Observation of Cooper pairs in a mesoscopic two-dimensional Fermi gas.

The formation of strongly correlated fermion pairs is fundamental for the emergence of fermionic superfluidity and superconductivity1. For instance, Cooper pairs made of two electrons of opposite spin and momentum at the Fermi surface of the system are a key ingredient of Bardeen-Cooper-Schrieffer (BCS) theory-the microscopic explanation of the emergence of conventional superconductivity2. Understanding the mechanism behind pair formation is an ongoing challenge in the study of many strongly correlated fermionic systems3. Controllable many-body systems that host Cooper pairs would thus be desirable. Here we directly observe Cooper pairs in a mesoscopic two-dimensional Fermi gas. We apply an imaging scheme that enables us to extract the full in situ momentum distribution of a strongly interacting Fermi gas with single-particle and spin resolution4. Our ultracold gas enables us to freely tune between a completely non-interacting, unpaired system and weak attractions, where we find Cooper pair correlations at the Fermi surface. When increasing the attractive interactions even further, the pairs gradually turn into deeply bound molecules that break up the Fermi surface. Our mesoscopic system is closely related to the physics of nuclei, superconducting grains or quantum dots5-7. With the precise control over the interactions, particle number and potential landscape in our experiment, the observables we establish in this work provide an approach for answering longstanding questions concerning not only such mesoscopic systems but also their connection to the macroscopic world.

Observation of Pauli Crystals.

The Pauli exclusion principle is a fundamental law underpinning the structure of matter. Because of their antisymmetric wave function, no two fermions can occupy the same quantum state. Here, we report on the direct observation of the Pauli principle in a continuous system of up to six particles in the ground state of a two-dimensional harmonic oscillator. To this end, we sample the full many-body wave function by applying a single atom resolved imaging scheme in momentum space. We find so-called Pauli crystals as a manifestation of higher order correlations. In contrast to true crystalline phases, these unique high-order density correlations emerge even without any interactions present. Our work lays the foundation for future studies of correlations in strongly interacting systems of many fermions.

Observing the emergence of a quantum phase transition shell by shell.

Many-body physics describes phenomena that cannot be understood by looking only at the constituents of a system1. Striking examples are broken symmetry, phase transitions and collective excitations2. To understand how such collective behaviour emerges as a system is gradually assembled from individual particles has been a goal in atomic, nuclear and solid-state physics for decades3-6. Here we observe the few-body precursor of a quantum phase transition from a normal to a superfluid phase. The transition is signalled by the softening of the mode associated with amplitude vibrations of the order parameter, usually referred to as a Higgs mode7. We achieve fine control over ultracold fermions confined to two-dimensional harmonic potentials and prepare closed-shell configurations of 2, 6 and 12 fermionic atoms in the ground state with high fidelity. Spectroscopy is then performed on our mesoscopic system while tuning the pair energy from zero to a value larger than the shell spacing. Using full atom counting statistics, we find the lowest resonance to consist of coherently excited pairs only. The distinct non-monotonic interaction dependence of this many-body excitation, combined with comparison with numerical calculations allows us to identify it as the precursor of the Higgs mode. Our atomic simulator provides a way to study the emergence of collective phenomena and the thermodynamic limit, particle by particle.

Quantum scale anomaly and spatial coherence in a 2D Fermi superfluid.

Quantum anomalies are violations of classical scaling symmetries caused by divergences that appear in the quantization of certain classical theories. Although they play a prominent role in the quantum field theoretical description of many-body systems, their influence on experimental observables is difficult to discern. In this study, we discovered a distinctive manifestation of a quantum anomaly in the momentum-space dynamics of a two-dimensional (2D) Fermi superfluid of ultracold atoms. The measured pair momentum distributions of the superfluid during a breathing mode cycle exhibit a scaling violation in the strongly interacting regime. We found that the power-law exponents that characterize long-range phase correlations in the system are modified by the quantum anomaly, emphasizing the influence of this effect on the critical properties of 2D superfluids.

High-Contrast Interference of Ultracold Fermions.

