Phase space crystals: a new way to create a quasienergy band structure

A novel way to create a band structure of the quasienergy spectrum for driven systems is proposed based on the discrete symmetry in phase space. The system, e.g., an ion or ultracold atom trapped in a potential, shows no spatial periodicity, but it is driven by a time-dependent field coupling highly nonlinearly to one of its degrees of freedom (e.g., ∼q(n)). The band structure in quasienergy arises as a consequence of the n-fold discrete periodicity in phase space induced by this driving field. We propose an explicit model to realize such a phase space crystal and analyze its band structure in the frame of a tight-binding approximation. The phase space crystal opens new ways to engineer energy band structures, with the added advantage that its properties can be changed in situ by tuning the driving field's parameters.