The year 2015 marks the 20th anniversary of the achievement of Bose–Einstein condensation (BEC) in atomic vapors. Atomic BEC, realized by Eric Cornell, Carl Wieman and Wolfgang Ketterle in 1995, commenced an exciting wave to explore quantum few- and many-body physics. Different from solid state matter and helium superfluids, atomic condensates in the gas phase come with drastically lower densities and temperatures, which place the systems in a unique regime where the atomic interactions are simple and precisely known. Furthermore, the condensates are typically weakly interacting: the interaction length scale of few nanometers is much shorter than the interatomic separation of hundreds of nanometers. Thus many early experimental observations can be well understood based on mean-field theory. One celebrated model is the Gross-Pitaevskii equation which provides a precise and quantitative description of weakly interacting Bose condensates [1]. Opportunities to extend cold atom research beyond the weak interaction regime emerged in the early 2000s. Several experimental schemes were developed to prepare atoms in the strong correlation and interaction regimes. One approach is loading atoms into an optical lattice, the interference pattern of laser beams. The lattice potential enhances the interactions of localized atoms and quenches their kinetic energy. A variety of lattice configurations are realized that promise novel quantum phases of ultracold atoms. Another route to strong interaction is based on Feshbach resonance, typically induced by magnetically shifting a molecular state near the scattering continuum of atoms. The resonant coupling between scattering atoms and the molecular state leads to a divergence of the atomic interaction length scale. Exactly on Feshbach resonance, the collision cross section reaches the quantum mechanical limit, and the system becomes strongly hydrodynamic with mean-free path as short as interparticle separation. Fascinating observations in the strong coupling regime soon followed: prominent examples include the superfluid-Mott insulator transition in optical lattices [2], resonantly interacting Fermi gases in the BEC–BCS (Bardeen–Cooper–Schieffer superfluid) crossover [3] and Efimov three-body states [4], and they lead to tremendous research interest into novel quantum states and quantum dynamics. One of the key motivations in these explorations is to uncover new, universal principles that underpin the physics of strongly correlated quantum systems. A well-studied universal behavior in many-body physics manifests in the critical phenomena near a continuous phase transition. Near the critical point, the system develops a diverging correlation length, which suppresses the dependence on other microscopic length scales. The systems are thus scale invariant and can be universally characterized by a few critical exponents. In cold atoms, divergence of correlation lengthes has been observed near the superfluid phase transition [5,6], as well as the superfluid-Mott insulator quantum phase transition in optical lattices. In the lattice experiments, scale invariance can be tested based on the equation of state measurements [7]. These works complement the transport measurements in condensed matter experiments. Has a new class of universal physics been discovered in atomic quantum gases? Yes, it turns out that a special class of strongly interacting quantum gas with ‘intrinsic’ scaling symmetry emerges near a Feshbach resonance [8]. Near the resonance, the scattering length scale diverges a → ∞ [9], unimaginable in classical systems, and the interaction length reaches the maximum value allowed by quantum mechanics as 1/k, called the ‘unitarity limit’ and k is the momentum of the scattering state. A famous example is the strongly interacting Fermi gas in the BEC–BCS crossover regime, where the resonant interactions lead to strong pairing and and universal hydrodynamic behavior. An exciting development in the study of the BEC–BCS crossover focuses on the Fermi gas exactly on Feshbach resonance. Here the system, also called a ‘unitary Fermi gas’, is the most complex and strongly correlated. The resonant interaction, however, implies a special scaling symmetry of the system. To understand the symmetry, we consider a scale transformation: x → Λx that stretches the coordinates of all atoms by a factor of Λ, and thus reduces the atomic density n by a factor of Λ3 in a 3D gas and the kinetic energy Ek ∝ −∇2 by Λ2. One can see that both kinetic energy Ek and interaction energy in the unitarity limit na → n/k are reduced by the same factor Λ2. The system is thus hypothesized to be invariant under the scale transformation [8]. Remarkably, similar scaling symmetry also holds for 2D quantum gas with a constant coupling g = const. [10]. Here stretching the size of the 2D gas by a factor of Λ also reduces both the kinetic