We are currently studying nonlinear atom-laser intearctions, such as multi-photon resonances and saturated absorption, as well as the dynamics of such systems. These processes arise through the nonlinearity of the atom-field coupling for an atom moving through counter propagating laser fields.
The interaction of a strong standing wave with an atomic system
has long been a topic of considerable interest for both theorists and experimentalists.
This system is at the heart of many atom optics problems ranging from the
workings of a gas laser  through saturation phenomena  to laser cooling
. The laser-atom interaction modifies the internal atomic state via
dependence on the atomic system parameters as well as those of the incident
light field. Balykin et al [4[ were the first to consider a multi-level
atom interacting with a resonant incident travelling wave laser via numerical
integration of the optical Bloch equations. This formalism treats the incident
light field classically and is a direct extension of the pioneering work
of Stenholm , Stenholm and Lamb, and Allen and Eberly . More recently
Farrell et al  have extended this work to a fully quantized light field,
and while finding some problems with the semi-classical model at high laser
intensities, have confirmed the accuracy of the optical Bloch equations
for laser intensities typically used in optical pumping experiments. These
models, however, deal with systems of linear first order differential equations
in which the atom is stationary and without spacial dependence.
With the introduction of a standing wave, and atomic motion, we now have a system of coupled, first order differential equations with harmonic coefficients . These atoms, moving with a thermal distribution of velocities, acquire nonlinear electric dipole moments under the action of the field. Analytic solutions are available for the case of a simple travelling wave interacting with a two-level system  and approximate perturbation solutions have been found for the strong-pump weak-probe case  in which the probe beam is considered a perturbation to the travelling wave solution. Recently Pegg and Schultz  have solved the case for a two-level system with relaxation in a standing wave field via transformation techniques. However, if relaxation terms are absent, there remains no analytic solution . Atoms commonly used in optical pumping experiments have many coupled levels. For example rubidium experiments typically involve 24 levels, leading to a 24 x 24 element density matrix, and a system of 576 coupled differential equations. It is therefore understandable there are no analytic solutions for a multi-level system corresponding to a real atom moving through a standing wave field, thus, approximation and numerical simulation are necessary.
Figure1: Two level atom modelled on the F=2,mf=2 -> F=3,mf=3 transition in atomic sodium. The atoms are fired directly down a standing wave beam at incremented velocities and the absorption calculated after 200 lifetimes..
We have developed a detailed model using the formalism of the optical Bloch equations(O.B.E's), solved using standard numerical techniques and Monte-Carlo methods. Initially we consider a single atom moving through a two dimensional Gaussian standing wave with left-hand circular polarization. As the atom travels through the beam its density matrix evolves according to the O.B.E's, allowing the calculation of parameters such as energy level populations, absorption, dispersion and multi-photon resonances. By selecting the path the atom takes through the beam, and firing a succession of atoms with gradually increasing velocities then measuring the density matrix as each atom exits the beam, we can build a picture of the role atomic trajectory plays in determining the nonlinear processes. Each new path and velocity dictates the rate at which the atom sees the nodes of the standing wave, thus it is not too surprising that variation in the angle the atom takes through the beam results in significant changes in the behavior of these nonlinear processes. This method allows, for the first time, calculation of the multi-photon resonances and their gradual saturation as the laser intensity is increased.
Figure2: Multi-photon resonances for a 20-level sodium model fired at an angle of Pi/8 through a standing wave field at 6mW/(cm)**2 and 54mW/cm**2 respectively. Note the emmissive 5-photon process as well as saturated primary and 3-photon process.
Reducing the strength of the relaxation terms and or increasing laser
intensity, corresponds to decreasing the amount of dissipation in the system,
which in turn increases the time the system takes to reach equilibrium.
The effects of such a reduction can be seen in a simple two-level atom
modelled on the F=2,mf=2 -> F=3,mf=3 transition for sodium. The system
initially enters into regular oscillatory behavior indicative of a limit
cycle and then undergoes a bifurcation process leading to fully chaotic
behavior as the dissipative terms are reduced and/or laser intensity increased.
The model is then extended to the multi-level cases of sodium and rubidium
with evidence that the populations of the energy sub-levels again oscillate
according a limit cycle, and with clear indication that even at saturation
intensity the limit cycle has undergone the first bifurcation on the period
doubling route to chaos. We can then compare these multi-level results
to a simulation of the Bloch vector equations and further demonstrate that
the system evolves into chaotic motion on the Bloch sphere.
Figure3: The excited state population of the multi-level sodium model contrasts the linear travelling wave case and nonlinear standing wave cases. The lower figure is the period 2 limit cycle describing the dynamics of the excited state population in the nonlinear case.
Figure4: The effect of a reduction in dissipative terms, as shown in the 2-level Bloch vector equations, is to destroy the stability of the limitcycle via a bifurcation process. This leads ultimately to chaotic motion on the Bloch sphere.
We next allow the atoms a Maxwell-Boltzmann distribution of velocities
and random trajectories through the beam. The laser is then set at incremental
detunings and the density matrix calculated and averaged for many atoms
at each respective detuning. This method allows the calculation of power
and Doppler broadened saturated absorption spectra that are in good qualitative
agreement with experiment. Finally we present preliminary results to demonstrate
the usefulness of this approach in calculating hyperfine structure in saturated
Figure5: The results for 10,000 sodium atoms per detuning, fired with random trajectories and a Boltzmann distribution of velocities. The laser intensity is set at 1W/cm**2 resulting in significant doppler and power broadening.
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Dr Robert Scholten
Dr.Peter Farrell (Optical Technology Research Laboratory, Victoria University)
Mirek Walkiewicz (MSc)
The rich structure of the dynamics of the nonlinear atom-field coupling
provides the opportunity for the application of a wide variety of theoretical
techniques as well as indicating new experimental directions.