Understanding quantum many-body systems with interactions is an important problem common to various fields of physics, such as atomic physics, condensed matter physics, nuclear physics, and particle physics.Among others, ultracold atoms have provided ideal grounds to study strongly interacting systems and to explore exotic quantum states because of their high controllability.We can widely control the strengths of the interactions from weak to strong as well as the geometry of the systems.In order to deepen our understanding of quantum manybody systems with interactions, we in this thesis investigate two types of one-dimensional (1D) systems realizable with ultracold atoms; bosons and spinless fermions near two-body resonances and bosons near a three-body resonance.One-dimensional systems near two-body resonances have been intensively studied because, in the homogeneous cases, they are exactly solvable systems.The energy spectra and thermodynamic quantities exactly obtained by the Bethe ansatz have greatly contributed to deepening our comprehension of many-body systems in 1D.On the other hand, the exact computations of correlation functions, which encode information about excitations of the systems, are in general much more complicated even in the integrable systems.Motivated by this background, we derive a series of exact relations for correlation functions in bosons and fermions near two-body resonances.These relations called universal relations originate from the universal properties of the resonant systems: Their properties depend on the interactions only through the scattering lengths characterizing the low-energy scattering.The universal relations are strong constraints on the systems because the relations hold for any particle number, scattering length, temperature, and with or without a trapping potential.The relations include the asymptotic behaviors of correlation functions in high-energy regime as well as the energy relations, in which the energies are expressed in term of the momentum distributions.The universal relations involve two-and three-body contacts, which are the integrals of local pair and triad correlations, respectively.Firstly we derive, in both bosons and fermions, universal relations for static correlation functions such as static structure factors and momentum distributions within the first-quantized formalisms.The power laws of the static correlation functions for large momentum and the energy relations are obtained.The coefficients in these power-law behaviors are proportional to the two-body contact.We clarify that the three-body contact makes no contribution to the bosonic energy relation, but it plays a crucial role in the fermionic one.Secondly, we derive universal relations for dynamic correlation functions for bosons.Using the operator product expansion (OPE) in the field theoretical formalism, we obtain the largeenergy-momentum behaviors of the dynamic structure factor, the dynamic current correlation, and the single-particle spectral density as well as the quasiparticle energy and the scattering rate in high-energy regime.While the behaviors of the dynamic structure factor and the current correlation are proportional to the two-body contact, that of the single-particle spectral density is proportional to the number density.Because of the interrelation between 1D bosons and fermions near two-body resonances, the results of the dynamic structure factor and the current i
Understanding quantum many-body systems with interactions is an important problem common to various fields of physics, such as atomic physics, condensed matter physics, nuclear physics, and particle physics.Among others, ultracold atoms have provided ideal grounds to study strongly interacting systems and to explore exotic quantum states because of their high controllability.We can widely control the strengths of the interactions from weak to strong as well as the geometry of the systems.In order to deepen our understanding of quantum manybody systems with interactions, we in this thesis investigate two types of one-dimensional (1D) systems realizable with ultracold atoms; bosons and spinless fermions near two-body resonances and bosons near a three-body resonance.One-dimensional systems near two-body resonances have been intensively studied because, in the homogeneous cases, they are exactly solvable systems.The energy spectra and thermodynamic quantities exactly obtained by the Bethe ansatz have greatly contributed to deepening our comprehension of many-body systems in 1D.On the other hand, the exact computations of correlation functions, which encode information about excitations of the systems, are in general much more complicated even in the integrable systems.Motivated by this background, we derive a series of exact relations for correlation functions in bosons and fermions near two-body resonances.These relations called universal relations originate from the universal properties of the resonant systems: Their properties depend on the interactions only through the scattering lengths characterizing the low-energy scattering.The universal relations are strong constraints on the systems because the relations hold for any particle number, scattering length, temperature, and with or without a trapping potential.The relations include the asymptotic behaviors of correlation functions in high-energy regime as well as the energy relations, in which the energies are expressed in term of the momentum distributions.The universal relations involve two-and three-body contacts, which are the integrals of local pair and triad correlations, respectively.Firstly we derive, in both bosons and fermions, universal relations for static correlation functions such as static structure factors and momentum distributions within the first-quantized