Cross-Kerr effective Hamiltonian for a non-resonant four-level atom

TL;DR

This paper derives an effective Hamiltonian describing cross-Kerr nonlinearities in a four-level atom system, revealing conditions for pure nonlinear effects with minimal absorption, relevant for quantum optics applications.

Contribution

It extends previous models by including linear and self-Kerr responses and identifies conditions where only pure cross-Kerr nonlinearity remains.

Findings

Large, pure cross-Kerr nonlinearity can be achieved with negligible absorption.

Linear and self-Kerr responses can vanish under specific resonance conditions.

The susceptibilities of probe transitions are characterized in detail.

Abstract

We derive a cross-Kerr type effective Hamiltonian for the four-level atom interacting with three electromagnetic fields in the N-configuration. When the atom has relaxed into the ground state a cross-Kerr nonlinearity arises between two weak probe fields. As a development on earlier work we show in general that the atom will also display a linear and self-Kerr response. However, if certain resonance conditions are satisfied then the linear and self-Kerr interactions will vanish. The electrical susceptibilities of the probe transitions are also explored and it is shown that a large, pure cross-Kerr nonlinearity can be generated with vanishing absorption of both probe fields.