3

Emerging Phenomena from Few- to Many-Body Systems

INTRODUCTION

In any typical gas at room temperature, the molecules are zooming around randomly and colliding more or less like billiard balls according to the laws of Newtonian mechanics. In this physical regime, the motion of the molecules is exemplified by chaos, which continues to be a subject of inquiry for atomic, molecular, and optical (AMO) researchers. But in order to display quantum mechanical behavior, the experimenter must cool the gas down to ultralow temperatures, below 1 microkelvin—that is, below one millionth of a degree above absolute zero. The use of lasers, magnetic fields, and other tools to reduce the temperature of a gas from room temperature down to this submicrokelvin range has become routine for many atomic species and even for some molecules. The figure of merit is the ratio between the average interparticle distance (d) and the quantum de Broglie wavelength (λ); when d/λ falls to the order of unity or less, a collection of particles is deep in the quantum regime in which the dual particle and wave natures of the system become important. In this fascinating regime, quantum phenomena that challenge our intuition become commonplace.

This chapter focuses on describing the advances in experimental control that have produced realizations of a broad range of quantum mechanical phenomena, for as few as two atoms, up to many-particle systems with thousands or even millions of atoms or molecules. These research activities have enabled a deeply satisfying intellectual pursuit to understand quantum complexity from the ground up, while at the same time providing a fertile playing field in which to connect

precisely controlled atomic and molecular systems. While this sometimes involves understanding the addition of one atom at a time, this research track ultimately connects with the interests of a number of subdisciplines in physics, including condensed-matter physics, precision measurement, chemical physics, and quantum information. Just as Chapter 2 describes the essential research tools based on the control of photons, this chapter provides an overview of how individual atoms, once brought under precise control, become both a fascinating subject for study and an enabling technology for a range of frontier research topics. Figure 3.1 gives an overview of the topics revolving around ultracold science that are discussed in this chapter.

All of the topics discussed in this report are challenging and have a frontier at the very edge of what is known and understood. Many of the phenomena being discussed are complex, but this is a word with multiple meanings. Some systems are complex because they consist of many particles, and because new phases emerge that have no counterpart whatsoever for small numbers of particles. On the other hand, even some systems of two particles can be tremendously complex, for instance if they involve atoms in the lower half of the periodic table, with electronic configurations that are not all closed shells, and with a huge number of internal

states that have nearly the same energy. And yet another meaning of complexity is what the chaos community identifies with and studies in all of its ramifications relating to randomness of trajectories or motions: in the classical Newtonian world chaotic complexity takes one form, but it has different manifestations in the quantum world.

Atoms come in two “flavors,” boson and fermion. Fermions cannot be in the same state at the same time—they must avoid each other. Bosons on the other hand have a well-known tendency to group together in the same quantum state. In the 1990s, bosonic atoms like rubidium were cooled down to the quantum regime to form a new phase of matter, the Bose-Einstein condensate (BEC), in which nearly all the atoms congregate into a single wave-like state. Soon afterward, fermionic atoms like the 6-nucleon isotope of lithium (⁶Li) were cooled down to a different quantum regime called the degenerate Fermi gas (DFG). In this regime, further cooling the fermionic atoms does not cause them to slow down—they have to keep moving so that they are all in different states. Following that period when these types of quantum degenerate gases were first created and initially explored, a fertile period of more than a decade followed, in which other types of mixtures were formed, and tunability using magnetic or electromagnetic fields proved that interactions between the constituent particles can be exquisitely controlled in the laboratory. This resulting remarkable level of interaction control enabled a further revolution in ultracold science, including the creation of dynamical phenomena such as solitons and vortices. Moreover, the advent of laser-formed optical lattices, in which light waves form a periodic array in space that traps atoms in those periodic arrays, resembles the physics of a crystalline environment seen by electrons in solids. During the past decade, this collection of experimental techniques has enabled access to an increasingly rich array of phenomena and the creation of novel phases of matter. In many cases, these phenomena are fascinating to study in their own right, while in other cases their value derives from the ability to simulate novel phenomena in condensed-matter and topological physics with an unprecedented level of control.

Recent years in the field of ultracold quantum gases have seen remarkable achievements. BECs and other types of quantum degenerate systems have reached such a high level of control that they can now be created in the laboratory across a rich variety of situations (in one-, two-, or three-dimensions, for instance, and in uniform quantum fluids or in the crystalline environment of an optical lattice), and for many different chemical elements; they can even be created on a satellite in outer space. The controllability of the deep quantum world continues to grow in leaps and bounds, spreading throughout the atomic periodic table of elements, and increasingly into the far richer and more diverse arena of ultracold molecules. The latter pose tremendous practical challenges, since the complex electronic, rotational, and vibrational modes of molecules are not nearly as well understood nor

as easy to control as the relatively simple modes of single atoms. This additional complexity makes molecules often less well suited to laser cooling and trapping (though rapid improvements in these techniques are ongoing).

Many of the experimental tools developed to understand and control atoms and molecules in ultracold environments have progressed to the level where they can now be used to simulate interesting and nontrivial quantum systems, such as the Fermi-Hubbard model that describes the behavior of quantum magnets and high-temperature superconductors. As discussed below in the section “Analog Quantum Simulation of Strongly Correlated Quantum Many-Body Systems,” this type of quantum simulation has the potential to simulate the behavior of systems that are currently beyond our ability to treat reliably using existing theoretical methods or computers. Such applications are at the forefront of current quantum information applications. Quantum simulation represents a promising science direction in its own right, while others appreciate developments in that subject as stepping-stones along the route to quantum computing applications, discussed at greater length in Chapter 4.

FROM FEW BODIES TO MANY BODIES: HOW COMPLEXITY BUILDS

Science has an innate tendency toward the discovery and understanding of more and more detailed phenomena, in virtually all areas. Examples of this abound, such as the fabrication and cataloguing of ever more specialized materials and metamaterials designed to affect light in increasingly specialized ways, or to unraveling the behavior of different types of quantum gases, consisting of different atoms, different molecules, different spin statistics (bosonic and fermionic), different topologies, or different spatial dimensions. At the same time, AMO physics shares a goal with many other disciplines—namely, to extract universal truths that persist across different systems, with a common physical origin, but differing in some cases by orders of magnitude in their length, mass, and energy scales.

The simplest example of such universality arises already with the interaction of a pair of particles, deep in the low-energy quantum mechanical regime where the quantum wavelength is the longest length scale in the problem. Pioneers of quantum physics such as Fermi, Bethe, and Schwinger showed that a single quantity, the scattering length, controls the way two particles interact at low energy, and in some cases, form a weakly bound state. The early successful theoretical descriptions of degenerate quantum gas experiments—for example, in the mid-1990s and afterward—made use of this fact because the behavior of quantum gases depended almost entirely on the scattering length, and simple scaling with mass or trapping potential strength. The tremendous chemical differences, say, between atomic helium and rubidium, turned out to be irrelevant for the description of their BECs, as it was only the atom-atom scattering length of the system that mattered. In the late 1990s, AMO physicists

demonstrated the ability to manipulate those scattering lengths at will, through the technique of magnetic Feshbach resonances. This technique ushered in a whole new era of controllable quantum gases that has continued robustly to this day.

This exquisite control of atom-atom interactions has launched studies in a strikingly rich and diverse array of systems. Experiments and theory relating to novel few-atom quantum states have seen tremendous progress during the past decade. For instance, the magnetic controllability of interactions enabled the first observation in 2006 of the intriguing, counterintuitive quantum states known as “universal 3-atom Efimov states,” and many subsequent experiments have similarly used magnetically tuned resonances to observe Efimov trimers in both homo-nuclear and heteronuclear cases. The word “universal,” used in this context, refers to the fact that any three-particle system with limited range but strong interactions (i.e., large scattering lengths) should have such states. Moreover, in the past few years, the first naturally occurring case of an Efimov resonance was observed experimentally for three very weakly bound helium atoms; that impressive experiment combined cold atom source and detection technology with ultrafast lasers to detect the size, shape, and binding energy of the delicate bound state. Box 3.1 gives some more details about this remarkable helium trimer system and its experimental observation. Additional, somewhat related families of universal quantum states consisting of four ground-state atoms have also been predicted theoretically and observed in experiments, with extensive progress and current interest in extensions of such studies to find universal states involving more than four atoms.

Microscopically, atoms interact predominantly in a pairwise manner, but multibody interactions can emerge in contexts such as three-body recombination processes, including for example Efimov states and other forms of universality associated with long-range interactions. Multibody interactions have also been observed in bosonic systems in optical lattice experiments. For fermions, a single impurity interacting with a few identical fermions has been studied (see Figure 3.2.1 in Box 3.2), and recently multibody interactions in high-spin fermions have been explored in individual lattice sites of an optical lattice.