Ek and interaction energy gσ by Λ2, where σ is the 2D density. Since the scaling symmetry in these systems comes directly from the form of the interactions, intrinsic scale invariance is robust at all temperatures T and chemical potentials μ and does not rely on the proximity to a phase transition. The scaling symmetry suggests that the only independent variables are the ratios of the length scales or the equivalent energy scales. In a grand canonical ensemble, the independent ratio can be chosen as μ/kBT. This realization greatly simplifies theoretical description as well as the experimental characterization of the system. Theoretically, a new set of universal thermodynamic relations are expected to be insensitive to the microscopic details; experimentally, measurements are expected to collapse into universal functions that only depend on μ/kBT. Measurements on unitary Fermi gas and the 2D Bose gas are consistent with the scaling hypothesis, shown in Fig. 1(a) and (b). Remarkable quantitative agreement between measurements and calculations has greatly advanced our understanding of the ‘unitary Fermi gas’ and 2D atomic gas. Ultracold atomic gases which are intrinsically scale invariant: (a) unitary Fermi gas, (b) 2D Bose gas and (c) unitary Bose gas. The scaling symmetry suggests that thermodynamic observables only depend on the ratio of chemical potential and thermal energy |$\tilde{\mu }=\mu /k_BT$|. In (a), thermodynamic pressure of a unitary Fermi gas is compared with calculations and other measurements. Figure courtesy of M. Zwierlein [11]. In (b), the collapse of the density measurement at various temperatures (inset) confirms the scaling symmetry. Figure adopted from [12]. Panel (c) shows the three-body loss spectrum in a 6Li133Cs mixture. The geometric progression of the Efimov resonance positions at B = B1, B2 and B3 toward the Feshbach resonance at B0 results from the discrete scaling symmetry of Efimov physics. Figure adopted from [13]. We may expect the same scaling symmetry for a resonantly interacting Bose gas. Here, however, the scaling symmetry is more subtle because of the presence of three-body physics. Predicted by Vitaly Efimov in the context of nuclear physics [14], an infinite number of three-body bound states exist for resonantly interacting bosons and their energies follow a discrete scaling law, breaking the otherwise continuous scaling symmetry. Identical fermions do not form such three-body states due to Pauli's exclusion principle. Theoretically the three-body bound states are supported by the ‘Efimov potential’ V(R)∝ − 1/R2, where R is the characteristic length scale of the three interacting atoms. While the Efimov potential does not break the continuous scaling symmetry, the presence of discrete three-body states does break the symmetry. An alternative picture based on renormalization group limited cycles [15] also explains the discrete scaling symmetry of the system, a scenario that rarely happens in electronic systems. Experimentally, Efimov states have been observed in cold gases of alkali atoms, as well as in helium-4. The geometric scaling symmetry of Efimov states is reported in a recent experiment, shown in Fig. 1(c). The same discrete scaling symmetry is expected for unitary quantum gas of bosons, which, if confirmed, will establish a unique and fascinating class of strongly interacting quantum systems. While in this article, we focus on atoms in optical lattices and unitary atomic gases as primary examples of universal physics that have been actively investigated, efforts in many new research frontiers have shown promising prospects to uncover new universal physics in the strongly correlated and interacting systems. Non-equilibrium dynamics is one of the hottest directions. Ultracold atoms are particularly suited for the study of quantum dynamics due to their slow evolution time scales of miniseconds to seconds. Dynamics near a quantum phase transition and the thermalization of isolated quantum systems are two research topics with many unanswered questions. Cold atom experiments may provide new universal insights that go beyond current theoretical framework. Another exciting direction is quantum many-body systems with long range or very strong interactions. Examples include trapped ions, and condensates of exotic atoms with huge magnetic dipole moments, polar molecules, as well as those with Rydberg excitations. These systems possess various long range interactions. Novel universal dynamics, for example, propagation of information and entanglement, are predicted to depend solely on the asymptotic form of the interaction and the dimension of the systems. The author thanks Yijun Wang for discussion and careful reading of the article. This work was supported by National Science Foundation (PHY-0747907) and Army Research Office-Multiple University Research Initiative (W911NF-14-1-0003).
