Floquet’s theorem says that any unitary, periodically driven system can be described by an effective time-independent Hamiltonian, where the effective Hamiltonian can have completely different properties than the static, undriven system. Floquet engineering makes use of this idea to simulate new Hamiltonians that would otherwise not be possible in the undriven case.  For interacting systems, this approach can be used to realize interesting correlated many-body states, but drive-induced heating  must be understood and mitigated. Cold atoms in optical lattices provide a controllable, well-isolated system in which these ideas can and have been realized. I describe research into two areas of Floquet engineering for interacting Bose-Einstein condensates in periodically driven optical lattices.

The first half of this thesis focuses on the study of heating mechanisms for condensates in periodically driven lattices. In the weakly interacting limit, one might expect that heating could be described with a Fermi Golden Rule approach. Parametric driving of fluctuations in the condensate, however, can lead to runaway heating that cannot be described perturbatively. We experimentally study heating in shaken 2D square lattices and demonstrate heating consistent with the theoretical predictions of parametric instabilities.

The second half of this thesis describes experiments that realize Floquet-induced effective staggered magnetic fields, and the relaxation dynamics of interacting particles subject to these fields. Interestingly, we observe pre-thermal relaxation dynamics, where an initially heated cloud suddenly subject to the effective Hamiltonian condenses into a state governed by the drive-induced effective Hamilton on a timescale faster than heating.