Quantum Kramers-Henneberger Transformation

Abstract

The classical Kramers-Henneberger transformation connects, via a series of unitary transformations, the dynamics of a quantum particle of mass $m$ located in a trap at position $α(t)$, with the dynamics of a charge $e$ moving in an electric field $e{\cal{E}}(t)=-m\ddotα(t)$ within the dipole approximation. In this paper, we extend the classical Kramers-Henneberger transformation to the quantum electrodynamic and quantum optical realm, by explicitly treating the trap location quantum mechanically, thus taking into account the quantum fluctuations of the time-dependent displacement force. Compared to the classical case, we show that quantum electrodynamic corrections appear, and we propose an optomechanical realization for the quantized position of the trap to show that such corrections can manifest in state-of-the-art experiments. These results open the path to novel quantum simulation of quantum electrodynamics and quantum optics of attoscience and ultrafast physics by using ultracold trapped atoms and ions.

Figures (2)

• Figure 1: (a) Average value of the electric field at different times of the optical cycle, for the classical input (black), and a first-order of quantum correction (color, see main text). (b) HHG yield as a function of different multiples of the fundamental frequency $\omega=0.057$ a.u. (corresponding to a laser wavelength $\lambda=800$ nm). Gabor transform of the harmonic yield for the classical input (c) and a correction strength, $\epsilon=0.1$ (d).
• Figure 2: Optomechanical scheme where the position $\hat{x}$ of an atomic cloud (red) inside a gaussian optical trap with mechanical frequency $\Omega$ is affected by the optomechanical interaction in Eq. \ref{['eq:hom']} with a cavity mode (green) populated with $\langle \hat{c}^\dagger \hat{c} \rangle$ average photons, and a decay rate $\kappa$.