Reduced density-matrix equations and superfluid hydrodynamics

Luca Salasnich

Liang Tong, Yu Chen

Abstract The growing demand to understand non-equilibrium dynamics in open quantum many-body systems—motivated by progress in quantum simulation and error correction—requires theoretical frameworks that avoid the exponential scaling of the full density matrix. In this work, we develop a first-principles approach to the dissipative hydrodynamical equations of a finite-temperature superfluid. We show that dominant external dissipation not only simplifies the description of intrinsic damping (Beliaev and Landau) but also stabilizes a hydrodynamic formulation. Starting from a density-matrix method aligned with the Zaremba–Nikuni–Griffin formalism, a closed set of dissipative quantum Liouville equations (DQLE) can be obtained. We then introduce a systematic hydrodynamic expansion beyond the conventional Wigner transform, yielding a closed set of dissipative hydrodynamic equations (DHE). Through detailed comparison with the full DQLE, we demonstrate that retaining only the zeroth-order term in the expansion of the anomalous density matrix offers the most accurate and stable DHE implementation, in close agreement with the DQLE under strong dissipation and weak interaction. Our results establish a stabilized hydrodynamic theory for dissipative quantum fluids and significantly reduce the computational complexity of modeling open many-body dynamics.

Christophe Coste

Лев П. Питаевский

I study vortex ring oscillations in a superfluid, trapped in an elongated trap, under the conditions of the Local Density Approximation. On the basis of the Hamiltonian formalism I develop a hydrodynamic theory, which is valid for an arbitrary superfluid and depends only on the equation of state. The problem is reduced to an ordinary differential equation for the ring radius. The cases of the dilute BEC and the Fermi gas at unitarity are investigated in detail. Simple analytical equations for the periods of small oscillations are obtained and the equations of non-linear dynamics are solved in quadratures. The results agree with available numerical calculations. Experimental possibilities to check the predictions are discussed.

H. Haug, K. Weiss

The equations of motion for the reduced density matrices of an interacting Bose system are treated using the concept of off-diagonal long-range order. The equations are decoupled on the one-particle level. With this approximation, the hydrodynamic equations for the densities and the velocities of the condensate and the depletion are derived. The interaction between the two fluid components depends on density variations and on the relative velocity. The latter interaction terms appear as kinetic pressure terms and a Magnus force term. Via the equation for the total momentum, a connection between the velocities of the condensate and the depletion, and the velocities in Landau's two-fluid model is derived. The excitation spectrum of the system is investigated in the phonon region. A comparison of the resulting velocities for the first and second sound with the experimentally determined ones shows that the functional dependence of the condensate density on temperature is similar to that of the superfluid component.