Ebook: Quantum Mixtures with Ultra-cold Atoms

The International School of Physics “Enrico Fermi” in Varenna: mixtures of people and ideas

Since its foundation in 1953, the International School of Physics “Enrico Fermi” has been the crucible of extraordinary ideas and of cutting-edge scientific perspectives. These were carried on with enthusiasm by outstanding scientists who initially as students and later as lecturers and Directors have fostered some of the most terrific advancements of modern physics, often certified by the Nobel Prize in Physics.

The Courses linked to the revolution produced by the laser have certainly been significant for the development of science and technology. Pioneers such as Isidor Rabi and Charles Townes contributed significantly to this history. Isidor Rabi, one of the most prominent and influential American physicists, ran a School in 1955 shortly before the beginning of the laser era. The School was dedicated to Enrico Fermi, with a special speech of Rabi, and there was also a lecturer in molecular structure named Townes. Charles Townes returned to Varenna as a lecturer in 1960 for a School on masers directed by Adriano Gozzini, the father of atomic physics in Italy. Three years later, in 1963, Townes directed the first School of quantum electronics, a field of research related to quantum optics and laser applications. In 1964, Charles Townes received the Nobel Prize in Physics for his contributions to the development of the maser and the laser. His research had an enormous impact on laser technology and applications in a wide range of fields.

The Varenna Schools organized in the following decades recognized the importance of lasers in different contexts, both as a spectroscopy instrument and, more recently, as the essential tool for the production and manipulation of ultracold atomic matter. For years, researchers in this field have constituted a family of people interacting with each other, mixing ideas, teaching as inspiration. In this respect, Varenna Schools represented key moments and meeting occasions to create and consolidate such a scientific community.

From the pioneering laser cooling techniques, reported in the 1991 edition Laser Manipulation of Atoms, and the first achievements and preliminary studies on Bose-Einstein condensation in Atomic Gases and Ultra-cold Fermi Gases, in the 1998 and 2006 Schools, to the demonstration of advanced studies on Quantum Matter at Ultralow Temperatures in the 2014 edition, Varenna Schools have always accompanied the field, fixing the main results and providing the participants with prestigious lecturers.

The 1998 Varenna School on Bose-Einstein condensation (BEC) took place just three years after the experimental realization of BEC and it played a crucial role in emphasizing the important perspectives in the emerging field of quantum-degenerate atomic gases. It is still regarded today as a historical School for ultracold quantum gases and the proceedings volume containing the contributions from the lecturers still represents the main reference for students who approach this field of research. This event provided an opportunity to celebrate the achievements of the three 1997 Nobel Laureates in laser cooling techniques, Claude Cohen-Tannoudji, William D. Phillips, and Steven Chu, and it also set the foundation for the recognition of the three 2001 Nobel Laureates for their work on achieving Bose-Einstein condensates: Eric A. Cornell, Wolfgang Ketterle, and Carl E. Wieman. Figure 1 shows a group photo with most of the lecturers of the 1998 School: among them, we can count six Nobel Prize winners.

In this photo, Lev P. Pitaevskii, scientific father and continuous source of inspiration for the whole community working on the theoretical aspects of Bose-Einstein condensation and superfluidity (sadly passed away in 2022, only a few weeks after the School on quantum mixtures), is also portrayed.

The discussions that took place in Varenna in 1998 opened prospects for producing atomic mixtures, which subsequently led to the first works on mixtures of fermions and bosons.

The School organized in 2006 addressed the new achievements on quantum-degenerate Fermi gases, and it was also the occasion to commemorate the 80th anniversary of the discovery of Fermi statistics. A key scientist in this field was Deborah (Debbie) S. Jin (sadly passed away in 2016), who played a crucial role in the study of quantum degeneracy of fermionic potassium, with the first realization of an ultracold Fermi gas in 1999 and pioneering investigations of its properties. This School marked a new direction in the field, showcasing the remarkable contributions of eminent scientists, in the production of ultracold, strongly correlated Fermi matter.

The 2014 School “Quantum Matter at Ultralow Temperatures” covered a range of different subjects, reflecting how far the frontiers of laser-based manipulation of atomic gases had expanded: quantum simulation with optical lattices, artificial gauge fields, topological quantum matter, long-range interacting systems, strongly correlated few- and many-body systems, quantum mixtures with coherent coupling between their components.

Given the recent proliferation of experiments involving different kinds of atoms in various configurations, which drastically increases the richness of the system and the physics that can be explored, the topic of the 2022 Varenna School focused on Quantum Mixtures of Ultra-cold Atoms.

International speakers addressed different types of quantum mixtures, from multi-component spin mixtures to mixtures of different atomic species, providing theoretical or experimental approaches for studying such powerful and promising platforms, characterized by ultimate control and able to show unprecedented features related to the interplay between two or more superfluid systems.

