# Ebook: Quantum Matter at Ultralow Temperatures

**Description**

The Enrico Fermi summer school on Quantum Matter at Ultralow Temperatures held on 7-15 July 2014 at Varenna, Italy, featured important frontiers in the field of ultracold atoms.

For the last 25 years, this field has undergone dramatic developments, which were chronicled by several Varenna summer schools, in 1991 on Laser Manipulation of Atoms, in 1998 on Bose-Einstein Condensation in Atomic Gases, and in 2006 on Ultra-cold Fermi Gases. The theme of the 2014 school demonstrates that the field has now branched out into many different directions, where the tools and precision of atomic physics are used to realise new quantum systems, or in other words, to quantum-engineer interesting Hamiltonians.

The topics of the school identify major new directions: Quantum gases with long range interactions, either due to strong magnetic dipole forces, due to Rydberg excitations, or, for polar molecules, due to electric dipole interactions; quantum gases in lower dimensions; quantum gases with disorder; atoms in optical lattices, now with single-site optical resolution; systems with non-trivial topological properties, e.g. with spin-orbit coupling or in artificial gauge fields; quantum impurity problems (Bose and Fermi polarons); quantum magnetism. Fermi gases with strong interactions, spinor Bose-Einstein condensates and coupled multi-component Bose gases or Bose-Fermi mixtures continue to be active areas. The current status of several of these areas is systematically summarized in this volume.

**Preface**

The Varenna summer school on Quantum Matter at Ultralow Temperatures featured important frontiers in the field of ultracold atoms. For the last 25 years, this field has undergone dramatic developments, which were chronicled by several Varenna summer schools, in 1991 on Laser Manipulation of Atoms, in 1998 on Bose-Einstein Condensation in Atomic Gases, and in 2006 on Ultra-cold Fermi Gases. The theme of the 2014 school demonstrates that the field has now branched out into many different directions, where the tools and precision of atomic physics are used to realise new quantum systems, or in other words, to quantum-engineer interesting Hamiltonians.

The topics of the school identify major new directions: Quantum gases with long range interactions, either due to strong magnetic dipole forces, due to Rydberg excitations, or, for polar molecules, due to electric dipole interactions; quantum gases in lower dimensions; quantum gases with disorder; atoms in optical lattices, now with single-site optical resolution; systems with non-trivial topological properties, e.g. with spin-orbit coupling or in artificial gauge fields; quantum impurity problems (Bose and Fermi polarons); quantum magnetism. Fermi gases with strong interactions, spinor Bose-Einstein condensates and coupled multi-component Bose gases or Bose-Fermi mixtures continue to be active areas. The current status of several of these areas is systematically summarised in this Volume.

The Varenna summer school attracted leaders in the field and students to spend nine days together in the beautiful surroundings of Lake Como. The unique spirit which can be traced back to Enrico Fermi and the unique atmosphere stimulated many scientific discussions about the current status and the future of the field. The many young active researchers, and the flurry of new ideas are a strong indication that in a few years time, it will be necessary again to review the new frontiers in ultracold atoms at a Varenna summer school.

On behalf of all the participants, we thank the Italian Physical Society and in particular Barbara Alzani, Ramona Brigatti and Marta Pigazzini for the perfect preparation and organization of a successful summer school, Monica Bonetti from the editorial office and Marcella Missiroli for editing the proceedings.

M. Inguscio, W. Ketterle, S. Stringari and G. Roati

**Abstract**

Simulating magnetic effects with cold gases of neutral atoms is a challenge. Since these atoms have no charge, one needs to create artificial gauge fields by taking advantage of the geometric phases that can result for instance from atom-light interaction. We review here some schemes that lead to the desired Hamiltonians, either in a bulk geometry or in a lattice configuration. We also detail the relations between some general concepts of magnetism, such as gauge invariance, Landau levels, topological bands, and the features that can be generated in cold atoms setups.

