We explored interaction cold atoms in new quantum regimes that have no prior analogue in condensed matter materials in three related main topics. First, motivated by recent advance in orbitally tuned Feshbach resonance experiments, we analyze the ground-state phase diagram and related low-energy excitation spectra of a high partial wave interacting Bose gas. Remarkably different from what was previously known in the p-wave case, the atomic superfluid is found to be momentum-independent in the present d-wave case. What is more, we study the quantum fluctuations in the condensates of a mixture of bosonic atoms and molecules with interspecies p-wave interaction. Our analysis shows that the quantum phase of coexisting atomic and molecular condensates is unstable at the mean-field level. The quantum Lee-Huang-Yang correction to the mean-field energy provides a remarkable mechanism to self-stabilize the phase. The correlated order spontaneously breaks a rich set of global U(1) gauge, atomic spin, spatial rotation and translation, and time-reversal symmetries. Second, we study the dynamics of a non-integrable spin chain model composed of two ingredients - a nearest neighbor Ising coupling, and an infinite range XX interaction. Unlike other fast scrambling many-body systems, this model is not known to be dual to a black hole. We demonstrate that our model exhibits fast scrambling for a wide parameter regime, accompanied by a fast growth of the entanglement entropy, as well as a swift change in the magnetization. Third, to extend the study of the highly chaotic many-body model, we analyze the ground state phases of the model with a nearest neighbor XXZ interaction and an infinite range XX interaction. By employing spin-wave theory, we find that there is a large parameter regime where the continuous U(1) symmetry of this model is spontaneously broken, which is not possible in the absence of the infinite range interaction by the Mermin-Wagner theorem. Furthermore, we demonstrate that in the U(1) symmetry broken phase, the half chain entanglement entropy violates the area law logarithmically. Our work demonstrates that the interplay of short and long range interactions can lead to novel quantum phases of matter.
This thesis studies bosonic and fermionic quantum gases. In the first part, a variational many-body theory is used to explain a Bragg scattering experiment carried out on a bosonic quantum gas [S.B. Papp et al. Bragg Spectroscopy of a Strongly Interacting 85Rb Bose-Einstein Condensate. Phys. Rev. Lett. 101, 135301 (2008)]. A key feature in the experiment was the use of a Feshbach resonance, which made large values of the scattering length accessible. Due to the large values of the scattering length, existing models such as the Beliaev model could not be used to explain the observations, and therefore the experiment was a particularly interesting one to analyze. In our first approach, we constructed ad hoc potentials that fitted the observed excitation spectrum, and later we improved our approach by using T-matrix formalism to describe the Feshbach resonant system. All in all, the phenomenological model we developed fits the observed excitation spectrum and yields correct molecular Feshbach resonance state energies in certain cases. The second part of this thesis studies fermionic quantum gases. We focus on studying a gas of spin-1/2 particles confined to a spin-dependent optical lattice. The lattice geometry is such that the up-spin component is loaded in a honeycomb lattice, and the down-spin component is confined to the underlying triangular lattice. We considered attractive on-site and nearest-neighbor interactions, and formulated the nearest-neighbor interaction term in such a way that it takes into account the possibility of spontaneous time-reversal symmetry breaking. Furthermore, we took into account the possibility of Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase, which breaks spatial symmetry. Within a mean-field approximation, we showed that the FFLO state is the ground state of the system in many instances. In addition, we found out that the system spontaneously breaks time-reversal symmetry if the nearest-neighbor interaction strength is large. Due to the time-reversal symmetry breaking, the system has topologically non-trivial phases characterized by nonzero Chern numbers. Finally, there were also cases where the time-reversal symmetry was broken in the FFLO phase, and thus we found a phase where spatial and time-reversal symmetries are simultaneously broken.
