  Hybrid lead halide perovskites, such as CH$_3$NH$_3$PbX$_3$ (X=I, Br), are
direct gap semiconductors that offer many superior optoelectronic properties
combined with extremely simple solution-processing fabrication methods. This
makes them very attractive for use in applications like solar cells or
light-emitting devices. Recently, also their nonlinear optical properties have
received increased attention due to reports of high nonlinear refraction in
thin films and nanoparticles. However, understanding of the underlying
mechanisms is poor and limited by the lack of knowledge of fundamental
parameters like the nonlinear refractive index of the bulk material. Here, we
measure both nonlinear absorption and nonlinear refraction in a
CH$_3$NH$_3$PbBr$_3$ single crystal using the Z-scan technique with femtosecond
laser pulses. At 1000 nm, we obtain values of 5.2 cm/GW and
9.5$\cdot$10$^{-14}$ cm$^2$/W for nonlinear absorption and nonlinear
refraction, respectively. Sign and magnitude of the observed refractive
nonlinearity are reproduced well by the two-band model. To our knowledge, these
measurements mark the first characterization of nonlinear refraction in any
metal halide perovskite single crystal and thus will serve as an important
reference for assessing the potential of this emerging material class for
nonlinear optical applications.