Many-body interference between indistinguishable particles can give rise to strong correlations rooted in quantum statistics. We study such Hanbury Brown-Twiss-type correlations for number states of ultracold massive fermions. Using deterministically prepared ^{6}Li atoms in optical tweezers, we measure momentum correlations using a single-atom sensitive time-of-flight imaging scheme. The experiment combines on-demand state preparation of highly indistinguishable particles with high-fidelity detection, giving access to two- and three-body correlations in fields of fixed fermionic particle number. We find that pairs of atoms interfere with a contrast close to 80%. We show that second-order density correlations arise from contributions from all two-particle pairs and detect intrinsic third-order correlations.

High-temperature pairing in a strongly interacting two-dimensional Fermi gas.

The nature of the normal phase of strongly correlated fermionic systems is an outstanding question in quantum many-body physics. We used spatially resolved radio-frequency spectroscopy to measure pairing energy of fermions across a wide range of temperatures and interaction strengths in a two-dimensional gas of ultracold fermionic atoms. We observed many-body pairing at temperatures far above the critical temperature for superfluidity. In the strongly interacting regime, the pairing energy in the normal phase considerably exceeds the intrinsic two-body binding energy of the system and shows a clear dependence on local density. This implies that pairing in this regime is driven by many-body correlations, rather than two-body physics. Our findings show that pairing correlations in strongly interacting two-dimensional fermionic systems are remarkably robust against thermal fluctuations.

Microscopy of the interacting Harper-Hofstadter model in the two-body limit.

The interplay between magnetic fields and interacting particles can lead to exotic phases of matter that exhibit topological order and high degrees of spatial entanglement. Although these phases were discovered in a solid-state setting, recent innovations in systems of ultracold neutral atoms-uncharged atoms that do not naturally experience a Lorentz force-allow the synthesis of artificial magnetic, or gauge, fields. This experimental platform holds promise for exploring exotic physics in fractional quantum Hall systems, owing to the microscopic control and precision that is achievable in cold-atom systems. However, so far these experiments have mostly explored the regime of weak interactions, which precludes access to correlated many-body states. Here, through microscopic atomic control and detection, we demonstrate the controlled incorporation of strong interactions into a two-body system with a chiral band structure. We observe and explain the way in which interparticle interactions induce chirality in the propagation dynamics of particles in a ladder-like, real-space lattice governed by the interacting Harper-Hofstadter model, which describes lattice-confined, coherently mobile particles in the presence of a magnetic field. We use a bottom-up strategy to prepare interacting chiral quantum states, thus circumventing the challenges of a top-down approach that begins with a many-body system, the size of which can hinder the preparation of controlled states. Our experimental platform combines all of the necessary components for investigating highly entangled topological states, and our observations provide a benchmark for future experiments in the fractional quantum Hall regime.

Quantum thermalization through entanglement in an isolated many-body system.

Statistical mechanics relies on the maximization of entropy in a system at thermal equilibrium. However, an isolated quantum many-body system initialized in a pure state remains pure during Schrödinger evolution, and in this sense it has static, zero entropy. We experimentally studied the emergence of statistical mechanics in a quantum state and observed the fundamental role of quantum entanglement in facilitating this emergence. Microscopy of an evolving quantum system indicates that the full quantum state remains pure, whereas thermalization occurs on a local scale. We directly measured entanglement entropy, which assumes the role of the thermal entropy in thermalization. The entanglement creates local entropy that validates the use of statistical physics for local observables. Our measurements are consistent with the eigenstate thermalization hypothesis.

Ultra-precise holographic beam shaping for microscopic quantum control.

High-resolution addressing of individual ultracold atoms, trapped ions or solid state emitters allows for exquisite control in quantum optics experiments. This becomes possible through large aperture magnifying optics that project microscopic light patterns with diffraction limited performance. We use programmable amplitude holograms generated on a digital micromirror device to create arbitrary microscopic beam shapes with full phase and amplitude control. The system self-corrects for aberrations of up to several λ and reduces them to λ/50, leading to light patterns with a precision on the 10-4 level. We demonstrate aberration-compensated beam shaping in an optical lattice experiment and perform single-site addressing in a quantum gas microscope for 87Rb.

Measuring entanglement entropy in a quantum many-body system.