formalisms.The power laws of the static correlation functions for large momentum and the energy relations are obtained.The coefficients in these power-law behaviors are proportional to the two-body contact.We clarify that the three-body contact makes no contribution to the bosonic energy relation, but it plays a crucial role in the fermionic one.Secondly, we derive universal relations for dynamic correlation functions for bosons.Using the operator product expansion (OPE) in the field theoretical formalism, we obtain the largeenergy-momentum behaviors of the dynamic structure factor, the dynamic current correlation, and the single-particle spectral density as well as the quasiparticle energy and the scattering rate in high-energy regime.While the behaviors of the dynamic structure factor and the current correlation are proportional to the two-body contact, that of the single-particle spectral density is proportional to the number density.Because of the interrelation between 1D bosons and fermions near two-body resonances, the results of the dynamic structure factor and the current i
One-dimensional spinless Bose and Fermi gases with contact interactions have the close interrelation via Girardeau's Bose-Fermi mapping, leading to the correspondences in their energy spectra and thermodynamics. However, correlation functions are in general not identical between these systems. We derive in both systems the exact universal relations for correlation functions, which hold for any energy eigenstate and any statistical ensemble of the eigenstates with or without a trapping potential. These relations include the large-momentum behaviors of static structure factors and of momentum distributions as well as energy relations, which connect the sums of kinetic and interaction energies to the momentum distributions. The relations involve two- and three-body contacts, which are the integrals of local pair and triad correlations, respectively. We clarify how the relations for bosons and fermions differ and are connected with each other. In particular, we find that the three-body contact makes no contribution to the bosonic energy relation, but it plays a crucial role in the fermionic one. In addition, we compute the exact momentum distribution for any number of fermions in the unitary limit.
Universality is a crucial concept in modern physics, which allows us to describe phenomena without referring to microscopic details of systems.The unitary Fermi gas is a prime example, where universal relations among physical observables and a universal equation of state have been developed and studied in detail.In this thesis, we extend these notions beyond the paradigm of the unitary Fermi gas and investigate universal effects of few-body correlations in resonantly interacting ultracold atomic gases.In particular, we discuss (i) universal relations in a spinless Fermi gas with an anisotropic p-wave resonance and (ii) universality of the few-body spectra and the ground-state properties of an impurity-boson system.Firstly, we investigate the universal relations in a spinless Fermi gas with an anisotropic p-wave resonance.We show that the high-momentum distribution and the short-range density-density correlation have characteristic power laws, whose coefficients define the p-wave contact tensor.We derive the adiabatic sweep theorem, which relates the p-wave contact tensor to the total energy.We also discuss the experimental scheme to measure the p-wave contact tensor and test our results.Secondly, we examine the ground states and lower excited states of an impurity resonantly interacting with one, two, and three non-interacting bosons.We calculate the energy spectra of this few-body system with two different models.The overall scale of length depends on the models.However, we find that once we fix the three-body parameter, which is a critical scattering length of the ground-state Efimov trimer, the few-body spectrum is independent of a specific choice of the models.This finding is corroborated by comparing our numerical results with the one obtained by Ardila and Giorgini [Phys.Rev. A 92, 033612 (2015)].Thirdly, we investigate the case in which the mass of an impurity is infinite.In this limit, we show that the model of a Feshbach resonance can be mapped to a bosonic extension of Anderson's single impurity model, which can be used to analytically solve the three-body problem.We find that the ground-state energy with a Feshbach resonance is drastically different from that in a potential-interaction system, and that the former approaches the latter logarithmically as the unitarity limit is taken.We argue This thesis would not have been possible without the help and support of many people.I would like to express my gratitude to my supervisor, Professor Masahito Ueda, for his guidance and sincere encouragement over the period of my PhD program.He was a great teacher; he taught me a lot of things, from the way of a good scientist to a key to happy married life.I would also like to thank my secondary supervisors, Professor Tetsuo Hatsuda and Professor Junji Yumoto, for valuable advice on my study, and sometimes on job hunting.They also organized meetings where I could learn studies of other students.Much of the work in this thesis started during my stay at Quantum Matter group in Monash University, where I greatly benefited from fruitful collaboration with the group members.I am deeply grateful to Professor Meera M. Parish