Another developing area has been looking at few-atom or few-molecule physics in optical tweezers. Optical tweezers are focused laser beams that attract atoms to the region of highest light intensity, thereby trapping them in a potential well. One interest is to create double-well potentials using a laser electromagnetic field, and then to observe entangled tunneling events, which yield insight into the effect of quantum entanglement on dynamical processes. (The concept of entanglement is discussed more fully in Chapter 4.) A second type of experiment observes an individual chemical reaction event of two molecules in a single potential well. In yet another kind of experiment, optical tweezers have been used to study entanglement physics involving highly excited Rydberg states of trapped atoms, as is discussed later in Chapter 4.

Ultracold physics has enabled the experimental observation of other peculiar types of few-body quantum states bound together into stable (or metastable) molecules via novel mechanisms. For example, once an electron in an atom is excited to a very high lying (Rydberg) state, it orbits slowly at a large distance from the nucleus, much as Pluto takes many Earth-years to orbit the Sun. It turns out that an atom in a Rydberg state can attract a nearby ground-state atom and create an ultra-long-range “Rydberg molecule.” Some of these quantum states are predicted to exhibit a striking electron probability distribution that resembles a trilobite fossil, while others show a resemblance to a butterfly. Over the past few years, experimental observations of Rydberg molecules with huge internuclear distances have multiplied, even including observation of the trilobite and butterfly molecules whose creation poses stringent challenges. Another type of diatomic molecule with still larger internuclear distances has been predicted and observed, a so-called macro-dimer, made of two Rydberg atoms bound to each other through very-long-range interactions. These interactions result from each of the slowly orbiting outer electrons strongly perturbing each other through the Coulomb force.

More detailed spectroscopy of such diatomic and analogous triatomic molecules has already revealed striking physics, such as huge electric dipole moments (thousands of times larger than in typical molecular ground states), which enable controllability by very small external fields. This area remains ripe for future explorations. Another direction for such studies comes from adding two, three, four, or even more ground-state atoms bound to a single excited Rydberg atom. For this class of systems, theory and experiment have begun to indicate the manner in which the bulk limit emerges, where very many ground-state atoms interact with, and in some cases even bind to, a single Rydberg atom.

ULTRACOLD PHYSICAL PROCESSES INVOLVING IONS

Charged particles experience far stronger forces in an electromagnetic field than the neutral atoms and molecules discussed in the preceding portion of this chapter. But in the blink of an eye, a neutral atom or molecule can be turned into a positively charged ion, simply by striking it with a laser beam of an appropriately

chosen frequency and intensity that ejects one or more of the atom’s electrons. A subject of fundamental interest that also has significant practical importance is how a system of positive ions and electrons behaves, and how it can be controlled and directed to behave under a wide variety of conditions. Such explorations are often viewed as the purview of the field of plasma physics, but they overlap significantly with atomic and molecular physics as well. During the past two decades, new insights into the behavior of plasmas have been derived by creating plasmas at ultracold temperatures by first cooling atoms to microkelvin temperatures and then using a laser to ionize a large fraction of the atom sample. Ultracold plasma physics has made several key strides during the past decade. Box 3.3 gives an indication for the progress in this area of research.

Ions of an atom or a molecule have numerous additional dynamical roles beyond those that arise in ultracold plasmas. One example that has been studied extensively by theory and experiment during the past decade is the system of one

or a few ions placed in an ultracold quantum gas of neutral atoms. This is an ultracold variant of a process that occurs continually in nature—for instance, when a cosmic ray particle ionizes an atom in living tissue or in a computer chip, and subsequently affects the nearby neutral molecules. On the face of it, immersing a single trapped ion in the midst of 100,000 or so atoms, say of a BEC, might seem to have only a minor effect on the atom cloud. In fact, however, a rather violent series of events occurs, because the atoms are attracted to an ion orders of magnitude more strongly than to each other. As a result, the atoms nearest to the ion are drawn in and collide with the ion. When two free atoms (e.g., Rb) collide simultaneously with the ion (e.g., Ba⁺), the ensuing three-body recombination process Rb + Rb + Ba⁺ => BaRb⁺ + Rb converts the atomic ion to a molecular ion, and in the process adds sufficient kinetic energy to the free Rb atom to eject it from the condensate. If the molecular ion remains trapped, as is usually the case, the three-body process can repeat and create a heavier molecular ion, BaRb₂⁺ while ejecting a second Rb atom from the condensate. This sequence of reactions can occur rather rapidly, and fundamentally change the initially quiescent Rb BEC into a roiling, highly excited, collection of residual atoms that are no longer even quantum degenerate. Studies in recent years have unraveled many of the details of this process, and shown various ways to control it—that is, how to either encourage such reactions or suppress them.

Another fascinating dynamical regime that has been achieved through separation of charge in a BEC occurs when a single atom in the middle of a large BEC is laser excited to a very high Rydberg state, whose size can be comparable to that of the entire BEC. The roaming Rydberg electron, as it moves through the three-dimensional (3D) atom cloud, can now interact with all the atoms in the BEC, producing energy shifts, while at the same time triggering the process of associative ionization—for example, Rb + Rb^* => Rb₂⁺ + e. This can also have a significant effect on the behavior of the rubidium atomic condensate, owing to the energy deposited into the cloud.

Cooling and trapping of ultracold molecular ions is another topic that has advanced rapidly in recent years, and this is being pursued with a number of different long-term goals. Some of them are focused on precision physics tests of fundamental symmetries of the universe, including measurements of the electron electric dipole moment or of the electron to proton mass ratio to explore the possibility of a time-dependent ratio. (More on such precision tests can be found in Chapter 6.) Other applications of ultracold molecular ions arise and are of interest in basic spectroscopic and chemical reaction studies with fully defined preparation and observation at the individual quantum level, and the study of ion-neutral collisions is important for sympathetically cooling the ions. Cold molecular ion reactions are also important for understanding classes of astrophysical processes.

DEVELOPMENTS WITH ATOMIC DEGENERATE QUANTUM GASES

Once atoms have been cooled down to temperatures in the range 1-100 nanoK, new states or phases of matter have been predicted to occur, and by now many novel phases have indeed been created and observed. Some of these are quantum versions of classical nonlinear wave phenomena, such as solitons (solitary waves) and vortices, and even neatly aligned arrays of vortices. The atoms carry spin degrees of freedom, which perform their own dance, either separately or in some cases in controlled concert with the motional degrees of freedom.

Interacting unexcited (ground state) atoms are the usual constituents of degenerate quantum gases—namely, BECs, DFGs, and mixed systems. Recent developments in the field of dilute quantum gases have included some of the first explorations of a BEC in the limit where the atom-atom interaction has been tuned

to be the strongest possible, which is the so-called unitary limit in which the scattering length vastly exceeds the typical interparticle spacing. While some significant theoretical understanding of this very challenging regime of many-particle physics has emerged, challenging puzzles remain, and a great deal more needs to be unraveled in future theoretical and experimental studies. See further discussion below in the section “Unitary Quantum Gases.”

One major development in the field of ultracold quantum gas physics has been a significant expansion in the types of atoms that can be controlled and brought into quantum degeneracy. In the early days of Bose-Einstein condensation studies, the preferred systems were elements with only one or two valence electrons—namely, the alkali atoms such as rubidium, cesium, sodium, and later hydrogen, metastable helium, ytterbium, and strontium. Within the past decade, it has become possible to treat heavy open-shell atoms as well, such as erbium, dysprosium, and thulium.

Systems of such heavy, open-shell atoms exhibit an incredibly rich array of magnetic resonances that initially looked astonishingly dense, chaotic, and challenging to treat theoretically and to control, but they have turned out to nevertheless enable a surprising degree of quantum control. Enlarging the space of controllable quantum degenerate species has the potential to open wider possibilities of exploration in multiple contexts, such as coexisting multiple spin states, mixed fermionic and bosonic isotopes, strong nonperturbative electric or magnetic dipole-dipole interactions, and anisotropic Van der Waals forces between the atoms. Van der Waals forces are long-range forces between atoms arising from quantum mechanical correlations between the positions of the electrons on different atoms. Such forces occur in many real materials, and their control could prove very fruitful.

