage with fewer auxiliary assumptions and for enabling future many-body simulations (e.g., Hubbard models) with programmable atom assemblies.

Prior boson sampling experiments have been predominantly photonic, demonstrating increasing scale but facing probabilistic single-photon sources, transmission losses, and often requiring modified protocols (e.g., scattershot or Gaussian boson sampling) or postselection. Foundational complexity arguments indicate computing permanents and sampling from the resultant distributions for indistinguishable bosons is #P-hard and believed classically intractable, while distinguishable-particle analogs are efficiently simulable. Ultracold atoms have a rich history as quantum-optics analogs, with demonstrations of two-atom interference and quantum gas microscopy enabling site-resolved preparation and detection. Techniques for deterministic atom rearrangement with optical tweezers, rapid high-fidelity cooling, and programmable lattice potentials provide a promising alternative to photonics, potentially avoiding exponential loss with circuit depth. The study builds on these advances and references extensive photonics work [4–18], theoretical complexity results [1,53–56,62–63], and atomic control/microscopy developments [21–26,33–41].

• Platform and state preparation: 88Sr atoms are loaded into a 16×24 optical tweezer array with 50–75% stochastic filling, implanted into a 2D tunnel-coupled optical lattice, and imaged with 99.8(1)% per-site fidelity. Atoms are rearranged into target patterns with typical per-atom success ~98% (up to 99.5% achievable). A second image enables postselection for perfect rearrangement. Atoms are then sideband cooled on the 1S0→3P1 transition to the 3D motional ground state.
• Dynamics: Atoms evolve under a single-particle Hamiltonian on the lattice with nearest-neighbor tunneling J/h ≈ 2×119 Hz and a position-dependent potential V (harmonic confinement; can be modified by tweezers). Evolution time t sets the unitary U = exp(−iHt/ħ). Single-particle loss is 5.0(2)% and independent of evolution time within studied ranges; no additional time-dependent loss was observed over millisecond dynamics.
• Detection: Final imaging measures site-resolved atom number parity due to light-assisted collisions (parity projection). For most experiments the probability of multi-occupancy per site is kept low; in small-n data, postselection on no lost or extra atoms suppresses parity-projection artifacts. Imaging error rates are calibrated (false negative ~0.2%; false positive ~10⁻⁵; day-to-day variation ~0.2%). Drift corrections align tweezers and lattice masks shot-to-shot.
• Controlling indistinguishability: Hidden degrees of freedom (electronic or out-of-plane motional states) can reduce visible bosonic behavior. By binning along y to create effective 1D walks along x, the y coordinate serves as a hidden label. Distinguishability is introduced via position labeling (different y) or time labeling (combining single-atom runs). The Hong–Ou–Mandel-like (HOM) interference is probed by varying t; a balanced beamsplitter analog occurs at t_HOM ≈ 0.96 ms.
• Indistinguishability estimation: Coincidence probabilities for labeled (distinguishable) vs unlabeled (nominally indistinguishable) data yield a ratio providing a lower bound on indistinguishability J, with calibrated corrections for loss and parity projection. Additional estimates come from models of the lattice potential and separate analyses without postselection.
• Characterizing U: A submatrix M (5×4) of U at t=1.46 ms is inferred via maximum-likelihood fits to one- and two-atom data, using a unitary parameterization (matrix exponential of a Hermitian expansion in generalized Gell-Mann matrices). Loss parameter and indistinguishability are incorporated; optimization (L-BFGS) yields M consistent within statistical variation with an independent spectroscopic model. U’s statistics are also characterized: near the center (15×15 sites), |U|² follows Porter–Thomas and phases are uniform, approximating Haar-like behavior.
• Interference tests beyond small n: For n up to 8 in 1D-binned data, compute coarse-grained observables—nearest-neighbor coincidences, full bunching (all n on one binned site), and clouding (all n on one half-array). For larger 2D arrays (up to 180 atoms over ~1015 sites), use generalized bunching: probability p_k that all n atoms fall within a subset of size k≥n (or k=1), averaged over subsets to obtain p̄_k, chosen with k≈m−m/n and m set by the number of significantly populated sites. Parity projection and detection errors are explicitly modeled; distinguishability is tuned via partitioning atoms into sub-ensembles (n_labels time labels) and combining data to simulate partial distinguishability. Simulations use exact permanents (small n) and Clifford–Clifford sampling (larger n), with thermal models for hidden motional excitations normal to the lattice.

