Source-linked AI summary
Tools for quantum simulation with ultracold atoms in optical lattices
Florian Schäfer, Takeshi Fukuhara, Seiji Sugawa, Yosuke Takasu, Yoshiro Takahashi
TL;DR
Quantum simulation with ultracold atoms offers access to solid-state models that remain difficult to simulate numerically, including the Fermi–Hubbard model away from half filling. This Technical Review organizes the experimental tools, techniques, and applications, while highlighting representative capabilities and the challenge of reaching the temperatures needed to study the underdoped regime.
Problem
Numerical methods cannot adequately simulate the Fermi–Hubbard model away from half filling, motivating experimental approaches to investigate this regime and its relevance to high-temperature cuprate superconductors.
Method
The review provides an accessible, structured compendium of optical-lattice quantum-simulation techniques, their methods, applications, and measurement protocols.
Results
The review surveys capabilities including topological Hofstadter-model realization, quantized Thouless pumping, Pomeranchuk cooling in SU(N) systems, and site-resolved imaging techniques.
Takeaways & Limitations
The compendium supports planning and upgrading ultracold-atom quantum-simulation experiments by clarifying available techniques, applications, advantages, and limitations.
Takeaways & Limitations
Reaching sufficiently low temperatures for fermionic atoms remains an important technical challenge for investigating the underdoped region of high-temperature cuprate superconductors.
Abstract
from arXiv · showhide
After many years of development of the basic tools, quantum simulation with ultracold atoms has now reached the level of maturity where it can be used to investigate complex quantum processes. Planning of new experiments and upgrading existing set-ups depends crucially on a broad overview of the available techniques, their specific advantages and limitations. This Technical Review aims to provide a comprehensive compendium of the state of the art. We discuss the basic principles, the available techniques and their current range of applications. Focusing on the simulation of varied phenomena in solid-state physics using optical lattice experiments, we review their basics, the necessary techniques and the accessible physical parameters. We outline how to control and use interactions with external potentials and between the atoms, and how to design new synthetic gauge fields and spin-orbit coupling. We discuss the latest progress in site-resolved techniques using quantum gas microscopes, and describe the unique features of quantum simulation experiments with two-electron atomic species.
Introduction
Optical-lattice quantum simulation uses controllable ultracold atoms to emulate many-body models that are difficult to study numerically or in complex solid-state materials. The review surveys lattice geometries, model realizations, experimental protocols, and techniques for controlling and probing these systems.
- The review focuses on simulating minimal models, especially the Fermi–Hubbard model, rather than reproducing every degree of freedom in real materials.The underdoped region is identified as especially important for understanding high-temperature cuprate superconductors.
- Ultracold atoms in optical lattices provide a controllable platform for studying condensed-matter many-body systems that are difficult to simulate conventionally.
- The review organizes technical guidance around lattice preparation, interaction control, perturbations, imaging, synthetic gauge fields, spin–orbit coupling, and two-electron atoms.It is specifically presented as an accessible technical reference for newcomers and includes exemplary applications.
- Optical lattice basics: Optical lattices can emulate Hubbard, Heisenberg, and Ising models across varied geometries, including square, honeycomb, Lieb, triangular, kagome, and superlattice configurations.Geometry controls dimensionality, energy bands, frustration, and accessible many-body phases.
- Optical lattice basics: Species-selective, mixed-dimensional, quasiperiodic, cavity, and holographically shaped lattices extend the systems and interactions available for simulation.Optical tweezers also enable defect-free atomic arrays in one, two, and three dimensions.
Controllable parameters.
Optical-lattice experiments offer precise control over Hamiltonian parameters and increasingly powerful measurement protocols. These tools connect global observables, spectroscopy, and site-resolved imaging to local correlations, topology, and energy measurements.
- Controllable parameters: The Hubbard hopping amplitude, on-site interaction strength, and their ratio can be precisely tuned through the optical-lattice depth and Feshbach resonances.Lattice shaking and Raman-assisted tunnelling can additionally create complex hopping amplitudes with Peierls phases.
- Controllable parameters: Filling, temperature, and the overall trapping potential are experimentally adjustable, while box traps and advanced light shaping reduce harmonic-trap inhomogeneity.
- Methods to diagnose optical lattice systems: Time-of-flight imaging measures coherence and momentum distributions, while band mapping accesses quasi-momentum populations in multiple Bloch bands.
- Methods to diagnose optical lattice systems: Spectroscopic methods probe band structures and interactions, with lattice modulation typically exciting states at the same quasi-momentum, Δk = 0.
- Methods to diagnose optical lattice systems: Quantum gas microscopes provide direct in-situ atom distributions, enabling site-resolved measurements such as entanglement entropy and contributing to internal-energy measurements.