Owing to their high tunability, ultracold atoms provide versatile platforms for investigating interacting systems.Here, the quantum statistics of particles is controlled by the choice of atomic species and the space geometry by the operation of optical lattices.In addition, the strength of the interparticle interaction can be tuned from weak to strong through the Feshbach resonance.One of the ultracold atomic systems which take maximum advantage of this tunability is a two-component Fermi gas near the two-body resonance, which is referred to as a resonant Fermi gas.Because the interaction is characterized by only one parameter, the s-wave scattering length, the interaction effect appears only through the scattering length near the resonance.Thus, the resonant Fermi gas shows universal behaviors independent of the details of the interaction and forms an ideal example of a strongly correlated quantum many-body system.This universality and the high tunability connect the study of ultracold atoms to various other fields of physics.In particular, a resonant Fermi gas in the unitarity limit, where the scattering length is tuned to infinity in three dimensions, shows remarkable universality because of the absence of length scales in the interaction.The emergence of the conformal invariance due to the absence of length scales is one representative example of the universality in the unitarity limit.The strong coupling, which is associated with conformality, motivates studies of the thermodynamic properties and the transport properties.For example, the system achieves an almost "perfect fluid" in the unitarity limit, where the bulk viscosity vanishes and the shear viscosity has a remarkably small ratio to the entropy density.Although the strong coupling leads to intriguing transport properties, it makes the calculations of transport coefficients difficult.This thesis investigates the transport properties of two-component Fermi gases near the two-body resonance in two and three dimensions.We particularly focus on their transport coefficients, such as the bulk viscosity, the shear viscosity, and the thermal conductivity, for an arbitrary scattering length.In order to carry out reliable analysis in the strongly correlated regime, such as near the unitarity limit, we need non-perturbative methods.One of the nonperturbative methods is to use the conformal symmetry that emerges in the unitarity limit.The application of this symmetry-based approach to hydrodynamics revealed the vanishing bulk viscosity in the unitarity limit.Another non-perturbative approach is an expansion in terms of the fugacity, called the quantum virial expansion.In the high-temperature and low-density regime, this expansion is valid because the fugacity is small.First of all, we focus on the bulk viscosity, which characterizes the dissipation caused by an expansion of the fluid volume.The vanishing bulk viscosity in the unitarity limit is intuitively understood because the entropy does not change before and after an isotropic expansion of the i List of Publications Publications relevant to this thesis 1.
In this thesis, we study the physical properties of several ultracold few-boson systems depending on the interactions between their constituents. Nowadays, experimentally, it is possible to have great control with high precision over the geometry and the interactions between the particles, making them an excellent setup to test directly the principles of quantum mechanics. A very interesting point is to study the evolution of their properties with the number of particles. The theoretical study of these systems pretends to microscopically understand the current experimental results and give support to new experimental developments.\n\n\t\t\t\t The method that will be used is the exact diagonalization of the Hamiltonian of the system. As we will see, in spite of the attempts to improve it, the method is limited by the fact that, in practice, it is only useful to study few-particle systems. The method has several advantages. First of all, one has access to both the ground and the excited states. In second place, the method is variational and converges to the exact solution as long as the Hilbert space in which we diagonalize is enlarged.\n\t\t\t\t Moreover, since we have access to the states of the system, it is possible to calculate any observable quantity of interest.\n\n\t\t\t\t First, we will study a system of spinless bosons trapped in a two-dimensional harmonic potential. The effect of the trap is to keep the system bound. It will be seen how the presence of a repulsive interaction changes the energy spectrum and other properties of the system. For instance, the density profile, which is usually measurable, and also the two-body distribution function, which is intimately related to the existence of correlations.\n\n\t\t\t\t Afterwards, the focus will be on the particular case of having only two bosons in the system interacting through a strong repulsive force. Inspired by the one-dimensional case where the fermionization phenomenon takes place in the strongly-interacting limit, we will study whether in two dimensions there is a resembling reminiscent effect. In other words, we will analyze if there are properties of the two strongly-interacting bosons in two dimensions that are like the ones of two noninteracting fermions.\n\n\t\t\t\t After that, we will tackle the localization phenomenon in a one-dimensional system that is caused by an external speckle potential that introduces disorder in the system. We will show that the localization is a robust phenomenon against repulsive contact interactions.\n\n\t\t\t\t Finally, we will study the influence of the spin-orbit coupling in a system of bosons with two possible pseudospin components, associated, for instance, to two hyperfine levels, confined in a two-dimensional harmonic trap. We will present an exhaustive analysis of the combined effects of the interaction and the spin-orbit coupling in the spectrum and the properties of the system. In particular we show the existence of a crossover in the ground state of the system susceptible to be experimentally identified.
