Abstract
We study the spatiotemporal structure of the biphoton entangled state generated by the four-wave mixing (FWM) process in a cold two-level atomic ensemble. We analyze, for the first time, the X-like shaped structure of the biphoton entangled state and the geometry of the biphoton correlation for different lengths and densities of the cold atomic ensemble. The propagation equations of the photon pairs generated from FWM process are derived in a spatiotemporal framework. By means of the input-output relations of the propagation equations, the biphoton amplitude function is obtained in a spatiotemporal domain. In the given frequency range, the biphoton amplitude displays an X-like shaped geometry, nonfactorizable in the space-time domain. Such an X-like shaped spatiotemporal structure is caused by the phase matching and the FWM gain. The former leads to the X-like shaped envelope of the biphoton correlation, while the latter gives rise to the oscillations around the X-like shaped envelope.
Highlights
Lz (b) amplitude shows an X-like shaped geometry structure, which is analogous to the X-shaped biphoton amplitude structure in SPDC9
By selecting an appropriate frequency range from ω j − 0.7Δp to ω j + 0.7Δp (j =s, as), we can get the X-like shaped structure of the two-photon amplitude, which is similar to the one in the spontaneous parametric down-conversion (SPDC) process, and we investigate how the X structure of the biphoton amplitude changes with the length and the density of the atomic medium
We have studied the spatiotemporal correlation of the biphoton created in a cold atomic medium with the four-wave mixing (FWM) process
Summary
Lz (b) amplitude shows an X-like shaped geometry structure, which is analogous to the X-shaped biphoton amplitude structure in SPDC9. It is important to emphasize that such an interesting X-like shaped correlation is nonseparable in space-time domain. We need to study this X-like shaped structure of biphoton amplitude in the complete spatiotemporal domain. To get insight into the nonseparable X correlation, on the one hand, we compare the X structure in the atomic medium (by FWM) with the one in BBO crystal (by SPDC); on the other hand, we analyze the evolution of X structure with different lengths and densities of the atomic medium. We realize that the X correlation is under control of the FWM gain and the phase-matching relation
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