This dissertation presents my research covering the field of ultracold atoms loaded in optical lattices. The static and dynamical properties of atoms in combined periodic and parabolic potentials are studied, with a focus on the strongly interacting regimes. Because parabolic magnetic and optical potentials are routinely used to confine atoms, the results of this research are directly relevant to ongoing experimental endeavours in atomic physics. After a review of the basic theory of atoms in homogeneous periodic potentials, the equilibrium and non-equilibrium properties of non-interacting and interacting atoms in periodic plus parabolic potentials are studied. The problem of the localization of the many-body wavefunction for systems with arbitrary peak onsite density is presented in Chapters 3 and 4. The physics pertaining to the experimental realization of Mott insulator states with one or more atoms per sites in inhomogeneous lattices is elucidated by introducing an intuitive model for strongly interacting bosons in one dimension. This model is then utilized to study the decay of the dipole oscillations of atomic ensembles subject to a small displacement of the parabolic potential. Good agreement is found with results of recent experiments. Chapters 5 and 6 are dedicated to the characterization of the Mott insulator state with unit filling, which plays a central role in proposed schemes for neutral atom quantum computation. The usefulness of Bragg spectroscopy to probe the excitation spectrum of the Mott state in homogeneous lattices is analyzed in Chapter 5, where the limits of validity of linear response theory in this strongly correlated regime are delimited. In Chapter 6 the effects of finite temperature on the confined Mott insulator state are studied, and a scheme is devised for possibly estimating the system's temperature, at energies of the order of the inter-particle interaction energy. Finally, in Chapter 7, a proposal is introduced to utilize the Mott state as a robust register for neutral atom quantum computation. Unwanted residual quantum coherences inherent to the Mott insulator ground state are eliminated by a judicious choice of the trapping potentials and a selective measurement on a molecular photo-associative transition.
This dissertation presents my research covering the field of ultracold atoms loaded in optical lattices. The static and dynamical properties of atoms in combined periodic and parabolic potentials are studied, with a focus on the strongly interacting regimes. Because parabolic magnetic and optical potentials are routinely used to confine atoms, the results of this research are directly relevant to ongoing experimental endeavours in atomic physics. After a review of the basic theory of atoms in homogeneous periodic potentials, the equilibrium and non-equilibrium properties of non-interacting and interacting atoms in periodic plus parabolic potentials are studied. The problem of the localization of the many-body wavefunction for systems with arbitrary peak onsite density is presented in Chapters 3 and 4. The physics pertaining to the experimental realization of Mott insulator states with one or more atoms per sites in inhomogeneous lattices is elucidated by introducing an intuitive model for strongly interacting bosons in one dimension. This model is then utilized to study the decay of the dipole oscillations of atomic ensembles subject to a small displacement of the parabolic potential. Good agreement is found with results of recent experiments. Chapters 5 and 6 are dedicated to the characterization of the Mott insulator state with unit filling, which plays a central role in proposed schemes for neutral atom quantum computation. The usefulness of Bragg spectroscopy to probe the excitation spectrum of the Mott state in homogeneous lattices is analyzed in Chapter 5, where the limits of validity of linear response theory in this strongly correlated regime are delimited. In Chapter 6 the effects of finite temperature on the confined Mott insulator state are studied, and a scheme is devised for possibly estimating the system's temperature, at energies of the order of the inter-particle interaction energy. Finally, in Chapter 7, a proposal is introduced to utilize the Mott state as a robust register for neutral atom quantum computation. Unwanted residual quantum coherences inherent to the Mott insulator ground state are eliminated by a judicious choice of the trapping potentials and a selective measurement on a molecular photo-associative transition.
