Title: Engineering Frequency-dependent Superfluidity in Bose-Fermi Mixtures,
Abstract: Unconventional superconductivity or superfluidity are among the most exciting
and fascinating quantum states in condensed matter physics. Usually these
states are characterized by non-trivial spatial symmetry of the pairing order
parameter, such as in $^{3}He$ and high-$T_{c}$ cuprates. Besides spatial
dependence the order parameter could have unconventional frequency dependence,
which is also allowed by Fermi-Dirac statistics. For instance, odd-frequency
pairing is an exciting paradigm when discussing exotic superfluidity or
superconductivity and is yet to be realized in the experiments. In this paper
we propose a symmetry-based method of controlling frequency dependence of the
pairing order parameter via manipulating the inversion symmetry of the system.
First, a toy model is introduced to illustrate that frequency dependence of the
order parameter can be adjusted by controlling the inversion symmetry of the
system. Second, taking advantage of the recent rapid developments of shaken
optical lattices in ultracold gases, we propose a Bose-Fermi mixture to realize
such frequency dependent superfluids. The key idea is introducing the
frequency-dependent attraction between Fermions mediated by Bogoliubov phonons
with asymmetric dispersion. Our proposal should pave an alternative way for
exploring frequency-dependent superconductors or superfluids with cold atoms.