- Methods to diagnose optical lattice systems: Measurement protocols combine local operations with subsequent measurements to separate spin components, resolve sublattice occupations, and access correlations and topology.Demonstrated applications include Stern–Gerlach imaging, non-local correlations, long-range coherence, Berry curvature, and topological invariants.
Controlled interatomic interactions
Ultracold-atom experiments provide several routes to engineer short-range interactions, including magnetic, optical, orbital, and confinement-induced Feshbach resonances. These approaches differ in tunability and applicability across atomic species and confinement settings.
- Feshbach resonances make short-range interactions adjustable by mixing an unbound entrance channel with a bound molecular closed channel.
- Optical Feshbach resonances bridge entrance and electronically excited bound states using near-resonant laser light.
- Orbital Feshbach resonances enable magnetic interaction control in two-electron atoms through differences in nuclear g-factors between ground and excited states.
- Confinement-induced resonances occur when the three-dimensional scattering length approaches the transverse confinement length in one-dimensional systems.The effect has also been demonstrated for two species in mixed dimensions.
- Magnetic Feshbach control is the most common and readily achievable method, while optical control offers fast switching and submicrometre-scale spatial control.
Anisotropic and long-range interactions.
Ultracold-atom platforms provide tunable isotropic, anisotropic, short-range, and long-range interactions for quantum simulation. The review describes magnetic and electric dipolar interactions, Rydberg-mediated effects, and related spin-model applications.
- Magnetic dipole–dipole interactions provide strong anisotropy, especially in atoms with large magnetic moments such as Cr, Dy, Er, and Ho.
- Feshbach resonances allow control over the relative strength of isotropic short-range and magnetic dipole–dipole interactions.
- Polar molecules provide another route to anisotropic interactions through electric dipole moments.
- Quantum-degenerate polar molecules have been realized by combining molecular association from ultracold atomic samples with further cooling techniques.
- Rydberg atoms provide electrostatic long-range interactions and building blocks including Rydberg blockade, dressing, and dipole spin-exchange interactions.
- Rydberg-state trapping and two-electron atomic species offer additional routes toward high-fidelity control and distinctive interaction physics.
136. Dissipation
Dissipation and controlled perturbations extend optical-lattice simulations beyond closed equilibrium systems. Experiments use dissipative processes, disorder, and quenches to study stabilization, localization, transport, and nonequilibrium dynamics.
- Dissipation: Two-body dissipation has been used to suppress phase-coherence growth and stabilize the Mott-insulator state in a dissipative Bose–Hubbard model.
- Dissipation: Dissipative Hubbard models have also been studied for fermions, with predictions of dynamical changes in spin correlations.
- Dissipation: Anomalous, subdiffusive momentum broadening has been observed due to dissipation.
- Disordered potentials: Random speckles, quasi-periodic potentials, and atomic impurities provide routes to disordered potentials and have supported studies of Anderson localization.
- Out-of-equilibrium dynamics: Quenches drive atomic systems out of equilibrium, enabling studies of time dynamics and non-equilibrium phenomena.
- Out-of-equilibrium dynamics: A transition from ballistic to diffusive expansion has been observed when interactions are introduced in fermionic quantum gases.
Quantum gas microscope
Quantum gas microscopes enable single-atom sensitivity and single-site resolution in optical lattices. They support measurements of correlations, ordering, entanglement, transport, and nonequilibrium dynamics, while imaging remains constrained by heating and early parity-projection limitations.
- Quantum gas microscopes enable observation and control of optical-lattice atoms with single-atom sensitivity and single-site resolution.
- Achieving high-resolution imaging in Hubbard-regime lattices is challenging because short lattice periods require substantial hopping and demanding imaging optics.
- Single-atom sensitivity and single-site resolution remain difficult because of limited scattering cross sections and heating from photon scattering.
- Early microscopes suffered parity projection and spin insensitivity, but spin-selective removal enabled spin-resolved and atom-number-sensitive measurements.
- Microscopes have measured particle–hole pairs, antiferromagnetic correlations, string orders, entanglement growth, and entanglement entropy.
Synthetic gauge fields
Synthetic gauge fields let optical-lattice experiments emulate magnetic phases, flux, and topological band structures that are unavailable from neutral atoms directly. Implementations include Raman coupling, lattice shaking, and synthetic dimensions.
- Implementations: Artificial gauge fields reproduce Peierls phases and Aharonov–Bohm-like fluxes in optical lattices.Raman-assisted tunnelling and lattice modulation provide distinct routes to complex tunnelling amplitudes.
- Topological models: The topological Hofstadter model was realized with a non-zero Chern number measured through anomalous-Hall-response-induced centre-of-mass motion.
- Implementations: Lattice shaking does not require an additional laser and has been extended to density-dependent gauge fields.These developments move toward gauge fields coupled dynamically to matter.
