Engineering Local optimality in Quantum Monte Carlo algorithms

Quantum Monte Carlo algorithms based on a world-line representation such as the worm algorithm and the directed loop algorithm are among the most powerful numerical techniques for the simulation of non-frustrated spin models and of bosonic models. Both algorithms work in the grand-canonical ensemble and have a non-zero winding number. However, they retain a lot of intrinsic degrees of freedom which can be used to optimize the algorithm. We let us guide by the rigorous statements on the globally optimal form of Markov chain Monte Carlo simulations in order to devise a locally optimal formulation of the worm algorithm while incorporating ideas from the directed loop algorithm. We provide numerical examples for the soft-core Bose-Hubbard model and various spin-S models.Comment: replaced with published versio

Spin and density resolved microscopy of Hubbard chains

This thesis reports on the microscopic investigation of antiferromagnetic order in Hubbard chains, realized with ultracold repulsively interacting fermions in an optical lattice. Extending a quantum gas microscope on spin resolution opened the possibility to Access the full charge and spin statistics of a strongly correlated fermionic many body system. Using this technique, we measured antiferromagnetic correlations over distances up to three lattice sites. Starting with a repulsive degenerate two component mixture of 6Li atoms trapped in a single plane of a vertical lattice, we loaded the Atoms into one-dimensional tubes of a transverse optical superlattice. By adiabatically ramping up the lattice potential along the tubes, we isentropically entered the strongly correlated regime of the one-dimensional Fermi-Hubbard Hamiltonian. We spatially separated the two spin states into opposite sites of local double wells orthogonal to the direction of the lattice before single atom and site sensitive imaging. In this way we extracted the spin information of every single lattice site and were able to evaluate spin-spin correlations. The two component degenerate gases were produced with sub-Poissonian atom number fluctuations by magnetically driven evaporative cooling in vicinity of a Feshbach resonance, used to set the scattering length. The dependence of antiferromagnetic correlations on the driving superexchange coupling was experimentally investigated by varying the interparticle scattering length before loading the atoms into the optical lattice. Taking advantage of the underlying trapping potential of the optical lattice, which shaped the filling and the entropy of the gases, the influence of density excitations and different entropies on spin correlations was observed. By post-selecting the Hubbard chains based on their average filling and local atom number fluctuations, spin correlations corresponding to 58% of the zero temperature predictions in the Heisenberg regime were measured. By comparing the obtained antiferromagnetic correlations to Quantum-Monte-Carlo predictions for our System the lowest measured entropy per particle could be stated as s = 0.51(4)kB, clearly below the critical value of s = ln(2)kB, which is required to form longer ranged correlations. Below this critical entropy, in contrast to local density fluctuations, these spin correlations strongly depend on the entropy of the system. In the future, such a spin thermometer can be used to benchmark novel cooling techniques, which are required to enter the temperature regime of d-wave superconducting states. The access to the full particle and spin statistics will enable us to read out multi-point correlation functions, which characterize such exotic states of matter.Diese Dissertation berichtet von der mikroskopischen Untersuchung antiferromagnetischer Ordnung in Hubbard Ketten, die mit ultrakalten, repulsiv wechselwirkenden Fermionen in optischen Gittern realisiert wurden. Durch die Erweiterung eines Quantengasmikroskops um Spin Auflösung,eröffnete sich die Möglichkeit, die komplette Dichte- und Spinstatistik eines stark korrelierten fermionischen Vielteilchensystems auszulesen. Durch Anwendung dieser Technik haben wir antiferromagnetische Korrelationen, die sich über drei Gitterplätze erstrecken,nachgewiesen. Angefangen mit einer entarteten zwei-komponentigen Mischung von Li-6 Atomen, die ineiner einzelnen Ebene eines vertikalen Gitters gefangen waren, laden wir die Atome in eindimensionale Subsysteme eines transversalen optischen Supergitters. Durch adiabatisches Anrampen des Gitterpotentials entlang des letzten verbliebenen Bewegungsfreiheitsgrades geht das System in das Regime des stark korrelierten ein-dimensionalen Fermi-Hubbard Hamiltonians über. Anschließend trennten wir die Atome unterschiedlichen Spins räumlich in entgegengesetzte Gitterplätze lokaler Doppelmulden senkrecht zur Gitterrichtung, bevor die Atome mit einzel-Atom und Gitterplatz sensitiv abgebildet wurden. Dadurch konnten wir den Spin jedes einzelnen Gitterplatzes auslesen und Spin-Spin Korrelatoren berechnen. Die zweikomponentigen entarteten Gase wurden mit sub-Poisson Atomzahlfluktuationen hergestellt, und zwar mithilfe von magnetisch unterstützter evaporativer Kühlung in Gegenwart einer Feshbach Resonanz, die benutzt wurde um die Streulänge einzustellen. Die Abhängigkeit der antiferromagnetischen Korrelationen von der Austauschwechselwirkung wurde experimentell untersucht indem die Streulänge zwischen den Atomen vor dem Laden in das optische Gitter eingestellt wurde. Das dem optischen Gitter zugrundeliegende Fallenpotential modellierte die Dichte- und Entropieverteilung, sodass der Einfluss von Dichteanregungen und verschiedenen Entropien auf Spin-Korrelationen gemessen werden konnten. Durch Selektion der Hubbard Ketten, basierend auf deren durchschnittlichen Besetzung und lokalen Atomzahlfluktuationen, konnten Spin Korrelationen, die 58% des vorrausgesagten Wertes im Rahmen des Heisenberg Modells bei T = 0 entsprechen, gemessen werden. Durch Vergleich der erhaltenen Korrelationen mit Quanten-Monte-Carlo Vorhersagen für unser System konnte die niedrigste Entropie pro Teilchen von s = 0.51(4)kB nachgewiesen werden, die damit deutlich unter der für länger-reichweitige Korrelationen notwendigen Entropie von s = ln(2)kB, liegt. Für Entropien unterhalb dieser kritischen Entropie hängen diese Spin Korrelationen stark von der Entropie des Systems ab, im Gegensatz zu lokalen Dichtefluktuationen. Daher könnte ein solches Spin-Thermometer in der Zukunft dazu genutzt werden, neuartige Kühlmethoden zu charakterisieren, welche nötig sind, um das Temperaturregime der d-Wellensupraleitung zu erreichen. Der Zugang zur kompletten Atomzahl- und Spinstatistik ermöglicht es uns Multipunkt-Korrelatoren zu berechnen, welche solche exotischen Materiezustände charakterisieren.Deutsche Übersetzung des Titels: Spin und Dichte aufgelöste Mikroskopie von Hubbard Kette

