This dissertation serves as a summary of my Ph.D. work numerically studying equilibrium and non-equilibrium properties of strongly-interacting one-dimensional (1D) boson systems. This work is motivated by the fact that 1D systems are realizable and highly controllable with ultracold atoms in optical lattice and atom chip experiments. We apply a recent worm algorithmic Monte Carlo approach developed for 1D continuous systems to study their equilibrium properties, both with and without an underlying lattice. We also apply an exact lattice approach based on the Bose-Fermi mapping to check our Monte Carlo results in the Tonks-Girardeau limit, and more importantly, to study far-from-equilibrium expansion dynamics of the systems.
We first study the scaling of one-particle correlations of the harmonically trapped Lieb-Liniger gas with changing temperature and interaction strength. Based on the universal behaviors of the density and momentum profiles, we are able to determine the effective parameters needed to fully characterize the system. We also find that the Tonks-Girardeau limit at low temperatures is the ideal regime for the experimental observation of the $1/k^4$ momentum tail.
An extra periodic lattice can drive the transition from superfluid to Mott insulator states. Exact and complete phase diagrams for such transitions are available only in the weak interacting and deep lattice limit, in which the system can be described using one-band Bose-Hubbard model. Beyond this limit, we use the worm algorithm in continuous space to map out the phase diagrams at various interaction strengths. We compare our phase diagrams with one-band Bose-Hubbard predictions and identify the regime where the one-band description breaks down. We introduce an inverse confined scattering solution to obtain effective Hubbard parameters, with which the Bose-Hubbard model provides correct results for strong interactions and deep lattices at unit filling.
In addition to the equilibrium properties, we also study the expansion dynamics of ultracold atoms in the hard-core limit. Experimentally, this is usually achieved by turning off confining potentials and letting atoms expand in optical lattices. Theoretical studies from initial ground states predicted the occurrence of fermionization of the momentum distribution after long expansion times. In addition, quasicondensation at finite momenta emerges when expanding from Mott insulating domains. Here, we develop a finite-temperature extension of the lattice approach for dynamics. We find the dynamical ferminoization of the momentum distributions at all temperatures. For expansion from initial Mott domains, we observe enhanced correlations reminiscent of dynamical quasicondensation. Surprisingly, we find the systems appear to cool down during the melting of the Mott domains. We use an emergent local Hamiltonian to understand these emergent phenomena.
This dissertation serves as a summary of my Ph.D. work numerically studying equilibrium and non-equilibrium properties of strongly-interacting one-dimensional (1D) boson systems. This work is motivated by the fact that 1D systems are realizable and highly controllable with ultracold atoms in optical lattice and atom chip experiments. We apply a recent worm algorithmic Monte Carlo approach developed for 1D continuous systems to study their equilibrium properties, both with and without an underlying lattice. We also apply an exact lattice approach based on the Bose-Fermi mapping to check our Monte Carlo results in the Tonks-Girardeau limit, and more importantly, to study far-from-equilibrium expansion dynamics of the systems.
We first study the scaling of one-particle correlations of the harmonically trapped Lieb-Liniger gas with changing temperature and interaction strength. Based on the universal behaviors of the density and momentum profiles, we are able to determine the effective parameters needed to fully characterize the system. We also find that the Tonks-Girardeau limit at low temperatures is the ideal regime for the experimental observation of the 1/k4 momentum tail.
An extra periodic lattice can drive the transition from superfluid to Mott insulator states. Exact and complete phase diagrams for such transitions are available only in the weak interacting and deep lattice limit, in which the system can be described using one-band Bose-Hubbard model. Beyond this limit, we use the worm algorithm in continuous space to map out the phase diagrams at various interaction strengths. We compare our phase diagrams with one-band Bose-Hubbard predictions and identify the regime where the one-band description breaks down. We introduce an inverse confined scattering solution to obtain effective Hubbard parameters, with which the Bose-Hubbard model provides correct results for strong interactions and deep lattices at unit filling.
