  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.