Probing correlated quantum many-body systems at the single-particle level

The detection of correlation and response functions plays a crucial role in the experimental characterization of quantum many-body systems. In this thesis, we present novel techniques for the measurement of such functions at the single-particle level. Specifically, we show the single-atom- and single-site-resolved detection of an ultracold quantum gas in an optical lattice. The quantum gas is described by the Bose-Hubbard model, which features a zero temperature phase transition from a superfluid to a Mott-insulating state, a paradigm example of a quantum phase transition. We used the aforementioned detection techniques to study correlation and response properties across the superfluid-Mott-insulator transition. The single-atom sensitivity of our method is achieved by fluorescence detection of individual atoms with a high signal-to-noise ratio. A high-resolution objective collects the fluorescence light and yields in situ `snapshots' of the quantum gas that allow for a single-site-resolved reconstruction of the atomic distribution. This allowed us to measure two-site and non-local correlation-functions across the superfluid-Mott-insulator transition. Non-local correlation functions are based on the information of an extended region of the system and play an important role for the characterization of low-dimensional quantum phases. While non-local correlation functions were so far only theoretical tools, our results show that they are actually experimentally accessible. Furthermore, we used a new thermometry scheme, based on the counting of individual thermal excitations, to measure the response of the system to lattice modulation. Using this method, we studied the excitation spectrum of the system across the two-dimensional superfluid-Mott-insulator transition. In particular, we detected a `Higgs' amplitude mode in the strongly-interacting superfluid close to the transition point where the system is described by an effectively Lorentz-invariant low-energy theory. Our experimental results helped to resolve a debate about the observability of Higgs modes in two-dimensional systems