  In this work we study new issues involving the type IIB superstring in a time
dependent plane wave background with a constant self-dual Ramond-Ramond 5-form
and a linear dilaton in the light-like direction. We construct a unitary
Bogoliubov generator which relates the asymptotically flat superstring Hilbert
space to the finite time Hilbert space. The time dependent vacuum is a
superposition of $SU(1, 1)\times SU(2)$ coherent states, which has a particular
structure of excitation, characterized by a condensation of right and left
moving supertring modes. We calculate the time dependent left/right
entanglement entropy and carry out the summation over the oscillator modes of
the superstring two-point function. We show that, close to the null
singularity, the entanglement entropy is well-behaved. In particular, for
asymptotically flat observers, the closed superstring vacuum close to the
singularity appears as superstring thermal vacuum, which is unitarily
inequivalent to the asymptotically flat vacuum. Actually, we show that close to
the singularity the superstring thermalizes and the entanglement entropy
becomes a thermodynamical entropy for a supersymmetric two-dimensional gas.
