  Using the coherent state functional integral expression of the partition
function, we show that the sine-Gordon model on an analogue curved spacetime
arises as the effective quantum field theory for phase fluctuations of a weakly
imperfect Bose gas on an incompressible background superfluid flow when these
fluctuations are restricted to a subspace of the single-particle Hilbert space.
We consider bipartitions of the single-particle Hilbert space relevant to
experiments on ultracold bosonic atomic or molecular gases, including, e.g.,
restriction to high- or low-energy sectors of the dynamics and spatial
bipartition corresponding to tunnel-coupled planar Bose gases. By assuming full
unitary quantum control in the low-energy subspace of a trapped gas, we show
that (1) appropriately tuning the particle number statistics of the
lowest-energy mode partially decouples the low- and high-energy sectors,
allowing any low-energy single-particle wave function to define a background
for sine-Gordon dynamics on curved spacetime and (2) macroscopic occupation of
a quantum superposition of two states of the lowest two modes produces an
analogue curved spacetime depending on two background flows, with respective
weights continuously dependent on the corresponding weights of the superposed
quantum states.
