We present a detailed study of the second order nonlinearity of 2D (mono-atomic layer) dichalcogenide MoS2, both in the visible and in the IR regime, and test its potential for spontaneous parametric down-conversion (SPDC), the amplification of vacuum fluctuations mediated by optical nonlinearity. We develop a model of SPDC from a deeply subwavelength nonlinear medium, where phase matching conditions are completely relaxed, and make predictions about the rate of emitted photons, their momentum, polarisation and spectrum. We show that detection in the visible spectral region is hindered by the strong photoluminescence background. Moving to the IR regime we observe indications of SPDC by performing polarization, power dependence and lifetime measurements around 1560 nm. We show that the signal from a single monolayer is qualitatively different from that generated by multi-layer MoS2. Finally, we characterize the latter as a new kind of photo-luminescence emission which is enhanced at the edges of multi-layer MoS2.
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