Physics on Ultracold Quantum Gases

The following sections are included:

• Preface
• Contents

Almost twenty years have passed since the first experimental observations of Bose–Einstein condensation (BEC) in ultracold atomic gases [1–3]. In these seminal experiments, up to 10⁴ ∼ 10⁵ neutral bosonic atoms of alkali were trapped using lasers and magnetic fields and were cooled down to temperatures below the microkelvin range, at which point the collective behavior of the gas becomes quantum mechanical. The realization of BEC in dilute atomic gases turns a new page in our understanding of Nature and provides us with a versatile platform, on which previously unsolved physical problems can be studied and many novel ideas can be tested. Coupled with other recently developed techniques, such as the Feshbach resonance, the optical lattice potentials and the synthetic gauge fields, and ultracold atomic gases — both bosonic and fermionic — are playing an increasingly important role in various fields of research, including quantum simulation, quantum computation and precision measurement, to name a few…

The study of cold atomic systems, from both experimental and theoretical perspectives, is crucially based on the understanding of atomic structures. Indeed, the abilities to confine neutral atoms in a trap, to cool them down to quantum degeneracy and to fine-tune the parameters within the system heavily rely upon the preparation and manipulation of atoms in various states. In this chapter, we discuss atomic structure of alkali-metal atoms. This choice is made based on two considerations. First, the electronic structure of an alkali-metal atom is fairly simple. There is only one valence electron while all other electrons form closed shells. Second, alkali-metal atoms are the most popular candidate for the current cold atom experiments due to their technical maturity and simplicity. However, we stress that the fundamental ideas discussed in this chapter are general and can be extended to other types of atoms.

The experimental achievements on cooling and trapping neutral atoms with laser light heavily rely on the understanding of atom-light interaction. Starting from the 1970s, remarkable progresses along this direction have witnessed the evolution of cold atom physics from a theoretical proposal to an astoundingly diverse field now involving multidisciplinary categories of physics, including atomic physics, nuclear physics and condensed matter physics.

One of the simplest problems involving the atom-light interaction is the coupling of a two-level atom with a single-mode light field. Although any real atom in practice acquires a complicated level structure as discussed in the previous chapter, a two-level description is valid provided that the two levels of interest are nearly resonant with the driving field, while all other levels are far detuned. In this chapter, we discuss some basics of this simplified but far-from-trivial problem, and introduce some fundamental physical properties of atom-light interacting process. To make a close connection to the most popular experimental situation, we focus on alkali-metal atoms in this chapter, and restrict our discussion to topics that are closely related to experimental schemes of laser cooling and trapping. Readers who are interested in the general theory of atom-light interaction are referred to specialized books on this subject, e.g., see Cohen-Tannoudji et al. [1].

The experimental realization of cooling and trapping neutral atoms with the assistance of laser light is one of the most important achievements in cold atom physics. One of the original motivations for this pursuit is to achieve precision measurement. As we have learned in Chap. 2, the transition frequency between two atomic levels (usually two hyperfine states) can be used as frequency standards. However, a direct application of this proposal using atoms at room temperature would be compromised with displacement and broadening of the spectral lines induced by the Doppler shift of a wide spread of atomic velocities. Thus, it is of great importance to reduce the velocity distribution of atoms, or equivalently, to reduce the temperature of an atomic ensemble. Besides, the high atomic velocities also limit the observation time, and hence the spectral resolution, in an apparatus with a finite size. If one would be able to trap atoms within a confined space, the performance of measurement will be significantly enhanced…

One of the most exciting areas of research on ultracold atomic gases is to emulate theoretical models and demonstrate general physics in multidisciplinary fields. Toward this goal, efforts are mainly devoted along two directions. The first one is to manipulate the single-particle properties of an atom. This includes adjusting the single-particle dispersion via an optical lattice and introducing a synthetic gauge field by Raman transitions. The other direction is to investigate the interaction effect. Along this line, the realization of the Bose–Hubbard model, the study of the crossover between the Bardeen–Cooper–Shrieffer (BCS) and the BEC limits in Fermi gases, and the demonstration of Efimov physics shed light on our understanding of the corresponding many-body and few-body physics…