The year 2015 marks the 20th anniversary of the achievement of Bose–Einstein condensation (BEC) in atomic vapors. Atomic BEC, realized by Eric Cornell, Carl Wieman and Wolfgang Ketterle in 1995, commenced an exciting wave to explore quantum few- and many-body physics. Different from solid state matter and helium superfluids, atomic condensates in the gas phase come with drastically lower densities and temperatures, which place the systems in a unique regime where the atomic interactions are simple and precisely known. Furthermore, the condensates are typically weakly interacting: the interaction length scale of few nanometers is much shorter than the interatomic separation of hundreds of nanometers. Thus many early experimental observations can be well understood based on mean-field theory. One celebrated model is the Gross-Pitaevskii equation which provides a precise and quantitative description of weakly interacting Bose condensates [1]. Opportunities to extend cold atom research beyond the weak interaction regime emerged in the early 2000s. Several experimental schemes were developed to prepare atoms in the strong correlation and interaction regimes. One approach is loading atoms into an optical lattice, the interference pattern of laser beams. The lattice potential enhances the interactions of localized atoms and quenches their kinetic energy. A variety of lattice configurations are realized that promise novel quantum phases of ultracold atoms. Another route to strong interaction is based on Feshbach resonance, typically induced by magnetically shifting a molecular state near the scattering continuum of atoms. The resonant coupling between scattering atoms and the molecular state leads to a divergence of the atomic interaction length scale. Exactly on Feshbach resonance, the collision cross section reaches the quantum mechanical limit, and the system becomes strongly hydrodynamic with mean-free path as short as interparticle separation. Fascinating observations in the strong coupling regime soon followed: prominent examples include the superfluid-Mott insulator transition in optical lattices [2], resonantly interacting Fermi gases in the BEC–BCS (Bardeen–Cooper–Schieffer superfluid) crossover [3] and Efimov three-body states [4], and they lead to tremendous research interest into novel quantum states and quantum dynamics. One of the key motivations in these explorations is to uncover new, universal principles that underpin the physics of strongly correlated quantum systems. A well-studied universal behavior in many-body physics manifests in the critical phenomena near a continuous phase transition. Near the critical point, the system develops a diverging correlation length, which suppresses the dependence on other microscopic length scales. The systems are thus scale invariant and can be universally characterized by a few critical exponents. In cold atoms, divergence of correlation lengthes has been observed near the superfluid phase transition [5,6], as well as the superfluid-Mott insulator quantum phase transition in optical lattices. In the lattice experiments, scale invariance can be tested based on the equation of state measurements [7]. These works complement the transport measurements in condensed matter experiments. Has a new class of universal physics been discovered in atomic quantum gases? Yes, it turns out that a special class of strongly interacting quantum gas with ‘intrinsic’ scaling symmetry emerges near a Feshbach resonance [8]. Near the resonance, the scattering length scale diverges a → ∞ [9], unimaginable in classical systems, and the interaction length reaches the maximum value allowed by quantum mechanics as 1/k, called the ‘unitarity limit’ and k is the momentum of the scattering state. A famous example is the strongly interacting Fermi gas in the BEC–BCS crossover regime, where the resonant interactions lead to strong pairing and and universal hydrodynamic behavior. An exciting development in the study of the BEC–BCS crossover focuses on the Fermi gas exactly on Feshbach resonance. Here the system, also called a ‘unitary Fermi gas’, is the most complex and strongly correlated. The resonant interaction, however, implies a special scaling symmetry of the system. To understand the symmetry, we consider a scale transformation: x → Λx that stretches the coordinates of all atoms by a factor of Λ, and thus reduces the atomic density n by a factor of Λ3 in a 3D gas and the kinetic energy Ek ∝ −∇2 by Λ2. One can see that both kinetic energy Ek and interaction energy in the unitarity limit na → n/k are reduced by the same factor Λ2. The system is thus hypothesized to be invariant under the scale transformation [8]. Remarkably, similar scaling symmetry also holds for 2D quantum gas with a constant coupling g = const. [10]. Here stretching the size of the 2D gas by a factor of Λ also reduces both the