The past decade has also seen remarkable growth in theoretical understanding and experimental realizations of polaron physics in dilute atomic gases at very low temperatures. In the ultracold atomic physics context, a polaron is an impurity immersed in, and interacting with, a gas of atoms that are either different species or else the same element but in a different internal spin state. Polaron physics in ultracold atomic systems mimics analogous physics in a number of condensed-matter systems, but with far greater control over the interactions between the impurity and the majority sea of atoms. The committee elaborates on this polaron physics below in the section “Polaron Physics with Ultracold Atoms.”

Unitary Quantum Gases

One of the most interesting topics being studied by experiments and theory 15 years ago was the two-component degenerate Fermi gas, as the atom-atom scattering length a was tuned through the unitary limit, a → infinity. The exploration of this regime, denoted the BCS-BEC crossover regime, was notable for showing how the many-body ultracold atomic Fermi gas can be controlled to sweep atoms in a degenerate Fermi gas into diatomic molecules that pair those atoms and form a BEC of molecules. Many remarkable results emerged from those studies, such as the measurement of an accurate equation of state (important for understanding the thermodynamics of such systems) and demonstration of an impressively small rate of destructive molecule-molecule collisions that was partly anticipated by theory and confirmed experimentally.

A next natural step would appear naively to do the same with a BEC formed from bosonic atoms, and similarly sweep the atom-atom scattering length a to this fascinating so-called unitary regime where a → infinity. The experimental investigation of the unitary limit of a degenerate Bose gas was resisted for many years, however, largely because three-body recombination losses (due to the process A + A + A → A₂ + A) have problematic loss rates that scale like a⁴ and would be catastrophic in the divergent |a| limit. Eventually, however, three-body losses

were understood to saturate, and not to diverge at the unitary limit. As a result, in recent years several groups have begun creating Bose gases whose atom-atom interactions are at the unitary limit. In order to minimize losses, these experiments are mostly carried out as “quench experiments,” where a stable BEC at small scattering lengths is jumped very quickly to infinite scattering length, followed by a time-dependent response of the quantum gas to this sudden change. In addition to the many-body dynamics that ensue after suddenly changing scattering lengths, highly interesting few-body dynamics also occur, such as turning a fraction of the atoms in the gas into universal dimer and Efimov trimer states. This subject is still in its comparative infancy and remains ripe for extensive future theoretical and experimental investigations.

Ultracold Gases with Strong Dipole-Dipole Interactions

When the atoms or molecules in a very low temperature system possess a strong electric or magnetic dipole moment, this adds a dominant new element to the possible equilibrium phases that can exist, and to the dynamics that can occur. While no ground-state atom possesses an electric dipole moment, they frequently do possess a magnetic dipole moment. Polar molecules, by definition, possess a permanent electric dipole moment, as do many Rydberg atoms or molecules (see the section below, “Quantum Simulation with Dipolar Interactions”). In contrast to typical van der Waals long-range interactions, which are always attractive for ground-state atoms, dipole-dipole interactions are anisotropic, and have regions of both attraction and repulsion, which makes their resulting few-body and many-body behavior more complicated, but also more interesting (see Figure 3.2). A 3D dipole-interacting system can self-assemble through the mutual attraction into nanodroplets, for instance, connecting with the rich geometrically intriguing behavior of ferrofluids (liquids filled with magnetic particles) studied in condensed-matter physics. On the other hand, if the dipole moments are all parallel and interacting along lines perpendicular to the dipoles in either two-dimensional (2D) pancakes or one-dimensional (1D) rows, the constituent interactions are all mutually repulsive, which can help to suppress undesirable inelastic collision or reactive processes.

Over the past decades, tremendous progress has been made in isolating specific quantum phenomena by focusing on basic systems of coherent light and quantum matter. The acquired knowledge calls now for the next challenge: to explore quantum matter with increasing complexity. Among the different scientific avenues and quantum systems, ultracold quantum gases of highly magnetic atoms constitute one of the most active and promising quantum platforms. A typical quantum gas is composed of about 100,000 atoms. Each of these magnetic atoms can be controlled in their external (motional) and internal (spin) degrees of freedom, and, as a whole many-body system, the particles interact with each other

via the ordinary short-range (“contact”) and magnetic dipole-dipole interactions. The latter has the key properties of being both long range and anisotropic, and it is expected to result in exciting physics, which is just starting to be unveiled at present. Even the interaction between two heavy open-shell atoms like Er or Dy, which have a huge number of nearly degenerate ground-state energy levels,

shows impressive complexity and difficulties for current theoretical methods to understand the atom-atom scattering and resonances.

An extensive amount of new physics can still be discovered and learned from lattice studies, but bulk quantum gases, in which magnetic atoms can move throughout large volumes, have their own mysteries to reveal. Here, novel macroscopic quantum phases of matter emerge, revealing an astonishing universality that connects the behavior of dense quantum fluids (like superfluid helium) to dilute dipolar gases such as those discussed above, through fundamental properties of quantum mechanics. Highlights of recent experimental work with dipolar gases include the observation of macro-droplets, droplet crystals, “roton” excitations analogous to those in superfluid helium, and many-body phases with supersolid properties. The latter is a rather paradoxical quantum phase of matter in which apparently antithetic properties—that is, crystalline solid and superfluid order—coexist. Although intensively searched for in superfluid helium, supersolidity remained a mere theoretical concept for almost five decades, until dipolar gases demonstrated spontaneously emerging states with the regular density patterns of a solid and the global phase coherence of a superfluid.

A major interest over the past two decades of explorations of gas-phase quantum degeneracy has been the study of multiple physical properties of the superfluid. This includes such phenomena as the formation and lifetime of vortices and vortex lattices, collective excitation frequencies, alternative damping mechanisms and sound wave propagation, as well as macroscopic behaviors such as turbulence and granulation. A recent example of granulation can be seen in Figure 3.3. Analogues

of classical fluid instabilities such as Taylor-Couette flow or Rayleigh-Bénard convection can also be studied in degenerate quantum gases.

Polaron Physics with Ultracold Atoms

Ultracold atomic systems have provided us with unique platforms to study the physics of quantum impurities interacting with a medium. The polaron is a quasiparticle that emerges when an impurity atom is coupled to the medium and becomes entangled with (“dressed” by) virtual quantum excitations of the medium. Such quasiparticles represent the cornerstones of most descriptions of many-body systems, as highlighted by Landau’s celebrated Fermi-liquid theory, which describes the strongly interacting electron system in terms of a weakly interacting liquid of quasiparticles. While the quasiparticle picture is generally well established, many open questions arise regarding its validity under extreme interaction conditions, or close to phase transitions to ordered states. With an unprecedented level of control, ultracold atomic systems serve as a unique test-bed to address such open questions, to benchmark theoretical descriptions, and to advance our understanding of emergent many-body phenomena.

In a quantum-gas experiment, the medium can have fermionic or bosonic character according to the choice of the particular atomic species. Impurity atoms immersed into the medium then form Fermi polarons or Bose polarons, respectively. To create impurities, one can for instance, excite atoms of the medium into other spin states; alternatively, the system can be doped with another atomic species. Accurate interaction control between the impurity and the medium is a key ingredient, which can be achieved by means of magnetically tuned Feshbach resonances that change the scattering length of the impurity when it interacts with the atoms of the medium. Various experimental tools have been developed to characterize the energy spectrum of the impurities, commonly based on radiofrequency spectroscopic methods. An advanced example is illustrated in Figure 3.4, where a time-domain interferometric method was employed to measure ⁴⁰K impurities immersed in a Fermi sea of ⁶Li atoms. Various theoretical tools have been developed, including variational methods, functional renormalization group approaches, and descriptions based on functional determinants. Thanks to advances in both experiment and theory, the energy spectrum of polarons is now widely understood in many basic situations, and for large ranges of the interaction strength. It is remarkable, and worth noting, that research on quantum gases has stimulated the discovery of new polaronic quasiparticles in more traditional solid-state semiconductor materials.

New frontiers of polaron physics arise in unexplored regimes, of which three examples are discussed here: (1) Current theoretical understanding of the effect of impurity motion is limited to simple “effective mass” approaches, and almost

no experiments have addressed the question of motion so far. The effective mass is a perturbative approach that assumes that the only effect of the medium on the impurity is to increase its apparent mass. This increase in effective mass represents the kinetic energy stored in the atoms of the medium as they “get out of the way” of the moving impurity. The effective mass approximation can be valid only if the quasiparticle motion remains much slower than the characteristic speed (Fermi speed) of the medium. For faster motion, the dressing cloud from the medium can no longer follow the impurity, and a complete breakdown of

the quasiparticle picture can be expected. (2) A way to conceptually construct new many-body states is to increase the concentration of polarons up to the point where their mutual interaction leads to new phenomena. While various theoretical predictions have been made for polaron-polaron interactions, experiments so far have shown only a few qualitative effects. Here it will be necessary to carry out new generations of quantitative experiments to test interactions between quasiparticles. (3) Few-body correlations in the medium have been considered theoretically, but in present experiments they have played only minor roles. To explore such correlations, new systems need to be introduced experimentally, such as systems with strong mass imbalance (i.e., consisting of different types of atoms with different masses), which provide access to new phenomena.