In the previous chapter, we have learned that the long-wavelength, low-energy scattering process between two distinguishable atoms is determined by the s-wave scattering length a_s. This single-value parameter thus governs the effective interaction strength between the two atoms via, e.g., the renormalization relation Eq. (5.50). It is then of great interest if one can find a way to tune a_s, especially toward the strongly interacting regime…

Following the experimental realization of BEC, a natural step forward is to bring fermionic atoms into quantum degeneracy. For neutral noninteracting bosonic atoms, the quantum degeneracy condition $n{\lambda }_d^3\sim 1$ leads to matter-wave interference and hence condensation. In contrast, such a phase transition does not exist for neutral noninteracting fermionic atoms, due to Fermi–Dirac statistics. This can be understood in the zero-temperature limit of noninteracting fermions, where atoms occupy all quantum states below a Fermi energy due to the Pauli blocking. This leads to a sharp Fermi surface in momentum space. It is well-known that the Fermi surface becomes unstable with the introduction of even an infinitesimally small attractive interaction between different atom species. Due to the so-called Cooper instability, the many-body ground state of a two-component Fermi gas with attractive interactions is not a simple Fermi sea, but rather a superfluid of correlated atom pairs, which are the analog of Cooper pairs of electrons in the BCS superconducting materials. On the other hand, the situation is more subtle for a Fermi gas with repulsive interactions. For example, when the repulsive interaction originates from attractive twobody scattering potentials, there exist different branches of many-body states that may lead to either a superfluid of two-body bound states or an effectively repulsive Fermi gas…

In this chapter, we introduce the mean-field theory of the BCS-BEC crossover as a natural extension of the standard BCS theory, while emphasizing their critical differences. Our introduction focuses on wide Feshbach resonances, which are studied by most of the experiments. For completeness, we also briefly discuss cases of narrow Feshbach resonance.

So far, our focus has been on the mean-field description of the BCS-BEC crossover, which can be regarded as an extension of the standard BCS theory. While in three dimensions, the BCS wave function provides a qualitatively correct picture for the BCS-BEC crossover at zero temperature and the mean-field theory we reviewed previously cannot be expected to yield quantitatively correct results in a strongly interacting system. On the other hand, the success of the standard BCS theory is largely due to the lack of considerable quantum fluctuations in the weak-interacting BCS limit, which is the case for the conventional superconducting materials. When the interaction strength is increased toward unitarity, the quantum fluctuation increases and must be taken into account at some point. At finite temperatures, one must also include the thermal fluctuations and the problem becomes more complicated. In the strongly interacting regime (−1 < (k_F a_s)⁻¹ < 1), the effective s-wave scattering length becomes very large and diverges at unitarity. Because of the lack of small parameters, the system cannot be characterized by a reliable perturbative theory. One therefore has to either resort to exact solutions via, for example, Quantum Monte Carlo methods, or rely on truncating fluctuation expansions based on physical intuitions…

In the previous chapters, we have discussed resonant s-wave superfluidity in Fermi gases with an equal population in the two spin states. In these systems, the Fermi surfaces of the different spin species overlap. In the weak-coupling limit, the Fermi surface becomes unstable due to the Cooper instability in the presence of a weak attractive interaction, and fermions with different spin species and on opposite sides of the Fermi surface tend to form Cooper pairs. On the other hand, for a Fermi gas where the Fermi surfaces of the two spin species do not overlap, e.g., in a spin-polarized Fermi gas or in a Fermi gas with atoms of different masses, the above scenario may fail to apply. Due to the mismatch of the Fermi surface, Cooper pairs are formed at an energy cost. In general, this implies that the ground state may be different from the standard BCS pairing state and a quantum phase transition is possible as the difference in the Fermi surfaces increases…

