In this work numerical methods for solving the problem of two ultracold correlated fermionic 6Li atoms in a one-dimensional optical dipole trap are described. The underlying quantum mechanical model contains the typical approximation of a pointlike interaction for ultracold gases and the exact shape of the trap potential, which is known from the experiment. Using these methods, in which solving the time-dependent Schrödinger equation plays a central role, it was possible to obtain the characteristical energy spectrum dependent on the interaction strength and the significant eigenstates. On the basis of these results typical features of a transition called “Fermionization“ have been observed and reference values could be determined. In addition latest tunneling experiments were simulated, which are used to prove the Fermionization. For this purpose the decay characteristics of the prepared two-particle states in the trap potential with a potenial barrier were observed.
In this work numerical methods for solving the problem of two ultracold correlated fermionic 6Li atoms in a one-dimensional optical dipole trap are described. The underlying quantum mechanical model contains the typical approximation of a pointlike interaction for ultracold gases and the exact shape of the trap potential, which is known from the experiment. Using these methods, in which solving the time-dependent Schrödinger equation plays a central role, it was possible to obtain the characteristical energy spectrum dependent on the interaction strength and the significant eigenstates. On the basis of these results typical features of a transition called “Fermionization“ have been observed and reference values could be determined. In addition latest tunneling experiments were simulated, which are used to prove the Fermionization. For this purpose the decay characteristics of the prepared two-particle states in the trap potential with a potenial barrier were observed.
In recent experiments, carried out at the University of Heidelberg, tunneling rates of ultracold distinguishable fermions out of an optomagnetical trap have been measured. The fermions interact by a tunable short-ranged interaction,\nand the trap is asymmetric making the trapped quantum system\neffectively one-dimensional. In this thesis, a method for calculating the energy levels and tunneling rates of one and two interacting particles out of a very general onedimensional potential well is devised. The method is based on expanding the Schrodinger equation of the system in a complex-momentum basis. This is done utilizing the so-called Berggren completeness relation. Ultimately,\nthe basis expansion leads to a complex symmetric non-Hermitian eigenvalue problem for a large, dense matrix.\nThe general method is applied to a system of trapped, ultracold fermionic atoms in a setup that closely resembles the Heidelberg experiments. The short-ranged interaction is modeled as a point-interaction, and the trap potential\nis regularized at large distances from the interesting region. The obtained energies and decay rates are contrasted to results obtained using the Wentzel-Kramers-Brillouin (WKB) approximation. Notable differences\ncan be observed, and these may be due to insufficiency of the WKB approximation to accurately describe the system in question.
Ultrakalte Atome in einer Dimension: von zwei hin zu vielen Teilchen: In dieser Dissertation werden ultrakalte bosonische und fermionische Systeme in Hinblick auf ihre stationaren sowie dynamischen Eigenschaften in eindimensionalen harmonischen Fallen und endlichen Gittern untersucht. Alle physikalischen Systeme werden im Rahmen dieser Arbeit mit einem “Bottom-up-Ansatz” behandelt, ausgehend von dem zugrunde liegenden Zwei-Korper-Problem. Die dem hier entwickelten “correlated-pair wavefunction” Ansatz zugrunde liegende Idee verwendet die exakte Losung des entsprechenden Zwei-Korper-Problems fur die Konstruktion der korrelierten Vielteilchenwellenfunktion. Fur Bosonen und Fermionen in der harmonischen Falle beschreibt der “correlated-pair wavefunction”-Ansatz sehr gut den Ubergang von schwacher zur starker Wechelwirkung. Auserdem werden bosonische Systeme mit zeitlich getriebener Wechselwirkung in Hinblick auf resonante Anregungsdynamik untersucht, wobei fur deren Analyse die Eigenschaften des Zwei-Korper-Spektrums und des entsprechenden Vielteilchenspektrums benutzt werden. Endliche Gitter mit kommensurabeler und inkommensurabeler Fullung werden mit Schwerpunkt auf “on-site” Effekten, welche aus starken Wechselwirkungen jenseits der Gultigkeit des “Single-Band-Ansatzes” resultieren, studiert. Die “correlated-pair wavefunction” wird auf Gitterfallen verallgemeinert, um solche Effekte behandeln zu konnen. Der “Interband”-Tunnelmechanismus tritt in endlichen Gittern auf, weil Resonanzen im Vielteilchenspektrum bei bestimmter Wechselwirkungstarke den “Self-Trapping”-Mechanismus uberwinden. Mit Hilfe der analytischen Funktionen werden Eigenschaften von Observablen berechnet und mit numerischen Ergebnissen der etablierten Berechnungsmethoden Multi-Configurational Time-Dependent Hartree und Quantum Monte Carlo verglichen. Experimentellen Daten gemessen nach deterministischen Praperation von wenigen Fermionen in einer Falle stimmen sehr gut mit den Ergebnisse dieser theoretischen Arbeit uberein.
