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Theoretical Study on Universal Few-Body Effects in Ultracold Atomic Gases

Shuhei Yoshida

Published Mar 22, 2018• 0 citations
Publisher

Abstract

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

DOI
10.15083/00077923

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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
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.
Few-body effects play an important role for the understanding of ultracold quantum gases. We make use of an effective field theory approach to investigate various aspects of universal few-body physics close to a Feshbach resonance. That is the regime where the scattering length is large compared to all other length scales of the system and thus determines the observables. It is also the regime where the Efimov effect whose main characteristic is the occurrence of a sequence of three-body bound states becomes important. At unitarity, i.e., for diverging scattering length, the ratio of binding energies of neighboring states approaches a constant. The existence of those trimers can be captured by a single three-body parameter and influences observables such as recombination rates. Starting from adequate Lagrangians, we derive atom-dimer scattering amplitudes that contain all information of interest about the three-particle systems. With this method, we first investigate the scattering of atoms and dimers at finite temperature in the presence of the Efimov effect. We calculate the dimer relaxation rate and obtain good agreement with available experimental data. Furthermore, heteronuclear mixtures exhibiting large interspecies scattering length are studied in detail. If bosons are the majority species, the Efimov effect occurs in an S-wave channel and we are able to compute three-body recombination and dimer relaxation rates. The results are compared to the outcome of two existing experiments. For mainly fermionic systems, the Efimov effect is only present in an overall P-wave and three-body recombination at threshold vanishes. However, in mixtures of atoms and dimers, scattering cross sections and dimer relaxation show the typical Efimov behavior of log-periodicity. We also consider two-dimensional Bose gases, where the Efimov effect does not occur. For those systems, we derive an equation including the first non-universal corrections and deduce three-body observables such as binding energies and atom-dimer scattering parameters.
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.
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 ppp-wave Feshbach resonance. The low-energy scattering in this system can be characterized by two parameters, that is, ppp-wave scattering length and effective range. At the unitarity limit where the ppp-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 sss-wave interaction in three dimensions. It is in contrast to a Fermi gas with the ppp-wave resonance in three dimensions in which the effective range is inevitably finite. We present the universal equation of state in this unitary ppp-wave Fermi gas within the many-body TTT-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 ppp-wave Fermi gas.
Ultracold atomic gases with tunable interactions offer an ideal platform for studying interacting quantum matter. While the few- and many-body physics are generally complex and intractable, the problem can be greatly simplified in an atomic gas by a controlled separation of relevant length and energy scales. Precise control of experimental parameters, via Feshbach resonances, optical potentials and radio-frequency radiation, enables deterministic measurements of few-body physics, including universal physics and the Efimov effect. In this thesis, I present our recent studies on precisely measuring two- and three-body physics in an ultracold Bose gas. I begin by describing our new apparatus used for generating and studying ultracold 39K gas samples. Then, I focus on our precision spectroscopy of Feshbach dimer binding energies, spanning three orders of magnitude and with sub-kilohertz resolution. These measurements enable us to locate a Feshbach resonance and determine scattering length values with unprecedented accuracy. Finally, I present our precise measurements locating an exotic three-body state, specifically the Efimov ground state. We find that the trimer state location significantly deviates from the value predicted by van der Waals universality. Due to small experimental and systematic uncertainties, our measurement is the strongest evidence of departure from the universal value and is the first observed deviation near a Feshbach resonance of intermediate strength.