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

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.

Figure 1

Mean field solutions for (a) the condensate amplitude $|{\phi }_0(z)|$ and (b) the condensate phase from Eq. (4) for $\overline{\eta }=1$ and $v/v_s=0.35$. The dashed line shows the continuation of each of the underlying dark soliton solutions.

Figure 2

The drag force on the impurity (in units $\sqrt{\gamma }g{n}_{\infty }^2$). (a) Drag force at $T=0$ as a function of velocity. $\overline{\eta }=0.3,1,3$ for diamond, square, circle, respectively, and each line terminates at their respective critical velocities $v_c/v_s=0.64,0.37,0.16$. (b)–(d) Drag force as a function of temperature. (b) $\overline{\eta }=0.3$ and velocities: blue diamond $\overline{v}=0.14$, blue square $\overline{v}=0.35$, blue circle $\overline{v}=0.56$. (c) $\overline{\eta }=1$ and velocities: red diamond $\overline{v}=0.08$, red square $\overline{v}=0.2$, and red circle $\overline{v}=0.3$. Panel (d) is data taken from (b) and (c), and compares the drag on a strong impurity moving at slow speed with that on a weak impurity moving at high speed. The strong impurity can have a larger drag in the quantum regime while the weak impurity will have the larger drag in the thermal regime.