Analogue stochastic gravity in strongly-interacting Bose–Einstein condensates

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.