Review of Biphoton Sources Based on the Double- Λ $\Lambda$ Spontaneous Four-Wave Mixing Process

This review article focuses on biphoton sources based on the double-$\Lambda$ spontaneous four-wave mixing (SFWM) process in laser-cooled as well as room-temperature or hot atomic ensembles. These biphoton sources have the advantage of providing stable frequencies, ultranarrow linewidths, and a tunability of the temporal biphoton width of more than one order of magnitude for high-bandwidth applications. Therefore, the generated photons can be efficiently interfaced to, e.g., atomic quantum memories. In contrast, solid-state biphoton sources typically require assistance by an optical cavity to operate at narrow linewidth that limits the tunability of the temporal width of the biphotons. Present state-of-the-art double-$\Lambda$ SFWM biphoton sources can achieve one of the following results: a spectral linewidth of 50 kHz (290 kHz) or a temporal width of 13 $\mu s$ (580 ns) with cold (hot) atoms, a detection rate of about 7$\times {10}^3$ cps, and a generation rate of ${10}^7$ cps at a duty cycle of 0.4% or of ${10}^5$ cps in the steady state. The theoretical background of these biphoton sources, experimental implementations with cold and hot atoms, and progress over the years, will be illustrated.