Polaron physics with ultracold atoms provides paradigmatic situations to approach many-body behavior. In future research, it will be important to complete our understanding on the emergence of many-body phenomena and to learn more about the connection to ordered states, such as novel superfluids. In a cross-disciplinary sense, it will be very fruitful to further build on the analogies of AMO-based systems with real condensed-matter systems.

MANY-BODY SYSTEMS WITH ULTRACOLD MOLECULES

In comparison with atoms, molecules have a much more complex level structure and many more quantum degrees of freedom, and their control is a major challenge in achieving ultracold molecular quantum gases. At the same time, the complexity of molecular systems provides new opportunities for engineering quantum many-body systems beyond the atomic counterparts. (See, for example, “Introduction to Ultracold Molecules: New Frontiers in Quantum and Chemical Physics,” Chemical Reviews, Volume 112, Issue 9 (Special Issue), September 12, 2012.)

Cooling and Trapping Molecules for Complete Quantum Control

Trapping and cooling of atoms are core techniques in atomic physics. Trapping provides time to measure and manipulate atoms with exquisite precision, while cooling reduces disorder in the system, and hence increases the certainty that the system starts in a specific, defined quantum state. As systems become larger and more complex, the number of possible states increases rapidly, and cooling to achieve complete quantum control becomes more important—but also more difficult.

Systems made of molecules rather than atoms have more potential to become complex. In addition to their overall motion, molecules have a richer internal structure that includes vibrations and rotations. Under sufficient control, these internal

states can be used as a powerful quantum resource—particularly when the overall motion is also cooled and the molecules are trapped. For example, the rotational structure of polar molecules can be used to amplify tiny energy-level shifts, indicating the existence of as-yet undiscovered fundamental particles—or, instead, to engineer tunable, long-range dipolar interactions between trapped molecules, with potential applications in quantum simulation and quantum information processing. The diversity of molecular species also opens the door to study of chemical processes; at ultracold temperatures, novel quantum effects can appear in chemical reaction dynamics.

These promising applications have motivated intense efforts to bring molecules under complete quantum control. Despite the challenges associated with controlling their complex internal structure, progress has been very rapid. In one of two primary methods, diatomic molecules are assembled from ultracold, trapped precursor atoms. This approach yields exceptionally low temperatures; it has enabled trapping of molecular arrays in optical lattices, and production of a quantum-degenerate gas of polar molecules. The range of molecular species that can be assembled in this way is limited, but sufficient for some envisioned applications.

In the meantime, methods for direct laser cooling and trapping of molecules also have been developed. This approach uses techniques similar to those now standard for atoms, and yields similar results. For example, laser-cooled molecules have been accumulated in magneto-optical traps, then transferred at ultracold temperature to dipole traps or tweezers—as needed for further cooling. This approach is proving applicable even to polyatomic molecules. Although not yet as advanced as the method of ultracold molecule assembly, laser cooling is proving far more general, and its outcomes are improving rapidly (see Figure 3.5).

Techniques for preparing ultracold molecules are new and rapidly improving, and their application promises a wide range of important science in the coming decade. As is discussed later in this chapter, it seems likely that molecules in optical lattices will be used to probe—and possibly detect—new fundamental physics, at energy scales far beyond those accessible with high-energy particle colliders. Physics at ultrahigh energy scales is accessible in table-top experiments through the extreme precision of AMO techniques, which can detect the tiny effects of high-energy phenomena on low-energy quantum states of atoms and molecules. Similar experiments can perform quantum simulations of highly correlated many-body systems such as spin lattice Hamiltonians with topological phases. Arrays of individual molecules in reconfigurable optical tweezer arrays may provide deterministic preparation of many-body systems along with high-fidelity detection, potentially opening new avenues in quantum computing and simulation. (See Chapter 4 for further discussion.)

Many-Body Systems Based on Ultracold Molecules

Samples of optically trapped ultracold polar molecules can now be produced in many laboratories. One of the most compelling applications for these molecules is to use them as elements in future programmable quantum simulators. In particular, the molecule frame electric dipole moment ordinarily rotates in the laboratory frame at microwave-scale (or lower) frequencies with a very long lifetime, making it natural to encode quantum information in this degree of freedom. The interaction between molecular electric dipoles is strong and long-ranged. Hence, even when molecules are separated by distances on the order of the wavelength of visible light, this coupling between molecules may dominate the dynamics of the system. This means that many-body systems of polar molecules in optical lattices or tweezer arrays can be engineered to become highly entangled, and to remain so for long times.

Many pioneering advances in this direction have been made in experiments using KRb molecules—for example, in the laboratories of Deborah Jin and Jun Ye at JILA. Here, out of K and Rb ultracold atoms loaded into a common optical

dipole trap, the JILA team assembled fermionic KRb molecules at a temperature of ~200 nK. These molecules are produced in the electronic-vibrational-rotational-hyperfine ground state, and a degenerate Fermi gas of molecules is born with a temperature just 30 percent of the Fermi temperature. These molecules have also been loaded in a 3D optical lattice, achieving a lattice filling of ~25 percent for ground-state KRb molecules. This is of the order of the “percolation threshold” in 3D, and surpasses it when as here every molecule is connected to all others via electric dipole-dipole interactions (see Figure 3.6). This team also simulated a direct, long-range spin-exchange coupling between two ground-state KRb molecules by encoding an effective spin into a pair of rotational states with dipolar interactions. This allowed observation of many-body spin dynamics with the spin and motional degrees of freedom completely decoupled.

In the meantime, ultracold molecule systems based on several other bi-alkali species have been developed in other laboratories around the world. For example, some new experiments use molecules that have much larger dipole moments, enabling stronger interactions. In some cases, these species, unlike KRb, cannot chemically react with each other; this mitigates losses while preparing the molecular samples. These and other experiments have demonstrated long coherence times of molecular spin superpositions, and improved rotational state coherence by better optimizing the optical-lattice parameters. These constitute early, but extremely promising first steps toward using ultracold polar molecules as a system for interesting, novel types of quantum simulations, as discussed below.

ANALOG QUANTUM SIMULATION OF STRONGLY CORRELATED QUANTUM MANY-BODY SYSTEMS

A typical BEC represents a weakly interacting many-body system, where a mean-field (average interaction) description is often sufficient to describe the underlying physics. In contrast, one of the fundamental challenges of quantum science is to understand the strongly correlated quantum many-body regimes where mean-field theory breaks down. These regimes generally have large critical fluctuations. Quantum phase transitions (for example, believed to underlie high-temperature superconductivity) are of particular interest. These sorts of many-body systems define one of the present frontiers of theoretical condensed-matter physics for both equilibrium and non-equilibrium phenomena. Such a strongly correlated many-body system can be built, for example, by loading a BEC or a degenerate Fermi gas into an optical lattice—that is, into a periodic array of microtraps generated by counterpropagating off-resonant laser light fields, with a lattice spacing given by half the wavelength of the light.

Atoms loaded into an optical lattice can hop between lattice sites via tunneling processes, reminiscent of electrons moving in a crystal lattice of solid-state physics, and these atoms interact with each other through molecular forces. Such atoms loaded into an optical lattice acquire an effective mass, which is tunable with the strength of the laser intensity defining the lattice structure, and can be much larger than the free-space mass. Another way to look at this is to note that the tunneling between neighbors can be enhanced or suppressed by lowering or raising the light intensity field forming the barriers between lattice sites. This provides control over “kinetic” versus (on-site) “interaction” energies, and thus allows one to tune these systems from weakly to strongly interacting. This particular system enables the realization of the Bose-Hubbard and Fermi-Hubbard models of condensed-matter physics—namely, the Hubbard model with bosonic or fermionic atomic or molecular components. Moreover, it opens the door toward realizing strongly correlated quantum phases in a laboratory setting, by tuning of the Hubbard parameters via external fields. Hubbard models built from ultracold atoms and molecules in an optical lattice provide us with a paradigmatic example of an “analog quantum simulator,” where a quantum many-body system can be realized in a controlled setting. The unique possibility of controlling AMO systems via electromagnetic (static, optical, and microwave) fields provides us with a toolbox, with which numerous many-body Hamiltonians of interest can be designed. These include models of superconductivity, of ferromagnetism and anti-ferromagnetism, of spin glasses and spin liquids, of Anderson and of many-body localization, and of complex competing phases including topological ones. Recent examples of extending the range of Hamiltonians that can be simulated include synthetic magnetic fields (or synthetic gauge fields, which can mimic magnetic fields or other types of force fields), or long-range interactions

provided by dipolar interactions from magnetic or electric dipoles, and of various forms of disorder. Engineering many-body Hamiltonians within a given set of control knobs is, of course, just one facet of simulating quantum many-body physics with atoms and molecules. Additional key elements are protocols for preparation of quantum phases, or initial states. Furthermore, atomic and molecular physics provides unique abilities to measure many-body observables and correlation functions: a seminal example is the quantum gas microscope for optical lattices, which allows single-site, particle- and spin-resolved readout in single-shot measurements.

