This can be achieved via the coil geometry or a current imbalance, with the 1D system oriented at 45 degrees to the coil axis.

If the second quarter-period oscillator has a different length $a_0^{\prime }$, the transformation also provides an overall magnification $(a_0^{\prime }/a_0{)}^2$.

See End Matter.

At small momenta, this is an approximation to the stereographic projection discussed in the End Matter.

Despite the near detuning required for alkali atoms, the losses should be small, as one only needs to make a small fractional rotation of the levels, as in Ref. [30].

The kinetic energy density can be written as $(-{\hslash }^2/2M)\psi ({\mathit{r}}^{*}){\nabla }^2\psi (\mathit{r})$ or $(+{\hslash }^2/2M)|\nabla \psi (\mathit{r}){|}^2$, which gives the same total kinetic energy under integration when $\psi$ vanishes on the boundary.

For a weakly interacting 2D BEC, $C\sim (\tilde{g}\overline{\rho }{)}^2$, so it contributes $⟨{\widehat{\rho }}_{\text{quartic}}⟩\sim {\tilde{g}}^2{\overline{\rho }}^2{p}_0^2/{\hslash }^2$. For $\overline{\rho }{p}_0^2/{\hslash }^2\gg 1$, this is larger than the contribution from the condensate $⟨{\widehat{\rho }}_{\text{quartic}}⟩\sim \overline{\rho }(\hslash /\xi {)}^4$ expected in the vicinity of a vortex or edge, with healing length $\xi \sim 1/\sqrt{\tilde{g}\overline{\rho }}$.