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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, $η$) 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 $δ$) and diagonal driving field with different frequencies (say $ω_1$ and $ω_2$ respectively) and a definite phase relationship $ϕ$ between them, one can obtain DL if (1) $η$ is a zero of the Bessel function of order 0 and $ϕ$ is an odd multiple of $π/2$ for equal and $\frac{ω_1}{ω_2}= odd integer$ driving frequencies, (2) $η$ is a zero of the Bessel function of order 0 and $ϕ$ is an integer multiple of $π$ including zero for $\frac{ω_1}{ω_2}= even integer \equiv m$, and (3) $ϕ= -\arcsin(\frac{J_0(η)}{δJ_m(η)})$ and $η$ is not a zero of the Bessel function of the even order $m$.