Dragg force

In one dimension the integration is straightforward. From (7.29) we find

$$F^{1D} = -\frac{imn {g_{i}}^2}{\pi\hbar^2} \int\limits_{-\in... ...^2-c^2}/\hbar} +\frac{1}{k-2m\sqrt{V^2-c^2}/\hbar} \right)\,dk$$ (7.38)

The integration over $\k$ gives $2\pi i$ if $V>c$ and zero otherwise. So, the force is

$$F^{1D} = \frac{2{g_{i}}^2n_{1D}m}{\hbar^2},$$ (7.39)

where $n_{1D} = N/L$ is the linear density. In a quasi one dimensional system (i.e. a very elongated trap or a waveguide) there are no excitations in the radial harmonic confinement and the coupling constant is obtained from (1.69) keeping in mind that the reduced mass equals to the mass of an incident particle $\mu = m$ for the scattering on a heavy impurity

$${g_{i}}=-\frac{\hbar^2}{mb_{1D}}$$ (7.40)

For the non-resonance scattering $b_{1D}=-a_{\perp}^2/b$, where $a_{\perp}=\sqrt{\hbar/m\omega_{\perp}}$. The expression of the force in terms of the scattering length reads as

$$F^{1D}=2n_{1D}\hbar^2/mb_{1D}^2$$ (7.41)

An interesting peculiarity is that the result does not depend on the velocity $V$ (where, of course, the velocity must be larger than the speed of sound). This phenomenon comes from particular properties of a $\delta$-potential, namely that the Fourier transformation of this potential is a constant. Numerical solutions by Pavloff[Pav02] for finite-range potentials in $1D$ show no friction for $V<c$, maximal friction for $V\ge c$ and smaller friction for $V\gg c$, although the constant result (7.41) was found for the $\delta$-potential.

In a 1D system energy dissipation is possible at $V<c$ due to creation of the ``gray solitons'' first considered in [Tsu71]. Non-linear calculations [Hak97] show that the critical velocity for this process decreases with increasing coupling constant ${g_{i}}$.

This theory can be checked in an experiment in a three-dimensional condensate. The impurity can be presented by a moving light sheet.