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arXiv:0904.0995 · Physical Review Letters

Drag Force on an Impurity below the Superfluid Critical Velocity in a Quasi-One-Dimensional Bose-Einstein Condensate

Andrew G. Sykes, Matthew J. Davis, David C K Roberts

Published Aug 18, 2009• 51 citations
arXivHTMLPDF

Abstract

The existence of frictionless flow below a critical velocity for obstacles moving in a superfluid is well established in the context of the mean-field Gross-Pitaevskii theory. We calculate the next order correction due to quantum and thermal fluctuations and find a nonzero force acting on a delta-function impurity moving through a quasi-one-dimensional Bose-Einstein condensate at all subcritical velocities and at all temperatures. The force occurs due to an imbalance in the Doppler shifts of reflected quantum fluctuations from either side of the impurity. Our calculation is based on a consistent extension of Bogoliubov theory to second order in the interaction strength, and finds new analytical solutions to the Bogoliubov-de Gennes equations for a gray soliton. Our results raise questions regarding the quantum dynamics in the formation of persistent currents in superfluids.

DOI
10.1103/physrevlett.103.085302

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The existence of frictionless flow below a critical velocity for obstacles moving in a superfluid is well established in the context of the mean-field Gross-Pitaevskii theory. We calculate the next order correction due to quantum and thermal fluctuations and find a nonzero force acting on a delta-function impurity moving through a quasi-one-dimensional Bose-Einstein condensate at all subcritical velocities and at all temperatures. The force occurs due to an imbalance in the Doppler shifts of reflected quantum fluctuations from either side of the impurity. Our calculation is based on a consistent extension of Bogoliubov theory to second order in the interaction strength, and finds new analytical solutions to the Bogoliubov-de Gennes equations for a gray soliton. Our results raise questions regarding the quantum dynamics in the formation of persistent currents in superfluids.
We study the drag force acting on an impurity moving through a 1D Bose-Einstein condensate in the presence of both quantum and thermal fluctuations. We are able to find exact analytical solutions of the partial differential equations to the level of the Bogoliubov approximation. At zero temperature, we find a nonzero force is exerted on the impurity at subcritical velocities, due to the scattering of quantum fluctuations. We make the following explicit assumptions: far from the impurity the system is in a quantum state given by that of a zero (or finite) temperature Bose-Einstein condensate, and the scattering process generates only causally related reflection/transmission. The results raise unanswered questions in the quantum dynamics associated with the formation of persistent currents.
It has been proposed in recent years that an impurity moving in a superfluid will experience drag forces even below the Landau critical velocity [1–3]. Quantum fluctuations that are present in superfluid systems at zero temperature can scatter from impurities present in the fluid. If the impurity is moving, there will be an asymmetric Doppler shift associated with the scattered waves, which will result in a net drag force. This challenges the long-accepted notion of dissipationless flow in superfluids. Analytical estimates of this drag force in superfluid Bose-Einstein condensates using Bogoliubov methods have found a non-zero drag force, resulting from the most sensible choice of boundary conditions in an infinite system. Through stochastic simulations of the 1D Bose gas under the truncated Wigner formalism, we verify the existence of this sub-critical drag force and quantify the drag forces present. We will also determine whether dissipationless flow can be recovered in finite sized systems.
We study the motion of a pointlike impurity in a Bose-Einstein condensate at T=0T=0T=0. By solving the Gross-Pitaevskii (GP) equation in a perturbative manner we calculate the induced mass of the impurity and the drag force on the impurity in three-, two-, and one-dimensional (1D) cases. The relationship between the induced mass and the normal mass of fluid is found, and coincides with the result of the Bogoliubov theory. The drag force appears for the supersonic motion of the impurity. In 1D the drag force is investigated also on the basis of the exact Lieb-Liniger theory, using the dynamic form factor, which has been evaluated by the Haldane method of the calculation of correlation functions. In this theory the force appears for an arbitrarily small velocity of the impurity. The possibility of measuring the form factor in existing experiments is noted.
Abstract We study the hydrodynamic forces acting on a small impurity moving in a two-dimensional Bose–Einstein condensate at non-zero temperature. The condensate is modelled by the damped-Gross Pitaevskii (dGPE) equation and the impurity by a Gaussian repulsive potential coupled to the condensate. For weak coupling, we obtain analytical expressions for the forces acting on the impurity, and compare them with those computed through direct numerical simulations of the dGPE and with the corresponding expressions for classical forces. For non-steady flows, there is a time-dependent force dominated by inertial effects and which has a correspondence in the Maxey–Riley theory for particles in classical fluids. In the steady-state regime, the force is dominated by a self-induced drag. Unlike at zero temperature, where the drag force vanishes below a critical velocity, at low temperatures the impurity experiences a net drag even at small velocities, as a consequence of the energy dissipation through interactions of the condensate with the thermal cloud. This dissipative force due to thermal drag is similar to the classical Stokes’ drag. There is still a critical velocity above which steady-state drag is dominated by acoustic excitations and behaves non-monotonically with impurity’s speed.
We discuss a force arising from zero-temperature quantum fluctuations in a weakly interacting Bose-Einstein condensate (BEC). We argue that there is a drag force on an object moving at speeds well below Landau's critical velocity, and motivate our argument by an analogy to the Casimir force in an electromagnetic vacuum. We sketch the calculation of this force for an object described by a general potential. We comment about the relevance of our results to the stability of superfluid flow, arguing that this effect is consistent with the observation of persistent currents and that this effect introduces new experimental observables. Finally, we describe a new hydryodynamic boundary condition for superfluid flow that takes into account the scattering of quantum fluctuations off a rough surface.
We investigate the drag force on a moving impurity in a spin-orbit-coupled Bose-Einstein condensate. We prove rigorously that the superfluid critical velocity is zero when the impurity moves in all directions but one, in contrast to the case of liquid helium and superconductor, where it is finite in all directions. We also find that when the impurity moves in all directions except the two special ones, the drag force has nonzero transverse component with a small velocity. When the velocity becomes large and the states of the upper band are also excited, the transverse force becomes very small due to opposite contributions of the two bands. The characteristics of the superfluid critical velocity and the transverse force are results of the order-by-disorder mechanism in spin-orbit-coupled boson systems.