  We examine the maximum negative energy density which can be attained in
various quantum states of a massless scalar field. We consider states in which
either one or two modes are excited, and show that the energy density can be
given in terms of a small number of parameters. We calculate these parameters
for several examples of superposition states for one mode, and entangled states
for two modes, and find the maximum magnitude of the negative energy density in
these states. We consider several states which have been, or potentially will
be, generated in quantum optics experiments.