The lectures provided a broad overview on the physics of atomic mixtures from the few-body regime, with the study of Feshbach resonances and the impurity problem, to the many-body regime, with the observation of new types of solitons and vortices and the realization of superfluid ferromagnetic systems. Several bosonic and fermionic mixtures showing complementary features were studied in different configurations, in harmonic, flat or lattice potentials, or in the presence of coherent coupling. Different kinds of interactions were explored, from short-range forces treatable within mean-field and beyond-mean-field schemes in spin mixtures, to long-range in dipolar gases and in hybrid systems made of charged ions immersed in a neutral gas.

We would like to express our sincere gratitude to all the participants that contributed to the stimulating scientific discussions in a truly friendly atmosphere. Special thanks go to the Italian Physical Society for organizing the event with the precious support of Barbara Alzani, Ramona Brigatti and Angela Di Giuseppe from the editorial office.

Rudolf Grimm, Massimo Inguscio, Giacomo Lamporesi and Sandro Stringari

The high degree of control on ultracold gases allows us to precisely manipulate their internal state. When the gas is made of atoms in two different internal states, it can be considered as a two-component spin mixture. Below a critical temperature, the gas becomes a superfluid mixture, never realized before with any other platform, and therefore interesting to study per se, but it also constitutes a promising and versatile platform for applications in spintronic devices or to study phenomena belonging to very different fields, such as magnetism, high-energy physics or gravitation. Here, I will revisit ground-state properties and excitations of a binary bosonic superfluid, and then introduce a coherent coupling between the states and treat the global state of the atoms as a spin in the presence of a variable external field.

In these lecture notes, we discuss the physics of a two-dimensional binary mixture of Bose gases at zero temperature, close to the point where the two fluids tend to demix. We are interested in the case where one of the two fluids (the bath) fills the whole space, while the other one (the minority component) contains a finite number of atoms. We discuss under which condition the minority component can form a stable, localized wave packet, which we relate to the celebrated “Townes soliton”. We discuss the formation of this soliton and the transition towards a droplet regime that occurs when the number of atoms in the minority component is increased. Our investigation is based on a macroscopic approach based on coupled Gross-Pitaevskii equations, and it is complemented by a microscopic analysis in terms of bath-mediated interactions between the particles of the minority component.

These lecture notes provide an introduction to Bose-Einstein condensates with Raman-induced spin-orbit coupling. Owing to the interplay between the peculiar single-particle dispersion, featuring a double-minimum structure, and the two-body interaction, these systems possess a complex phase diagram. Three different quantum phases can be observed, i.e., a stripe, a plane-wave, and a single-minimum phase, each characterized by different broken symmetries. The condensate dynamics is also significantly affected by the spin-orbit coupling, as revealed by the behavior of the Bogoliubov spectrum in infinite systems and the behavior of the discretized collective mode frequencies in trapped configurations, especially close to the phase transitions. In turn, the superfluid and rotational properties are also deeply modified by the coupling between the motional and spin degree of freedom. Finally, special attention is devoted to the stripe phase and its supersolid character, which can be clearly revealed by the study of its dynamic features.

Atomic ultracold mixtures have been realized with very different species or with different internal states of one species. The latter system offers a unique possibility to coherently interchange the constituents of the mixture offering a new degree of freedom which is under pristine experimental control. By interpreting a mixture of two components as a pseudo-spin-1/2 system, one can realize in good approximation an XXX-Heisenberg model with single-ion anisotropy and precisely controllable transverse field. This is a model system exhibiting a reach phase diagram including an experimentally accessible quantum phase transition. The experimental system is especially interesting for the study of dynamics which is theoretically very challenging to be predicted. In this lecture the fundamental properties of coherently coupled two-component gases will be discussed and the results on probing the quantum critical point with dynamics after a fast quench will be presented.

Spinor Bose-Einstein condensates offer a rich playground for the study of symmetry breaking, topological excitations and the interplay between superfluidity and magnetism. An updated overview of these subjects is given.

These lecture notes contain an introduction to the physics of quantum mixtures of ultracold atoms trapped in multiple internal states. I will discuss the case of fermionic isotopes of alkaline-earth atoms, which feature an intrinsic SU(N) interaction symmetry and convenient methods for the optical manipulation of their nuclear spin. Some research directions will be presented, with focus on experiments performed in Florence with nuclear-spin mixtures of ¹⁷³Yb atoms in optical lattices.

Thanks to the unique control of physical parameters in laser-cooled atomic systems, recent experiments with mixtures of bosons and fermions have observed dual Bose-Fermi superfluid mixtures. This breakthrough is the culmination of an effort started with the first observation of superfluidity in ³He. In this chapter we describe the techniques that enabled this discovery. We discuss the collisional stability of the mixture and we show how the study of inelastic losses can be used as a probe of short range correlations in ultracold gases. We will also focus on the dynamics of Bose-Fermi superfluid counterflow and we will explore the origin of dissipation in this system.

