  We present a non-perturbative quantization of gravitational null initial
data. Our starting point is the characteristic null initial problem for
tetradic gravity with a parity-odd Holst term in the bulk. After a basic review
about the resulting Carrollian boundary field theory, we introduce a specific
class of impulsive radiative data. This class is defined for a specific choice
of relational clock. The clock is chosen in such a way that the shear of the
null boundary follows the profile of a step function. The angular dependence is
arbitrary. Next, we solve the residual constraints, which are the Raychaudhuri
equation and a Carrollian transport equation for an $SL(2,\mathbb{R})$
holonomy. We show that the resulting submanifold in phase space is symplectic.
Along each null generator, we end up with a simple mechanical system. The
quantization of this system is straightforward. Our basic strategy is to start
from an auxiliary Hilbert space with constraints. The physical Hilbert space is
the kernel of a constraint, which is a combination of ladder operators. The
constraint and its hermitian conjugate are second-class. Solving the constraint
amounts to imposing a simple recursion relation for physical states. On the
resulting physical Hilbert space, the $SL(2,\mathbb{R})$ Casimir is a Dirac
observable. This observable determines the spectrum of the two radiative modes.
The area at the initial and final cross sections are Dirac observables as well.
They have a discrete spectrum, which agrees with earlier results on this topic.
