We discuss in this paper ground‐state and nonequilibrium properties of ultracold atoms in optical lattices in the strongly correlated limit. We review fermionic Mott‐insulators studied on the basis of quantum Monte Carlo simulations, where local quantum criticality is displayed in one dimension. We continue with exact results for hard‐core bosons in one dimension, showing their universal properties in equilibrium, and their nonequilibrium dynamics. Here we show that starting from a Fock state, a quasi‐condensate emerges at finite momentum during free expansion. On the other hand, the free evolution of an initially confined quasi‐condensate of hard‐core bosons leads to a bosonic gas with a Fermi edge, and hence a fermionization that can only be obtained out of equilibrium.
We discuss in this paper ground‐state and nonequilibrium properties of ultracold atoms in optical lattices in the strongly correlated limit. We review fermionic Mott‐insulators studied on the basis of quantum Monte Carlo simulations, where local quantum criticality is displayed in one dimension. We continue with exact results for hard‐core bosons in one dimension, showing their universal properties in equilibrium, and their nonequilibrium dynamics. Here we show that starting from a Fock state, a quasi‐condensate emerges at finite momentum during free expansion. On the other hand, the free evolution of an initially confined quasi‐condensate of hard‐core bosons leads to a bosonic gas with a Fermi edge, and hence a fermionization that can only be obtained out of equilibrium.
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.
We summarize in the present work exact results obtained for Tonks-Girardeau gases on one-dimensional optical lattices both for the ground state and nonequilibrium dynamics. On the theoretical side, impenetrable bosons offer the opportunity to study strongly interacting systems in one-dimensional lattices exactly, by means of the Jordan-Wigner transformation, and hence contribute to the topic of strong correlations at the center of interest in both condensed matter physics and quantum gases. This motivation is further enhanced by recent experimental realizations of such systems with ultracold atoms. After having shown their universal properties in equilibrium, we concentrate on their nonequilibrium dynamics. It will be shown that, starting from a pure Fock state, quasi-long-range correlations develop dynamically and lead to the formation of quasicondensates with a momentum determined by the underlying lattice. We expect this effect to be relevant for atom lasers with full control of the wavelength. Then, we will show that the free evolution of an initially confined Tonks-Girardeau gas leads to a momentum distribution that approaches at long times that of the equivalent fermionic system, giving rise to a bosonic gas with a Fermi edge, and hence a fermionization that can only be obtained out of equilibrium. Remarkably, although the momentum distribution function of the Tonks-Girardeau gas becomes equal to the one of the fermions, no loss in coherence is observed in the system, as reflected by a large occupation of eigenstates of the one-particle density matrix.
Using quantum Monte Carlo (QMC) simulations we study the ground state properties of the one‐dimensional fermionic Hubbard model in traps with an underlying lattice. We show that this model displays quantum critical behavior at the boundaries of the local Mott‐insulating regions. A local compressibility defined to characterize the Mott‐insulating phase has a non‐trivial critical exponent. Both the local compressibility and the variance of the local density show universality with respect to the confining potential. We study the momentum distribution function and we find that it is not appropriate to characterize the transitions in the system.
We study the free expansion of Tonks‐Girardeau gases on one‐dimensional lattices after removing the confining potential. We find that for low initial densities the momentum distribution of the expanding gas rapidly approaches the one of noninteracting fermions. In the opposite limit, i.e, when the initial system is a pure Mott insulator with one particle per lattice site, the expansion leads to the emergence of quasi‐condensates at finite momentum. These traveling quasi‐condensates develop quasi‐long range one‐particle correlations similar to the ones shown to be universal in the equilibrium case.
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.
We study properties of ultracold bosonic atoms in one-, two-, and three-dimensional optical lattices by large scale quantum Monte Carlo simulations of the Bose-Hubbard model in parabolic confining potentials. Our results indicate that local properties of the atoms can be accessed by probing the system's response to local potential perturbations. Furthermore, we show how the formation of Mott insulating regions is reflected in the momentum distribution of the atoms, amenable to experimental detection. We disprove previous claims concerning the relevance of fine structure in the momentum distribution function. Furthermore, we discuss limitations of local density approximations for confined systems, and demonstrate the absence of quantum criticality due to the inhomogenous potential. Instead, we show that quantum critical behavior can be observed in flat confining potentials. Our results indicate that the experimental detection of the Mott transition in moderately sized optical lattices would be significantly eased in flat confinement potentials.