  Impressive progress in the control of atomically thin crystals is now
enabling the realization of gated structures in which two electrodes are
separated by atomic scale distances. The electrical capacitance of these
structures is determined by phenomena that are not relevant in capacitors with
larger electrode separation. With the aim to analyze these phenomena, we use
linear response theory to develop a systematic description of capacitance for
two coupled electron liquids, accounting for the wave nature of electrons, as
well as for the effect of both intra and interlayer Coulomb interactions. Our
theory leads to a general expression for the electrical capacitance in terms of
both intra and interlayer electronic polarizabilities. The intralayer
polarizability is directly related to the conventional expression for the
quantum capacitance, whereas the interlayer polarizability term accounts for
interaction-induced correlations between charges hosted by opposite capacitor
plates. We refer to this latter term as to the cross quantum capacitance. We
discuss the implications of the general expression for the capacitance, show
that it leads to established results when the effect of interlayer correlations
is negligible, and that the intra and interlayer polarizabilities play a
comparable role for capacitors with very small electrode separation. Using two
different approaches, we calculate the capacitance in specific cases, and find
that the interlayer polarizability can be either positive or negative, so that
the cross quantum capacitance can either increase or decrease the total
capacitance. We conclude by showing that the cross quantum capacitance term can
lead to a non-monotonic evolution of the total capacitance with increasing
separation between the capacitor plates, which would represent an unambiguous
manifestation of the cross quantum capacitance if observed experimentally.