In addition to the equilibrium properties, we also study the expansion dynamics of ultracold atoms in the hard-core limit. Experimentally, this is usually achieved by turning off confining potentials and letting atoms expand in optical lattices. Theoretical studies from initial ground states predicted the occurrence of fermionization of the momentum distribution after long expansion times. In addition, quasicondensation at finite momenta emerges when expanding from Mott insulating domains. Here, we develop a finite-temperature extension of the lattice approach for dynamics. We find the dynamical ferminoization of the momentum distributions at all temperatures. For expansion from initial Mott domains, we observe enhanced correlations reminiscent of dynamical quasicondensation. Surprisingly, we find the systems appear to cool down during the melting of the Mott domains. We use an emergent local Hamiltonian to understand these emergent phenomena.
Bosonic atoms confined in optical lattices are described by the Bose-Hubbard model and can exist in two different phases, Mott insulator or superfluid, depending on the strength of the system parameters. In the vicinity of the phase boundary, there are degeneracies that occur between every two adjacent Mott lobes. Because of this, nondegenerate perturbation theory fails to give meaningful results for the condensate density: it predicts a phase transition in a point of the phase diagram where no transition occurs. Motivated by this, we develop two different degenerate perturbative methods to solve the degeneracy-related problems. Moreover, we study the one-dimensional repulsively interacting Bose gas under harmonic confinement, with special attention to the asymptotic behavior of the momentum distribution, which is a universal k−4 decay characterized by the Tan's contact. The latter constitutes a direct signature of the short-range correlations in such an interacting system and provides valuable insights about the role of the interparticle interactions. We investigate the system constituted of N interacting particles in the strongly interacting limit. In such a regime, the strong interparticle interaction makes the bosons behave similarly to the ideal Fermi gas. Because of the difficulty in analytically solving the system for N particles at finite interaction, the Tonks-Girardeau regime provides a favorable scenario to probe the contact. Therefore, we are able to provide an analytical formula for the Tan's contact. Furthermore, we analyze the scaling properties of the Tan's contact in terms of N in the high-temperature regime as well as in the strongly interacting regime. Finally, we compare our analytical calculations of the Tan's contact to quantum Monte Carlo simulations and discuss some fundamental differences between the canonical and the grand-canonical ensembles.
Experimental advances have made ultracold atoms in optical lattices a favourable setting to study out-of-equilibrium phenomena and attracted considerable attention in recent years.These systems are highly versatile in that experimental parameters can be tuned over a wide range of values in real time.This facilitates the study of quantum quenches, in which parameters in the corresponding Hamiltonian are varied in time faster than the system can respond adiabatically.Such protocols open the door to a rich range of many-body physics and have been studied intensely both theoretically and experimentally.The Bose-Hubbard model (BHM) has been shown to describe interacting ultracold bosons in an optical lattice, allowing the opportunity for experiments to probe the out-of-equilibrium dynamics of the model.The BHM is a particularly convenient context for studying quantum quenches as it displays a quantum phase transition between superfluid and Mott-insulator phases.In this thesis, we develop a strong-coupling approach that allows the study of correlations in space and time in both the superfluid and Mott-insulating phases of the BHM.Specifically, we obtain a two-particle irreducible (2PI) effective action within the contourtime formalism that allows for a description of both equilibrium and out-of-equilibrium phenomena.We derive equations of motion for both the superfluid order parameter and the single-particle many-body Green's functions.First, we assess the accuracy of our formalism by studying the equilibrium solution for the homogeneous BHM and comparing our results to existing strong-coupling methods as well as to exact methods where possible.We then consider homogeneous systems that are initially thermalized in the Mott phase, and which are then subjected to quenches.We solve numerically the equations of motion for this scenario and calculate the single-particle density matrix.We demonstrate a Lieb-Robinson-like maximal propagation velocity for the spreading of single-particle correlations in one, two, and three dimensions.We study the dependence of the maximal propagation velocity on the quench protocol, chemical potential, temperature, and dimensionality.We compare our results to exact methods, existing strong-coupling approaches, and experiments where possible.Lastly, we extend our strong-coupling approach to the disordered BHM and derive equations of motion for the disorder-averaged single-particle Green's function.We discuss how these equations of motion can be used to study the phase stability of many-body localization in the disordered BHM for dimensions higher than one.