kinetic Ek and interaction energy gσ by Λ2, where σ is the 2D density. Since the scaling symmetry in these systems comes directly from the form of the interactions, intrinsic scale invariance is robust at all temperatures T and chemical potentials μ and does not rely on the proximity to a phase transition. The scaling symmetry suggests that the only independent variables are the ratios of the length scales or the equivalent energy scales. In a grand canonical ensemble, the independent ratio can be chosen as μ/kBT. This realization greatly simplifies theoretical description as well as the experimental characterization of the system. Theoretically, a new set of universal thermodynamic relations are expected to be insensitive to the microscopic details; experimentally, measurements are expected to collapse into universal functions that only depend on μ/kBT. Measurements on unitary Fermi gas and the 2D Bose gas are consistent with the scaling hypothesis, shown in Fig. 1(a) and (b). Remarkable quantitative agreement between measurements and calculations has greatly advanced our understanding of the ‘unitary Fermi gas’ and 2D atomic gas. Ultracold atomic gases which are intrinsically scale invariant: (a) unitary Fermi gas, (b) 2D Bose gas and (c) unitary Bose gas. The scaling symmetry suggests that thermodynamic observables only depend on the ratio of chemical potential and thermal energy |μ~=μ/kBT|. In (a), thermodynamic pressure of a unitary Fermi gas is compared with calculations and other measurements. Figure courtesy of M. Zwierlein [11]. In (b), the collapse of the density measurement at various temperatures (inset) confirms the scaling symmetry. Figure adopted from [12]. Panel (c) shows the three-body loss spectrum in a 6Li133Cs mixture. The geometric progression of the Efimov resonance positions at B = B1, B2 and B3 toward the Feshbach resonance at B0 results from the discrete scaling symmetry of Efimov physics. Figure adopted from [13]. We may expect the same scaling symmetry for a resonantly interacting Bose gas. Here, however, the scaling symmetry is more subtle because of the presence of three-body physics. Predicted by Vitaly Efimov in the context of nuclear physics [14], an infinite number of three-body bound states exist for resonantly interacting bosons and their energies follow a discrete scaling law, breaking the otherwise continuous scaling symmetry. Identical fermions do not form such three-body states due to Pauli's exclusion principle. Theoretically the three-body bound states are supported by the ‘Efimov potential’ V(R)∝ − 1/R2, where R is the characteristic length scale of the three interacting atoms. While the Efimov potential does not break the continuous scaling symmetry, the presence of discrete three-body states does break the symmetry. An alternative picture based on renormalization group limited cycles [15] also explains the discrete scaling symmetry of the system, a scenario that rarely happens in electronic systems. Experimentally, Efimov states have been observed in cold gases of alkali atoms, as well as in helium-4. The geometric scaling symmetry of Efimov states is reported in a recent experiment, shown in Fig. 1(c). The same discrete scaling symmetry is expected for unitary quantum gas of bosons, which, if confirmed, will establish a unique and fascinating class of strongly interacting quantum systems. While in this article, we focus on atoms in optical lattices and unitary atomic gases as primary examples of universal physics that have been actively investigated, efforts in many new research frontiers have shown promising prospects to uncover new universal physics in the strongly correlated and interacting systems. Non-equilibrium dynamics is one of the hottest directions. Ultracold atoms are particularly suited for the study of quantum dynamics due to their slow evolution time scales of miniseconds to seconds. Dynamics near a quantum phase transition and the thermalization of isolated quantum systems are two research topics with many unanswered questions. Cold atom experiments may provide new universal insights that go beyond current theoretical framework. Another exciting direction is quantum many-body systems with long range or very strong interactions. Examples include trapped ions, and condensates of exotic atoms with huge magnetic dipole moments, polar molecules, as well as those with Rydberg excitations. These systems possess various long range interactions. Novel universal dynamics, for example, propagation of information and entanglement, are predicted to depend solely on the asymptotic form of the interaction and the dimension of the systems. The author thanks Yijun Wang for discussion and careful reading of the article. This work was supported by National Science Foundation (PHY-0747907) and Army Research Office-Multiple University Research Initiative (W911NF-14-1-0003).