This thesis covers most of my work in the field of ultracold atoms loaded in optical lattices. It makes a route through the physics of cold atoms in periodic potentials starting from the simple noninteracting system and going into the many-body physics that describes the strongly correlated Mott insulator regime. Even though this thesis is a theoretical work all the chapters are linked either with experiments already done or with ongoing experimental efforts. This thesis can be divided into four different parts. The first part comprises chapters 1 to 3. In these chapters, after a brief introduction to the field of optical lattices I review the fundamental aspects pertaining to the physics of systems in periodic potentials. The second part deals with the superfluid weakly interacting regime where standard mean field techniques can be applied. This is covered in chapters 4 and 5. Specifically, chapter 4 introduces the discrete nonlinear Schroedinger equation (DNLSE) and uses it to model some experiments. In chapter 5 I go one step further and include the small quantum fluctuations neglected in the DNLSE by studying quadratic approximations of the Bose-Hubbard Hamiltonian. Chapters 6 to 8 can be grouped as the third part of the thesis. In them I adopt an effective action formalism, the so called two particle irreducible effective action (2PI) together with the closed time path (CTP) formalism to study far-from-equilibrium dynamics. The many-body techniques discussed in these chapters systematically include higher order quantum corrections, not included in the quadratic approximations of the Hamiltonian, which we show are crucial for a correct description of the quantum dynamics outside the very weakly interacting regime. Finally, chapter 9 to 11 are devoted to study the Mott insulator phase. In these chapters using perturbation theory I study the Mott insulator ground state and its excitation spectrum, the response of the system to Bragg spectroscopy, and propose a mechanism to correct for the residual quantum coherences inherent to the Mott insulator ground state. Even though small these are not ideal for the use of neutral atoms in optical lattice as a tool for quantum computation.
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.
In the present thesis we study several schemes where the detection of the emitted light from an atomic ensemble gives information about the state of the atoms. In chapter 1 we review the basic theory for the light matter interaction in second quantized form. In chapter 2 we show how conservative potentials for the atoms, such as dispersive optical lattices, can be realized experimentally due to the dispersive interaction with a far detuned laser We then review some basic properties of the physics of a particle in a periodic potential. We show that the ground state and the lowest lying excitations of a Bose-Einstein condensate in a dispersive optical lattice can be described by the Bose-Hubbard Hamiltonian. In chapter 3 we study the photonic bandstructure of a chain of fixed pointlike atoms in a biperiodic configuration. We calculate the photonic spectra and the probe transmission if the atoms are in free space and inside a standing wave optical resonator. In chapter 4 we consider light scattering from ultracold atoms in an optical lattice where we calculate the photonic scattering cross section as a function of energy and direction of emission along the Mott-insulator-superfluid phase transition. We take into account the finite tunneling rate for the atoms, when evaluating the scattering cross section of the photons and the effect of photon recoil onto the atoms. In chapter 5 we consider a gas of ultracold atoms trapped at two spatially separated regions in space. Using a two Raman lasers to couple atoms out of the systems we show that it is possible to measure the mean value of the atomic field operator of the two systems by measuring the scattered light intensity of one of the Raman lasers. We show how this setup might be used to measure the temperature of a Bose-Einstein condensate and to monitor the superfluid to Mott-insulator phase transition for a gas of ultracold atoms in an optical lattice.
We discuss interacting and noninteracting one dimensional atomic systems trapped in an optical lattice plus a parabolic potential. We show that, in the tight-binding approximation, the noninteracting problem is exactly solvable in terms of Mathieu functions. We use the analytic solutions to study the collective oscillations of ideal bosonic and fermionic ensembles induced by small displacements of the parabolic potential. We treat the interacting boson problem by numerical diagonalization of the Bose-Hubbard Hamiltonian. From analysis of the dependence upon lattice depth of the low-energy excitation spectrum of the interacting system, we consider the problems of ``fermionization'' of a Bose gas, and the superfluid-Mott insulator transition. The spectrum of the noninteracting system turns out to provide a useful guide to understanding the collective oscillations of the interacting system, throughout a large and experimentally relevant parameter regime.
