Effective tunneling coefficient of a coupled double-well system modulated by anharmonic periodic potentials

We numerically study coherent tunneling oscillations of the particles between two levels in a double-well potential in the presence of anharmonic periodic potentials. Extremely short driving pulses modify the tunneling coefficient to kappa(eff)=kappa cos A, where kappa is the bare tunneling coefficient without the driving field and A is the pulse area of the driving wave form. The modulation amplitude of the kappa(eff) gradually decreases as the driving wave form becomes broad and is given by kappa(eff)=kappa J(0)(A) for the sinusoidal modulation, where J(0)(x) denotes the ordinary Bessel function of order zero. Theoretical derivation of the effective tunneling coefficient kappa(eff)=kappa cos A is also shown for a periodic delta kick with alternating sign by means of the transfer matrix formula.