Ultracold atoms in optical lattices provide an extremely clean and controllable setting to explore quantum many-body phases of matter. Here we give a brief review of the strong-correlation physics that has been realized using such ultracold atoms in optical lattices ranging from the realization of Hubbard models to studies of quantum magnetism and the detection of single atoms with lattice site resolution. All this has opened up fundamentally new opportunities for the investigation of quantum many-body systems. Introduction Over the past years, ultracold atoms in optical lattices have emerged as versatile new systems to explore the physics of quantum many-body systems. On the one hand, they can be helpful in gaining a better understanding of known phases of matter and their dynamical behavior; on the other hand, they allow one to realize completely novel quantum systems that have not been studied before in nature [1, 2, 3]. Commonly, the approach of exploring quantum many-body systems in such a way is referred to as “quantum simulations.” Examples of some of the first strongly interacting many-body phases that have been realized both in lattices and in the continuum include the quantum phase transition from a superfluid to a Mott insulator [4, 5, 6], fermionic Mott insulators [7, 8], the achievement of a Tonks- Girardeau gas [9, 10], and the realization of the Bose-Einsten condensate (BEC)– Bardeen-Cooper-Schrieffer (BCS) crossover (see also Chapter 12) in Fermi gas mixture [11] using Feshbach resonances [12]. In almost all of these experiments, detection was limited to time-of-flight imaging or more refined derived techniques that mainly characterized the momentum distribution of the quantum gas [2]. For several years, researchers in the field have therefore aspired to employ in situ single-particle detection methods for the analysis of ultracold quantum gases. Only recently has it become possible to implement such imaging techniques, marking a milestone for the characterization and manipulation of ultracold quantum gases [13, 14, 15, 16, 17]. In our discussion, we will focus on one of the most successful of these techniques based on high-resolution fluorescence imaging. Despite being a rather new technique, such quantum gas microscopy has already proven to be an enabling technology for probing and manipulating quantum many-body systems. For the first time, controllable and strongly interacting many-body systems, as realized with ultracold atoms, could be observed on a local scale [17, 16].
At the time of the first workshop in this series in 1993, the only experimentally realized Bose condensate (at least in the simple sense conjectured by Einstein) was liquid 4 He. In the intervening twenty-plus years, much has happened in the world of Bose-Einstein condensation (BEC). Probably the most exciting development has been the attainment of condensation in ultracold bosonic atomic gases such as 87 Rb and 23 Na in 1995, followed a few years later by the achievement of degeneracy and eventually Bardeen-Cooper-Schrieffer (BCS) pairing in their fermionic counterparts, and the experimental realization of the theoretically long-anticipated “BEC-BCS crossover” by using the magnetic field degree of freedom to tune the system through a Feshbach resonance. One particularly fascinating aspect of the latter has been the realization of a “unitary gas” at the resonance itself – a system which prima facie has no characteristic length scale other than the interparticle separation, and is therefore a major challenge to theorists. Other systems in which BEC has been realized, sometimes transiently, include exciton-polariton complexes in semiconducting microcavities and, at least in a formal sense, the magnons in a magnetic insulator, as well as ultracold gases with a nontrivial and sometimes large “spin” degree of freedom. As compared with our “traditional” Bose condensate, liquid 4 He, these new systems typically have many more (and more rapidly adjustable) control parameters, and have therefore permitted qualitatively new types of experiment. One particularly fascinating development has been the use of optical techniques to generate “synthetic gauge fields” and thus mimic some of the topologically nontrivial systems which have recently been of such intense interest in a condensed-matter setting. At the same time, there remain long-standing issues from helium physics, such as the nature and consequences of “spontaneously broken U(1) symmetry,” the “Kibble-Zurek” mechanism, and more generally the relaxation of strongly nonequilibrium states to equilibrium; in some cases, the new systems have been used to address these more quantitatively than was possible with 4 He. The chapters in this volume address all of these questions and more, and should be of intense interest to both the experimental and the theoretical sides of the BEC community.
