  The double-slit experiment is the most direct demonstration of interference
between individual quantum objects. Since similar experiments with single
particles and more slits produce interference fringes reducible to a
combination of double-slit patterns it is usually argued that quantum
interference occurs between pairs of trajectories, compactly denoted as
second-order interference. Here we show that quantum mechanics in fact allows
for interference of arbitrarily high order. This occurs naturally when one
considers multiple quantum objects interacting in the presence of a
nonlinearity, both of which are required to observe higher-order interference.
We make this clear by treating a generalised multi-slit interferometer using
second-quantisation. We then present explicit experimentally-relevant examples
both with photons interacting in nonlinear media and an interfering
Bose-Einstein condensate with particle-particle interactions. These examples
are all perfectly described by quantum theory, and yet exhibit higher-order
interference based on multiple particles interacting nonlinearly.
