Abstract We revisit the properties of the two-component Fermi gas with short-range interactions in three dimensions, in the limit where the s -wave scattering length diverges. Such a unitary Fermi gas possesses universal thermodynamic and dynamical observables that are independent of any interaction length scale. Focusing on trapped systems of N fermions, where N ≤ 10 , we investigate how well we can determine the zero-temperature behavior of the many-body system from published few-body data on the ground-state energy and the contact. For the unpolarized case, we find that the Bertsch parameters extracted from trapped few-body systems all lie within 15% of the established value. Furthermore, the few-body values for the contact are well within the range of values determined in the literature for the many-body system. In the limit of large spin polarization, we obtain a similar accuracy for the polaron energy, and we estimate the polaron’s effective mass from the dependence of its energy on N . We also compute an upper bound for the squared wave-function overlap between the unitary Fermi system and the non-interacting ground state, both for the trapped and uniform cases. This allows us to prove that the trapped unpolarized ground state at unitarity has zero overlap with its non-interacting counterpart in the many-body limit N → ∞ .
Abstract We revisit the properties of the two-component Fermi gas with short-range interactions in three dimensions, in the limit where the s -wave scattering length diverges. Such a unitary Fermi gas possesses universal thermodynamic and dynamical observables that are independent of any interaction length scale. Focusing on trapped systems of N fermions, where N ≤ 10 , we investigate how well we can determine the zero-temperature behavior of the many-body system from published few-body data on the ground-state energy and the contact. For the unpolarized case, we find that the Bertsch parameters extracted from trapped few-body systems all lie within 15% of the established value. Furthermore, the few-body values for the contact are well within the range of values determined in the literature for the many-body system. In the limit of large spin polarization, we obtain a similar accuracy for the polaron energy, and we estimate the polaron’s effective mass from the dependence of its energy on N . We also compute an upper bound for the squared wave-function overlap between the unitary Fermi system and the non-interacting ground state, both for the trapped and uniform cases. This allows us to prove that the trapped unpolarized ground state at unitarity has zero overlap with its non-interacting counterpart in the many-body limit N → ∞ .
We elucidate universal many-body properties of a one-dimensional, two-component ultracold Fermi gas near the p-wave Feshbach resonance. The low-energy scattering in this system can be characterized by two parameters, that is, p-wave scattering length and effective range. At the unitarity limit where the p-wave scattering length diverges and the effective range is reduced to zero without conflicting with the causality bound, the system obeys universal thermodynamics as observed in a unitary Fermi gas with contact s-wave interaction in three dimensions. It is in contrast to a Fermi gas with the p-wave resonance in three dimensions in which the effective range is inevitably finite. We present the universal equation of state in this unitary p-wave Fermi gas within the many-body T-matrix approach as well as the virial expansion method. Moreover, we examine the single-particle spectral function in the high-density regime where the virial expansion is no longer valid. On the basis of the Hartree-like self-energy shift at the divergent scattering length, we conjecture that the equivalence of the Bertsch parameter across spatial dimensions holds even for a one-dimensional unitary p-wave Fermi gas.
I perform lattice Monte Carlo studies of universal four-component fermion systems in one spatial dimension. Continuum few-body observables (i.e., ground-state energies and integrated contact densities) are determined for both unpolarized and polarized systems of up to eight fermions confined to a harmonic trap. Estimates of the continuum energies for four and five trapped fermions show agreement with exact analytic calculations to within approximately one percent statistical uncertainties. Continuum many-body observables are determined for unpolarized systems of up to 88 fermions confined to a finite box, and 56 fermions confined to a harmonic trap. Results are reported for universal quantities such as the Bertsch parameter, defined as the energy of the untrapped many-body system in units of the corresponding free-gas energy, and its subleading correction at large but finite scattering length. Two independent estimates of these quantities are obtained from thermodynamic limit extrapolations of continuum extrapolated observables. A third estimate of the Bertsch parameter is obtained by combining estimates of the untrapped and trapped integrated contact densities with additional theoretical input from a calculation based on Thomas-Fermi theory. All estimates of the Bertsch parameter and its subleading correction are found to be consistent to within approximately one percent statistical uncertainties. Finally, the continuum restoration of virial theorems is verified for both few- and many-body systems confined to a trap.
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
We consider a Fermi gas with two components of different masses, with the s-wave two-body interaction tuned to unitarity. In the range of mass ratio 8.62<M/m<13.6, it is possible for a short-range interaction between heavy fermions to produce a resonance in a three-body channel. The resulting system is scale invariant and has universal properties, and is very strongly interacting. When M/m is slightly above the lower limit 8.62, the ground state energy of a 2ratio1 mixture of heavy and light fermions is less than 2% of the energy of a noninteracting gas with the same number densities. We derive exact relationships between the pressures of the unitary Fermi gases with and without three-body resonance when the mass ratio is close to the critical values of 8.62 and 13.6.
We treat the trapped two-component Fermi system, in which unlike fermions interact through a two-body short-range potential having no bound state but an infinite scattering length. By accurately solving the Schrödinger equation for up to N=6 fermions, we show that no many-body bound states exist other than those bound by the trapping potential, and we demonstrate unique universal properties of the system: Certain excitation frequencies are separated by 2variant Planck's over 2piomega, the wave functions agree with analytical predictions and a virial theorem is fulfilled. Further calculations up to N=30 determine the excitation gap, an experimentally accessible universal quantity, and it agrees with recent predictions based on a density functional approach.
Ultracold atomic systems have been of great theoretical and experimental interest in the past few decades.They provide ideal paradigms to study quantum phenomena, simulate condensed matter systems, and could potentially become a platform for quantum computations.This thesis focuses on the theoretical investigation of universal and non-universal properties of ultracold few-atom systems.The goal of this thesis is to answer or provide insights into several interesting open questions.First, what is the role of the non-universal three-body parameter in weakly-interacting harmonically trapped Bose gases and at which order does it kick in?Second, how large is the ground state energy of small two-component Fermi gases with infinitely strong interactions compared to their non-interacting counterparts?Lastly, what is the interplay between the spin-orbit coupling and the s-wave two-body interaction and how does it affect the energy spectrum of the two-atom system?In order to answer these questions, we develop and employ different analytical and numerical techniques.A perturbative effective field theory treatment to study the weaklyinteracting trapped Bose gas is employed.The treatment is extended to up to fourth-order perturbation theory and techniques to regularize the diverging sums are developed.A logarithmic divergence in the three-body sector is identified and the need for a non-universal three-body parameter is shown.v The explicitly correlated Gaussian method is used to calculate the ground state energy of small trapped two-component Fermi gases with up to ten particles.A code is developed from scratch and parallelized.An improved optimization scheme is developed.A novel technique to remove the finite-range dependence is proposed and tested.Our calculations serve as benchmarks for other numerical techniques.The explicitly correlated Gaussian method is extended to treat systems with periodic boundary conditions for the first time.Standard perturbation theory is used to treat the spin-orbit coupling term in trapped two-atom systems.Our treatment is based on the exact two-body zero-range solution in the trap.The interplay between the spin-orbit coupling and the two-body s-wave interaction is investigated and shown to have a great impact on the energy spectrum for certain parameter combinations.