This thesis concerns the physics of ultracold bosons in quasicrystalline lattices, studied by means of quantum simulation using ultracold 39K atoms loaded to an eight-fold rotationally symmetric optical quasicrystal. Quasicrystalline and quasiperiodic systems, thanks to their long-range order and lack of periodicity, exhibit a wealth of interesting properties such as a dense and self-similar momentum space, and a localization transition to Anderson insulator or Bose glass states (in non-interacting or interacting cases respectively) reminiscent of that in disordered systems, whilst lacking the rare regions that typically exist in such systems. Here we will cover experiments into this system’s behaviour in both equilibrium and non-equilibrium settings. First, we focus on mapping out its phase diagram in both weakly and strongly interacting regimes. We observe consistency with numerical predictions for the superfluid-Bose glass phase transition using coherence measurements, as well as finding a region of roughly zero compressibility around where the numerics predicts the existence of a Mott insulator ground state. We also study which regimes of the phase diagram can be ramped through adiabatically, finding that entropy is created upon traversing the Bose glass phase irrespective of ramp duration, which we attribute to its localized nature. Next we look into the dynamics of the system after a quench of the lattice depth in the weakly interacting regime. Upon quenching from the superfluid regime to the Bose glass regime, we observe a quantum walk to higher order diffraction peaks in momentum space whose maximum order is unbounded by kinetic energy thanks to the dense, self-similar nature of the quasicrystal’s momentum space. Quenching in the opposite direction we observe instead a walk between sites in real space for short times by looking at the position (as opposed to momentum) space autocorrelation function of the cloud, finding an expansion rate in agreement with an intuitive estimate based on the Lieb-Robinson bounds of a periodic system. We also look into the timescales of these dynamics in the context of what we might expect from the underlying Hamiltonian, before finally investigating the loss of coherence exhibited by the system after a slower quench of the lattice depth from an initially superfluid cloud. We observe that coherence is lost sooner after the quench for deeper final lattices, with the loss becoming almost instantaneous for quenches that cross the localization transition across the whole cloud. After this we explore the system’s transport dynamics by means of a quench to an excited state at the same time as an additional confining potential is switched off. We observe that unlike in the case of a periodic square lattice, where ballistic expansion that scales with the tunnelling strength occurs in the non-interacting case before becoming slower upon the introduction of interactions thanks to diffusion, in the quasicrystal the transport rate no longer scales with tunnelling strength. Instead we observe a drop relative to the tunnelling rate by 2 − 3 orders of magnitude as the lattice depth is increased from 1Er to 5Er with little dependence on the interaction strength. We also note that the finite rate measured for the deeper lattices is potentially an overestimate since the finite loss rate of atoms makes mass transport indistinguishable from density-dependent losses at that point. Finally, we discuss use of a quantum gas magnification scheme with which to magnify the real space density distribution of our atoms such that it can be measured with single-site resolution. Some preliminary results are presented regarding how the density distribution changes with lattice depth in the weakly interacting regime, in which we observe that as lattice depth increases a higher proportion of atoms occupy deeper sites close to the centre of the quasicrystal’s configuration space, as we might expect.
