Phase controllable dynamical localization: a generalization of the Dunlap-Kenkre result

Dunlap-Kenkre result states that Dynamical Localization (DL) of a field driven quantum particle in a discrete periodic lattice happens when the ratio of the field magnitude to the field frequency (say, $\eta$) of the diagonal sinusoidal drive is a root of the ordinary Bessel function of order 0. This has been experimentally verified. A generalization of the Dunlap-Kenkre result is presented here. We analytically show that if we have an off-diagonal driving field (with modulation $\delta$) and diagonal driving field with different frequencies (say $\omega_1$ and $\omega_2$ respectively) and a definite phase relationship $\phi$ between them, one can obtain DL if (1) $\eta$ is a zero of the Bessel function of order 0 and $\phi$ is an odd multiple of $\pi/2$ for equal and $\frac{\omega_1}{\omega_2}= odd integer$ driving frequencies, (2) $\eta$ is a zero of the Bessel function of order 0 and $\phi$ is an integer multiple of $\pi$ including zero for $\frac{\omega_1}{\omega_2}= even integer \equiv m$, and (3) $\phi = -\arcsin(\frac{J_0(\eta)}{\delta J_m(\eta)})$ and $\eta$ is not a zero of the Bessel function of the even order $m$.