In this thesis, we address the long-standing question of time-reversal protocols for the quantum mechanical wave function for continuous degrees of freedom. We propose two new mechanisms - quantum time mirrors - one for two-band systems, the other one for systems described by the nonlinear Schrödinger equation like Bose-Einstein condensates. In both cases, a homogeneous pulse is applied to the system that is supposed to flip the velocity direction and thus invert the motion of the system. In a two-band system, the pulse induces a transition from one band to the other. For bands with opposite group velocity directions, e.g. Dirac cone systems like graphene, the band-transition directly leads to an inversion of the motion. By generalizing the results for arbitrary bands and additionally investigating the effects of perturbations like disorder or electro-magnetic fields, we are able to determine for any (effective) two-band system at hand with a given pulse whether it is a suited candidate for the quantum time mirror.
In the Bose-Einstein condensates, the inversion of motion is achieved by quickly varying the nonlinear term. The phases in the wave function acquired during the pulse affect the velocity and for certain parameters it switches sign. Although echoes are induced by this mechanism, the fidelity is rather low. However, time lenses, i.e. refocusing the wave packet by a time-dependent pulse, can be achieved with high fidelity in this setup. Even solitary waves can be created whose refocus fidelity stay above 99% for more than 100,000 pulses.
In the last part of the thesis, we investigate zitterbewegung in two-band systems with time-dependent potentials. In the driven setup, we find persistent, i.e. undamped, modes of the zitterbewegung that would decay for a wave packet in a static environment. Moreover, using the quantum time mirror pulses described above, echoes of the zitterbewegung can be achieved that are in some sense similar to the well-known and highly exploited spin echo