- Synthetic dimensions: Synthetic dimensions can use time, internal states, or momentum space as additional lattice directions.Quantized centre-of-mass transport per cycle was observed for bosonic and fermionic Thouless pumps, and chiral edge currents were observed in internal-state lattices.
- Synthetic dimensions: Synthetic-dimension systems impose hard-wall boundaries with limited artificial-site numbers and non-local interactions along the artificial direction.
Spin–orbit coupling.
The review describes engineered spin–orbit coupling and the distinctive capabilities of two-electron atoms. These systems support topological bands, SU(N) physics, specialized cooling, and precise interaction control.
- Spin–orbit coupling: Spin–orbit coupling can be engineered through Raman coupling of internal states or Raman-laser-assisted tunnelling.The mechanism relies on spin–momentum locking.
- Spin–orbit coupling: Optical Raman lattices have realized two-dimensional spin–orbit coupling with topological bands and a three-dimensional semimetal.
- Two-electron atoms: Two-electron atoms provide access to SU(N) symmetry and two-orbital systems unavailable with alkali-metal atoms.The review addresses preparation and detection methods for these systems.
- SU(N) systems: SU(N) symmetry eliminates spin-exchange collisions, stabilizing populations of individual spin components and benefiting synthetic-dimension implementations.
- SU(N) systems: Pomeranchuk cooling uses enlarged spin symmetry to lower the temperature of atoms in an optical lattice.At unity filling, entropy stored in localized spin degrees of freedom enables cooling, confirmed by doublon production-rate measurements.
Two-orbital systems.
Two-orbital systems exploit long-lived electronic states and narrow optical transitions to create new platforms for spin–orbital physics and interaction control. Their orbital structure also enables specialized Feshbach-resonance techniques.
- Two-orbital systems: Long-lived metastable states and clock transitions enable occupancy-resolved spectroscopy when on-site collisional shifts exceed the spectral linewidth.
- Interaction control: The 3P2 state enables tuning of interatomic interactions through magnetic Feshbach resonances involving isotropic and anisotropic interactions.
- Interaction control: Optical Feshbach resonances in two-electron atoms can be enhanced, with narrow linewidths enabling efficient control with only small losses.
- Two-orbital systems: The 3P0 state provides an experimental platform for a two-orbital 1S0 + 3P0 SU(N) system.Such systems have been theoretically studied for rich quantum phases and spin–orbital phenomena including the Kondo effect.
- Interaction control: Orbital degrees of freedom and interorbital nuclear-spin exchange produce an SU(N)-symmetric orbital Feshbach resonance.This is analogous to magnetic Feshbach resonances in alkali atoms, within the scope described by the review.
Outlook
The review surveys optical-lattice realizations of theoretical models and their applications to numerically hard or conceptually important problems. It identifies lower T/t for fermions as a central challenge for reaching the underdoped Fermi–Hubbard regime.
- Applications: Ultracold atoms in optical lattices realize Hubbard, Heisenberg, and Ising models relevant to condensed-matter physics.
- Applications: The review describes tools and applications for simulating numerically hard and conceptually important problems.
- Challenges: T/t is about 0.1 for ultracold atoms, compared with typically below 10^-4 for electrons in solids.The review identifies reaching sufficiently low fermionic temperatures as important for investigating the underdoped high-Tc cuprate regime.
- Future directions: Future targets include unconventional high-Tc superfluids, quantum computing with Rydberg tweezer arrays, and noisy intermediate-scale quantum devices.
- Future directions: Quantum-simulation tools may also support precision measurements with a Fermi-degenerate optical lattice clock and searches for new particles.
Competing interests
The paper presents optical lattices as a versatile framework for quantum simulation, connecting lattice design, Hubbard models, interactions, and spin models. It declares no competing interests.
- The authors declare no competing interests.
- Optical lattice toolbox: Optical lattices provide periodic potentials that structure atomic clouds and support quantum simulations analogous to solid-state lattice systems.Standing-wave lasers can form one-dimensional lattices, while orthogonal superpositions create three-dimensional cubic lattices.
- Hubbard models: Deep optical lattices realize Hubbard-model descriptions in which hopping and on-site interactions compete to produce quantum phase transitions.The fermionic formulation assumes spin-1/2 atoms occupy a single lattice band.
- Spin models: At half filling and strong interactions, the Fermi–Hubbard model reduces to a Heisenberg model with super-exchange coupling J=4t^2/U.The coupling is antiferromagnetic for J > 0 and ferromagnetic for J < 0.
- Spin models: A tilted Bose–Hubbard model can emulate the Ising model by mapping occupation numbers to spins and enabling observation of paramagnetic-to-antiferromagnetic transitions.
S1. Formation of ultracold atomic gases
Ultracold atomic-gas preparation combines atomic loading, laser and evaporative cooling, and transfer into magnetic or optical traps. These steps produce Bose–Einstein condensates or Fermi-degenerate gases suitable for optical-lattice experiments.