• Prepared and measured on-demand Fock states up to 180 atoms distributed over ~1015 output sites with high repetition (~1 Hz), low and time-independent loss (5.0(2)%), and high detection fidelity (99.8(1)% per site).
• Indistinguishability: HOM-style measurements give a lower bound J = 97.1±1.5% and, incorporating lattice modeling, an estimate J = 99.5±0.5%. The dominant imperfection is attributed to residual out-of-plane motional excitations; elastic and inelastic interactions are negligible on experimental timescales.
• Two- to eight-atom 1D quantum walks: Measured bunching and clouding agree with indistinguishable-boson theory across evolution times and input spacings (NN and NNN). Full-bunching enhancement up to five atoms is observed, consistent with the n! bosonic scaling over distinguishable particles. Clouding signals for up to eight atoms show clear separation between unlabeled (bosonic) and labeled (distinguishable) cases, with no observed degradation as n increases.
• Single-particle unitary characterization: Maximum-likelihood reconstruction of a 5×4 submatrix of U from one- and two-atom data agrees (within statistical variation) with a spectroscopic Hamiltonian model. In a central 15×15 region the distribution of |U|² is close to Porter–Thomas, indicating near-Haar behavior locally; across the larger lattice, harmonic confinement structures U’s connectivity.
• Generalized bunching at larger scales: For n = 4–36 atoms (m ≈ 500 populated modes) the averaged generalized bunching p̄_k robustly separates indistinguishable from distinguishable behavior; trends align with a thermal model for hidden motional excitations, consistent with low effective temperatures (⟨n_max⟩ ≈ 0). Measurements are sensitive to calibrated loss and detection errors and are modeled accordingly.
• 180-atom regime: With square n×n input patterns at next-nearest-neighbor spacing, generalized bunching and atom-survival histograms vary systematically with the number of time labels (n_labels), interpolating between bosonic and distinguishable limits. For sufficiently many labels, simulations match data; unlabeled data show the largest bunching/survival shifts, consistent with bosonic interference. Effective on-demand per-atom success probability (prepare, evolve without loss, detect) is ~92%.

The results demonstrate that ultracold atoms in a tunnel-coupled optical lattice can implement boson sampling at scales challenging for classical verification, while maintaining key advantages over photonics: deterministic Fock-state preparation, high-fidelity detection, and loss that does not grow with evolution depth. A suite of targeted tests—HOM-like indistinguishability, bunching, clouding, generalized bunching, and unitary calibration—collectively substantiates bosonic interference across small to very large atom numbers. Agreement with theory for simulable regimes (e.g., with time labels or few atoms) and consistent trends under tunable distinguishability bolster confidence in the unsimulable regimes. The main source of residual distinguishability is attributed to thermal excitations in a hidden motional degree of freedom; interactions are negligible. While the applied unitaries are not fully Haar-random due to harmonic confinement, central regions exhibit near-random statistics, and the framework supports future programmable unitaries via tweezers. These findings reduce the assumptions needed for hardness compared with several photonic approaches and open pathways to certify high-order interference, explore sampling-complexity phase transitions, and transition toward interacting many-body simulations.

This work presents a new atomic platform for boson sampling that integrates rapid and programmable Fock-state assembly, coherent non-interacting evolution with low and time-independent loss, and high-fidelity site-resolved detection. The system samples from configurations of up to 180 atoms across ~1015 lattice sites, and a comprehensive set of diagnostics validates indistinguishability and interference signatures from few to many particles. A maximum-likelihood method directly infers submatrices of the single-particle unitary from one- and two-atom data, agreeing with spectroscopic models and enabling future programmable-unitary studies. Looking ahead, implementing fully programmable (e.g., Haar-averaged) unitaries with optical tweezers, developing more efficient calibration protocols for large systems, and leveraging mid-circuit measurements for feedforward can strengthen certification and broaden capabilities toward universal linear-optical computation with atoms. Extending these tools to interacting platforms promises rapid assembly and study of Hubbard-model states, probing complexity in interacting dynamics, and exploring hardware-efficient fermionic computing architectures.

• Direct verification of full boson-sampling distributions is infeasible at the largest system sizes; certification relies on targeted tests and comparisons in simulable regimes.
• Detection measures parity per site, not full number resolution; mitigation via low multi-occupancy and 1D binning is not applicable to all configurations.
• The applied unitaries are from Hamiltonian evolution with harmonic confinement, not fully Haar-random; potential classical structure could, in principle, be exploited without Haar averaging.
• Full-bunching probabilities decay exponentially with particle number, limiting direct observation beyond ~5 atoms at current data rates.
• Generalized bunching and survival histograms are sensitive to calibration of single-particle loss and imaging errors; robust modeling is required.
• Direct unitary inference is limited by experimental trial overhead and slow drifts, constraining the number of parameters that can be characterized at constant precision.
• Residual distinguishability arises from thermal excitation in hidden motional DOFs; while small, it sets the dominant preparation imperfection.