Introduction

One of the most important peculiarities of Landau theory superfluidity is the existence of a finite critical velocity. If a body moves in a superfluid at $T=0$ with velocity $V$ less then $v_c$, the motion is dissipationless. At $V>v_c$ a drag force arises because of the possibility of emission of elementary excitations. However, both theoretical and experimental investigation in superfluid $^4$He are difficult. The critical velocity in $^4$He is related to creation of rotons, for which one has no simple theoretical description. Further, an important role is played by complicated processes involving vortex rings production.

The situation in low-density weakly-interacting Bose-Einstein condensed (BEC) gases is simpler. The Landau critical velocity in this case is due to Cherenkov emission of phonons which can be described by mean-field theory. Due to the presence in the theory of an intrinsic length parameter, the correlation length $\xi$, the friction force for a small body does not depend on its structure. Vortex rings in the BEC cannot have radius less then $\xi$ (see, e.g., [JR82]) and it is reasonable to believe that probability of their creation by a small body is small. Thus quantitative investigation of critical velocities in BEC are very interesting and can be used to probe the superfluidity of a quantum gas.

Recently existence of the critical velocity in a Bose-Einstein Condensed gas was confirmed in a few experiments. At MIT a trapped condensate was stirred by a blue detuned laser beam [RKO+99] and the energy of dissipation was measured. The critical velocity was found to be smaller than the speed of sound due to emission of vortices. The diameter of the laser spot in this experiment was of a macroscopic size and was large compared to the healing length. An improved technique allowed measurement of the drag force acting on the condensate in a subsequent experiment [ORV+00].

The analytical study of flow of the condensate over an impurity is highly nontrivial due to the intrinsic nonlinearity of the problem arising from the interaction of the particles in the condensate. In one dimension the dissipation could occur at velocities smaller than predicted by Landau's approach due to emission of solitons [Hak97]. The dependence of the critical velocity on the type of the potential was studied both by using a perturbative approach and numerical integration in [LP01,Pav02]. The effective two dimensional problem was considered in [KM00]. In this work generation of excitations in the oscillating condensate in a time dependent parabolic trap in the presence of a static impurity was studied analytically. A three-dimensional flow of a condensate around an obstacle was calculated numerically by integration of the Gross-Pitaevskii (GP) equation and emission of vortices was observed [FPR92,WMA99].

There are different definitions of the superfluidity. It is possible to make following experiment. Move a small body through the system. According to Landau if there is no normal part (we consider zero temperature or small enough) no dissipation will happen if the speed is smaller than the speed of sound. Our goal is to calculate effect of such a probe, a small impurity moving through a condensate which is described by the Gross-Pitaevskii equation.

We want to find an answer to a question which is rather complicated. From one side we know from usual considerations in the mean field regime that the system is superfluid. From the other side we know that in the Tonks-Girardeau regime the system is mapped on the fermions, which are, definitely, not superfluid. Indeed what we find is that the situation is somewhere between.