The study of out-of-equilibrium dynamics in ultra-cold gases is a new and exciting field, driven largely by the recent experimental advances in controlling and imaging cold clouds. The experimental and theoretical work thus far has been somewhat exploratory and largely numerical in nature, as the very paradigms for thinking about these systems are not well established. In this thesis I consider several different scenarios of ultra-cold bosonic and fermionic gases driven out of equilibrium and study their properties. In Chapter 1, I provide an overview of the phenomenology of ultra-cold gases, highlighting the timescales governing these systems and how the experimentalist can tune them. I discuss how cold gases can be cooled and trapped and discuss the basic physics behind optical lattices. I also discuss experimental probes of these gases, in particular the new high resolution imaging techniques developed recently at Chicago, Munich and Harvard. In Chapter 2, I discuss an early experiment (circa. 2008) which observed long lived spin dynamics in a thermal spin-1/2 Fermi gas. This experiment is an nice illustration of interesting physics resulting from the separation of timescales between spin and collisional dynamics. In my opinion, it is an excellent example of why cold gases are naturally suited to studying non-equilibrium dynamics. I simulate the experiment numerically using a collisionless Boltzmann equation and explain the observed spin dynamics both qualitatively and quantitively. In Chapter 3, I continue the discussion of spin waves in thermal gases by extending previous works on spin-1/2 gases to spin-1 Bose gases. In contrast to Chapter 2, the bulk of the work in this Chapter is analytic in nature. In particular, I find a spin wave instability in the thermal spin-1 Bose gas, which is the high temperature analog of the polar to ferromagnetic transition in a spin-1 Bose Einstein condensate. In Chapter 3, I turn my attention to bosonic systems and briefly review the the Bogoliubov mean-field theory. I calculate the momentum distribution and density-density correlation function of an interacting Bose gas within the Bogoliubov framework. Then I consider bosons in an optical lattice, and introduce the Bose Hubbard model. I calculate the mean-field phase diagram of the Bose Hubbard model and then consider fluctuations about the mean field, and derive the excitation spectrum of the lattice gas in the superfluid and insulating regimes. In Chapter 4, I ask what we learn by studying the dynamics of correlation functions following a sudden change in the interactions in a superfluid. Using the Bogoliubov theory developed in Chapter 3, I will show how the underlying excitation spectrum influences the long and short time behavior of the correlation functions. By considering a lattice dispersion, I study the analogous problem in a weak optical lattice and discuss how the lattice dispersion leads to additional features in the correlation functions. I will also discuss the timescale governing the revival of the condensate fraction in a quantum depleted gas. In Chapter 5, I derive equations of motion governing the dynamics of one and two body correlation functions in the single-band Bose Hubbard model, applicable to bosons in deep lattices. I then consider a simple quench from a Mott insulating initial state to a weakly interacting final state and produce analytic expressions describing the dynamics of correlations following such a quench. I discuss the timescale for the development of long range order following such a quench. I study the problem of chapter 4 using an equations of motion approach. This approach complements the Bogoliubov approach of Chapter 4. First, I derive exact expressions for a quench to a non-interacting state. I then consider how interactions redistribute quasi-momentum to first order in perturbation theory in different dimensions. In Chapter 6, I calculate the relevant timescales for local and global dynamics in trapped lattice Bose gases, a work done in collaboration with Dr. Kaden R. A Hazzard. Using a time-dependent Gutzwiller mean-field theory, I show that the timescale for local equilibration in these systems is fast in experimental terms. I then show that due to the spatial inhomogeneities inherent to cold gases, achieving global equilibrium can be quite complicated, sometimes taking longer than the lifetime of the experiment, an issue of practical importance to current day experiments. I continue this discussion in Chapter 7 which is a collaborative work with experimentalists David McKay and Prof. Brian DeMarco from the University of Toronto and the University of Illinois, Urbana Champaign. Using experimental and numerical methods, we show that the rapid timescales for local dynamics in interacting systems invalidates a frequently used cold atom technique for mapping out the momentum distribution of atoms in an optical lattice.
