Christoph Dauer

• This thesis studies resonant behavior in periodically driven ultracold quantum gases and optical waveguide arrays. Motivated by resonance phenomena from classical mechanics and the wealth of exotic physics that has been found in periodically driven systems, the goal of this thesis is to investigate intriguing resonance-induced behavior in these physical systems. In order to reach this goal, each of the three main parts of this thesis discusses a different type of resonance in a suitable physical model. For the theoretical description of the time-periodic systems we use Floquet theory. In particular, Floquet theory generalizes the concept of an eigenbasis to time-periodically driven Hamilton operators. Floquet eigenstates, which are also named Floquet steady states, have properties that are tunable by the periodic drive. In the first part we show with the methods of Floquet scattering theory that the periodic driving of a short range potential between ultracold atoms induces Feshbach resonances. We develop the Floquet-Feshbach resonance theory which is capable of calculating the properties of the drivinginduced Feshbach resonances. As a result, highly tunable resonance properties are found. The real part of the scattering length can be adjusted to large positive and negative values, while the imaginary part of the scattering length, describing atom loss, stays relatively small. In an ultracold gas experiment, a time-dependent interatomic potential can be achieved by modulating a magnetic field in the vicinity of a magnetic Feshbach resonance. In this thesis, we model this case by a multichannel description of atom scattering with time-periodic parameters. We find that the periodic drive both engineers the parameters of the magnetic Feshbach resonance, that already exists in the static case, and induces new resonances. The Floquet-Feshbach resonance theory also predicts in this model that the properties of the resonances can be tuned by the periodic drive. The second part remains in the realm of ultracold quantum gases, and investigates resonant behavior in interacting quantum many-body systems. In such systems, nontrivial correlations are present due to the interaction and entanglement between the particles. In one dimension, the Tomonaga-Luttinger liquid serves as a universal low-energy description of a wide class of quantum many-body systems. In this thesis, we formulate a Floquet-Bogoliubov theory which yields Floquet steady states of time-periodic Hamiltonians that are quadratic in bosonic operators. The Floquet-Bogoliubov theory can be applied to a plurality of physical models, in particular to the periodically driven Tomonaga-Luttinger liquid. However, Floquet solutions turn out to only exist in certain parameter regions. The absence of a Floquet solution is traced back to the phenomenon of parametric resonance and is seen as a divergence in certain expectation values. A finite lifetime in the Tomonaga-Luttinger description regularizes these divergences to peaks with finite maximum and enables us to find a Floquet steady state of a periodically driven quantum many-body system. Combining all the results, we predict a standing density-wave pattern for a one-dimensional Bose gas subject to a time-periodic modulation of the interaction strength. In the last part we stay in one dimension and introduce time-dependent dissipation as a directiondependent filter for a Hamiltonian quantum ratchet. Based on the details of a surface plasmonpolariton waveguide array experiment, we investigate dynamic transport phenomena in a periodically driven Su-Schrieffer-Heeger ratchet model. Directed transport with the velocity of one unit cell per driving period is found at certain resonant frequencies in this model. A Floquet-Bloch analysis relates these transport properties to a nontrivial topology. However, the direction and the magnitude of the current depends on the initial state. The direction-dependent filter circumvents this issue by breaking the time-reversal symmetry. The filter absorbs states that move in a certain direction while leaving the states moving in the other direction unimpaired. The properties of the Filter are analyzed with a Floquet S-matrix theory, which we derive for the case of a non-hermitian, time-periodic impurity operator and a time-periodic bulk. Our findings agree with the results of the waveguide experiment.