While most of our discussion below will focus on analog quantum simulators as emulating the dynamics of an isolated quantum many-body system in equilibrium and non-equilibrium dynamics, one can extend this notion to treat an open system as well, where the coupling to a bath is designed by quantum reservoir engineering. This includes in particular driven-dissipative systems, where the many-body system approaches a dynamical equilibrium, with the potential to observe new non-equilibrium phases and phase transitions. In addition, this can be developed into a tool for preparing interesting entangled states, as relevant for quantum information applications (see Chapter 4). Such simulations give deep insight into the role of entanglement in quantum thermalization (as discussed further in Chapter 7).

Quantum Simulation of Fermi-Hubbard Models in Various Spatial Dimensions

As described above, quantum simulation of the Hubbard model with ultracold fermionic atoms in optical lattices is a prime example of emerging quantum systems. Study of this particular Hamiltonian enables AMO physicists to address crucial open questions in many-body quantum mechanics and materials science. This iconic model is, at the same time, both computationally extremely challenging and highly relevant. It is believed to capture the essential physics of high-temperature superconductors and other quantum materials; however, no first principles understanding of this “deceivingly” simple model exists. Hence, in the upcoming years, quantum simulations have the potential to exert a large impact on the understanding of this physics, and potentially to help us design new high-temperature superconductors.

Recent experiments, in which fermionic atoms in optical lattices mimic electrons hopping in a crystal lattice, are a virtually perfect realization of the Hubbard model (see Figure 3.7). Quantum gas microscopy enables ultimate control for readout and manipulation, on a single-site, single-atom level. Parameters can be widely tuned, and with the observation of anti-ferromagnetism, experiments are now starting to explore the low-temperature phases of the model. Doping away from half filling brings the system into poorly understood terrain, and phases such as strange metals, pseudogap phases, and the important high-temperature superconducting phase are hoped to be seen. The complexity of these phases stems from the intricate

interplay between spin and charge degrees of freedom, and the fermionic nature of the particles. Experiments have started to explore the doped Hubbard model, probing string-like excitations and allowing theories to be benchmarked. Particularly useful are experiments on transport and non-equilibrium physics, which are doubly hard numerically. Recent experiments, for example, study transport in the doped regime, where measuring resistivity that varies linearly with temperature is the hallmark for strange metal behavior. Other experiments directly observe the microscopic motion of holes in the antiferromagnetic environment.

The recent experimental breakthroughs in fermionic quantum simulation, in which experiments are demonstrating a genuine quantum advantage over simulations on classical computers in particular in 2D and 3D, provide a unique opportunity for discovery during the coming years. Among the key challenges is measurement of the complete phase diagram of the 2D Hubbard model with repulsive onsite interactions, as a paradigmatic model for high critical temperature (T_c) superconductivity; this is a problem for which computational complexity provides a fundamental limitation to fermionic quantum Monte Carlo simulations. Quantum gas microscopy plays a particularly important role in this. It enables experimenters to take a complete quantum “snapshot” of a complex strongly correlated many-body quantum state, which is unprecedented in physics. This capability is

complementary to the kind of observables available in condensed-matter physics. Snapshots enable the direct detection of patterns and of high-order multiparticle correlation functions. This gives the possibility to find “hidden order,” which can even be facilitated by machine learning. By measuring correlations between the motion of charge carriers, and simultaneously registering microscopic fluctuations in the surrounding magnetic environment, it may be possible to answer open questions concerning the nature of the superconducting pairing mechanism in the Hubbard model, and to shed light onto the pseudogap regime. At temperatures a few times lower than currently achieved, Cooper pairs may form a superfluid phase, which is a likely analogue of the superconducting phase in high-temperature superconductors. A key goal will be to answer outstanding questions, such as whether the superconducting phase is competing with, or emerging from, the antiferromagnetic phase.

Will such simulation of the Hubbard model hold the key to high-temperature superconductivity? There are many reasons to believe that the plain Hubbard model contains most—but not all—of the ingredients necessary for understanding high-temperature superconductivity in the cuprates. A full explanation is likely to require additional ingredients in the model although there is no consensus about which, if any, are important. But that is just where the remarkable control over ultracold atoms in an optical lattice comes into its own—the idea is to approach the problem by realizing the bare Hubbard model first and then adding other ingredients, as needed, in a controlled way.

By adding additional terms to the 2D fermionic Hubbard Hamiltonian—for example, offsite interactions, long-range hoppings or synthetic gauge fields, or by simply changing the lattice geometry, quantum simulation of a broad range of phenomena could be enabled. Further expanding the Hamiltonian will help shed light on other open questions such as frustrated quantum magnets, topological phases, spin glasses and spin liquids, localization, as well as provide new insight into non-equilibrium physics. Such phases will be uniquely realized and studied with fermions in optical lattices, while quantum gas microscopy provides detection (“state tomography”) on a genuinely many-body quantum level.

Quantum Simulation with Dipolar Interactions

Dipolar interactions provide anisotropic and long-range couplings between particles, as compared to short-range contact potentials. In the toolbox of quantum simulation, this provides new elements to engineer and explore long-range interactions. There is a rich literature of theoretical proposals for how to engineer strongly correlated quantum many-body systems with ultracold magnetic atoms and ultracold polar molecules, and these phenomena are now beginning to be explored in the laboratory, both with highly magnetic atoms in optical lattices (such

as chromium, dysprosium, and erbium) and with polar molecules where electric dipole moments can provide even stronger dipolar couplings.

For magnetic atoms, recent experimental highlights include both the realization of extended Bose-Hubbard models and spin lattice models, in which the superfluid-to-Mott-insulator phase transition and the spin dynamics are largely determined by the dipolar coupling between particles occupying different lattice sites (see Figure 3.8).

The unique feature of polar molecules in designing quantum simulators is, first of all, their strong electric dipoles and corresponding strong dipolar interactions. In addition, molecules offer new methods for control—for example, microwave dressing to couple rotational states to internal spin states, and further cooling for reduced ensemble entropy—that will likely bring to fruition many of the theoretical ideas around the paradigm of dipolar interactions. Some proposed examples include observing a novel self-stabilized crystalline phase, simulating nontrivial quantum magnets, producing symmetry-protected topological phases or phases with true topological order such as fractional Chern insulators, and so on. In additional to polar molecules and dipolar atoms, many-body quantum dynamics of dipolar systems are also being explored with programmable Rydberg simulators and spin defects in diamond, as discussed in Chapter 4. These developments represent a broad new frontier in quantum science.

Artificially Engineered Gauge Potentials

One of the unexpected developments during the past decade has been a strong growth in the field of spin-orbit coupled quantum gases. Whereas the coupling between the orbital and spin magnetic moments of atomic electrons has been studied in atomic spectroscopy for more than a century, a very different type of spin-orbit coupling (SOC) can be created for an atom in a laser field. In this situation, the intrinsic spin degrees of freedom are coupled to the linear momentum of the atom. This has been demonstrated experimentally by many groups already, and it connects with studies of so-called Rashba or Dresselhaus spin-orbit coupling that arises extensively for electrons in condensed matter. The use of lasers to drive Raman transitions can also engineer unusual dispersion relations such as linear Dirac cones in the momentum dependence of the atomic energy landscape. Interestingly, such a Dirac cone provides a way to mimic or simulate the fundamental high-energy quantum mechanical equation (Dirac equation) in a low-energy system. This opens up connections with the physics of topological band structures of tremendous current interest in condensed-matter physics. This line of research with spin-orbit coupling in atomic gases has added significantly to our ability to controllably simulate emerging and challenging phenomena in condensed-matter physics using ultracold atomic systems.