Scale invariance is conventionally discussed in the context of critical phenomena near a continuous phase transition. In atomic gases, a new class of scale invariant systems emerges, which we call “intrinsic scale invariance.” In these systems, the scaling symmetry does not rely on a phase transition, but comes from the balance of kinetic and interaction energy at different length scales. Three examples are discussed here: three-dimensional unitary Fermi gas, two-dimensional dilute Bose gas, and unitary Bose gas. The first two are invariant under continuous scaling transformation, and thus they share the same universal thermodynamics. Unitary Bose gas is a special case with expected discrete scaling symmetry because of Efimov three-body physics.We show that the discrete scale invariance can be captured by a complex scaling dimension and suggest a universal form for the observables. Introduction The achievement of atomic Bose-Einstein condensation (BEC) in 1995 initiated a new adventure to explore quantum many-body phenomena in the gas phase, a journey that has yielded much excitement and is still progressing fast. Because of their diluteness, ultracold atomic gases can be precisely modeled and characterized with high accuracy. In addition, new opportunities have emerged in the past decade to prepare cold atoms in the strong coupling regime. By loading the samples into optical lattices [1] or tuning them near a Feshbach resonance [2], for instance, cold atoms offer a unique platform to investigate strongly correlated quantum phenomena. Examples include the superfluid-Mott insulator transition [3] (see also Chapter 13), BEC-BCS (Bardeen-Cooper-Schrieffer) crossover [4, 5, 6, 7] (see also related overview in Chapter 12), Efimov trimer states [8], and Berezinskii- Kosterlitz-Thouless transition in two dimensions [9] (see also Chapter 10). One key motivation in these explorations is to uncover new, universal laws that underpin the behavior of strong-coupled few- and many-body quantum systems. In this chapter, we discuss “scale invariance” as a generic attribute in various cold atom systems, including all of the aforementioned examples. While scale invariance is conventionally discussed in the context of critical phenomena, the cold atom systems that interest us belong to a new class, which we call “intrinsic scale invariance.” In the following, we first overview the main features of different types of scale invariance and discuss related examples in cold atom systems.