- Experiments first trap the required atomic species and cool the atoms to the ultra-low temperatures needed for quantum simulation.Light-induced forces are central to both trapping and cooling, supported by developed theoretical and experimental tools.
- Atomic sources: Atomic beams begin with species-dependent vapour sources, whose temperatures range from slightly above room temperature for rubidium to above 1000°C for erbium or dysprosium.
- Laser cooling: Resonantly counter-propagating laser light slows atoms from several hundred metres per second to about 10 m/s, allowing trapping and laser cooling.
- Initial cooling: A magneto-optical trap can be loaded from an atomic vapour and cools atoms below milli-kelvin temperatures.
- Evaporative cooling: Atoms are transferred to magnetic or optical traps for forced evaporative cooling, typically reaching below micro-kelvin temperatures.
- Quantum-degenerate gases: Further cooling produces Bose–Einstein condensates for bosons or Fermi-degenerate gases for fermions, with Raman-sideband cooling also demonstrated for Bose–Einstein condensation.
S2. Measurement of spin-correlations in optical lattices
Spin correlations can be measured through coherent singlet–triplet oscillations in optical superlattices or through selective removal and band mapping. These methods have enabled observations of spin correlations in ultracold atoms.
- Without quantum gas microscopy, measuring spin correlations is usually difficult.
- Singlet–triplet oscillations: Spin correlations can be detected by inducing singlet–triplet oscillations in double wells of an optical superlattice.
- Singlet–triplet oscillations: The protocol freezes tunnelling, applies a spin-dependent potential gradient, and induces coherent oscillations between singlet and triplet states.
- Readout: Neighbouring pairs are merged into single sites with a superlattice before detecting the resulting state populations.
- Selective removal: Photoassociation light or a magnetic-field sweep across an s-wave Feshbach resonance can selectively remove singlet-correlated atom pairs.The initial atom loss corresponds to the number of initially prepared singlet pairs.
- Experimental demonstrations: Spin correlations have been observed using singlet-selective loss, while singlet–triplet oscillations have also been measured through band mapping after merging neighbouring sites.
S3. Technical aspects of quantum gas microscopes
Site-resolved imaging requires near-diffraction-limited optics, careful management of vacuum-window aberrations, and cooling schemes that yield enough fluorescence while limiting heating. Several optical designs and atom-transport methods address these constraints.
- Resolution requirements: Site-resolved detection requires imaging resolution comparable to lattice spacings of several hundred nanometres and therefore close to the diffraction limit.
- Experimental constraints: Ultracold-atom imaging is challenging because fragile samples in ultra-high vacuum cannot tolerate probe-beam heating and must be viewed through thick vacuum windows.Windows thicker than a millimetre cause serious aberrations for high-numerical-aperture objectives and increase the imaging distance.
- Optical solutions: Custom long-working-distance objectives can correct for thick window glass and reduce imaging aberrations.
- Optical solutions: A 200-micrometre sapphire window or an in-vacuum solid immersion lens provides alternative ways to improve high-resolution imaging.
- Atom positioning: Because combined imaging optics have limited working distance, atoms must be prepared near the vacuum window or solid immersion lens.
- Atom positioning: Atoms can be transported from the laser-cooling region to the imaging region using magnetic forces or moving optical lattices.
S4. Comparison of synthetic gauge fields and spin-orbit coupling implementations
The review compares experimental implementations of synthetic gauge fields and spin–orbit coupling in real space, organizing them by physical scheme and coupled atomic states. It also catalogs the atomic species and mixtures used in quantum-degenerate systems.
- Comparison scope: The review compares experimentally implemented synthetic gauge fields in real space and focuses on three major implementation methods.The comparison includes lattice-based schemes and spin–orbit-coupling implementations in one and two dimensions.
- Reference tables: The review’s comparison tables cover synthetic gauge fields and spin–orbit couplings in real space alongside atomic species and mixtures brought to quantum degeneracy.The supplementary comparison includes atomic-species combinations and publication references.
- Synthetic gauge fields: Lattice shaking implements a Peierls substitution in a one-dimensional optical lattice and can produce staggered effective magnetic flux in triangular lattices.The listed examples also include the Haldane model in honeycomb lattices.
- Spin–orbit coupling: One-dimensional spin–orbit coupling implementations include Raman coupling, rf+Raman dressing, clock transitions, magic-wavelength optical lattices and ultra-narrow transitions.The table associates these schemes with hyperfine, electronic or nuclear spin states in several atomic species.
- Spin–orbit coupling: Two-dimensional spin–orbit coupling implementations include two-dimensional optical Raman lattices, additional coupling and topological lattice bands.The listed systems include 87Rb and 173Yb, with a nodal-line semimetal phase identified for the latter.