Another recent creative growth area for the field of ultracold atomic systems has been the introduction of techniques to create a “synthetic dimension.” Typically, cold quantum gases are designed to exist in a 3D trap, or in some cases in reduced dimensional traps—for example, of a pancake-type 2D geometry or a cigar-shaped 1D geometry. The idea of a synthetic dimension arises when one couples the atomic states into a spin degree of freedom with several states that behave very analogously to a true spatial dimension. Consider for example an atom moving in one dimension by hopping (tunneling) between adjacent wells of a cigar-shaped optical lattice. If the atom has, say, five distinct internal spin states, one can arrange for hopping transitions to occur among the neighboring spin states in such a way that the atom appears to be moving in quasi-2D optical lattice strip of width 5. Quantum gas systems with a synthetic dimension have been used to explore a version of the famous quantum Hall effect in condensed-matter systems, except that in the cold atom version of this effect, the motion of electrically neutral atoms mimics the behavior of charged electrons in a strong magnetic field.

The unique properties of alkaline-earth atoms and recently developed precision clock-spectroscopy capabilities make such cold atom systems an ideal platform to engineer and study exotic forms of quantum matter—ideal because of the extensive controllability of the relevant parameter spaces involved. For example, fermionic ⁸⁷Sr atoms can be prepared in long-lived ¹S₀ and ³P₀ electronic orbitals, each with 10 nuclear spin sublevels (⁸⁷Sr has nuclear spin I = 9/2), and trapped in

an optical lattice with 1D, 2D, or 3D geometries. For example, in a 3D lattice with the Sr gas cold enough to be Fermi degenerate, all degrees of freedom, including the electronic, nuclear, and motional quantum states, are fully controlled, and the resulting few- to many-body properties and dynamics can be probed with extremely high precision—that is, with sub-Hertz spectral resolution.

Theoretical investigations of correlated quantum many-body systems have proceeded at an astonishing pace. However, experimental progress on such correlated systems embodied in atomic quantum gases has been hampered by the extremely low entropy (essentially temperature) required to observe novel quantum phases. A complementary approach to study some of these phenomena is to focus instead on the non-equilibrium many-body spin dynamics. This allows precise investigations and observations of intrinsic signatures for collective spin interactions, including strongly modified spectral shapes, spin dephasing and decoherence, interaction-induced frequency shifts, and correlated quantum spin noise. Describing these observations requires a genuine many-body theory reaching beyond the conventional mean-field framework.

Another surprising development is the use of the tremendous precision of optical lattice clocks to study an important topic in many-body physics—spin symmetry and its effect on many-body phenomena. In alkaline-earth atoms, the nuclear spin can be in N distinct states, which makes atoms in different states distinguishable in principle. However, because the electrons have total spin and orbital angular momentum J = 0, the nuclei in the interior of the atoms have no effect on the exterior shell of electrons. Thus, the atom-atom interactions (determined by the electrons) are almost completely independent of the nuclear states. This special symmetry of the atom interactions (known as SU[N] symmetry) leads to novel many-body effects. When N nuclear-spin sublevels are populated in this system, one can directly and precisely probe atomic interactions under the SU(N) symmetry. Recent clock spectroscopy results provided the first direct observation of SU(N) symmetry for alkaline-earth atoms and the related two-orbital SU(N) magnetism. Non-equilibrium spin-orbital dynamics in SU(N) magnetism was directly measured (for N ≤ 10) via Ramsey spectroscopy. The unique approach enabled by the exquisite energy resolution of optical lattice clocks paves the way for future exploration of the fascinating consequences of SU(N) symmetry in spin-orbital lattice models, as well as test-beds for high-energy lattice gauge theories.

By allowing the atoms to tunnel in an optical lattice, exotic new phenomena governed by the competition between interactions and correlated spin motion can be probed. A single neutral Sr atom tunneling in the lattice, when interrogated by the clock laser, can behave as a charged electron moving in an ultrastrong magnetic field. This synthetic field arises from a simple act of having the clock laser imprint a local quantum phase that varies from one lattice site to the next. Hence, when an atom traverses a closed loop, it accumulates a net phase as if it were a charge

that was encircling magnetic flux. The effect of the laser on the atom can also be arranged to generate a coupling between the atomic motion and internal states—that is, SOC—as every time an atom changes its internal level by absorbing or emitting a photon, it experiences a momentum kick. With a quantum-degenerate ⁸⁷Sr Fermi gas loaded into a 3D optical lattice, clock spectroscopy can directly probe SOC in higher dimensions. The system can be modeled using few- and many-body approaches; these studies seek new physics induced by the interplay between SOC and interactions such as those that occur in topological superfluids, which host protected Majorana modes, a vehicle of interest for topologically protected quantum computation.

Topological Matter with Cold Atoms

Topology is the field of mathematics that studies the most fundamental properties associated with the shapes of objects—namely, those properties that do not change when the object is continuously deformed. The number of holes in an object is one example of a topological invariant. Thus, a donut and a coffee cup, although they look quite different, can be continuously deformed into each other because they both have one hole. In recent decades, topology has been discovered to lie at the heart of many striking physical phenomena, ranging from topological defects (e.g., magnetic monopoles and vortices) to topological states of matter. These phenomena are robust, here not against changes in the shape of the sample, but rather against smooth deformations of the geometry of the quantum states themselves. This in turn implies that the phenomena are robust under continuous deformations (or errors) in the values of the parameters describing the energy and interactions of the particles in the system. Since their discovery in the solid state, topological states have attracted the interest of a wide scientific community for their unusual transport properties, suggesting promising technological applications. For example, one can readily imagine spintronics-based devices that are robust against small errors in device fabrication. In addition, certain classes of topological states host exotic excitations (so-called non-Abelian anyons), which, if properly manipulated, could be used as building blocks for fault-tolerant quantum computations. Today, in parallel to the intense effort that is dedicated to the search for novel topological materials, a substantial research activity concerns the realization of topological states using engineered quantum systems, such as ultracold gases trapped in optical lattices. Indeed, these engineered systems could offer the possibility of accessing a wide variety of topological phenomena in a highly controllable environment, an ideal setting to explore and manipulate topological matter in regimes that are hardly accessible in the solid state.

Major steps have been achieved in the past decade in terms of creating topological states of matter in ultracold gases. The first topological phenomena have

recently been reported by several experimental groups worldwide. These achievements concern the realization of topological energy level structures (so-called “Bloch bands”) for neutral atoms, which can be designed through artificial magnetic fields and spin-orbit couplings (as described in the previous section). Also of interest are the development of novel probes for geometric and topological band properties, which can, for instance, enable assessments of the robustness of the topological protection of various quantum mechanical states against decoherence. Within the past 5 years, this progress led to the measurement of the topologically invariant Chern number of Bloch bands, to the extraction of the local Berry curvature, and to the realization of topological pumps in atomic gases. Very recently, ultracold gases revealed novel topological effects; these include the observation of topological Anderson insulators in the presence of engineered disorder, the detection of higher order Chern numbers, and the observation of quantized circular dichroism. Such phenomena are at least for now inaccessible in traditional materials. Such topological phenomena are important in condensed-matter systems such as the fractional quantum Hall effect, and the advantage of realizing them with ultracold gases is that they can now be probed and controlled in exquisite detail by realizing them in a system of laser-trapped atoms.

Cold-atom experiments exploring topological phenomena are currently restricted to operating in the noninteracting (or weakly interacting) regime, where the properties of interest can be deduced from single-particle band structures. The strongly correlated regime of topological matter could be reached by enhancing the interactions between the atoms, which can be easily achieved in these setups. However, current experiments have shown that severe heating and instability mechanisms result when combining the techniques that are used to generate the topological band structures (i.e., artificial gauge fields) with strong interactions. As a result, a major challenge today concerns the elimination of heating processes and the stabilization of strongly interacting atomic gases in the presence of artificial gauge fields. An interesting route toward this goal involves the preparation of target topological states through engineered dissipation. Once stabilized in the laboratory, it will be crucial to develop proper probing schemes to detect the intriguing properties of these interacting topological states. A significant question concerns the detection of non-Abelian anyons (such as individual zero modes of Majorana—and Majorana fermions formed by pairs of such zero modes) in this atomic context. This would be an important step toward exploration of new science, and such phenomena are also believed to hold promise for technologically important applications in quantum information processing.

