The study of pattern formation in nonlinear optical systems is ideal for probing aspects of fundamental many-body physics and nonlinear dynamics. Pattern formation in warm atoms has given rise to sensitive methods for all-optical devices, but single-photon sensitivities in pattern-forming systems have yet to be realized. In order to obtain single-photon nonlinearities, the threshold power required for pattern formation must be lowered. This can be achieved by using cold atoms, which have a larger nonlinear susceptibility than warm atoms. I report an ultra-low threshold for pattern formation in cold atoms that is 500 times lower than that observed in warm atoms. I describe the discrepanices between current models for pattern formation and my experimental system, which involve the dependence of the nonlinear susceptibility on both the frequency detuning from the atomic resonance and the intensity required of the input optical fields. I then discuss the wave-mixing processes that give rise to pattern formation and the current theoretical models of pattern formation in both warm and cold atoms. I also explore the differences between the predictions of current models and my experimental findings, and I describe what must be done to form a complete understanding of pattern formation in cold atoms.
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