Ultracold atoms in one dimension: from two to many: In this thesis ultracold bosonic and fermionic systems are studied with respect to their stationary and dynamical properties in one-dimensional harmonic traps and finite lattices. All physical systems explored within this work, are treated with a bottom-up approach, inspired from the underlying two-body problem. This perspective is at the most illustrated from the correlated-pair wavefunction, an Ansatz proposed here to describe the crossover from weak to strong interactions for bosons and fermions trapped in a one-dimensional harmonic potential. The underlying idea of the Ansatz is to employ the exact solution of the corresponding two-body system for the construction of a many-body correlated function. Moreover, bosonic systems under driven interaction strengths are studied with respect to resonant excitation dynamics, using for the analysis the properties of the two-body spectrum and the corresponding many-body ones. Furthermore finite lattices loaded with commensurate and incommensurate filling are investigated, with a focus on on-site effects induced from strong interactions beyond the validity regime of standard single-band approaches. A generalization of the correlated-pair wavefunction to lattice potentials is performed to cover these effects. Interband tunneling dynamics in finite lattices occur due to resonances in the many-body spectrum at certain typically strong interaction strengths breaking the self-trapping mechanism. The properties of the observables resulting from the analytical functions proposed here are compared with corresponding ones calculated by established computational methods: Multi-Configurational Time-Dependent Hartree and Quantum Monte Carlo. Recent experimental data obtained after deterministic preparation of few-fermion ensembles in a trap are in very good agreement with theoretical calculations performed in this work.
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.
Ultracold bosonic gases in optical lattices are strongly correlated quantum systems similar to solids. The strong correlation between the electrons in a solid on the one hand, and the bosonic atoms in optical lattice on the other, exhibit various quantum phenomena like insulation, conductivity, localization of electrons and atoms, respectively. Controlled by the intensity of the lattice laser, the ultracold bosonic gas can be transferred from a regime with superfluid character for shallow lattices into a regime of strong correlations, the Mott insulator. As an additional external parameter besides the lattice depth, one can generate spatial inhomogeneities by superimposing an additional standing wave (so-called two-color superlattices), which gives rise to localization effects or the formation of a Bose-glass phase. In the present work, numerical simulations are employed in order to investigate characteristic signatures of the quantum phases in the low-energy excitation spectrum of one-dimensional systems. We simulate temporal small amplitude modulations of the optical lattice in analogy to experiments, and evaluate the response of the system from the time-evolved initial state. The lattice systems are described in the framework of the Bose-Hubbard model. For the evaluation of the time-evolved state, we employ several numerical methods. We analyze systems of small size (6 particles on 6 sites) using an exact time-evolution by integration of the time-dependent Schrödinger equation. The formulation of an importance truncation scheme enables us to retain only the relevant components of the model space in the strongly correlated regime and, thus, allows for the investigation of systems with 10 particles on 10 sites using exact time-evolution. Based on this method, we present results of the Mott-insulating regime as well as for the Bose-glass phase. Furthermore, we employ particle-hole methods, which allow for the treatment of systems with experimental lattice sizes and particle numbers. Starting from the equation of motion method we adapt the Tamm-Dancoff approximation as well as the random-phase approximation for the occupation number representation of the Bose-Hubbard model. We present results of simulations of up to 50 particles on 50 sites and discuss the impact of the lattice depth on the low-energy excitations (U-resonance). Moreover, the impact of a two-color superlattice and the variation of its amplitude is investigated.
Ultracold quantum gases are a powerful platform for quantum simulation due to their versatility and the ability to precisely engineer interactions between species. In recent work, we have observed and characterized novel fermion-mediated interactions in an ultracold 6Li-133Cs mixture, opening the door to the simulation of mediated interactions relevant to solid state and high energy physics. In our current experimental setup, atoms are confined in a quasi-one-dimensional geometry using a single, red-detuned laser beam with a simple Gaussian intensity profile. This geometry limits the space of phenomena we can simulate, and the different gravitational sag of heavy Cs and light Li makes overlapping the two species complicated. Here, we report on the design and progress towards a blue detuned dipole trap produced using a 635 nm laser with a TEM01 intensity profile. The trap will provide asymmetric confinement, with stronger confinement in the z-direction which will allow us to compress samples into a 2-dimensional disc, opening the doors to exciting new physics such as vortices and droplet phases.
We introduce a numerical method to solve the two-dimensional time-dependent Schrödinger equation, which characterizes a system of two atoms with a finite-range interaction potential confined within a harmonic oscillator trap. We choose a Gaussian-shaped potential for the interaction potential. Such a system has been previously studied analytically, except that a zero-range interaction potential was used instead. We observe a strong agreement between the results for the two types of interactions. Also, we investigate the one-dimensional time-dependent Schrödinger equation for the relative motion and compute the ground state energy level as a function of the coupling strength.