  Collective modes propagating in a moving superfluid are known to satisfy wave
equations in a curved space time, with a metric determined by the underlying
superflow. We use the Keldysh technique in a curved space-time to develop a
quantum geometric theory of fluctuations in superfluid hydrodynamics. This
theory relies on a 'quantized' generalization of the two-fluid description of
Landau and Khalatnikov, where the superfluid component is viewed as a
quasi-classical field coupled to a normal component -- the collective
modes/phonons representing a quantum bath. This relates the problem in the
hydrodynamic limit to the 'quantum friction' problem of Caldeira-Leggett type.
By integrating out the phonons, we derive stochastic Langevin equations
describing a coupling between the superfluid component and phonons. These
equations have the form of Euler equations with additional source terms
expressed through a fluctuating stress-energy tensor of phonons. Conceptually,
this result is similar to stochastic Einstein equations that arise in the
theory of stochastic gravity. We formulate the fluctuation-dissipation theorem
in this geometric language and discuss possible physical consequences of this
theory.