The realization of strongly correlated topological states of matter using ultracold atomic gases is deeply connected to another active field of research: the quantum simulation of lattice gauge theories, for probing fundamental particle physics. Not only are topology and lattice gauge theories deeply related at the theoretical

level, the proposed schemes to realize dynamical gauge fields (and the required matter-gauge field couplings) in the laboratory are reminiscent of those currently exploited to create topological band structures. First, experimental achievements along these lines include the cold-atom implementation of a minimal toric-code Hamiltonian, as well as the engineering of density-dependent gauge fields leading to a minimal Z2 lattice gauge theory. These developments suggest that intense pluridisciplinary activities can be expected in the next few years, hence further connecting the condensed-matter, high-energy, and quantum optics (AMO physics) communities. The report will return to quantum simulation of dynamical gauge fields—that is, where the gauge field becomes a dynamical variable—in Chapter 4.

Non-Equilibrium Quantum Many-Body Dynamics

Understanding the non-equilibrium dynamics of strongly interacting quantum systems represents a central challenge that connects AMO physics to a number of other disciplines including condensed-matter, high-energy, and quantum information science. This challenge stems in part from the fact that quantum systems can be taken out of equilibrium in a multitude of different ways, each with its own set of expectations and guiding intuition. The various strategies for exploring non-equilibrium quantum physics might be broadly summarized into six categories:

1. Quench initial conditions;
2. Periodic (Floquet) driving;
3. Strong disorder leading to the breakdown of statistical mechanics (this phenomenon is related to the effect that is sometimes referred to as “many-body localization (MBL)”);
4. Dissipative coupling to an environment;
5. Critically slow dynamics; and
6. Prethermalization.
7. Quantum- and nano-thermodyanmics

The diversity of expectations from each of these different microscopic scenarios is readily apparent. For example, under a sudden quench (controlled change of system parameters), one typically expects a many-body system to quickly evolve toward a new local thermal equilibrium. At first sight, this suggests a simple description. However, capturing both the microscopic details of short-time thermalization as well as the crossover to late-time hydrodynamics (in which the local density of quantities protected by conservation laws relax very slowly) remains extremely difficult. Alternatively, a many-body system can also be taken out of equilibrium via Floquet driving (via periodic modulation of external control parameters in the system). In this case, the non-equilibrium system is generically expected to absorb

energy from the driving field (although in some cases it may temporarily achieve pseudo-equilibrium on short time scales).

Some of the most unique and stunning physical phenomena sit precisely at the interface between these various non-equilibrium strategies; these include, for example, questions regarding fundamental speed limits on the propagation of quantum information as well as the dissipative stabilization of entanglement and localized quantum memories.

Perhaps one of the most surprising insights to come out of the study of non-equilibrium AMO systems this past decade is the discovery of new phases of intrinsically non-equilibrium quantum matter. Indeed, previous classifications of matter relied on thermodynamic equilibrium. In particular, from the viewpoint of many-body physics, all phases of matter should satisfy certain properties that embody the notion of rigidity. First, the system should have many locally coupled degrees of freedom so that a notion of spatial dimension and thermodynamic limit can be defined. Second, within a well-defined thermodynamic phase, the system’s ordering should be robust against a wide range of perturbations of both the initial state and the equations of motion.

These topics are currently being explored in a wide variety of AMO systems, including ultracold gases in a quantum simulator setting and trapped-ion analog quantum simulators (see also Chapter 4). Below, the committee discusses how the techniques of quantum gas microscopy are being used to experimentally study many-body localization—a very active topic in many-body physics. Quantum entanglement also plays a key role in non-equilibrium dynamics, and is discussed in the following chapters.

Observation of Many-Body Localization

Far from equilibrium, settings pose some of the most challenging and complex problems in quantum many-body physics. They often involve a rapid and extensive growth of entanglement in the quantum system, making it quickly intractable to an attempted numerical simulation on modern classical supercomputers. Nevertheless, these settings pose fundamental open questions concerning the emergence of thermalization in closed quantum systems, transport properties, and the susceptibility of these systems to classical noise or driving. In equilibrium thermodynamics, one assumes the existence of an infinitely large “bath” that can exchange energy (and in some cases particles) with the system under study. Weak interactions with the bath bring the system into equilibrium at the same temperature as the bath. In contrast, atomic systems can be nearly perfectly isolated from the environment—for example, sub-microKelvin temperatures are routinely achieved via laser cooling in an apparatus sitting at room temperature. For an isolated system, the particle number and total energy are fixed, and the only “bath” the particles see

are each other. How (and if) such a closed quantum system reaches equilibrium following a sudden quench of parameters is an exciting research frontier at present. These questions are also deeply linked to emerging quantum technologies, whose protocols involve dynamical control of quantum many-body systems in one way or another. Analog AMO quantum simulators can provide experimental insight into these fundamental and challenging problems (see Figure 3.9).

One of the most prominent topics in this context is the question of if and how localization and non-ergodicity can emerge in a generic interacting many-body setting. Such very unfamiliar nonthermal behavior is called “many-body localization (MBL),” a phenomenon in which long-range transport of extensive quantities (e.g., energy or particles) is absent, and quantum properties of the macroscopically large system can be preserved over very long times.

The theoretical treatment of localization phenomena in quantum systems dates back to the 1950s, when Anderson wrote his celebrated article about localization of electrons in disordered lattices. It took many years until the effect of interactions could be added into this picture, with Mott for example early on finding localization in an interacting but not disordered metal; and only in 2006, based on new theoretical results, has localization been predicted to emerge in a disordered and

interacting system, at finite-energy densities and in one dimension. (The stability of the localized phase in higher dimensions has not been firmly established.) This defined the notion of MBL, and triggered many theoretical works aimed to understand the mechanisms and implications of localization, the dynamical and static properties of the localized phase, the nature of the phase transition between the ergodic and localized phases, and many more aspects of this novel stable nonthermal phase of quantum matter. Interestingly, and perhaps not surprisingly due to the fact that MBL is a true quantum phenomenon, it has turned out that quantum information theoretical concepts such as entanglement entropy are well suited to further characterize this phenomenon.

Even though localization and transport measurements are traditional topics of condensed-matter physics, MBL is difficult to observe in real materials. This is owing to the fragility of this quantum effect to decoherence, due to coupling of the quantum many-body system to the outside world. In solid-state materials, such a coupling is often unavoidable, and caused by phonons, which interact with the electronic degrees of freedom. It turns out that AMO systems, primarily neutral atoms and ions, but also impurity spins in diamonds, are ideal systems to study many-body localization experimentally, owing to the intrinsic absence of phonons, and these systems’ excellent isolation from the environment. Also, thanks to the exquisite control and observability in these synthetic quantum systems, experiments at high energy density and far from equilibrium can be realized by starting with a sudden quench of the Hamiltonian parameters. Most of the experiments performed so far, including the first clear observation of MBL in 2015, were realized in 1D systems, and probed localization by the emergence of a nonthermal steady state of the atomic density distribution or of the magnetization profile. However, later experiments using the newly developed quantum gas microscope platform, with access to individual components of the many-body system, extracted quantum entropies, or explored localization in two dimensions, in a regime where no existing theoretical method is able to predict the system behavior. The latter experiments are a prototypical example where AMO quantum simulators provide truly new insight into a difficult quantum many-body problem, and thus offer a practical quantum advantage over classical simulations.

The study of non-equilibrium quantum phenomena like MBL defines an interesting challenge to both theory and experiment. It provides a setting where new theoretical methods are required to predict even the qualitative behavior of quantum many-body systems, especially close to the transition between the ergodic (i.e., self-thermalized) and MBL phases, as well as in higher dimensions, and where experimental results are available to test such new approaches. Experimentally, even better isolation of the quantum systems from the environment is needed to reach a clearer separation of short- and long-time dynamics, and to unambiguously identify MBL in certain regimes. At the same time, the system sizes need to be scaled

up to enable experimental finite-size scaling tests as a way to certify the results in regimes where no theory prediction is available. With the availability of larger systems and longer evolution times, experiments may also allow one to answer open questions about the stability of MBL to ergodic regions in higher dimensions and in the presence of interfaces, both prominent topics of current scientific debate. At the same time, better control and faster data rates are desired, for example, to scale up techniques to measure the entanglement entropy or related quantities in theoretically inaccessible regimes.

In this view, AMO experiments provide the most advanced analog quantum simulators; and one important future challenge of experiment and theory alike is to quantify the quantum advantage of these simulators. On the topic of many-body localization and far from equilibrium dynamics in general, these synthetic many-body systems provide the only available, or the most advanced, experimental platform to explore the dynamic behavior of quantum matter.