The quantum many-body problem is at the heart of some of the most formidable open problems in modern physics, such as high-T
c superconductivity or the behaviour of neutron stars. Ultracold atomic systems can now be used to simulate model Hamiltonians of condensed matter or nuclear physics, in very well-controlled environnement. In this thesis, we have developed a general method to probe the thermodynamics of homogeneous quantum systems using trapped atomic gases. These measurements are directly compared to the predictions of theories of the quantum many-body problem. We have applied this technique to the spin-
1/2 Fermi gas and the Bose with short-range interactions. Using fermionic
6Li, we explored a part of the wide parameter space by changing the interaction strength, the spin-population imbalance or the temperature of the gas. This system exhibits remarkably rich physics, such as normal/superfluid phase transitions (that can be of thermal or quantum character) or a Fermi liquid-type behaviour of the normal phase. We have also used this method to probe a Bose gas of
7Li atoms close to a Feshbach resonance. We have measured the Equation of State of the Bose gas as a function of interactions at very low temperature. For the first time, we measured quantitatively the Lee-Huang-Yang beyond mean-field correction to the ground-state energy of the system, first predicted in 1957. We compared the experimental results to Quantum Monte-Carlo calculations. We have extended this study using out-of-equilibrium measurements of the Bose gas in the strongly interacting regime, which gives a first hint on the properties of the hypothetical unitary Bose gas.
Advances in experiments on dilute ultracold atomic gases have given us access to highly tunable quantum systems. In particular, there have been substantial improvements in achieving different kinds of interaction between atoms. As a result, utracold atomic gases oer an ideal platform to simulate many-body phenomena in condensed matter physics, and engineer other novel phenomena that are a result of the exotic interactions produced between atoms. In this dissertation, I present a series of studies that explore the physics of dilute ultracold atomic gases in different settings. In each setting, I explore a different form of the inter-particle interaction. Motivated by experiments which induce artificial spin-orbit coupling for cold fermions, I explore this system in my first project. In this project, I propose a method to perform universal quantum computation using the excitations of interacting spin-orbit coupled fermions, in which effective p-wave interactions lead to the formation of a topological superfluid. Motivated by experiments which explore the physics of exotic interactions between atoms trapped inside optical cavities, I explore this system in a second project. I calculate the phase diagram of lattice bosons trapped in an optical cavity, where the cavity modes mediates effective global range checkerboard interactions between the atoms. I compare this phase diagram with one that was recently measured experimentally. In two other projects, I explore quantum simulation of condensed matter phenomena due to spin-dependent interactions between particles. I propose a method to produce tunable spin-dependent interactions between atoms, using an optical Feshbach resonance. In one project, I use these spin-dependent interactions in an ultracold Bose-Fermi system, and propose a method to produce the Kondo model. I propose an experiment to directly observe the Kondo effect in this system. In another project, I propose using lattice bosons with a large hyperfine spin, which have Feshbach-induced spin-dependent interactions, to produce a quantum dimer model. I propose an experiment to detect the ground state in this system. In a final project, I develop tools to simulate the dynamics of fermionic superfluids in which fermions interact via a short-range interaction.
Motivated by recent advances in orbitally tuned Feshbach resonance experiments, we analyze the ground-state phase diagram and related low-energy excitation spectra of a d-wave interacting Bose gas. A two-channel model with d-wave symmetric interactions and background s-wave interactions is adopted to characterize the gas. The ground state is found to have three interesting superfluid phases: atomic, molecular, and atomic–molecular. In great contrast to what was previously known about the p-wave case, the atomic superfluid is found to be momentum-independent for the d-wave case discussed here. The Bogoliubov spectra above each superfluid phase are obtained both analytically and numerically.
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).
In this thesis we theoretically study low-dimensional, strongly corrleated systems of cold atoms, which are not in an equilibrium situation. This is motivated by recent experimental progress, which has made it possible to study quantum many-body physics in a controllable and clean setting; and parameters can be changed during the experiment. In Chapter 2 and 3 we study phases and quantum phase transitions of ‘tilted ’ Mott insulator of bosons. We analyze a variety of lattices and tilt directions in two dimensions: square, decorated square, triangular, and kagome. We show that there are rich possibilities for correlated phases with non-trivial entanglement of pseudospin degrees of freedom encoded in the boson density. For certain configurations three-body interactions are necessary to ensure that the energy of the effective resonant subspace is bounded from below. We find quantum phases with Ising density wave order, with superfluidity transverse to the tilt direction, a quantum liquid state with no broken symmetry. We also find cases for which the resonant subspace is described by effective quantum dimer models. In Chapter 4 we study spin 1/2 chains with a Heisenberg interaction which are