We introduce a grating assisted tunneling scheme for tunable synthetic magnetic fields in photonic lattices, which can be implemented at optical frequencies in optically induced one- and two-dimensional dielectric photonic lattices. We demonstrate a
conical diffraction pattern in particular realization of these lattices which possess Dirac points in $k$-space, as a signature of the synthetic magnetic fields. The two-dimensional photonic lattice with grating assisted tunneling constitutes the realization of the Harper-Hofstadter Hamiltonian.
The asymptotic form of the wave functions describing a freely expanding Lieb-Liniger gas is derived by using a Fermi-Bose transformation for time-dependent states, and the stationary phase approximation. We find that asymptotically the wave functions
approach the Tonks-Girardeau (TG) structure as they vanish when any two of the particle coordinates coincide. We point out that the properties of these asymptotic states can significantly differ from the properties of a TG gas in a ground state of an external potential. The dependence of the asymptotic wave function on the initial state is discussed. The analysis encompasses a large class of initial conditions, including the ground states of a Lieb-Liniger gas in physically realistic external potentials. It is also demonstrated that the interaction energy asymptotically decays as a universal power law with time, $E_mathrm{int}propto t^{-3}$.
