Time Independent Description of Rapidly Oscillating Potentials

The classical and quantum dynamics in a high frequency field are found to be described by an effective time independent Hamiltonian. It is calculated in a systematic expansion in the inverse of the frequency ($\omega$) to order ${\omega }^{-4}$. The work is an extension of the classical result for the Kapitza pendulum, which was calculated in the past to order ${\omega }^{-2}$. The analysis makes use of an implementation of the method of separation of time scales and of a quantum gauge transformation in the framework of Floquet theory. The effective time independent Hamiltonian enables one to explore the dynamics in the presence of rapidly oscillating fields, in the framework of theories that were developed for systems with time independent Hamiltonians. The results are relevant, in particular, for exploring the dynamics of cold atoms.

Figure 1

The lowest quasienergy resonance of the oscillating Gaussian (3), $E_0$ (solid line) and ${\Gamma }_0$ (dashed line), as a function of the driving frequency, compared to the lowest resonance of the effective Hamiltonian (16), $E_0$ (diamonds) and ${\Gamma }_0$ (circles), for $\gamma =9$ and $\beta =0.02$ (in “atomic units” $\hslash =m=e=1$). Full symbols correspond to the effective Hamiltonian truncated at order ${\omega }^{-4}$ while empty symbols to order ${\omega }^{-2}$. The effective potential (7) is depicted in the inset, for $\omega =1.5$.