  In [13], a new quasi-local energy is introduced for spacetimes with a
non-zero cosmological constant. In this article, we study the small sphere
limit of this newly defined quasi-local energy for spacetimes with a negative
cosmological constant. For such spacetimes, the anti de-Sitter space is used as
the reference for the quasi-local energy. Given a point $p$ in a spacetime $N$,
we consider a canonical family of surfaces approaching $p$ along its future
null cone and evaluate the limit of the quasi-local energy. The optimal
embedding equation which identifies the critical points of the quasi-local
energy is solved in order to evaluate the limit. Using the optimal embedding,
we show that the limit recovers the stress-energy tensor of the matter field at
$p$. For vacuum spacetimes, the quasi-local energy vanishes to a higher order.
In this case, the limit of the quasi-local energy is related to the
Bel-Robinson tensor at $p$.
