Quasi-one-dimensional Rydberg atoms exposed to alternating positive and negative electric field pulses (kicks) are an example of a chaotic atomic system. Chaotic ionization is predicted in this system via a phase space turnstile mechanism, and we have explored this experimentally. Turnstiles form a general transport mechanism for numerous chaotic systems, and this study is the first to explicitly illuminate their relevance to atomic ionization. Two experiments are presented. In the first we show that the ionization of the electron depends not only on the initial electron energy, but also on the phase space position of the electron with respect to the turnstile--that part of the electron packet inside the turnstile ionizes quickly, after one period of the applied field, while that part outside the turnstile ionizes after multiple kicking periods. In the second experiment we show the signature of the turnstile manifests itself in the step-function-like behavior of the ionization fraction as a function of the kick strength. This behavior persists for different values of kicking periods and starting electron energies.