Finite systems of bosons and/or fermions described by the Hubbard model can be realized using ultracold atoms confined in optical lattices. The ground states of these systems often exhibit a coexistence of compressible superfluid and incompressible Mott insulating domains. We analyze such systems by studying the out-of-equilibrium dynamics following a weak sudden quench of the trapping potential. In particular, we show how the temporal variance of the site occupations reveals the location of spatial boundaries between compressible and incompressible regions. The feasibility of this approach is demonstrated for several models using numerical simulations. We first consider integrable systems, hard-core bosons (spinless fermions) confined by a harmonic potential, where space-separated Mott and superfluid phases coexist. Then, we analyze a nonintegrable system, a $J\text{\ensuremath{-}}V\text{\ensuremath{-}}{V}^{\ensuremath{'}}$ model, with coexisting charge density wave and superfluid phases. We find that the temporal variance of the site occupations is a more effective measure than other standard indicators of phase boundaries such as a local compressibility. Finally, to make contact with experiments, we propose a consistent estimator for such temporal variance. Our numerical experiments show that the phase boundary is correctly spotted using as little as 30 measurements. Based on these results, we argue that analyzing temporal fluctuations is a valuable experimental tool for exploring phase boundaries in trapped atom systems.
Finite systems of bosons and/or fermions described by the Hubbard model can be realized using ultracold atoms confined in optical lattices. The ground states of these systems often exhibit a coexistence of compressible superfluid and incompressible Mott insulating domains. We analyze such systems by studying the out-of-equilibrium dynamics following a weak sudden quench of the trapping potential. In particular, we show how the temporal variance of the site occupations reveals the location of spatial boundaries between compressible and incompressible regions. The feasibility of this approach is demonstrated for several models using numerical simulations. We first consider integrable systems, hard-core bosons (spinless fermions) confined by a harmonic potential, where space-separated Mott and superfluid phases coexist. Then, we analyze a nonintegrable system, a J\ensuremath-V\ensuremath-V\ensuremath′ model, with coexisting charge density wave and superfluid phases. We find that the temporal variance of the site occupations is a more effective measure than other standard indicators of phase boundaries such as a local compressibility. Finally, to make contact with experiments, we propose a consistent estimator for such temporal variance. Our numerical experiments show that the phase boundary is correctly spotted using as little as 30 measurements. Based on these results, we argue that analyzing temporal fluctuations is a valuable experimental tool for exploring phase boundaries in trapped atom systems.
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.
Confinement of atoms by laser technology has opened up a new field of physics. Following the realization of Bose– Einstein condensation (BEC) of bosonic atoms, fermionic atom systems and even boson–fermion mixture systems were successfully trapped and cooled. Application of a periodic potential, so called optical lattice, could also unveil interesting nature of the atoms. One of the interesting phenomena of the atoms on optical lattice is Mott transition. Although there is a long history of the study on bosons or fermions on lattice, coexistence of bosons and fermions on lattice is a rather new topic to study. In particular, we have little knowledge about strongly-correlated bosons and fermions on lattice. In our previous works we performed numerical simulations of Bose–Fermi mixtures to clarify the phase diagram and the effect of the confinement potential. When the total number of bosonic and fermionic atoms per site takes an integer value and interatomic interactions are sufficiently strong, the system would undergo Mott transition where the local atom density is fixed to the integer value. A characteristic of this Mott insulating phase is that a boson and fermion located next to each other can exchange their positions by going through a virtual doubly-occupied state with the atom exchange energy t ’ tbtf=Ubf where tb and tf are the hopping energy of the bosons and fermions respectively and Ubf denotes the on-site interaction between the boson and the fermion. Although there still remain a number of open questions on the Mott state of Bose–Fermi mixtures, the property of superfluid state of the strongly-correlated bosons and fermions has yet to be revealed. In this short note we would like to focus on strongly-correlated Bose–Fermi mixtures at commensurate filling in a three-dimensional periodical potential and see characteristic behaviors, if any, of the superfluid phase close to the phase boundary to the Mott insulating phase. In the following analysis we assume uniformity of the system and thus neglect the possibility of the emergence of so-called supersolid where superfluidity and structural long-range order coexist. For simplicity we ignore all the internal degree of freedom of the atoms and the influence of the trap potential which in fact creates a nonuniform density profile of the bosons and fermions. For the system at commensurate filling on optical lattice, we assume that the interatomic interactions are weak enough to avoid formation of Mott state but strong enough to allow us to expect that there are only few empty or doublyoccupied sites and most sites are singly occupied. Namely we analyze the system that stays in a superfluid phase but lies close to the boundary to the Mott phase. Since a boson and fermion located next to each other can exchange their positions as stated above, the energy of the superfluid state of the mixture system is given mostly by the kinetic energy of the bosons and fermions exchanging their positions. We therefore express the Hamiltonian in the following form, ignoring the kinetic energy of the atoms at the empty sites and the interaction energy at the doubly-occupied sites:
We study the ground state properties of ultracold fermions trapped in a one-dimensional double-well optical lattice by using the density-matrix renormalization group method. The system is described by an extended Hubbard model with alternating hopping integrals and an external harmonic confining potential. We clarify the characteristic features of the metal–insulator phase transition originating from the double-well structure of the lattices. Several different types of insulating regions coexist when the number of atoms at each site is an integer or a half integer. We found that each insulating phase except for a band insulator exhibits rather large local density fluctuations, reflecting the strong dimerization of atoms within the unit cells of double-well lattices. The phase characteristics are elucidated in detail by investigating the profiles of the local density of atoms, the local density (spin) fluctuations, the double-occupancy probability and the spatially extended sp in correlations.
We characterize the Mott insulating regime of a repulsively interacting Fermi gas of ultracold atoms in a three-dimensional optical lattice. We use in situ imaging to extract the central density of the gas and to determine its local compressibility. For intermediate to strong interactions, we observe the emergence of a plateau in the density as a function of atom number, and a reduction of the compressibility at a density of one atom per site, indicating the formation of a Mott insulator. Comparisons to state-of-the-art numerical simulations of the Hubbard model over a wide range of interactions reveal that the temperature of the gas is of the order of, or below, the tunneling energy scale. Our results hold great promise for the exploration of many-body phenomena with ultracold atoms, where the local compressibility can be a useful tool to detect signatures of different phases or phase boundaries at specific values of the filling.
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.
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. Since due to the confining potential the density is space dependent, Mott-insulating domains always coexist with metallic regions, such that global quantities are not appropriate to describe the system. We define a local compressibility that characterizes the Mott-insulating regions and analyze other local quantities. It is shown that the momentum distribution function, a quantity that is commonly considered in experiments, fails in giving a clear signal of the Mott-insulating transition. Furthermore, we analyze a mean-field approach to these systems and compare it with the numerically exact QMC results. Finally, we determine a generic form for the phase diagram that allows us to predict the phases to be observed in the experiments.