From time to time in physics, single technical achievements have spawned an explosion of research activity and have led to the genesis of entirely new avenues of investigation. Precisely this occurred in the field of quantum gases - the topic of this Focus Issue - following the achievement of Bose-Einstein condensation in a dilute alkali gas in 1995. These experiments built on the foundational work on quantum fluids in dense systems, such as superfluid helium, and the technical achievements of laser cooling in atomic physics. Dilute quantum gases have already proved to be an incredibly fertile ground for exploring a wide range of macroscopic quantum phenomena. The rapid pace of experimental discovery and theoretical advance is evident and every year continues to bring significant achievements. A great advantage is that microscopic observations can be made directly and compared with the predictions of many-body theoretical formulations founded on interaction parameters which are completely determined - something which is not possible in their dense fluid counterparts. This Focus Issue brings together papers which cover many of the topical aspects of quantum gases: optical lattices, quantum degenerate fermions, quantum atom optics, molecular Bose-Einstein condensation, vorticity, and quantum fluctuations in systems of reduced dimensionality. One pervasive theme is the flexibility which arises from the ability to control the interactions, even dynamically if necessary. In quantum gases, this can be effected in two ways, either by changing the internal nature of the pairwise potentials by a Feshbach resonance, or by changing the nature of external potentials in which the atoms are confined and supported against gravity. The external potential may be used to spin gases to high vorticity, imprint on the system a periodic optical lattice potential, or even change the effective dimensionality. Optical lattices are particularly well represented in this Focus Issue. This is where there is the strongest connection with the physics of solids and strongly correlated fluids. Interest in optical lattices is partly due to the recent experimental demonstration that in these systems one can study the quantum phase transition from a Mott insulator, where the atoms are localized to individual sites of the optical potential, to a superfluid phase with long-range order. In contrast to a condensed matter system, the transition can be easily dynamically controlled since it is induced by simply varying the depth of the optical potential. Another area of focused interest is in fermionic gases where there is an ongoing effort, both experimentally and theoretically, to observe and characterize the high-temperature superfluid transition that should occur with enhanced pairwise interactions. If realized, this quantum system would allow experimental investigation of the crossover from Bose-Einstein condensation of dilute weakly-interacting composite bosons to the superconducting-like transition that occurs in a system of weakly interacting fermions. The topics of this Focus Issue represent well the building blocks of quantum gas studies which should allow the exploration of a wide range of macroscopic quantum phenomena in a highly controllable and clean system in which the effects of disorder may be completely absent. Given the recent progress, it will be intriguing to see where this field leads in the future. Focus on Quantum Gases Contents Anisotropic dissipation of superfluid flow in a periodically dressed Bose-Einstein condensate D M Stamper-Kurn Rapidly rotating Bose-Einstein condensates in anharmonic potentials G M Kavoulakis and Gordon Baym Feshbach resonances and collapsing Bose-Einstein condensates J N Milstein, C Menotti and M J Holland Three-body recombination of cold fermionic atoms H Suno, B D Esry and Chris H Greene Bogoliubov spectrum and Bragg spectroscopy of elongated Bose-Einstein condensates C Tozzo and F Dalfovo Quantum atom optics with Bose-Einstein condensates Klaus Mølmer Creation of effective magnetic fields in optical lattices: the Hofstadter butterfly for cold neutral atoms D Jaksch and P Zoller The Josephson frequency of resonantly coupled atomic and molecular condensates R A Duine and H T C Stoof Superfluid current disruption in a chain of weakly coupled Bose–Einstein condensates F S Cataliotti, L Fallani, F Ferlaino, C Fort, P Maddaloni and M Inguscio Bright matter wave solitons in Bose–Einstein condensates K E Strecker, G B Partridge, A G Truscott and R G Hulet Spin dynamics for bosons in an optical lattice Juan José García-Ripoll and Juan Ignacio Cirac Mean field ground state of a spin-1 condensate in a magnetic field Wenxian Zhang, Su Yi and Li You Atom resonant tunnelling through a moving barrier M L Chiofalo, M Artoni and G C La Rocca Local correlations in a strongly interacting one-dimensional Bose gas D M Gangardt and G V Shlyapnikov Shift of the molecular bound state threshold in dense ultracold Fermi gases with Feshbach resonance R Combescot Landau damping in trapped Bose condensed gases B Jackson and E Zaremba An exact reformulation of the Bose–Hubbard model in terms of a stochastic Gutzwiller ansatz Iacopo Carusotto and Yvan Castin Chemical potential standard for atomic Bose–Einstein