Jonathan Leto, University of Central Florida


Area Theorems for Optical Pulses

Abstract:  The phenomena of Self-Induced Transparency (SIT) can be described as a "short coherent pulse traveling anomalously long distances, at anomalously low velocities through resonant absorbers" [1].  According to the Optical Bloch equations [1], a square pulse with an area of $\pi$ will excite the atoms into their excited state. If followed by another $\pi$ pulse the atoms are de-excited and returned to the ground state. Hence, the total effect of a $2\pi$ pulse is that the first half excites the atoms and the second half returns the atoms to their original state.  Under a Two-Level atom approximation these pulses were theoretically predicted and experimentally verified by McCall & Hahn in the late 1960's at Bell Labs [2]. An area theorem, which takes the form of a simple ordinary differential equation, describes the evolution of the area under the electric field envelope. It has also long been known that by the method of nonlinear moments of integrable systems, one can derive the classical area theorem, which corresponds to the zeroth nonlinear moment [3].

Modern optical devices such as quantum gates (which are being developed to implement "qubits", quantum equivalents to transistors) or atom interferometers [4] utilize three atomic levels and two transitions, which has lead to much research into three-level atoms. Thus it would be useful to have an Area Theorem for three-level systems. The method of nonlinear moments mentioned above should be able to obtain the equivalent of the McCall-Hahn area theorem for those three-level systems which are integrable.  We will present our results toward a three-level area theorem in the limit when one of the two incident pulses is weak.

[1] Allen & Eberly - Optical Resonance and Two-Level Atoms (1987)
[2] McCall & Hahn - Physical Review Letters 18 pp908 (1967)
[3] D.J. Kaup - Physical Review A pp704 (1977)
[4] Eckert, Mompart, Corbalan, Lewenstein, Birkl - LANL arXiv.org quant-ph/0511195 "Three Level atom optics in dipole traps and waveguides"

Advisor: David Kaup (UCF)