Motivated by the recent rapid development of the field of quantum gases in optical lattices, we present a comprehensive study of the spectrum of ultracold atoms in a one-dimensional optical lattice subjected to a periodic lattice modulation. Using the time-dependent density matrix renormalization group method, we study the dynamical response due to lattice modulations in different quantum phases of the system with varying density. For the Mott-insulating state, we identify several excitation processes, which provide important information about the density profile of the gases. For the superfluid, the dynamical response can be well described in a local density approximation. This simplification can be valuable in understanding the strong-correlated superfluid in a slow-varying harmonic potential. All these spectroscopic features of an inhomogeneous system can be used as a test for the validity of the Bose-Hubbard model in a parabolic trapping potential.
We study properties of ultracold quantum gases trapped in 1D and under the influence of an underlying optical lattice. We analyze both fermionic and bosonic systems following different numerically exact approaches, like quantum Monte Carlo for soft-core bosons and two-species fermions, and a recently developed exact approach for HCB. The exposition is organized as follows. In Chap. 2, we study single species degenerated fermions on 1D lattices. We show that a splitting of the system takes place in a region of the spectrum with eigenstates that have a non-vanishing weight only in a fraction of the system. We also show that if on top of the lattice an alternating potential is introduced, doubling the original periodicity, an additional slicing' of the system can be achieved. The width and number of such regions can be controlled in a given energy range by the amplitude of the new modulation. By filling these systems the ground state may exhibit insulating regions due to the presence of local gaps, which can be observed in the local density of states. As a complement to these 1D results we present in Appendix A results obtained in 2D lattices. In Chap. 3 we review in some detail the two quantum Monte Carlo (QMC) techniques that are relevant to our study of confined soft-core bosons and two-species fermions on optical lattices, the worldline QMC and the zero temperature projector QMC, respectively. The basis of the classical Monte Carlo method are also presented, included a proof of the Central Limit theorem (Appendix B) that establishes its foundations. Properties of soft-core bosons confined on optical lattices are presented in Chap. 4. QMC simulations show that due to the presence of the slowly varying confining potential the concept of commensurability, typical for periodic systems, loses its meaning. Mott domains appears for a continuous range of fillings and always coexist with superfluid phases, as it was mentioned before. The latter feature is reflected by the behavior of the global compressibility, which never vanishes. A local compressibility, proportional to the variance of the local density, is defined in order to characterize the local Mott-insulating phases. Finally, a phase diagram for trapped systems is also presented. In Chap. 5 we study the ground state properties of two-components fermions confined on optical lattices. Like for the bosonic case, Mott domains also appear for a continuous range of fillings and always coexist with compressible phases. We define a local order parameter (that we denominate local compressibility), which characterizes the Mott insulating phases in an unambiguous way. By means of this local compressibility, we study in detail the interphase between the metallic and insulating region finding that critical behavior sets in revealing a new critical exponent. Furthermore, the behavior of the local compressibility and the variance of the density are found to be universal in this case, independently of the confining potential and the strength of the interaction, as usual in critical phenomena. We present in Chap. 6 a recently developed exact numerical approach that allows to study ground state properties of HCB confined on 1-D lattices. This exact treatment is applied to study the off-diagonal behavior of the one-particle density matrix (OPDM) and related quantities in the equilibrium case. We find that the OPDM decays as a power-law x^-1/2 for large-x, irrespective of the confining potential chosen. The power-law above is shown to determine the scaling of the occupation of the lowest natural orbital in the thermodynamic limit. This scaling and its finite size corrections are also studied for arbitrary powers of the confining potential. In addition, we find a power-law decay of the NO occupations at low densities, and show that its exponent is also universal. The low density limit in the lattice, equivalent to continuous systems, is also analyzed in detail. The approach above is generalized in Chap. 7 to study the non-equilibrium dynamics of HCB in 1-D configurations with an underlying lattice. The presence of the lattice enables the creation of pure Fock states of HCB (a HCB per lattice site) where there is no coherence in the system. We show that quasi-long range correlations develop in the equal-time-one-particle density matrix when such states are allowed to evolve freely, and that they lead to the formation of quasi-condensates of HCB at finite momentum. In addition,we obtain an universal power-law describing the population of the quasi-condensate as a function of time. Finally, we discuss how such systems can be used to create atom lasers with a wave-length that can be controlled through the lattice parameter.