The recent experimental realization of a synthetic gauge field in dilute gases of neutral atoms has greatly extended the horizon of quantum simulation in these systems [1–7]. By engineering the atom-laser interaction, one may realize both Abelian or non-Abelian synthetic gauge potentials in ultracold atomic gases. It is then possible to simulate, for example, the behavior of charged particles in an electromagnetic field in a system of neutral atoms [2, 3]. Due to the extraordinary tunability of the synthetic magnetic field in ultracold atoms, it has been suggested that quantum Hall effects can be observed in these systems. Another important example is the implementation of spin-orbit coupling, a non-Abelian gauge field, in cold atomic gases, where the internal degrees of freedom of the atoms are coupled to the center-of-mass motional degrees of freedom [4]. Interesting phenomena like the quantum spin Hall effect, topological insulator and topological superfluidity, where spin-orbit coupling plays a key role, may then be investigated in ultracold atomic gases [8–10]. Furthermore, it is now possible to study the novel effects induced by spin-orbit coupling in an ultracold Bose gas, which do not have counterparts in systems of condensed matter [11, 12]. In this chapter, we present a brief review of the recent experimental and theoretical progresses in this direction, with the emphasis on the novel effects induced by synthetic spin-orbit coupling in an ultracold Fermi gas.

Strongly correlated many-body physics is currently one of the central problems in physics. There are many important physical phenomena, such as high temperature superconductivity (high-T_c), concerned with strongly correlated many-body physics. Some seminal mechanics, such as the Hamiltonian, have been proposed to study these phenomena. For example, it is widely believed that the Fermi–Hubbard model plays a key role in high-T_c material. Unfortunately, the real physics underlying these phenomena is still unresolved; the main obstacle is that there is no efficient method to solve these Hamiltonian equations yet. For these strongly correlated manybody models, traditional analytic methods, such as the mean-field theory and perturbation theory, do not work. Powerful numerical methods, such as the Quantum Monte Carlo (QMC) and Density Matrix Renormalization Group (DMRG), also cannot give reasonable results for many of the relevant systems. The QMC method suffers from a sign problem for both frustrated systems and fermionic systems, and the DMRG method is limited to solving 1D and quasi-1D systems. To understand these key models and the strongly correlated physics, we need new ideas and technologies. The optical lattice, as a new platform, provides an opportunity to investigate these models. It serves as the bridge between physical theories and the physical phenomena of real materials…

The following sections are included:

• Introduction to the Bose–Hubbard model
• Simulation of the Bose–Hubbard model in optical lattices
• References

Optical lattices provide a platform to continuously tune the system parameters. This characteristic makes it very convenient to investigate the dynamic process that cannot be achieved in conventional material. We will introduce two related processes that have been experimentally studied in the optical lattice: the quench dynamics near the phase transition and thermalization processes.

As an ubiquitous factor in solid-state systems due to random defects and dislocations, disorder introduces various exotic many-body phenomena that have attracted great attention in condensed matter physics. The most famous example is the Anderson localization in a free electron system [1]. In non-interacting systems, it is widely believed that localization occurs in one and two dimensions for an arbitrarily small disorder, whereas there is a critical value for the strength of disorder in a three-dimensional system. However, there is little knowledge about the interplay between disorder and interaction, in particular strong interaction, in many-body systems. In addition, there are several difficulties hampering experimental investigation on this subject using conventional condensed matter systems [2]. First, disorders cannot be easily controlled or experimentally fine-tuned. Second, samples prepared using the same method acquire some distribution of disorders. To characterize their properties, we need to perform measurements for as many samples as possible and average the results. This requires a large number of samples, which is quite demanding in time and labor, and usually very expensive. Third, in some disordered systems, there may be a large number of low-energy excitations, which makes a quantitative investigation more subtle and difficult. Besides, the most important problem is to investigate the interplay between disorders and interactions in a many-body system. This means that we need to control the disorder and the interaction together, which is almost impossible in conventional materials…

The following sections are included:

• General phases of spin systems
• Simulate spin systems in an optical lattice
• References