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.