"This thesis presents research on a variety of dynamical phenomena in strongly correlated systems. We use 87Rb atoms trapped in a disordered optical lattice to realize the 3D Bose Hubbard model, with independently, dynamically tunable lattice depth and disorder strength. The equilibrium behavior of the Bose Hubbard and disordered Bose Hubbard models are understood, but there are many outstanding questions about dynamics. One area of interest is in dynamical behavior near quantum phase boundaries, such as the generation of quench-induced excitations in the presence of disorder. Another area concerns heat flow, relaxation, and thermalization. The primary focus of this thesis is driving a gas across a disorder-induced quantum phase transition from the Bose glass to the superfluid phase via quenching the disorder. We measured excitations generated during the quench and related them to crossing the equilibrium phase transition determined by quantum Monte Carlo simulations of the trapped system. The behavior we observe is reminiscent of the quantum Kibble Zurek mechanism, where the relaxation timescale diverges and local fluctuations are ""frozen"" into the system near the phase boundary. Understanding how disorder impacts the Kibble Zurek paradigm about behavior near a phase boundary provides insight into the dynamics of strongly interacting, disordered systems. The three other works discussed in this thesis concern relaxation and energy transfer in clean optical lattices. Understanding these dynamical properties of lattice gases is key to developing tools such as cooling in a lattice. In the first result, a gas comprised of two spin states was loaded into a spin-dependent optical lattice, such that one spin state experienced the lattice, and the other did not. We found that when the lattice gas was heated, energy transfer to the non-lattice gas did not occur at higher lattice depths due to the mismatch in the dispersion relations. In the second result, lattice atoms were excited to a higher band and found to decay to the ground state via two mechanisms. One decay channel involved generating excitations in the non-lattice atoms, while the other channel involved collisions between lattice atoms, where one atom decayed to the ground state and the other was excited to a higher band. The final result measured relaxation timescales by removing atoms with low quasimomentum and measuring rethermalization. The observed timescales are faster than the tunneling time and interaction time of the lattice. Proof-of-principle cooling was also demonstrated by removing atoms with higher quasimomentum and allowing the remaining atoms rethermalize to a cooler temperature."
This thesis describes experiments focused on investigating out-of-equilibrium phenomena in the Bose-Hubbard Model and exploring novel cooling techniques for ultracold gases in optical lattices. In the first experiment, we study quenches across the Mott-insulator-to-superfluid quantum phase transition in the 3D Bose-Hubbard Model. The quench is accomplished by continuously tuning the ratio of the Hubbard energies. We observe that the degree of excitation is proportional to the fraction of atoms that cross the phase boundary, and that the amount of excitations and energy produced during the quench have a power-law dependence on the quench rate. These phenomena suggest an excitation process analogous to the mechanism for defect generation in non-equilibrium classical phase transitions. This experiment constitutes the first observation of the Kibble-Zurek mechanism in a quantum quench. We have reported our findings in Ref. [1]. In a second experiment, published in Ref. [2], we investigate dissipation as a method for cooling a strongly interacting gas. We introduce dissipation via a bosonic reservoir to a strongly interacting bosonic gas in the Mott-insulator regime of a 3D spin-dependent optical lattice. The lattice atoms are excited to a higher energy band using laser-induced Bragg transitions. A weakly interacting superfluid comprised of atoms in a state that does not experience the lattice potential acts as a dissipative bath that interacts with the lattice atoms through collisions. We measure the resulting bath-induced decay using the atomic quasimomentum distribution, and we compare the decay rate with predictions from a weakly interacting model with no free parameters. A competing intrinsic decay mechanism arising from collisions between lattice atoms is also investigated. The presence of intrinsic decay can not be accommodated within a non-interacting framework and signals that strong interactions may play a central role in the lattice-atom dynamics. The intrinsic decay process we observe may negatively impact the success of cooling via dissipation because a fraction of intrinsic decay events can deposit a large amount of energy into the lattice gas. In a third experiment, we develop and carry out the first demonstration of cooling an atomic quasimomentum distribution. Our scheme, applied in a proof-of-principle experiment to 3D Bose-Hubbard gas in the superfluid regime, involves quasimomentum-selective Raman transitions. This experiment is motivated by the search of new cooling techniques for lattice-trapped gases. Efficient cooling exceeding heating rates is achieved by iteratively removing high quasimomentum atoms from the lattice. Quasimomentum equilibration, which is necessary for cooling, is investigated by directly measuring rethermalization rates after bringing the quasimomentum distribution of the gas out of equilibrium. The measured relaxation rate is consistent at high lattice depths with a short-range, two-particle scattering model without free parameters, despite an apparent violation of the Mott-Ioffe-Regel bound. Our results may have implications for models of unusual transport phenomena in materials with strong interactions, such as heavy fermion materials and transition metal oxides. The cooling method we have developed is applicable to any species, including fermionic atoms. Our results are available in Ref. [3]