Quantum interference in the classically forbidden region: A parametric oscillator

We study tunneling between period-2 states of a parametrically modulated oscillator. The tunneling matrix element is shown to oscillate with the varying frequency of the modulating field. The effect is due to spatial oscillations of the wave function and the related interference in the classically forbidden region. The oscillations emerge already in the ground state of the oscillator Hamiltonian in the rotating frame.

Figure 1

(Color online) The scaled effective Hamiltonian of the oscillator in the rotating frame $g(Q,P)$, Eq. (5), for $\mu =0.5$. The minima of $g(Q,P)$ correspond to the period-2 vibrations. The eigenvalues of $\widehat{g}$ give scaled oscillator quasienergies.

Figure 2

(Color online) The wave function of the ground state in the left well ${\psi }_l\left(Q\right)$ in the oscillation region for $\lambda =0.09$ and $\mu =0.5$. The solid line shows explicit expressions (13, 14, 15), the dashed line shows numerical results. Inset: ${\psi }_l\left(Q\right)$ near its second zero with higher resolution.

Figure 3

(Color online) Scaled matrix element of tunneling between period-2 states as a function of the scaled detuning of the modulation frequency from twice the oscillator eigenfrequency. The solid lines show explicit expression (16), the dashed lines show the result of numerical calculations. Inset: a higher-resolution plot of $\mid \delta g\mid ∕2$ vs $\mu$ near the zero of $\delta g$ at $\mu =6\lambda$. The data refer to $\lambda =0.09$.