On the hydrodynamics of nonlinear gauge-coupled quantum fluids

Abstract

By constructing a hydrodynamic canonical formalism, we show that the occurrence of an arbitrary density-dependent gauge potential in the meanfield Hamiltonian of a Bose-condensed fluid invariably leads to nonlinear flow-dependent terms in the wave equation for the phase, where such terms arise due to the explicit dependence of the mechanical flow on the fluid density. In addition, we derive a canonical momentum transport equation for this class of nonlinear fluid and obtain an expression for the stress tensor. Further, we study the hydrodynamic equations in a particular nonlinear fluid, where the effective gauge potential results from the introduction of weak contact interactions in an ultracold dilute Bose gas of optically-addressed two-level atoms. In the Cauchy equation of mechanical momentum transport of the superfluid, two non-trivial terms emerge due to the density-dependent vector potential. A body-force of dilation appears as a product of the gauge potential and the dilation rate of the fluid, while the stress tensor features a canonical flow pressure term given by the inner-product of the gauge potential and the canonical current density. By numerical simulation, we illustrate an interesting effect of the nonlinear gauge potential on the groundstate wavefunction of a superfluid in the presence of a foreign impurity. We find that the groundstate adopts a non-trivial local phase, which is antisymmetric under reversal of the gauge potential. The phase profile leads to a canonical-flow or phase-flow dipole about the impurity, resulting in a skirting mechanical flow. As a result, the pressure becomes asymmetric about the object and the condensate undergoes a deformation.

Gauge transformations and Galilean covariance in nonlinear gauge-coupled quantum fluids

“…As a consequence, the fluid pressure also inherits a flow-dependent term. This may be seen by expressing the hydrodynamic equations (5) and (6) in the reference frame of the fluid, which yield [23], respectively…”

“…where η and A are nonlinear effective potentials which depend on the density of the fluid. After performing a Madelung transformation, ψ = √ ρe iθ/ , on the macroscopic condensate wavefunction ψ, it is possible to show [23] that the dynamics of the field components ρ and θ, may be expressed in the Hamiltonian form…”

“…From a dynamical perspective, perhaps the most striking feature of a density-dependent gauge potential is the occurrence of an exotic flow-dependent nonlinear term in the wave equation for the quantum phase. This is exemplified notably by the stress tensor of the fluid, which features a "gauge pressure" term related to the overlap of the gauge potential and the gauge-covariant current [23]. One would expect such a term to carry important implications for the symmetry properties of the fluid.…”

“…We work within a hydrodynamic canonical formalism and consider transformations at the level of the field equations, rather than the wave equations proceeding from these. Furthermore, although the motivation for our study stems from a well-established microscopic model proposed in [11], we follow a general approach adopted from a previous paper [23], which is not tied to any particular model. Here, we simply assume that a density-dependent gauge potential A (ρ) enters the Hamiltonian functional, whose components take the form of arbitrary density-dependent functions.…”

Few-to-many vortex states of density-angular-momentum coupled Bose-Einstein condensates

“…The rotational properties of the theory present an opportunity to understand the vortex solutions and associated superfluidity in two-dimensional homogeneous [61,62] and trapped configurations [63], as well as the simulation of curved spacetime for the excitations of the ground state [64]. Recent work has also examined the theories mathematical structure from a hydrodynamical perspective [65,66], and proposals have now appeared that generalize the theory to support gauge theories such as those with a topological Chern-Simons structure [67].…”