The topic of the present lecture notes are two-species quantum mixtures composed of a deeply degenerate Fermi gas and a second component, the latter being fermionic or bosonic. A key ingredient is the possibility to tune the s-wave interaction between the different species by means of magnetically controlled Feshbach resonances, which allow us to investigate regimes of strong interactions. In two case studies, we review our experiments on mixtures of ⁶Li fermions with fermionic ⁴⁰K or bosonic ⁴¹K atoms and on mixtures of fermionic ¹⁶¹Dy with ⁴⁰K atoms. We cover various topics of fermionic quantum many-body physics, ranging from impurity physics and quasiparticles over phase separation to the formation of ultracold molecules and progress towards novel superfluids.

These lecture notes give a brief introduction to the so-called Fermi-polaron problem, which explores the behaviour of a mobile impurity introduced into an ideal Fermi gas. While this problem has been considered now for more than a decade in ultracold atomic gases, it continues to generate surprises and insights as new quantum mixtures emerge, both in atomic gases and in the solid state. Here we summarise the basic theory for the Fermi polaron with a focus on the three-dimensional case, although the results can be straightforwardly generalised to two dimensions. Our aim is to provide a pedagogical treatment of the subject and we thus cover fundamental topics such as scattering theory and renormalisation. We discuss the ground state of the Fermi polaron and how it is connected to the phase diagram of the spin-imbalanced Fermi gas, and we also give a brief overview of the energy spectrum and non-equilibrium dynamics. Throughout, we highlight how the static and dynamic behaviour of the Fermi polaron is well described using intuitive variational approaches.

In these notes I provide an overview of ongoing theoretical and experimental research on ultracold atomic mixtures composed of two different fermionic species. First, I describe a general and simple framework that should allow also a non-expert reader to understand the rich few-body phenomena connected with such systems, and their possible impact at the many-body level. I then move to discuss the specific combination of fermionic lithium (⁶Li) and chromium (⁵³Cr) atoms, currently investigated in our lab, highlighting its peculiar properties with respect to other Fermi mixtures available nowadays. Finally, I summarize recent experimental progress achieved in producing and characterizing this novel system, providing an outlook for future studies based on ultracold ⁶Li-⁵³Cr Fermi mixtures.

The discovery of ultracold dilute liquids has significantly elevated our interest in various phenomena which go under the name of beyond-mean-field (BMF) physics. In these lecture notes we give an elementary introduction to the quantum stabilization and liquefaction of a collapsing weakly interacting Bose-Bose mixture. A detailed derivation of the leading BMF correction, also known as the Lee-Huang-Yang (LHY) term, in this system is presented in a manner suitable for further generalizations and extensions. Although the LHY term is a nonanalytic function of the density n, under certain conditions the leading BMF correction becomes analytic and can be expanded in integer powers of n, effectively introducing three-body and higher-order interactions. We discuss why and how well the Bogoliubov approach can predict these few-body observables.

Over the last decade there has been rapid progress in studying ultracold mixtures of trapped ions and neutral atoms. These systems are particularly interesting for studying quantum impurity physics and quantum chemistry, and may find applications in buffer gas cooling of trapped ions. In these lecture notes I review some of the recent progress and give relevant background information that may be useful for students who are considering working in this research area, but already have a firm knowledge of quantum and atomic physics. The notes are written from the Amsterdam perspective and therefore are not meant as a comprehensive review. I introduce the interaction potential between atoms and ions as well as some relevant atomic physics background. Then I explain how ion traps work. I treat the problem of a trapped ion in a neutral buffer gas and under what circumstances the gas can cool the ion. Next, I explain how the motional state of trapped ions can be measured. For applications in quantum technology, it will be of vital importance to be able to control the interactions between ion and atoms. I treat the case of using Rydberg dressing or excitation to boost the range and strength of the atom-ion interactions. Finally, I treat a number of applications in quantum chemistry that were studied in the group in Amsterdam, followed by a brief outlook.

Interactions play a crucial role in ultracold gases. At the moment, most experiments deal with particles that interact via a short-range isotropic potential. In this lecture, we consider gases in which particles interact significantly or even dominantly via dipole-dipole interactions. We first discuss the basics and the mean-field treatment, and the interesting new physics resulting from quantum fluctuations. We then illustrate some new possibilities opened by dipolar mixtures.

We review the theoretical description of the role of quantum geometry in superfluidity and superconductivity of multiband systems, with focus on flat bands where quantum geometry is wholly responsible for supercurrents. This review differs from previous ones in that it is based on the most recent understanding of the theory: the dependence of the self-consistent order parameter on the supercurrent is properly taken into account, and the superfluid weight in a flat band becomes proportional to the minimal quantum metric. We provide a recap of basic quantum geometric quantities and the concept of superfluid density. The geometric contribution of superconductivity is introduced via considering the two-body problem. The superfluid weight of a multiband system is derived within mean-field theory, leading to a topological bound of flat band superconductivity. The physical interpretation of the flat band supercurrent in terms of Wannier function overlaps is discussed.