**Abstract**

The experimental realization of stable, ultracold Fermi gases near a Feshbach resonance allows to study gases with attractive interactions of essentially arbitrary strength. They extend the classic paradigm of BCS into a regime which has never been accessible before. We review the theoretical concepts which have been developed in this context, including the Tan relations and the notion of fixed points at zero density, which are at the origin of universality. We discuss in detail the universal thermodynamics of the unitary Fermi gas which allows a fit free comparison between theory and experiment for this strongly interacting system. In addition, we address the consequences of scale invariance at infinite scattering length and the subtle violation of scale invariance in two dimensions. Finally we discuss the fermionic excitation spectrum accessible in momentum-resolved RF-spectroscopy and the origin of universal lower bounds for the shear viscosity and the spin diffusion constant.

**Abstract**

These lecture notes review the universal thermodynamics of strongly interacting Fermi gases, experimentally realized with ultracold atoms near Feshbach resonances. These gases serve as a pristine model system for fermionic matter with contact interactions. Over the recent years, their equation of state has been measured to an ever-increasing precision that allows distinguishing between different theoretical approaches to the many-fermion problem. In the spin-balanced, resonant case, the equation of state only depends on temperature and density. The superfluid transition is signaled by a lambda-like feature in the specific heat of the gas. For non-resonant interactions, the scattering length introduces a conjugate extensive thermodynamic quantity, the contact. It encodes the probability to find two particles in close proximity and thus governs the interaction energy of the gas, the tails of the momentum distribution, the wings and the mean transition frequency of radiofrequency spectra, the probability of photoassociation and other experimental quantities. Introducing spin imbalance allows addressing a fifty year old question on the fate of fermionic superfluidity when there are more up spins than down spins and pairing cannot be complete. Phase separation between the superfluid and a mixed normal phase, as well as the eventual breakdown of superfluidity at the Pauli or Clogston-Chandrasekhar limit, have been directly observed. The mixed normal phase is identified as a Fermi liquid of Fermi polarons, dressed quasi-particles with a modified effective mass and energy. Prospects of observing an inhomogeneous superfluid state, the Fulde-Ferrell-Larkin-Ovchinnikov state of mobile Cooper pairs, are briefly discussed.

**Abstract**

In a spinor Bose-Einstein gas, the non-zero hyperfine spin of the gas becomes an accessible degree of freedom. At low temperature, such a gas shows both magnetic and superfluid order, and undergoes both density and spin dynamics. These lecture notes present a general overview of the properties of spinor Bose-Einstein gases. The notes are divided in five sections. In the first, we summarize basic properties of multi-component quantum fluids, focusing on the specific case of spinor Bose-Einstein gases and the role of rotational symmetry in defining their properties. Second, we consider the magnetic state of a spinor Bose-Einstein gas, highlighting effects of thermodynamics and Bose-Einstein statistics and also of spin-dependent interactions between atoms. In the third section, we discuss methods for measuring the properties of magnetically ordered quantum gases and present newly developed schemes for spin-dependent imaging. We then discuss the dynamics of spin mixing in which the spin composition of the gas evolves through the spin-dependent interactions within the gas. We discuss spin mixing first from a microscopic perspective, and then advance to discussing collective and beyond-mean-field dynamics. The fifth section reviews recent studies of the magnetic excitations of quantum-degenerate spinor Bose gases. We conclude with some perspectives on future directions for research.

**Abstract**

Ultracold atoms in optical lattices have proven to provide an extremely clean and controlled setting to explore strongly interacting quantum many-body phases of matter. Imaging of atoms in such lattice structures has reached the level of single-atom sensitive detection combined with the highest resolution down to the level of individual lattice sites. This has opened up fundamentally new opportunities for the characterization and the control of quantum many-body systems including quantum magnetism. These lecture notes cover selected parts of the course given at the Varenna Summer School on “Quantum Matter at Ultralow Temperatures”.