Open System Quantum Simulation: Photonic Crystal Waveguides

The quantum simulation the committee has discussed so far refers to engineering many-body Hamiltonians and dynamics of isolated systems with ultracold atoms and molecules. In experiments, isolated systems are inevitably coupled to an environment causing decoherence, dissipation, and heating. On the other hand, physical realizations of many-body systems in quantum optical setups, where atoms are coupled to laser fields in cavities and photonic nanostructures, are intrinsically (but controllably) open systems. Here the coupling to the environment provides input and output channels: input channels provide a way of driving the system of interest with light, while output channels allow one to monitor the system by observing the emitted light in photon or homodyne detection. In a broader context, one can engineer the coupling of the many-body system of interest to quantum reservoirs, as a novel tool to generate new non-equilibrium quantum phases and as a way of obtaining entangled quantum states of interest in driven-dissipative dynamics.

An outstanding example is provided by coupling atoms or atom-like solid-state systems to photonic crystal waveguides. This area has been progressing rapidly in the past few years, and it is simultaneously at the heart of AMO science, while also touching on other disciplines, including quantum measurement theory, quantum control, materials physics, and of course quantum optics. The concept, as stressed in a recent review article, is to engineer periodic materials with a band structure that can dramatically enhance the interaction of photons with a single atom or with multiple atoms. This modified photon-atom interaction can also affect the atom-atom interaction mediated by photon modes. The modification extends control to other aspects, such as the radiative lifetime of an excited atom in such a wave-

guide. Figure 3.10 depicts such a waveguide with modifications of the light-atom interactions. The key challenge in this field involves strong coupling of coherent quantum emitters to nanophotonic systems. Recently, major advances have been made in coupling coherent atom-like emitters in diamond to nanophotonic cavities and waveguides. In particular, Silicon-vacancy (SiV) color centers have been coupled to nanoscale diamond waveguides with cooperativities approaching 100 (meaning that the color center almost exclusively exchanges energy with the waveguide optical modes and not with other modes); see also Chapter 4. Such systems have been used to realize strong optically mediated interactions between two color centers, which is a key building block of the system illustrated in Figure 3.10. Such systems are emerging as leading candidates for realization of quantum networks, discussed in Chapter 4.

From Analog Quantum Simulation to Quantum Information Science

Analog quantum simulators, as mappings of quantum many-body Hamiltonians of interest to the “natural” Hamiltonians provided by a particular AMO platform, are, of course, not restricted to atoms and molecules in optical lattices

representing Hubbard models. Other examples are spin models implemented with trapped ions or Rydberg tweezer arrays. The committee discusses these systems as programmable analog quantum simulators in Chapter 4. There exists an increasing number of control elements, such as single-site control and addressing, being added to many-body lattice systems, while keeping as a unique feature the potential scalability to a large number of particles. This positions programmable analog quantum simulators as quantum devices between the traditional quantum simulator, which is scalable to large particle numbers but with limited control, and the universal, fully programmable (digital) quantum computer. Illustrations and applications of these systems will be given in Chapter 4.

SUMMARY OF OPPORTUNITIES AND RECOMMENDATIONS

The committee has only scratched the surface with this overview of emerging phenomena in many-body systems that can be built “from the ground up,” expanding the exquisite control of single atoms (and photons) to small groups of particles and then further toward larger scales. All of the developments discussed in this chapter point toward rich opportunities for future investigations. In the few-body limit, it is of continuing interest to identify the scope of universality in quantum states, for their own intellectual interest, for their connections with many-body physics, and for the potential to add new types of controllability at both the few-body and many-body level. Ultracold molecules are starting to form a more diversified research platform, where molecular quantum gases promise to tackle a rich set of many-body phenomena, chemically interesting cold molecules will provide new insights to fundamental reaction processes, and molecules chosen for specific precision measurement targets are being brought under increasingly sophisticated levels of quantum control. Trapped ion systems, and sympathetically cooled ion-plus-atom hybrid systems, are of continuing interest, for quantum computing and simulation, and for physical chemical dynamical processes. AMO science continues to develop and extend our understanding of deeper questions such as thermalization and many-body localization, with new insights already emerging. Recent observation of a chemical reaction event within a single laser tweezer, in addition to explorations of multiple other correlated or entangled phenomena such as multiparticle tunneling between potential wells, suggest that this field is still in its early stages, and promises to grow rapidly in the coming decade.

Much of the progress in emerging quantum many-body physics with AMO has relied on extending the techniques of trapping and cooling to atoms and molecules with an increasingly complex internal-level structure, thus providing opportunities for the design of increasingly complex quantum many-body systems and exploration of novel many-body phenomena. It will be key to continue developments to extend these techniques to an increasing number of atomic and molecular species.

A unique feature of atomic and molecular quantum many-body systems is our ability to microscopically understand and characterize the properties and interactions of atoms and molecules—that is, the many-body Hamiltonians of AMO systems are in principle derivable from a microscopic theory. This is in contrast to the (often) phenomenological modeling in solid-state physics as a starting point to describe many-body dynamics. Thus, developing the theoretical tools to quantitatively predict the behavior of increasingly complex atoms and molecules is a key element for further development and leadership of AMO in designing quantum many-body systems, and achieving ultimate quantum control.

AMO quantum simulators provide us with a novel tool to study in a controlled setting equilibrium and non-equilibrium quantum many-body physics, including regimes not accessible to classical computations. This provides us first with new opportunities for discoveries in many-body physics. On the other hand, quantum simulators allow us also to generate entangled states with immediate application in quantum sensing. These connections between quantum simulation and entanglement-enhanced precision measurement are again unique to AMO physics.

While AMO provides both bosonic and fermionic Hamiltonians, such as Hubbard models, to implement analog quantum simulators, the “natural” fermions provided by fermionic atomic and molecular species are a unique feature to these simulators, where fermions are implemented on the “quantum hardware level.” This is an important resource distinguishing it from both classical and quantum hardware for which complex algorithms and complex quantum gates are required in a universal digital (quantum) computer to effectively reproduce Fermi statistics. Thus, there are unique opportunities to address the problems of fermionic quantum many-body systems for both higher dimensional systems and in quantum chemistry. The 2D Fermi-Hubbard model with repulsive interactions is one of the paradigmatic examples where quantum simulators can demonstrate a regime of genuine quantum advantage. The recent breakthrough of the quantum gas microscope, possibly combined with machine learning algorithms, provides a unique AMO tool for analyzing and discovering highly correlated equilibrium quantum phases such as high-Tc superconductivity, or exploring entanglement in non-equilibrium systems. This work is closely related to the efforts in quantum information science, discussed in the following chapter. Last, the “natural” fermions provided by fermionic species in atomic quantum simulation also offer unique opportunities, as discussed in Chapter 4 in context of variational quantum simulation with Fermi-Hubbard models as the basis of programmable quantum simulators.

Finding: Few-body physics continues to be of continuing interest to identify and test the scope of quantum universality, for its intrinsic intellectual interest, its connections with many-body physics, and to strengthen the controllability of both few-body and many-body quantum systems. Developing theoretical tools able to quantitatively predict the behavior and interactions of increas-

ingly complex atoms and molecules is crucial for further developments in these areas.

Finding: Due to recent theoretical and experimental breakthroughs, ultracold molecules now constitute a very promising research platform able to tackle diverse many-body phenomena and explorations of fundamental reactive processes, with certain molecules yielding viable targets for precision measurement science.

Finding: Trapped ion systems, neutral atoms, systems with long-range interactions (such as those based on molecules and Rydberg atoms) and ion-neutral hybrid systems are leading candidates for quantum information processing and simulation, and for studying chemical dynamical processes.

Recommendation: The atomic, molecular, and optical science community should aggressively pursue, and federal agencies should support, the development of enhanced control of cold atoms and molecules, which is the foundational work for future advances in quantum information processing, precision measurement, and many-body physics.

Finding: Quantum gases of atoms and molecules enable controlled exploration of equilibrium and non-equilibrium many-body physics and the generation and manipulation of entangled states applicable to quantum information processing and quantum metrology, and further developing our understanding of deep questions such as the nature of thermalization, many-body localization, and stable quantum matter away from equilibrium.

Recommendation: Federal funding agencies should initiate new programs to support interdisciplinary research on both highly correlated equilibrium phases and non-equilibrium many-body systems and novel applications.

Finding: AMO-based quantum simulators have the ability in the short term to demonstrate genuine quantum advantage over classical computational devices, without requiring the mastery of complex quantum gates required for a universal digital quantum computer. These systems can provide unique insights into complex models from condensed-matter and high-energy physics, and lead to development and testing of useful quantum algorithms.

Recommendation: Federal funding agencies should initiate new programs involving development, engineering, and deploying the most advanced programmable quantum simulator platforms, and make these systems accessible to the broader community of scientists and engineers.