Transport properties and nonequilibrium dynamics in strongly correlated materials are
typically difficult to calculate. This holds true even for minimalistic model Hamiltonians
of these systems, such as the fermionic Hubbard model.
Ultracold atoms in optical lattices enable an alternative realization of the Hubbard
model and have the advantage of being free of additional complications such as phonons,
lattice defects or impurities. This way, cold atoms can be used as quantum simulators of
strongly interacting materials. Being thermally isolated systems, however, we show that
cold atoms in optical lattices can also behave very differently from solids and can show a
plethora of novel dynamic effects.
In this thesis, several out-of equilibrium processes involving interacting fermionic atoms
in optical lattices are presented. We first analyze the expansion dynamics of an initially
confined atomic cloud in the lowest band of an optical lattice. While non-interacting atoms
expand ballistically, the cloud expands with a dramatically reduced velocity in the presence
of interactions. Most prominently, the expansion velocity is independent of the attractive
or repulsive character of the interactions, highlighting a novel dynamic symmetry of the
Hubbard model.
In a second project, we discuss the possibility of realizing negative absolute temperatures in optical lattices. Negative absolute temperatures characterize equilibrium states
with an inverted occupation of energy levels. Here, we propose a dynamical process to realize equilibrated Fermions at negative temperatures and analyze the time scales of global
relaxation to equilibrium, which are associated with a redistribution of energy and particles
by slow diffusive processes.
We show that energy conservation has a major impact on the dynamics of an interacting
cloud in an optical lattice, which is exposed to an additional weak linear (gravitational)
potential. Instead of ‘falling downwards‘, the cloud diffuses symmetrically upwards and
downwards in the gravitational potential. Furthermore, we show analytically that the
radius R grows with the time t according to R ∼ t^1/3, consistent with numerical simulations
of the Boltzmann equation.
Finally, we analyze the damping of Bloch oscillations by interactions. For a homogeneous system, we discuss the possibility of mapping the dynamics of the particle current to
a classical damped harmonic oscillator equation, thereby giving an analytic explanation for
the transition from weakly damped to over-damped Bloch oscillations. We show that the
dynamics of a strongly Bloch oscillating and weakly interacting atomic cloud can be discribed in terms of a novel effective “stroboscopic” diffusion equation. In this approximation,
the cloud’s radius R grows asymptotically in time t according to R ∼ t^1/5