**Abstract**

The Frehlich polaron model describes a ubiquitous class of problems concerned with understanding the properties of a single mobile particle interacting with a bosonic reservoir. Originally introduced in the context of electrons interacting with phonons in crystals, this model found applications in such diverse areas as strongly correlated electron systems, quantum information, and high energy physics. In the last few years this model has been applied to describe impurity atoms immersed in Bose-Einstein condensates of ultracold atoms. The tunability of microscopic parameters in ensembles of ultracold atoms and the rich experimental toolbox of atomic physics should allow to test many theoretical predictions and give us new insights into equilibrium and dynamical properties of polarons. In these lecture notes we provide an overview of common theoretical approaches that have been used to study BEC polarons, including Rayleigh-Schroedinger and Green's function perturbation theories, self-consistent Born approximation, mean-field approach, Feynman's variational path integral approach, Monte Carlo simulations, renormalization group calculations, and Gaussian variational ansatz. We focus on the renormalization group approach and provide details of analysis that have not been presented in earlier publications. We show that this method helps to resolve the striking discrepancy in polaron energies obtained using mean-field approximation and Monte Carlo simulations. We also discuss applications of this method to the calculation of the effective mass of BEC polarons. As one experimentally relevant example of a non-equililbrium problem we consider Bloch oscillations of Bose polarons and demonstrate that one should find a considerable deviations from the commonly accepted phenomenological Esaki-Tsu model. We review which parameter regimes of Bose polarons can be achieved in various atomic mixtures.

**Abstract**

These notes are a short summary of the properties of interacting quantum one-dimensional systems, both clean and disordered. They are centered on the cold atom realization of such systems. They discuss the basic theories used to describe such systems, as well as several experimental examples. Finally some of the open questions and wishes for the future are discussed.

**Abstract**

We review some historic and some recent spectroscopic measurements on Rydberg states in dense media. The historic measurements of the density shift turn into the physics of ultra-long-range molecules at ultracold temperatures and allow even the preparation and investigation of individual electronic impurities in a quantum fluid.

**Abstract**

In this lecture I review some basic concepts and results regarding two-component Bose gases with a coherent coupling between the components. The lecture treats mainly weakly interacting gases. Ground state, Bogoliubov modes, topological excitations are considered. Results in the presence of an optical lattice are also discussed at the end of the lecture.

**Abstract**

Statistical mechanics is one of the most comprehensive theories in physics. From a boiling pot of water to the complex dynamics of quantum many-body systems it provides a successful connection between the microscopic dynamics of atoms and molecules and the macroscopic properties of matter. However, statistical mechanics only describes the thermal equilibrium situation of a system, and there is no general framework to describe how equilibrium is reached or under which circumstances it can be reached at all. This problem is particularly challenging in quantum mechanics, where unitarity appears to render the very concept of thermalization counterintuitive. With the rapid experimental progress in the control and probing of ultracold quantum gases this question has become within reach of detailed experimental investigations. In these notes we present a series of experiments with ultracold one-dimensional Bose gases, which provide novel insights into this fundamental question.

**Abstract**

Current understanding of correlations and quantum phase transitions in many-body systems has significantly improved thanks to the recent intensive studies of their entanglement properties. In contrast, much less is known about the role of quantum non-locality in these systems. On the one hand, standard, “theorist- and experimentalist-friendly” many-body observables involve correlations among only few (one, two, rarely three ...) particles. On the other hand, most of the available multipartite Bell inequalities involve correlations among many particles. Such correlations are notoriously hard to access theoretically, and even harder experimentally. Typically, there is no Bell inequality for many-body systems built only from low-order correlation functions. Recently, however, it has been shown in Tura J. et al., Science 344, (2014) 1256 that multipartite Bell inequalities constructed only from two-body correlation functions are strong enough to reveal non-locality in some many-body states, in particular those relevant for nuclear and atomic physics. The purpose of this lecture is to provide an overview of the problem of quantum correlations in many-body systems—from entanglement to non-locality—and the methods for their characterization.

**Abstract**

This paper provides an introduction to Majorana fermions in many-body systems as non-Abelian anyons. Considering a particular realization of Majorana fermions as edge states in topological quantum wires and assuming an implementation in systems of ultracold atoms and molecules, I also discuss the protocol for braiding the Majorana fermions and possible ways to demonsrate their non-Abelian statistics and their usage for quantum computation.