Entanglement is one of the most intriguing features of quantum mechanics. It describes non-local correlations between quantum objects, and is at the heart of quantum information sciences. Entanglement is now being studied in diverse fields ranging from condensed matter to quantum gravity. However, measuring entanglement remains a challenge. This is especially so in systems of interacting delocalized particles, for which a direct experimental measurement of spatial entanglement has been elusive. Here, we measure entanglement in such a system of itinerant particles using quantum interference of many-body twins. Making use of our single-site-resolved control of ultracold bosonic atoms in optical lattices, we prepare two identical copies of a many-body state and interfere them. This enables us to directly measure quantum purity, Rényi entanglement entropy, and mutual information. These experiments pave the way for using entanglement to characterize quantum phases and dynamics of strongly correlated many-body systems.

Strongly correlated quantum walks in optical lattices.

Full control over the dynamics of interacting, indistinguishable quantum particles is an important prerequisite for the experimental study of strongly correlated quantum matter and the implementation of high-fidelity quantum information processing. We demonstrate such control over the quantum walk-the quantum mechanical analog of the classical random walk-in the regime where dynamics are dominated by interparticle interactions. Using interacting bosonic atoms in an optical lattice, we directly observed fundamental effects such as the emergence of correlations in two-particle quantum walks, as well as strongly correlated Bloch oscillations in tilted optical lattices. Our approach can be scaled to larger systems, greatly extending the class of problems accessible via quantum walks.

Orbital excitation blockade and algorithmic cooling in quantum gases.

Interaction blockade occurs when strong interactions in a confined, few-body system prevent a particle from occupying an otherwise accessible quantum state. Blockade phenomena reveal the underlying granular nature of quantum systems and allow for the detection and manipulation of the constituent particles, be they electrons, spins, atoms or photons. Applications include single-electron transistors based on electronic Coulomb blockade and quantum logic gates in Rydberg atoms. Here we report a form of interaction blockade that occurs when transferring ultracold atoms between orbitals in an optical lattice. We call this orbital excitation blockade (OEB). In this system, atoms at the same lattice site undergo coherent collisions described by a contact interaction whose strength depends strongly on the orbital wavefunctions of the atoms. We induce coherent orbital excitations by modulating the lattice depth, and observe staircase-like excitation behaviour as we cross the interaction-split resonances by tuning the modulation frequency. As an application of OEB, we demonstrate algorithmic cooling of quantum gases: a sequence of reversible OEB-based quantum operations isolates the entropy in one part of the system and then an irreversible step removes the entropy from the gas. This technique may make it possible to cool quantum gases to have the ultralow entropies required for quantum simulation of strongly correlated electron systems. In addition, the close analogy between OEB and dipole blockade in Rydberg atoms provides a plan for the implementation of two-quantum-bit gates in a quantum computing architecture with natural scalability.

Photon-assisted tunneling in a biased strongly correlated Bose gas.

We study the impact of coherently generated lattice photons on an atomic Mott insulator subjected to a uniform force. Analogous to an array of tunnel-coupled and biased quantum dots, we observe sharp, interaction-shifted photon-assisted tunneling resonances corresponding to tunneling one and two lattice sites either with or against the force and resolve multiorbital shifts of these resonances. By driving a Landau-Zener sweep across such a resonance, we realize a quantum phase transition between a paramagnet and an antiferromagnet and observe quench dynamics when the system is tuned to the critical point. Direct extensions will produce gauge fields and site-resolved spin flips, for topological physics and quantum computing.

Quantum simulation of antiferromagnetic spin chains in an optical lattice.

Understanding exotic forms of magnetism in quantum mechanical systems is a central goal of modern condensed matter physics, with implications for systems ranging from high-temperature superconductors to spintronic devices. Simulating magnetic materials in the vicinity of a quantum phase transition is computationally intractable on classical computers, owing to the extreme complexity arising from quantum entanglement between the constituent magnetic spins. Here we use a degenerate Bose gas of rubidium atoms confined in an optical lattice to simulate a chain of interacting quantum Ising spins as they undergo a phase transition. Strong spin interactions are achieved through a site-occupation to pseudo-spin mapping. As we vary a magnetic field, quantum fluctuations drive a phase transition from a paramagnetic phase into an antiferromagnetic phase. In the paramagnetic phase, the interaction between the spins is overwhelmed by the applied field, which aligns the spins. In the antiferromagnetic phase, the interaction dominates and produces staggered magnetic ordering. Magnetic domain formation is observed through both in situ site-resolved imaging and noise correlation measurements. By demonstrating a route to quantum magnetism in an optical lattice, this work should facilitate further investigations of magnetic models using ultracold atoms, thereby improving our understanding of real magnetic materials.