condensates Sigmund Kohler and Fernando Sols The short-distance first-order correlation function of the interacting one-dimensional Bose gas Maxim Olshanii and Vanja Dunjko The relation between the Gross–Pitaevskii and Bogoliubov descriptions of a dilute Bose gas A J Leggett Superfluidity of Bose–Einstein condensate in an optical lattice: Landau–Zener tunnelling and dynamical instability B Wu and Q Niu Density-optimized efficiency for magneto-optical production of a stable molecular Bose–Einstein condensate Matt Mackie, Anssi Collin, Kalle-Antti Suominen and Juha Javanainen A two-atom picture of coherent atom–molecule quantum beats Bogdan Borca, D Blume and Chris H Greene Superfluid dynamics of a Bose–Einstein condensate in a periodic potential C Menotti, A Smerzi and A Trombettoni Phase diagram of two-component bosons on an optical lattice Ehud Altman, Walter Hofstetter, Eugene Demler and Mikhail D Lukin Murray J Holland University of Colorado at Boulder, USA
Since the realization of Bose-Einstein condensation in 1995, experimentalists in ultracold atomic gas community have developed a set of entirely new methods to study quantum gases. In ultracold atomic gases, implementation of synthetic gauge fields, the spin-orbit coupling, optical lattices and Feshbach resonances, provides a clean and highly controllable platform to tune the parameters of the system, which are typically fixed for real materials. Motivated by the rapid developments of experimental progresses, I investigate the ultracold atomic gases with synthetic gauge fields. \n \nIn this thesis, a bosonic gas in optical lattices in the presence of an effective staggered magnetic flux will be studied, using a generalized Bose-Hubbard model. Phase transition from the Mott insulating phase to the superfluid phase will be discussed. Novel superfluid phases including the stripe phase and the plane wave phase will be discussed. Furthermore, the existence of a tricritical point in the phase diagram is predicted. \n \nTo study Fermi systems with synthetic gauge fields, a spin-1=2 Fermi gas with Rashba spin-orbit coupling is investigated by studying the structure of vortex line through the Bardeen-Cooper-Schrieffer to Bose-Einstein condensation (BCSBEC) crossover. Due to the spin-orbit coupling, the s-wave interacting Fermi gas behaves like an effective p-wave paired system. A discussion about the vortex line structure will be provided by calculating the spatial distribution of the gap function, the density, the particle current, and also the elementary excitations. It has been found that the most robust superfluid occurs around the unitary point, indicated by the smallest healing length and the largest critical current. \n \nFinally, motivated by the recent progress of p-wave resonantly Fermi gas in experiment, an extension of Nozières-Schmitt-Rink method to the p-wave case will be given. An investigation of the normal state properties will be given by calculating the equation of state and the p-wave contacts, which have been measured by the experiment. Furthermore, the superfluid transition temperature with realistic parameters appropriate for experiment is calculated and is found to be within experimental reach.
Advances in trapping and cooling of ultracold gases over the last several decades have made it possible to test many formerly outstanding predictions from disparate branches of physics. This thesis touches on three historical problems that have found new life recently in the context of ultracold Bose gases of alkali atoms. The first problem revolves around an outstanding prediction from Boltzmann over a century and half old that the breathing mode of a isotropically trapped classical gas should oscillate indefinitely. I analyze recent experimental results [Nat. Phys. 11, 1009 (2015)], and attribute observed damping sources to trap imperfections. The second question is about the analogue of first and second sound modes from liquid helium in trapped dilute gases. I present the results of a joint theoretical/experimental investigation of the breathing mode of a finite temperature Bose-Einstein condensate (BEC), attributing a striking collapse revival behavior of the resultant oscillation to in-phase and out-of-phase normal modes of the thermal cloud and condensate. The third problem is that of the formation of Borromean ring-like three-body bound states, referred to as Efimov trimers, in strongly-interacting few-body systems. I extend the predicted spectrum of Efimov states into the realm of many degenerate internal levels, and investigate the difficult three-body elastic scattering problem.\nThese questions are part of the broader theme of this thesis: How can our understanding of few-body physics in the ultracold limit be translated into statements about the bulk behavior of an ultracold gas? For weakly-interacting Bose gases, this translation is well-known: the many-body properties of the gas are well-described by the tracking just the one and two particle correlations. I analyze a generalization of this procedure to higher order correlations, the general connection between few-body physics and correlations in a dilute gas, and results for the emergence of Efimov physics in the magnetic phase of the strongly-interacting Bose gas.