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Microscopy of the interacting Harper-Hofstadter model in the few-body limit

The interplay of magnetic fields and interacting particles can lead to exotic phases of matter exhibiting topological order and high degrees of spatial entanglement. While these phases were discovered in a solid-state setting, recent techniques have enabled the realization of gauge fields in systems of ultracold neutral atoms, offering a new experimental paradigm for studying these novel states of matter. This complementary platform holds promise for exploring exotic physics in fractional quantum Hall systems due to the microscopic manipulation and precision possible in cold atom systems. However, these experiments thus far have mostly explored the regime of weak interactions. Here, we show how strong interactions can modify the propagation of particles in a 2xN, real-space ladder governed by the Harper-Hofstadter model. We observe inter-particle interactions affect the populating of chiral bands, giving rise to chiral dynamics whose multi-particle correlations indicate both bound and free-particle character. The novel form of interaction-induced chirality observed in these experiments demonstrates the essential ingredients for future investigations of highly entangled topological phases of many-body systems.

Experimental realization of a long-range antiferromagnet in the Hubbard model with ultracold atoms

Many exotic phenomena in strongly correlated electron systems emerge from the interplay between spin ordering and motional degrees of freedom. For example, doping an antiferromagnet is expected to give rise to interesting phases including pseudogap states, stripe-ordering and incommensurate spin order. Ultracold fermions in optical lattices offer the potential to answer open questions on the low-temperature regime of the doped Hubbard model, which is thought to capture essential aspects of the cuprate superconductor phase diagram but is numerically intractable in that parameter regime.

We have observed antiferromagnetic long-range order in a repulsively interacting Fermi gas of Li-6 atoms on a 2D square lattice containing about 80 sites. At our lowest temperature of T/t=0.25, the ordered state is directly detected from a peak in the spin structure factor and a diverging correlation length of the spin correlation function. When doping away from half-filling into a numerically intractable regime, we find that long-range order extends to doping concentrations of about 15%. Our results open the path for a controlled study of the low-temperature phase diagram of the Hubbard model.

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.

Site-resolved observations of antiferromagnetic correlations in the Hubbard model

Quantum many-body systems exhibiting magnetic correlations underlie a wide variety of phenomena. High-temperature superconductivity, for example, can arise from the correlated motion of holes on an antiferromagnetic Mott insulator. Strongly correlated many-body systems can be realized using ultracold fermionic atoms in optical lattices with a tunability that is unparalleled in conventional solid-state systems.

Using our Fermi Gas Microscope with Li-6 atoms, we have observed antiferromagnetic correlations in a Hubbard-regime optical lattice with single-site resolution. Our detection technique relies on a selective spin-removal technique and subsequent site-resolved imaging. This allows measuring the spin correlation function, which is found to decay exponentially. The extraordinary detection capabilities of the microscope also allow us to study lattice loading dynamics affect and make comparisons to theory at an unprecedented level of detail. Our temperatures are the lowest reported in a Hubbard model system with cold atoms and approach the limits of available numerical techniques. Our results demonstrate that quantum gas microscopy is a powerful tool for studying fermionic quantum magnetism.

Greiner Lab members take home multiple awards at the 2016 ICAP Poster Session!

Anton and Matthew win awards for their posters on Probing Antiferromagnetic Ordering and Dynamics with the Fermi Gas Microscope and Quantum Thermalization in an isolated many-body system, respectively.

Site-resolved imaging of a fermionic Mott insulator

Conventional band theory predicts an insulating behavior of a solid if the freely moving electrons occupy every possible quantum state in the highest energy band, whereas the state is conducing otherwise. This simple picture of band insulating and conducting metallic states is altered in the presence of strong interactions, which can lead to an insulating behavior even in the presence of a half-filled energy band. These Mott insulators, named after the British physicist Sir Nevill F. Mott, are one of the conceptually simplest examples of many-body systems, where strong correlations lead to surprising phenomena.

We have observed fermionic Mott insulators, metals and band insulators with single-site resolution by trapping a repulsively interacting, two-component mixture of Li-6 atoms in a square optical lattice. We observe large, defect-free 2D Mott insulators containing more than 400 atoms with an average entropy per particle of 1.0kB.

The images above show single-shot picture of the atomic distribution for varying interactions U/t, along with the deconvolved images in the bottom row. The metallic state (left picture) is characterized by a large occupation variance, the band insulating region appears as a core of empty sites (middle picture) and the Mott insulator is signaled by an extended region of constant occupation with one particle per site and strongly reduced occupation variance (right picture and ring in middle picture).

Measuring entanglement entropy in a quantum many-body system

Entanglement in the ground state of the Bose–Hubbard model. We study the transition from Mott insulator to superfluid with four atoms on four lattice sites in the ground state of the Bose–Hubbard model.