We study ultracold gases of alkali-metal atoms in the quantum degenerate regime. The interatomic interactions in these type of systems can be tuned using resonances induced by magnetic or electric fields. The tunability of the interactions, together with the possibility of confining the atoms with several kinds of external potentials, allows for a completely novel approach to study basic problems in many-body physics, and moreover, allows to enter regimes which have never been accessible in condensed matter or nuclear physics. For example, this has led to the experimental demonstration of an intimate relation between two types of superfluidity: the crossover from Bose-Einstein condensation of tightly bound molecules to the superfluid behavior related to weakly bound BCS-like pairs. Another experimental landmark was reached when the existence of universal Efimov three-body bound states was proven in experiments with ultracold bosonic cesium atoms. In this thesis, we study several aspects of these strongly interacting and ultracold atomic gases. We develop an analytical model that encapsulates all of the relevant scattering physics in atomic systems where open-channel shape resonances and closedchannel Feshbach resonances give rise to complicated and non-trivial scattering properties. This model provides lots of physical insight and is shown to describe important quantities, such as the molecular energies and scattering phase shifts, with a high level of accuracy. The model is compared to full numerical coupled-channel calculations in two atomic systems: rubidium and lithium. We study the BCS-BEC crossover using a many-body description of the ultracold gas that includes the non-trivial energy dependence of the scattering model. We show that it gives rise to superfluid behavior associated with the formation of BCS-like pairs while the low-energy interactions are repulsive in character. The energy dependence of the interactions is crucial, as it gives rise to attractive interactions at the Fermi energy, necessary for the formation of Cooper pairs. We demonstrate new ways of controlling the interatomic interactions using a combination of magnetic and electric fields. This leads to experimental control of, for instance, the three-body parameter in the context of Efimov physics and of non-universal behavior in the BCS-BEC crossover in fermionic gases. Using a four-body method based on first principles, we solve the molecule-molecule scattering problem to calculate several important properties of bosonic and fermionic Summary 145 molecules that consist of light and heavy atoms. These type of molecules are of current experimental interest, and we predict several exciting relations between three- and four-body observables in these type of systems.
In this thesis, cold gases as well as the electron gas are analyzed with two expansion techniques that have, in essence, the same aim: To describe parameter regimes where the few-body aspect dominates over the collective many-body behavior.The Operator Product Expansion (OPE) of quantum field theory is used to extract high-frequency properties of the correlation function that determines the radio-frequency (RF) response of a two-dimensional Fermi gas with zero-range interactions.Exact sum rules that relate the zeroth and first moment of the radio-frequency spectrum to the particle number and the contact of the gas, are derived from these properties.A quantum cluster (virial-) expansion is employed to study the properties of the spectral function of the two-dimensional Fermi gas at small to moderate degeneracy.From the spectral function, observables like the density of states and the momentum distribution are calculated and are found to be in excellent agreement with Tan's universal relations.In the second part of the thesis, the cluster expansion is applied to both bosonic and fermionic three-dimensional quantum gases.Virial coefficients, contacts and momentum distribution are calculated up to third order in the fugacity.This includes three-particle correlations exactly.The results are are in excellent agreement with known universal relations.In the third part of this thesis, an OPE is applied to the problem of the interacting electron gas in both two and three dimensions.A power-law decay of the momentum distribution and the static structure factor at high momenta is found, compatible with earlier findings by J. Kimball.