We directly measure entanglement entropy in a quantum many-body system, leveraging our single-site-resolved control over bosonic atoms in an optical lattice afforded by the quantum gas microscope. We observe the growth of spatial entanglement between subsystems as the atoms, initially localized in a Mott insulator phase, tunnel between subsystems establishing quantum correlations between them.

In our experiments, entanglement between subsystems of a quantum many-body system is quantified by entanglement entropy. The entanglement entropy of a subsystem originates from the loss of information contained in its correlations with the remainder of the system, when we trace over the remainder. For entangled subsystems, the measured entropy of either of them exceeds that of the full system - a fact that cannot be explained by classical correlations alone.

We measure second-order Renyi entanglement entropy by preparing two independent and identical copies of a quantum many-body state of four bosons on four lattice sites. The copies are then interfered with each other, and the resulting atom numbers (odd or even) at each site are measured with our high resolution microscope. Analogous to the celebrated Hong-Ou-Mandel interference of two identical single photon states on a beam splitter, two identical many-body bosonic states interfere to produce even number of bosons in the output ports. Thus, the resulting average atom number parity quantifies the purity of the many-body state. This quantum purity is directly related to the second-order Renyi entropy.

These experiments pave the way for investigating quantum many-body physics using entanglement entropy. For large systems, entanglement entropy gives valuable information about the underlying quantum phase of matter, and signals the existence of quantum phase transitions. Entanglement entropy is also a valuable tool to investigate non-equilibrium states of quantum matter.

Strongly Correlated Quantum Walks in Optical Lattices

Using a novel single-site addressing scheme, we study the quantum walks of individual and interacting bosonic atoms in an optical lattice. Top: An individual particle, initially localized to one lattice site, performs a free quantum walk and delocalizes over 20 lattice sites (left). An applied gradient reveals the excellent coherence of this motion, forcing the particle to undergo Bloch oscillations in position space (right).

Bottom: We deterministically add more particles to the system, and observe their dynamics with single-particle resolution. We directly detect bosonic bunching in a Hanbury Brown-Twiss experiment of two bosons, and observe the emergence of fermion-type correlations when the repulsive interactions become strong. Our work opens the door to the study of condensed matter problems using quantum walks of interacting particles.

"Probing the superfluid to Mott insulator transition at the single atom level" wins the 2011 AAAS Newcomb Cleveland Prize!

Markus is now a Professor of Physics!

Orbital Excitation Blockade and Algorithmic Cooling in Quantum Gases

Nature DOI: 10.1038/nature10668

Markus wins a MacArthur Fellowship!

Quantum simulation of antiferromagnetic spin chains in an optical lattice

Nature DOI: 10.1038/nature09994

Quantum Magnetism in an Optical lattice. (a) Two spin chains, one paramagnetically ordered (back), and one antiferromagnetically ordered (front). The background is a single-shot image from the quantum gas microscope, demonstrating the scalability of this approach. (b) Strong magnetic interactions and fast dynamics are realized by mapping the position degree of freedom of each atom onto the state of a single spin 1/2 particle. By adiabatically tuning the tilt (E/U) of the lattice, we drive a phase transition from a paramagnetically ordered (Mott) state, to an antiferromagnetically ordered state with density-wave (DW) ordering. Because the DW ordered state is contains sites with occupations of only n=2,0, it appears dark (podd~0), while the Mott state appears bright (podd~1). This first demonstration of quantum magnetism in a lattice should open new perspectives for studies of criticality, and high temperature superconductivity, with cold atoms. The work uses a theoretical proposal by Prof. Subir Sachdev and collaborators, PRB 66, 075128 (2002).

Probing the superfluid to Mott insulator transition at the single atom level

Science DOI: 10.1126/science.1192368

News: Single atom imaging of Mott insulator selected as a top breakthrough of this year by Science

Mott insulator (MI) in a Quantum Gas Microscope. (a) Sketch of Quantum Gas Microscope, enabling high fidelity single lattice site imaging. (b) Mott insulator shell structure with n=1 MI (bright ring), surrounding a n=2 MI core (dark). (c) near perfect n=1 Mott insulator.

Experimentally observed Mott shells (averaged) for increasing atom number.

The Quantum Gas Microscope enables high fidelity detection of single atoms in a Hubbard-regime optical lattice, bringing ultracold atom research to a new, microscopic level. We investigate the Bose-Hubbard model using space- and time-resolved characterization of the number statistics across the superfluid - Mott insulator quantum phase transition. Site-resolved probing of fluctuations provides us with a sensitive local thermometer, allows us to identify microscopic heterostructures of low entropy Mott domains, and enables us to measure local quantum dynamics, revealing surprisingly fast transition timescales. Our results may serve as a benchmark for theoretical studies of quantum dynamics, and open new possibilities for realizing and probing quantum magnetism.